Half-space solutions with $7/2$ frequency in the thin obstacle problem

TL;DR

This paper investigates the structure of solutions to the 3D thin obstacle problem with 7/2 homogeneity, proving the isolated nature of half-space solutions and establishing convergence rates and free boundary regularity.

Contribution

It demonstrates that half-space solutions with 7/2 frequency are isolated and provides convergence rates and regularity results for solutions near these profiles.

Findings

Half-space solutions with 7/2 frequency are isolated in 3D.

Established convergence rates to the blow-up profile.

Proved regularity of the free boundary near contact points.

Abstract

For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence to this profile. As a consequence, we obtain regularity of the free boundary near such contact points.