Boundary regularity for the fractional heat equation

Abstract

In this dissertation we present an introduction to nonlocal operators, and in particular, we study the fractional heat equation, which involves the fractional Laplacian of order 2s. In the first chapters we make a review of known classical results in the topic. After that, we introduce modern results on the elliptic problem for the fractional Laplacian and we use them to derive the main original result of the dissertation. We show that a solution "u" of the homogeneous fractional heat equation on a bounded domain U fulfills that u is in C^s(R^n) and that u/d^s can be extended Hölder continuously up to the boundary of the domain, where d(x) is the distance between x and the boundary of U. Furthermore, we are able to discuss the non-homogeneous case and obtain a similar result when the non-homogeneous term is time independent. Finally, we show an application and an extension of the result obtained. We are able to show that the Pohozaev identity holds for the solution of the fractional heat equation for positive times, and we extend the main result obtained for the fractional Laplacian to other nonlocal stable operators under certain conditions.