Hölder regularity of stable solutions to semilinear elliptic equations up to $\mathbf{\mathbb{R}^9}$: full quantitative proofs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866908458108846080 |
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| author | Cabre, Xavier |
| author_facet | Cabre, Xavier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_11352 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hölder regularity of stable solutions to semilinear elliptic equations up to $\mathbf{\mathbb{R}^9}$: full quantitative proofs Cabre, Xavier Analysis of PDEs 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. |
| title | Hölder regularity of stable solutions to semilinear elliptic equations up to $\mathbf{\mathbb{R}^9}$: full quantitative proofs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2205.11352 |