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.