论文标题
hölder的稳定解决方案的稳定性半线性椭圆方程式最多$ \ mathbf {\ mathbb {r}^9} $：完整的定量证明
Hölder regularity of stable solutions to semilinear elliptic equations up to $\mathbf{\mathbb{R}^9}$: full quantitative proofs
论文作者
论文摘要
本文涉及[Cabré，Figalli，Ros-Oton和Serra，Acta Math中获得的结果。 224（2020）]，它在最佳尺寸$ n \ leq 9 $的最佳范围内建立了稳定解决方案的Hölder规律性。出于说明性目的，我们提供了所有结果的独立证明。它们仅涉及基本分析工具，并且旨在为PDE专家以外的更广泛的数学受众访问。 2020年文章中的两个结果依赖于紧凑性论点。相反，我们在这里介绍了最近的论文[Cabré，出现在Amer中的定量证明。 J. Math，Arxiv：2211.13033]。他们允许量化Hölder规律性指数，并显着简化边界规律性的处理。 我们还对相关方程式的类似进度和开放问题发表评论。
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 $n\leq 9$. 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 [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.
