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Let $\Lambda$ be a Lagrangian submanifold of $T^{*}X$ for some closed manifold Let $S(x,\xi )$ be a generating function for $\Lambda$ which is quadratic at infinity, and let 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 is Lipschitz continuous on smooth on $X\setminus X_0$ and $(x,\partial W/\partial x(x\left)\right)\in \Lambda$ for $X\setminus X_0.$ Let for $(x,p)\in \Lambda$. Then 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 Viterbo refers to as a variational solution. We prove that...