# Showing there exists a solution to a variational inequality

I'm working through a book that provides the following exercise which I'm having trouble with. They cite this paper which I can't find (and even if I could, I can't read italian).

The problem:

Let $$\Omega$$ be a bounded open domain in $$\mathbb{R}^n$$ and let $$E$$ and $$F$$ be closed nonempty subsets of $$\Omega$$. Let $$\phi \in H^1(\Omega)$$ and denote by $$K$$ the convex set: $$K = \{v \in H^1(\Omega)\ : v \geq \phi\text{ on } E \text{ and } v \leq \phi\text{ on } F \}$$ Let $$a(u,v) = \int_{\Omega}\sum_{i=1}^nu_{x_i}v_{x_i}\,dx \quad \forall u,v \in H^1(\Omega)$$ Show that there exists a solution to the variational inequality: $$u \in K: a(u, v - u) \geq 0 \quad \forall v \in K$$

• The task is, apparently, to minimize $\int_\Omega|\nabla u|^2$ and to show that the minimizer exists, which more or less amounts to showing that $K$ is closed in $H^1$. (BTW, the condition $u\ge\varphi$ on $E$ should be understood with certain care because point evaluation is not continuous in $H^1$ when $n\ge 2$). May 12 '19 at 15:48