Graph selectors and viscosity solutions on lagrangian manifolds

McCaffrey, David

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 795-815.

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 $W\left(x\right)$ 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 $(x,\partial W/\partial x(x\left)\right)\in \Lambda$ for $X\setminus X_0.$ Let $H(x,p)=0$ for $(x,p)\in \Lambda$. Then $W$ is a classical solution to $H(x,\partial W/\partial x(x\left)\right)=0$ 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 $H(x,p)$ is not convex in $p$.

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