The free boundary of variational inequalities with gradient constraints

In this paper we prove that the free boundary of the minimizer of I(v)≔∫U12|Dv|2−ηvdx, subject to the pointwise gradient constraint |Dv|p≤1,|Dv|p≤1, is 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 pp-norm to the boundary of the domain.