# American Institute of Mathematical Sciences

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

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. Citation: Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167  [1] Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. 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