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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 |
(*)$ \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.
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