Princípio de Harnack de fronteira para o operador p-Laplaciano

Abstract

The aim of this work is to present and prove the Boundary Harnack Principle for p-Laplacian in smooth domains. The author talks about the results of H. Aikawa and N. Shanmugalingam, presenting fundamental details previously omitted, in addition to also presenting preparatory and/or related results. For that, the properties of the p-Laplacian and the p-harmonic functions will be presented, among which are the Harnack Inequality and the Comparison Principle. Next, a geometrical characterization for C1.1 domains is proved in detail. More precisely, every bounded domain of class C1,1 satisfies the ball condition and vice versa. This is a well-known result, but it is almost never demonstrated. Finally, the author combines all the previous results with the Carleson Estimate to prove the Boundary Harnack Principle. This theorem guarantees that, in a bounded domain, any two p-harmonic functions that vanish in a part of the boundary of this domain deteriorate at the same rate as they approach a smaller portion of the boundary. His proof is based on the use of the inside/outside conditions of the ball, and the results cited above, to prove that the p-harmonic functions are uniformly comparable with the function d_D(.), which assumes, at each point x of D, the value of the distance from x to the boundary of D.