Submitted Paper

Inserted: 21 oct 2026
Last Updated: 21 oct 2026

Year: 2026

Abstract:

We consider the weighted $p$-Laplacian associated with a measure $\mu$ having a continuous positive density with respect to the Lebesgue measure on an open convex 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 the weighted Sobolev space of functions vanishing on the relative boundary of an open set $\Omega\subset X$, defined as the closure of Lipschitz functions supported away from it. This yields Faber--Krahn-type inequalities for the first $(p,q)$-eigenvalue of the weighted Dirichlet $p$-Laplacian. We also show that the assumptions of Talenti's theorem force a rigid structure on the function driving the symmetrization: its level sets are quantitatively separated, the modulus of its gradient is constant on each of them, and, under a mild additional assumption, they are parallel hypersurfaces, so that the function depends only on the distance either from a hyperplane or from a focal set. We present several examples fitting this abstract framework, in which the isoperimetric sets are balls or half-spaces.

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