Hölder regularity of stable solutions to semilinear elliptic equations up to ${\mathbb{R}}^9$: full quantitative proofs

Abstract

This article concerns the results obtained in [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)], which established the Hölder regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n\le 9$. For expository purposes, we provide self-contained proofs of all results. They involve only basic Analysis tools and are intended to be accessible to a broader mathematical audience beyond PDE specialists. Two of the results in the 2020 article relied on compactness arguments. Here we present, instead, quantitative proofs from the more recent paper [Cabré, to appear in Amer. J. Math, arXiv:2211.13033]. They allow to quantify the Hölder regularity exponent and simplify significantly the treatment of boundary regularity. We also comment on similar progress and open problems for related equations.

To appear in "Bulletin of the American Mathematical Society"