The free boundary for a superlinear system

In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $${\int }_{\mathrm{\Omega }}(\mathrm{\mid }\mathrm{\nabla }\mathbf{u}{\mathrm{\mid }}^2+\frac{2}{p}\mathrm{\mid }\mathbf{u}{\mathrm{\mid }}^p),0<p<1,$$ but solutions can be also understood in an ad hoc viscosity way. First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the $C^{1,\alpha }$-regularity of the ``flat’’ part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity.