On the free boundary for thin obstacle problems with Sobolev variable coefficients

TL;DR

This paper develops a quasi-monotonicity formula for thin obstacle problems with Sobolev coefficients, leading to new regularity and structural insights into free boundaries, including rectifiability and Minkowski content finiteness.

Contribution

It introduces a novel quasi-monotonicity formula for variable coefficient thin obstacle problems, advancing understanding of free boundary regularity and structure.

Findings

Proves rectifiability of free boundary with Lipschitz coefficients

Establishes local finiteness of Minkowski content of free boundary

Provides regularity results at points with finite contact order

Abstract

We establish a quasi-monotonicity formula {for an intrinsic frequency function related to solutions to} thin obstacle problems with zero obstacle driven by quadratic energies with Sobolev $W^{1,p}$ coefficients, with $p$ bigger than the space dimension. From this we deduce several regularity and structural properties of the corresponding free boundaries at those distinguished points with finite order of contact with the obstacle. In particular, we prove the rectifiability {and the local finiteness of the Minkowski content} of the whole free boundary in the case of Lipschitz coefficients.