Conjuntos finos e potencial não-linear de Wolff-Hedberg estacionário e parabólico

Abstract

In this work we have two goals, one of them is to demonstrate the Wiener criterion for Wolff’s potential. The second one is to extend this result to the parabolic version of Wolff’s potential, using the concept of thinn sets. To achieve that, we studied capacities associated with lower semi-continuos and positive kernels and defined the Havin- Mza ya s non-linear potential and in which conditions it admits capacitary measures. After that, we demonstrated an equivalence between Havin-Mza ya s non-linear potential and Wolff’s potencial. That equivalence was mandatory so we could acomplish our first goal. After fulfilling our first objetive we started working on the parabolic case. We used the fundamental solutions for the heat equation as our lower semi-continuous and positive kernel in the capactity theory that we have developed in this work. We showed the Parabolic version of the Riesz Kernel and proved that it returns the fundamental solution of the heat equation, in certain conditions. Then, we defined the parabolic version of Wolff’s potential associated with the parabolic Riesz’s Kernel, demonstrated the equivalence between it’s continuos version and it’s diadic version. And finally, we were able to show our main duty, that it was to demonstrate the Wiener’s criterion for the heat equation making use of the thinn sets theory and Wolff’s potencial. This same result returns the classical paper of Evans and Gariepy, where they showed the Wiener’s criterion for the heat equation.