Isoperimetric sets for weighted twisted eigenvalues

In tis paper we prove an isoperimetric inequality for the first twisted eigenvalue ${\lambda }_{1,\gamma }^T(\mathrm{\Omega })$ of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space $H_0^1(\mathrm{\Omega },d\gamma )$ and have weighted mean value equal to zero in $\mathrm{\Omega }$. We are interested in positive measures $d\gamma =\gamma (x)dx$ for which we are able to identify the isoperimetric sets, namely, the sets that minimize ${\lambda }_{1,\gamma }^T(\mathrm{\Omega })$ 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).