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Graph selectors and viscosity solutions on Lagrangian manifolds

Published online by Cambridge University Press:  11 October 2006

David McCaffrey
David McCaffrey*
Affiliation:
University of Sheffield, Dept. of Automatic Control and Systems Engineering, Mappin Street, Sheffield, S1 3JD, UK; david@mccaffrey275.fsnet.co.uk
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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.

Keywords

Viscosity solutionLagrangian manifoldgraph selector.
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

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