Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity

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January 2003, 9(1): 167-192. doi: 10.3934/dcds.2003.9.167

Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity

1.	Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730
2.	Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706

Received May 2002 Revised October 2002 Published November 2002

Abstract
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The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with convex Hamiltonians $H=H(Du)$ is established, provided the discontinuous initial value function $\varphi(x)$ is continuous outside a set $\Gamma$ of measure zero and satisfies

(*)$ \qquad\qquad \varphi(x)\ge\varphi_{\star \star}(x) \equiv \lim$inf$_{y\rightarrow x, y\in\mathbb R^d\backslash\Gamma}\varphi(y).

The regularity of discontinuous solutions to Hamilton-Jacobi equations with locally strictly convex Hamiltonians is proved: The discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time. The $L^1$-accessibility of initial data and a comparison principle for discontinuous solutions are shown. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, $L$-solutions, minimax solutions, and $L^\infty$-solutions is also clarified.

Keywords: viscosity solutions, accessibility of initial data., minimax solutions, stability, $L^\infty$ solutions, Hamilton-Jacobi equations, discontinuous solutions, regularity after finite time, semicontinuous solutions.

Mathematics Subject Classification: 35F25, 49L99, 35D10, 35B35, 35D0.

Citation: Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167

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