I need to check whether the following characterization of the minimizer of a convex functional is valid.
Let $X$ be a reflexive Banach space (think $W^{1,p}(\Omega)$ with $\Omega \subset \mathbb R^n$ smooth and bounded), $\Psi: X \to \mathbb R \cup \{ + \infty \}$ a convex functional, and $A = \partial \Psi$. Assume that $u$ satisfies
$$ \langle A v , v - u \rangle \ge 0 $$
for all $v \in V$, where $V$ is a dense subspace of $X$ such that $\Psi|_V$ and $A$ are continuous in a norm of $V$ and $A$ is univalued (think $V = W^{1,p} \cap L^\infty$). Assume, furthermore, that
$$ \inf_V \Psi = \inf_X \Psi $$
Question: Is $u$ a minimizer of $E$ on $X$? The proof is, of course, trivial if $u \in V$.
In particular my convex functional is of the form $$ \frac{1}{p} \int_{\Omega} |\nabla v|^p + \int_{\Omega} \Phi (v) $$ where $\Phi$ is a convex function. Of course, in this case $ X = W^{1,p}$ and $V = W^{1,p} \cap L^\infty$.
