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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.29721 |
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Table of 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.