Almost minimizers for the thin obstacle problem with variable coefficients

TL;DR

This paper investigates the regularity and structure of almost minimizers in the thin obstacle problem with variable coefficients, extending classical monotonicity formulas to handle non-constant cases.

Contribution

It generalizes Weiss- and Almgren-type monotonicity formulas to variable coefficient settings and analyzes regularity and structure of almost minimizers in this context.

Findings

Established $C^{1,eta}$ regularity of almost minimizers

Proved optimal growth of almost minimizers under quasisymmetry

Provided a structural theorem for the singular set

Abstract

We study almost minimizers for the thin obstacle problem with variable H\"older continuous coefficients and zero thin obstacle and establish their $C^{1,\beta}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.