Almost minimizers for a sublinear system with free boundary

Abstract

We study vector-valued almost minimizers of the energy functional $$\int_D\left(|\nabla\mathbf{u}|^2+\frac2{1+q}\left(\lambda_+(x)|\mathbf{u}^+|^{q+1}+\lambda_-(x)|\mathbf{u}^-|^{q+1}\right)\right)dx,\quad0<q<1.$$ For H\"older continuous coefficients $\lambda_\pm(x)>0$, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of "regular" free boundary points.