Regularity theory for obstacle problems and boundary Harnack inequalities

Abstract

[eng] This thesis is dedicated to the study of elliptic and parabolic Partial Differential Equations, both local and nonlocal. More specifically, this work concerns the regularity properties of some obstacle problems. Obstacle problems are prototypical examples of free boundary problems, that is, PDE problems where the unknowns are not only a function, but also a subdivision of the domain into different regions, and the PDE satisfied in each region is different. Free boundary problems are a very active field of research. On the one hand, free boundaries are a good model for interfaces in real-world settings, with applications in Physics, Biology, Finance and Engineering. On the other hand, they have been a source of interesting mathematical challenges, motivating the fine analysis of solutions to elliptic and parabolic equations. This Thesis is divided into two Parts. Part I is devoted to the study of several different obstacle problems. In Chapter 1, we study the obstacle problem for parabolic nonlocal operators, in the supercritical regime s < 1/2. We establish the optimal C^{1,1} regularity of solutions, which is surprisingly better than in the elliptic problem, and we also show that the free boundary is globally C^{1,α}. Our main difficulties are the lack of monotonicity formulas, and the supercritical scaling of the equation, that is, the fact that the highest order of differentiation corresponds to the time derivative. Chapter 2 is devoted to the generic regularity properties of the free boundary in the thin obstacle problem. Since there are many pathological examples of solutions to free boundary problems, often the goal is instead of proving regularity for all solutions, proving regularity for most solutions in an appropriate sense. In our work, we show that, for one-parameter monotonous families of solutions, for almost every solution, the free boundary is smooth outside of a set of codimension 2 + α (in the free boundary). In particular, this means that in R^3 and R^4, the free boundary is generically smooth. We conclude Part I with Chapter 3, where we use a nonlocal analogue of the Bernstein technique to establish semiconvexity estimates for a wide class of nonlinear nonlocal elliptic and parabolic equations, including obstacle problems. As a consequence, we extend the known regularity theory for nonlocal obstacle problems in the full space to problems in bounded domains. In Part II, we extend the boundary Harnack inequality to (local) elliptic and parabolic equations with a right-hand side. The boundary Harnack is a classical result that states that if u and v are positive harmonic functions that vanish on part of the boundary of a regular enough domain, then u/v is bounded and Hölder continuous up to the boundary. Boundary Harnack inequalities are used in the proof of the smoothness of free boundaries in several obstacle problems, in the key step of seeing that if a free boundary is flat Lipschitz, then it is C^{1,α}. The goal of our work was to extend the regularity theory of obstacle problems to the fully nonlinear setting. To do so, we developed boundary Harnack inequalities for equations in non-divergence form with a right-hand side. Chapter 4 concerns elliptic equations and Chapter 5 is about parabolic equations. The techniques used are different. In the elliptic setting, it is enough to use barriers, scaling arguments and a standard iteration to deduce the Hölder regularity of the quotient. However, in the parabolic world, the proofs are much more involved and they are based on a delicate contradiction-compactness argument.

[cat] Aquesta tesi se centra en l'estudi d'Equacions en Derivades Parcials el·líptiques i parabòliques, tant locals com no locals, concretament en les propietats de regularitat d'alguns problemes d'obstacle. Els problemes d'obstacle són exemples de problemes de frontera lliure, és a dir, problemes d'EDP on les incògnites són una funció i una partició del domini en diferents regions, i l'EDP satisfeta en cada regió és diferent. Els problemes de frontera lliure són un camp de recerca molt actiu, tant per les seves aplicacions al món real, com pels reptes matemàtics que suposen. Aquesta Tesi està dividida en dues parts. La Part I està dedicada a l'estudi de diversos problemes d'obstacle. Al Capítol 1, estudiem el problema de l'obstacle per a operadors parabòlics no locals, en el règim supercrític s < 1/2. Establim la regularitat C^{1,1} òptima de les solucions i demostrem que la frontera lliure és globalment C^{1,α}. El Capítol 2 està dedicat a les propietats de regularitat genèriques de la frontera lliure en el problema de l'obstacle prim. En particular, veiem que a R^3 i R^4, la frontera lliure és diferenciable gairebé per tota solució. Concloem la Part I amb el Capítol 3, on utilitzem un anàleg no local de la tècnica de Bernstein per, entre altres aplicacions, estendre la teoria de regularitat coneguda per a problemes d'obstacle no locals a problemes en dominis fitats. A la Part II, estenem la desigualtat de Harnack de frontera a equacions (locals) el·líptiques i parabòliques amb un terme independent. La desigualtat de Harnack de frontera diu que si u i v són funcions harmòniques positives que s’anul·len en part de la frontera d'un domini prou regular, llavors u/v està fitada i és C^α fins la vora. Al nostre treball, desenvolupem desigualtats de Harnack de frontera per a equacions en forma de no-divergència amb terme independent. Com a conseqüència, obtenim la regularitat C^{1,α} de la frontera lliure a alguns problemes d’obstacle completament no-lineals. El Capítol 4 tracta sobre equacions el·líptiques i el Capítol 5 parla d'equacions parabòliques