Almost minimizers for a sublinear system with free boundary

We study vector-valued almost minimizers of the energy functional $${\int }_D(\mathrm{\mid }\mathrm{\nabla }\mathbf{u}{\mathrm{\mid }}^2+\frac{2}{1+q}({\lambda }_{+}(x)\mathrm{\mid }{\mathbf{u}}^{+}{\mathrm{\mid }}^{q+1}+{\lambda }_{-}(x)\mathrm{\mid }{\mathbf{u}}^{-}{\mathrm{\mid }}^{q+1}))dx,0<q<1.$$ For Hölder 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.