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Topics in singular boundary value problems, evolution equations and mass transport Kang, Xiaosong 2003

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TOPICS IN SINGULAR B O U N D A R Y V A L U E PROBLEMS, EVOLUTION EQUATIONS A N D MASS TRANSPORT by XIAOSONG K A N G MSc, The Chinese University of Hong Kong 1999 BSc, Wuhan University 1994 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF  PHILOSOPHY in  T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Mathematics We accept this thesis as conforming _Lojhe required standard  T H E U N I V E R S I T Y OF BRITISH C O L U M B I A April 2003 © Xiaosong Kang, 2003  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Mathematics The University of British Columbia Vancouver, Canada  Abstract In the first part of this thesis, the Hardy-Sobolev critical semilinear equations are studied via variational methods. Unlike the non-singular case s = 0, or the case when 0 belongs to the interior of a domain in EJ , we show that the value and the attainability of the best Hardy-Sobolev constant on a smooth domain 0, 1  p (ty s  :=  inf |  J \Vu\ dx; u € 2  HQ(Q) and  J  = 1  j  when 0 < s < 2, 2*(s) = ^Ef, and when 0 is on the boundary dfi, are closely related to the properties of the curvature of dfi, at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:  u~ - A M = —— + f(x,u) in ft c R , \x\ v  x  n  s  where / is a lower order perturbative term at infinity and f(x,0) = 0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0. This is joint work with Nassif Ghoussoub. In the second part of this thesis, we prove, using comparison principles, the strict localization in the Cauchy problem for unbounded solutions to a porous medium type equation with a a N source term, ut = V • (u Vu) + u@, x G R , N > 1, 8 > a + 1, a > 0, in the case of arbitrary compactly supported initial functions u . A n estimate on the support in terms of suppuo and the blow-up time T is also derived. Our result extends the well-known one dimensional case and solves an open problem in this field. 0  This is joint work with Changfeng Gui. In the last part of the work, using the Monge-Kantorovich theory of mass transport, we establish an inequality for the relative total energy - internal, potential and interactive of two arbitrary probability densities, their Wasserstein distance, their barycenters and their entropy production functional. This inequality is remarkably encompassing as it implies most known geometrical - Gaussian and Euclidean - inequalities as well as new ones, while allowing a direct and unified way for computing best constants and extremals. As expected, such inequalities also lead to exponential rates of convergence to equilibria for solutions of FokkerPlanck and McKean-Vlasov type equations. The proposed inequality also leads to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker-Planck type equations. This is joint work with Martial Agueh and Nassif Ghoussoub.  ii  Table of Contents Abstract  ii  Table of Contents  i  Acknowledgements  Introduction  i  i  iv  v  Chapter 1. Hardy-Sobolev Critical Elliptic Equations with Boundary Singularities 1 1.1 1.2 1.3 1.4 1.5 1.6  Introduction Best Sobolev-Hardy Constants Blow-up Analysis and Attainability of Best Constants Least Energy Solution to The Perturbed Dirichlet Problems The Neumann Problem Sign-changing Solution to The Neumann Problem Via Duality  2 5 12 21 26 36  Chapter 2. Localization Properties For a Porous Medium Type Equation in Higher Dimensions 43 2.1 2.2 2.3 2.4  Background Self-similar Representations and Self-similar Solutions Proof of Theorem 2.1 Effective Localization in a Special Case  43 47 51 61  Chapter 3. Geometric inequalities via a general comparison principle for interacting gases 64 3.1 3.2 3.3  Introduction Main inequality between two configurations of interacting gases Optimal Euclidean Sobolev inequalities 3.3.1 Euclidean Log-Sobolev inequalities 3.3.2 Sobolev and Gagliardo-Nirenberg inequalities 3.4 Optimal geometric inequalities 3.4.1 HWBI inequalities 3.4.2 Gaussian inequalities 3.5 Trends to equilibrium 3.6 A remarkable duality  Bibliography  65 76 82 82 84 86 86 90 94 97  103  iii  Acknowledgements The ultimate emergence of this thesis would not have happened without the support of my adviser Nassif Ghoussoub, from whom I have learned a lot about mathematics. This work has benefitted a great deal from conversations and collaboration with Nassif. On the other hand, I was also given a great deal of freedom in my research, but never lacked for help when it was needed. It is a great pleasure to thank Professor Changfeng Gui, who providently pointed me into the direction of the blow-up problems on porous medium equations with source term, and the many questions they posed. His encouragement and vital remarks will not be forgotten. I owe many thanks to Martial Agueh, for fruitful and stimulating discussions and collaboration; the last part of this thesis is joint work with Martial and Nassif. Thanks should also go to Professor Victor Galaktionov for his interest and advice; he generously shared with me his expertise on blow-up problems and pointed out several important references. I thank my teachers Professors Hua Chen, Jingyi Chen, Richard Froese, Kee Lam and Juncheng Wei, especially Minyou Qi; they helped me at different stages of my career. I want to say that I have tremendously appreciated and deeply admired the friendship of Mihail Cocos, Izak Grguric and their families, also Catalin Dochitoiu in Ottawa. We had so many good times together, it is hard to imagine living in Vancouver without these friends. A special word of thanks goes to my aunt Hongjin Kang for her beautiful gifts. Funding was partially provided by U B C Graduate Fellowship and a grant from the Natural Sciences and Engineering Research Council of Canada. Most of all, this thesis is dedicated to my parents, Hong-kui Kang and Zhen-shou X u , with love and respect. They have always been there for me, through thick and through thin.  X V  Introduction This thesis consists of three separate parts. H a r d y - S o b o l e v C r i t i c a l E l l i p t i c Equations. In the first part of this thesis, we consider the value of the best Hardy-Sobolev constant on a domain 0 of M , n  H {tt) := inf <^ / \Vu\ dx;ue 2  Un  s  H£(Sl) and / J-f-r— = 1  Jn \ \ x  and the corresponding ground state solutions for  on dQ,  0 when 0 < s < 2, and 2*(s) =  Unlike the non-singular case and assuming 0 is on the boundary of the domain 0, we show that these problems are closely connected to the curvature of the boundary <9f2 at 0. This is in sharp contrast with the non-singular context s = 0, or when 0 belongs to the interior of a domain f2 in JR , where it is well known that ^ (0) = rio(M ) for any domain O and that Hsity is never attained unless cap(J? \ f2) = 0. The case when <9Q has a cusp at 0 has already been shown by Egnell [26] to be quite different from the non-singular setting. Indeed, by considering open cones of the form C = {x E R \ x = r6,6 £ D and r > 0} where D is a connected domain of the unit sphere S of JRJ , Egnell showed that yu (C) is actually attained for 0 < s < 2 even when C ^ M . The case where 50 is smooth at 0 turned out to be also interesting as the curvature at 0 gets to play an important role. Indeed, we shall show that the positivity of the sectional curvature at 0 is needed for problems with Dirichlet boundary conditions, while the Neumann problems require the positivity of the mean curvature at 0. n  n  s  n  n  71-1  1  n  s  In fact we have the following result : Let O be a C -smooth domain in R 2  n  with 0 G dQ, then  < /x (I?+). Moreover, s  1) If T(Q) C 1?" for some rotation T (in particular, if fl is convex, or if O is star-shaped around 0), then /x (0) = /i (JR") and it is not attained unless O is a half-space. s  s  2) On the other hand, when n > 4, and if the principal curvatures of d£l at 0 are negative, then < ^ (I?™), the best constant p (Cl) is attained in HQ(£V) and the corresponding Euler-Lagrange equation has a positive solution on f2.  // (fi) s  s  S  In the following Dirichlet problem, the same concavity condition around the origin will play a key role.  v  Let 0 be a bounded domain in R  n  |  with C boundary and consider the Dirichlet problem 2  -An  =  ^  |  +  ^  A  in ft.  M  u —0  on dVt.  for 0 < s < 2. Assume that 0 £ ffl andthat the principal curvatures of dfl are non-positive in a neighborhood of 0. If n > 4 and if 0 < A < Ai (the first eigenvalue of - A on #o(ft)), then the above equation has a positive solution. For the Neumann problem, it is the positivity of the mean curvature at 0 that is needed. Let O be a bounded domain in R with C boundary and consider the Neuman problem n  f  - A M  [  D^u  2  + Xu inn. on dVt.  l ^ p i  =  =0  for 0 < s < 2. Assume that 0 G 50 and that the mean curvature of dVl at 0 is positive. If n > 3 and A < 0, then the Neumann problem has one positive solution.  Localization Properties For Porous Medium Type Equations. The second part of our work is concerned with the space localization properties of the blow-up solutions in the Cauchy problem for a porous medium equation with source of the form: iH  = A(u )  + u in R  m  p  n  x [0, T),  where p > 1, m > 1. This equation usually appears as a model equation in gas dynamics, laser fusion and combustion theory. It is well-known that the solution of the above equation generically blows up in finite time. Since the elliptic operator involved is degenerate, unlike the classical heat equation, this equation describes a process with finite speed of propagation, that is, for compactly supported initial data, the solution remains compactly supported in the space variable at all time. The behavior of the spatial support near the blow-up time is not a priori obvious. If p < m, the solution could blow up everywhere even for compactly supported initial data, while if p — m, it has been proved recently that the support is uniformly bounded, in other words, the support does not blow up. For p > m, similar localization phenomenon have been observed numerically by a group of Russian applied mathematicians in 70's. In 1985, Galaktionov justified this fact in one space dimension, by utilizing a clever sturmian argument. Since that comparison technique used is essentially of O D E nature, it is impossible to extend to higher dimensions. The main goal in chapter two is to prove this expected localization in the case p > m. We establish the following theorem: Suppose that u is the solution to the problem f ut = V • {u Vu)+u> in R ( u(x, 0) = UQ(X) in R , c  3  N  N  vi  x [0,T)  with a compactly supported initial datum u and that T is the blow-up time, where ft > a +1. Then there exists a K > 0, depending on a, ft, N, u and T , such that for all t 6 [0, T ) , Q  0  suppu(-.t) C 5(0, K). The strategy of the proof is to apply Alexandroff's reflection principle on the solutions expressed in self-similar variables, along with several important properties of self-similar solutions, to reduce the original problem to an auxiliary initial-boundary value problem on a ball. By an easy but important observation on the symmetry properties of the solution to that initialboundary value problem, we get the desired result through a standard averaging technique.  Geometric Inequalities Via the Monge-Kantorovich Theory. The last part of this thesis is about geometric inequalities, based on recent advances in the theory of mass transport. The main idea is to try to describe the evolution of a generalized energy functional along an optimal transport that takes one configuration to another, taking into account the - relative - entropy production functional, the transport cost (Wasserstein distance), as well as their centres of mass. Once this general comparison principle is established, then various - new and old - inequalities follow by simply considering different examples of - admissible - internal energies, of confinement and interactive potentials. Here is our framework: Let F : [0, oo) —> R be a differentiable function on (0, oo), V and W be C -real valued functions on R and let f2 C R be open and convex. The set of probability densities over Q is denoted by Va(ri) — {p : O —> R; p> Oand fnp(x)dx = 1}. The associated Free Energy Functionalis then defined on Va(Q,) as: 2  n  n  which is the sum of the internal energy  the potential energy  and the interaction energy  Of importance is also the concept of relative energy of po with respect to p\ simply defined as:  H^(polPi) : = H ^ U ) - H r ( P i ) vii  where p0 and pi are two probability densities. The to pv is normally defined as  relative entropy production'of p with respect 2  h(p\pv) = f P V{F (p) + V + W*p)) fa  dx  ,  in such a way that if p  v  is a probability density that satisfies  V(F(p ) v  + V + W*p ) = 0 a.e. v  then  h{p\Pv)= f p\V(F'(p)-F'(p )  +  v  W*(p- )\ dx. 2  Pv  Our notation for the density p reflects this paper's emphasis on its dependence on the confinement potential, though it obviously also depends on F and W. v  We need the notion of Wasserstein distance W between two probability measures p and pi on R , defined as: 2  0  n  W (A),Pi):=  inf  2  2  /  7er(p ,Pl) 0  \x - y\ d~f(x,y), 2  jRn R  n  x  where T(p , pi) is the set of Borel probability measures on R respectively.  xR  n  n  0  The  with marginals p and pi, 0  barycentre (or centre of mass) of a probability density p, denoted b(p)  :  / xp(x)dx  =  will play a role in the presence of an interactive potential. In this part, we shall also deal with non-quadratic versions of the entropy. call  Young function, any strictly convex C^-function c : R  n  limia-i^oo  For that we  —> R such that  c(0) =  0  and  = oo. We denote by c* its Legendre conjugate defined by  c*(y) =  sup{yz-c(z)}.  zeR  n  p on 0, we define the generalized relative entropy production-type function of p with respect to pv measured against c* by For any probability density  TAP\PV)-= which is closely related to the  I pc*(-V(F'(p) +  V  + W*p)) dx,  generalized relative entropy production function of p with respect  to pv measured against c* defined as: I . (p\ ) = f V (F'(p) + V + W • p) • Vc* (V (F'(p) + V + W * p)) dx. fa c  Pv  :  P  viii  Indeed, the convexity inequality c*(z) < z • Vc*(z) satisfied by any Young function c, readily implies that lc*(p\pv) < Ic*(p\pv)- Note that when c(x) = IAP\PV)  c_[_  f  we have  2~  =• h{p\pv) = f p V(F'(p) + V + W*p)  dx = 21 *(p\pv),  2  c  Jn and we denote Ic*(p\pv) by X2(p\pv). The following general inequality is the main result of this part: Let ft be open, bounded and convex subset of R , let F : [0, oo) — > R be differentiable function on (0, oo) with F(0) — 0 and x H-> x F(x~ ) convex and non-increasing, and let PF{X) := xF'(x) — F(x) be its associated pressure function. Let V : R —*• R be a C confinement potential with D V > XI, and let W be an even C -interaction potential with D W > vi where A, v 6 R, and / denotes the identity map. Then, for any Young function c : R —* R, we have for all probability densities po and p\ on ft, satisfying supp po Cftand n  n  n  n  2  2  2  2  n  PF(PO) e  w^M,  OpolPi) + ^ V  2  (p , 0  ) - ^lb(po) - b ( ) | < H - ; ^ : ( P o ) + 2  P l  P l  2  c  W  lAPolPv).  When c is quadratic, the above inequality reads as the following HBWI inequlity  H £ % o | p i ) < W2(p0,p1Wl2(po\pv)  - ^ V ( p o , P i ) + ^lb(po) - b ( ) | . 2  2  2  P l  This extends the HWI inequality obtained by Otto and Villani, hence it implies many Gaussian inequalities.  When both confinemnet and interaction potentials are zero, we apply the main inequality to various functionals to revisit various inequalities and to determine their corresponding best constants. In particular, we show how it readily implies various Log-Sobolev inequalities, the Gagliardo-Nirenberg inequalities, and in particular the Sobolev inequalities. For example, it yields the optimal Euclidean p-Log Sobolev inequality for any p > 1, which extends corresponding results of Beckner (p = 1) and Del Pino-Dolbeault (1 < p < n).  ix  Chapter 1 Hardy-Sobolev Critical Elliptic Equations w i t h B o u n d a r y Singularities  Abstract Unlike the non-singular case s = 0, or the case when 0 belongs to the interior of a domain Q in ]R , we show that the value and the attainability of the best Hardy-Sobolev constant on a n  smooth domain O,  fjk (Q) := inf j f \Vu\ dx; u e H^{n) and f 2  a  when 0 < s < 2, 2*(s) = ^ f ^ , and when 0 is on the boundary dVt are closely related to the properties of the curvature of dVt at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:  u~ p  -Au =  —  l  —  \x\  s  + f(x, u) in Q c M , n  where / is a lower order perturbative term at infinity and f(x, 0) = 0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0.  1  1.1 Introduction We consider the value of the best Hardy-Sobolev constant on a domain 0 of R , n  // (0) := inf | J  \Vu\ dx; u G H^(n) and J  = 1j  2  s  (1.1.1)  and the corresponding ground state solutions for  (1.1.2)  u =0 when 0 < s < 2, and 2*(s) = ^ J 2  s j 2  on 5 0  . Unlike the non-singular case and assuming 0 is on the  boundary of the domain 0, we show that these problems are closely connected to the curvature of the boundary 50 at 0. This is in sharp contrast with the non-singular context s = 0, or when 0 belongs to the interior of a domain O in R , where it is well known that /x (0) = n  s  ^{R ) 71  for any domain O and that At (0) is never attained unless cap(i? \ O) = 0. n  s  The case when 50 has a cusp at 0 has already been shown by Egnell [26] to be quite different from the non-singular setting. Indeed, by considering open cones of the form C = {x G R ; x — n  r9,9 G D and r > 0} where D is a connected domain of the unit sphere 5 showed that n (C) is actually attained for 0 < s < 2 even when C ^ s  n _ 1  of R , Egnell n  R. n  The case where 5 0 is smooth at 0 turned out to be also interesting as the curvature at 0 gets to play an important role. Indeed, we shall show that the positivity of the sectional curvature at 0 is needed for problems with Dirichlet boundary conditions, while the Neumann problems require the positivity of the mean curvature at 0. More precisely, assume that the principal curvatures  a ,a _i x  of 5 0 at 0 are finite. The  n  boundary 50 near the origin can then be represented (up to rotating the coordinates if necessary) by: ^ n— 1 x  n  = - ^2  = h(x')  a  i i x  +  °(\ '\ )> x  2  1=1  where x' = (x\,£ -i) n  € B(0,5) n {x = 0} for some 5 > 0 where 5(0,6) is the ball in R  centered at 0 with radius S.  n  n  If we assume the principal curvatures at 0 to be negative, that is maxi<j< _i a n  < 0, then  t  the sectional curvature at 0 is positive and therefore dQ -viewed as an (n - 1)-Riemannian is strictly convex at 0 ([32]). The latter property means that there exists  submanifold of R n  a neighborhood U of 0 in dQ, such that the whole of U lies on one side of a hyperplane H that is tangent to dQ at 0 and U n H = {0}. In our context, we specify the orientation of dQ in such a way that the normal vectors of dQ are pointing inward towards the domain fl. The \ O at 0.  above curvature condition then amounts to a notion of strict local convexity of M  N  Indeed, setting  P  = {x = [x\ x ) G R ~ x R : x > n  1<s  l  + ... + xl_ )} n 5(0, 8),  1  n  n  x  then, with the above orientation of dQ, the condition that the principal curvatures are negative, yields the existence of 8 > 0 and 7 < 0 such that P  C 0, up to a rotation. If the principal  7 i 5  curvatures of dCl are only non-positive on a neighborhood of 0, then we simply have that Po,<5  C  il for some 8 > 0. The following result will be established in sections 2 and 3.  Theorem 1.1. Let O be a C -smooth domain in R 2  n  with 0 G dVt, then fj, (Q) < // (i?+). a  s  Moreover,  1) IfT(Q) C JR" for some rotation T (in particular, if Vt is convex, or if £1 is star-shaped around 0), then p (Q) = p (R+) and it is not attained unless Q is a half-space. s  s  2) On the other hand, when n > A, and if the principal curvatures of dQ at 0 are negativ (i.e., i/maxi< < _ i  n  1  Q!j <  0), then p (Q) < p (R \), r  s  s  the best constant p (Q) is attained in s  HQ(Q) and (1.1.2) has a positive solution on O.  The "global convexity" assumption on Q in 1) can be contrasted with the hypothesis on the principal curvature in 2) which, as discussed above, can be seen as a condition of  local strict  concavity of the boundary at 0 when viewed from the interior of O. However, we shall see that the latter is not a necessary condition for the existence of solution for equation (1.1.2), since  3  p, (Q) < p (lR!\), even though 5ft is "flat at zero".  we will exhibit domains ft where  s  s  Such an analysis is relevant to the study of elliptic partial differential equations with singular potentials of the form  -Au =  + f(x, u) in ft C R , n  under both Dirichlet and Neumann boundary conditions. Here / is to be seen as a lower order perturbative term at infinity and f(x, 0) = 0. We shall see that in both Neumann and Dirichlet problems, our existence results depend on conditions on the curvature of the boundary near 0. The following two statements summarize the situation. Slightly more general results will be established later. In the following Dirichlet problem, the same concavity condition around the origin will play a key role.  Theorem 1.2. Let ft be a bounded domain in ¥t with C boundary and consider the Dirichle n  2  problem _  = M ' ;;- " a  A  u  (  \u  a  +  in ft. (1.1.3)  | a |  u  = 0  on <9ft.  for 0 < s < 2. Assume that 0 G 5ft and that the principal curvatures of 5ft are non-positive in a neighborhood o/O. If n > 4 and if 0 < A < Ai (thefirsteigenvalue of —A on  HQ(Q)),  then (1.1.3) has a positive solution.  For the Neumann problem, it is the positivity of the mean curvature at 0 that is needed.  Theorem 1.3. Let ft be a bounded domain in M with C boundary and consider the Neuma n  2  problem -Au D-yU  = ^  W  -  =0  + Xu in 0.  (  L  L  4  )  on 5ft.  for 0 < s < 2. Assume that 0 G 5ft and that the mean curvature o/5ft at 0 is positive (i.e.,  Y^i=i di> 0). Ifn>2> and A < 0, then (1.1-4) has one positive solution. 4  1.2  Best Sobolev-Hardy Constants  The best Hardy-Sobolev constant of a domain fi, C R  is defined as:  n  := inf  ti (Q) a  |y  |V«| dx;M G ^(0)  and ^  2  = l |  (1.2.5)  where 0 < s < 2, 2*(s) = 2te=£l. In the non-singular case s = 0, this is nothing but the best Sobolev constant of fi and it is well known that po(fi) =  u- (R ) for any domain fi and that f/,o(fi) is never attained unless n  0  cap(R \ fi) = 0. n  Similar results hold in the singular case (0 < s < 2) provided 0 belongs to the interior of the domain fi. Indeed, as noticed by several authors [37], the best constant in the HardyLittlewood-Sobolev inequality is not attained on those domain fi containing 0 and satisfying cap(J? \ fi) ^ 0, while it is attained on R n  by functions of the form  n  =  (  a  . ( n - , ) ( „ - 2 ) ) ^  (a+\x\ ~ )^ 2  s  for some a > 0. Moreover, the functions y are the only positive radial solution to a  u  -Au  2*(«)-l  =  , ,  in R ,  (1.2.7)  n  \x\  s  hence, by denoting  fj, := fj, (R ), we have: f 177 I 2 n  s  s  Ms ( / JR  n  2 ( s )  ^ T ^ ) ^ 177 l * ( ) 2  FI  S  2  =  /" IT/ I W 2  IIVl/ ||5 = //" 7 l7- = 17/ | * ( ) i  a  2  Jii  n  FI  In this section, we deal with the more interesting case when 0  s  n=2  •  2=2  (1-2-8)  belongs to the boundary of the  domain fi. We shall see that the situation is completely different as it very much depends on the smoothness and the curvature of the boundary at 0 The case when dfi is not smooth at 0 has been well analysed by Egnell [26]. Starting with the case where O is a half-space JR" or more generally an open cone of the form C = {x E R ; x = n  r9,0 G D and r > 0} where D is a connected domain on the unit sphere S " 1  -1  of i ? , Egnell n  [26] showed that p (C) is actually attained for 0 < s < 2 even when C ^ R , and therefore n  s  there exists a positive solution for  Au  =  w u(x)  I  u satisfies - A M  G  on  dC,  (1.2.9)  = o(|a;| ) as \x\ —• oo in C. 2_n  p (C) ^ p (R ) n  s  s  whenever  R \C is non-negligeable. n  H$(C), u > 0 in C, which attains p (R ). n  Such a solution  s  | 2 * ( s ) —2  = A" ^ 1  u  i n C,  = 0  A consequence of Egnell's result is that For otherwise, we can find a  .  | x |  1  u s  in R , where A > 0 is a Lagrange multiplier. n  By the strong  maximum principle u > 0 in R , which is a contradition. One obtains in particular that, n  i (R \_) > fi (R ), r  r  n  s  s  and more generally that  ti (C ) > p (C ), s  x  s  (1.2.10)  2  whenever Ci are cones such that C\ C CiThe main ingredient in this analysis comes from the fact that the quantities HVuH^^n) and  J  Rn ^ j | x  ( ) S  dx are invariant under scaling u(x) i—• r  Ii  2 'u(r2;). This means that whenever 0 G 2  dQ,  we have /i (fi) = // (Afi) for any A > 0. It is also clear that / / is invariant under rotations. s  s  s  fj, (^2) if  These observations combined with the fact that // (Oi) >  s  s  C 0 , yield that th 2  best constant for any finite cone (that is, the intersection of an infinite cone with a bounded connected open set) is the same as the best constant for the corresponding infinite cone.  In the sequel, we deal with the distinct and more interesting case where 0  is a smooth point  of the boundary of the domain f2 as stated in Theorem 1.1. In contrast to Egnell's result on pointed cones, we have in particular the following examples which give a totally different picture when the "cones" are smooth at 0.  Proposition 1.4. For each 7 G R, define the open paraboloid P = { = (x',x ) l  x  n  GR~ n  6  l  x  R : x > 7k'| }2  n  1) Ifl>  0, then p (P ) = p {R%). s  7  s  2) If-y< 0, then p (P ) = s  7  p (R ). n  s  It follows that ^ (P ) is not attained unless P = R  n  S  7  7  or R\.  Proof: (1) follows from monotonicity and the general easy fact (which will be proved later),  that p (il).< ii (R l)  whenever dfl is smooth at 0.  r  s  s  For (2), notice that for A > 0, A P = Pr. On the other hand, if 7 < 0, then M := R \{x n  7  (0,x ); xn  < 0}  n  f \Vu \  Choose ue G  U <A<IAP . 0  7  such that  Crf(M),  < fi (M) + e. There exists 5 > 0, such that for all  2  M  =  e  a  implies that /^(Pi) < // (M) + e.  5,u  A <  £  A  C$°{Pz), which  —  s  7  Since  fJ> (M).  s  n  s  = 1, and  M  S  G  = p (P ) by scaling invariance, and since obviously p (R )  s  ^* dx  M  It leads immediately to inf A* (P2) <  S  inf\fi (Pi)  J  =  a  faiM), we get  our claim.  Behind these examples lies a more general phenomenon summarized in Theorem 1.1 whose proof will be given in various parts throughout this section. First, we prove that  // (ft) s  <  Ps(RD- Note that fi (K%.) = ii (B ) for all 5 > 0, where s  a  B  6  6  = {x = (x\ x ) n  G  R ;  \x'\ + {x - 5) < 6 }.  n  2  +  2  2  n  Indeed, since Bs C JR", we have that ris(R!\.) < PsiBs) for all 5 > 0, hence ^ (i?+) < s  m£ li (Bs). s  J  On the other hand, choose ue G C Q ° ( P " ) , such that §Rn ^^-dx  s  = 1, and  I V u | < /ii (iR™) + e. There exists m G N , such that for all m > m , u G C o ° ( P ) , which 2  Rn  e  s  implies that ps(Bm)  e  e  M  e  < p (R l) + e. This leads immediately to inf^ n (Bs) < /i (JR!Ji), hence r  s  s  s  to equality. Since \B$ = B\$ for all A, 8 > 0, we get the conclusion from scaling invariance. Now by the smoothness assumption on the domain O, there exists -modulo a rotation- a ball B  e  C Q centered at (0, e).  This means that p (0) < ri (B ) s  s  e  = p (J?+). s  Assertion (1) of  Theorem 1.1 is then obtained by monotonicity and by the rotation invariance of /x ($7). s  7  If the principal curvatures ofdQ. at 0 are negative, and ifn>4, then /x (f2) <  T h e o r e m 1.5.  s  p (Rl). s  As seen in the introduction, if the principal curvatures of dQ at 0 are negative, then there is 7 < 0 and 8 > 0 such that the set  P , = {x = (x', x ) G R '  71 1  7  5  n  xR:x >  7(x + ... + x ^)} n 5(0, 8), 2  n  2  is included in Q, up to a rotation. We also note that if the principal curvatures of d£l are non-positive on a neighborhood of 0, then Po.tf C 0. By Egnell's result [26], the problem  -Au = , t ]uf  (  2u  in RI  N  (1.2.11)  +  u G Hl(R ),u> n  +  0  has a positive solution </>, which, up to a multiplier, also attains the best constant may assume that  cp  G  Hl(R ), that J n  +  = 1, and  Rn  p (R i ). We 1  s  r  \\V(f>\\l = p {RX). s  We also have the following estimates (see ([26], or appendix in [43]):  To prove the theorem, it is sufficient to find a function  u G Hr)(£l) such that  Following Jannelli and Solimini [43], we shall "bend", cut-off and rescale (f), to get it into 0 while still controlling its various norms. Indeed, denote For any a > 0, the change of variables 9 (x) a  words, if  x' = ( x i , x  n  _ i , 0), while  x = x' + x e . n  n  = x - ^\x'\ e is measure-preserving, in other 2  n  Je is the Jacobian matrix related to 9 , then \det(J )\ = 1. Define the bending a  a  8  6a  4>^(x) = 4>(8o-(x)). By direct computations, we know that \x\*  JRP  JR™ \e« (x)\°  -  l  -  a  x  JRI | + a | * ' | a | . x  ux  Cn  = l - C 2+ o ( 0 .  Consider the functional IQ(V) — \ J  \\7v\ dx —  J  2  RN  ^^r~dx.  RN  By a variant of Pohozaev  identity [27], one has  £[Io(<Kx - ek\xfe ))) = n  "  e  0  |W| V P = Ck,  = £ f  e  J{x =0}  Z  n  where C > 0, therefore, for sufficiently large a > 0, we have  W ) a)  = \»s(RD  -^U) l^ +C  °( " )( J  (1.2.12)  1  Combining the above two identities, we obtain / J  | V ^ |  Rn  2  = / " J  Rn  \Vcj>\ + C^ + o(-) = p (R )+C^ a a a 2  n  s  Note that for 7 = 0, we have ^  +  + o(-). a  (1.2.13)  = </>, which means that there is no any error term in the  above estimates. Define now a cut-off function ib , such that 1/v = 1 for |x| < \8o and ib = 0 for |x| > 8a, ip a  a  is radially symmetric, and \ip' (r)\ < C\.  Through some careful but elementary calculations,  a  we obtain, when n > 4,  Similarly,  /  l^' '" i- 2-o(l). r  =  |x|  C  cr  s  a  Set now ^ (x) a  a  =  o-^^ \ax)ip (ax). a  a  9  and note that supp(</> ) C P «5 C CT  7>  invariant under the scaling  u(x) ^  Q for every a > 0. Since H V u H ^ ^ ) and j  \x\  }  rV  u  are  the following estimates then hold:  (ra),  (1.2.14)  (1.2.15) Combining (1.2.14), (1.2.15) with the curvature condition near 0, i.e, 7 < 0, and letting a go to infinity, we arrive at the conclusion that  p (£l) < /u (vR"). s  s  E x t e r i o r Domains: The "strict concavity of 0, at 0" (implied by the strict negativity of the principal curvatures of <9Q at 0) is not necessary for the existence of the solution to (1.1.2), since there are domains Q that are flat at 0, yet satisfying /i (f2) < s  p (R ] ). These examples are based on the following r  s  r  observations:  Iffl. is an exterior domain with 0 G dfi, then p (ty =  P r o p o s i t i o n 1.6.  s  Indeed, the hypothesis means that R \Q, n  p (R ) n  s  = U < A < i A a Because 0  = p (R \{0}).  p {n) = s  n  s  n  s  is connected and bounded. In this case, we have  n  R \{0}  p (R ).  C%°(R \{0}) n  is dense in  H^lff ) 1  for  n > 2, we also have  Combining these two facts with scaling invariance, yields easily that  p (R ). n  s  The above remark allows the construction of various interesting examples. be any exterior domain with 0 G dQ and define Q  r  Indeed, let Qo  := fio l~l v3(0,r), where B(0, r) is the  standard Euclidean ball with radius r > 0, centered at 0. Obviously dQ, is smooth at 0 and r  Ms(^ri) < A*s(^r ) ^ i > 2 - We have the following r  r  2  There exists r > 0 such that r —> p (fl )  P r o p o s i t i o n 1.7.  s  0  decreasing on (r , +00). In particular, p (R ) n  0  s  10  < Ms(^r)  r  < fi>s(R+)  is left-continuous and strict for all r G (r , +00). 0  Proof.  Using similar arguments as above (scaling invariance and approximation of smooth  functions), combined with the smoothness assumption on dflo, one can easily observe that:  Ms(^o) = inf ri (fir) s  and  p {Rl) = s  Now we claim that for all r > 0, /J (fi ) > i (R ). r  r  s  r  Indeed otherwise, by Corollary 1.10,  n  s  sup// (fi ).  s  there is some r* > 0, such that /z (£2 .) = /j, (R ) is attained by some function u G ifo(fi ») n  s  s  r  r  u > 0. In other words, n (R ) is also attained by this function u, hence u satisfies the n  with  9  corresponding Euler-Lagrange equation in the whole space, while by the Strong Maximum Principle, we know u > 0 in R , which is a contradiction. n  The argument for the left-continuity of // (O ) goes like this: For a fixed r > 0 and arbitrarily s  r  small e > 0, one can always choose a function u G Co°($7 ), such that r  and  J \Vu\ dx < /i (fl ) + e, 2  n  s  r  ^ j | ' dx = 1. Since supp(w) is compact, the distance dist(9B(0, r), s u p p ( « ) ) =: 5 > 0. (  x  S  It follows that supp(w) C fi ', where r—5 < r' <r, hence ^ (^V) < Ms(^r)+e, for r—8 < r' r  which means that p, (Q ) s  that  S  <r,  is left-continuous. This implies that there must be some r > 0, such  r  p {RX) > ii (n ) > l^ (R ). Now define r := inf{r > 0;p (R ) < /x (ft ) < p (Rl)}. s  s  n  r  n  s  s  0  s  s  r  It is clear that for every r > r , ns(R ) < / z ( f i ) ' < ixs(R\). Suppose now there exist r > T\ > ro, but /v, (f2 ) = ps(nr2). Using Corollary 1.10 again, there exists a nonnegative n  2  0  s  function  r  n  u\ G i?o(O ), where ri  s  /i (fi s  r 2  is attained. Hence U\ satisfies the corresponding Euler-  )  Lagrange equation in fi , and again this violates the Strong Maximum Principle, hence the r2  strict monotonicity. R e m a r k 1.8. In the above situation, both cases r > 0 and TQ = 0 could happen. Indeed, 0  a) If R \Tlo  = 5(0, r*) nR^, then r >r*.  n  0  Notice that in this case, we have  fx (fi ) < ^s(R-l) s  r  whenever r > r , and therefore there exists a solution to (1.1.2), though dQ, is flat near 0  0. b) If  R \W = B := {x = (x', x ) G R : (x - 5f + \x'\ < S }, then r = 0. n  n  0  s  n  2  n  11  2  0  1.3  Blow-up Analysis and Attainability of Best Constants  In this section, we show that some aspects of the well known blow-up techniques are still valid in our context. The novelties here -when there is a singularity at 0 G 5ft- are the fact that the energies are not translation invariant, and that the limiting case is the half-space RJ\. as opposed to all of R . n  Consider the Dirichlet problem  -Au  = Au^  + ^  in ft  H  (1.3.16)  |a|  u = 0  on - 5ft  where ft is a bounded domain in R , 0 G ft, 0 < s < 2 < n and 2 < q < 2*(0) = n  Here  A > 0, if q > 2, but we can take A G R, if q = 2. The following discussion applies to the cases where 0 € ft and where 0 G 5ft, a boundary that is smooth near the origin. The "limiting problem" will be: -Au  = l£g?i  on  u(x) —> 0  M  (1.3.17)  as  —»• oo,  where  f R,  if  0 G ft,  { Rl,  if  0 G 5ft.  n  M=  {  The energy functional for (1.3.16) is well defined on HQ(0) by  while (1.3.17) corresponds to the functional I  Q  1  defined on D ' (M), 1 2  V JM \ \  JM  x  which is the closure of CQ°(M)  under the norm  ||M||DI,2(M)  = /  |Vit| . 2  JM  In view of Egnell's result, both limiting problems have a solution corresponding to a critical  12  point of IQ. The following is a direct extension of the known case when s — 0 , established by Struwe. T h e o r e m 1.9. Suppose {um)m  0 strongly in  is a sequence in HQ(Q) that satisfies I\(u ) —»• c and I'\{u ) —> m  as m —> oo. Then, there is an integer k > 0, a solution U° of  (1.3.16) in Hl(Q), solutions U\...,U  of (1.3.17) in 7 > ( M ) c D > (R ),  k  radii r ^ , . . . , 1) w  m  > 0  —> f / °  m  such that for some subsequence m —» oo  weakly in  l  2  ;  —>  2  n  sequences of  and  0  HQ(Q),  k  2) \\U -U°-  J2( m)^ (( L)~ -)\\ rJ  m  UJ  r  0 ; ^ e r e || • || zs tfie norm  l  in D > (R ), l  2  n  3=1  k  is the norm in .^(M  s  dx),  j=o k  4)h(Um)->h(U°)+Y I (U'). l  0  3=1  Recalling that a functional / is said to have the Palais-Smale condition at level c (P-S) , if any c  sequence (um)m  in HQ(Q) that satisfies I\(um) —• c and I'\{um) —• 0 in H~ (Vi) as m —» oo, is l  necessarily relatively compact in 5 Q ( 0 ) , we can immediately deduce from the above theorem that I\ satisfies {PS)C for any c < 2 ~^^ps(M)'^. 2  This implies the following:  C o r o l l a r y 1.10. Suppose that 0 G dVt and that dtt is smooth near the origin.  1) If p (tt) < p (R \), 1  s  s  then p (Cl) is attained. s  2) If the principal curvatures of dVt at 0 are negative, and if n > 4 , then there is a positive solution to (1.1.2).  Proof: The above theorem yields that I  0  p (Sl) < p (R i), v  s  s  satisfies (PS)  C  then  13  for any c < ^0^p (R%)^. s  1) If  where  P={peC° That (3  =  2  0  can be proved using the similar argument for s — 0 [56, p. 178]. The  A*s(0) ^  (n-s)  ([0,1] : H*(tt)) : p(0) = 0, I (p(l)) < 0} .  mountain pass theorem yields a sequence Uk £ #o(ft) such that 2-s  Io(uk)  and d/o(«fc)  A* (ft)^  2(n - s)  s  The (P-S) condition yields that uk - » u in #o(ft), is /  |V^/| = / 2  n  n  J (M) 0  0  in  2^)^s(^)  =  #  _ 1  (ft).  ^  and dl (w) = 0; that 0  g r , so that / | V u | - - f T%\= Io(u). p Jn \x\s J  s u p i o ( » = sup  t>o  2  t>o [2 Jn  But  L iv«|  n—s 2-s  5  sup J (*u) = 7w r t>o 2(n - s)  , \2/P / r 1H  0  p  which implies that Jn  |V"I  2  = ^s(O),  (in & ) is attained at u. For 2), it is enough to combine assertion 1) with Theorem 1.5.  The proof of Theorem 1.9 requires several lemmas, some of which are quite standard, like the following Brezis-Lieb type lemma (when s = 0) L e m m a 1 . 1 1 . Assume {un} C #o(ft) ^ °h  that u —> u a.e. on ft and u -> u weakly in  su  Then,  H^(Q).  1) v  n  f K r ! _ f l"n-tt|" Jn \x\ Jn \x\ s  /  s  |Vu | - / \Vu 2  n  n  n  . r ]«|£ Jn |xi as n —> oo s  - V u | -> / 2  n  n  |Vu|  2  14  as n -> oo.  n  3)-  Ifu ^u  weakly in D > (R ), then " j * | > l  n  2  n  | u  \ ^ - n \ ^ { u  n  J ^ i n  - u  Proof: The first two assertions are standard. Here is a proof of 3).  H~ (R ). 1  n  By the mean value  theorem, we have  u\  u  p  \u -u\  2  n  (u -u)  p  n  2  n  \x\  \x\  s  For R > 0 and w € D(R ), n  |p-2  I J\x  \u \ p  +  2  n  2  - ^  a:  -u))w  n  MP'  2  \U\\W\  X  \Un\  y  IP-2  \U\\W\  . S(p~2)  \x\  ,8,8  P  v  + c  v  (/  I  +  u\ \u\\w p2  X\  P  P P  ' \J  )\X\>R\A ) S  " \J  N V  \x\°)  l/p  \J\x\>R  <  -(u  X  n  \%\  < C  | « i r  we get from Holder's inequality:  -u \p-2  x\>R \  < ( p - l ) [K| +  n  s  ""(/EE  Here we have used the Hardy-Sobolev inequality:  as)  We also have that  /  \u\ ~ u -w X p  to  2  u x\>R  J\x\>R  |£| ' S  lp-1  (  1  ~P  w  l/p  u\ \ ~^ p  x l/p  <C  W  dx  \X\P  )  19)  ( /  By the dominated convergence theorem, for every e > 0, there exists R > 0 and k > 0 such that for all n > A;, we have / |w | w p_2  n  \x\>R  V I-!* 2  \u - u\ ~ (u - u) p  2  n  n  I-!* 2  15  |«| ~ p  \Y X  2>  w < e\\w\\.  As in [37, Lemma 4.3], we have on B(0, R),  I  \u - u\ ~ (u - U) p  2  n  n  x  J\x\<R  and  w  —> 0  n —> oo  as  f \ \ -w~\ U P  L  UW  u\  2  p  n  n  \x\<R  \  J\x\<R \  x  as  '  x S  n —> oo.  Hence P-2 J\x\<R '|a:|<i? V  x  x  7I.IP-  \u\ ~ u -w, x p  2  \  J  J\x\<R  2  which completes the proof, L e m m a 1.12. Consider (un)n in HQ(Q) such that I\(un) —> c, and dl\(un) —> 0 in H  1  (f2).  n-2  For (X ) G (0, oo) with X —> 0, assume that the rescaled sequence v (x) := X n  n  n  such that v —• v weakly in D ' (M ) 1 2  n n  u (X x) is n  n  and v —• v a.e. on Ft .  n  n  n  n  Then, dIo(v) = 0 and the sequence w (x) := u (x) - X n  n  v  2 n  satisfies Io(w ) —> c - IQ(V), dl (w ) —> 0 in H (Q) and \\w \\ = \\u \\ - \\v\\ + o(l). 1  n  0  2  n  2  n  2  n  Proof. Easy computations yield the dilation invariance:  V (X  U (X x)^  2 n  n  n  dx JR  R  n  n  _ \u (X x)\  p  [K JR n  IRP  PI"  JR  n  s  n  n  PI"  n  \Ur.  dx JR"  p|"  dx,  therefore Io(vn) — Io(un), i-e., the functional To is invariant under dilation. Since vn D ' (R ), l 2  n  it is clear that ||w„||  2  =  (Vw , Vw )zi n)  =  \\vnf + INI - 2(Vv, Vv ) = K | | - H  =  \\u \\ - ||u|| + o(l),  n  n  (R  = {Vv - V v , Vv - Vv) n  2  2  n  2  n  2  n  16  2  + 0(1)  vm  Since vn —> v weakly in D (R ), 1,2  I (w ) 0  dIo(u ) —> 0 in H (f2), Lemma 1.11 leads to  n  x  n  = I {v )-I{v)  n  0  + o(l) = I (u )-I {v)  n  = c-I (v)  0  n  + o(l)  0  + o(l)  0  Since A —> 0, we have dIo(v) = 0, and again by Lemma 1.11, we finally obtain n  dl {w ) = dl (u ) - dl (\^rv{--)) 0  n  0  n  + o(l) = o(l).  0  We also need the following:  L e m m a 1.13. Ifu G D > (R ) l  2  andve C^{R ),  n  ) l7/.|p  r  2  Proof. By Holder's inequality,  \ 1_ )  u  v  \  2  ,2s  ' .  2  < ~ J(fJxW- (f \ \ y \X\  J ~W rv \u\ 2  then  n  v  \X\ P  P'  _ r \u\ ~  p  p  uv p  HfV * -  V-'supp^ l^l  /  \J  \\  X S  J  Now apply the Hardy-Sobolev inequality.  P r o o f of T h e o r e m 1.9: Let u  n  in H~ (Vl). 1  —> u be in Hl(Vt) such that I\(u ) -> c, and dl\(u ) n  n  —> 0  That such a (PS)-sequence is bounded, is well known and can be found in [37,  Lemma 4.4]. Note that when q = 2, A can be chosen to be any real number. There exists therefore a subsequence, still denoted by (u ) such that for some UQ G #o(Q), u n  n  — > UQ weakly  in HQ(0,) and V u —>• VUo a.e. A n easy consequence of Lemma 1.11 is that dI\(Uo) = 0. n  Moreover, the sequence u\ := u — UQ satisfies n  IKII = I W - ' I W + o ( i ) 2  2  (*)< dl {u )^0  in  l  0  k  n  I (ui)^c-I (U ). Q  x  17  0  H-\Q)  Case (1): If u n l  0 in D>(n, | x | - d x ) , then (dl0(un),u^)  d / « ) -> 0. It follows that un -> 0 in Case (2): If <  2  n  a  and we are done.  ^(fi),  0  = f | V < | - J Jgjf - 0, since  0 in # ( 0 , \x\- dx), then from s  Jn  Ja  x  and i  \p\ /P 2  / |Vutf>,u.(W / l ^ j V./n  in we have  J Jn  for large n and we may therefore assume that  SRU  \X\  S  for some 0 < 5 < (J^j  > 5  Define an analogue of Levy's concentration function,  =/  Q n { r )  bP  JB(p,r)  \ \ x  Since Q (0) = 0 and Q(oo) > S, there exists a sequence A* > 0 such that for each n, n  S  5= [ Define vn(x) := ( A ^ ) ^ M ^ ( A ^ a ; ) . Since D ' (JR ) weakly, 1  2  n  Define ft = n  = \\un\\ is bounded, we may assume vn —> U\ in  -»• t/i a.e. on I T and 5 = J and let /  n  l  Jn„ ^9n  ^  dx. We now show that Ui ^ 0.  G i?o(^) be such that for any h G HQ(Q,), we have (dIo(U^), h) =  In V / • V / i . Then g (x) := (\ )^ f {\ x) n  B ( 0 1 )  n  n  n  n  satisfies  |V^ | = / 2  n  n  |V/ |  2  n  and < d / ( t £ ) , ft) = 0  • V / i for any h G i7"o(fi ). n  If C/j = 0, then TJ* -»• 0 in L (B(0,l), p  loc  \x\~ dx). s  18  Choosing h G C%°(R ) n  such that supp  h C B(0,1), we get from Lemma 1.13,  |V(K)|  /  2  =/•  V ^ - V ( ^ ) + o(l)  =/  ^  fact that J ^ ^ B  oc  2  —  <\j 2  / |v(K)| + °(i)  A*-C-«T*) ^-^^^  —:  Hence V v £ ^ 0 in L  + / v ^ - V ( ^ ) + o(l)  | V « ) |  + o(l).  2  ( S ( 0 , 1 ) ) and ^ -»• 0 in If(B(0,1),  \x\~ dx), which contradicts the s  = 5 > 0. Thus we have proved that U\ ^ 0.  Q  Since ft is bounded, we can assume that A* —>• A ^ > 0. If A ^ > 0, the fact that u\ —>• 0 weakly in H^tt)  will imply that v*(x) := (X^)^  contradicts that Ui^O,  ul(\ x)  -+ 0 weakly in D - ( i R ) , which 1  n  2  n  and therefore A* —• 0.  (*) and Lemma 1.12, dI0(Ui) = 0, and E/i is a weak solution of  By  = H  -Au  w —> 0 where M = R  n  can  in  M  as  |x| —• oo  if 0 e ft and where M = JR™ if 0 G 5ft. Indeed, to show the latter case, we  assume without loss of generality that dR\  = {x  n  = 0} is tangent to 5ft at 0, and that  —e = ( 0 , — 1 ) is the outward normal to 5ft at that point. For any compact K c R™., n  n K = 0, as A^ —*• 0. Since suppwfc C  have for k large enough, that  iR , it follows that i> = 0 a.e. on K, and therefore suppw C re  The  sequence u^x)  : =  u,n2l II  (^J^J  (X^f^Ui  —  2  _  a  ^  s o  satisfies  «n|| -|W -||^l|| 2  2  2*  /o(<)-c-/ (Ub)-Wi) A  d/ (w )^0 2  0  in  F-Hft). 19  R\.  2  + 0(l)  and v  k  we  —> v a.e. in  Moreover, any nontrivial critical point u of I on H$(M) satisfies 0  so that  By iterating the above procedure, we construct similarly sequences (Uj), (A") with the above properties. Since for every j > 1, IQ{UJ) > c*, the iteration must necessarily terminate after a finite number of steps. R e m a r k 1.14. This type of blow-up result also holds for domains fi with a conic singularity at 0. More precisely, consider an infinite open cone of the form C = {x € M ;x D and r > 0} where D is a connected domain of the unit sphere 5  n _ 1  n  = r6,6 G  of M ,  and assume  n  the domain fi satisfies C D fi = C fl (fi D f? ) for every ball /3 (centered at 0) with radius r  r  r < ro, where TQ is some positive number (i.e., Q, has a conic singularity at 0), then Theorem 1.9 remains true, with M - i n this case- being the corresponding infinite cone C.  Another consequence is the following corollary obtained by combining Corollary 1.10 with Egnell's analysis, which imply that lim ^ + Ms(^ H B ) = fJ, {R+ )• r  C o r o l l a r y 1.15.  r  0  s  Suppose that 0 G dQ and that dQ is C at 0. If p (Q) is not attained, then 2  s  there exists ro > 0 such thatfifl  B  ro  ^ 0  and p, (fi) 3  = // (fi fl B ) s  r  for every r G (0, rn).  Note that Theorem 1.1 implies that /z (fi) is not attained whenever fi is star-shaped around 0, s  and therefore there is no ground-state solution for (1.1.2). The following standard Pohozaevtype identity, gives a stronger result: P r o p o s i t i o n 1.16.  If the domain fi is star-shaped around 0, then problem (1.3.18)  has no non-trivial solution. 20  Proof. The assumption fi is star-shaped around 0 simply means that x • 7 > 0 on <9fi \ {0}, where 7 is the outward unit normal to dfi. Multiply the equation (1.3.18) by x • Vu on both sides and integrate by parts, we obtain  - / \Vu\ x • d a + — - / \Vu\ dx = —- / JdQ. 2 Ja 2*{s) J 2  -dx x  2  7  z  s  Q  On the other hand, multiplying the equation by u and integrating, we have  Combining these two identities, one gets J  \Vu\ x-^ydo = 0, which concludes the proposition. 2  dQ  R e m a r k 1.17. Unlike the case s = 0 , we can have solutions to (1.1.2) for star-shaped domains. Indeed, consider a bean-shaped domain with vertex at 0. Since the principal curvatures are strictly negative at 0, there exists a solution to (1.1.2). Note that this is not contradictory to Proposition 1.16, since the domain is not star-shaped at 0, though it is star-shaped at some other point.  1.4  Least Energy Solution to The Perturbed Dirichlet Problems  Throughout this section, we assume that fi is a bounded domain in ]R and that 0 € <9fi, dQ, n  is Lipschitz continuous, <9fi is C at the origin. Consider the functional 2  on H^(Q), where 2 < q < 2* := We shall deal first with the case of linear perturbations. T h e o r e m 1.18.  Let fi be a bounded domain in JR! with Lipschitz boundary and consider th 1  Dirichlet problem (1.4.19)  21  for 0 < s < 2 and n > 4. Assume that 0 G dft and that <9ft is C -smooth at 0. If dQ has non2  positive principal curvatures on a neighborhood ofO (in particular, if 89. has negative principal curvatures at 0), then for any 0 < A < A (1.4-19) has a positive solution. 1 ;  Proof: The results of the last section give that I satisfies the Palais-Smale condition (PS) q  for any c < (n-s)^(^P+)^• 2  C  S°> we need to find a critical level below that threshold, for the  functional  on the space HQ ( f t ) . To use a mountain-pass argument, note that since A < Ai, then 0 is clearly a strict local minimum for I. The condition on the curvature at 0 implies that -modulo a rotation- there is some P s C ft, where 7 < 0 and 5 > 0. Since p (Cl) < p (RJ\_), we only need to consider 7j<  s  s  two cases:  Case 1: ^ ( f t ) <  p {R\). s  By Corollary 1.10, there exists then a function  w G i^o(ft), such that  J  |Vu;| = 2  Q  /^ (ft) s  and  dx = 1. Without loss of generality we can assume that w is nonnegative by replacing w with \w\. Since A is positive, we have the following inequality:  f 1 1 v* sup I{tw) < sup J{tw), where J(v) = / {-|V?;| }dx. t>o t>o fn 2 2*(s) \x\ 2  2  (s)  +  s  Since s u p  i > 0  J(tw) = ^ " 1 ^ / X g ( f t ) ^ , the conclusion follows.  Case 2: ^ (ft) = s  2  p {R\). s  This means that 7 = 0 in view of Theorem 1.5. In this case, we will closely follow the strategy used in Theorem 1.5 where we start from an extremal function  <f>  G  HKR ^), 1  and through  cutting and scaling, we get a test function 4> on ft, whose various norms are controllable a  perturbations of those of <f>. Note that bending is not required here, therefore we only need to pay the cost of the scaling and of the cut-off. 22  As mentioned in Theorem 1.5, the decays estimates on <> / and tp are: \(/>(x)\ ~  jzfi=2,  |V0(a;)  i^pr and \S7ib (x)\ ~ ^ . Since no bending is required, direct computations show that a  /  ,  IWIUI --^, 2  J\X\>±5<T  °  r  c  |0 | *w f f  \x\>±8a J\x\>hi  2  \\ X  N  S  We therefore have the following estimates: / |V^|  2  Ja  = ^(J£)  + 0(-^), a  f \4>o = Ca Ja  2  9  +0(0  /" \6 \|2*(«) *^  ^  2  'n ~^' Ja  _  n  )  5  1 1  ^  L  2  n  _  s  For 2 < g < 2*, we obtain  = a  q J  ^~  j \^\x)^ {x)\^dx  n  Rn  q(n-2)  = Ca 2  n  + o(a  a  q(n-2)  2  n  ).  Notice that when q = 2, the order of 0" is —2 and the above estimates, combined with the assumption fj,s(Ci.) = /^(J?™) give, for n > 5,^  = f ( ^ ) +O ( ^ ) ) -  Pl'Kifa + O(^))  - XCt ^ 2  + 0(4,).  Since A > 0, then for cr large, the minimum is attained in a uniformly bounded interval, and achieves its maximum at t M , where  it is easy to see that sup 1(t<f>) t>0  a  t  = Hs(^)^  =s  M  23  - C a  -2  Substituting the value into the expression of I(t<f> ) and noticing that ^ a  is bounded when  a —> oo, it eventually leads to  J~ / x ( » ) ^ - Co' + o(a~ ), z{n — s) S  sup/(t0 ) =  t>o  a  2  2  8  where C > 0 is independent of a. From the above identity we can see that for sufficiently large cr,  2  —  SUpI(t(f) ) = — a  t>o  S  2(n — sj  n—s  r/X (fi)^, s  and we are done. The case n = 4 could be treated similarly, with the help of the stronger estimate  / \<p \ ~CoJn 2  f Jhi  2  a  and the proof is complete. Now we deal with the Dirichlet problem with a non-linear perturbative term. T h e o r e m 1.19.  Let Vi be a bounded domain in M with Lipschitz boundary. Assume also th n  0 G dQ and that dVl is C -smooth at 0. If n> 4, then equation 2  -Au  = l " ! ' ^ " + Add-?" 2  2  | x |  1  u = 0  1  in ft. (1.4.20)  1  on dfl.  with A > 0 has one positive solution under one of the following conditions: 1) ^ <q<2*,where2* 2  1  = 2*(0) = ^ ,  2) 2 < q < 2* and dtt has non-positive principal curvatures in a neighborhood o/O.  Proof: The idea again is to try to find a critical point for the functional C  1  1  24  2'W ^ '  \  in HQ(Q) through a mountain-pass argument, by using that I satisfies {PS) for any c < Q  C  2 ( r a " - s ) M s ( ^ ) ^ - As above, we need to deal with two cases. S  r7  ^(fi)  Case 1:  s  < p {Rl). s  As before, there exists by Corollary 1.10, a positive function w G HQ($Y), such that f A* (fi) s  a  n  (  l JQ ^ \x\e W  f 1 < sup J(tw), where J(v) = / {TT|Vi;| -  q  t>0  J{tw) =  1 v* 2  2 2 7  JQ  Z  (s)  r-rTjVl^i  2  t>0  t > 0  =  = 1- Since A is positive, we have:  }  supl (tw)  while s u p  \Vw\  2  Q  \)  A  S  Fl  ^y^ (^)^s  Case 2: /x,(fi) = /x (J^). s  Again, as in Theorems 1.5 and 1.18, from an extremal function <f> G H^(RJl),  one gets through  bending, cutting-off and scaling, a function <p on fi, with the following estimates:  /  a  /  |V ^ |  = fi (R )  2  + C^ + o(-),  n  s  +  cr  Q JQ  |^|« = C o - ^ -  n  / - ^  + o(a^- ),  for 2 < g < 2*,  n  =  1  _^  + 0 (  (1.4.21)  a  (1.4.22)  I).  ( 1  . . 4  2 3 )  Now we estimate the mountain-pass value. By (1.4.21), (1.4.22), (1.4.23), and the assumption /x (fi) = id (R \_), we obtain r  s  s  = £ Mn) + ci)  - ci)  -1 ^(^ 2  -  C  A  *  ^  Since —1 < ^ ~ ^ — n < 0, sup I (t (f> ) achieves its maximum at 9  n  2  a  t>0  t  M  =//  s  (fi)TO  Ca  _  q J  tM,  + o(a-*^-). where  ^' . n  Substituting the value into the expression of I(t(j) ) and noticing that tM is bounded when a  a — > oo, this eventually leads to sup/(r^) = t>o  2(n -  r//  s)  s  fi)a- + O(70-  25  1  - CAo-  2  n  + o(a  2  n  ).  Hence for a is sufficiently large, without any restriction on 7, the range of q in 1) guarantees that 2—  '  s  SUp I[t<f> ) <  n—s  rfJL (Q)  a  a  t>o  ~<  2  2{n-s)  In part 2) now we only need to deal with 7 = 0 (since 7 < 0 belongs to Case 1, which has been discussed). As in the proof of Theorem 1.18, no more bending is required, therefore we only need to pay the cost of the cut-off and scaling, hence we have  J  /  Jn  .  M = ^(") + o(-^j), 2  IQ  a  Jn ,  10  b \  q  a  /  ^  9  1.5  2 )  q(n-2)  2  \6 l * ™  =l +  2  n  + o(a  2  n  )  ;  1 0(—).  ( s )  m  In Jn  We require ^ ~  9(n-2)  = Ca  ~^ ~ '  s  n  s  - n > —n + 2, hence the conditions q > 2 and n > 4 are sufficient.  The Neumann Problem  When <9ft G C , it is easy to see that the embedding i f (ft) < — > LP(Q, \x\~ dx) is continuous, 2  1  s  where p is the Sobolev-Hardy exponent. Just as in the non-singular case, problem (1.1.4) has a variational structure. It is easy to check that the positive solution of (1.1.4) corresponds to the nonzero critical points of the functional  r 1 Jr. 2  1 1  1  u  r{s)  2*(s) \x\  s  1  2  J  defined on i f (ft) and the norm ||it||j/i(n) := | | V u | | i 2 + ||w||x,2 is equivalent to 1  \u\ = ( f (\Vu\ + Xu )dx)^ Jn 2  2  H  The following lemma deals with the relative compactness of Palais-Smale sequences. T h e o r e m 1.20.  Let (UJ) be a sequence in i f (ft) such that J(UJ) 1  — •  c and J'iuj)  —>  0  H~ (Sl) as j —> 00. / / the value l  c<-^-u. (R )^, n  4(n —  s) s  26  (1-5.24)  in  then there is a non-zero u G H (£l)  such that J(u) < c and J'{u) = 0.  l  We first state one lemma which is similar to those in [64]. L e m m a 1.21. Denote B = -Bin{x > h(x')}, where Bi = B(0,1) is the unit ball in R , h(x') n  n  a C function defined in {x' G i ? 1  For any u G H (Bf)  n _ 1  , \x'\ < 1} with h, Dh vanishing at 0'.  with supp(u) C B\, we have the following:  l  (i) lfh = 0, then  r  p 2*(s)  / \Vu\ dx>2^p (  (1-5.25)  2  ^  s  JB  -  7  ^  )  ^  JB F T  (ii) For all e > 0, there exists 8 > 0, so that if |Vft| < 8, then  r  r 2*(s)  / \Vu\ dx>2^^ - )(  )2%y  2  s  6  JB  P r o o f of T h e o r e m 1.20: Consider a sequence (UJ) C H ^) 1  J'iuj)  —> 0 in H~ (Vt)  as  l  (1.5.26)  JB F l  j —> oo,  such that J(UJ) —> c and  that is,  J( ) = j f ( i ^  (1.5.27)  U i  /  r  (J'(WJ-),  \2*(s)-l  = / (VujVp) -  ^ + A « ^ ) c f a -> 0.  in  F r  Let ip = Uj, then C =  (I_  1 ) / ^Ll  dx  (1),  + 0  2 2*{s) J \x\ . so that \UJ\H < C*. Extract a subsequence, still denoted by Uj such that K  J  s  W  Q  Wj  u weakly in ^ ( f i ) and weakly in ( L * ( f i , |a;|~ )Gfo))*, 2  27  (s)  s  (1.5.28)  and  2n  Uj -> u strongly in L (Q), q < 2* = g  and strongly in L (d£l). 2  7% ~~~ £  Passing to the limit in (1.5.28) we see that u is a critical point of J. Thus it suffices to verify that u^O.  Supposing not, we get that:  / u dx —> 0, as j  Jn Let  2  oo.  —y  (1.5.29)  e be a small positive constant to be determined later, and let  (<p )a=i a  be a unity partition  on 0 with diam(supp<pQ,) < 5 for every a. Here e,8 > 0 are related as in Lemma 1.21. Since dfl G C , from Lemma 1.21, we have 2  / Wwpatfdx > (2^p Jn  s  a = 1,N,  - e){ f Jn  ^ l dx}^,  lu  \\ x  a iS)  u G i? (l2), and 5 small enough. Thus 1  <(2^u. -e)-iZLJnWujvh\ dx 2  s  < (2^  -  s  e )  -i{(i +  J  e )  n  \ .\2 Du  +  C e  J \utfdx} Q  = ( 2 ^ - e)- {l + e) f \Dutfdx + o(l). l  s  a  From (1.5.27), one has  f (\V \  2  n  Uj  - ^  + \u )dx = o(l) 3  (1-5.30)  2  Combining (1.5.27), (L5.28), (1.5.29), (1.5.30), we get  ( ); ] In \ \ 2  2(n-s)  is  Uj  x s  2-s  c.  Hence  ,2(n  —  s) .a  (^3^c)  P  s  -2  < (2^  ,  M s  - )-\l e  28  . 2(n —s) + e)-^r7 ' i c  namely, c >  { ( 2 » - » p - e)/(l + e)} 2-'. Choose e > 0 sufficiently small, the above contras  dicts (1.5.24). Thus u £ 0. Finally we show that J (it) < c. Set TJ, = Uj - u, hence t>,-  0 in H ^). 1  In the same spirit as  in the Brezis-Lieb lemma (see [11]), the following holds  r (u-) * 2  Ja  r (v) *  (s)  \x\  Ja  s  2  (s)  Fl  s  ru* 2  Ja N  {s)  s  {  )  and /  | V u / = J \VVJ\ +  / |V?j| dx + o(l),  2  Ja  2  Ja  Ja  therefore from (1.5.27), (1.5.30), we have  and  Ja  Consequently, we get  Ja  Fl  J(u)=c + o(l)- J ~  A  n  [ \Vvj\ dx - ) Ja S  2  s  which implies that J(u) < c. R e m a r k 1.22. If we define  / ,  f \Vu\ dx  „ . ,I  + A L\u\ dx  2  2  n  2  g  then Theorem 1.20 is equivalent to the statement: If p ,x(0) s  < 2~  S  f i , then fJ, ,\(ty s  s  is achieved.  We now complete the proof of Theorem 1.3 by finding a least energy solution to (1.1.4). Since the boundary dVt is C , and the mean curvature of 30, at 0 is positive, the boundary near the 2  origin can be represented (up to rotating the coordinates if necessary) by: j  Tl—l  x = h(x') =-^cxiX + 2  n  1  »=i  29  o(\x'\ ), 2  where x' =  ...,x _i) G D(0,8) for some 5 > 0 where D(0,5) = B(0,8) D {x = 0}. Here n  n  a _ i are the principal curvatures of dtt at 0 and the mean curvature  i  a  n  the proof below, it costs no generality to assume that all OJ* > 0, % = 1 , n  u (x) = e 2 ^ ) ( e + e  >  0- For  — 1. Set  \x\ ~ )^. 2  s  Under the above assumptions, problem (1.1.4) possesses a positive solutio  T h e o r e m 1.23.  provided n > 3. Proof. For notational convenience, we denote 2*(s) by p throughout the proof. The solutions of (1.1.4) corresponds to the nonzero critical points of the functional ,P  J(u) = f [l\Vu\ - - ^ f - -\u )dx. 2  '  K  J2 l  ]  2  p\x\*  1  n  2  J  Set  c = inf sup J(ib(t)), ^ *t6(0,l) 6  {tp G C([0,1], H ^));  tp(0) = 0, J(^(l)) < 0}.  1  the so-called mountain-pass level, where * = We also set.  c* =  inf {sup J(tu);u > 0, u ^ 0}.  ueif(n)  t>o  It is easy to see that c < c*. In view of Lemma 1.20, we need to prove c* <  2 4  ~j^/x|~ . We 3  n  claim that n—s  Y = suVt>0J(tue)<^pr  (1-5.31)  £  for e > 0 sufficiently small. Denote n—1  K {e)= l  ! \Vu \ , 2  Jn  e  K {e)=!^dx 2  Jn  and x  ^5>x . 2  <?(*') = 1  i=1  The proof is divided into two cases. Case 1: n > 4. One then has  K^e)  =J  \Vu \ dx - J 2  Rn+  e  ' = \Ki ~ f -, dx' Jt'  ]  Rn  dx' ff* > \Vu \ dx + O ( e ^ ) 2  D m  €  n  \Vu \ dx - D(0,S) f „ dx'Jg(x') \Vu \ dx + O(e^). 2  e  2  n  30  D(0  a  X  e  n  where  K = f  \Vu \ dx = (n - 2) / 2  X  2  l  e  JR  ^—^dy,  (1 + | y | 2 - s ) - 2 T i -  JR  n  n  which is independent of e. Observing that  =J -idx'f ' \Vu \*dx 9{x }  Rn  =  Q  e  n  (n-2) e^f _ dx'f{ 2  Rn  2(n-s)  1  (e+\x\ ~ )^^ 2  =  (n-2) J _ dy'J^\^^  s  2  Rn  1  2(n-s)  )  we note that  '  2_  lxf 2S9(X  l i m - ^ / ( c ) = (n - 2) / 2  C  3)  dx',  which implies that /(e) = O ( e ^ ) . Moreover,  -\(n-  2) e^ f  f '  2  < r(n  ^  h{x ]  f  dx'\  \h(x')- (x')\dx' a  where C depends only on 5, n. Since  h(x') = g(x') + o(\x'\ ), it follows that 2  V  C  T  >  0, 3C(a) > 0 such that  \h(x') - g(x')\ <c\x'\ + C((r)\x'\% 2  and  g\x'\ + C{o)\x'\^ 2  h (e) < Ce^ /  JD\ >D(p,S)  (e+  2(n-s) |rr'| - )"^7 2  < Ce2^(cr + C(a)e2(2^)),  which implies ii(e) = o(e2^) as e — > 0.  31  s  Thus we obtain K {e) = \K x  - 1(e) + o ( e ^ ) .  X  (1-5.33)  On the other hand,  K^)  =J  Rn+  $dx - J  = \K, - t . ,  dx' J  D m  ^ dx  h{x,) 0  dx' Jf  s  $dx  1  n  + 0(e&)  n  - f  dx'  D m  ^dx  + 0(e&).  n  where K  — f  ^2  Mn-s)  M-—-F  e  JR.™ \X\* ~  f  C  JR  p(,»-2)  2(n-s) |x| (e+|x| - )^=^  n  s  2  s  dy  JR™ R  „  N  2  , „  dx  = fl-s  It is well known (see [37]) that K\, K  An-o  JR"  satisfy  K /iqr=n  :=  a  x  p (R ). n  s  Since  dx' [  9(X  //(e) := / JR™-  1  }  ^-dx  n  \\  JO  = [ JR"-*  X  dy  dy' [  n  M(l + M - ) ^ s  Jo  2  S  (1.5.34)  this means,  11(e) = 0(e&). Similarly,  dx' f h(x')^r-dx \ \x\ JD(O,5) Jg{x') f  s  n  — o(e -«). 2  Therefore,  K (e) = \K 2  2  11(e) + o(e&).  Moreover, careful calculations lead to 0(e5=i),  K:• (e) := K / = < 0(|e^lne|),  Jn  [ 0(e&), 32  n = 3 n= 4  n>5  (1.5.35)  Let t > 0 be a constant that e  J(t u ) = Y = sup J{tu ) t>o £  e  e  e  = supAi^e) + t>o 2  XK (e))t - -K (e)t*>}. p 2  3  (1.5.36)  2  For n > 4, 7^3 (e) = o ( e ^ ) , hence  Y = e  J(z^) < supine)* t>o  2  -  2  +  -/v (6)^] 2  p  =^^[_^L_]^f  o(6^)  2(n-s)  (K (e))^  +o(e^).  2  We claim that 2  2  1  1  iCi(e)/(iC (e))^ < 2 - ^ p 2  s  1  2  1  + o(e^) = - K i / ( - i ^ ) ^ + 2  o(e&),  which will lead to our conclusion. By (1.5.33), (1.5.38), the above is equivalent to  {\K, - I(e))(lK )&  < \K (\K  2  X  - 7/(6) + o{e&))&  2  - \K {{\K )& X  + o(e^)  - n=l lK )^II{e)}  2  {  + o(e^).  2  It follows that: Tim^o^y  =lim ^o^ £  l  n  J JK«-1  2(n-s) (l+|j/l " ) 2-s f , g(v')dy' JH™-1 2(n-s) lw'l (l+lw'l - ) ~ Z  2  2  s  fn-CM ^  2  f°° JO  r  +  2  s  dr 2(n-a) (l+r ~ ) ~ r ~ dr 2(n-s) r a  2  oo  s  s  n  2  2 3  s  2  3  s  Integrating by parts, one has for 2 < (5 < 2(n — s) — 1,  r°°  rP- dr  _ 2n-2-s  2  (l + ^ - s j i ^ -  P-1  1  f°° io  r^~ dr s  (l + r ~ ) 2  s  2 3  Observing that  I  7o  2(n-s)  ,  (1 -|- ~ ) 2  r  s  2-s  /  ./O  2(n-s)  (1 -f  i 1  33  /  Jo  .  (l + r  „ 2(n-s) 2 — s  ) ~ 2  s  hence  ^~ dr  8-1 f°° ~ 2n-p-l-sJ 2n-(3-l-sJ  s  Jo  n  (l  +  +  2 2 -  r r  s  s  )  ^ ^  P- dr 2  r  0  0  (  1 +  r  2  _  s  )  ^ l -  Therefore one has ^  n + l + £ n-3  ™ 11(e)  , ' '  1  and n-2Kx  (n-2f  n-siv,  Z*  r  00  I J JO  n-s  0  n + 1  -  2 f l  dr  Z  0 0  ~, Z . 2(n-s) / / 2 - ) ^ ./o (1 + (1 4- 2 s ) 2-s J O 7  (  1 +  r  s  r"- "^ 1  r ^ ) ^  r  (n — 2) n — s 3  =  n —s n— 2  (n-2f.  We get //(e)  n-sK  2  Case 2. n = 3. Careful calculations lead to Ki(e) < \K  - Ce&\ lne| + o ( e ^ ) for some C > 0,  x  (1.5.37)  K (e) = ^K -0(e^). 2  Let J(t u ) = Y = s u p 6  6  e  (1.5.38)  2  J(tw ). From (1.5.37), (1.5.38) and the estimate K (t) for n = 3  t>0  e  z  before, £ are uniformly bounded for e G (0, eo) for some eo > 0. e  Thus  Y < sup [f/^(e)* - \K (e)V) + O ( e ^ ) 2  e  =  t>0  2  2 ^ ) [ ^ - ] Z  [  U  S  > (if (£))n=5 L  F  5  + 0 ( ^ ) .  F  J  2  Consequently if iCi(e)/(tf (e))^ < 2 = % - O ( e ^ )  (1.5.39)  -  2  a  then (1.5.31) follows. By (1.5.37) and (1.5.38), (1.5.39) reduces to ]-K  x  - Ce2-s|lne| < 2 ^p [^K s  2  34  - 0(e*-°)]*-*.  +0(e*-*)  =  ^p Kp  +0(e^).  s  n-2  Since  Ki/K^"  8 —  p , we get (1.5.39) immediately. The proof of the theorem is complete. s  R e m a r k 1.24. Working in the framework of [1], the above calculation leads to \x\  un  s  I  as e —> 0, and where p ^ > 0 is a constant depending on n, s, H is the mean curvature of <9fi n  a  2  2 2_  at 0, £ = e~s for n > 4 (and e ^ In - for n = 3). Hence our theorem can also be derived from 2  s  e  e  the above formula and Remark 1.22 R e m a r k 1.25. As noticed in [53] (there s = 0), if fi is an exterior domain, the mean curvature at 0 (when seen from inside) is negative, then there exists a least-energy solution. The proof is almost the same as above. While if J ? \ f i is close to a ball in some sense, then for A > 0, n  (1.1.4) has no least energy solution.  One may of course replace the nonlinearity in (1.1.4) with a more general nonlinear term and obtain similar results. The same arguments also apply to get the following extension of Theorem 1.23, T h e o r e m 1.26.  Suppose that the mean curvature of 60 at 0 is positive, then the problem  \VU\P- VU-U 2  u  = 0  on 80,  >0  in fi.  (1-5-40)  has a solution. Here v is the outward unit normal to dVt, A > 0,1 < p < n,0 < s < p,p*(s) = ^ , A  = div(|VMr V ). 2  p  W  U  35  1.6  Sign-changing Solution to The Neumann Problem Via Duality p, and p (R ) n  For notational convenience, we again denote 2*(s) by  s  by  p throughout this s  section. Assume that 0 is bounded and the mean curvature of 80 at 0 is positive. Here we are concerned with changing sign solutions to (1.1.4). First we shall establish a similar result for 2 < q < p, the subcritical case _ A M  ^  M  ^  I  H  -  M  A  inO  (1.6.41)  N  £ To this end, for given  =0  on  u ^ 0,u £ L (0, \x\~ ), denote by q  s  on.  (vi[u), vi(u)) the first eigen-pair for  the eigenvalue problem:  ' -Av  = v^^-\v  g where v £ R.  infi  = 0  on  (1.6.42)  30.  Since A > 0, the variational characterization of the eigenvalues gives that  ui(u) > 0 and vi(u) cannot change sign in 0. Namely ,  £ H\Q), J  U!(u) = inf | | | V H l 2 + MHllw and vi{u) £ H (0) l  = 1, w ± oj  satisfies:  „ s _ l|V^(«)Hi + A||^(«)||i (  / r  |u[g- t>i(u) \; 2  \Ja  |x|  2  J  s  The eigenfunction Vi(u) is uniquely determined under the normalization:  r l^l " ^N 9  2  > o m n.  2 = x a n dU i ( u )  In fact, it is easy to see that the map u —> Vi(u) is a continuous map from L (0,\x\~ ) q  H\0),  s  to  f o r g e (2,p).  T h e o r e m 1.27. For n > 6 and A > 0,  there exists a nontrivial solution u of (1.1.4) satisfying:  r \u\ - uvi(u) p  Ja  2  M  s  in particular u changes sign in O. 36  =  Previous results on changing sign solutions for Neumann problem involving the critical Sobolev exponent has been established in [20], while for equation (1.1.4), similar multiplicity results have been discussed in [58] for s = 0 and in [37] for s > 0,0 G ft, through the duality method. The proof we sketch here is similar to that in [20]. First we need Definition 1.28.  Let X be a Banach space and B be a closed subset of X. We say that a class  T of compact subsets of X is a homototy-stable family with boundary B provided that (1) every set in T contains B, and (2) for any set A in J and any n G C([0,1] x X;X) satisfying w(t,x) — x for all (t,x) G 7  ({0} x  X) U ([0,1] x B) we have n({l} xA)zF.  ,  We say that a closed set M is dual to the family T if  M n B = 0 and M n A  0 for all  A G T.  We need the following weakened version of the Palais-Smale condition Definition 1.29.  A C -functional E on Banach space X satisfies the Palais-Smale condil  tion at level c and around the set M (in short, (PS)M,C)> if every sequence (u ) satisfying n  lim  n  n  E(u ) — c, lim | | £ " ( u ) | | = 0 and lim dist(u , M) = 0, has a convergent subsequence. n  n  n  n  n  We need the following P r o p o s i t i o n 1.30.  ([36]) Let E be a C -functional on X and consider a homotopy stable 1  family T of compact subsets of X with a closed boundary B. Let M be a dual set to T such th inf E = c := c(E, J ) = inf m a x £ ( x ) . 7  M  AeT xeA  ff E satisfies (PS) ,c, then M V\K ^§, M  C  where K is the set of all critical points of E at level c  c. 37  Consider now the ^-functional on  H ^) 1  = [[\\Vu\  Jn  -  2  A  + \\u ]dx. Q \\ * 2  x  where 2 < q < p = 2*(s). Define the mountain-pass class to be  H = {7 € C([0MH\n)) (0) = 0,7(1) n  ^ 0 and 4(7(1)) < 0},  which is homotogy-stable with boundary B = {J < 0}, and an appropriate dual set of T\ is q  M\ = {u e H\ty, u^0,<  J' (u),u >= 0, } q  the Nehari manifold. For q < p, the dual property is well know since J satifies (PS) condition, q  while q = p is just our result in section 4. Consider now the following homotopy-stable class F = {Ac  H\n);  2  A closed symmetric with i(h(A) n Si) > 2, V7i EH}  where 8 = {u e H (tt),\\u\\ l  = 1}, the unit sphere of H ^),  and H = {h : H ^)  1  1  ->  F/ (ft); h even homeomorphism}. Here 7 is the so-called Krasnoselski's genus, which is defined 1  as: 7(A) = inf{fc; there exists / : A —» i? \{0} odd and continuous}. fc  Set  Ml = Ml n{ue  H\S}), j \ \ ~ ^ ) u  Jn Ci  >q  =  inf  sup J , i — 1,2, q  9  \\ x  and  o},  u  d x  =  s  Q =  Q  I P  .  Note that ci is the mountain-pass value defined in section 4, so C\ = i n f M f is dual to ^  Ml  J. We will see that  by the following lemma  L e m m a 1.31. For q £ (2,p) #iere exists a nontrivial solution u of (I.6.4I) satisfying ;  J (u) = c , and / q  2 q  Jn  T-T  = 0,  \\ x  where vi(u) is the first eigenfunction for the eigenvalue problem (1.6.42) 38  Proof. Notice that h : u \—•  defines an even homeomorphism from M\ to S, since A > 0,  therefore for every A G F  we have ~f(A D M\) > 2. Furthermore, the map tp : A D Mf —> R  given by : tb(u) = J  ^H ^  2  ^  u  somewhere.  1  is odd and continuous . Since vi(u)  That is, M f is dual to T .  The duality implies that  2  establish the reverse inequality, for  > 0, u must vanish  c  2>q  > inf  A e M  | JQ  To  u G M. denote by (v (u), v (u)) the second eigen-pair for 2  2  2  (1.6.42). Hence  = mf{"  and z/ (w) < ^ y j " 2  "  J  ,  ;w ± 0, /  112  1  , ;  1  = 0},  ;  span{i>i(u),v (u)} we derive that, A G  = 1. Thus, if we let A =  2  •^2 and ^ " ^ _ i | ^ < l,Viu G A,w ^ 0. In turn, SUPAJQ > c . Moreover, the supremum is fn Tip obtained at some point wo G A, which in particular satisfies , (J'q(wo),wo) = 0. This yields, V  2  2  IIV^oll^ + AlKH  2  T  >  >  ||VM||1 + Un  \x\  s  _(Jn gf)  A|HH  Un  |x|  s  1  /  .  Un  a f  "  |x| / s  That is,  Since here q is subcritical, J satisfies (PS), the rest is merely an application of Proposition g  5.4. The above argument does not establish the duality for q = p since p is a critical exponent, here we follow [20] to resort to a limiting process. The idea is to consider the second solutions u,  2 q  of the subcritical problems and try to find a strong limit as q —* p.  Lemma 1.32. Ifn>6, then (1) d  >g  —>  Ci(i = 1,2) as q -* p.  (2) There exist a > 0 and <5 > 0 such that for every 0 < \q — p\ < 5Q, 0  2  c% < ci,, + 77 q  g  n—3  rpt 4(n — s) 39  3  - cr.  We shall omit the tedious details, interested readers can find similar arguments in [20] or [37]. The idea is to compute J (A) where A is the linear subspace spanned by the first solution u\, Q  q  and a truncate extremal u defined in section 5, here one can use the precise estimates on the e  truncated extremal functions (appendix in [37]). P r o o f of the theorem: For e > 0 small, let pe — p — e. Set  C2  :=  >e  J (u ), Pe  where ue is the  e  solution obtained by Lemma 1.31. It is easy to check that C2, is bounded uniformly in e and e  u satifies (1.6.41) for q = p , in addition, there is K > 0 such that for e small, e  e  \\Vuf\\ <K, 2  where,  u+ = m a x K , 0} G H (Q)\{0} l  and  u~ = max{-w , 0} €  H\ty\{0}.  £  Thus, for a sequence e —> 0 we can find u ,u~  G H (Q) such that,  +  l  n  uf —^  weakly in  n  To shorten notations, set u± = ufn,ciin We claim that u  +  = Ci ,i  H (£l). l  = 1,2,p = p ,I n  tPen  €n  ^ 0 and u~ ^ 0. To see this, notice that  n  = h and M f n  £ n  = Mn.  G M , thus J (u^) > c^ . On n  n  n  the other hand, for n large one has 2 JniK)  +  Jn(u~) = J (u ) n  n  =  C ,n < 2  Ci,  n  +  £  n—s  _ ^fif  ^  ~ O  Hence  for n large. A variant of Cherrier's inequality (see [20]) gives that, for every r > 0 there exists a constant M  T  > 0 such that, ("fe" - T)( / S ) " < IIV«ll + 2^37 J |x| a  s  n  40  M |M| T  2  (1.6.43)  for all u e H ^). 1  We derive,  livtijii! +  \\\t£\\l = f  <  fexdivnjiil  + AiinJii!)*  Ja \ \ x  k'i > 0, and therefore for large n: /  > A; > 0.  (1.6.44)  2  Ja m  8  Suppose not. Assume for example that u  = 0. This implies,  +  1  „  I" 17; \Pn  1  0  <?  n  ~  s  and  \\V<\\1- f^=o{l)  (1-6-45)  Ja \ \ x  Consequently, ,1  1,  (-  [ \u \  2— S  P  n  -)  /  V T "  <  -  n-s  r f l  7  8  ^  -<T  +  o(l)  Now fix r > 0 in (1.6.43) such that 0  2-3  (  Un  i * ! * ^  Un  J  -  f  e  >  Un  —  -  £  (1-6-46)  £ Ja \ \> x  = ||Vii+||l + o(l) - *b)(J ^  >  )* - ^ o I K H l + o(l).  n  Since s < 2 < n, the first factor in LHS is positive. While one has (1.6.44) and the assumption l l i t n l b —>  0, it leads to  Ja \ \ x  2 -» n  That is, 2ll  V U  nll2  P  n  Ja  \x\  s  ~ 2(n-s) Ja  \X\  S  ^ \ ) U  L  >^(^fe—o)e o(i). +  which contradicts (1.6.46). Similarly u~ ^ 0.  41  Set u = u  +  - u~ ^ 0. Clearly u  -± u weakly in i f (ft) and u is a changing sign solution to 1  n  equation (1.1.4). We claim that un converges (up to a subsequence) strongly to u G H (9). l  As before set u = u + w , w n  n  0 weakly in H ^). 1  n  l,n + 4 4<fc^W ' °" (n—s)  C  _  >Jn{u  Since J(u) > c  we obtain  u  + W) n  = J( ) + | | | V t i , | | l - i / ^ + o(l) B  tt  >  C l  n  I||V«, ||l-i/ J^p+o(l), B  +  n  that is, 1  1  /" ?/)+l  9 — <?  Pn  »-«  (1.6.47)  Furthermore, + \Pn  o=< j;(w„),«„ >=< j'(w),u > +||v«; ||l - — / \w. Pn Jn n m n  +  o(l),  (1.6.48)  or,  1  f  K Pn  +0(1).  As above, one see that (1.6.47), (1.6.49) can hold simultaneously only if  lim  ||Vtu ||2 = 0 n  Moreover one has  \Un\ UnVl{u ) Pn  1  n  /  f \u\ ^ V ^ u )  Jn  This completes the proof.  42  p  x  (1.6.49)  Chapter 2 Localization Properties For a Porous M e d i u m Type Equation i n Higher Dimensions  2.1  Background  In this chapter we consider the Cauchy problem for a porous medium equation with a source term  ut = V • (u Vu) + u in R a  13  ' in  V  u(x, 0) =  UQ(X)  x [0, T)  N  '  L  (2.1.1)  R, N  where (3 > a + 1, a > 0 are constants, N > 1. This equation has been widely used as a simple model for a nonlinear heat propagation in a reactive medium, where u denotes temperature. We assume that un is a non-negative not identically zero function. By a solution of (2.1.1) we will always mean a non-negative continuous function u(x,  t) satisfying u  a+l  G C([0, T),  Hl (R )), N  oc  which also satisfies (2.1.1) in the weak sense, namely  / Jo for all 0 G C Q  0  0  /  [u<k - u Vu • V 0 c  +  P(j)]dxdt +  U  JRN  f  u (x)(f>{x, 0)dx = 0 0  (2.1.2)  JRN  ^ x [0,T)).  Under the above assumptions, local existence in time, uniqueness and pointwise comparison can  be proved for (2.1.1), and the function  u(x,t) is a classical solution at any point where  u > 0, see for instance [45]. It is well known [55] that when a > 0 , 43  l < / 3 < c r + l + ^ , the  solution of (2.1.1) always blows-up in finite time, while for (3 > a + 1 +  the blow-up occurs  if wo is large enough. In the latter case there also exist small solutions which live globally in time. Here and in what follows by a finite blow-up time T > 0 we mean ||u(-,t)||  00  is finite for  all t G [0,T), but limsup _ -||?i(-,t)|| t  >r  =+00.  00  In our study we shall restrict attention to unbounded blow-up solutions to (2.1.1) with /3 > <7+l.  First we recall some known facts briefly, more details can be found in surveys [46], [31], or [55] for a comprehensive treatment. Like the case of the porous medium equation ut = V • (u Vu), a  due to its degeneracy at the interface u = 0, a basic feature is that (2.1.1) describes a process with a finite speed of propagation of disturbances, i.e., if the initial data UQ is compactly supported, then for all t G (0, T ) ,  suppiz(-, t) = {x G R \u(x,t) N  > 0}  is a bounded set. The finite speed of propagation is one of the most important facts about porous medium type equations and makes them very interesting objects of study. Some definitions are in order. A n unbounded solution of problem (2.1.1) is called strictly  localized if n (u ) = {xG R \u(x,T~) N  L  0  = \imsupu(x,t) > 0}  is bounded, effectively localized ii BU(UQ) is bounded, where BU(UQ) = {x G JR^I there exist t —> T~,x n  n  —> x with u(x ,t ) —> +oo}, n  n  the blow-up set of u. Observe that by definition of the blow-up T , the blow-up set BU(u ) 0  is  a non-empty closed set. Similar to the porous medium equation [59], for the solution u to (2.1.1), the retention property s u p p u ( - , £ i ) C suppu(-,i ) holds for every 0 < ti < t 2  2  44  < T, by the barrier method. Hence  strict localization happens only if u has compact support. For convenience we will include a 0  proof of retention property in section 2. It is well known that the localization is determined by the competition between the diffusion and reaction terms. For 1 < (3 < a + 1, blow-up could happen in the whole space R , hence N  there is no localization. The absence of strict localization can be easily proved by so called "methods of stationary states", one may find a similar argumentation in [55, Theorem 7, p. 463], though in a different setting.  For j3 = a + 1, in dimension one or higher, BU(UQ) is a  bounded region, so it is called regional blow-up. In this case, one can establish the symmetry of blow-up set [41], and describe the blow-up profile precisely [24]. The situation (3 > a + 1 is different.  Here one reasonably could expect the case should be similar to the semilinear  one where a = 0. In the latter case, it has been shown [61] that the blow-up set has locally finite (N — l)-dimensional Hausdorff measure, thus in particular its Lebesgue measure is zero. In some special cases, as UQ is radially decreasing, single-point blow-up has been proven [65], [28].  Note that for semilinear heat equations, the asymptotic blow-up profile is quite well  understood [38], at least for 1 < (3 <  (^§)+.  In one spatial dimension case, for a > 0 and (3 > a+l, Galaktionov has demonstrated in [29] the strict localization for initial data with compact support, hence the blow-up is at most regional. His argument is based on some Sturmian type theorems (intersection comparison technique or lap numbers), which are essentially of O D E nature. The approach can be extended to deal with radially symmetric solutions in arbitrary dimensions. However, in higher dimensions the problem of spectrum of nonsymmetric blow-up remains open, as emphasized in [31]. Little seems known such as strict or effective localization, or any kind of Hausdorff measure estimate. The fact is partly because, as many researchers pointed out, though the intersection comparison technique has shown to be a strong tool in analyzing nonlinear diffusions in one space variable, it seems hard to extend to higher dimensions unless  45  an assumption like radial symmetry is made. The main purpose of this chapter is to prove the strict localization in higher dimensions for arbitrary compactly supported initial data,  without any symmetry condition, see Theorem 2.1  below. We also obtain an estimate for the size of the support, hence it provides an affirmative answer to an open problem in [55, Question 5, p. 315]. Theorem 2.2 is about the effective localization, where u is not necessarily of compact support but satisfies some extra symmetry 0  and decay conditions.  This result provides some insight on another open problem in [55,  Question 6, p. 315].  Theorem  2.1. Suppose that u is the solution to the problem (2.1.1) with a compactly suppo  initial datum UQ and that T is the blow-up time, where (3 > o + 1. Then there exists a K depending on o, j3, N, u and T, such that for all t £ [0, T), 0  suppu(-,t) C Theorem  2.2. Suppose UQ(X)  —  B(0,K).  uo(r), where r — \x\, and uor < 0. Suppose in addition tha  2  Mo < C\x\ ' -( + ), as \x\ —> oo, where 0 < C < C . Then u will decay to 0 uniformly in al 3  ff  1  s  t € (0, T). Hence in this case the effective localization holds. Note that C is a positive constant which will be defined in section 2. s  Our approach is inspired by [24], where among many other things, the localization for (3 = o + 1 is proved. However the situation here is quite different from the (3 = o + 1 case. The difficulty lies in the fact that , when (3 > o + 1, there are no compactly supported self-similar solutions, though there are some small self-similar supersolutions with compact support in more restrictive o, (3 ranges. It seems difficult to compare these self-similar (super)solutions with our solutions directly. Our strategy is , first, to establish the uniform bound for u when \x\ is large, then to compare the solution with the self-similar solution for f3 = o + 1 as \x\ large, since at infinity our solution satisfies a parabolic differential inequality. Another difference 46  is that our similarity transformed operator is not translation invariant, hence one must use the original u and the transformed one (which is globally defined in time) simultaneously, for different purposes. In fact, we will apply comparison technique mainly on the original function, while derive a bound estimate on the transformed solution. To obtain such an estimate, extra work is needed to overcome a hurdle, where a monotonicity argument plays a key role. Our study begins with a section devoted to self-similar representation and self-similar solutions, these results will be used in section 3 where Theorem 2.1 is proved. Section 4 is the proof of Theorem 2.2, some related issues are also discussed.  2.2  Self-similar Representations and Self-similar Solutions  First we prove the following statement, which of course is known for porous medium equation with source.  Suppose that u is the solution to (2.1.1). Then  P r o p o s i t i o n 2.3.  suppw(-,ti) C suppw(-,t2)  for all  0<t <t <T. l  2  Proof. Set w to be the solution of the following Cauchy problem for a porous medium equation:  w = V • (w Vw) in R a  x (0, oo)  N  t  VJQ(X)  By the comparison principle one has  x £ R  N  such that  = u(x,tx) in R : N  w(x,t) < u(x,t + ty), for all 0 < t < T — t\. Suppose  u(x,ti) > 0, that is w(x,0) > 0. By the retention property for porous  medium equation [59, Proposition 16, p.  384], as  t - h > 0, w(x,t - t ) > 0. Hence 2  u(x,t ) > w(x t — ti) > 0. Hence the proof is complete. 2  ) 2  47  2  x  •  Let us introduce the similarity representation of the solution u to (2.1.1) in the following way:  v(Z,T) = (T-t)&u(x,t), where f =  G M, N  have f = T - e m  r m  x , T -t  r = - l n ^ e [0, oo), m =  (2.2.1) > 0 for (3 > a + 1. Note that we  = Te~ . T  Direct calculation leads to the following Cauchy problem for v  v = V • (v Vv) - mVv • f - ^ a  T  v(f, 0) = T^u (^T ) m  0  Notice that if suppu C B(0,RQ), 0  + ^ in iR^ x (0, oo),  ^  in i R . w  then suppt>(-,0) C B(0, RoT~ ), and vice versa. As usual m  B(0, R) is a ball in JR^, centered at zero with radius R. When v is r independent, i.e., the stationary solution to (2.2.1), u is called self-similar solution, hence u(x, t) = (T - t)~0^9(^), where £ = ( ^ , T  v • (^v^)  - -^-y  m  0 satisfies  - mve  • £ + e = o. p  (2.2.3)  It is important that asymptotic blow-up properties of the solutions are usually governed by the nontrivial self-similar solutions, hence self-similar solutions with the same blow-up times act as "attractors" in some sense, see Chapter IV in [55] and an extended list of references therein. In the sequel we only use one-dimensional self-similar solutions and their well-known properties, hence we just collect some facts without providing details, they can be found in [55, Chapter IV]. Notice that we will use different notations.  Fact l.([55, p. 195]) For 3 > a + 1, N = 1, the equation (2.2.4) has a positive radial solution  48  G R , which is strictly decreasing and  ®s =  1  " Ws)'  - -ti + = o  -  6,(0) = 6 >e 0  =  H  ( ^ ) ^  (2.2.4)  [ 0^(O) = O , 0 - > O a s £ i - > o o . a  Here  is the spatially homogeneous solution of equation (2.3) and the fact 9Q > 9H will play  some role in later proofs. The proof of this proposition is based on a shooting type argument. Hence if we define  (T - t)-^9 ,  u (xi,t) = (T — t)~'^ 9s(Ci), it solves (u ) = (u%u )' + u^,u (0,t) = [  s  s  s  s  ^,m=  where & =  0  t  JT  In addition 9 has the following asymptotics [55, p. 190]. S  O.iti) = C ^ ~' 0  (1 + 1/(6)),  a+1)  s  - 0 as  - +oo,  where C = C (a, (3) > 0 is a constant. s  s  Formally one can see that when  x\ —> oo, automatically we will have  ( -\)m T  = 6 -*  0 0  uniformly for £ G (0, T), so one can expect 1  i x . ( , t) -> gl  7*1  2  C (T - t)-v=*\ _\ \~^^ S  )m  2  =  C \x \-v=™ a  x  as X\ —> oo uniformly for £ G (0, T), so one concludes that u uniformly (in t) decays to zero s  as x\ goes to infinity. More importantly, this argument is not just a formal one. Due to a surprising property of self-similar solutions, one has ^  u (xi,t) < u (x T-) s  s  u  = Cslxil'^+v^x  G  R\xi ^ 0,  which implies  sup u (xi,t) —• 0, as |xi| —> oo. s  0<t<T  49  > 0 for all t G (0, T), thus (2.2.5)  In the literature this property is called "criticality", one might find a proof for it in [55, pp. 197-198]. . This fact will help us get the desired uniform decay for our solution later.  Fact 2. ([55, p. 180]) There is a function of the following form  u {x t) = ( T - t ) - ? 0 ( x i ) , x  u  which satisfies  Mt  = (OirK)')'  + (ui)  a+1  in  R x (0,T), 1  where  •&(xi) = {  M < T + 2 )  L s  3 L = ~{a + s  2  if|xi|>^  and further i? satisfies the following O D E  --& +  = 0.  a  The important thing here is that u\ has compact support, this will help us establish the desired localization property.  Fact 3.([55, p. 219]) For (3 > a + 3, there exist some small positive numbers A, a such that the globally defined (in time) function  satisfies  . W t > K M ' ) ' + «2. where 6 = ^  m =  </>(6) = A(l 50  §)l  As mentioned before, the global N-dimensional solutions exists only i f / 3 > o - + l + -^, so<7 + 3 is the optimal lower bound for f3 to guarantee the existence of a supersolution when the spatial dimension N = 1.  Similar to m, the support of u is compact, but unlike m, its support is time dependent and 2  suppw (-,*) C 2  B(0,a(T+l) ) m  for all  t G (0,T).  We will use this supersolution to establish a better estimate on the size of the support, when  P > CT  3.  +  Among the above 3 functions u ,Ui,u , s  2  either has the same blow-up time T as u does, or it is  globally defined in time, so one can expect to use comparison technique on our solution with these functions. That is exactly what we plan to do. In the following, without loss of generality we may assume that u is a classical solution. In the case of degeneracy one can approximate the weak solution  u(x, t) by a sequence {u (x, t)} n  of classical positive solutions satisfying uniformly parabolic problems, then pass to the limit. This kind of typical treatment can be found, for example, in Vazquez's notes [59], [60].  2.3  Proof of Theorem 2.1  In this section we always assume that UQ is compactly supported, suppito C B(0,Rn), m — P-(<r+l) . 2(13-1) ^  n U  -  Several lemmas are in order.The first is essentially contained in [3]. The proof is based on Alexandroff's reflection principle and comparison, we omit its elegant but elementary proof. L e m m a 2.4.  Assume that supptto C 5(0, Rn). Then for all R> 0, one has sup  x€dB(0,R+2Ro)  u(x,t) <  inf u(x,t) for allt G (0, T),  x£B(0,R)  51  i.e., P ZedB(0,(R+2Ro)T- e )  V(£,T)<  S U  m  inf V(£,T), ^B(Q,RT~ e )  Tm  m  for all T > 0.  Tm  fn particular, since m > 0, we have sup £edB(0,(R+2Ro)T-™e™)  V(£,T)  <  inf V(£,T), SeB(0,RT-™)  (2.3.1)  for all r > 0.  The next lemma tells us that, roughly speaking, uniform boundedness yields uniform decay. L e m m a 2.5.  Assume that there exists Ri > 0 such that v(£,0) = 0 for | £ T | > Ri, and m  vt£, r) < 9 whenever \£T e- \ m  > R for all r > 0. Then  Tm  Q  x  sup u(x,t) —• 0, as \x\ —> oo. 0<t<T  Proof. By the definition of v, we know u(£,0) = 0for | £ H > i ? i if and only if UQ(X) = 0 for  \x\ > Ri,  and V{£,T)  < 8 for \£T e- \ m  >R r>0  Tm  0  h  if and only if  u{x, t) < 9 (T - f T ^ T for \x\>R t> 0  0.  u  Set w(x, t) = u (xi — Ri, t), where x = (xi, x') G M , x\ G fR (u is defined in Fact 1, section N  l  s  s  2). So by the assumption we obtain  w = V • (w Vw) + w a  t  0  in {x G R \x N  1  > i?i} x [0, T)  w(x, 0) > u(x, 0) = 0 if xi > i?i [ w(x,t) = e (T-t)~^ Q  > u(x,t) on {xi = 52  x [0,T).  By comparison, it follows that  u(x, t) < w(x, t) — u (x\ s  for all  — Ri,  t)  Xi>R 0<t<T. u  According to (2.2.5),  u(x,t) < C \xi s  as long as xi > R ,t x  R^e-^+v  G [0, T). Hence sup u(x, 0<t<T  £) — >0  as  —>•  Xi  +00.  Since this argument can be repeated in every direction, the lemma is proved.  L e m m a 2.6. If (3 > o + l , s u p p «  C  0  5(0, Ro),swp u(x,t) 0<t<T  —• 0 as \x\ —>  exists a K > 0, such that suppu(-,t) C B(0,K) for edit G [0,T).  Proof. Recall from Fact 2 in section 2 that  u (x t) = 1  (T-t)-*d(x )  u  1  satisfies (ui)t = ( ( « i ) > i ) ' ) ' + K )  C T + 1  m JR x (0, T ) 1  and s u p p u i ( - , £ ) C {x\ G ^ffS JII 1  < -7^}-  By assumption, there exists R > 0, R > RQ such that 2  2  u(x,t) < min{r ^(0), 1} for all \x\ > R , t G [0,T). _  2  53  •  00. Then there  Consider  wi(x, t) = ui(xi - R , t), x G JR . N  2  Then we have  (wi) = V • ( W V w i ) +  in {x G iR^lxi >  t  R } x (0,T). 2  We also have  u = V • (u'Vu) + d < V • (u Vu) + u ^ u  3  1  t  in {x\xi > R } 2  x [0, T ) , since 0 < u < 1 there.  On the parabolic boundary, one has  wi(x, 0) > u(x, 0) = 0 in {x\x > R }, x  2  and Wi(x, t) = (T-  fT-tf(O) > T ^ ( 0 ) > w(ar, t) in {x\x _  x  By comparison again, ^(x,*) >  u(x,t) in  >  = R } x [0, T ) . 2  R } x [0,T). 2  Hence suppu(-,t) C {x\xi < R + 2  Repeating this argument in every direction, we know that suppit(-,£) C B(0, K), where K = i?2 +  •  ¥ •  Hence if one can prove the hypothesis of Lemma 2.5, the main theorem will be a simple consequence of the above two lemmas. R e m a r k . Before going through the whole proof of the theorem, we would like to estimate the support size, i.e., the size of K in Lemma 2.6.  . 54  Let us accept the hypothesis in Lemma 2.5 for a moment (we will see later that the proof of this fact does not depend on Lemma 2.5 and Lemma 2.6). Thus from the proof of Lemma 2.5 one knows  (x,t)<cx': ^ j:  W)  u  for Xi > R, i = 1 , J V , where R > Ri, C > 0 are constants, and C is independent of T . For example, one can choose R = kR  1:  C = 2C for some k = k(a, (3) > 0. S  In Lemma 2.6 R is chosen to satisfy 2  u(x,t) < CT~°  when |x| > R . 2  Notice that here and in what follows, C represents a generic constant that depends on cr, (3, N only, it may change from line to line. Hence we can choose R > R so that 2  Later on, we will see that Ri could be chosen as Ri = 2RQ + CT , m  Ro, T, thus R  2  where C is independent of  is the form  R = Co + C T  m  2  x  +  Ctf*^,  where Ci is independent of T , i = 0,1,2 and only Co depends on RQ.  Now we are going to prove the theorem. The tool employed in the argument, besides differential inequalities and comparison principle, is the monotonicity of the solution to an auxiliary equation.  Proof of Theorem 2.1  55  Lemma 2.5 and 2.6 reduce the proof to finding a number R\ > RQ such that V(£,T) < 0  0  whenever | £ T e m  r m  | > R for all r > 0. X  To this end, we consider, for fixed numbers R > 0, Q > 0, to > 0, the initial-boundary value problem ' Pt = V • (p'Vp) - m V p • £ - ^ p(x, t) = 0  in £ ( 0 , fl) x (t„,oo)  on 55(0, i2) x [to, +oo)  (2.3.2)  [ p(x,t ) = Q on 5(0,7?) 0  Let A# denote the first eigenvalue of the Laplacian under Dirichlet condition on 5(0, R). Then AR = j f c . We make the following claim. C l a i m . If R  2  > Ai and  Q > Q*(#) =  a  +  (  1  V  X  _  r  1  i  1  '  .a + l - A f l V + l '  O  \  1 0-1  1 /  (3-1  then the solution of problem (2.3.2) blows-up in finite time. Let us accept for the moment the validity of this claim and conclude the proof of theorem. Set Ri = 2RQ + RT  m  where R is to be chosen later. We claim that  sup  (2.3.3)  V(£,T)<Q*(R)  feaB(0,i?iT- e ) m  rm  for all r > 0. Otherwise Lemma 2.4, in particular (2.3.1), implies the existence of a t > 0 such that 0  M. v((, ,)>Q'(R).  (  Then  R  tt  v is a supersolution of (2.3.2) for this to and some Q > Q*(R), from the claim the solution  to that problem blows up in finite time, hence v also does, a contradiction which proves the validity of (2.3.3).  56  Finally, it is easy to check that we can find an R = R(o,j3,N) > 0 so large that for all  R > R(cr, (3, N), we have 6 > Q*(R), since Q*(R) -> 6 = (^j)*^ < 0 as R -»• oo, by Fact 1  0  H  0  1 in section 2. Therefore the theorem is concluded by (2.3.3), lemma 3.2 and 3.3. It remains to prove the claim. The key ingredients are the symmetry of p and its monotonicity. Observe that for all A E O(N), the orthonormal group in ]R , p(A • x,t)  is also a solution  N  to (2.3.2). By uniqueness,  p(A • x, t) = p(x,t), for all t > t . Hence p(x,t) = p(r,t), where 0  r = \x\. Sop (0,t) = 0 for all t. r  Now the problem is reduced to one space-variable, we can use a trick in [28].  Set z{r,t) = r^-VrCr,t),(r,t) € [0, R) x [t ,oo). 0  One has z(r,  t ) = 0, for r € (0, R) 0  and 2(0,  Since  t) = 0 for all t e [t , +oo). 0  p > 0 in B(0,R) x [t ,+oo),p = 0 on dB(0,R) x [i ,+oo), by the Strong Maximum Q  0  Principle or Hopf's Lemma, it yields that for all t > to, z < 0 when r = R. In addition z satisfies the following linear P D E ,  zt = (P z)rr c  in (0,  - ^r— +  (/V + m(N -1  2) -  —L-)* -  p — 1  mrz ,  (2.3.4)  r  R) x (to, oo).  Notice that in the P D E above, the coefficient of z is bounded on (0, R) x [t , 0  as long as ti  is smaller than the existing time of p, so we can apply the maximum principle for (2.3.4) to conclude z < 0. Hence p (r, t) < 0 for all r < R, t > to as long as p does not blow up. r  57  Now set  S(t) = /  p(x,t)cf)(x)dx,  JB(p,R)  where R  > Ai, 0 is the first Dirichlet eigenfunction of the Laplacian, normalized so that  2  SB(O,R) t (  -  )=1  I  n  f a c t  ' ^  ^( )-  =  r  For notational convenience, in the following we denote J5(0, R) by B. So using equation (2.3.2) one has  a + lJ  F  a + lJ  r  B  -m J {V-p • x)<f> - J~i = ~ X T  o" +1  fP  a +  JB  dn  y aB  f  +  V - m / (Vp • x)<f> -  f  JB  P - I J B  p<(>+ f pV JB  Since p < 0, we have r  / Wp-xcf)dx— /  cfo" / p (r,t)<p(r)r dr < 0 N  r  and  / p +V= /  p V+ /  CT  <  / J{p>i}nB  P V  a+  P 4>+ I  a+  p4>< I P 4>+ I  0  0  J{p<i}nB  JB  JB  P<j>-  Hence  S\t)  > (1 -  *  ( 1  a +1  - ^TT  /  J  p>t -  B  ) 5 / 3 ( t )  ( - ^ + cr + 1  - ^TT (  +  j^-r) p-l  /FT  (7+1  p— 1  V + l  /3-1'  cr + l  by Jensen's inequality. On the other hand, <r + l '  ™  a-^-(^fl^)«>o. 58  fJB ( 2  - 3  5 )  by the condition in the claim. Integrating (2.3.5) we obtain the conclusion, that is, S(t) —• oo as  T-, where T < T, =  f  ~  < oo.  •  It will be instructive to compare our result with the analysis for N = 1. As mentioned in the Introduction, in that case the method of intersection comparison is applied on exact self-similar solutions with the same blow-up time, so the spatial structure of unbounded solution close to the final blow-up time could be analyzed quite precisely. Hence there hold some important estimates on the size of the support of unbounded solutions. For example, the estimate measw(T ) < m e a s « + CT -  m  0  holds, where C > 0 is a constant which depends only on a, j3. It seems hard to get such a precise estimate in high dimensions, for one does not have Sturmian type comparison. But as we will see below, if j3 > a + 3, one can get a similar estimate. It is quite reasonable to believe that this kind of estimates should also be true for all (3 > a + 1, unfortunately we haven't found a proof yet, since the argument strongly relies on Fact 3, section 2.  Proposition 2.7. If (3 > a + 3, then suppu(-,T-) C  where  B(0,K'),  Ci,i = 0,1, are positive constants independent of RQ,T.  K' = CQRQ + C\T , m  Proof. Recall from Fact 3 in section 2 that u {x t) = 2  u  (  1  satisfies  59  6  Xi  (l + t) ' ro  where ^(6)  = ^(1 -  Since the hypothesis of lemma 3.3 holds, we can choose R3 > i? so that u(x,t) < (T+1) ^ -  0  1  A  for all |x| > Rz,t > 0. Set w2(x, t) = u2(x\ — i? , t). 3  Notice that w2(x,t) = (t + 1)~^A  > (T + 1 ) ~ ^ M > u(x,t) on {x\xi = R } x [0,T), 3  w (x, 0) > u(x, 0) if xi > R . Since 2  3  [w ) > V • « V ™ ) + w$ \ +  2  2  t  by comparison again, it leads to w2(x,t) > u(x,t) on {x\xx > R3} x [0, T ) . So suppw(-,T~) C {x\x! < R + 3  a(T+l) }. m  As before, one concludes suppu(-.T-) C B{Q,R3 + a{T + l) ). m  Under the same consideration as in the Remark, we know that the number R > R\ only needs 3  to satisfy  while we know R  has the form R  R = CRo + CT ,  the proposition is proved.  x  m  3  = 2RQ + RT  m  x  for some big R independent of RQ, T, so •  Note that for fixed Rn, if T > 1, the estimate above is better than what we obtained in the Remark of Lemma 2.6. The proposition above also provides another proof for Theorem 2.1, in the case (3 > a + 3.  60  2.4  Effective Localization in a Special Case  The proof is pretty much the same as that for Theorem 2.1. Hence in the following we just give a sketch, the interested reader can easily provide the details.  Proof.  Since  u is not necessarily compactly supported, the Alexandroff's reflection principle 0  (Lemma 2.5) does not hold a priori. But the extra symmetry assumption recovers the lemma. Observe that by uniqueness, u(x,t)  has the same symmetry as UQ. Take the derivative of the  equation (2.1.1) with respect to r, we get U  = ^rj(^  rt  then notice that z := r  N _ 1  u  r  A r _ 1  ^  < 7  «r)rr ~ ^  ^ {r ~ U U ) N  N  l  +  a  r r  f3u ~ U , l3  1  r  satisfies the following good linear P D E :  Zt = (U°z)rr ~ ^ ^ ( U r  0  z )  +  r  (pU^Z.  Using the comparison principle on the half line, since z(0, t) = 0, z(r, 0) = r ~ UQr N  l  < 0, we  have z < 0, that is, u < 0 for r > 0, t > 0. But that means the solution is radial nonincreasing r  , so the conclusion of Lemma 2.5 is trivially true, by replacing 2Ro with any positive number. Hence one can find a R such that  u(x,t)  <9 (T-t)'^,  ior \x\>  0  R,t>0.  Notice that as in Theorem 2.1, the proof for this fact only depends on the validity of the conclusion of Lemma 2.5. Following the argument in Lemma 2.5, by asymptotic behavior of UQ, there exists R\ > 0 such that u(x, 0) <  w(x, 0) if x >R , 1  where  i  w(x,  t) = u {x s  x  -  Ri,t),  u  So the same comparison arguments imply immediately that supo Tu(x,t) <t<  —• 0,  61  as |x| —>  +oo.  s  is defined in section 2.  In particular, it implies that the blow-up set is compact.  •  Remarks. 1. It is instructive to compare this result with a result, essentially contained in [63], which says: If N > 2,0 < u  Q  < L\x\~^+D  i  j R ^ and ^  n  <  < Pc, then u exists globally in time(  hence the blow-up set is empty). 2  The function L\x\ 0-( +I) is the so-called singular solution, for except at the origin it satisfies ct  the equation pointwise, where L, p are constants depending on N, a, (3 only. Notice that in c  this case j3 > a + 1 + — holds automatically. So our result can be viewed, in some sense, as an extension of the above under extra symmetry assumptions, for we require an upper bound on initial data only at infinity. Note that our equation (2.1.1) and the semilinear equation u = Au + u ,p p  t  =  have the  same nontrivial stationary solutions up to a suitable change of variables, so the argument in [63] (see also [42] for further studies) can be modified to get the above conclusion. In our case since there is no semigroup representation formula for the solution of (2.1.1), it is not easy to get estimates on u, hence for most qualitative properties obtained in semilinear cases it seems hard to obtain these properties in our quasilinear situation. 2. In the proof above the symmetry condition is only required to ensure the Alexandroff's reflection principle. It will be interesting to find more general initial data to make some kind of comparison preserved for all t > 0. If so, with the same asymptotic assumption, we can conclude the effective localization. It is conjectured in [55] that the effective localization should be true if we only assume that u$ decays to zero at infinity. Our radial symmetry case could be probably studied by the intersection comparison technique [29] or Friedman-McLeod method  62  [28], hence the above result might be known for specialists. However, we can't find the exact theorem in the literature and our proof seems simpler than intersection comparison method, while for Friedman-McLeod method, which is powerful for the Dirichlet problem, it might not be so easy to deal with the Cauchy problem. 3. We shall expect the single-point blowup in this radial case [30], which we are unable to completely solve.  63  Chapter 3 Geometric inequalities via a general c o m p a r i s o n principle for interacting gases  Abstract Using the Monge-Kantorovich theory of mass transport, we establish an inequality for the relative total energy - internal, potential and interactive - of two arbitrary probability densities, their Wasserstein distance, their barycenters and their entropy production functional. This inequality is remarkably encompassing as it implies most known geometrical - Gaussian and  Euclidean - inequalities as well as new ones, while allowing a direct and unified way for  computing best constants and extremals.  As expected, such inequalities also lead to expo-  nential rates of convergence to equilibria for solutions of Fokker-Planck and McKean-Vlasov type equations. The proposed inequality also leads to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker-Planck type equations The article is written in a self-contained fashion as it offers a streamlined, unified and compact approach to a substantial number of inequalities originating in disparate areas of analysis and geometry. Some of the ideas presented here are known to the experts and may already be in the literature. They are included for the same pedagogical reasons that motivated the survey style of the paper.  64  3.1 Introduction The recent advances in the Monge-Kantorovich theory of mass transport have - among other things - led to new and quite natural proofs for a wide range of geometric inequalities. Most notable are McCann's generalization of the Brunn-Minkowski's inequality [48], Otto-Villani's [52] extension of the Log Sobolev inequality of Gross [40] and Bakry-Emery [4], as well as Cordero-Nazaret-Villani's proof of the Gagliardo-Nirenberg inequalities [21]. While this paper continues in this spirit, we however propose here a basic framework - already present in McCann's thesis [47] - to which most geometric inequalities belong, and a general inequality from which most of them follow. Besides the obvious pedagogical relevance of a streamlining approach, we find it interesting and intriguing that most of these inequalities appear as different manifestations of one basic principle in the theory of interacting gases that compares the different types of - internal, potential and interactive - energies of two states of a system after one is transported "at minimal cost" into another. The main idea is to try to describe the evolution of a generalized energy functional along an optimal transport that takes one configuration to another, taking into account the - relative - entropy production functional, the transport cost (Wasserstein distance), as well as their centres of mass. Once this general comparison principle is established, then various - new and old - inequalities follow by simply considering different examples of - admissible - internal energies, of confinement and interactive potentials. Here is our framework: Let F : [0, oo) —> JR be a differentiable function on (0, oo), V and W be C -real valued functions 2  on JR and let ft C JR be open and convex. The set of probability densities over ft is denoted n  by P ( f t ) a  n  =  {p  :  then defined on  —>•  ft  JR;  p > 0 and / p(x)dx = 1}. The associated Free Energy Functional is n  V (£l) as: a  65  which is the sum of the internal energy  H (p) := / F(p)dx, Jn F  the potential energy  R (p) •= f pVdx Jn v  and the interaction energy  H (P) :=\i  p(W*p)dx.  W  Of importance is also the concept of relative energy of p with respect to pi simply defined as: 0  E ' (p \p ):=E^ (p )-E^(p ). F  w  v  w  0  1  0  1  where po and pi are two probability densities. The relative entropy production of p with respect to pv is normally defined as  h{p\pv) = f P V(F'(p) + V + W*p) Jn in such a way that if p  dx  2  is a probability density that satisfies  v  V{F(p )  + V + W*p )) = 0 a.e.  v  v  then  h(p\ )= Pv  f p\V(F'(p)-F'( ) Jn  +  Our notation for the density p  Pv  v  W*(p-p )\ dx. 2  v  reflects this paper's emphasis on its dependence on the con-  finement potential, though it obviously also depends on F and W. We need the notion of Wasserstein distance W% between two probability measures po and pi on R , n  defined as:  W (Po,Pi) := 2  2  inf  /  7er(p ,Pl) 0  \x-y\ d>y{x,y), 2  jRn Rn x  where r ( p , pi) is the set of Borel probability measures on R  n  0  x R  n  with marginals po and pi,  respectively. The barycentre (or centre of mass) of a probability density p, denoted  b(p) := / xp(x)dx JR™  66  will play a role in the presence of an interactive potential.  In this paper, we shall also deal with non-quadratic versions of the entropy. call  For that we  Young function, any strictly convex C -function c : Ft —• Ft such that c(0) = 0 and n  x  liniia-i^oo  = oo. We denote by c* its Legendre conjugate denned by  c*(y) = sup {y • z - c(z)}. zeR  n  For any probability density  p on ft, we define the generalized relative entropy production-type  function of p with respect to pv measured against c* by [ pc (-V(F'(p)  which is closely related to the  + V + W^p)) dx,  k  IAP\PV)-=  generalized relative entropy production function of p with respect  to pv measured against c* defined as: I , (p\ ) = f V (F'{p) + V + W • p) • Vc* (V (F'(p) + V + W * p)) dx. Jn c  Pv  :  P  Indeed, the convexity inequality implies that  c*(z) < z • Vc*(z) satisfied by any Young function c, readily  l *(p\p ) < I *(p\pv)- Note that when c(x) = c  v  c  we have  lAp\pv) =: h{p\pv) = I p\ V(F'(p) + V + W*p) * dx = 21 Ap\pv), and we denote  l *(p\pv) by X (p|pv)c  2  The following general inequality -established in section 2- is the main result of this paper. It relates the free energies of two -almost- arbitrary probability densities, their Wasserstein distance, their barycenters and their relative entropy production functional. The fact that it yields many admittedly powerful geometric inequalities is remarkable.  I. Basic comparison principle for interactive gases: Let ft be open, bounded and convex subset of FT , let F : [0, oo) —> Ft be differentiable 1  67  function on (0, oo) with F(0) = 0 and x H-> x F(x~ ) n  PF(X) '•= xF'(x)  convex and non-increasing, and let  n  — F(x) be its associated pressure function. Let V : R  — > M be a C -  n  2  confinement potential with Z ) V > XI, and let W be an even C -interaction potential with 2  2  > ul where X,v G R, and / denotes the identity map. Then, for any Young function  DW 2  c :R  n  —> 1?, we have for all probability densities po and p\ on fi, satisfying supp po C fi and  w^(n),  PF(PO) G  KZ(PoW)  + ^ W  2 2  ( p  0  ,  P  - ^|b(po) - b ( ) |  )  l  < KlvvT {P*)  2  (3.1.1)  +1APO\PV).  W  P l  Furthermore, equality holds in (3.1.1) whenever po = pi = Pv+c where the latter satisfies  V(F'(p ) v+c  + V + c + W*pv+c)  = 0  a.e.  (3.1.2)  The above equation simplifies considerably when c is a quadratic Young functional of the form c(x) := c (x) = ^ \ x | for a > 0, since then we have the identity: 2  a  J . (po|pv) + J C + ^ w ( P o ) = X . ( p | p y J = | / ( p o | p y J , W  c  c  0  + c  2  + c  which means that inequality (3.1.1) can be written as:  II. Quadratic comparison principle for interactive gases: For all probability densities po and pi on fi, satisfying supppo C fi, and PF(PO) £ W  1,00  (fi),  we have for any a > 0,  H ^ ( p o l P i ) + i ( A + i/ - - ) W | ( p , Pi) - T > ( P O ) - b(pi)| < f J (po W . z o~ z z 2  0  2  Minimizing the above inequality over a > 0 then yields the HBWI gases:  inequality  for  (3-1.3) interactive  For all probability densities p and pi on fi, satisfying suppp C fi, and PF(PO) € 0  H f % o | p i ) < W (p , )^I (p \p ) 2  0 Pl  2  0  v  0  - ^ W  68  2  ( p  0  ,  P  l  )  + ||b(p ) - b ( ) | . 2  0  P l  (3.1.4)  This extends the HWI inequality established in [52] and [17], with the additional "B" referring to the new barycentric terms. In the remainder of this introduction, we describe various particular cases of inequalities (3.1.1) and (3.1.3) and show how they easily yield various - new and old - geometric inequalities.  Systems with no potential nor interaction energy - Euclidean geometric inequalities: We start with the most basic system - where no potential nor interaction energies are involvedsince it already contains many important features of the approach and their applications. Assume that V — W = 0 and that F is differentiable on (0,oo), F(0) = 0 and that x  H*  x F(x~ ) is convex and non-increasing. Let PF(X) := xF'(x) — F(x) be its associated pressure n  n  function and let c : M  N  —» M be any Young function. Inequality (3.1.1) gives that for any  probability density p e W  1,00  0  Q, and any pi € V (ty,  ( f t ) with support in  -K *(p )  + / p c* ( - V ( F ' o p )) dx,  F+nP  - H f (pi) <  a  Q  Q  (3.1.5)  0  and subsequently, -Hf ( ) < -H  ^ ( p ) + MpolPoo)  (3.1.6)  where p^ is a probability density such that V(F'(poo)) = 0 a.e.  Moreover, equality holds  F +  Pl  0  whenever po — pi = p where p is a probability density on 0 such that V(F'(p ) + c) = 0 a.e. c  c  c  Applying the above inequality with any Po = p and pi = p , we obtain the remarkably simple c  inequality: H  F+nP  F  {  p  )  <  f  p  c  *  (  _  V  (  F 0  p  )  )  d  x  _  H  P  F  {  p  c  )  +  K  c  f  (3.U)  Jn  where K is the unique constant determined by the equation c  F'(p ) + c = K and f p =l. c  c  c  (3.1.8)  Jn Applied to various - displacement convex - functionals F, one recovers several known inequalities. 69  For example, by taking F(x) — ^ y , where 1 ^ 7 > 1-^, which satisfies the above assumptions, and by letting c(x) = -^\ x \ where r G (0, ^ ) we get that r  ( ^ T  2  +  ")  L'  1 r  -  T1  "' *• -  1 V / 1 2 HF  t+ e)  (3 L9)  for all / G C%°(]R ) such that || / |[ = 1. A standard scaling argument on / now yields the n  r  Gagliardo-Nirenberg inequalities (See Corollary 3.5). By taking F(x) = x\n(x) then PF(X) = x, and if c : M  N  that c* is p-homogeneous for some p > 1, then p =  where a =  c  (3.1.7) then yields for any f  —> M is any Young function such e~ ^ dx. Inequality c  c  peV {R ) n  a  p\npdx<  [  pc* (-—)  dx-n  -In ( /  e~ - dx) , ci  x)  (3.1.10)  with equality when p = p . This time around, a scaling argument on the Young function c c  (Corollary 3.3) yields the Euclidean p-Log Sobolev inequality for any p > 1 / plnpdx<-lnf —j- [ pc*(-^-)dx). JR^ V KneP- ^ JR» V P J  (3.1.11)  V  n  1  1  J  Such inequalities were first established by Beckner in [6] for p = 1, and by Del-Pino and Dolbeault [25] for 1 < p < n. The case where p > n was established recently and independently by I. Gentil [35] who used the Prekopa-Leindler inequality and the Hopf-Lax semi-group associated to the Hamilton-Jacobi equation. Motivated by the recent work of [21], one can see that (3.1.5) yields a stronger statement of the following type sup{J(p);  [ p(x)dx = 1}< inf{1(f); f tl>(f(x))dx = 1}, Jo, Jn  (3.1.12)  where Hf) = f [ c * ( - V / ( x ) ) -Gtyo Jn  f(x))} dx-  (3.1.13)  and J(p) = ~ f l (p(y)) + c(y) (y)]dy Jn F  P  70  (3.1.14)  with G(x) = (1 - n)F(x) + nxF'(x) and where ip satisfies | ipp(F' o ip)' \ = 1. Here we have assumed that c* is p-homogeneous.  Moreover, we have equality in (3.1.12) whenever there  exists / (and p = tp(f)) that satisfies the first order equation: o V0'(/)V/(:r) = Vc(ar) a.e.  -(F  1  (3.1.15)  In this case, the extrema are achieved at / (resp. p = ib(f)). The latter is therefore a solution for the quasilinear (or semi-linear) equation div{Vc*(-V/)} - (G o V0'(/) = W )  (3.1.16)  since it is essentially the L -Euler-Lagrange equation of / on the manifold 2  {/eCo°°(fi); / ^ ( / ( x ) ) d x = l > . Jn  Equally interesting is the fact that tb(f) is also a stationary solution of the (non-linear) FokkerPlanck equation: ^  = d i v { « V ( F ' ( t i ) + c)}  (3.1.17)  since J is nothing but the Free Energy functional on V {ft), whose gradient flow with respect a  to the Wasserstein distance is precisely the evolution equation (3.1.17). In other words, this points to a remarkable correspondence between Fokker-Planck evolution equations and certain quasilinear or semi-linear equations which appear as Euler-Lagrange equations of the entropy production functionals. Behind this correspondence lies a non-trivial "change of variable" that is given by the solution of the Monge transport problem. It essentially maps the solutions of the evolution equation associated to (3.1.13) to those ofthe Fokker-Planck equations (3.1.17). A typical example is the correspondence between the "Yamabe" equation - A / = | / | * - / on R , 2  where 2* = ^  ^ ^ s  e  2  n  (3.1.18)  critical Sobolev exponent, and the non-linear Fokker-Planck equation  71  OU  , 1  — = Au  ,  (3.1.19)  n -f div(rr.'u),  which -after appropriate scaling- reduces to the fast diffusion equation:  du = Au ». di  (3.1.20)  1 _  The correspondence was motivated by the work of [21] where mass transport is used to establish Sobolev-type inequalities.  Solutions of (3.1.18) can be obtained by minimizing the energy  functional on the unit sphere of L *, that is: 2  Using mass transport, they show that the above infimum is equal to the supremum of the functional (3.1.22) over the space of probability densities. Cordero et al. also deal with the Gagliardo-Nirenberg inequalities and obtain best constant results that Del Pino-Dolbeault had obtained earlier by carefully analyzing porous media evolution equations [25]. The link between the two methods becomes much clearer via the above correspondence. More details in section 6.  Systems with non-trivial potential but no interaction energy - Gaussian-type inequalities: Assume now that F is as above but that W = 0, while V : R  n  potential with D V 2  > XI, where A G R, and that c : R  n  —> R is a C -confinement 2  —>• R is again a Young function. Our  basic inequality then yields: for all probability densities po and pi on fi, satisfying supppo C fi,  72  Po > 0 a.e. on fl and P {po) <= W  1>00  F  (fi),  - v (Pi) + ^ (Po,Pi) < - H ^ v ( p o ) + c  where p  v  +^(Pok)  2  H  2  is defined by V ( F ' ( p T y ) + y ) = 0 a.e. Furthermore, equality holds in (3.1.23) whenever  Po = Pi = pv+c  where the latter satisfies  ( '(Pv ) + V + c)  V  F  +c  a.e.  = 0  (3.1.24)  In particular, we have for any probability density p such that supp p C O and PF(P) € W ' ^ ) , 1  K T  where Kv+C  P X  : M  + \n  (p,Pvj  < ?AP\PV)  0 0  n n )+K  -  p  Pv+c  V+C  is the unique constant such that  F'iPvJ  +V + c= K  while / pv+c  v+C  =  1.  (3.1.26)  Ja If V is a convex potential (i.e., A > 0), then the term involving the Wasserstein distance can be omitted, and if V is strictly convex, then we have the identity V[x) - x • W(x)  =  —V*(VV(x)  in such a way that a correcting "moment" appears in the inequality:  l C £ > ) < IAP\PV)  ~ ^ (PVJ  + Kv c  Pf  +  (3.1.27)  Again, the pressure Pp is always positive and we obtain the inequality:  < 1AP\P ) + Kv .  KIZ (P)  V  V)  +C  (3.1.28)  If we now consider the quadratic case (i.e., inequality (3.1.3)), we then get for any a > 0, H£(PO|PI) +  By letting po =  py,  k\  - -)W ( ,Pi) 2  2  Po  < ^ (po|py). 2  (3.1.29)  this already gives a generalized Talagrand's inequality : If V is uniformly  convex (i.e., A > 0), then for any probability density p on Q,  W (p,p )<f-^M, 2  v  73  (3.1.30)  which in the case where F{x) — x\rvx gives  < ^J^f\n(f)p dx.  W (fp ,p ) 2  v  v  (3.1.31)  v  where here pv is the normalized Gaussian  and oy = J  e~ dx. v  Rn  Back to (3.1.29) and after minimization over a > 0, one gets the HWI inequality: A. - ^W (p , ) .  H£(po|pi) < W (po, )VHPOW) 2  (3.1.32)  2  Pl  2  0 Pl  This inequality, first established by Otto-Villani [52] contains many Gaussian inequalities. For example, it yields:  The generalized Log-Sobolev inequality : If V is uniformly convex (i.e., A > 0), then for all pi on ft with supppo  probability densities p and 0  W °(£l), 1,0  C ft and PF(PO) £  H£(p |pi) < ^/ (po|pv) .  (3-1.33)  2  0  2  which in the case where F(x) = x\nx yields the Log-Sobolev inequality of Gross [40] and Bakry-Emery [4]: for any function  g such that g py 2  W '°°(R ) 1  G  n  and  f g pvdx 2  = 1, we  Rn  have  /  g Hg )p dx< 2  \ ! \Vg\ p dx.  2  2  v  (3.1.34)  v  A JRn  JRn  The general case of non-trivial confinement and interaction potentials: Let F be as above, let again V : R  n  —> R be a C -confinement potential with D V > XI, 2  2  but let now W be an even C -interaction potential with D W > ul where X,v e R (not 2  2  necessarily positive). In this case, the general inequality (3.1.1) applied with for any probability density  p such that suppp F+nP ,W-2xVW  TJ  F  n _J V  .  A+l/  0 /  V  < 1AP\PV)  l  \  (P) + -^-w (p,  )  2  Pv+c  - H ' (p c) PF w  v+  74  +  K  v+C  c  W '°°(fl),  C ft and PF{P) £  .  p\ = pv+ yields  V  \  --|b(p)  \|2  -b(p  v + c  )| (3.1.35)  where Ky+  C  is such that  F'( ) Pv+c  +V +c +W• p  =K  v+c  and /  v+C  p  v+c  = 1.  (3.1.36)  Jn  If A + v > 0, then the term involving the Wasserstein distance can be omitted from the equation. If W is convex, then the barycentric term can also be omitted, and if V is strictly convex, then we have H. . ; v  (  (p)<J .(HO c  v )  + #v+c-  (3.1.37)  On the other hand, the HWBI inequalities (3.1.4) obtained in the quadratic case have many interesting consequences. For example,  The generalized Log-Sobolev inequality with interaction potentials: If V+W is uniformly convex (i.e., A + v > 0), then for all probability density functions p and pi on fi with supppo C fi 0  and P { ) e W - ^), 1  F  00  Po  Hf (po|pi) - ^|b(po) - b ( ) | W  P l  2  < 2p-^y/ (po|p^),  (3.1.38)  2  The generalized Talagrand's inequality with interaction potentials: If  is uniformly convex  (i.e., A + v > 0), then for any probability density function p on fi,  ^W (p,p )-^\b(p)-b(p )\ <E^ (p\p ). 2  2  2  v  (3.1.39)  w  v  v  In addition, if W is convex (i.e., v > 0), we obtain in particular:  H ^ ( p o l P i ) < ^h(po\pv).  (3-1.40)  and  iy (P,/v)<v  -  2Hr (pM  2  A  (3-1.41)  Finally, these inequalities combined with the following energy dissipation equation  | Y$  w  (p(t)\p ) = -h (p(t)\ v), v  75  P  (3.1.42)  provide rates of convergence to equilibria for solutions to the McKean-Vlasov type equation ' §f = div{pV(F'(p)  + V + W*p)}  in ( 0 , o o ) x l ?  ri  (3.1.43) [ p{t = 0) = po  in {0} x RJ . 1  One can then recover the recent results of Carillo, McCann and Villani in [17], which state that if V + W is uniformly convex and if W is also convex then K  (P(t)\pv) < e-^ ( \ ),  W  (3.1.44)  w  Po  Pv  v  W  and W ( (t), p ) < - ^ 2  P  v  m  e  ^ y \  (3.1.45)  If on the other hand, V + W is uniformly convex, while the barycenter b (p(t)) of any solution p(t,x) of (3.1.43) is invariant in t, then K$  w  (p(t)\p ) < e-^K$ (p \pv),  (3.1.46)  w  v  0  and W (p(t), ) 2  Pv  < -^)tJ v™(Po\pv)_ 2E  e  y  A+  {  Throughout this paper, suppp denotes the support of p 6 V (fl), a  {iGfi:  3  1  A  7  )  v that is, the closure of  p ^ 0}, |fi| is the Lebesgue measure of fl C R , and q > 1 stands for the conjugate n  index of some p > 1, - + - = 1.  3.2  Main inequality between two configurations of interacting gases  T h e o r e m 3 . 1 . Let fl be open, bounded and convex subset of M , let F : [0,oo) —> Ft be n  differentiable function on (0, oo) with F(0) — 0 and x (-> x F(x~ ) n  n  convex and non-increasing,  and let PF{X) := xF'(x) — F(x) be its associated pressure function.  Let V : RJ —> R be a  C -confinement potential with D V > XI, and let W be an even C -interaction 2  2  2  76  1  potential with  D W > ul where \,u £ Ft, and I denotes the identity map. Then, for any Young function 2  c : Ft —> Ft, we have for all probability densities po and p\ on ft, satisfying supppo Cftan n  P (p ) £ F  0  W^°°(ft),  H^(Ai'lPi) + ^ W ( 2  P o  ,  P l  )  - ||b(p ) - b ( ) | 0  2  P l  < H ^ r ' ^ o ) + / Poc* ( - V (F(po) + V + W*p0)) dx.  (3.2.48)  Furthermore, equality holds in (3.2.48) whenever po = Pi = Pv+ , where the latter satisfies C  V{F'(p )  + V + c + W*p )  v+c  = 0 a.e.  v+c  (3.2.49)  fn particular, we have for any probability density p on ft with suppp C ft and PF(P) £  E  v x  v  (p) + - y - W ( P , Pv c)  v  ~ g l (Po) - b ( p y ) | b  +  2  < / pc* ( - V (F(p) + V + W * p)) cfc - H ^ ' ( p w  Jo.  where Ky  +C  2  +c  y + C  ) + iT  y + c  ,  (3.2.50)  is such that F'(p ) v+C  + V + c+ W *p  y + c  - X  y + C  while / p y  + c  = 1.  (3.2.51)  Jn  The proof is based on the recent advances in the theory of mass transport as developed by Brenier [10], Gangbo-McCann [33], [34], Caffarelli [13] and many others. For a survey, see Villani [62]. Here is a brief summary of the needed results. Fix a non-negative C , strictly convex function d : Ft 1  n  —> Ft such that d(0) = 0. Given two  probability measures p and u on Ft , the minimum cost for transporting p onto v is given by n  W (p,u):= d  inf 7€r(/*,l/)  / j R n  77  d(x-y)dj(x,y), x  R  n  (3.2.52)  where T(p, v) is the set of Borel probability measures with marginals p and u, respectively. When d(x) = | x | , we have that W  = W , where W is the Wasserstein distance. We say  2  2  d  that a Borel map T : R  n  J3 C M . n  2  2  —• J?" pushes /J forward to ^, if p(T~ (B)) = u(B) for any Borel set 1  The map T is then said to be d-optimal if  W (fj.,u)= j  d(x - Tx)dp{x) = inf /  d  where the infimum is taken over all Borel maps S : R  d(x - Sx)dfi(x),  (3.2.53)  —• i R that push p forward to i/.  n  n  For quadratic cost functions d(z) = | | ^ | , Brenier [10] characterized the optimal transport map 2  T as the gradient of a convex function. A n analogous result holds for general cost functions d, provided convexity is replaced by an appropriate notion of d-concavity. See [33], [13] for details. Here is the lemma which leads to our main inequality (3.2.48). It is essentially a compendium of various observations by several authors. It describes the evolution of a generalized energy functional along optimal transport. The key idea lying behind it, is the concept of displacement  convexity introduced by McCann [48]. For generalized cost functions, and when V = 0, it was first obtained by Otto [51] for the Tsallis entropy functionals and by Agueh [2] in general. The case of a nonzero confinement potential V and an interaction potential W was included in [23], [17]. Here, we state the results when the cost function is quadratic, d(x) = \ x | . 2  L e m m a 3.2. Let Q, C M  be open, bounded and convex, and let po and p\ be probability  n  densities on Q, with suppp  0  C ft,  and PF{PO) G W '°°(Q). l  Let T be the optimal map that  pushes po G "P(ft) forward to p G 7^ (ft) for the quadratic cost d(x) = \x\ . Then 2  a  x  • Assume F : [0, oo) -»• R is differentiable on (0, oo), F(0) = 0 and x H + x F(x' ) n  n  is  convex and non-increasing, then the following inequality holds for the internal energy: E( ) F  Pl  - R (p ) > f (T - /) • V (F'(po)) dx. F  0  Po  78  (3.2.54)  • Assume V : R —>• R is such that D V > XI for some X G R, then the potential ener n  2  satisfies E { )-E {p )> v  Pl  v  / po(T-I)-VVdx  Q  + ^W (p , ).  (3.2.55)  2  0 Pl  —> R is even, and D W > vi for some v G R, then the interactio  • Assume W : R  n  2  energy satisfies ' R{ ) w  Pl  - E( )  >  w  Po  f po{T-I)-V{W*po)dx Jn +^(Wi(po,P,)-\b(po)-b( )\ ).  (3.2.56)  2  Pl  Proof: If T is the optimal map that pushes p -= ^ ( O ) forward to p\ G 7^(0) for the quadratic 0  cost d(x) = \x\ , 2  define a path of probability densities joining them, by letting p be the pusht  forward measure of p by the map T = (1 — t)l + tT. It is known from the correspondence 0  t  between Lagrangian and Eulerian coordinates that - at least for smooth p - the trajectory T t  t  satisfies  ' ff  =  U (t,T ) Pt  t  To = In, where the velocity U  Pt  is such that ' ^  + div(p t/ ) = 0 t  pt  Pt=o = Po(1) Under the above assumptions on F, it turns out (see McCann [48]) that the function t » E (p ) is convex on [0,1], which essentially leads to (3.2.54) via the following inequality F  t  for the internal energy:  R( ) F  Pl  - E( ) F  Po  > [jH (p )] F  t  t=Q  = - j f F'(p ) div (p (T - /)) dx. 0  0  (2) As noted in [23], the fact that D V > XI, which means that 2  V{b) - V{a) > W ( a ) • (b - a) + ^| a - b\  2  79  (3.2.57)  for all a, b G R ,  easily implies (3.2.55) via the following inequality for the corresponding  n  potential energy:  Mpo)  Hv(pi) -  ^jT|(r-/)(s)| A)(^ 2  >  [|Hv(ft)]^ +  =  - ^ l / d i v ( p ( T - / ) ) d x + ^ (po,Pi).  (3.2.58)  2  0  2  (3) The proof of (3.2.56) can be found'in Cordero-Gangbo-Houdre [23]. But for completeness, we repeat the argument of these authors here. Indeed, following [23], we write the interaction energy as follows:  H^(pi)  = \f  W(x-y)p (x)p (y)dxdy 1  = \l  l  W(T(x)-T(y)) (x)p (y)dxdy Po  ^  0  JnxQ  =  \l  W(x-y  >  \ I  [W(x -y) + VW(x - y) • ((T - I)(x) - (T - I)(y)) p (x)p (y)] dxdy  Po  0  \(T ~ 1)0) - (T - I)(y)\ po(x)po(y)dxdy 2  = R ( o) + l [  VW{x - y) -((T- I)(x) - (T - I)(y)) p (x)p (y) dxdy  w  P  -t  0  |(T-I){x)-(T-I)(y)\ po(x)p (y) 2  V 4  (T-I)(x)-(T-I)(y)) (x)po(y)dxdy 0  +J l  +  +  0  dxdy,  0  (3.2.59)  JnxQ  where we used above that D W 2  > ul.  The last term of the subsequent inequality can be  written as:  f  \(T - I)(x) - (T - I)(y)\ o(x) (y) dxdy 2  P  Po  JClxQ,  = 2 [ \{T-I)(x)\ (x)dx-2\ 2  Po  [ (T - I)(x) o(x) dx P  R  n  = 2 f\(T-  I)(x)\ p (x) dx - 2|b( ) - b(p )| . 2  2  0  Pl  Jn  80  0  (3.2.60)  And since VW is odd (because W is even), we get for the second term of (3.2.59)  ! [ Jnxn  V  W  {  x  _ j,) .  _  ( ( T  _  (  T  _  ]  ( ) (y)  1){y))  Po  x  Po  dxdy  = 2/ VW(x - y) • (T - I)(x)po(x)p (y) dxdy Jnxn 0  = 2 f  p (T-I)-V{W*po)dx.  (3.2.61)  0  JQxQ  Combining (3.2.59) - (3.2.61), we obtain that  H ( ) - E (p ) > / p (T-I)-V(W*p )dx Jnxn w/  w  Pl  Q  0  0  + ^-( f | ( T - / ) ( x ) | p ^ - | b ( p o ) - b ( p i ) | y \Jn J 2  2  0  z  This proves (3.2.56). P r o o f of T h e o r e m 3.1: Adding (3.2.54), (3.2.55) and (3.2.56), one gets  Hf%o)  -  +^W  < f(x-Tx)in  2 2  (  P  o  , P i ) - ||b( ) - b( )| P o  2  P l  (3.2.62)  p V ( F ( p ) + V + W * po) dx. 0  0  Since p V(F'(p )) = V (Pp(p )), we integrate by part j^PoV (F'(p )) • xdx, and obtain that 0  0  0  0  / x • V(F'(p ) + V + W * p ) p o =  Jn  0  0  Hr v' v  2xVW  (Po).  This leads to  E ( ) F w v  Po  <  - ny ( )  + ^Wl( , )  Kvv ™  (Po) ~ [ PoV(F'(po) + V + Jn  F  W  Pl  2X  - -\b( ) - b ( ) | U  Po Pl  Po  2  P l  (3.2.63)  W*p )-T(x)dx. 0  Now, use Young's inequality to get  - V (F' ( (x)) + y(x) + (W* p )(a;)) • T(x) Po  (3.2.64)  0  < c (T(x)) + c* ( - V (F'(p (x)) + V(x) + 0  81  (W*p )(x))), 0  and deduce that  (3.2.65)  Finally, use again that T pushes p forward to pi, to rewrite the last integral on the right hand 0  side of (3.2.65) as  J c(y)p (y)dy to obtain (3.2.48).  Now, set po = pi :=  n  1  pv+c in (3.2.63). We have that T = I, and equality then holds in (3.2.63).  Therefore, equality holds in (3.2.48) whenever equality holds in (3.2.64), where T(x) = x. This occurs when (3.2.49) is satisfied. (3.2.50) is straightforward when choosing po := p and pi := pv+c in (3.2.48).  3.3  Optimal Euclidean Sobolev inequalities  3.3.1 Euclidean Log-Sobolev inequalities The following optimal Euclidean p-Log Sobolev inequality was established by Beckner [6] in the case where p = 1, by Del Pino- Dolbeault [25] for 1 < p < n, and independently by Gentil for all p > 1. Corollary 3.3. (General Euclidean Log-Sobolev inequality^)  Let ft C R  n  be open bounded and convex, and let c : R —> R be a Young functional s n  that its conjugate c* is p-homogeneous for some p > 1. Then,  L  plnpdx < — In( P  ,  P  f  pc* (-YR)  dx)  (3.3.66)  for all probability densities p on R , such that suppp C ft and p G W °(R ). n  J  1,0  e~ dx. Moreover, equality holds in (3.3.66) if p(x) — K\e~ ^ c  Rn  X9c  82  n  Here, a := c  for some A > 0, where  Proof: Use F(x) = xln(x) and V = W = 0 in (3.2.50). Note that P {x)  = x, and then,  H (p) = 1 for any p G V {R ).  n  F  PF  n  a  So, p (x) =  We then have for p G V (M ) D  c  a  W ^{R ) l  n  such that supp p C fi,  Jp\iipdx<J  pc*(-^j  dx-n-ln^J  e-^ dx^,  (3.3.67)  x)  with equality when p = p . c  Now assume that c* is p-homogeneous and set Y — f  pc* ( ~ )  c  p  Rn  dx. Using c\(x) := c(Ax)  in (3.3.67), we get for A > 0 that /  plnpdx<  /  pc* (—^  ] dx + nlnA — n — lncr ,  for all p G P ( ^ ) satisfying suppp C fi and p G W n  1>00  a  p (x) = A  (/^e-'<*>cb)~V'->>. Ac  Ac  (3.3.68)  c  ( f i ) . Equality holds in (3.3.68) if  Hence  pInpdx < —n — Ina c + inf (GJX)),  / where  G , ( A ) = „]n(A) + 1 £  pc« (-^)  - „ln(A) +  The infimum of G (X) over A > 0 is attained at A = (^T ) ^. 0  P  1  p  J  plnpdx  Hence  <  G (X )  =  ^ l n (*!«,)+ = - » - ] » ( , . ) p \n / p  P  P  p  ^  n — ln(a )  —  c  r  for all probability densities p on R , such that suppp C fi, and p G n  W °°(R ). 1,  n  Corollary 3.4. (Optimal Euclidean p-Log Sobolev inequality,)  /  \f\ \n(\f\ )dx<-ln(c P  JR  p  n  p  83  \Vf\ dx),  f  p  \  JR  p  n  J  (3.3.69)  holds for allp> 1, and for all f G W > (R ) such that \\ f \\ = 1, where l  p  n  p  r(f+i)  if P>1,  a v •'  (3.3.70)  I *[r(f + i ) p and c/ is the conjugate of p  if P = 1,  + i = 1).  equality holds in (3.3.69) for f(x) = Ae~ *  For p > 1,  « ^ /or  A  where K = (f  e'^^  G  R  dx)' '.  9  Rn  some X > 0 and x  1/p  Proof: First assume that p > 1, and set c(x) = (p - 1)| x \ and p = \ f \ in (3.3.66), where 9  /  G C™{R )  f  n  and-1|=  1. We have that c*(x) =  p  , and then, f^  pc* (-^)  dx =  | V / | d x . Therefore, (3.3.66) reads as p  Rn  /  | / r i n ( | / n d x < - l n f  JR*  P  P  —  r  KneP-i-oi  1  |V/|*dz  f  (3.3.71)  JR"  Now, it suffices to note that  o- := /  e  -(p-DI*l«cLr  c  JR  N  **r(* + i)  (3.3.72)  (p-i)tr(f + I)  To prove the case where p = 1, it is sufficient to apply the above to p = 1+e for some arbitrary e  e > 0. Note that  " r(! + i) r(^ + i)  n so that when e go to 0, we have limC  3.3.2  Pe  =  1  ny/n  Sobolev and Gagliardo-Nirenberg inequalities  C o r o l l a r y 3.5. (Gagliardo-Nirenberg inequalities,)  Let 1 < p < n and r G ^0,  such that r ^ p. Set 7 :=- + r  84  where + - = 1. Then, for 1  q'  p  q  '  J  n  any f e W *{R ) we have l  n  •||/||r < C ( p , r ) | | V / | | ; | | / | | ^ ,  (3.3.73)  where 9 is given by - = ^ + ^  P*  =  ,  (3.3.74)  r p* r~f ^Zp- d where the best constant C(p,r) > 0 can be obtained by scaling. an  Proof: Let F(x) =  n^)  ^  -  o r t n  *  s  v a  where 1 ^ 7 > 1 - i , which follows from the fact that p ^ r e  l  u e  °f  7' the function F satisfies the conditions of Theorem 3.1. Let  c(x) — y I x \ so that c*(x) = p ^ p - i | % | , and set V = W = 0. Inequality (3.2.50) then gives q  p  for all / e C °°(I? ) such that || / || = 1, n  c  r  [ |/r<^/ where  V7 - 1 — h ^ satisfies  J  P  JRn  |V/| -i¥^(p p  0 O  ) + C . o o  (3.3.75)  Jn R  1  P o o  -V/i (ar) = 00  and where  insures that J  h  r OQ  x\ x \ - hp(x) a.e., q  2  (3.3.76)  = 1. The constants on the right hand side of (3.3.75) are not  easy to calculate, so one can obtain 9 and the best constant by a standard scaling procedure. Namely, write (3.3.75) as r7l|V/||? P  / '  1  P ll/li: V7-1  , +n  Ml/ \n S > ^ ( A » ) - C » = : C ,  for some constant C. Then apply (3.3.77) to f\(x)  (3.3.77)  = /(Ax) for A > 0. A minimization over A  gives the required constant.  The limiting case where r is the critical Sobolev exponent r = p* — ^  (and then 7 = 1 - ^ )  leads to the Sobolev inequalities: C o r o l l a r y 3.6. (Sobolev inequalities,)  Ifl<p<n,  then for any f £ W ' {R ), 1 p  | | / | | p ,  n  <C(p,n)||V/|| 85  p  (3.3.78)  for some constant C(p,n) > 0.  Proof: It follows directly from (3.3.75), by using 7 = 1 — ^ and r = p*.  Note that the scaling argument cannot be used here to compute the best constant C (p, n) in (3.3.78), since || V / \\ = \ ~ \ \ V / ||J and || f\ \\ p  v  n  A  v r  f \\ scale the same way in (3.3.77). P  = \P- \\ n  Instead, one can proceed directly from (3.3.75) to have that  V  ^  =  O0  ^U[^(P=c)-Coo]J  l | V / | l  -{p[H^{p )-C ])  1  O0  L  |  V  /  L  L  P  '  which shows that C( , n) = ( P  \np[H ^{p ) p  00  where p ^ —  = ^ | x\  q  -  ) ^ ,  (3-3.79)  - Coo]) }  is obtained from (3.3.76), and Coo can be found using  that Poo is a probability density, p/n  n  Coo = (1 - n) R n  3.4 3.4.1  +  \nq  dx  (3.3.80)  Optimal geometric inequalities H W B I inequalities  We now establish HWBI inequalities relating the total energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their entropy production functional, and we deduce extensions of various powerful inequalities by Gross [40], Bakry-Emery[4], Talagrand [57], Otto-Villani [52], Cordero[22] and others. T h e o r e m 3.7. (HWBI inequality;  Let ft be an open, bounded and convex subset of Ft . Let F : [0, oo) n  function on (0, oo) with F(0) = 0 and x — i > x F(x~ ) n  86  n  Ft be a differentiable  convex and non-increasing, and let  PF{X) := xF'(x) — F(x) be its associated pressure function. Let U : R  —> R be a C -  n  2  confinement potential with D U > pi, and let W be an even C -interaction potential with 2  2  D W > ul where p, u £ R.  Then we have for all probability densities po and p\ on fl  2  satisfying supp  C fl and Pp(po) G W '°°(fl), 1  Po  H£%o|pi) <  Wiipo^Whtolpu)  - ^W (p , )  + ||b(p ) - b( )| .  2  (3.4.81)  2  0 Pl  0  Pl  The proof of Theorem 3.7 relies on the following proposition.  Under the above hypothesis on fl and F, let U,W : JR —>R be C -function n  P r o p o s i t i o n 3.8.  2  with D U > pi and D W > ul, where p,u G R, and W is even. Then for any a > 0, we have 2  2  for all probability densities po and pi on fl, satisfying supppo C fl, and H^(polPi) +  \(» + v ~ \)W (p«, 2  p ) - | | b ( ) - b ( ) | < \h{ o\Pul P o  P  P l  1 00  (3.4.82)  2  x  W ' ^),  PF{PO) £  Proof: Use (3.2.48) with c(x) = ^ | x \ , V = U - c and A = p - \ to obtain 2  H^(polPi)  +  <  \{p + v-  V  Z  <J  2 2  ( P o , P i ) + ^|b(po)-b( )|  (3.4.83)  2  P l  z  K^viu-cJ-TiPo) + Jq Poc* ( - V (F'(po) + U-C + W* po)) dx.  By elementary computations, we have / p c*(-V(F o ,  Jn  0  P o  + C / - c + W*po)) dx  = ^ [ Po V(F'(po) + U + W*po) d x + ^ - [ p \x\ dx2 Jn 2(7 Jn 2  2  0  / p x • VU dx  Jn  0  -  Jn  / p x-V(F'(p ))dx Jn 0  0  p x • \7(W * po) dx, 0  and H ^ > ^ ( p o ) = -H ^(p ) + J n P  0  Pox • V{W * p ) dx + J 0  87  p x • W dx - ~ Q  J  \ x | p dx. 2  0  By combining the last 2 identities, we can rewrite the right hand side of (3.4.83) as  c7vSf-™(Po)  H  + J Poc* ( - V ( F ' o po + U - c + W * po)) dx  = £ / Pol V(F'( )  + U + W*po) \ dx-  = ? f Po\V{F'(p )  + U + W*p ) \ dx+ [ div(p x)F'(po)dx- / n P ( p ) d x  z  Po  Jn  0  * Jn  2  f M ' V ( F ' o p o ) dx - / n P ( p ) d x  Jn  0  2  0  0  Jn  Po  2 Jn  Jo.  F  Jn  = ? f Po V(F'(po) + -7 + W*Po) ' d x - f - n f F'{ )dx -  F  Jn  Po  0  + f x • V F ( p ) dx 0  Jn  / nPjp(p )dx Jn 0  Po \7(F'(Po) = ^Jpo\v{F'{po)  + U + W* )  dx+  + U + W*p )  dx.  Po  0  dxx + n \/ F J x-VF(p )d Jn Jn n 0  o  po dx (3.4.84)  Inserting (3.4.84) into (3.4.83), we conclude (3.4.82).  P r o o f of T h e o r e m 3.7: To establish the HWBI inequality (3.4.81), we rewrite (3.4.82) as H ^ ( p o l P i ) + ^ V | ( P o , P i ) - ||b(po) - b ( ) |  2  P l  < ^ W ( p , P i ) + ^h(polPu):  (3.4.85)  2  2  0  then minimize the right hand side of (3.4.85) over a > 0. The minimum is obviously achieved a t  &  =  m , ) P0 P1  _  T  h  i  g  i e l d s  /  3 4  )  gl  Setting W = 0 (and then v = 0) in Theorem 3.7, we obtain in particular, the following HWI inequality first established by Otto-Villani [52] in the case of the classical entropy F(x) = x lnx, and extended later on, for generalized entropy functions F by Carillo, McCann and Villani in [17]. Corollary 3.9. (HWI inequalities [17])  Under the hypothesis on Q and F in Theorem 3.7, let U : R —> R be a C -function wit n  88  2  D U > pf, where p G Ft. Then we have for all probability densities p and p\ onft,satisfying 2  0  supppo C ft, and P (po) G W  1,00  F  (n)  f  Hg(po|pi) < W ( , )^I(p \ ) 2  PQ  Pl  0  - |w (p ,Pi).  (3.4.86)  2  Pu  2  0  If U + W is uniformly convex (i.e., p + z^ > 0) inequality (3.4.82) yields the following extensions of the Log-Sobolev inequality: Corollary 3.10. (Log-Sobolev inequalities with interaction potentials^  fn addition to the hypothesis onft,F, U and W in Theorem 3.7, assume p + u > 0. Then for all probability densities p and p onft,satisfying suppp C ft, and PF{PO) £ W °°(Q,), we l,  0  x  0  have H5%o|pi) - | | b ( ) - b ( ) | < ^ - ^ I ^ p y ) .  (3.4.87)  2  P o  fn particular, if 6(po) = b(pi)>  w  e  h  ave  Pl  that (3-4.88)  K U (PO\PI)<^^UPO\PU). F  W  Furthermore, if W is convex, then we have the following inequality, established in [17] tff (po\Pi) W  <Yp  l M p u )  -  Proof: (3.4.87) follows easily from (3.4.82) by choosing a =  (  3  A  8  9  )  and (3.4.89) follows from  (3.4.87), using v = 0 because W is convex.  In particular, setting W = 0 in Corollary 3.10, one obtains the following generalized LogSobolev inequality obtained in [18], and in [23] for generalized cost functions. Corollary 3.11. (Generalized Log-Sobolev inequalities [18], [23])  Assume thatftand F satisfy the assumptions in Theorem 3.7, and that U : R —> Ft is a C n  89  uniformly convex function with D U > pi, where p > 0. Then for all probability densities p 2  0  and pi on fl, satisfying suppp C fl, and 0  we have  G W °(fl), 1,0  PF(PO)  Hg(pobi) < ^h(po\pu).  (3.4.90)  One can also deduce the following generalization of Talagrand's inequality. We note in particular that when W = 0, the result below is obtained previously by Blower [7], Otto-Villani [52] and Bobkov-Ledoux [8] for the Tsallis entropy F(x) = x\nx, and by Carillo-McCann-Villani [17] for generalized entropy functions F. C o r o l l a r y 3.12. ('Generalized Talagrand Inequality with interaction potentials^  In addition to the hypothesis on fl, F, U and W in Theorem 3.7, assume p + u > 0. Then all probability densities p on fl, we have -^W {p, )  V  - ||b(p) - b{ )\  2  2  Pu  Pu  < R (p\pu). FW  (3.4.91)  In particular, ifb(p) = b(pu), we have that  W (p, )<J 'y \ mFu  2  pu  (3.4.92)  Pu  Furthermore, ifW is convex, then the following inequality established in [17] holds: W (p, 2  P u  )<J  m  Proof: (3.4.91) follows from (3.4.82) if we use po  v  W { p M  A*  .  (3.4.93)  :— pu, Pi := p, notice that I {pu\Pu) — 0, 2  and then let a go to oo. (3.4.93) follows from (3.4.91), where we use v = 0 because W is convex.  3.4.2  Gaussian inequalities  Proposition 3.8 applied to  F(x) = xlnx when W = 0, yields the following extension of Gross'  Log-Sobolev inequality established by Bakry and Emery in [4]. First, we state the following . 90  HWI-type inequality from which we deduce Otto-Villani's HWI inequality [52], and the LogSobolev inequality of Gross [40] and Bakry-Emery [4]. Corollary 3.13. Let U : R  n  —>• R be a C -function with D U > pi where p G R, and denote 2  by pu the normalized Gaussian  2  where au = j  e~ dx. Then for any a > 0, the following u  Rn  holds for any nonnegative function f such that fpu G W °°(R ) l,  f fHf)Pudx  + \(ji--)W^Upu,Pu)<^  f  a  Z jRn  A  JRP  and J  n  Rn  ^-Pudx.  fpu dx = 1:  (3.4.94)  J  Proof: First assume that / has compact support, and set F(x) = xlnx, po = fpu, Pi — Pu and W = 0 in (3.4.82). We have that  HEifPu\ u) P  + \{P ~ -)W (fPu,Pu) 2 a 2  2  <% f 2J  + U "fpudx.  fpu  Rn  (3.4.95)  By direct computations,  v(/^)_v/ fpu  and  (  <  KU UPV\PV) W  f  [fpuHfP^  W j  (3A96)  + Ufpu-pulnpu-Up^dx  (3.4.97)  JPJ  1  =  I (fpuIn/) dx + \nou /  =  f  (pu ~ fpu) dx  fHf) dx. Pu  JR  n  Combining (3.4.95) - (3.4.97), we get (3.4.94). We finish the proof using a standard approximation argument. Corollary 3.14. (Otto-Villani's HWI inequality [52];  Let U : R —»• R be a C -uniformly convex function with D U > pi, where p > 0, and denote n  2  2  by pu the normalized Gaussian  where au = f  function f such that fpu G W °°(R ) l,  /  fHf) dx Pu  and f  n  Rn  Pu  JR  - ^W (f ,Pu), 2  Pu  A  n  91  Then, for any nonnegative  fpu dx = 1,  < W ( Jpu)Vl(fPu\pu) 2  &~ dx. u  Rn  (3.4.98)  where I(fpu\pu)=  ^^-pudx.  f  J  JR  n  Proof: It is similar to the proof of Theorem 3.7. Rewrite (3.4.94) as  J  fHf)Pudx  + ^W (f ,Pu) 2  Pu  <  ^-W (fpu, )^\l{fPu\pu), 2  Pu  JR  n  a > 0 of the right hand side is attained at a =  and show that the minimum over  ?V ' ),  w  pu pu  \/I{fpu\pu)  Now, setting / := g and a := ^ in (3.4.98), one obtains the following extension of Gross' [40] 2  Log-Sobolev inequality first established by Bakry and Emery in [4]. Corollary 3.15. ("Original Log Sobolev inequality [4], [40])  Let U : Ft —> Ft be a C -uniformly convex function with D U > pi where p > 0, and deno n  2  2  by pu the normalized Gaussian that g p e W^iEJ ) 2  1  v  and f  where ou = J  g  u  dx = 1, we have  2  Rn  e~ dx. Then, for any function g such  Rn  Pu  f g Hg ) dx<2  f \Vg\ dx.  2  2  Pu  Pu  P  JR  n  (3.4.99)  JR  n  As pointed out by Rothaus in [54], the above Log-Sobolev inequality implies the Poincare's inequality. Corollary 3.16. (Poincare's inequality^  Let U : FL — > Ft be a. C -uniformly convex function with D U > pi where p > 0, and denote n  2  2  by pu the normalized Gaussian ~ , where ou = J  e~ dx. Then, for any function f such u  Rn  that fpu e W^iFT)  and J  Rn  f p dx = 0, we have v  f f Pudx<-  f \Vf\ dx.  2  JRn  (3.4.100)  2  Pu  P  JR  n  Proof: From (3.4.99), we have that [ JRn  fMfe)  Pu dx < ±- [ 92  *P  ^^ JRn  P u  Je  dx,  (3.4.101)  where f = 1 + ef for some e > 0. Using that f e  fp  Rn  /  dx — 0, we have for small e,  v  fMfe)Pudx= ^  f  6  *  JR  n  f dx  + o(e%  Pu  (3.4.102)  JR  n  and  /  ^—^Pudx^e  [  2  Je  JR  n  \Vf\ dx  + o(e ).  2  (3.4.103)  3  Pu  JRn  We combine (3.4.101) - (3.4.103) to have that  /  fpudx<-  f P  JR  n  \Vf\ dx  + o(e).  2  Pu  (3.4.104)  JR  n  We let e go to 0 in (3.4.104) to conclude (3.4.100).  If we apply Corollary 3.12 to F(x) = x l n x when W = 0, we obtain the following extension of Talagrand's inequality established by Otto and Villani in [52]. C o r o l l a r y 3.17. (Original Talagrand's inequality [57], [52])  Let U : M —> Ft be a C -uniformly convex function with D U > pi where p > 0, and denote n  2  2  Pu the normalized Gaussian f such that J  Rn  where ou = f  e~ dx. Then, for any nonnegative function u  Rn  f pu dx = 1, we have W<(fpu,Pu)<J-[  fHf) dx.  (3.4.105)  Pu  y P  JR™  In particular, if / = ^gy for some measurable subset B of Ft , where dj(x) = pu(x)dx and n  FB is the characteristic function of B, we obtain the following inequality in the concentration of measures in Gauss space, first proved by Bobkov and Gotze in [9]. Corollary 3.18. (Concentration of measure inequality [9])  Let U : Ft —> Ft be a C -uniformly convex function with D U > pi where p > 0, and deno n  2  2  by 7 the normalized Gaussian measure with density pu =  where au = J  e~ dx. Then u  Rn  for any e-neighborhood B of a measurable set B in Ft , we have 1  e  7(5 ) > 1 - e ^ W M ^ y ) ) , 6  93  (3.4.106)  where e>  ^ l n ( ^ ) .  Proof: Using / = f  B  = ^ § y in (3.4.105), we have that  W (f p ,p )< j^Q KB))' 2  B  u  u  ]  E  and then, we obtain from the triangle inequality that  (^) ^(^^j.  W , ( W i , « * > < But since | x — y | > e for all (x,  (3.4.107)  +  £ i? x (R \B ),  we have that  n  e  W (/ py,^)>c. 2  (3.4.108)  B  We combine (3.4.107) and (3.4.108) to deduce that  ml  /l>g  *  e -  .i-7(i**\a ); - 2 V  4  e  * h /  1  U B )  which leads to (3.4.106).  3.5  Trends to equilibrium  We use Corollary 3.11 and Corollary 3.12 to recover rates of convergence for solutions to equation  % = div{pV{F'{p) + V + W*p)}  in ( 0 , o o ) x i R  n  (3.5.109)  ( p(t = 0) = po  in {0} x R , n  recently shown by Carillo, McCann and Villani in [17]. Here we consider the case where V + W is uniformly convex and W convex, and the case when only V + W is uniformly convex but the barycenter b(p(t)) of any solution p(t,x) of (3.5.109) is invariant in t. For a background and 94  other cases of convergence to equilibrium for this equation, we refer to [17] and the references therein. Corollary 3.19. (Trend to equilibrium)  Let F : [0, oo) —> Fi be strictly convex, differentiable on (0,oo) and satisfies F(0) = 0, lim ^oo x  = oo,  and x i—• x F(x~ ) n  n  is convex and non-increasing. Let V, W : M —> [0, oo n  be respectively C -confinement and interaction potentials with D V > XI and D W > vi, 2  2  2  where X,v € Ft. Assume that the initial probability density po hasfinitetotal energy. Then (i) . IfV + W is uniformly convex (i.e., X + v > 0) and W is convex (i.e. v > 0), then, for any solution p of (3.5.109), such that Ry (p(i)) < oo, we have: W  (p(t)\p ) < e- H^(poM, 2At  E'  F W  (3.5.110)  v  and  " W (p(t), ) < e - ^ A  2  2 H g  Pv  '^  P 0 | p y )  .  (3.5.111)  (ii) . If V + W is uniformly convex' (i.e., X + v > 0) and if we assume that the barycenter  b(p(i)) of any solution p(t,x) of (3.5.109) is invariant in t, then, for any solution p of (3.5.109) such that Ry' (p(t)) < oo, we have: w  E  FW  (p(t)\pv) < e- ^H^(p 0 |pv), 2  (3.5.112)  and W (p(t),p ) < 2  v  -  e  2  (  ^  ^  ^  .  (3.5.113)  Proof: Under the assumptions on F, V and W in Corollary 3.19, it is known (see [17], and references therein) that the total energy H y  w  - which is a Lyapunov functional associated  with (3.5.109) - has a unique minimizer py defined by  p V{F'(p ) v  v  + V + W*p )=0 v  95  a.e.  Now, let p be a - smooth - solution of (3.5.109). We have the following energy dissipation equation  j E y (p(t)\p ) = -h{p{t)\p ). F  W  t  v  (3.5.114)  v  Combining (3.5.114) with (3.4.89), we have that  |  E'  F W  (p(t)\ ) < -2XH$  W  Pv  (p(t)\p ).  (3.5.115)  v  We integrate (3.5.115) over [0,t] to conclude (3.5.110). (3.5.111) follows directly from (3.4.93) and (3.5.110). To prove (3.5.112), we use (3.5.114) and (3.4.88) to have that  jK y F  W  (p(t)\pv) < -2(A + u)K$  w  (p(t)\ ).  (3.5.116)  Pv  We integrate (3.5.116) over [0,t] to conclude (3.5.112). As before, (3.5.113) i s a consequence of (3.5.112) and (3.4.92).  Below, we apply Corollary 3.19 to obtain rates of convergence to equilibrium for some equations of the form (3.5.109) studied in the literature by many authors.  Examples: • IfW  = 0 and F(x) = x Inx in which case (3.5.109) is the linear Fokker-Planck equation  | | = Ap + div(pVl/), Corollary 3.19 gives an exponential decay in relative entropy of solutions of this equation to the Gaussian density pv = ~-, rate 2A when D V 2  cry — f  e~ dx, at the v  Rn  > XI for some A > 0, and an exponential decay in the Wasserstein  distance, at the rate A. • If W = 0, F(x) = £^  where 1 ^ m > 1 - ±, and V(x) = X f lj  for some A > 0, in which  case (3.5.109) is the rescaled porous medium equation (m > 1), or fast diffusion equation 96  (1 - £ < m < 1), that is | f = A p + div(Axp), Corollary 3.19 gives an exponential m  decay in relative entropy of solutions of this equation to the Barenblatt-Prattle profile  p {x) = v  (c+®g*\ \>y x  (where C > 0 is such that j  Rn  p(x) dx = 1) at the  rate 2A, and an exponential decay in the Wasserstein distance at the rate A.  3.6  A remarkable duality  In this section, we apply Theorem 3.1 when V = W = 0, to obtain an intriguing duality between ground state solutions of some quasilinear PDEs and stationary solutions of FokkerPlanck type equations. Corollary 3.20.  LetQ, C Ft be open, bounded and convex, letF : [0, oo) —• Ft be differentiable n  on (0, oo) such that F(0) = 0 and x t—• x F(x~ ) n  n  be convex and non-increasing. Let ip : Ft —>  [0, oo) differentiable be chosen in such a way thattp(0) = 0 and \ ibp(F'otf))' \ = K where p > 1, and K is chosen to be 1 for simplicity. Then, for any Young function c with p-homogeneous Legendre transform c*, we have the following inequality: sup{-  f F(p) + cp;peV (tt)}<M{[ Jci a  Jn  c*(-V/)-GW(/);/eQ°(^),  f W ) = i} Jn (3.6.117)  where Gp{x) := (1 — n)F(x) + nxF'(x). Furthermore, equality holds in (3.6.117) if there exists f (and p — ip(f)) that satisfies - ( F ' o ^ ) ' ( / ) V / » = Vc(x) a.e.  (3.6.118)  Moreover, f solves d i v { V ^ ( - V / ) } - ( G o ^ ) ' ( / ) = A^(/)  in ft  Vc*(-V/) •v = 0  on 5ft,  F  for some A G Ft, while p is a stationary solution of  97  fe = div{pV (F'(p)  + c)}  pV (F'(p) + c) • 1/ = 0  in (0, oo) x 0  (3.6.120)  on (0, oo) x <9ft.  Proof: Assume that c* is p-homogeneous, and let Q"(x) = x « F " ( x ) . Let  J(p) : = - / [F(p(y)) + c(i/)p(i/)]dy and  J(p) : = - / ( F + nP )(p(x))dx+ / F  Jn  Jn  c*(-V(Q'(p(x)))dx.  Equation (3.2.48) (where we use V = W = 0, and then A = v = 0) then becomes  J(pi) < J(po)  (3.6.121)  for all probability densities po,Pi on ft such that supppo C ft and PF(PQ) £ W  1,00  ( f t ) . If p  satisfies  -V(F'(p(x))) = Vc(x) a.e, then equality holds in (3.6.121), and p is an extremal of the variational problems sup{ J(p);  p G Pa(O)} = inf{J(p);p G P (ft),suppp C ft,P (p) G W - ^ ) } . 1  a  0 0  F  In particular, p is a solution of  div{„V(F'  W +  c)} = 0 inn  pV(F'(p) + c) • i/ = 0  (  3  6  i  2  2  )  on 5ft.  Suppose now tp : Ft -> [0, oo) differentiable, ^(0) = 0 and that / G C ^ f t ) satisfies ~(F' o ^)'(/)V/(x) = Vc(x) a.e. Then equality holds in (3.6.121), and / and p = ip(f) are extremals of the following variational problems  inf{/(/); / G  C °°(ft), f V>(/) = 1} 0  jn  98  = sup{J(p);p G  P (ft)} a  where  /(/) = JT>(/)) = - / [ F ° ^ + n P W ] ( / ) + / Jn  Jn  c * ( - V ( Q ' o </,(/))).  If now ib is such that | ipp(F' otb)'\ = 1, then | (Q' oib)'\ = l and '(/) = - /[Po</> + n P W ] ( / ) + / c * ( - V / ) ) , Jn Jn because c* is p-homogeneous.  This proves (3.6.117).  The Euler-Lagrange equation of the  variational problem = l]  i n f { ^ * ( - V ( / ) ) - [ P o ^ + n P o V ] ( / ) ; Jj(f) C  F  reads as  d i v { V c * ( - V / ) } - ( G o ^ ) ' ( / ) = A W ) in ft F  Vc*(-V/)-i/ = 0 where  Ae  on 5ft  1? is a Lagrange multiplier, and G(x)  — (1 - n)F(x) + nxF'(x).  This proves  (3.6.119). To prove that the maximizer p of sup{- /" (P(p) + cp) dx; p <= 7> (ft)} Jn a  is a stationary solution of (3.6.120), we refer to [44] and [50]. Now, we apply Corollary 3.20 to the functions F(x) — xlnx, ip(x) — \ x \ and c(x) = (p p  p  1)| px \ , with p > 0 and c*(x) = \ ^ q  and ^ + - = 1, to derive a duality between stationary q  solutions of Fokker-Planck equations, and ground state solutions of some semi-linear equations. We note here that the condition | ipp(F'  Corollary 3.21. Letp>l that ||  oip)' \ = K holds for K — p. We obtain the following:  and let q be its conjugate (± + ± = 1). For all f e W^iJR ), such 71  = I, any probability density p such that f  p{x)\x\ dx < oo, and any p > 0, we q  Rn  have JM  < J (/), M  99  (3-6.124)  where JM  •=-  f Pin (p) dy-ip-1)  [ \py \"p(y)dy,  JR  JRn  n  and W)'-=-  f  l/| ln(|/r)+ p  [ JR  Furthermore, if h e W *(R ) l  — n.  JR  n  n  is such that h>0, \\h \\ = 1, and  n  p  Vh(x) = -p x\x \ ~ h{x) a.e., q  Q  2  then  JM ) = WP  Therefore, h (resp., p = h ) is an extremum of the variational problem: p  sup{ J^p): p e W '\M ), l  n  || P 111 = 1} = inf { / „ ( / ) : / € W *{R ), \\ f \\ = 1}. l  n  v  It follows that h satisfies the Euler-Lagrange equation corresponding to the constraint minimization problem, i.e., h is a solution of ^ A  p  / +  p/|/r ln(|/|) 2  =  A / | / r  2  ,  (3.6.125)  where A is a Lagrange multiplier. On the other hand, p = h is a stationary solution of the p  Fokker-Planck equation:  du = Au + div(pp \x\ xu) 8i q  q  2  (3.6.126)  We can also apply Corollary 3.20 to recover the duality associated to the Gagliardo-Nirenberg inequalities obtained recently in [21]. Corollary 3.22.  Let 1 < p < n, and re  i + ^ = 1. Then, for f e W ' (R ) l p  n  (o, ^  such that r ^ p. Set 7 := \ + ±, where  such that || / || = 1, for any probability density p and for r  all p > 0, we have JM  < W) 100  (3-6.127)  where UP) : = — - r  f P  1  • - = - ( - * - ; + ")  f  l  - ^f  \y\ p(y)(y)dy, q  and W )  \f\  n  iv/r  + —f p  V7 - 1 ) JRn PPP J n Furthermore, if h G W ' (M ) is such that h>0, \\h \\ = 1, and R  1 p  n  r  V/i(x) = -p x\ x \ - hp(x) q  g  a.e.,  2  then j,(h ) =  W-  r  Therefore, h (resp., p = h ) is an extremum ofthe variational problems r  sup{  J  M  : p G W^(R ), n  || p |K =  1} = inf { / „ ( / )  :/ G  W ^ f J T ) , || / | | = 1}. r  Proof: Again, the proof follows from Corollary 3.20, by using now ib(x) = \ x \ and F(x) = r  where 1 ^ 7 > 1 —  which follows from the fact that p ^ r G ^0,  . Indeed, for this  value of 7, the function F satisfies the conditions of Corollary 3.20. The Young function is  now c(x) =-^\px|«, that is, c*(x) =  J  r  , and the condition \tbv(F' oib)'\ = K holds  with K = rj. Moreover, if h > 0 satisfies (3.6.118), which is here,  -Vh(x) = p x\ x \ ~ hp(x) a.e., 9  9  2  then h is extremal in the minimization problem defined in Corollary 3.22.  As above, we also note that h satisfies the Euler-Lagrange equation corresponding to the constraint minimization problem, that is, h is a solution of p-?A f p  + (-^  + n) / | / P 101  2  = A/| / T , 2  (3.6.128)  where A is a Lagrange multiplier. On the other hand, p = h is a stationary solution of the r  evolution equation:  — = Au + div( p \x\ - xu). 1  q  q  2  n  Example: In particular, when / / = l , p = 2 , 7 = l — £ and then r = 2* =  (3.6.129)  is the critical  Sobolev exponent, then Corollary 3.22 yields a duality between solutions of (3.6.128), which is here the Yamabe equation: -A/ = A/|/f- , 2  (where A is the Lagrange multiplier due to the constraint || / H2* = 1), and stationary solutions of (3.6.129), which is here the rescaled fast diffusion equation:  du . ,1 [2n-2 \ — = Au » + div —xu . at \ n- 2 J  102  Bibliography [1] Adimurthi, G . Mancini, The Neumann problem for elliptic equations with critical nonlinearity, Nonlinear Analysis, a tribute in honor of G. Prodi, Edi. by Ambrosetti et al.,Scoula  Norm. Sup. Pisa. (1991), 9-25. Existence of solutions to degenerate parabolic equations via the MongeKantorovich theory. Preprint, 2002.  [2] M . Agueh.  [3] D. Aronson, L . Caffarelli, The initial trace of the solution of the porous medium equation, Trans. Amer. Math. Soc. 280(1983), 351-366 [4] D. Bakry and M . Emery. Springer (1985), 177-206.  Diffusions hypercontractives. In Sem. Prob. XIX, L N M , 1123,  [5] G . I. Barenblatt, Similarity, Self-similarity, Intermediate Asymptotics, versity Press. 1996, reprinted 1997 [6] W . Beckner. Geometric asymptotics 11 (1999), No. 1, 105-137. [7] G . Blower.  Cambridge Uni-  and the logarithmic Sobolev inequality. Forum Math.  The Gaussian isoperimetric inequality and the transportation. Preprint.  [8] S. G . Bobkov and M . Ledoux.  From Brunn-Minkowski to Brascamp-Lieb and to Loga-  rithmic Sobolev inequalities. Geom. Funct. Anal. 10 (2000), 1028-1052. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J . Funct. Anal. 163, 1 (1999), 1 - 28.  [9] S. Bobkov and F . Gotze.  [10] Y . Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 4 (1991), 375 - 417. [11] Ft. Brezis, E . Lieb, A relation between point convergence of functions and convergence  of functionals, Proc. Amer. Math. Soc 88(1983), 486-490. [12] Ff. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical exponents, Comm. Pure Appl. Math 36(1983), 437-477.  Allocation maps with general cost function, in Partial Differential Equations and Applications (P. Marcellini, G. Talenti and E. Vesintin, eds). pp. 29 - 35. Lecture  [13] L. Caffarelli.  notes in Pure and Appl. M a t h , 177. Decker, New-York, 1996. [14] L . Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compositio. Math. 53(1984), 259-275 [15] P. Caldiroli, R. Musina, On the existence of extremal functions for a weighted Sobolev embedding with critical exponent, Calc. Var. 8(1999), 365-387 103  [16] P. Caldiroli, R. Musina, On a variational degenerate elliptic problem,  Nonlinear Diff.Eqns  Appl. 7(2000), 187-199.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. To appe  [17] J. Carrillo, R. McCann, and C . Villani. in Revista Matematica Iberoamericana.  Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monat  [18] J . A . Carillo, A . Jiingel, P.A. Markowich, G . Toscani, A.Unterreiter. Math. 133 (2001), no. 1, 1-82.  [19] F . Catrina, Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and non existence) and symmetry of extremal functions, Comm. Pure Appl. Math 2(2001) 229-258 [20] M . Comte, G . Tarantello, A Neumann problem with critical Sobolev exponent, Houston Journal of Mathematics. 18(1992), 279-294  [21] D. Cordero-Erausquin, B. Nazaret, and C . Villani. A mass-transportation approach to  sharp Sobolev and Gagliardo-Nirenberg inequalities. Preprint 2002. [22] D. Cordero-Erausquin. Some applications of mass Arch. Rational Mech. Anal. 161 (2002) 257-269.  transport to Gaussian-type inequalities  [23] D. Cordero-Erausquin, W . Gangbo, and C. Houdre. Inequalities for generalized entropy and optimal transportation. To appear in Proceedings of the Workshop: Mass transportation Methods in Kinetic Theory and Hydrodynamics[24] C . Cortazar, M . Del Pino and M . Flgueta, On the blow-up set for ut Indiana Univ. Math. J 47(1998), 541-561 [25] M . Del Pino, and J . Dolbeault. The optimal To appear in J. Funct. Anal. (2002).  = Au +u , m  m  m > 1,  euclidean IP-Sobolev logarithmic inequality.  [26] H . Egnell, Positive solutions of semilinear equations in cones, Tran. Amer. Math. Soc 11(1992), 191-201. [27] I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 207-265 [28] A . Friedman, B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J 34(1985), 427-447 [29] V . A . Galaktionov, Proof of the localization of unbounded solutions of the nonlinear c parabolic equation ut = [u ux)x +u@, (in Russian) Differ Equations 21(1985), 751-758 [30] V . A . Galaktionov, e-mail communication  104  [31] V . A . Galaktionov, J . Vazquez, The problem of blow-up in nonlinear parabolic equations,  preprint, 1999 [32] S. Gallot, D . Hulin and J . Lafontaine, Riemannian geometry, [33] W . Gangbo and R. McCann. 2, (1996), 113 - 161.  Springer-Verlag, 1987  The geometry of optimal transportation. Acta Math. 177,  [34] W . Gangbo and R. McCann. Optimal in the Monge's mass transport . Acad. Sci. Paris, t. 321, Serie I, pp. 1653 - 1658, (1995).  problem. C . R.  The general optimal LP-Euclidean logarithmic Sobolev inequality by HamiltonJacobi equations. Preprint 2002.  [35] I. Gentil.  [36] N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge  Univ. Press, 1993 [37] N. Ghoussoub, C . Yuan, Multiple solutions for quasi-linear PDEs involving the critical  Sobolev and Hardy exponents, Tran. Amer. Math. Soc 12(2000), 5703-5743. [38] Y . Giga, R. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38(1985), 297-319 [39] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order. Second  edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin-New York, 1983. [40] L . Gross. 1083.  Logarithmic Sobolev Inequalities. Amer. J. Math. Vol 97, No. 4 (1975), 1061 -  [41] C. Gui, Symmetry of the blow-up set of a porous medium equation,  Comm. Pure Appl.  Math. 48(1995), 471-500 [42] C. Gui, W . Ni and X . Wang, Further study on a nonlinear heat equation, 169(2001), 588-613  J. Diff. Equat.  [43] E . Jannelli, S. Solimin, Critical behaviour of some elliptic equations with singular po-  tentials, preprint 1996. [44] R. Jordan, D. Kinderlehrer and F. Otto. The variational formulation ofthe equation. SIAM J . Math. Anal. Vol 29, No.l, pp. 1 - 17 (1998).  Fokker-Planck  [45] A . S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys 42(1987), 169 -222 [46] H . Levine, The role of critical exponents in blow-up problems, 262-288.  105  SIAM Review 32(1990),  R. McCann. A convexity theory thesis, Princeton Univ., 1994. R. McCann. - 179.  for interacting gases and equilibrium crystals. Ph.D  A convexity principle for interacting gases. Adv. Math 128, 1, (1997), 153  W . - M . Ni, I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm Pure Appl. Math 44(1991),819-851.  F. Otto. The geometry of dissipative evolution equation: the porous medium equation Comm. Partial Differential Equations. 26 (2001), 101-174. F. Otto. Doubly degenerate diffusion equations as steepest descent. Preprint. Univ. Bonn, (1996).  Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 2 (2000), 361 - 400.  F. Otto and C . Villani.  X. Pan, X . Wang, Semiliner Neumann problem in exterior domains, 31(1998), 791-821  Nonlinear Anal.  O. Rothaus. Diffusion on compact Riemannidn manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. Anal. 42 (1981), 102 - 109. A. Samarskii, V . Galaktionov, V . Kurdyumov and A . Mikhailov, Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987; English translation: Walter de Gruyter, Berlin/New York, 1995 M . Struwe, Variational methods, Springer-Verlag, Berlin-Heidelberg-New York, 1990. M . Talagrand. Transportation cost Funct. Anal. 6, 3 (1996), 587 - 600.  for Gaussian and other product measures. Geom.  G. Tarantello, Nodal solutions of semilinear elliptic equations with critical exponent, Integ. Equ, 5(1992) 25-42  Diff  J. L . Vazquez, A n introduction to the mathematical theory of the porous medium equation, Shape Optimization and Free Boundaries, M . C. Delfour e d . Mathematical and Physical Sciences, Series C, vol 380, Kluwer Ac. Publ. Boston and Leiden; 1992 J. L . Vazquez, Asymptotic behaviour for the porous medium equation: in the whole space, Notas del Curso de Doctorado " Metodos Asintoticos en Ecuaciones de Evolucion", to appear J. Velazquez, Estimates on (N-l)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, fndiana Univ. Math. J 42(1993), 445-476  C. Villani. Topics in mass transportation. Lecture notes 2002. 106  [63] X . Wang, O n the Cauchy problem for raction-diffusion equations, Soc. 337(1993), 549-589  Trans. Amer. Math.  [64] X . - J . Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Diff. Eqns 93(1991), 671-684. [65] F . Weissler, Single point blow-up for a semilinear initial value problem, 55(1984), 204-224  107  J. Differ. Equal  

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