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

Abstract

We solve a class of weighted isoperimetric problems of the form $$\min \left\{\int \_\partial Ew{e}^{V}dx:\int \_E{e}^{V}dx=\mathrm{constant}\right\}$$ 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.