Almost minimizers for a sublinear system with free boundary

Abstract

AbstractWe study vector-valued almost minimizers of the energy functional $$\begin{aligned} \int _D\left( |\nabla \textbf{u}|^2+\frac{2}{1+q}\left( \lambda _+(x)|\textbf{u}^+|^{q+1}+\lambda _-(x)|\textbf{u}^-|^{q+1}\right) \right) dx,\quad 0<q<1. \end{aligned}$$ ∫ D | ∇ u | 2 + 2 1 + q λ + ( x ) | u + | q + 1 + λ - ( x ) | u - | q + 1 d x , 0 < q < 1 . For Hölder continuous coefficients $$\lambda _\pm (x)>0$$ λ ± ( x ) > 0 , we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of “regular" free boundary points.