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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.13033 |
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| _version_ | 1866912447751782400 |
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| author | Cabre, Xavier |
| author_facet | Cabre, Xavier |
| contents | We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is much more delicate. It requires the function to be a stable solution of a semilinear elliptic equation with a nonnegative, nondecreasing, and convex nonlinearity. As an application, our estimates provide quantitative proofs of two results established by contradiction-compactness arguments in [Cabre, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)]. We recall that this work proved the Hölder regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n \leq 9$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_13033 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Estimates controlling a function by only its radial derivative and applications to stable solutions of elliptic equations Cabre, Xavier Analysis of PDEs We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is much more delicate. It requires the function to be a stable solution of a semilinear elliptic equation with a nonnegative, nondecreasing, and convex nonlinearity. As an application, our estimates provide quantitative proofs of two results established by contradiction-compactness arguments in [Cabre, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)]. We recall that this work proved the Hölder regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n \leq 9$. |
| title | Estimates controlling a function by only its radial derivative and applications to stable solutions of elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2211.13033 |