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Quantitative comparison results for first-order Hamilton-Jacobi equations

Vincenzo Amato, Luca Barbato

TL;DR

This work addresses the stability of a classical symmetry result for the first-order Hamilton-Jacobi equation $| abla u|=f(x)$ under domain symmetrisation. By coupling rearrangement theory with quantitative inequalities (isoperimetric, Pólya–Szegő, Hardy–Littlewood) and introducing pseudo-rearrangements, the authors prove that the $L^1$ deficit ${\|u^G\|}_1-{\|u\|}_1$ controls both the domain's Fraenkel asymmetry and the distance of $u$ from radial symmetry, yielding a quantitative stability version of the Giarrusso-Nunziante inequality; they also obtain partial control of the source term via a rearrangement $f_u$. The approach hinges on gradient rearrangements, rescaling arguments, and sharp estimates linking asymmetry propagation along level sets to the deficit, culminating in explicit exponents and constants. These results advance shape-optimization analysis for Hamilton-Jacobi equations and provide robust, quantitative measures of how domain geometry and data symmetry influence solutions.

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.

Quantitative comparison results for first-order Hamilton-Jacobi equations

TL;DR

This work addresses the stability of a classical symmetry result for the first-order Hamilton-Jacobi equation under domain symmetrisation. By coupling rearrangement theory with quantitative inequalities (isoperimetric, Pólya–Szegő, Hardy–Littlewood) and introducing pseudo-rearrangements, the authors prove that the deficit controls both the domain's Fraenkel asymmetry and the distance of from radial symmetry, yielding a quantitative stability version of the Giarrusso-Nunziante inequality; they also obtain partial control of the source term via a rearrangement . The approach hinges on gradient rearrangements, rescaling arguments, and sharp estimates linking asymmetry propagation along level sets to the deficit, culminating in explicit exponents and constants. These results advance shape-optimization analysis for Hamilton-Jacobi equations and provide robust, quantitative measures of how domain geometry and data symmetry influence solutions.

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.
Paper Structure (13 sections, 18 theorems, 103 equations)

This paper contains 13 sections, 18 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Distribution function and rearrangements
  4. Pseudo-rearrangements
  5. Rearrangement of the gradient
  6. Some quantitative inequalities
  7. Rescaling properties
  8. Asymmetry of the domain
  9. Almost radiality of the solutions
  10. Property of the source term
  11. Funding
  12. Data Availability
  13. Competing Interests

Key Result

Theorem 1.1

Let $\Omega$ be a bounded open set of $\mathop{\mathrm{\mathbb{R}}}\nolimits^n$, $n\geq 2$, and let $f\in L^\infty(\Omega)$ be a non-negative function with Let $u$ be a solution to original_u and let $u^\text{G}$ be the unique spherically decreasing solution to original_v. Then there exist positive constants $\theta=\theta(n)$ and $C_1:= C_1(n, {\left|\Omega\right|}, m,M),\, C_2:= C_2(n, {\left|

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 2.1: Coarea formula
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.2
  • Definition 2.4: Pseudo-rearrangement
  • Lemma 2.3: AT
  • ...and 22 more