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.

我们考虑 Anzellotti 类型的几乎最小化器，用于具有零薄障碍物的薄障碍物（或 Signorini）问题，并在薄流形的任一侧建立它们的\(C^{1,\beta }\)正则性，最佳增长远离自由边界，自由边界规则部分的\(C^{1,\gamma }\)正则性，以及奇异集的结构定理。自由边界的分析基于对能量方法的成功改编，例如 Weiss 型单调性公式的单参数族、Almgren 型频率公式以及用于解决薄障碍问题的表观和对数表观不等式.