A weighted isoperimetric inequality and applications to symmetrization

We prove an inequality of the form integral(partial derivative Omega) a(\x)Hn-1 (dx) greater than or equal to integral(partial derivative B) a()Hn-1 (dx), where Omega is a bounded domain in R-n with smooth boundary, B is a ball centered in the origin having the same measure as Omega. From this we derive inequalities comparing a weighted Sobolev norm of a given function with the norm of its symmetric decreasing rearrangement. Furthermore, we use the inequality to obtain comparison results for elliptic boundary value problems