On the symmetric rearrangement of the gradient of a Sobolev function

Abstract

In this paper, we generalize a classical comparison result for solutions to Hamilton–Jacobi equations with Dirichlet boundary conditions, to solutions to Hamilton–Jacobi equations with non-zero boundary trace.

As a consequence, we prove the isoperimetric inequality for the torsional rigidity (with Robin boundary conditions) and for other functionals involving such boundary conditions.