Free boundary regularity for almost minimizers of the parabolic Signorini problem

TL;DR

This paper investigates the smoothness of the free boundary in the parabolic Signorini problem with zero obstacle, establishing new monotonicity and frequency formulas to prove optimal regularity results.

Contribution

It introduces Weiss-type and Almgren-type formulas for almost minimizers, advancing understanding of free boundary regularity in the parabolic Signorini problem.

Findings

Established Weiss-type monotonicity formula for almost minimizers.

Derived Almgren-type frequency formula for growth analysis.

Proved regularity of the regular free boundary set.

Abstract

In this paper, we study the regularity of the "regular" part of the free boundary for almost minimizers in the parabolic Signorini problem with zero thin obstacle. This work is a continuation of our earlier research on the regularity of almost minimizers. We first establish the Weiss-type monotonicity formula by comparing almost minimizers with parabolically homogeneous replacements and utilizing conformal self-similar coordinates. Subsequently, by deriving the Almgren-type frequency formula and applying the epiperimetric inequality, we obtain the optimal growth near regular free boundary points and achieve the regularity of the regular set.