Published Paper

Inserted: 22 jul 2024
Last Updated: 25 oct 2026

Journal: Interfaces and Free Boundaries
Volume: 28
Number: 2
Pages: 191--235
Year: 2026
Doi: DOI 10.4171/IFB/537

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.

Keywords: Rectifiability, free boundary, Thin obstacle, Sobolev coefficients

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