Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity

Andrea C.G. Mennucci

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 3, page 426-451
  • ISSN: 1292-8119

Abstract

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We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number of n - 1 dimensional manifolds, but for a n - 1 negligeable subset. These results can be applied to the cutlocus of a C2 submanifold of a Finsler manifold.

How to cite

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Mennucci, Andrea C.G.. "Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity." ESAIM: Control, Optimisation and Calculus of Variations 10.3 (2010): 426-451. <http://eudml.org/doc/90737>.

@article{Mennucci2010,
abstract = { We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of \{H=0\} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a $\{\{\mathcal H\}\}^\{n-1\}$ negligeable subset. These results can be applied to the cutlocus of a C2 submanifold of a Finsler manifold. },
author = {Mennucci, Andrea C.G.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamilton-Jacobi equations; conjugate points.; Hamilton-Jacobi equation; manifold; min solution; viscosity solution; geodesics; regularity; conjugate points; Finsler manifold},
language = {eng},
month = {3},
number = {3},
pages = {426-451},
publisher = {EDP Sciences},
title = {Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity},
url = {http://eudml.org/doc/90737},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Mennucci, Andrea C.G.
TI - Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 3
SP - 426
EP - 451
AB - We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${{\mathcal H}}^{n-1}$ negligeable subset. These results can be applied to the cutlocus of a C2 submanifold of a Finsler manifold.
LA - eng
KW - Hamilton-Jacobi equations; conjugate points.; Hamilton-Jacobi equation; manifold; min solution; viscosity solution; geodesics; regularity; conjugate points; Finsler manifold
UR - http://eudml.org/doc/90737
ER -

References

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