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Graph selectors and viscosity solutions on Lagrangian manifolds
Published online by Cambridge University Press: 11 October 2006
Abstract
Let $\Lambda $ be a Lagrangian submanifold of $T^{*}X$
for some closedmanifold X. Let $S(x,\xi )$
be a generating function for $\Lambda $
whichis quadratic at infinity, and let W(x) be the corresponding graph selectorfor $\Lambda ,$
in the sense of Chaperon-Sikorav-Viterbo, so that thereexists a subset $X_{0}\subset X$
of measure zero such that W is Lipschitzcontinuous on X, smooth on $X\backslash X_{0}$
and $(x,\partial W/\partialx(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 $H(x,\partial W/\partialx(x))=0$
on $X\backslash X_{0}$
and extends to a Lipschitz function on thewhole 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, includingcertain commonly occuring cases where H(x,p) is not convex in p.
- Type
- Research Article
- Information
- ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 4 , October 2006 , pp. 795 - 815
- Copyright
- © EDP Sciences, SMAI, 2006
References
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