The peak of the solution of elliptic equations

TL;DR

This paper presents a counterexample challenging the inheritance of convexity in elliptic PDE solutions and addresses the hot spot problem, providing partial insights using local Pohozeav identities.

Contribution

It offers a counterexample to convexity inheritance and introduces a local Pohozeav identity approach to partially solve longstanding elliptic PDE problems.

Findings

Counterexample to convexity inheritance

Partial solutions to hot spot problem

Use of local Pohozeav identities

Abstract

A counter example of inheritance of convexity of domain of positive solution of Dirichlet boundary value problem and the hot spot problem that proposed by J. Rauch is given. The difficulty of these two problems is that the critical points of the solutions is not singleton but a level curve. However, using Pohozeav identity locally, partial answer to both problems can be derived.