On the shape of the free boundary of variational inequalities with gradient constraints

TL;DR

This paper provides estimates on the number of local maxima and connected components of the free boundary in certain variational inequalities with gradient constraints, advancing understanding of their geometric structure.

Contribution

It introduces new bounds on the complexity of the free boundary in variational inequalities with gradient constraints, a topic not extensively explored before.

Findings

Bound on the number of local maxima of the free boundary

Estimate on the number of connected components of the free boundary

Enhanced understanding of free boundary geometry in variational inequalities

Abstract

In this paper we derive an estimate on the number of local maxima of the free boundary of some variational inequalities with pointwise gradient constraints. This also gives an estimate on the number of connected components of the free boundary.