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Level sets of solutions to the stationary Hamilton-Jacobi equation are John regular

Elisa Davoli, Ulisse Stefanelli

TL;DR

The work addresses the regularity of sublevel sets $U_t$ of the unique nonnegative viscosity solution to the external Hamilton–Jacobi problem $H(x,\\nabla u)=0$ in $\mathbb{R}^n\setminus K$ with $u=0$ on $K$. It introduces a geometric, Finsler-based framework to show that all sublevels $U_t$ are John domains with a uniform constant, under general convexity/nondegeneracy conditions on $H$, and it furnishes sharp counterexamples illustrating the limits of interior-ball and interior-cone regularity. The analysis extends to quantify when a uniform John constant can be obtained if $K$ is itself a John domain and demonstrates the sharpness of these findings through carefully constructed $K$ and $H$-dependent configurations. The results have implications for related functional inequalities and the regularity of the evolving sets $U_t$ in optimal control and front-propagation contexts, with a robust geometric underpinning via the induced Finsler metric.

Abstract

Let $u$ be the unique nonnegative viscosity solution of the Hamilton-Jacobi equation $H(x,\nabla u)=0$ in the external domain ${\mathbb R}^{ n} \setminus K$ with $u=0$ on $K$. Under general conditions on $H$, we prove that all sublevels of $u$ are John domains. Moreover, if $K$ itself is a John domain, we provide a uniform lower bound on the John constant of all sublevels. We exhibit counterexamples showing that John regularity is sharp in this setting.

Level sets of solutions to the stationary Hamilton-Jacobi equation are John regular

TL;DR

The work addresses the regularity of sublevel sets of the unique nonnegative viscosity solution to the external Hamilton–Jacobi problem in with on . It introduces a geometric, Finsler-based framework to show that all sublevels are John domains with a uniform constant, under general convexity/nondegeneracy conditions on , and it furnishes sharp counterexamples illustrating the limits of interior-ball and interior-cone regularity. The analysis extends to quantify when a uniform John constant can be obtained if is itself a John domain and demonstrates the sharpness of these findings through carefully constructed and -dependent configurations. The results have implications for related functional inequalities and the regularity of the evolving sets in optimal control and front-propagation contexts, with a robust geometric underpinning via the induced Finsler metric.

Abstract

Let be the unique nonnegative viscosity solution of the Hamilton-Jacobi equation in the external domain with on . Under general conditions on , we prove that all sublevels of are John domains. Moreover, if itself is a John domain, we provide a uniform lower bound on the John constant of all sublevels. We exhibit counterexamples showing that John regularity is sharp in this setting.
Paper Structure (8 sections, 3 theorems, 67 equations, 6 figures)

This paper contains 8 sections, 3 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}
  3. Sharpness of Theorem \ref{['thm:main']}
  4. Sharpness of the assumptions on $K$
  5. No lower bound on the interior-ball radius
  6. No interior-ball regularity
  7. No interior-cone regularity
  8. Proof of Proposition \ref{['prop:regu']}

Key Result

Theorem 1.1

Let $H:{\mathbb R}^n \times {\mathbb R}^n \to {\mathbb R}$ be such that Let $K\subset {\mathbb R}^n$ be compact and connected and $u$ be the unique nonnegative viscosity solution to eq:0. Then, the sublevels $U_t$ are John domains for all $t>0$. Moroever, if $K$ is a John domain with respect $x_0\in K^\circ$ with John constant $\kappa_0>0$, all sublevels $U_t$ are John For $t> 2 \sup\{r>0\,:\, B

Figures (6)

  • Figure 1: The construction for the proof of Theorem \ref{['thm:main']}
  • Figure 2: The example of Section \ref{['sec:sharpK']}
  • Figure 3: The example of Section \ref{['sec:counter1']}
  • Figure 4: The example of Section \ref{['sec:counter3']}
  • Figure 5: A two-dimensional representation of the sets $V_\beta$ and $Q_\beta$
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 1.1: John regularity
  • Proposition 1.2: Regularity of the map $t \mapsto U_t$
  • Lemma 4.1