Uniqueness of absolute minimizers for $L^\fz$-functionals involving Hamiltonians $H(x,p)$

TL;DR

This paper proves the uniqueness of absolute minimizers for certain $L^z$-functionals involving Hamiltonians with specific convexity and growth conditions, extending previous results to more general $x$-dependent cases.

Contribution

It generalizes the uniqueness theorem for absolute minimizers to a broader class of Hamiltonians with $x$-dependence, confirming an open question.

Findings

Established uniqueness of absolute minimizers under new conditions.

Extended previous theorems to Hamiltonians with $x$-dependence.

Connected absolute subminimality with convexity of Hamilton-Jacobi flow.

Abstract

For a bounded domain $U\subset\rn$, consider the $L^\fz$-functional involving a nonnegative Hamilton function $H:\overline U\times\rn\to [0,\fz)$. In this paper, we will establish the uniqueness of absolute minimizers $u\in W^{1,\fz}_\loc(U)\cap C(\overline U)$ for $H$, under the Dirichlet boundary value $g\in C(\partial U)$, provided \noindent (A1) $H$ is lower semicontinuous in $\overline U\times\rn$, and $H(x,\cdot)$ is convex for any $x\in\overline U$. \noindent (A2) $\displaystyle H(x,0)=\min_{p\in \rn}H(x,p)=0$ for any $ x\in \overline U$, and $\displaystyle\bigcup_{x\in \overline U}\big\{p: H(x,p)=0\big\}$ is contained in a hyperplane of $\rn$. \noindent (A3) For any $\lz>0$, there exist $\displaystyle 0<r_\lz\le R_\lz<\fz$, with $\displaystyle\lim_{\lz\to\fz}r_\lz=\fz$,such that $$B(0,r_\lz)\subset \Big\{p\in\rn\ |\ H(x,p)< \lz\Big\}\subset B(0,R_\lz)\ \forall\ \lz> 0\ \mbox{and}\ x\in \overline U.$$ This generalizes the uniqueness theorem by \cite{j93, jwy, acjs} and \cite{ksz} to a large class of Hamiltonian functions $H(x,p)$ with $x$-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by {\cite{jwy}}. The proofs rely on geometric structure of the action function $\mathcal L_t(x,y)$ induced by $H$, and the identification of the absolute subminimality of $u$ with convexity of the Hamilton-Jacobi flow $t\mapsto T^tu(x)$