Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity

Gui-Qiang Chen; Bo Su

doi:10.3934/dcds.2003.9.167

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2003, Volume 9, Issue 1: 167-192. Doi: 10.3934/dcds.2003.9.167

This issue Previous Article Interaction estimates and Glimm functional for general hyperbolic systems Next Article Rigidity of partially hyperbolic actions of property (T) groups

Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity

Gui-Qiang Chen^1, and
Bo Su^2,

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: January 2003

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Abstract

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.

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Mathematics Subject Classification: 35F25, 49L99, 35D10, 35B35, 35D05.

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