Almost minimizers for a sublinear system with free boundary

Abstract

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

39 pages. arXiv admin note: text overlap with arXiv:2112.00676