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Discontinuous solutions of a Dirichlet-type boundary-value problem for the first-order partial differential equation

Discontinuous solutions of a Dirichlet-type boundary-value problem for the first-order partial... - Generalized (possibly, non-smooth and discontinuous) solutions of a boundary-value problem for the first-order partial differential equation are defined. Upper (lower) minim ax solutions are considered. Their definition is based on the weak invariance property of the epigraph (hypograph) with respect to a characteristic differential inclusion. The existence and uniqueness of the generalized solution is proved. A generalized solution is defined as a function that is approximated by upper and lower solutions. The differential pursuit-evasion game is considered as an example. It is shown that the value function of the game coincides with the generalized solution of the boundary-value problem for the appropriate Bellman-Isaacs equation. A comparison with the notion of a discontinuous viscosity solution, proposed by H. Ishii, is made. 1. INTRODUCTION In this paper we consider the first-order partial differential equation (PDE) F(x,u,Du) = Q, with the boundary condition u(x) = a(x), xedG. (1.2) Here G is an open set in Rn, dG is the boundary of G, u(x) is a real function defined on clG:=Gu3G, and Du:= (du/dxv...,du/dxj is the gradient of the function u. The real functions (x,z9s)*-+F(x,z9s) and ^() are defined on G x R x Rn and dG, respectively. Assumptions on these http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Russian Journal of Numerical Analysis and Mathematical Modelling de Gruyter

Discontinuous solutions of a Dirichlet-type boundary-value problem for the first-order partial differential equation

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References (8)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0927-6467
eISSN
1569-3988
DOI
10.1515/rnam.1993.8.2.145
Publisher site
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Abstract

- Generalized (possibly, non-smooth and discontinuous) solutions of a boundary-value problem for the first-order partial differential equation are defined. Upper (lower) minim ax solutions are considered. Their definition is based on the weak invariance property of the epigraph (hypograph) with respect to a characteristic differential inclusion. The existence and uniqueness of the generalized solution is proved. A generalized solution is defined as a function that is approximated by upper and lower solutions. The differential pursuit-evasion game is considered as an example. It is shown that the value function of the game coincides with the generalized solution of the boundary-value problem for the appropriate Bellman-Isaacs equation. A comparison with the notion of a discontinuous viscosity solution, proposed by H. Ishii, is made. 1. INTRODUCTION In this paper we consider the first-order partial differential equation (PDE) F(x,u,Du) = Q, with the boundary condition u(x) = a(x), xedG. (1.2) Here G is an open set in Rn, dG is the boundary of G, u(x) is a real function defined on clG:=Gu3G, and Du:= (du/dxv...,du/dxj is the gradient of the function u. The real functions (x,z9s)*-+F(x,z9s) and ^() are defined on G x R x Rn and dG, respectively. Assumptions on these

Journal

Russian Journal of Numerical Analysis and Mathematical Modelling – de Gruyter

Published: Jan 1, 1993

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