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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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Submitted on : Sunday, August 18, 2013 - 7:19:54 AM
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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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