Metric entropy for Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian

TL;DR

This paper investigates the BV regularity of viscosity solutions to Hamilton-Jacobi equations with uniformly directionally convex Hamiltonians, providing metric entropy bounds and counterexamples for general convex cases.

Contribution

It establishes BV bounds for the slope of backward characteristics and quantifies the metric entropy of the solution map for such Hamilton-Jacobi equations.

Findings

BV bound on the slope of backward characteristics

Quantification of metric entropy in ${f W}^{1,1}_{ ext{loc}}$

Counterexample showing BV failure for general convex Hamiltonians

Abstract

The present paper first aims to study the BV-type regularity for viscosity solutions of the Hamilton-Jacobi equation \[ u_t(t,x)+H\big(D_{x} u(t,x)\big)~=~0\qquad\forall (t,x)\in ]0,\infty[\times\mathbb{R}^d \] with a coercive and uniformly directionally convex Hamiltonian $H\in\mathcal{C}^{1}(\mathbb{R}^d)$. More precisely, we establish a BV bound on the slope of backward characteristics $DH(u(t,\cdot))$ starting at a positive time $t>0$. Relying on the BV bound, we quantify the metric entropy in ${\bf W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^d)$ for the map $S_t$ that associates to every given initial data $u_0\in{\bf Lip}\big(\mathbb{R}^d\big)$, the corresponding solution $S_tu_0$. Finally, a counter example is constructed to show that both $D_xu(t,\cdot)$ and $DH(D_xu(t,\cdot))$ fail to be in $BV_{\mathrm{loc}}$ for a general strictly convex and coercive $H\in\mathcal{C}^2(\mathbb{R}^d)$.