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

Abstract : It 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.
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Calculus of Variations and Partial Differential Equations, Springer Verlag, 2014, 49 (3-4), pp.1061-1074. 〈10.1007/s00526-013-0611-y〉

https://hal.inria.fr/hal-00851752
Contributeur : Helene Frankowska <>
Soumis le : dimanche 18 août 2013 - 07:19:54
Dernière modification le : jeudi 11 janvier 2018 - 06:12:14

### Citation

Piermarco Cannarsa, Hélène Frankowska. From pointwise to local regularity for solutions of Hamilton-Jacobi equation. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2014, 49 (3-4), pp.1061-1074. 〈10.1007/s00526-013-0611-y〉. 〈hal-00851752〉

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