The Singular Set of Higher Dimensional Unstable Obstacle Type Problems

TL;DR

This paper investigates the singular points in a higher-dimensional unstable free boundary problem involving a Laplacian equation with a characteristic function, aiming to understand the structure of solutions and their singularities.

Contribution

It provides a detailed analysis of the singular set in an unstable obstacle problem in higher dimensions, advancing understanding of free boundary regularity and singularity structure.

Findings

Characterization of singular points in the problem

Identification of conditions for singularity formation

Insights into the structure of the singular set

Abstract

In this paper we will investigate the singular points of the following unstable free boundary problem: {equation}\label{Eq} \Delta u= -\chi_{\{u>0\}} \quad\quad\textrm{in} B_1(0) {equation} where $\chi_{\{u>0\}}$ is the characteristic function of the set $\{u>0\}$.