Almost minimizers for the thin obstacle problem

Abstract

We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their \(C^{1,\beta }\) regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the \(C^{1,\gamma }\) regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperimetric inequalities for the solutions of the thin obstacle problem.