Um estudo sobre regularidade de soluções fracas de EDPs e Q-minimizantes

Abstract

In this dissertation we propose a study on the regularity of weak solutions of PDEs in divergent form and also on regularity of the Hölder type for Qminimizers of non-linear functionals whose integrand satisfies a certain polynomial growth condition. The first part is dedicated to the classical study that consists of obtaining regularity of weak PDE solutions in divergent form with class C1 coefficients. Next, we address the infamous De Giorgi method, which consists of obtaining local limitation and Hölder-type regularity for weak solutions of PDEs of divergent form with only measurable coefficients satisfying an ellipticity condition. In the third and final part, we present the classic results of Giusti and Gianquinta on the Hölder-type regularity theory for Qminimizers. We close the presentation of the dissertation with comments highlighting the differences between the three approaches studied, as well as the presentation of some examples.