Some minimum principles for a class of nonlinear elliptic problems in divergence form

TL;DR

This paper establishes convexity properties and minimum principles for solutions to a class of nonlinear elliptic divergence form problems, leading to a priori estimates based on boundary mean curvature.

Contribution

It introduces new minimum principles for a specific $P$-function and links solution estimates to boundary mean curvature, advancing understanding of nonlinear elliptic problems.

Findings

Solutions are convex in strictly convex domains.

A new minimum principle for a $P$-function is developed.

A priori estimates depend on boundary mean curvature.

Abstract

In this paper we study a general class of nonlinear elliptic problems in divergence form. First, we prove that the solutions to these problems satisfy a convexity property when the given domain is strictly convex. Then, making use of this convexity property, we develop some minimum principles for an appropriate $P$-function, in the sense of L.~E.~Payne. Finally, this new minimum principle is applied to find a priori estimates for the solutions, in terms of the mean curvature of the boundary of the underlying domain.