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 $$ \mathrm {min}\left{\int\_{\partial E}w e^V,dx:\int\_E e^V,dx=\mathrm {constant}\right} $$ 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.