Graph selectors and viscosity solutions on lagrangian manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, p. 795-815

Let $\Lambda$ be a lagrangian submanifold of ${T}^{*}X$ for some closed manifold $X.$ Let $S\left(x,\xi \right)$ be a generating function for $\Lambda$ which is quadratic at infinity, and let $W\left(x\right)$ 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 $X\setminus {X}_{0}$ and $\left(x,\partial W/\partial x\left(x\right)\right)\in \Lambda$ for $X\setminus {X}_{0}.$ Let $H\left(x,p\right)=0$ for $\left(x,p\right)\in \Lambda$. Then $W$ is a classical solution to $H\left(x,\partial W/\partial x\left(x\right)\right)=0$ on $X\setminus {X}_{0}$ 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 $H\left(x,p\right)$ is not convex in $p$.

DOI : https://doi.org/10.1051/cocv:2006023
Classification:  49L25,  53D12
Keywords: viscosity solution, lagrangian manifold, graph selector
@article{COCV_2006__12_4_795_0,
author = {McCaffrey, David},
title = {Graph selectors and viscosity solutions on lagrangian manifolds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {12},
number = {4},
year = {2006},
pages = {795-815},
doi = {10.1051/cocv:2006023},
zbl = {1114.49030},
mrnumber = {2266819},
language = {en},
url = {http://www.numdam.org/item/COCV_2006__12_4_795_0}
}

McCaffrey, David. Graph selectors and viscosity solutions on lagrangian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, pp. 795-815. doi : 10.1051/cocv:2006023. http://www.numdam.org/item/COCV_2006__12_4_795_0/

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