Diferenciabilidade e aproximação por funções C1: para funções Lipschitz, de Sobolev e variação limitada.

Abstract

In the scope of the research in Applied Analysis as in Partial Differential Equations, a mathematician tends to deal with functions that are not continuous. In this way, this work aims to present regularity and approximation results by more regular functions, for functions that are first only integrable in the Lebesgue sense and/or enjoy a preestablished property. However, in order to establish such results, it is necessary to consolidate a series of fine results into real-values of the Real Analysis in order to obtain more sophisticated tools. In particular, will be exported results on Differentiability Lp*, Approximate Differentiability and Differentiability q.t.p. for Lipschitz functions, for Sobolev functions whose weak gradient are functions in L p and for functions of Bounded Variation whose gradient is a vector measure of Radon. Also will be exposed regularity results for convex functionsWe will show that such functions are locally Lipschitz and that for almost every point of the domain of a convex function there are derived from the second order. This is the famous Aleksandrov’s Theorem. In addition, we will establish the approximation results by functions C 1 for functions Lipschitz, Sobolev and Bounded Variation where we have control over the smallness of the Lebesgue measure of the set where the chosen function and its gradient differ, respectively, from the functions and its gradient.