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From pointwise to local regularity for solutions of Hamilton-Jacobi equations.
- Source :
- Calculus of Variations & Partial Differential Equations; Mar2014, Vol. 49 Issue 3/4, p1061-1074, 14p
- Publication Year :
- 2014
-
Abstract
- It is well-known that solutions to the Hamilton-Jacobi equation 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 09442669
- Volume :
- 49
- Issue :
- 3/4
- Database :
- Complementary Index
- Journal :
- Calculus of Variations & Partial Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 94629411
- Full Text :
- https://doi.org/10.1007/s00526-013-0611-y