Almost minimizers for a singular system with free boundary

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.

arXiv comment25 pages