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

For a bounded domain $U\subset \text{rn}$, consider the $L^{\text{fz}}$-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 $u\in W_{\text{loc}}^{1,\text{fz}}(U)\cap C(\bar{U})$ for $H$, under the Dirichlet boundary value $g\in C(\mathrm{\partial }U)$, provided \noindent (A1) $H$ is lower semicontinuous in $\bar{U}\times \text{rn}$, and $H(x,\cdot )$ is convex for any $x\in \bar{U}$. \noindent (A2) ${\displaystyle H(x,0)=\underset{p\in \text{rn}}{\min }H(x,p)=0}$ for any $ x\in \overline U$,and$\displaystyle\bigcup_{x\in \overline U}\big{p: H(x,p)=0\big}$iscontainedinahyperplaneof$\rn$. \noindent (A3) For any $\text{lz}>0$, there exist $\displaystyle 0<r_\lz\le R_\lz<\fz$,with$\displaystyle\lim_{\lz\to\fz}r_\lz=\fz$,suchthat$$B(0,r_\lz)\subset \Big{p\in\rn\ |\ H(x,p)< \lz\Big}\subset B(0,R_\lz)
\forall\ \lz> 0\ \mbox{and}\ x\in \overline U.$$ 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 $t\mapsto T^tu(x)$