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From pointwise to local regularity for solutions of Hamilton-Jacobi equation

By Piermarco Cannarsa and Hélène Frankowska

Abstract

International audienceIt is well-known that solutions to the Hamilton-Jacobi equation $$\u_t(t,x)+H\big(x,\u_x(t,x)\big)=0$$ fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be differentiable at a given point $(t,x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t,x)$? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t,\cdot)$ at $x$ is nonempty. Our approach uses the representation of $u$ as the value function of a Bolza problem in the calculus of variations, as well as necessary optimality conditions for such a problem

Topics: Hamilton-Jacobi equations, semiconcave functions, Bolza problem, Riccati equations, [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]
Publisher: Springer Verlag
Year: 2014
DOI identifier: 10.1007/s00526-013-0611-y
OAI identifier: oai:HAL:hal-00851752v1
Provided by: Hal-Diderot
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