Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917543423246336 |
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| author | Bartoli, Giulio Saracco, Giorgio |
| author_facet | Bartoli, Giulio Saracco, Giorgio |
| contents | We consider the weighted $p$-Laplacian associated with a measure $μ$ that is absolutely continuous with respect to the Lebesgue measure on an open connected subset $X\subset\mathbb{R}^N$. We prove that Talenti's weighted Pólya--Szegő inequality -- originally established for Lipschitz functions on $X$ -- extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets $Ω\subset X$. This yields Faber--Krahn-type inequalities for the first $(p,q)$-eigenvalue of the weighted Dirichlet $p$-Laplacian. We present several examples fitting this abstract framework, including classical Euclidean and Gaussian cases alongside new results for homogeneous weights in convex cones, anisotropic Gaussians, and log-concave Gaussian perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29721 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization Bartoli, Giulio Saracco, Giorgio Analysis of PDEs Spectral Theory 35P15 We consider the weighted $p$-Laplacian associated with a measure $μ$ that is absolutely continuous with respect to the Lebesgue measure on an open connected subset $X\subset\mathbb{R}^N$. We prove that Talenti's weighted Pólya--Szegő inequality -- originally established for Lipschitz functions on $X$ -- extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets $Ω\subset X$. This yields Faber--Krahn-type inequalities for the first $(p,q)$-eigenvalue of the weighted Dirichlet $p$-Laplacian. We present several examples fitting this abstract framework, including classical Euclidean and Gaussian cases alongside new results for homogeneous weights in convex cones, anisotropic Gaussians, and log-concave Gaussian perturbations. |
| title | Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization |
| topic | Analysis of PDEs Spectral Theory 35P15 |
| url | https://arxiv.org/abs/2605.29721 |