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Dimension of the singular set in the parabolic obstacle problem

Alejandro Martínez, Xavier Ros-Oton

Abstract

In this paper we study the singular set in the parabolic obstacle problem for general obstacles $\varphi \in C^{2,1}$. We prove that the singular set has parabolic Hausdorff dimension at most $n-1$. Prior to our result, this was only known when $Δ\varphi \equiv -1$. Our approach combines a truncated parabolic frequency formula and monotonicity estimates with an iterative argument showing that the frequency is saturated for all values of the truncation parameter between $2$ and $3$.

Dimension of the singular set in the parabolic obstacle problem

Abstract

In this paper we study the singular set in the parabolic obstacle problem for general obstacles . We prove that the singular set has parabolic Hausdorff dimension at most . Prior to our result, this was only known when . Our approach combines a truncated parabolic frequency formula and monotonicity estimates with an iterative argument showing that the frequency is saturated for all values of the truncation parameter between and .
Paper Structure (12 sections, 40 theorems, 164 equations)

This paper contains 12 sections, 40 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. The singular set
  3. Ideas of the proof
  4. Organization of the paper
  5. Acknowledgments
  6. Notation and Preliminary Results
  7. Notation and definitions
  8. Preliminary results
  9. Weiss and frequency formulae
  10. Variants of the Lemmas with extra assumption:
  11. The 2nd Blow-up in the Top Stratum
  12. Proof of main result

Key Result

theorem 1

Let $\Omega \subset \mathbb{R}^n$ be any open set, let $u \in L^\infty(\Omega \times (-T, T))$ solve the parabolic obstacle problem eq:parabolic_obstacle_problem with $f\in C^{0, 1}(\bar{\Omega} \times [-T, T])$ and $f > 0$. Let $\Sigma \subset \Omega \times (-T, T)$ be the set of singular points. T where $\dim_\text{par}$ denotes the parabolic Hausdorff dimension.

Theorems & Definitions (68)

  • theorem 1
  • lemma 2
  • proof
  • lemma 3: FRS24, Lemma 4.6
  • theorem 4
  • lemma 5
  • lemma 6
  • lemma 7
  • proposition 8
  • lemma 9
  • ...and 58 more