Dimension of the singular set in the parabolic obstacle problem

TL;DR

This paper investigates the size of the singular set in the parabolic obstacle problem with general obstacles, establishing that its parabolic Hausdorff dimension is at most n-1, extending previous results.

Contribution

It introduces a novel approach combining a truncated parabolic frequency formula and monotonicity estimates to prove the dimension bound for general obstacles.

Findings

Singular set has parabolic Hausdorff dimension at most n-1

Method applies to general obstacles in C^{2,1} class

Extends previous results from special to general obstacles

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 $\Delta \varphi \equiv -1$. 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 $3$.