The Hopf--Lax operator and the Hopf--Lax formula

Let $n\ge 1$, let $H:{\mathbb{R}}^n\to \mathbb{R}$ be convex and superlinear, ${\lim }_{\mathrm{\mid }p\mathrm{\mid }\to \mathrm{\infty }}\frac{H(p)}{\mathrm{\mid }p\mathrm{\mid }}=+\mathrm{\infty },$ let $L$ be its Legendre transform (The Legendre transform of a finite-valued convex Hamiltonian), and let $u_0:{\mathbb{R}}^n\to \mathbb{R}$ be bounded and uniformly continuous (Uniform continuity of a map of metric spaces: one $\delta$ serving every point, Lower bound, bounded below, bounded set). For $t>0$ and $x\in {\mathbb{R}}^n$ define $Q_t{u}_0(x):={\inf }_{y\in {\mathbb{R}}^n}\{u_0(y)+tL(\frac{x-y}{t}\left)\right\},$ the infimum being computed in $\mathbb{R}\cup \{+\mathrm{\infty }\}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined, Greatest lower bound (infimum)) over the extended-real values $u_0(y)+tL((x-y)\mathrm{/}t)$; the term $tL((x-y)\mathrm{/}t)$ is $+\mathrm{\infty }$ exactly when $L((x-y)\mathrm{/}t)=+\mathrm{\infty }$. Set $Q_0{u}_0:=u_0$.

The Hopf--Lax operator with Lagrangian $L$ is the family $(Q_t{)}_{t\ge 0}$, and the function $(x,t)\mapsto Q_t{u}_0(x)$ is the Hopf--Lax formula for the Cauchy problem $u_t+H(Du)=0$, $u(\cdot ,0)=u_0$. The infimum is an extended-real expression at this point: finiteness and the confinement of near-minimisers are proved in Finiteness, superlinearity of the Lagrangian and localisation of Hopf--Lax near-minimisers, where superlinearity makes $L$ real-valued everywhere.

• What is fixed and what is postponed. The definition fixes the autonomous Hamiltonian $H:{\mathbb{R}}^n\to \mathbb{R}$, convex and superlinear; the datum class, bounded and uniformly continuous $u_0$; the infimum over all $y\in {\mathbb{R}}^n$ of $u_0(y)+tL((x-y)\mathrm{/}t)$ for $t>0$, read in the extended reals; and the value $Q_0{u}_0=u_0$. Neither the attainment of the infimum nor its finiteness is asserted here, and no assertion that $L$ is real-valued is smuggled into the definition; the value $+\mathrm{\infty }$ is kept visible until Finiteness, superlinearity of the Lagrangian and localisation of Hopf--Lax near-minimisers proves the opposite under superlinearity.
• Time scaling. The velocity variable in the formula is $(x-y)\mathrm{/}t$, the average velocity of a straight path from $y$ at time $0$ to $x$ at time $t$; the factor $t$ multiplies the Lagrangian density. This normalisation is the one for which the dynamic-programming identity $Q_{t+s}{u}_0=Q_t(Q_s{u}_0)$ of The Hopf--Lax operators form a semigroup (dynamic programming) holds with the coefficient $t+s$. No choice principle is used in the definition.
• Bounded data without continuity. The same pointwise infimum formula defines $Q_{t}f(x)$ for any bounded function $f$, even when $f$ is not uniformly continuous. Since $L(v)\ge -H(0)$ and the competitor $y=x$ is finite, these values are real. This extension is used for the nonexpansiveness estimate in The Hopf--Lax operator is a contraction in the supremum norm; continuity conclusions such as The Hopf--Lax operator preserves a modulus of continuity retain their stated hypotheses.