Quantitative comparison results for first-order Hamilton-Jacobi equations

TL;DR

This paper provides a quantitative stability analysis for first-order Hamilton-Jacobi equations, linking domain asymmetry and solution deviation to the comparison deficit, thus refining classical symmetrisation results.

Contribution

It introduces a stability version of the Giarrusso-Nunziante inequality, connecting the comparison deficit to domain asymmetry and solution deviation.

Findings

The deficit controls domain asymmetry and solution deviation.

A stability inequality for first-order Hamilton-Jacobi equations is established.

The results refine classical symmetrisation inequalities.

Abstract

In this paper, we study a quantitative refinement of a classical symmetrisation result for first-order Hamilton-Jacobi equations. We prove that the deficit in the comparison result, established by Giarrusso and Nunziante, controls both the asymmetry of the domain and the deviation of the solution and data from radial symmetry. This yields a stability version of the Giarrusso-Nunziante inequality.