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 [Cabre, 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 only involve basic Analysis tools and can be understood by mathematicians who are not experts in PDEs. Two of the results in the 2020 article relied on compactness arguments. Here we present, instead, quantitative proofs from the more recent paper [Cabre, 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.