On strong solutions of quasi-variational inequalities (Q1374502)

The author studies the quasi-variational inequality: \(u \in Su\) and \((Au-f(u),v-u) \geq 0\) for all \(v \in S(u)\) with \(Au \in \partial_u \varphi (p,u) |_{p=Lu}\) and \(f(u) \in F(u)\). Here \(H\) is a real Hilbert space, \(S\) an upper semicontinuous set-valued mapping of \(H\) into the closed convex subsets of \(H\) with convex domain \(D(S)\), \(\varphi\) is a lower semicontinuous mapping of \(\mathcal H \times H\) into \(\mathbb{R}^+\) with \(\varphi (p,\cdot)\) being a convex function of \(H\) into \(\mathbb{R}^+\) and \(D(\varphi) = \mathcal H \times H\) where \(V\) is a Hilbert space dense in \(H\), \(L\) is a bounded linear mapping \(L\) of \(V\) into the Hilbert space \(\mathcal H\), \(F(u)\) is a set-valued upper semicontinuous mapping of \(V\) into the closed convex subsets of \(H\). The existence of a strong solution \(u\) is established in Section 2. It allows the author to deduce some global regularity properties. The optimal control problem is considered in Section 3. In the last section there are given some applications: A quasi-variational inequality in the theory of control of free surfaces, quasi-variational inequality arising from a control problem with impulses, and constrained non-cooperative games with quadratic loss functions.