Graph selectors and viscosity solutions on Lagrangian manifolds

David McCaffrey

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 4, page 795-815
  • ISSN: 1292-8119

Abstract

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Let Λ be a Lagrangian submanifold of T * X for some closed manifold X. Let S ( x , ξ ) be a generating function for Λ which is quadratic at infinity, and let W(x) be the corresponding graph selector for Λ , in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset X 0 ⊂ X of measure zero such that W is Lipschitz continuous on X, smooth on X ∖ X 0 and ( x , ∂ W / ∂ x ( x ) ) ∈ Λ for X ∖ X 0 . Let H(x,p)=0 for ( x , p ) ∈ Λ . Then W is a classical solution to H ( x , ∂ W / ∂ x ( x ) ) = 0 on X ∖ X 0 and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions. We also prove that these conditions are satisfied in many cases, including certain commonly occuring cases where H(x,p) is not convex in p.

How to cite

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McCaffrey, David. "Graph selectors and viscosity solutions on Lagrangian manifolds." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 795-815. <http://eudml.org/doc/249664>.

@article{McCaffrey2006,
abstract = { Let $\Lambda $ be a Lagrangian submanifold of $T^\{*\}X$ for some closed manifold X. Let $S(x,\xi )$ be a generating function for $\Lambda $ which is quadratic at infinity, and let W(x) be the corresponding graph selector for $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset $X_\{0\}\subset X$ of measure zero such that W is Lipschitz continuous on X, smooth on $X\backslash X_\{0\}$ and $(x,\partial W/\partial x(x))\in \Lambda $ for $X\backslash X_\{0\}.$ Let H(x,p)=0 for $(x,p)\in \Lambda$. Then W is a classical solution to $H(x,\partial W/\partial x(x))=0$ on $X\backslash X_\{0\}$ and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions. We also prove that these conditions are satisfied in many cases, including certain commonly occuring cases where H(x,p) is not convex in p. },
author = {McCaffrey, David},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Viscosity solution; Lagrangian manifold; graph selector.; viscosity solution; graph selector},
language = {eng},
month = {10},
number = {4},
pages = {795-815},
publisher = {EDP Sciences},
title = {Graph selectors and viscosity solutions on Lagrangian manifolds},
url = {http://eudml.org/doc/249664},
volume = {12},
year = {2006},
}

TY - JOUR
AU - McCaffrey, David
TI - Graph selectors and viscosity solutions on Lagrangian manifolds
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 795
EP - 815
AB - Let $\Lambda $ be a Lagrangian submanifold of $T^{*}X$ for some closed manifold X. Let $S(x,\xi )$ be a generating function for $\Lambda $ which is quadratic at infinity, and let W(x) be the corresponding graph selector for $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset $X_{0}\subset X$ of measure zero such that W is Lipschitz continuous on X, smooth on $X\backslash X_{0}$ and $(x,\partial W/\partial x(x))\in \Lambda $ for $X\backslash X_{0}.$ Let H(x,p)=0 for $(x,p)\in \Lambda$. Then W is a classical solution to $H(x,\partial W/\partial x(x))=0$ on $X\backslash X_{0}$ and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions. We also prove that these conditions are satisfied in many cases, including certain commonly occuring cases where H(x,p) is not convex in p.
LA - eng
KW - Viscosity solution; Lagrangian manifold; graph selector.; viscosity solution; graph selector
UR - http://eudml.org/doc/249664
ER -

References

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