Generalized Hopf formulas for the nonautonomous Hamilton-Jacobi equation
The main result of the paper is Theorem 2.1 stating that if \(\varphi (.)\) is convex and \(H(.,.)\) satisfies certain rather complicated assumptions, then the function \(V(.,.)\) given by the formula: \[ V(t,x):=\displaystyle \sup_{s\in \mathbb{R}^n}\left[\langle s,x\rangle +\int_t^TH(\tau,s)d\tau-\varphi^* (s)\right], \;(t,x)\in \overline G, \] is a continuous minimax (hence also a viscosity) solution of the problem: \[ {{\partial V}\over {\partial t}}(t,x)+H\left(t,{{\partial V}\over {\partial x}} (t,x)\right)=0, \;(t,x)\in G:=(0,T)\times \mathbb{R}^n, \;V(T,x)\equiv \varphi (x). \] The author uses the definitions and properties of the minimax solutions in [\textit{A. I. Subbotin}, ``Generalized solutions of first-order PDEs. The dynamic optimization perspective'' (1994; Zbl 0820.35003)] and notes that a similar generalization of the Hopf-Lax formula may be obtained for more general problems of the form: \[ {{\partial V}\over {\partial t}}(t,x)+H\left(t,V(t,x),{{\partial V}\over {\partial x}}(t,x)\right)=0, \;(t,x)\in G:=(0,T)\times \mathbb{R}^n, \;V(T,x)\equiv \varphi (x), \] which are nonautonomous versions of the case considered in [\textit{E. N. Barron, R. Jensen} and \textit{W. Liu}, Comm. Partial Differ. Equations 22, 1141-1160 (1997; Zbl 0887.35032)].
scientific article; zbMATH DE number 1568670