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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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