# Graph selectors and viscosity solutions on Lagrangian manifolds

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

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

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topMcCaffrey, 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 -

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