Almost minimizers for the thin obstacle problem with variable coefficients

Abstract

We study almost minimizers for the thin obstacle problem with variable Hölder 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.