Fatou theorems for some nonlinear elliptic partial differential equations.

摘要

We consider the p-Laplacian equation, div({dollar}vertnabla{dollar}u{dollar}vertsp{lcub}rm p-2{rcub}nabla{dollar}u) = 0, and other related nonlinear elliptic partial differential equations. We prove that if D is a Lipschitz domain and u a nonnegative solution of the p-Laplacian equation there exits {dollar}beta >{dollar} 0, depending on p, n and the Lipschitz character of D, such that the set E{dollar}sb{lcub}rm u{rcub}{dollar} of points of the boundary, {dollar}partial{dollar}D, where u has nontangential limit has Hausdorff dimension {dollar}{lcub}geq{rcub}beta{dollar}. Examples by Wolff and Lewis show that E{dollar}sb{lcub}rm u{rcub}{dollar} could have zero surface measure; our result ensures that E{dollar}sb{lcub}rm u{rcub}{dollar} is far from being empty. In the case of smooth domains we also obtain that {dollar}beta{dollar} tends to n {dollar}-{dollar} 1 as p tends to 2. (The natural thing to expect since the 2-Laplacian is just the Laplacian).; The method to prove these theorems requires the use of some properties of linear uniformly elliptic equations in nondivergence form. In the first half of the thesis we deal with this linear part and develop a potential theory for these equations and their adjoints, obtaining the results we will need later in the nonlinear situation. The coefficients of the linear operators are assumed to be smooth, but more importantly for our applications, all the constants involved in the basic estimates do not depend on the smoothness of the coefficients.