Regularidade Hölder em equações elípticas na forma divergente

Abstract

Elliptic partial differential equations are essential objects of study for modern Mathematics, particularly in the area of analysis, but also in Physics. We initially aim to study the weak solutions of such equations. For this we will define such solutions and obtain a minimum condition for them to be studied. We will analyze, before delving into the solutions of such equations, the Hölder continuity of functions from the local growth of its integral. Then we will obtain the John-Nirenberg Inequality through the study of Diadic Cubes together with the Calderon-Zygmund Lemma. Having finished the study of the bounded mean oscillation functions, we will in fact turn to the solutions of the homogeneous equations, thus passing through the Caccioppoli Inequality and also approaching the Harmonic Functions. Using these estimates we will arrive at Hölder continuity of the solutions and their gradient, assuming the coefficients of the equations are at least continuous. Then we will approach more general coefficients, and for that we will initially obtain the local limitation of the subsolutions of the equation by the approach of De Giorgi. Having done that, we will analyze both the subsolutions and the supersolutions of the equation in the homogeneous case, passing through Density and Oscillation Theorems, and finally arriving at De Giorgi’s Theorem, from which it is also possible to obtain the Hölder continuity of the solutions. Finally, we will approach the Weak Harnack Inequality and enunciate some consequences of it, among which the Moser’s Harnack Inequality, the Hölder continuity of Solutions, and the Liouville Theorem.