Generalized solutions of partial differential equations of the first order. The invariance of graphs relative to differential inclusions
type: Item
instance of: scholarly article
title: Generalized solutions of partial differential equations of the first order. The invariance of graphs relative to differential inclusions
zbMATH Open document ID: 0861.49007
DOI: 10.1007/BF02363859
author: A. I. Subbotin
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-04-22T00:00:00Z
review text: The paper is devoted to the determination of minimax solutions for first order partial differential equations of the form \[ F(x,u,Du)=0, \qquad x\in G\subset \mathbb{R}^n, \] by means of techniques from differential game theory. The paper has more a review character pointing out some approaches to the notion of generalized (minimax) solution. One defines the minimax, upper and lower solutions for the above equation. The minimax solution turns out to be in the same time an upper and lower solution. The connection between \(u\)-stable resp. \(v\)-stable functions and upper resp. lower solutions for the Hamilton-Jacobi-Bellman-Isaacs equation is shown. Also, the relation between minimax and viscosity solutions in the sense of Lions is largely discussed. The last two paragraphs contain results regarding discontinuous solutions for the Dirichlet boundary value problem: \[ H(x,Du)- u=0, \quad x\in G\subset \mathbb{R}^n, \qquad u(x)= \sigma(x), \quad x\in\partial G. \] One defines the upper, lower, and minimax solutions and one proves, by means of Lyapunov function, the uniqueness of a minimax solution.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 49J24
Mathematics Subject Classification ID: 35F15
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 903434
zbMATH Keywords: minimax solutions
zbMATH Keywords: first order partial differential equations
zbMATH Keywords: Hamilton-Jacobi-Bellman-Isaacs equation
zbMATH Keywords: viscosity solutions
zbMATH Keywords: discontinuous solutions
zbMATH Keywords: Dirichlet boundary value problem
zbMATH Keywords: uniqueness
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 903434