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arXiv · 2605.29721

Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization

Giulio Bartoli·Giorgio Saracco
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Abstract

We consider the weighted $p$-Laplacian associated with a measure $\mu$ 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 P\'olya--Szeg\H{o} inequality -- originally established for Lipschitz functions on $X$ -- extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets $\Omega\subset X$. 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.

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Giulio Bartoli, Giorgio Saracco. 2026-05-28. Spectral inequalities for weighted $p$-Laplacians via Talenti symmetrization. https://arxiv.org/abs/2605.29721

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