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UBC Theses and Dissertations

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

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 Ω in IRn, we show that the value and the attainability of the best Hardy-Sobolev constant on a smooth domain Ω, μs(Ω) := inf [Chemical Equation] when 0 < s < 2, 2*(s) = [Chemical Equation], and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form: -Δu = [Chemical Equation] where f 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 source term, [Chemical Equation] , in the case of arbitrary compactly supported initial functions u0. An estimate on the support in terms of supp u0 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. 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 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. This is joint work with Martial Agueh and Nassif Ghoussoub.

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