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

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( gu_xi )_xi = lgu textin W ì mathbbRⁿ - {left( {gamma u_{{x_{i} }} } right)}_{{x_{i} }} = lambda gamma u,{text{in}},Omega subset mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in mathbbRⁿ mathbb{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.