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Regularity of minimal surfaces : a self-contained proof Mather, Kevin 2015

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Regularity of Minimal SurfacesA Self-contained ProofbyKevin MatherB.Sc., The University of Manitoba, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)The University Of British Columbia(Vancouver)April 2015c© Kevin Mather, 2015AbstractIn this thesis, a self-contained proof is given of the regularity of minimal surfaces via viscos-ity solutions, following the ideas of L. Caffarelli,X.Cabre´ [2], O.Savin[11][12], E.Giusti[7] andJ.Roquejoffre[8], where we expand upon the ideas and give full details on the approach. Basicallythe proof of the program consists of four parts: 1) Density and measure estimates, 2) Viscositysolution methods of elliptic equations , 3) a geometric Harnack inequality and 4) iteration of theDe Giorgi flatness result.iiPrefaceThe topic of this thesis and the methodology used for proving the results were suggested by mysupervisors Dr. Ailana Fraser and Dr.Young-Heon Kim. This thesis surveys a collection of knownresults and while not original the author has tried to present and organize it in a way unique to theauthor. Moreover the author has provided more details where needed as well as a few modificationsto make things more clear than they were presented in their original form.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Minimal Sets And The Monotonicity Formula . . . . . . . . . . . . . . . . . . . . 112 Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Boundary Defining Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Minimal Surfaces Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Curvature Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Viscosity Solutions For The Minimal Surface Equation . . . . . . . . . . . . . . . 244 Regularity of Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1 Flatness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Regularity Of Minimal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43ivChapter 1IntroductionThis thesis focuses on the study of regularity of minimal surfaces. There are many results con-cerning the regularity of minimal surfaces. There is the work of De Giorgi which he presented hisfirst results in 1952 developing the ideas of Caccioppoli. The De Giorgi[7] approach to studyingminimal surfaces is to view them as boundaries of sets or as the interfaces between two fluids.A full proof that more closely resembles De Giorgi’s original proof can be found in the book byGiusti [7] and the style of techniques used from Federer book[5]. Here we opt for a somewhatmore modern approach which makes use of a geometric Harnack inequality and viscosity solutionsbased on the works of Caffarelli and Co´rdoba[3],Savin[11] and Roquejoffre[8]. The main resultsare the Gauss-Green formula for functions in the class of bounded variation, a geometric Harnackinequality for solutions to a quasi linear elliptic equation and the regularity of flat solutions to theminimal surface equation. In this chapter we provide an introduction to the topic of this thesis, andestablish the framework in which one can work with general sets, allowing us to then set up thespace of admissible sets in which the minimal sets can be found.1.1 PreliminariesThe purpose of this first chapter is to introduce the preliminary definitions, conventions and notationas well as some of the basic theory of minimal surfaces and functions of bounded variation. Wefollow [4], [8], [11], [12] as our main references.Definition 1. Let U be an open subset of Rn. A function f ∈ L1(U) is said to have boundedvariation in a region U ifsup∣∣∣∣∫Uf divφdx∣∣∣∣≤ ∞,1where the supremum is taken over all test function φ in C1c (U ;Rn) with ‖φ‖∞ ≤ 1. We write BV (U)to denote the collection of functions with bounded variation in U.Since we are primarily interested in presenting a regularity theory of minimal surfaces in the contextof minimal boundaries, we are more interested in the case when f is the indicator function of somemeasurable set E. Therefore we define the perimeter of E in an open set U to be the total variationof E in U .Definition 2. For a Lebesgue measurable subset E ⊂Rn, we say that E has finite perimeter in U ifP(E;U) = sup∣∣∣∣∫Edivφdx∣∣∣∣< ∞,where the supremum is taken over all test functions φ ∈C1c (U) with ‖φ‖∞ ≤ 1.By the classical form of the Gauss-Green theorem, we have that for a set E with C1 boundary,P(E;U) coincides with the classical notion of perimeter of ∂E, i.e H n−1(∂E).Here we are able to gain some insight into the structure of sets and functions of boundedvariation, and are able to associate a measure and “normal vector” to a given set E or function f .Theorem 1 (Representation theorem for BVloc functions). Let U be a open subset of Rn. Then forf ∈ BVloc(U), there exists a Radon measure µ on U and a µ-measureable function σ : U → Rnsuch that(i) |σ(x)|= 1 µ-a.e. and(ii)∫U f divφ dx =−∫U φ ·σ dµfor all φ ∈C1c (U ;Rn).For f ∈ BVloc(U) we write ‖D f‖ or ‖∇ f‖ to denote the measure µ given in the above theorem, andD f or ∇ f to denote the vector-valued measure given by the measure with density σ with respectto ‖D f‖. Now, if f = 1E and E is a set of locally finite perimeter in U , we will write ‖∂E‖ for themeasure µ and νE =−σ , where we think of νE as the outward normal on E.We endow the space BV (U) with the norm ‖ · ‖BV given by‖ f‖BV = ‖ f‖L1(U)+‖∇ f‖(U),2which makes the set BV (U) into a Banach space. The fact that ‖ · ‖BV is a norm follows from thefact that ‖ ·‖L1 is a norm and that∫U |∇ · | is a seminorm. All that is left to prove is that the space iscomplete with this metric.Theorem 2 (BV is a Banach Space). Let U be an open subset of Rn. Then the set of functionsBV (U) with the norm‖ f‖BV = ‖ f‖L1(U)+‖∇ f‖(U),is a Banach Space.Proof. Let fi be a Cauchy sequence in BV (U). Then from the definition of ‖·‖BV , we have that thesequence is also Cauchy in the L1-norm and hence, by completeness of L1(U), there is a candidatelimit function f ∈ L1(U) such that ‖ fi− f‖L1(U)→ 0 as i→∞. Now, since the sequence fi is Cauchyin the BV-norm, we have that ‖ fi‖BV is bounded which gives an upper bound on the variationmeasures. Now by theorem 3 (see below), f ∈ BV (U), so all that is left to show is that the ficonverge to f in the BV-norm. Since fi converges to f in L1, it is enough to show that‖∇( fi− f )‖(U)→ 0.Let ε > 0 and let N be large enough so that for n,m > N we have‖ fm− fn‖BV < ε.This implies that‖∇( fm− fn)‖(U) < ε.Since fn→ f in L1 we have that ( fm− fn)→ ( fm− f ) in L1. Thus, by theorem 3, we have that‖∇( fm− f )‖(U)≤ liminf‖∇( fm− fn)‖(U)≤ ε.Hence fi converges to f in the BV -norm, hence BV (U) is a Banach space.Theorem 3 (Semicontinuity of measures). Suppose that U is an open subset of Rn and that wehave a sequence of functions fk ∈ BV (U) and f ∈ L1(U) such that fk→ f in L1loc(U). Then f is inBV (U) and we have the following estimate on its variation measure:‖∇ f‖(U)≤ liminfk→∞‖∇ fk‖(U).3Proof. Take any φ ∈C1c (U ;Rn) such that ‖φ‖∞ ≤ 1. Then, from theorem 1, we have that∫Uf divφdx = limk→∞∫Ufkdivφdx=− limk→∞∫Uφ ·d‖∇ fk‖≤ liminfk→∞∫Ud‖∇ fk‖= liminfk→∞‖∇ fk‖(U).Thus taking supremum over all φ in C1c gives the desired result.We will make use of the following approximation theorem which we will take on faith (see [4]section 5.2.2).Theorem 4 (Approximation by smooth functions). For f ∈ BV (U), there exist functions { fk}∞k=1⊂BV (U)∩C∞(U) such that1. fk→ f in L1(U) and2. ‖∇ fk‖(U)→‖∇ f‖(U) as k→ ∞.Since there is a well-known coarea formula for Lipschitz functions, one might ask if this resultcan hold for functions of bounded variation. We can extend this result to the space of BV functionssince, in some sense, the space of BV functions is the closure of C1 functions under the BV-normwith the variation measure being its derivative. The coarea formula will be needed in the proofof the ABP estimate in chapter 3. For f : U → R and t ∈ R, we define the superlevel set of thefunction f , Et , byEt = {x ∈U | f (x) > t}.The following theorem is from [4] with the proof modified to make use of the preexisting coareaformula for Lipschitz functions.Theorem 5 (The coarea formula for BV Functions). Let U be an open subset of Rn and supposethat f ∈ BV (U). Then‖∇ f‖(U) =∫ ∞−∞‖∂Et‖(U)dt.4Proof. By making use of the coarea formula for Lipschitz functions we can quite easily extend theresult to the space of BV functions by taking a smooth approximating sequence. For f ∈ BV (U),take a smooth approximating sequence fn as in theorem 4. For this sequence, we know that eachfn is Lipschitz, hence the coarea formula will hold. If we denote the superlevel set of each fn byEnt = {x ∈U | fn(x) > t}, then we have that‖∇ f‖(U) = limn‖∇ fn‖(U)= limn∫U|∇ fn|dx= limn∫ ∞−∞‖∂Ent ‖(U)dt≥∫ ∞−∞‖∂Et‖(U)dt by Fatou’s Lemma.Now, to establish the other inequality we need to work with the definition of the variation measure.Take φ ∈C1c (U ;Rn) such that ‖φ‖∞ ≤ 1. Then we have that∫U f divφdx =∫ ∞−∞∫Et divφdxdt, fromwhich we see that we get the other inequality.First suppose that f ≥ 0, so thatf (x) =∫ ∞01Et (x)dt (a.e x ∈U).Thus,∫Uf divφdx =∫U(∫ ∞−∞1Et (x)dt)divφdx=∫ ∞0∫U1Et (x)divφ(x)dxdt=∫ ∞0∫Etdivφdxdt.Likewise, if f ≤ 0,f (x) =∫ 0−∞(1Et (x)−1)dt,5and hence∫Uf divφdx =∫U(∫ 0−∞(1Et (x)−1)dt)divφ(x)dx=∫ 0−∞∫U(1Et −1)divφ(x)dxdt=∫ 0−∞∫Etdivφdxdt.The general case follows if we express f in its positive and negative pieces as f+− f−. So we havethat Et has finite perimeter a.e.. Now, for φ as above, we have∫Uf divφdx≤∫ ∞−∞‖∂Et‖(U)dt.Hence,‖∇ f‖(U)≤∫ ∞−∞‖∂Et‖(U)dt,which gives the other inequality and therefore giving equality, and the desired result is proved.Here we recall the classical notion of perimeter as the Hausdorff measure and compare it to ourcurrent notion of perimeter (see [9] or [1] for more details on this topic).Definition 3. Let (X ,d) be a metric space. For any subset U ⊂ X, let diam U denote the diameterof the set U, that isdiam U = sup{d(x,y) | x,y ∈U} .Then, for s and δ positive, define the (s,δ )- dimensional outermeasure, H sδ , byH sδ (U) =ωs2sinf{∞∑i=1(diam Ui)s |U ⊂∞⋃i=1Ui, diamUi < δ},Where ωs =pis/2Γ(s/2+1). Finally, the s-dimensional Hausdorff measure H s is defined by taking δto zero, that isH s(U) = supδ>0H sδ (U) = limδ→0H sδ (U).At first glance it would seem that trying to define the perimeter of a set E via the Hausdorffmeasure would restrict the definition to a smaller class of sets, but in fact this is not the case. Later6we will state a result which tells us that the perimeter measure associated with a set E is nothingmore than the restriction of the n− 1 dimensional Hausdorff measure to the reduced boundary ofthe set E. In fact, when working in general metric measure spaces where there is no smooth orvector space structure like in Euclidean space, the perimeter measure of a set E is defined in termsof the restriction of the Hausdorff measure to the measure theoretic boundary of E.In analysis and PDE, the integration by parts formula are both useful and important. Here wegive results that can help extend the Gauss-Green formula to the class of BV functions. With theintegration by part formula, we are able to validate some calculations using BV functions. First,we start by stating a series of density and measures estimates which can be used to show that theperimeter measure associated with a set is just n− 1 dimensional Hausdorff measure restricted toits reduced boundary. The dimension of the Hausdorff measure makes sense since the perimetershould be a quantity of codimension 1. The presentation of the results below is based on that givenin the book of Evans and Gariepy[4], with appropriate modifications made when needed.Definition 4. Let U be an open set in Rn and let E have finite perimeter in U. Then we say that apoint x ∈U ∩ supp(‖∂E‖) is in the reduced boundary of E if the following hold(i) νE(x) := limr→0∇E(B(x;r))‖∂E‖(B(x;r)) exists and(ii) |νE(x)|= 1.We denote the reduced boundary of E by ∂ ∗E, and note that the vector νE(x) is the normal to ∂ ∗Eat x.One can think of the reduced boundary as the set of points where it makes sense to have a welldefined unit normal vector νE to the boundary of the set E.Lemma 6. There exist positive constants A1, . . . ,A5, depending only on n, such that, for eachx ∈ ∂ ∗E,(i) liminfr→0L n(B(x;r)∩E)rn> A1 > 0,(ii) liminfr→0L n(B(x;r)\E)rn> A2 > 0,(iii) liminfr→0‖∂E‖(B(x;r))rn−1> A3 > 0,(iv) limsupr→0‖∂E‖(B(x;r))rn−1≤ A4,7(v) limsupr→0‖∂ (E ∩B(x;r))‖(Rn)rn−1≤ A5.Results (i) and (ii) say that, for a given point x ∈ ∂ ∗E, the infinitesimal densities are nonzero.Results (iii) and (iv) say that the perimeter measure associated with E is comparable to n− 1dimensional Hausdorff measure.The following theorem says that ∂ ∗E has a very nice structure and that it is almost a smoothsurface.Theorem 7. Assume that E has locally finite perimeter in Rn.(i) Then∂ ∗E =∞⋃k=1Kk∪N,where‖∂E‖(N) = 0and Kk is a compact subset of a C1-hypersurface Sk. Furthermore,(ii) νE |Sk is normal to Sk and(iii) ‖∂E‖=H n−1|∂ ∗E .Now, if ∂ ∗E is most of ∂E, then this would strongly suggest that ∂E is smooth in some sense.For example, if we were to randomly sample from E and then do some curve fitting on the data,then almost surely we would have that the fitted surface would be smooth. In theorem 12 we willshow that for a minimal set E, in fact ∂ ∗E is most of ∂E.Definition 5. Let x ∈ Rn. We say that x ∈ ∂∗E, the measure theoretic boundary of E, iflimsupr→0L n(B(x;r)∩E)rn> 0andlimsupr→0L n(B(x;r)\E)rn> 0.Then next lemma says that, for a BV set, the two types of boundaries, the reduced boundaryand the measure boundary, are essentially the same. Here we present an enhanced and completeproof based on the one presented in the book of Evans and Gariepy[4].Lemma 8. (i) ∂ ∗E ⊂ ∂∗E.8(ii) H n−1(∂∗E \∂ ∗E) = 0Proof. Assertion (i) follows from Lemma 6. The proof of assertion (ii) is very similar to Theorem12, found in the next section, the only real difference is to establish a uniform lower bound on ameasure estimate of ‖∂E‖ over balls. Since the mappingr 7→L n(B(x;r)∩E)rnis continuous, if x ∈ ∂∗E, then there exists 0 < α < β < 1 and r j→ 0 such thatα ≤L n(B(x;r j)∩E)ωnrnj< β .Thusmin{L n(B(x;r j)∩E),Ln(B(x;r j)\E)} ≥min{α,1−β}ωnrnj ,so the isoperimetric inequality (see Evans and Gariepy[4] section 5.6.2 theorem 2) giveslimsupr→0‖∂E‖(B(x;r))rn−1> 0.For x ∈ ∂∗E, letc(x) := limsupr→0‖∂E‖(B(x;r))rn−1> 0.Let S = ∂∗E\∂ ∗E and defineLk = {x ∈ S : c(x) >12k}.We have S =⋃∞k=1 Lk, and ‖∂E‖(Lk) = 0. Let ε > 0. Since ‖∂E‖ is a Radon measure, it is outerregular, so there is an open set Aεk ⊂ Rn containing Lk such that‖∂E‖(Aεk)≤ ‖∂E‖(Lk)+ ε2−k = ε2−k.Let δ > 0 and define the setsFk :={B(x;r) |x ∈ Lk, B(x;r)⊂ Aεk , 0 < r <δ10, ‖∂E‖(B(x;r))≥ rn−12k}.Fk is a covering of Lk, so, by the Vitali covering lemma, there exists a countable disjoint collection9{B(xi;ri)}∞i=1 ⊂Fk such thatLk ⊂∞⋃i=1B(xi;5ri).As diamB(xi;5ri)< δ , we see that this covering is one of the coverings which the infimum is takenover in the definition of H n−1δ (Lk), soH n−1δ (Lk)≤ ωn−1∞∑i=1(5ri)n−1 (1.1)≤ ωn−15n−12k∞∑i=1‖∂E‖(B(xi;ri)) (1.2)= ωn−15n−12k‖∂E‖(∞⋃i=1B(xi;ri)) (1.3)≤ ωn−15n−12k‖∂E‖(Ak) (1.4)≤ ωn−15n−1ε. (1.5)Letting ε → 0 we get H n−1δ (Lk) = 0 and thereforeH n−1δ (S) =Hn−1δ (∞⋃k=1Lk) = limk→∞H n−1δ (Lk) = 0.Hence taking limits in δ gives H n−1(S) = 0.Here we present the Gauss-Green formula from [4].Theorem 9 (Generalized Gauss-Green Theorem). Let E ⊂ Rn have locally finite perimeter.(i) Then H n−1(∂∗E ∩K) < ∞ for each compact set K ⊂ Rn.(ii) Furthermore, for H n−1 a.e. x ∈ ∂∗E, there is a unique measure theoretic unit outer normalνE(x) such that ∫Edivφdx =∫∂∗Eφ ·νEdH n−1for all φ ∈C1C(Rn;Rn).10Proof. By theorem 1 we have that∫Edivφdx =∫Rnφ ·νEd‖∂E‖.But, since the support of the measure is concentrated on ∂ ∗E, we have that ‖∂E‖(Rn \ ∂ ∗E) = 0,and so from lemma 7 we find that‖∂E‖=H n−1|∂∗E .Thus the Gauss-Green formula holds for domains that are of bounded variation.1.2 Minimal Sets And The Monotonicity FormulaIn geometry and PDE, the monotonicity formulas have been proven to lead to very useful resultssuch as the Morawetz and virial identities from nonlinear dispersive equations. Here we develop amonotonicity formula for minimal sets and use it to show that most of the content of ∂E is containedin ∂ ∗E. But first, to justify some calculations, we must introduce the trace class for functions ofbounded variation. One can extend the notion of the trace operator from Sobolev functions to BVfunctions. To do this, assume that U is an open bounded set with Lipschitz boundary. Since ∂Uis Lipschitz, it has a unit outer normal vector H n−1-a.e by Rademacher’s theorem. In this settingwe can define the trace class for functions of bounded variation. We follow [7] as a guide for thissection.Theorem 10. Let U be an open bounded set with Lipschitz boundary. Then there exists a boundedlinear operatorT : BV (U)→ L1(∂U ;H n−1)such that for f ∈ BV (U) and for almost every x ∈ ∂U we havelimr→0∫−B(x;r)∩U| f −T f (x)|= 0.The function T f , which we will also denote by tr( f ), is called the trace of f on ∂U , and is uniqueup to sets of H n−1|∂U measure zero.Let E be a set of BV in B1 with 0 ∈ ∂E. Define the functionφE(r) = r1−n∫Br|∇E|.11Now this quantity is invariant under the scalingφE(r) = φ 1r E(1)since, for any g ∈C10(B1), if we define h(rx) = g(x), then we get∫B11 1r Ediv(g)dx = r∫B11E(rx)div(h)(rx)dx= r1−n∫Br1Ediv(h)dx.Hence, taking the supremum over all test functions gives the invariance. In fact, if E is scaleinvariant, i.e. E is a cone, then φ is constant.Below is a precise definition based on the informal one presented in [3]Definition 6. We say that a set E is a minimal surface in Ω if, for every open set A ⊂ Ωwhich isrelatively compact in Ω, we haveP(E;A)≤ P(F ;A)whenever E and F coincide outside a compact set included in A.The Theorem below is from [7].Theorem 11. If E has minimal perimeter in B1 and 0 ∈ ∂E, then φE is a increasing function.Further φE is constant if and only if E is a cone.Proof. Let f be a smooth approximation to E in B1 and leta(r) =∣∣∣∣∫Br|∇ f |−∫Br|∇E|∣∣∣∣ , b(r) =∫∂Br|tr( f )− tr(E)|.Let J(r) =∫Br |∇ f | and h(x) = f (x/|x|) be the radial inward extension of f from ∂B1.Now, if we let φ(x) = x/|x| and let ∇T f denote the tangential component of ∇ f , then we have thefollowing calculation between the gradient of h and f ,12(n−1)∫B1∇h(x) = (n−1)∫ 10∫∂Br∇( f ◦φ)= (n−1)∫ 10∫∂Br(∇ f ◦φ)Dφ= (n−1)∫ 10∫∂Br(∇ f )(x|x|) ·(δi j|x|−xix j|x|3)i j= (n−1)∫ 10∫∂Br1|x|∇ f (x|x|)−1|x|〈∇ f ,x|x|〉 ·x|x|= (n−1)∫ 10∫∂Br1|x|∇T f (x|x|)= (n−1)∫ 10∫∂B11r∇T f (x)rn−1=∫∂B1∇T f (x).Hence we have that(n−1)∫B1|∇h|=∫∂B1|∇T f |.From this we can infer thatJ′(1) = (n−1)∫B1|∇h|+(∫∂B1|∇ f |− |∇T f |)≥ (n−1)(∫B1|∇ f |− (a(1)+b(1)))+(∫∂B1|∇ f |− |∇T f |)= (n−1)J(1)+∫∂B1(|∇ f |− |∇T f |)− (n−1)(a(1)+b(1)).Now, making use of the simple inequality 1−√1− t ≥ t/2 for |t| ≤ 1 with t =(∇ f · x)2|∇ f |2, we getthatJ′(1)≥ (n−1)J(1)+∫∂B1|∇ f |2(∇ f · x)2|∇ f |2− (n−1)(a(1)+b(1)).13Now, the left hand side is nothing more then just φ ′(1) whereφ(r) = r1−n∫Br|∇ f |.Doing a rescaling gives usφ ′(r)≥ 12∫∂Br(∇ f · x)2|∇ f |rn+1−C(n,r)(a(r)+b(r)).Integrating this last expression from r1 to r2 and applying the Cauchy-Schwartz inequality givesφ(r2)−φ(r1)≥12(∫Br2\Br1∇ f · x|x|n)2∫Br2\Br1|∇ f ||x|n−1−C(n,r1,r2)∫ r2r1(a(r)+b(r))dr.Now, if fnL1−→1E and∫B1 |∇ fn|→∫B1 |∇E|, then we have that∫Br |∇ fn|→∫Br |∇E|, so an(r)→ 0for a.e. r. As well, we have that bn(r)→ 0 by the L1 convergence of fn to E with the fact thatfn = tr( fn) and 1E = tr(1E) for a.e. r. We are able to conclude thatφE(r2)−φE(r1)≥12(∫Br2\Br1∇E · x|x|n)2∫Br2\Br1|∇E||x|n−1≥ 0,and so φE is monotone.Now, with this new tool we can show that, for a minimal set E,the boundary of E is almost thegraph of a function. The theorem below is a combination of one presented in [7] which makes useof ideas from [4].Theorem 12. If E is a minimal set in B1, thenH n−1(∂E \∂ ∗E) = 0.Proof. Let K be any compact set contained in ∂E \ ∂ ∗E. Then, since |∇E| is supported on ∂ ∗E,we have that14∫K|∇E|= 0.Since |∇E| is a Radon measure, it is outer regular, so for ε > 0 there exists an open set Aε containedin B1 containing K such that∫Aε|∇E| ≤ ε.For δ > 0 and x ∈ K, there exists r < δ such that B(x,5r)⊂ Aε . Hence, by the Vitali coveringlemma we can choose a countable subset of xi such that1. B(xi,ri)∩B(x j,r j) = /0 if i 6= j.2. K ⊂ ∪∞i=1B(xi,5ri).This gives∞∑i=1∫B(xi,ri)|∇E| ≤∫Aε|∇E|< εNow, using Theorem 11 and the estimate from Lemma 6 with A3 being is the constant fromthat Lemma, we have that∫B(xi,ri)|∇E| ≥ A3rn−1i .This givesA3∞∑i=1rn−1i ≤ ε.Hence we have bounds on the δ -Hausdorff measure of the set KH n−1δ (K)≤ 5n−1 A3ωn−1ε,and so the taking limits in δ and then in ε givesH n−1(K) = 0.Now, since H n−1 is an outer measure and Rn is strongly Lindelo¨f, we can find a countable15cover of compact subsets of ∂E \∂ ∗E. Thus, by subadditivity, we get thatH n−1(∂E \∂ ∗E) = 0., we know that for a minimal set E, ∂ ∗E makes up most of ∂E, and from Theorem 7, ∂E ismade up of pieces of C1 hypersurfaces. This gives strong evidence that ∂E is at least C1. Later wewill show that it is in fact much smoother.16Chapter 2Viscosity SolutionsA viscosity solution of a given elliptic PDE is a form of weak solution to the equation. The role thatthey play can be thought of being somewhat analogous to how distributions solve a given equation.Here we give a brief overview of viscosity solutions based on the definitions outlined in the bookof Caffarelli and Cabre´[2].In this section we consider elliptic equations of the formF(D2u,Du) = f , (2.1)where u and f are functions defined on a bounded domain Ω in Rn and F(M, p) is a real valuedfunction defined on S ×Ω, where S is the space of n×n symmetric matrices.A continuous function u in a domain Ω is a viscosity supersolution of (4.1) if for every x0 ∈Ω andfor any φ ∈C2(Ω) such that u−φ has a local minimum at x0, we haveF(D2φ(x0),Dφ(x0))≤ f (x0).We say that u is a viscosity subsolution if for every x0 ∈ Ω and for any φ ∈C2(Ω) such that u−φhas a local maximum at x0, we haveF(D2φ(x0),Dφ(x0))≥ f (x0).In the case of the minimal surface equation, we have that f = 0 andF(M, p) =tr(M)√1+ |p|2−pT Mp(1+ |p|2)3/2.17We will work in this more general framework of elliptic equations in Rn rather than just workingwith the minimal surface equation directly so as to make the proofs simpler. Let S again denotethe set of symmetric n×n matrices. Then we will be looking at equations of the formF :S ×Rn→ RwithF(D2u,Du) = 0, u : B1→ R.We shall assume that F satisfies the following properties:1. F is elliptic i.e. if N is a symmetric positive semidefinite matrix;(written N ≥ 0) thenF(M+N, p)≥ F(M, p) (2.2)2. F is uniformly elliptic in a neighborhood of (0,0) ∈S ×Rn. This means that there existsδ > 0 and λ ,Λ such that if |p| ≤ δ and N ≥ 0 is a symmetric positive semidefinite matrixthenλ‖N‖ ≤ F(M+N, p)−F(M, p)≤ Λ‖N‖ if ‖N‖,‖M‖ ≤ δ . (2.3)3. Planes are solutions to the equation, that isF(0, p) = 0. (2.4)4. F is linear in its first argument:F(M+N, p) = F(M, p)+F(N, p). (2.5)Here ‖M‖ = max{|λi| | λi is an eigenvalue of M}. Further, from the above properties it will beshown that we have the following bound on Fλ‖M+‖−Λ‖M−‖ ≤ F(M, p)≤ Λ‖M+‖−λ‖M−‖ i f ‖M‖, |p| ≤ δ . (2.6)We will prove (2.6) in the special case of the minimal surface equation case and that the minimalsurface equation satisfies the above four properties. The general case follows by replacing theminimal surface equation by a general F . For the minimal surface equation we have that F is given18byF(M, p) =tr(M)√1+ |p|2−pT Mp(1+ |p|2)3/2.Now, since the trace of a matrix is linear, and a bilinear form is linear in its matrix argument, wehave that the minimal surface equation is linear in its matrix argument. Now to show that it isuniformly elliptic in a neighborhood, we clearly have that F(N, p) ≤ n‖N‖. For the lower boundwe make use of the fact that N is self-adjoint:F(N, p) =tr(N)√1+ |p|2−pT N p(1+ |p|2)3/2≥‖N‖√1+ |p|2−|p|2p|p|T N p|p|(1+ |p|2)3/2≥‖N‖√1+ |p|2−|p|2‖N‖(1+ |p|2)3/2= ‖N‖(1(1+ |p|2)3/2)≥ ‖N‖(1(1+ |δ |2)3/2).Since N is positive definite, we see that F(N, p) ≥ 0. Hence this shows that the minimal surfaceequation is elliptic and also establishes that F is uniformly elliptic in a neighborhood of p = 0. Forthe third property it is clear that F(0, p) = 0.Now, we want to write M as M+−M− where M+ and M− are both positive semidefinite matricesand M+M− = 0.We can do this by defining M+ and M− as follows: if we orthogonally diagonalizethe matrix M asM = OT DO,where O is an orthogonal matrix and D is a diagonal matrix and let D′ be the diagonal matrixobtained by replacing each negative entry of D with a zero, then defineM+ = OT D′O.Similarly, if we let D′′ be the matrix obtained by replacing each positive entry of D with a zero then19we defineM− = OT D′′O.Also, we make note that, in general, the representation of M into two positive semidefinite matricesis not unique. However if we also add the condition that M+M− = 0, then the representation isunique and with this representation we have that ‖M+‖,‖M−‖ ≤ ‖M‖. Making use of the fact thatthe minimal surface equation is quasi-linear we have thatF(M, p) = F(M+−M−, p) = F(M+, p)−F(M−, p),and applying (2.3) gives usλ‖M+‖−Λ‖M−‖ ≤ F(M, p)≤ Λ‖M+‖−λ‖M−‖ i f ‖M‖, |p| ≤ δ .2.1 Boundary Defining FunctionBefore we can define what it means for an abstract set to satisfy the minimal surface equation, wemust first try to define a function that encapsulates the main essence of ∂E. Here we make theassumption that E contains everything below a height −ε , that is {x | xn ≤ −ε} ⊂ E, or, at leastlocally, E should contain everything below some height in a small ball.We now define the boundary defining function u of E. For this we will introduce some more no-tation. For a measurable subset A, we say that A ⊂λ E if λ n(A∩E) = λ n(A), where λ n is n-dimensional Lebesgue measure. Observe that if B ⊂ A and A ⊂λ E, then we have that B ⊂λ Esinceλ n(B)+λ n(A\B) = λ n(A) = λ n(E ∩A) = λ n(E ∩B)+λ n(E ∩A\B)≤ λ n(B∩E)+λ n(A\B).Hence the last inequality is an equality giving us that B⊂λ E. Let Cr(x′,xn) be the cylinder defineby Cr(x′,xn) := B′r(x′)× [−δ ,xn]. Notice that if for some r0 > 0 we have that Cr0(x′,xn)⊂λ E, thenfor every r < r0 we have that Cr(x′,xn)⊂λ E.Define the function u(x′) byu(x′) = sup{xn| there exists an r > 0 such that Cr(x′,xn)⊂λ E}.Then u is lower semi-continuous and (x′,u(x)) is in the measure theoretic boundary of E. To see20that u is lower semi-continuous let {x′k} be a sequence of points that converge to x′ and let ε > 0.Then there exists r > 0 such that Cr(x′,u(x′)−ε)⊂λ E. Since x′k converges to x′, we have that thereexists some k0 such that, for k > k0, xk ∈ B′r(x′). Let rk = r− |x′− x′k| (i.e. rk = d(x′k,∂B′r(x′))).Then we have that Crk(x′k,u(x′)−ε)⊂λ E, and so this gives u(x′k)≥ u(x′)−ε by definition of u(x′).Thus taking limits in k gives that liminfy′→x′u(y′)≥ u(x′), i.e. u is lower semi-continuous. We also notethat it is enough to show that u takes values in ∂E since, for a minimal set, we have that ∂E = ∂ ∗E.To show that (x′,u(x′)) is in the measure theoretic boundary, we just need to show that, for everyr > 0, we have that λ n(E ∩Br((x′,u(x′)))) > 0 and λ n(EC ∩Br((x′,u(x′)))) > 0. Let r > 0. Thenthere exists r′ such that Cr′(x′,u(x′)− r/3) ⊂λ E. If r′′ = min{r,r′}, we have that Cr′(x′,u(x′)−r/3)∩Br(x′,u(x′))⊃ B′r′′(x′)× [u(x′)− r/2,u(x′)− r/3], and so λ n(E ∩Br((x′,u(x′)))) > 0.We get the other measure estimate by contradiction. Suppose that there is an r > 0 such thatλ n(EC ∩Br((x′,u(x′)))) = 0. Then λ n(E ∩Br((x′,u(x′)))) = λ n(Br((x′,u(x′)))), so B′r/100(x′)×[u(x′)− r/100,u(x′) + r/100] ⊂λ E. Then we can find an r′ so that Cr′(x′,u(x′)− r/99) ⊂λ E.Finally, let r′′ = min{r,r′}. Then, for this choice of r′′, we have that Cr′′(x′,u(x′)+ r/100) ⊂λ E,but this is a contradiction since then we would have that u(x′)+ r/100 ≤ u(x′). This shows thatλ n(EC ∩Br((x′,u(x′))) > 0 and so (x′,u(x′)) is in the measure theoretic boundary of E.Now we are in a position to make sense of what it means for an abstract set to satisfy theminimal surface equation.Definition 7. Let u be the boundary defining function of ∂E. Then we say that the boundary of aset ∂E satisfies the minimal surface equationF(D2u,Du) :=∆u√1+ |∇u|2−(∇u)T D2u∇u(1+ |∇u|2)3/2u = 0in the viscosity sense if for any function φ ∈C2 which touches u from below,we have thatF(D2φ ,Dφ)≤ 0,and for any function φ ∈C2 which touches u from above we haveF(D2φ ,Dφ)≥ 0.If a set only satisfies the first inequality, we say that it is a supersolution, and if it satisfies thesecond one, then it is a subsolution.21It should be noted that if u is C2, then we can write the minimal surface equation in divergent formasF(D2u,Du) = div(∇u√1+ |∇u|2).This definition makes sense since we have shown that a minimal set is “almost” a graph of afunction in a measure theoretical sense. Hence, the defining function u captures all relevant “infor-mation” of ∂E. Now, since we do not know whether u is smooth or not, we will take u and producea new function which is close to u and has better regularity via a convolution.Definition 8. We define the inf-convolution of u byuε(x′) = infy′∈B′1{u(y′)+1ε |y′− x′|2}It can be shown that uε is C2 except on a set of measure zero, that it is semiconcave and that uε → uuniformly on compact sets of B′1 as ε → 0. To see this, note thatuε(x′)−|x′|2ε = infy′(u(y′)+|y′− x′|2ε −|x′|2ε)= infy′(u(y′)+|y′|2ε −2x′ · y′ε).Now this function is concave and upper semi-continuous as it is an infimum of continuous linearfunctions. By Alexandrov’s theorem, this implies that there exists a set Z of measure zero so thatfor x′0 ∈ B′1 \Z, there exists a matrix D2uε(x′0) and a vector Duε(x′0) for which the following holdsin a neighborhood of x′0uε(x′) = uε(x′0)+Duε(x′0) · (x′− x′0)+12(x′− x′0)T D2uε(x′0)(x′− x′0)+o(|x′− x′0|2).In the next chapter we will show that if u is a viscosity supersolution to the minimal surface equationin the viscosity sense, then so is its inf-convolution.22Chapter 3Minimal Surfaces ResultsIn this chapter, we introduce and prove all the necessary tools needed to prove the regularity ofminimal sets, which will be done in the final chapter. The approach is based on [8], [10],[11], [12]and [3]. First, we need to recall the basics of curvature for C2 objects in Euclidean space. Theseideas will help us to use smooth approximation to extract important estimates for the minimal sets.3.1 Curvature ResultsWe introduce some of the basic results on curvature taken from Gilbarg and Trudinger[6] whichwill be of use later. Let Ω be a open bounded simply connected subset of Rn having non-emptyC2 boundary ∂Ω. The distance function d is defined by d(x) := dist(x,∂Ω). Now, for y ∈ ∂Ω,let nˆ(y) denote the unit normal to ∂Ω at y. The curvatures of ∂Ω at a fixed point y0 ∈ ∂Ω aredetermined as follows. Applying a rotation if necessary we may assume that the xn direction liesin the same direction as nˆ(y0) in some neighborhood U of y0. Then ∂Ω can be expressed as thegraph of a function given by xn = φ(x′) where x′ = (x1, . . . ,xn−1) with φ ∈C2 and Dφ(y′0) = 0. Thecurvatures of ∂Ω at y0 are then defined by the eigenvalues of the Hessian matrix D2φ at y′0, denotedby κ1, . . .κn−1, and are called the principal curvatures of ∂Ω at y0. The mean curvature of ∂Ω at y0is given by the Cesa´ro mean of the principal curvaturesH(y0) =1n−1n−1∑i=1κi =1n−1∆φ(y′0).The unit normal nˆ(y) at a point y = (y′,φ(y′)) ∈U ∩∂Ω is given by23nˆi(y) =−Diφ(y′)√1+ |Dφ(y′)|2, i = 1, . . . ,n−1, nˆn(y) =1√1+ |Dφ(y′)|2.Also, in a principal coordinate system we have that the derivatives of the normal at y0 are given byD jnˆi(y′0) =−κiδi j, i, j = 1, . . . ,n−1.Lemma 13. Let Ω be bounded and ∂Ω ∈ Ck for k ≥ 2. Then there exists a positive constant µdepending on Ω such that d ∈Ck(Γµ), where Γµ = {x ∈ Rn||d(x)|< µ}.Lemma 14. If Ω and µ are as above and x0 ∈ Γµ , then we have that∆d(x0) =−n−1∑j=1κi1−κid(x0),where κi is the i th principal curvature at the point y0 ∈ ∂Ω, where y0 realizes x0’s distance to∂Ω. In particular, when we have x0 ∈ ∂Ω and ∂E is the graph of a function a C2 function φ , then∆d(x0) =−∑κi = divDφ√1+ |Dφ |2(x0).3.2 Viscosity Solutions For The Minimal Surface EquationIn this section we make use of viscosity solution methods to develop some theory on minimalsets, the most important of which is theorem 18. This will set the stage in which one can prove ageometric Harnack inequality.Theorem 15. If E is minimal and u is the defining function of ∂E, then u is a viscosity supersolutionof div Du√1+|Du|2= 0 in B1.The following proof is based on the original given by Roquejoffre[8] but has been altered tomake use of theorem 9 to make it simpler and shorter.Proof. Suppose that u is not a viscosity supersolution of the MSE. Then there exists φ ∈ C2(B′1)such that u−φ has a maximum at x′0 ∈ B′1, butdivDu√1+ |Du|2(0) > 0.We may assume that x′0 = 0 and that u(0)− φ(0) = 0. Let Γ denote the graph of φ . Then, fromlemma 14 we have that24∆d(·,Γ)|x=0 =−divDφ(0)√1+ |Dφ |2(0).Now, since the map x 7→ d(x,Γ) is smooth in a neighborhood of 0, we can find a δ > 0 such that1. (Γ+δen ≥ ∂ ∗E over B′r)subgraph∪E is a perturbation of E in B1 for some r ∈ (0,1). Sinceu−φ is lower semi-continuous, for any r ∈ (0,1) we could take any δ < infBr\Br/2(u−φ)(x′).That is, we perturb E by pushing φ upwards through it.2. (Γ+δen)subgraph∩Ec has nonempty interior by lemma 6.3. divDφ√1+ |Dφ |2> 0 in Ω= {d(x,Γ)< δ}∩Ec and, since we also have div Dφ(0)√1+ |Dφ |(0)2=−∆d(·,Γ)|x=0, by further restricting δ we have that −∆d > 0 in Ω.Now, by integrating over Ω and using Theorem 9 we get that0 >∫Ω∆d =∫Ωdiv(Dd)=∫∂Γ+δenDd ·ndH n−1 +∫∂EDd ·νEdH n−1=∫∂Γ+δen1H n−1 +∫∂EDd ·νEdH n−1.Hence we have that ∫∂ΓdH n−1 <−∫∂EDd ·νedH n−1 ≤ P(E ∩Ω),but this contradicts the fact that E was minimal. Hence u is a supersolution to the minimal surfaceequation.Corollary 16. If E is minimal and u is the defining function of ∂E, then u is a viscosity subsolutionof the minimal surface equation.Proof. All that was used in the above proof was that u took values in ∂ ∗E so that we could applylemma 6. So, if instead we assume that u is not a viscosity subsolution, then we can find a φ thattouches u from above. Then, as above φ would give a perturbation of ∂E downwards. Applyingthe Gauss-Green formula to this perturbation gives the wrong inequality about the minimality of∂E, which implies that u is a viscosity subsolution.25Now that we know that u satisfies the minimal surface equation in the viscosity sense, we willshow that its approximation uε , the inf-convolution of u, will be a supersolution to the minimalsurface equation. This is quite important since the proof of the ABP estimate for u will depend onthe regularity of uε . Recall that uε(x) = infy∈B(u(x)+1ε |x− y|2).Lemma 17. For all ε > 0 small we have that uε is a super solution to the MSE.The following proof is based on the original given by Roquejoffre[8] but has been altered tohave a more geometric flavor than the original algebraic proof.Proof. Since u is lower semi-continuous, it realizes its minimum on compact sets. Let yε(x) be theminimum for any x and ε in the definition of uε(x). Then we have that uε(x) = u(yε(x))+ 1ε |x−yε(x)|2.Now, looking at uε , if we touch the graph from below by a C2 function φ at some point x0 andnotice that uε lies below u, then by forming a translated copy of the graph of u by the mapping(x,u(x)) 7→ (x0−x∗+x,u(x)+φ(x0)−u(x∗)) where x∗ is the point yε(x0). Now this mapping is anisometry and hence the translated surface satisfies the minimal surface equation since the originalgraph did. Hence if a C2 function ψ touches this translation from below at a point, then we havethat divDψ√1+ |Dψ|2< 0. Now, since φ touched uε from below and the translation lies above uε ,and this same φ also touches the the translate at this point, we have that div Dφ(0)√1+ |Dφ |2(0)< 0,hence uε is a supersolution to the minimal surface equation in the viscosity sense.Now we need to establish an ABP measure estimate. Here we will do this in the general frameworkof elliptic equations in Rn that we gave in the previous chapter to make things simpler. The fol-lowing proof of the ABP estimate is a based on a combination of the proofs given from Savin andRoquejoffre[8], with additional details on the estimates of the eigenvalues for the viscosity relation.Proposition 18. [ABP Estimate] Let u be a viscosity supersolution of the minimal surface equationin B′1. Let B⊂ B′1 be a set of nonzero measure. For a≤ δ/2, let A be the set of contact points of uby sliding paraboloids of opening −a and centers y′ ∈ B from below until they touch u for the firsttime. Then there exists a q > 0 that depends only on δ such that|A| ≥ q|B|provided that A⊂⊂ B′1. Here δ is chosen so that 2.6 is satisfied.26Proof. Since uε is semi-concave, by Alexandrov’s theorem, we have that uε is C2 almost every-where. That is, there exists a set Z of measure zero such that, for all x′0 ∈ B′1 \Z,uε(x′) = uε(x′0)+Duε(x′0) · (x′− x′0)+12(x′− x′0)T D2uε(x′0)(x′− x′0)+o(|x′− x′0|2)= P(x′,x′0)+o(|x′− x′0|2).From the above we see that P(x′,x′0)−η2 |x′− x′0|2 touches uε from below at x′0 for any η > 0.Aassume that x′0 ∈ B \Z. Then, from the normal conditions, we have thatDuε(x′0) = D(−a2|x′− y′|2)|x′=x′0=−a(x′0− y′),which tells us that |Duε(x′0)| ≤ 2a≤ δ andD2uε(x′0)≥−aI.Now, since uε is a viscosity super-solution, by applying the viscosity relation to P(x′,x′0)−η2 |x′−x′0|2 at x′0 we getF(D2uε(x′0)−ηI, p,uε(x′0),x′0)≤ 0.Now we want to show that we can bound the eigenvalues of D2uε(x′0) from above by λ (x′0)≤Ca,where C is a universal constant. Suppose that there exists a unit eigenvector e with eigenvaluebigger then Ca. This combined with the previous matrix inequality gives that D2uε(x′0)≥Cae ·eT −aI. Then, if Λ and λ denote the local elliptic constants in the δ -neighborhood of (0,0)∈S ×Rn−1,we have that0≥ F(D2uε(x0)−ηI,Duε)= F(D2uε(x0)−Cae · eT +aI +Cae · eT ,Duε)≥ F(Cae · eT −ηI−aI,Duε) (by 2.2)≥ λ (Ca−a−η)− (n−1)Λa, (by 2.6)27which is greater then zero if C is large enough, and this contradicts the face that uε is a supersolu-tion. Hence we have that D2uε ≤CaI.Now, if we look at the contact condition of P and uε , i.e. the condition that the normal vectorscoincide, we see that−Duε(x′0)√1+ |Duε(x′0)|2=−D(a/2|x′0− y′|2)√1+ |D(a/2|x′0− y′|2|2= ax′0− y′√1+a2|x′0− y′|2.So Duε(x′0) = −a(x′0 − y′) and so y′ = x′0 +1a Duε(x′0). The map φ : A 7→ B given by φ(x0) =x′0 +1/aDuε(x′0) is a surjection from A to B. From the bounds above we have that the differentialmap of φ is given by Dφ = I + 1a D2uε , and so ‖Dφ‖ ≤ 1a‖D2uε‖+1 ≤C+1. Also, we have that0≤ Dφ ≤ (C+1)I, so applying the coarea formula (theorem 5) we get that|B|=∫φ(A)dx≤∫A|detDφ |dx≤ (C+1)|A|.This establishes the result for uε , and taking limits gives us the general result. If Aε denotes the setof contact point for uε , then we have thatlimsupA1/k = ∩∞m=1∪∞k=m Ak ⊂ A.Now each set ∪∞k=mAk has measure at least q|B|, and since the sequence is decreasing by continuityof measure ,we have that |A| ≥ q|B|.For a > 0, let Da denote the set of interior contact points with u, i.eDa = {z′ ∈ B′1| ∃ y′ ∈ B′1 such that u(z′)+a2|z′− y′|2 ≤ u(x′)+a2|x′− y′|2 ∀x′ ∈ B′1}.Here we want to somehow increase the constant set obtained from the ABP estimate. Thephilosophy is that if we can touch the graph at one point with a paraboloid of curvature 1/a,28then, by changing the curvature a bit, we will touch the graph of u by larger amount. The proofbelow is based on the one given in [11] with more details on the eigenvalues computations and setcontainment argument.Lemma 19. (Bootstrap contact lemma) There exists positive universal constants C and c such thatif a≤ δ/C andDa∩B′r(x′0) 6= /0for some ball B′r(x′0)⊂ B′1, then|DCa∩B′r/8(x0)| ≥ crn−1.Proof. Let z′1 ∈ B′r(x′0)∩Da be a contact point and let by y′1 ∈ B′1 be the center of the paraboloidP(x′,y′1) = −a2|x′− y′1|2 +u(z′1)+a2|z′1− y′1|2 that touches u from below at z′1. First we try to finda point x′1 ∈ B′1/16(x′0) such that, for some universal constant C,u(x′1)−P(x′1,y′1)≤Car2.Let φ : B′1→ R+ be the radially symmetric functionφ(x′) =1α (|x′|−α −1), 1/16≤ |x′| ≤ 11α (16α −1), |x′| ≤ 1/16.where α is a large constant. Letψ(x′) = P(x′,y′1)+ar2φ(x′− x′0r).We will show that ψ is a strict subsolution in the annular region r16 < |x′− x′0|< r.We have that ‖Dψ‖ ≤ ‖DP‖+ar‖Dφ‖ ≤ ‖DP‖+a‖Dφ‖ ≤Ca≤ δ . For r/16 < |x′− x′0|< r wehave that the hessian of ar2φ(x′− x′0r) in a basis (er,eh1 , . . . ,ehn−2), where er is a radial vector andthe ehi is an orthogonal frame to er, is given by29D2ar2φ(x′− x′0r) =aφ ′′(|x′−x′0|r)00ar|x′− x′0|φ ′(|x′−x′0|r)In−2=a(α+1)(|x′−x′0|r)−α−200 −ar2(|x′−x′0|r)−α−2In−2 .In this form we can directly read off the eigenvalues. So, by choosing α large enough, we have that‖ψ+‖= a((α+1) |x′−x′0|r−α−2−1) , ‖ψ−‖= a+ar2(|x′−x′0|r)−α−2, andF(D2ψ,Dψ) = F(−aI +aD2φ ,Dψ)≥ a(λ ((α+1)|x′− x′0|r−α−2−1)−(a+ar2(|x′− x′0|r)−α−2)Λ> 0.Now we slide the graph of ψ from below until it touches u for the first time. That is, we are lookingfor the point x′1 that obtains the minimum ofinfB′r(x′0)(u−ψ).We have that this minimum is negative sinceu(z′1)−ψ(z′1) = P(z′1,y′1)−ψ(z′1) =−ar2ψ(z′1− x′0r) < 0.Since for x′ ∈ ∂B′r(x′0) we have thatu(x′)−ψ(x′)≥ P(x′,y′1)−P(x′,y′1)≥ 0,we see that x′1 6∈ ∂B′r(x′0). Now the minimum can’t be in B′r(x′0) \B′r/16(x′0) because u is a super-solution and so we would have that F(D2ψ,Dψ)≤ 0, but we just saw that F(D2ψ,Dψ) > 0. Thisshows that the minimum is attained in B′r/16(x′0). Since x′1 is the minimum and its value is negativewe get that30u(x′1)≤ ψ(x′1) = P(x′1,y′1)+ar2φ(x′− x′0r)≤ P(x′1,y′1)+Car2.Now, by sliding the family of paraboloidsP(x′,y′1)−C′ a2|x′− y′|2 + cy′ , y′ ∈ B′r/64(x′1),from below until they touch the graph of u for the first time, where C′ is a large constant whichwill be determined later, we will show that the set of contact points occur in B′r/16(x′0). Using theprevious estimate can get an upper bound on cy′ sinceu(x′1)−P(x′1,y′1)+C′ a2|x′1− y′|2− cy′ ≥ 0.Rearranging this gives us thatcy′ ≤Car2 +C′a2(r/64)2.Now if |x′− x′1| ≥ r/16 we have thatu(x′)−P(x′,y′1)+C′ a2|x′− y′|2− cy′ ≥C′ a2|x′− y′|2− cy′≥C′a2(|x′− x′1|− |x′1− y′|)2−Car2−C′a2(r/64)2≥C′a2(3r64)2−Car2−C′a2(r/64)2which is positive if C′ is chosen to be large enough. So, we have that all the contact points oc-cur inside B′r/16(x′1) ⊂ B′r/8(x′0). Now all that is left to show is that the centers of the family of31paraboloids given above are compactly contained in B′1. We have thata2|x′− y′1|2 +C′a2|x′− y′|2 = (1+C′)a2|x′|2−ax ·(y′1 +C′y′)+a2(|y′1|2 +C′|y′|2)= (1+C′)a2(|x′|2−2x ·(y′11+C′+C′1+C′y′))+a2(|y′1|2 +C′|y′|2),and so the centers arey′11+C′+C′1+C′y′with openings of−(C′+1)a. As y′ ∈ B′r/64(x′1) we have that, if r is small enough, the centers of theparaboloids range over a ball of radius cr where c = C′64(1+C′) . This ball will be compactly containedin B′1 if r is small enough. We are now able to apply the previous ABP estimate which gives us|D(C′+1)a∩B′r/8(x0)| ≥ crn−1.The following is a covering lemma that we will use to prove another measure estimate. Thefollowing lemma roughly says that if we continue to carve more and more out of a ball in a certainfashion, then we have to be carving measure out at an exponential rate. The proof provided belowis modified from Savin[11].Lemma 20. Assume the closed sets Fk satisfyF0 ⊂ F1 ⊂ F2 ⊂ . . .Fk ⊂ . . .B′1/3with F0 6= /0. Suppose further for any r,x′ withB′r/8(x′)⊂ B1/3, B′r(x′)⊂ B′1,thatFk∩B′r(x′) 6= /0implies that|Fk+1∩B′r/8(x′)| ≥ crn−132Then we have that|B′1/3 \Fk| ≤ (1− c1)k|B′1/3|for some constant c1 depending on c.Proof. For x′ ∈ B′1/3 \Fk let rx′ = dist(x′,Fk). Since Rn−1 is a strongly Lindelo¨f space, making useof Vitali covering lemma we can find a countable subset I ⊂ B′1/3 \Fk such that1. B′1/3 \Fk =⋃x′∈I B′rx′/5(x).2. For x′ 6= y′ we have that B′rx′/40(x′)∩Bry′/40(y′) = /0.This collection satisfies the hypothesis, so|Fk+1 \Fk|= |∪x′∈I(Fk+1∩B′rx′/5(x′))|≥ |∪x′∈I(Fk+1∩B′rx′/40(x′))|= ∑x′∈I|Fk+1∩B′rx′/40(x′)|≥ c∑x′∈I|Brx′/5(x′)|= c|B1/3 \Fk|.This estimate gives the desired estimate of the lemma since|B′1/3 \Fk|= |B′1/3∩FCk |= |B′1/3∩FCk+1|+ |B′1/3∩FCk \FCk+1|= |B′1/3 \Fk+1|+ |Fk+1 \Fk|.Rearranging the above gives|B′1/3 \Fk+1|= |B′1/3 \Fk|− |Fk+1 \Fk| ≤ (1− c)|B′1/3 \Fk|.Now we only need one more measure estimate before we are able to prove the Harnack inequality.The proof is from the work of Savin[12], which we include for completeness.33Theorem 21. Let u : B′1→ R, u≥ 0 be a viscosity supersolution to F(D2u,Du) = 0. Given µ > 0,there exists ε > 0 and M depending on n,δ ,λ ,Λ such that ifu(0)≤ εthenH n−1({u > Mu(0)}∩B′1/3) < µ.Here δ is as in from 2.3 and λ , Λ are the corresponding elliptic constants.Proof. Let a = 18u(0) and define the setFk = DCka∩B′1/3,where C is the constant from Lemma 19. Since u ≥ 0 in B′1, the paraboloid of opening −a andcenter 0 touches the graph of u for the first time in B′1/3. To see this let z′ be the contact point. Thenwe have0≤ u(x′)−P(x′,z′) = u(x′)+a2‖x′‖2−a2‖z′‖2 +u(z′).Rearranging and setting x = 0 we find that‖z′‖2 ≤2au(0) =19.Thus we have F0 6= /0 and so from lemma 19 we see that the sets Fk satisfy the conditions of lemma20 as long as Cka < δ . HenceH n−1(B′1/3 \DCka)≤ (1− c1)kH n−1(B′1/3), if Cka < δ .Now, for z′ ∈ DCka there exists a y′ ∈ B′1 such thatu(z′)+Cka2‖z′− y′‖2 ≤ u(x′)+Cka2‖x′− y′‖2 ∀x′ ∈ B′1,which gives u(z′)≤ u(0)+2Cka≤ 3Cka :=M. Given µ , choosing k large enough so that (1−c1)k≤µ and ε so that 18Ckε < δ , we get the desired result.We make a note that these results only make use of the fact that u\∂E is a super solution to the34minimal surface equation we can also use the fact that it is a subsolution to get similar results whichwill be needed in the proof of the Harnack inequality in the next section.35Chapter 4Regularity of Minimal SurfacesIn this section, we make use of all the tools developed in the last chapter to prove the desiredregularity of minimal sets. We start by proving a geometric Harnack inequality, one of the majortools used to prove the De Giorgi flatness result. The De Giorgi flatness result says that if yourset is contained in a flat strip on some scale, then, on a fixed smaller scale relative to the originalscale in some new coordinate system, the set is contained in an even flatter strip. The proof of theHarnack inequality is based on the proof given by Savin in [12]. We have removed an implicititerative covering argument that was in the original by taking r0 to be small enough so that onlytwo applications of Theorem 21 are needed.We are now at the point where we can prove the Harnack inequality.Theorem 22. Assume that E is minimal in B1 and that∂E ∩B1 ⊂ {|xn| ≤ ε}with ε ≤ ε0(n). Then∂E ∩Br0 ⊂ {|xn| ≤ ε(1−η)},where r0 =12∗11+3√2, and η and ε0(n) are small universal constants.Proof. For this we assume that the result does not hold inside Br0 ∩ ∂E, that is, ∂E is not con-strained by a slightly smaller set of planes. Then there exists points x1 = (x′1,y1)),x2 = (x′2,y2) ∈∂E ∩Br0 such that y1 = u(x′1) > ε(1−η) and y2 = u(x′2) < −ε(1−η). To apply theorem 21,we need to shift our coordinates system so that they are centered at x1 and x2 respectively. For36the point x2, we just shift up by ε and apply the theorem to the function u( 11−r0 (x′− x′2)) + ε .For the other point x1, we shift down by ε and then reflect to apply the theorem to the func-tion −u( 11−r0 (x′− x′1))+ ε). Now theorem 21 gives us that H n−1({u > Mηε − ε}∩B′(x′2;(1−r0)/3)) < C(r0)µ and H n−1({u > ε −Mηε}∩B′(x′1;(1− r0)/3)) < C(r0)µ . Now we want thetwo regions {u > ε −Mηε}∩B(x′1;(1− r0)/3) and {u > Mηε − ε}∩B(x′2;(1− r0)/3) to eachhave more than half the mass of B′r0 . This will give us that the regions overlap which will lead toa contradiction. We compare the perimeter of ∂E inside the cylinder C = B′r0× [−1,1]∩B1. Sincethe projection map is Lipschitz and has Lipschitz constant one, we have thatH n−1(Πen(∂E ∩C))≤ P(E;C).If Mη < 1, we have that the two sets are disjoint and by applying theorem 21 we find that each hasH n−1-mass greater than(1+C2(r0))2H n−1(B′r0)−C(r0)µinside B′r0 by the choice of r0. SoP(E;C) > (1+C2(r0))Hn−1(Br0)−2C(r0)µOn the other hand, if we preform a surgery on E and add the set F = B′r0 × [−ε,ε] to it, we get acompetitor to E. Since E contains everything below {xn < −ε}, we can view this surgery as justadding a cap to the top side of ∂E over B′r0 , and, by the minimality of E we find thatP(E;C)≤ P(E ∪F ;C) =H n−1(B′r0)+Cε ≤Hn−1(B′r0)+C1(r0)ε0(n).This gives a contradiction if we choose µ small and ε0(n) small enough so thatC1(r0)ε < C2(r0)H n−1(B′r0).4.1 Flatness TheoremHere we prove the final obstacle to showing the regularity of flat solutions to the minimal surfaceequation: the De Giorgi flatness theorem. Repeated applications of the De Giorgi flatness theoremto ∂E will imply that a solution must have a certain amount of smoothness. Our main reference for37this section is [12].Theorem 23 (Flatness theorem). Assume that E is minimal in B1,0 ∈ ∂E and∂E ∩B1 ⊂ {‖xn‖ ≤ ε}with ε ≤ ε0(n). Then there exists a unit vector ν1 such that∂E ∩Br0 ⊂ {‖x ·ν1‖ ≤ε2r0},where ε0 and r0 are the constants from theorem 22.Proof. We prove this by contradiction. Assume that there is a sequence of minimal surfaces ∂Eksuch that0 ∈ ∂Ek∩B1 ⊂ {|xn|< εk},with εk→ 0, for which the conclusion does not hold. Now, for each point x0 ∈ ∂Ek∩Br, we applythe Harnack inequality at x0 and obtain that in the xn direction, the oscillation of ∂E ∩Br(x0) isless then 2εk(1−η/2). We repeatedly apply the Harnack inequality as long as the hypothesis issatisfied, that is, as long as m satisfies 2mεk(1−η/2)m−1 ≤ ε0(n), and obtain that in ∂E ∩Brm(x0),the oscillation is less than 2εk(1−η/2)m in the xn direction. We can let m go to infinity as εk goesto zero. Now, by dilating the situation in the xn direction by a factor of 1/εk, we have that the setsAk := {(x′,xnεk)| (x′,xn) ∈ ∂E ∩B1}are included in {‖xn‖ ≤ 1}. A compactness result with these functions gives us some flat limitfunction. For m as above we have that the oscillations of Ak inside ‖x′− x′0‖ < r−2m is less than2(1−η/2)m. Thus, each function fk representing Ak is Ho¨lder continuous with Cα norm boundedabove by 2, and so, by Arzela-Ascoli, by passing to a subsequence if necessary, we get that thefk converge to a Ho¨lder continuous function w(x′) inside {‖x′‖ ≤ r}. Next we will show that w isharmonic by showing that it is harmonic in the viscosity sense. Assume that P(x′) is a quadraticpolynomial that touches w from below at some point (x′t ,w(x′t)). Then φτ(x′) = P(x′)+cm−τ2‖x′t−x′‖2 will touch fk from below at some point (x′t,k, fk(x′t,k)), and by the convergence of fk to w wehave that x′t,k→ x′t . Since φτ touches Ak from below, εkφτ touches ∂Ek from below, and since ∂Eksatisfies the minimal surfaces equation in the viscosity sense, by applying the viscosity relation tothe touching paraboloids we have that38εk(∆P− τ)√1+ εk|∇P− τ(x′− x′t,k)|2− ε3k(∇P− τ(x′− x′t,k))T (D2P− τI)(∇P− τ(x′− x′t,k))(1+ εk|∇P− τ(x− xt,k)|2)3/2≤ 0.Now by dividing by εk and letting k→ ∞, τ → 0 we get that ∆P≤ 0 at xt . We also get that ∆P≥ 0at xt since if we touch w from above by P, then we would get that φτ(x′) = P(x′)+cm−τ2|x′t +x′|2would touch fk from above, so εkφτ would touch ∂Ek from above. Finally applying the viscosityrelation givesεk(∆P+ τ)√1+ εk|∇P+ τ(x′− x′t,k)|2− ε3k(∇P+ τ(x′− x′t,k))T (D2P+ τI)(∇P+ τ(x′− x′t,k))(1+ εk|∇P+ τ(x− xt,k)|2)3/2≥ 0.Likewise taking limits gives ∆P ≥ 0, and so ∆w = 0 in the viscosity sense. However functionswhich are harmonic in the viscosity sense are harmonic in the classical sense, so we have that w isharmonic.Now w(0) = 0 since for every k we have that 0 ∈ ∂Ek. Doing a Taylor expansion we have‖w(x′)− x′ ·∇w(0)‖ ≤r04if ‖x′‖ ≤ 2r0provided that r0 is chosen small and universal.Now since fk converge to w, using the compactness of Arzela´-Ascoli we can make ‖ fk(x′)−w(x′)‖ ≤ ρ for any ρ > 0 on ‖x′‖ ≤ 2r0 if k is large enough. So,‖ fk(x′)− x′ ·∇w(0)‖ ≤ ‖ fk(x′)−w(x′)‖+‖w(x′)− x′ ·∇w(0)‖ ≤ ρ+ r04,and choosing ρ < r0/4 and recalling that uk = εk fk, multiplying the above inequality by εk givesus that‖(x′,uk(x′) · (εk∇w(0),1)‖ ≤ εkr02.39By normalizing the above vector, this gives us that ∂Ek satisfies the result of the theorem, i.e.∂Ek∩Br0 ⊂ {‖x ·ν‖ ≤εk2r0,}which is a contradiction.4.2 Regularity Of Minimal SetsNow we are at the point where we can show the regularity of ∂E. We present expanded details onthe regularity process presented in [12]. By iterating Theorem 23, we obtain unit vectors vk suchthat∂E ∩Brk0 ⊂ {‖x · vk‖ ≤ε2krk0}.Now the vk converge to a unique limit v(0), for if there were two or more limit points, say ν1 andν2, then we could take k > l large enough so that vk ∼ ν1 and vl ∼ ν2. Then on Brk we would havethat a portion of the constraining planes generated by vk and vl would be disjoint. Since ∂E mustlive in both of these regions, we get that it is not the graph of a function, a contradiction. Hencethis gives uniqueness of the limiting normal vector. We want to see just how fast the unit vectorsconverge. To go from vk to the next iteration we would have that ∂E is contained in both of theconfined planes corresponding to vk and vk+1. To find the angle between the two vectors we canuse the vertical length of both sets of constraining planes. The sum of these vertical lengths will bea very good approximation to the arc length of a circle formed by the intersection of the ball andthe plane spanned by the two vectors. Since ∂E must be contained in both sets of planes, in theworst case we would have that the lower corner of one of the planes touches the corner of the otherplane. Now the arc length of circle is given by rk+10 θk, where θk is the angle between vk and vk+1.This gives us thatrk0θk =rk02kε+rk+102k+1ε,and so θk =ε2k+r02k+1ε . This allows us to get an estimate on the tightness of vk to vk+1:‖vk+1− vk‖2 = 2−2cosθk≤ θ 2k40so‖vk+1− vk‖ ≤ θk≤C(r0)ε2k.Now, since ∑∞k=1 vk+1−vk is absolutely summable by the above estimate and Rn is a Banach space,∑∞k=1 vk+1− vk exists and we have that v(0) = v1 +∑∞k=1 vk+1− vk . This gives us that the rate ofconvergence of vk to v(0) is‖v(0)− vk‖ ≤∞∑n=k‖vn+1− vn‖ ≤ 2C(r0)ε2k,and so ‖x · v(0)‖ ≤ ‖x · vk‖+‖v(0)− vk‖rk0 ≤ (C(r0)+1)εrk02k = (C(r0)+1)εrk(1− log2logr0)0 ,which gives us the C1,α regularity of ∂E since∂E ∩Brk0 ⊂ {‖x · v(0)‖ ≤ (C(r0)+1)εrk(1− log2logr0)0 .}Finally, we can apply the theory of DeGiorgi-Nash-Moser[13] to conclude that u ∈C2,α . This to-gether with classical Schauder theory, gives us that u∈C∞ by bootstrapping, since we can applyingSchauder interior estimates to the solution of the minimal surface equation as found in Gilbarg andTrudinger PDE book [6].The necessary estimate and interior regularity result are as follows.Theorem 24 (Interior Schauder Estimates). Let α ∈ (0,1). Suppose Ω is a domain and u∈C2,α(Ω)solves the elliptic equationLu = ai jDi ju+biDiu+ cu = f in Ω, (4.1)where ai j,bi,c are in Cα(Ω). Then for Ω′ ⊂⊂Ω,‖u‖C2,α (Ω′) ≤C(‖u‖C(Ω)+‖ f‖Cα (Ω)), (4.2)where C =C(n,α,L,Ω′,Ω) ∈ (0,∞).Theorem 25 (Interior Regularity). Let k≥ 0 be an integer and α ∈ (0,1). Let Ω⊂Rn be open and41suppose that u ∈C2(Ω) satifiesLu = ai jDi ju+biDiu+ cu = f in Ω (4.3)for some elliptic operator L with coefficients ai j,bi,c ∈ Ck,α(Ω) and some f ∈ Ck,α(Ω). Thenu ∈Ck+2,α(Ω)We see that, by applying theorem 25 to u for the elliptic equation induced by the minimalsurface equation on u, we find that the ai j and bi are the derivatives of u. Since u is C2,α , we havethat they are Cα , so the conditions of the theorem hold, hence u is in fact C4,α . Now we have thatthe coefficient’s are C2,α so we can apply the result again and get that u is C6,α . By proceedinginductively we do the so called bootstrapping procedure and find that u is Ck,α for every k ∈N, andhence u is C∞.42Bibliography[1] Tom Apostol. Mathematical Analysis. Oxford Mathematical Monographs. Pearson, 1974.→ pages 6[2] Luis A. Caffarelli and Xavier Cabre´. Fully Nonlinear Elliptic Equations. ColloquiumPublications. American Mathematical Society, Providence, Rhode Island, 1995. → pages ii,17[3] Luis A. Caffarelli and Antonio Co´rdoba. An elementary regularity theory of minimalsurfaces.Differential and Integral Equations. An International Journal for Theory and Applications,6(1):1–13, 1993. → pages 1, 12, 23[4] Lawrence C. Evans and Ronald F. Gariepy. Measure theory and fine properties of functions.Studies in advanced mathematics. CRC Press, Boca Raton, 1992. → pages 1, 4, 7, 8, 9, 10,14[5] Herbert Federer. Geometric Measure Theory. Classics in Mathematics. Springer, 1996. →pages 1[6] David Gilbarg and Neil S. Trudinger. Elliptic Partial Differential Equations of Second Order.Classics in Mathematics. Springer, Berlin Heidelberg New York, 2001. → pages 23, 41[7] Enrico Giusti. Minimal Surfaces and Functions of Bounded Variation, volume 80 ofMonographs in Mathematics. Birkhauser, Boston, 1984. → pages ii, 1, 11, 12, 14[8] Roquejoffre Jean-Michel. Regularity of minimal surfaces: a viscosity solutionsapproach,CIRM Lecture Notes, 2009. → pages ii, 1, 23, 24, 26[9] Nicola Fusco Luigi Ambrosio and Diego Pallara. Functions of Bounded Varition and FreeDiscontinuity Problems. Oxford Mathematical Monographs. Oxfords Science Publications,2000. → pages 6[10] Ovidiu Savin. Small Perturbation Solutions for Elliptic Equations. Communications inPartial Differential Equations, 32(4):557–578, 2007. → pages 2343[11] Ovidiu Savin. Phase Transitions, Minimal Surfaces, and a Conjecture of De Giorgi. CurrentDevelopments in Mathematics, pages 59–113, 2009. → pages ii, 1, 23, 29, 32[12] Ovidiu Savin. Minimal Surfaces and Minimizers of the Ginzburg Landau energy.Contemporary mathematics (American Mathematical Society), 526:43–58, 2010. → pagesii, 1, 23, 33, 36, 38, 40[13] J.Mathews P.Rockstroh T.Begley, H.Jackson. Existence and Regularity of Solutions to theMinimal Surface Equation, 2013. → pages 4144

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