Hölder regularity of stable solutions to elliptic equations up to ℝ⁹: Full quantitative proofs

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 9"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>9</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\leq 9</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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: <ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://arxiv.org/abs/2211.13033">2211.13033</ext-link> ]. They allow us to quantify the Hölder regularity exponent and significantly simplify the treatment of boundary regularity. We also comment on similar progress and open problems for related equations.