Gradient Estimates for Solutions of Nonlinear Elliptic and Parabolic Equations

Abstract

This chapter presents a general outline of the maximum principle method for obtaining a priori estimates for the gradient of solutions of elliptic or parabolic equations of second order. It is well known that such gradient estimates are a fundamental element in the proof of existence theorems and it is this fact that has supplied the main motivation for the considerable efforts that have gone into obtaining these results. The maximum principle technique was originally introduced only for equations involving two independent variables and with the main ideas not fully worked out. The chapter illustrates the method in its simplest form by considering a solution u of the Laplace equation Δu = 0 in a domain Ω of n-dimensional Euclidean space. The maximum principle technique has been used independently by Ladyzhenskaya and Uraltseva and by James Serrin to obtain gradient estimates for non-uniformly elliptic equations with more than two variables. The technique was employed in an important paper of Ladyzhenskaya and Uraltseva to obtain interior gradient estimates for non-uniformly elliptic and parabolic equations. The chapter provides a self-contained exposition of the main ideas described, and at the same time, simplifying the treatment and determining more or less optimal hypotheses concerning the structure of the equation.