In this thesis we study variational inequalities with gradient constraints. We consider the questions of regularity of their solutions, and regularity of their free boundaries. We also study the relation between variational inequalities with gradient constraint and the obstacle problem. In particular we consider their relation in the vector-valued case.
In Chapter 2 we generalize some known results about the equivalence of variational inequalities with gradient constraint and variational inequalities with constraint on the so- lution, by allowing less regular functionals and constraints. In Chapter 3 we prove that some class of vector-valued variational inequalities with gradient constraint is equivalent to variational inequalities with constraint on the solution.
In Chapter 4 we prove the optimal regularity for a class of variational inequalities with gradient constraints. The regularity and the shape of the free boundary of this problem is the subject of Chapters 5 and 6. We prove that the free boundary is as regular as the part of the boundary that parametrizes it. In order to prove this, we generalize the notion of ridge, by replacing the Euclidean norm by other norms. We also consider the singularities of the distance function (in the new norm) to the boundary of a domain.