Variational and quasivariational inequalities with first order constraints

We study the existence of solutions of stationary variational and quasivariational inequalities with curl constraint, Neumann type boundary condition and a pp-curl type operator. These problems are studied in bounded, not necessarily simply connected domains, with a special geometry, and the functional framework is the space of divergence-free functions with curl in Lp and null tangential or normal traces.The analogous variational or quasivariational inequalities with gradient constraint are also studied, considering Neumann or Dirichlet non-homogeneous boundary conditions. The existence of a generalized solution for a Lagrange multiplier problem with homogeneous Dirichlet boundary condition and the equivalence with the variational inequality is proved in the linear case, for an arbitrary gradient constraint.