Symmetry of solutions of semilinear PDEs on Riemannian domains

TL;DR

This paper investigates symmetry properties of solutions to semilinear PDEs on Riemannian domains, including manifolds with density, providing a general framework and results for stable solutions.

Contribution

It introduces a comprehensive framework for symmetry analysis of semilinear PDE solutions on Riemannian domains, encompassing weighted Laplacians and stable solutions.

Findings

Framework for symmetry in semilinear PDEs on Riemannian domains

Results for stable solutions and manifolds with density

Evidence of the naturalness of the proposed framework

Abstract

This paper deals with symmetry phenomena for solutions of the Dirichlet problem involving semilinear PDEs on Riemannian domains. We shall present a rather general framework where the symmetry problem can be formulated and provide some evidence that this framework is completely natural by pointing out some results for stable solutions. The case of manifolds with density, and corresponding weighted Laplacians, is inserted in the picture from the very beginning.