  In tis paper we prove an isoperimetric inequality for the first twisted
eigenvalue $\lambda_{1,\gamma}^T(\Omega)$ of a weighted operator, defined as
the minimum of the usual Rayleigh quotient when the trial functions belong to
the weighted Sobolev space $H_0^1(\Omega,d\gamma)$ and have weighted mean value
equal to zero in $\Omega$. 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 $\lambda_{1,\gamma}^T(\Omega)$ 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).
