This article is cited in 4 papers

Semi-concave Singularities and the Hamilton–Jacobi Equation

Patrick Bernard<sup>ab

a</sup> École normale supérieure – Paris, 75230 Paris Cedex 05, France
^b Université Paris-Dauphine – CEREMADE (UMR 7534), Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

Abstract: We study the Cauchy problem for the Hamilton–Jacobi equation with a semiconcave initial condition.We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one.

Keywords: Hamilton–Jacobi equations, viscosity solutions, variational solutions, calculus of variations.

MSC: 49L25, 37J05

Received: 31.07.2013
Accepted: 08.10.2013

Language: English

DOI: 10.1134/S1560354713060075

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