Submitted Paper

Inserted: 21 oct 2026
Last Updated: 10 nov 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 Pólya–Szegő 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 present several examples fitting this abstract framework, in which the isoperimetric sets are balls, half-spaces, or complements of balls.

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• BS-Talenti-Faber--Krahn.pdf