On Global Discontinuous Solutions for Hamilton-Jacobi Equations
Authors: Gui-Qiang Chen and Bo Su
Title:
On Global Discontinuous Solutions for Hamilton-Jacobi Equations
Abstract
The uniqueness of classical semicontinuous viscosity solutions
of the Cauchy problem for Hamilton-Jacobi equations is established
for globally Lipschitz continuous and convex Hamiltonian $H=H(Du)$,
provided the discontinuous initial
value function $\varphi(x)$ is continuous outside a set $\Gamma$ of
measure zero and satisfies
$$
\varphi(x)\geq \varphi_{**}(x)
:=\liminf _{y\rightarrow x, y\in \RR^d\backslash\Gamma}\varphi(y).
(*)
$$
We prove that the discontinuous solutions with almost everywhere continuous
initial data satisfying $(*)$ become Lipschitz continuous after finite time
for locally strictly convex Hamiltonians.
The $L^1$-accessibility of initial data and a comparison principle
for discontinuous solutions are shown for a general Hamiltonian.
The equivalence of semicontinuous viscosity solutions,
bi-lateral solutions, $L$-solutions, minimax solutions,
and $L^\infty$-solutions is clarified.
This article has appeared in:
Comptes Rendus des Sciences de l'Acad\'{e}mie des Sciences.
S\'{e}rie I. Math\'{e}matique, Paris
vol. ??,
pages ?? (2002)
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Author Address
Gui-Qiang Chen
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu