The free boundary of variational inequalities with gradient constraints

TL;DR

This paper investigates the regularity of the free boundary in certain variational inequalities with gradient constraints, linking it to the geometric properties of a generalized ridge related to the domain.

Contribution

It establishes the regularity of the free boundary by analyzing the generalized ridge and its relation to the boundary's tangent bundle.

Findings

The free boundary's regularity matches that of the boundary's tangent bundle.

Introduces a generalized notion of ridge for planar domains.

Connects the singularities of the p-norm distance function to boundary regularity.

Abstract

In this paper we prove that the free boundary of some variational inequalities with gradient constraints 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 singularities of the distance function in the p-norm to the boundary of the domain.