A note on the singular set of the no-sign obstacle problem

TL;DR

This paper proves the uniqueness of blowups at singular points in the no-sign obstacle problem, addressing a previously open question and advancing understanding of the problem's singular set structure.

Contribution

It establishes the uniqueness of blowups at singular points for the no-sign obstacle problem, providing a positive answer to an open problem from prior literature.

Findings

Proves uniqueness of blowups at singular points.

Addresses an open problem in the literature.

Enhances understanding of the singular set in obstacle problems.

Abstract

In this note, we prove the uniqueness of blowups at singular points of the no-sign obstacle problem $\Delta u=\chi_{_{B_1\backslash \{u=|Du|=0\}}}\ \text{in}\ B_1,$ thus give a positive answer to a problem raised in \cite[Notes of Chaper 7, page 149]{PSU12}.