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From pointwise to local regularity for solutions of Hamilton–Jacobi equations

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Abstract

It is well-known that solutions to the Hamilton–Jacobi equation

$$\begin{aligned} u_{t}(t,x)+H(x,u_{x}(t,x))=0 \end{aligned}$$

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 conditions for such a problem.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 , Rome, Italy

    P. Cannarsa

  2. CNRS and Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, case 247, 4 place Jussieu, 75252 , Paris, France

    H. Frankowska

Authors
  1. P. Cannarsa
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  2. H. Frankowska
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Correspondence to P. Cannarsa.

Additional information

Communicated by L. Ambrosio.

This work was co-funded by the European Union under the 7th Framework Programme “FP7-PEOPLE-2010-ITN”, grant agreement number 264735-SADCO.

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Cannarsa, P., Frankowska, H. From pointwise to local regularity for solutions of Hamilton–Jacobi equations. Calc. Var. 49, 1061–1074 (2014). https://doi.org/10.1007/s00526-013-0611-y

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