Isoperimetric Sets for Weighted Twisted Eigenvalues

In this paper we prove an isoperimetric inequality for the first twisted eigenvalue λ _1,γ^T(Ω ) of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space H_0^1(Ω ,dγ ) and have weighted mean value equal to zero in Ω . We are interested in positive measures dγ =γ (x) dx for which we are able to identify the optimal sets, namely, the sets that minimize λ _1,γ^T(Ω ) among sets of given weighted measure. In the cases under consideration, the optimal sets are given by two identical and disjoint copies of the isoperimetric sets (for the weighted perimeter with respect to the weighted measure).