Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions

TL;DR

This paper reviews classification results for stable solutions to various nonlinear elliptic equations, including minimal cones, the Allen-Cahn equation, and reaction-diffusion problems, highlighting techniques and open problems in regularity theory.

Contribution

It provides detailed proofs and insights into the stability and minimality of solutions across three key elliptic problems, connecting techniques and identifying open questions.

Findings

Proved minimality of the Simons cone in high dimensions.

Discussed the De Giorgi conjecture for the Allen-Cahn equation.

Presented regularity results for stable solutions in low dimensions.

Abstract

These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar. The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J. Simons on the flatness of stable minimal cones in low dimensions. Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E. De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone. The third problem was raised by H. Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic. Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces.