Upper estimates for a class of non-linear partial differential equations

Abstract

The main purpose of this paper is to obtain upper estimates for a certain cross-sectional measure of the solutions of a Dirichlet and an initial-boundary value problem associated with the Monge-Ampère equation in the plane—problems which are shown to be relevant to the two-dimensional motion of a perfect fluid of constant density, for example. The estimates obtained are valid irrespective of the character of the equation (elliptic, hyperbolic, or whatever); indeed, its character may vary within the plane region. The estimates easily yield pointwise estimates, together with criteria governing data which give rise to non-existence of a solution. The analysis is based on a particularly compact identity for the second derivative of the cross-sectional measure referred to above, and on comparison principles. The identity is also used to discuss estimates for a Dirichlet problem associated with a non-linear extension to the harmonic equation in the plane.