Stable solutions of nonlinear fractional elliptic problems

• Autores: Tomás Sanz Perela
• Directores de la Tesis: Xavier Cabré Vilagut (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2019
• Idioma: español
• Tribunal Calificador de la Tesis: Yannick Sire (presid.) , Rubió Joan Solá Morales (secret.) , Ángel Calsina Ballesta (voc.) , Albert Mas Blesa (voc.) , Maria Colombo (voc.)
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• Resumen

  • This thesis is devoted to study integro-differential equations. This type of equations constitutes nowadays a very active field of research which has important applications in modeling real-life phenomena where nonlocal interactions appear. The most canonical example of integro-differential operators is the fractional Laplacian, which is the infinitesimal generator of a radially symmetric Lévy process.

    We are interested in the regularity and qualitative properties of stable solutions to semilinear equations. Such solutions are those at which the linearized operator associated to the equation is nonnegative. They correspond to steady states of a system which are stable under small perturbations.

    The thesis is divided in two parts. In the first one we study the boundedness of stable solutions to a semilinar problem for the fractional Laplacian in a bounded domain. We consider the case when the domain is a ball, and we establish a condition on the power of the Laplacian and the dimension for stable solutions to be bounded. In particular, in dimensions between 2 and 6, both included, we prove that stable solutions are bounded for all powers of the fractional Laplacian.

    To establish these results we use the extension problem for the fractional Laplacian. This is an important technique that relates a fractional problem with a local one in a half-space of one more dimension.

    In the second part of the thesis we are focused on the study of saddle-shaped solutions to the integro-differential Allen-Cahn equation. These solutions, defined only when the dimension is even, are doubly radial, odd with respect to the Simons cone, and vanish only on this set. The importance of studying saddle-shaped solutions is due to their relation with the theory of nonlocal minimal surfaces and a fractional version of a conjecture by De Giorgi. They are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.

    First we consider the problem where the operator appearing in the equation is the fractional Laplacian, and we use the extension problem. Our results establish the uniqueness of the saddle-shaped solution in all even dimensions, and its stability in even dimensions greater or equal than 14. Before this work, it was known that these solutions are unstable in dimensions 2, 4, and 6. Thus, after our result, the stability remains an open problem only in dimensions 8, 10, and 12.

    In dimensions greater or equal than 14, our result leads to the stability of the Simons cone as a nonlocal minimal surface. This is the first analytical proof of a stability result for the Simons cone in the nonlocal setting.

    We also study, for first time in the literature, saddle-shaped solutions to more general integro-differential Allen-Cahn equations with an operator which is rotation invariant and uniformly elliptic, but not the fractional Laplacian. Since the extension problem is no longer available, some new nonlocal techniques are developed in this thesis.

    We establish an appropriate setting to develop a theory of existence and uniqueness for the saddle-shaped solution. More precisely, we characterize the kernels for which we can carry out such a theory, by finding a necessary and sufficient condition on the convexity of the kernel. These results are achieved by writing the operator acting on a doubly radial odd function as a new integro-differential operator acting on functions defined only at one side of the Simons cone.

    Under the previous assumption on the kernel, we establish existence, uniqueness, and asymptotic behavior of the saddle-shaped solution to the integro-differential Allen-Cahn equation in all even dimensions. For this, we prove, among others, an energy estimate for doubly radial minimizers, a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in "narrow" sets.