The Hamilton--Jacobi Cauchy problem and its classical solutions

Let $n\ge 1$, let $O\subseteq {\mathbb{R}}^n$ be nonempty and open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement), let $T>0$, and let $H:O\times [0,T]\times {\mathbb{R}}^n\to \mathbb{R}$ be continuous (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form). Write $Z:=O\times (0,T),\bar{Z}:=\bar{O}\times [0,T],\mathrm{\Gamma }:=(O\times \{0\})\cup (\mathrm{\partial }O\times [0,T])$ for the open space--time cylinder, its closure in ${\mathbb{R}}^{n+1}$, and the parabolic boundary. Given a continuous $u_0:O\to \mathbb{R}$, the Hamilton--Jacobi Cauchy problem is $u_t+H(x,t,Du)=0\text{ in }Z,u=u_0\text{ on }O\times \{0\},$ where $Du$ denotes the spatial gradient and $u_t$ the time derivative of the unknown function (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder).

A classical solution of this problem is a function $u\in C^0(\bar{Z})\cap C^1(Z)$ ($C^k$ maps and multi-index derivative notation in Euclidean space) with $u(x,0)=u_0(x)$ for every $x\in O$ and $u_t(x,t)+H(x,t,Du(x,t))=0\text{for every }(x,t)\in Z\mathrm{.}$ The continuity requirement $u\in C^0(\bar{Z})$ ensures continuous attainment of the initial datum and also continuity on the lateral and terminal faces; no condition on the lateral face $\mathrm{\partial }O\times [0,T]$ is imposed unless it is stated explicitly.

Stationary specialization. Suppose that $H$ does not depend on $t$ and that $v\in C^1(O)$ satisfies $H(x,Dv(x))=0$ for every $x\in O$; such a $v$ is called a classical solution of the stationary Hamilton--Jacobi equation $H(x,Dv)=0$. If in addition $v$ extends continuously to $\bar{O}$, then $u(x,t):=v(x)$ is a classical solution of the Cauchy problem with datum $u_0:=v{\mathrm{\mid }}_O$, because $u\in C^0(\bar{Z})\cap C^1(Z)$ and $u_t\equiv 0$ on $Z$.

• What the definition fixes. The open cylinder $Z=O\times (0,T)$, its closure, the parabolic boundary $\mathrm{\Gamma }$, the initial face $O\times \{0\}$ on which the datum is read, the Hamiltonian domain $O\times [0,T]\times {\mathbb{R}}^n$ with its continuity, and the regularity class $C^0(\bar{Z})\cap C^1(Z)$ of a classical solution are the data used by every viscosity notion on this page. In particular, "the initial datum is attained" means continuous attainment on the initial face, not a merely pointwise boundary value on a larger set.

• No lateral condition, no choice. The definition imposes no condition on $\mathrm{\partial }O\times [0,T]$; statements about bounded domains add whatever boundary comparison they need explicitly. Nothing is selected anywhere in the definition, so no choice principle is used.