Generalized Hopf formulas for the nonautonomous Hamilton-Jacobi equation
type: Item
instance of: scholarly article
title: Generalized Hopf formulas for the nonautonomous Hamilton-Jacobi equation
zbMATH Open document ID: 1020.49023
DOI: 10.1007/BF02359303
author: I. V. Rublev
published in: Computational Mathematics and Modeling
publication date: 2003-09-14T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 35F25
zbMATH DE Number: 1568670
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: Hopf formulas
zbMATH Keywords: viscosity solution
zbMATH Keywords: minimax solution
zbMATH Keywords: contingent derivative
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1568670