Max-min representation of solutions of Hamilton-Jacobi equation in the quarter plane
type: Item
instance of: scholarly article
title: Max-min representation of solutions of Hamilton-Jacobi equation in the quarter plane
zbMATH Open document ID: 0874.35025
author: K. T. Joseph
author: G. D. Veerappa Gowda
published in: Differential Equations and Dynamical Systems
publication date: 1997-09-09T00:00:00Z
review text: The monotone difference scheme \(u_k^n:=u(k\Delta x,n\Delta t)\), \(k=0,\pm 1,\pm 2,\dots\), \(n=0,1,\dots\), \(\Delta x=L\Delta t\rightarrow 0\), is used to obtain max-min representations of generalized solutions of Hamilton-Jacobi equations of the form: \(u_t+H(u_x)=0\) defined by Hamiltonians \(H:\mathbb{R}^n\rightarrow\mathbb{R}\) that are assumed to be \(L\)-Lipschitzian. Iterating the recurrence formula: \[ u_k^n=\max_{\alpha\in\mathbb{R}}\min_{\beta=\pm 1}[u_{k-\beta}^{n-1}+\Delta t(L\alpha\beta-H(\alpha))], \] the authors obtain \(u_k^n\) as a max-min function of the data \(u_k^0:=u^0(k\Delta x)\), \(u_0^n:=\lambda(n\Delta t)\) and therefore a max-min representation of the limit function: \(u(x,t):=\lim_{\Delta t\rightarrow 0}u_k^n\) if \(x=k\Delta x=kL\Delta t\), \(t=n\Delta t\), which is considered a generalized solution of the initial value problem: \(u(x,0)=u^0(x)\), \(x\in\mathbb{R}\), \(t>0\), of the Dirichlet problem: \(u(x,0)=u^0(x)\), \(u(0,t)=\lambda(t)\), \(x,t>0\) and, respectively, of the Neumann problem: \(u(x,0)=u^0(x)\), \(u_x(0,t)=\lambda(t)\), \(x,t>0\).
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 1014374
zbMATH Keywords: max-min representations of generalized solutions
zbMATH Keywords: difference scheme
zbMATH Keywords: initial value problem
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Neumann problem
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1014374