Regularity of almost minimizers for the parabolic thin obstacle problem

TL;DR

This paper investigates the regularity properties of almost minimizers in the parabolic thin obstacle problem, establishing new Hölder regularity results for these minimizers and their gradients, including variable coefficient cases.

Contribution

It provides the first regularity results for almost minimizers in the parabolic Signorini problem with zero obstacle and variable Hölder coefficients.

Findings

Established $H^{\sigma,\sigma/2}$-regularity for almost minimizers.

Proved Hölder regularity of spatial gradients on either side of the thin space.

Extended results to cases with variable Hölder coefficients.

Abstract

In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their $H^{\sigma,\sigma/2}$-regularity for every $0<\sigma<1$, as well as $H^{\beta,\beta/2}$-regularity of their spatial gradients on the either side of the thin space for some $0<\beta<1$. A similar result is also obtained for almost minimizers for the Signorini problem with variable H\"{o}lder coefficients.