The weighted isoperimetric inequality and Sobolev inequality outside convex sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918189774929920 |
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| author | Chen, Lu Lan, Jiali |
| author_facet | Chen, Lu Lan, Jiali |
| contents | In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $λ_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $α$, such that $w^{1/α}$ is concave on $\R^n$.
Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted Pólya-Szegö principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The weighted isoperimetric inequality and Sobolev inequality outside convex sets Chen, Lu Lan, Jiali Analysis of PDEs In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $λ_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $α$, such that $w^{1/α}$ is concave on $\R^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted Pólya-Szegö principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}. |
| title | The weighted isoperimetric inequality and Sobolev inequality outside convex sets |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.01647 |