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

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 Pólya–Szegő inequality – originally established for Lipschitz functions on $X$ – extends to Sobolev functions with zero boundary trace on arbitrary Borel subsets $\mathrm{\Omega }\subset X$. 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.