Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problem

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - (gamma u(xi)) (xi) = lambda gamma u in Omega subset of R-n with Dirichlet boundary condition, where gamma is the normalized Gaussian function in R-n. To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.