On the symmetric rearrangement of the gradient of a Sobolev function

TL;DR

This paper extends classical comparison results for Hamilton-Jacobi equations to cases with non-zero boundary conditions, leading to new isoperimetric inequalities for torsional rigidity and related functionals.

Contribution

It generalizes comparison principles to non-zero boundary traces and establishes new isoperimetric inequalities for torsional rigidity with Robin boundary conditions.

Findings

Generalized comparison results for Hamilton-Jacobi equations with non-zero boundary trace

Proved isoperimetric inequalities for torsional rigidity with Robin boundary conditions

Extended classical inequalities to broader boundary conditions

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.