Soluciones viscosas de problemas de Hamilton - Jacobi.

Abstract

In this paper we study existence, uniqueness and stability results for a parabolic problem involving non-local operators of Lévy type. We begin developing the notion of viscosity solution in our context and stating its first properties, from which we can deduce a stability result. Later, via the concept of discontinuous viscosity solution, we prove the Perron’s Method, which give us sufficient conditions to obtain a discontinuous viscosity solutions for our problem. After that, we focus in the Comparison Principle, for what we divide the study in two cases: without considering an explicit dependence on time variable, and doing it. This result helps us to recover continuity and ensure uniqueness of the viscosity solutions given by Perron’s Method. Finally, we construct sub and super-viscosity solutions to our problem and proceed to apply results obtained before. In this way, we ensure there exist an unique continuous viscosity solution for the problem we are interested in.