Regularity of stable solutions to reaction-diffusion elliptic equations

TL;DR

This paper proves that stable solutions to reaction-diffusion elliptic equations are smooth in dimensions up to 9, resolving a long-standing open problem and highlighting the critical dimension for regularity.

Contribution

It establishes the optimal dimension (9) for the regularity of stable solutions, answering a question posed by Brezis in the 1990s.

Findings

Stable solutions are smooth in dimensions up to 9.

Existence of unbounded stable solutions in dimensions 10 and higher.

Addresses an open problem in the regularity of extremal solutions.

Abstract

The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue question in differential geometry: the regularity of stable minimal surfaces.