We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar and vector-valued case.u ∈ C 1,1 loc (U ; R N ). This problem is a generalization to the vector-valued case of the elastic-plastic torsion problem, which is the problem of minimizing J η (v) :=ˆU |Dv| 2 − ηv dx for some η > 0, over {v ∈ H 1 0 (U ) | |Dv| ≤ 1 a.e.}.

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Abstract. In this paper we prove that the free boundary of the minimizer ofsubject to the pointwise gradient constraintis as regular as the tangent bundle of the boundary of the domain. To this end, we study a generalized notion of ridge of a domain in the plane, which is the set of singularity of the distance function in the p -norm to the boundary of the domain.

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In this paper we derive an estimate on the number of local maxima of the free boundary of the minimizer ofsubject to the pointwise gradient constraint |Dv|p ≤ 1.This also gives an estimate on the number of connected components of the free boundary.