On a class of weighted Gauss-type isoperimetric inequalities and applications to symmetrization

We solve a class of weighted isoperimetric problems of the form $$\min \{\int \mathrm{\_}\mathrm{\partial }Ew{e}^{V}dx:\int \mathrm{\_}E{e}^{V}dx=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\}$$ where $w$ and $V$ are suitable functions on ${\mathbb{R}}^d$. As a consequence, we prove a comparison result for the solutions of degenerate elliptic equations.