For a standard linear programming problem, let $V$ be a real Hilbert space, $v\in V$ being fixed. $C$ a convex subset of $V$. What is the condition we have to impose on $u$ and $C$, so that the following linear programming admits a unique solution?$u^\ast = \arg\min_{u\in C}\langle u,v\rangle_V$.

Or more specifically in my case, Let $V=[H_0^1(\Omega)]^3, C=\lbrace u\in V:\|u\|_{L^\infty(\Omega)}\leq 1\rbrace, u\in L^2(\Omega)$. Is the uniqueness of a minimizer to the problem

$\varphi^\ast =\arg\min_{\varphi \in C}\int_\Omega u\textrm{div}\varphi dx$

assured?

Thanks.