Concavity of solutions to semilinear equations in dimension two

TL;DR

This paper establishes a sufficient condition for the concavity of solutions to semilinear equations in two-dimensional convex domains, using comparison with ellipses, and extends concavity propagation results to all dimensions.

Contribution

It introduces a new sufficient condition for solution concavity based on ellipse comparison and generalizes concavity propagation to higher dimensions.

Findings

Solution concavity is guaranteed under specific geometric conditions.

Concavity propagates from the boundary in all dimensions.

The method uses comparison with ellipses inspired by Kosmodem'yanskii.

Abstract

We consider the Dirichlet problem for a class of semilinear equations on two dimensional convex domains. We give a sufficient condition for the solution to be concave. Our condition uses comparison with ellipses, and is motivated by an idea of Kosmodem'yanskii. We also prove a result on propagation of concavity of solutions from the boundary, which holds in all dimensions.