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

This article concerns the results obtained in [Cabr\'e, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)], which established the H\"older regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n\leq 9$. 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 [Cabr\'e, to appear in Amer. J. Math, arXiv:2211.13033]. They allow to quantify the H\"older regularity exponent and simplify significantly the treatment of boundary regularity. We also comment on similar progress and open problems for related equations.