Almost minimizers for a singular system with free boundary

TL;DR

This paper investigates the regularity of vector-valued almost minimizers for a specific energy functional involving gradient and magnitude terms, and analyzes the structure of the free boundary using advanced mathematical tools.

Contribution

It introduces new regularity results for almost minimizers and the free boundary in a singular system, employing Weiss-type monotonicity and epiperimetric inequalities.

Findings

Established regularity for minimizers and free boundary

Developed Weiss-type monotonicity formula for analysis

Applied epiperimetric inequality to energy minimizers

Abstract

In this paper we study vector-valued almost minimizers of the energy functional $$ \int_D\left(|\nabla\mathbf{u}|^2+2|\mathbf{u}|\right)\,dx . $$ We establish the regularity for both minimizers and the "regular" part of the free boundary. The analysis of the free boundary is based on Weiss-type monotonicity formula and the epiperimetric inequality for the energy minimizers.