Variational inequality

Remarks

If $K = H$, then taking $v = u \pm w$ shows that the inequality holds exactly when $F(u) = 0$, so the problem contains the equation $F(u) = 0$. If $F = \nabla f$ for a convex differentiable function $f$, then $u$ solves the problem exactly when $u$ minimizes $f$ on $K$. Stampacchia's theorem treats the linear case. Let $a$ be a bounded coercive bilinear form on $H$ and let $\ell$ be a continuous linear functional on $H$. Then there is exactly one $u \in K$ with $a(u, v - u) \ge \ell(v - u)$ for all $v \in K$. When $a$ is symmetric, this $u$ minimizes $\tfrac12 a(v, v) - \ell(v)$ over $K$. For $H = \R^n$, $K$ compact and $F$ continuous, a solution always exists, by the Hartman–Stampacchia theorem, whose proof uses the Brouwer fixed point theorem. For a continuous monotone $F$ the solution set is closed and convex, and a strictly monotone $F$ has at most one solution.