On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data

TL;DR

This paper investigates stable solutions of boundary reaction-diffusion equations with nonlinear diffusion and Neumann conditions, providing geometric inequalities, classification results, and applications to nonlocal problems, including a counterexample for fractional Laplacians.

Contribution

It introduces new geometric Poincaré-type inequalities and classification results for stable solutions, extending the analysis to nonlocal problems and fractional Laplacians.

Findings

Derived geometric Poincaré-type inequality for stable solutions

Classified stable solutions under nonlinear boundary conditions

Provided a counterexample for fractional Laplacian framework

Abstract

We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.