On the eigenvalues of the biharmonic operator with pointwise gradient constraint

We study an eigenvalue problem for some fourth order variational inequalities, which result from a pointwise constraint on the gradient of the unknown function u. We use a nonsmooth variational approach to find existence and multiplicity (under symmetry assumptions) for solutions of the problem. A major issue to be faced is how to exclude tangency between the set of u's satisfying the contraint condition and the L2-sphere.