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Free boundary regularity for almost minimizers of the parabolic Signorini problem

Seongmin Jeon, Arshak Petrosyan

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.

Free boundary regularity for almost minimizers of the parabolic Signorini problem

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.

Paper Structure

This paper contains 15 sections, 30 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Solutions of parabolic Signorini problem
  3. Almost minimizers
  4. Main results
  5. Proofs of Theorems \ref{['mainthm:par-Weiss']}--\ref{['mainthm:par-Clg-reg-set']}
  6. Notation and Preliminaries
  7. Notation
  8. Preliminaries
  9. Weiss- and Almgren-type monotonicity formulas
  10. Almgren rescalings and blowups
  11. Growth estimates
  12. $3/2$-Homogeneous blowups
  13. Regularity of the regular set
  14. Existence of weak solutions
  15. Examples of almost minimizers

Key Result

Theorem A

Fix $\kappa_0>2$, $0<\delta<2$ and $0<\varepsilon\le\alpha<1$. For $z_0\in\Gamma(u)\cap Q_{1/2}'$, let $u\in \mathcal{F}_{z_0}$ satisfy the weighted almost parabolic Signorini property at $z_0$. For $0<\kappa<\kappa_0$, we set where $a=a(\kappa,\alpha)>0$ and $b=b(\kappa,\varepsilon)>0$ are as in Theorem thm:par-weiss. Then $W_{\kappa,\alpha,\varepsilon,\delta}(r,u,z_0)$ is nondecreasing in $r$

Theorems & Definitions (63)

  • Definition 1.1: unweighted version
  • Definition 1.2: weighted version
  • Definition 1.3
  • Theorem A
  • Theorem B: Almgren-type monotonicity formula
  • Theorem C: Optimal growth near regular free boundary
  • Theorem D: Regularity of the regular set
  • Theorem 2.1
  • Lemma 2.2: Complementarity condition
  • Lemma 3.1
  • ...and 53 more