On a class of variational inequalities involving gradient operators (Q801288)

Let H be a real Hilbert space and K a nonempty closed convex subset of H. This paper deals with the solvability of the variational inequality: Find u in K such that (F(u),v-u)\(\geq 0\) for all \(v\in K\), where F:K\(\to H\) is a gradient operator, which means that there exists a functional \(\phi\) whose Fréchet derivative satisfies \(\phi '(u)=F(u)\). Here it is not assumed that \(\phi\) is bounded from below, but an assumption is made that the restriction of \(\phi\) to certain affine subspaces of finite codimension attains its infimum. Applications of the abstract results to some elliptic operators are also given.