Qualitative properties of solutions to integro-differential elliptic problems

Resumen

The thesis is devoted to the analysis of elliptic PDEs and related problems. It is mainly focused on the study of qualitative and regularity properties of solutions to integro-differential equations. The study of such equations has attracted much attention recently since they arise naturally in different areas when dealing with processes where long range interaction phenomena appear. The canonical example of integro-differential operators is the fractional Laplacian, which is translation, rotation, and scale invariant. The thesis is divided into three parts. Part I concerns the study of uniqueness and regularity properties of solutions to integro-differential linear problems. First, we prove, by following a nonlocal Liouville-type method, the uniqueness of solutions in the one-dimensional case, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to semilinear problems of Allen-Cahn type. Next, we establish the first boundary regularity result for the Neumann problem associated to the fractional Laplacian. We prove that weak solutions are Hölder continuous up to the boundary by developing a delicate Moser iteration with logarithmic corrections on the boundary. We also establish a Neumann Liouville-type theorem in a half-space, which is used together with a blow-up argument to show higher regularity of solutions. Part II of the thesis is focused on the study of the saddle-shaped solution to the integro-differential Allen-Cahn equation. These solutions, whose zero level set is the Simons cone, are expected to be the simplest minimizer which is not one-dimensional to the local and nonlocal Allen-Cahn equation in high enough dimensions. It plays, thus, the same role as the Simons cone in the theory of minimal surfaces. First, we study the saddle-shaped solution for the fractional problem by using the extension problem. We establish its uniqueness and, in dimensions greater or equal than 14, its stability. As a byproduct, we give the first analytical proof of a stability result for the Simons cone in the nonlocal setting for such dimensions. The key ingredient to prove these results is a maximum principle for the linearized operator. Next, we study saddle-shaped solutions for any rotation invariant and uniformly elliptic integro-differential operator. In this scenario, we need to develop some new nonlocal techniques since the extension approach is not available. In this respect, our main contribution is a characterization of the kernels for which one can develop a theory of existence and uniqueness of saddle-shaped solutions. Under these assumptions, we establish an energy estimate for doubly radial odd minimizers and some properties of the saddle-shaped solution, namely: existence, uniqueness, asymptotic behavior, and a maximum principle for the linearized operator. Finally, in Part III we develop a nonlocal Weirstrass extremal field theory. In analogy to the local theory, we construct a calibration for the nonlocal functional in the presence of a foliation made of solutions when the nonlocal Lagrangian satisfies an ellipticity condition. The model case in our setting corresponds to the energy functional for the fractional Laplacian, for which such a calibration was still unknown. The existence of such a calibration allows us to prove that any leaf of the foliation is automatically a minimizer for its own exterior datum, with no need to have an existence result of minimizers, neither to know their regularity.