Stable solutions to reaction-diffusion elliptic problems

TL;DR

This paper investigates the regularity, existence, and nonexistence of stable solutions to various reaction-diffusion elliptic PDEs, including classical and fractional Laplacians, revealing new results and open problems in the field.

Contribution

It proves stable solutions are smooth up to dimension 9 for the classical Laplacian and explores the failure of certain stability results for boundary reactions, introducing new theories and open questions.

Findings

Stable solutions are smooth up to dimension 9 for the classical Laplacian.

No nonconstant stable solutions exist in convex domains with zero Neumann boundary conditions for interior reactions.

Boundary reactions can admit nonconstant stable solutions, contrary to interior cases.

Abstract

We are concerned with stable solutions to reaction-diffusion elliptic PDEs. We begin with regularity questions, first addressing the classical Laplacian. In joint work with Figalli, Ros-Oton, and Serra, we proved that stable solutions are smooth up to the optimal dimension 9, thereby solving an open problem posed by Brezis in the mid-1990s. We describe this result and also discuss related progress and open problems for the fractional Laplacian -- arising naturally in boundary reaction problems -- , the $p$-Laplacian, and minimal surfaces. We then turn to existence questions, starting with the Casten-Holland and Matano theorem for interior reactions, which states that no nonconstant stable solution exists in convex domains under zero Neumann boundary conditions. We present a recent result with Consul and Kurzke (forthcoming) establishing that the analogous statement fails for boundary reactions. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line, for which we present new results and open problems.