Discontinuous Solutions in $L^{\infty}$ for Hamilton-Jacobi Equations
Author: Gui-Qiang Chen and Bo Su
Title:
Discontinuous Solutions in $L^{\infty}$ for Hamilton-Jacobi Equations
Abstract
We introduce an approach to construct global discontinuous solutions
in $L^{\infty}$ for Hamilton-Jacobi equations. Our approach allows the
initial data only in $L^{\infty}$ and applies to the equations with
nonconvex Hamiltonians. The profit functions are introduced to formulate
the notion of discontinuous solutions in $L^\infty$.
The existence of global discontinuous solutions in $L^{\infty}$ is
established.
Our solutions in $L^{\infty}$ coincide with the viscosity solutions
and the minimax solutions, provided that the initial data are continuous.
A prototypical equation is analyzed to examine the $L^{\infty}$ stability
of our $L^{\infty}$ solutions. Our analysis also shows that global
discontinuous solutions are determined by the topology in which the
initial data are approximated.
This article has appeared in:
Chinese Annals of Mathematics , 21B(2) , pages 165-186 (2000)
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A closely related paper is Change me.
Author Address
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu