Symmetry properties of stable solutions of semilinear elliptic equations in unbounded domains

TL;DR

This paper investigates whether classical symmetry results for stable solutions of semilinear elliptic equations in bounded convex domains extend to unbounded convex domains, providing new conditions under which solutions remain constant or symmetric.

Contribution

It extends known symmetry results to unbounded convex domains for stable solutions, identifying specific geometric conditions that ensure solutions are constant or exhibit symmetry.

Findings

Stable non-degenerate solutions are constant in unbounded convex domains.

Solutions are symmetric if the domain satisfies certain growth conditions.

Asymptotic symmetry results are established for domains with specific geometric properties.

Abstract

We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland [20] and Matano [44] states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain $\Omega$ further satisfies $\Omega$ $\cap$ {|x| $\le$ R} = O(R^2), when R $\rightarrow$ +$\infty$. If the domain is a straight cylinder, an additional natural assumption is needed. These results can be seen as an extension to more general domains of some results on De Giorgi's conjecture.As an application, we establish asymptotic symmetries for stable solutions when the domain satisfies a geometric property asymptotically.