A weighted isoperimetric inequality and applications to symmetrization

Abstract

We prove an inequality of the form , where is a bounded domain in with smooth boundary, is a ball centered in the origin having the same measure as . 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.

Keywords

• Weighted isoperimetric inequality
• Weighted Sobolev norm
• Symmetric decreasing rearrangement
• Comparison theorem