Almost minimizers for a sublinear system with free boundary

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

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