Solução de viscosidade da equação eikonal em variedades Riemannianas

Abstract

The aim of this dissertation is to study the existence and uniqueness of viscosity solutions of the eikonal equation where the is a bounded open subset of a riemannian manifold M. For this, firstly, We present some basic notions such as: Banach-Finsler Manifolds and uniformly bumpable Finsler manifold. Moreover, we present a detailed proof that every Finsler manifold in the sense of Neeb-Upmaier K-weak uniform (in particular, every riemannian manifold) is uniformly bumpable, as discussed by Jiménez and Sanchez [11]. Next, we study the notions of subdierential calculus in riemannian manifolds and we present a detailed proof of some results obtained by Azagra, Ferrera and López [1], among which the smooth variational principle on riemannian manifolds, the perturbed minimization principle for the difference of two functions and the Deville's mean value inequality. Finally, we apply these results to show existence and uniqueness of viscosity solutions to eikonal equation.