Energy stability for a class of semilinear elliptic problems

TL;DR

This paper investigates how the energy of solutions to semilinear elliptic problems varies under domain deformations within unbounded Lipschitz domains, providing stability and instability results for specific geometric configurations.

Contribution

It introduces a variational framework to analyze energy behavior under domain variations and applies it to cone and cylinder geometries, revealing stability properties and connections to overdetermined problems.

Findings

Energy stability depends on domain shape and variation type.

Stability and instability results are established for cone and cylinder domains.

Connections are made to overdetermined boundary value problems.

Abstract

In this paper, we consider semilinear elliptic problems in a bounded domain $\Omega$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $\Omega$ inside $\mathcal C$. Once a rigorous variational approach to this question is set, we focus on the cases when $\mathcal C$ is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.