On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds

TL;DR

This paper studies the existence and stability of nonconstant solutions to semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds, highlighting geometric influences and applications in biology.

Contribution

It provides new insights into how geometry and nonlinearity affect solution stability, especially on symmetric manifolds and surfaces of revolution.

Findings

Stability depends on Ricci curvature and boundary geometry.

Existence of solutions varies with manifold symmetry.

Refined results for surfaces of revolution and spherical symmetry.

Abstract

We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.