Equilibrium points of a singular cooperative system with free boundary

We study maps minimising the energy $${\int }_D(|\mathrm{\nabla }\mathbf{u}{|}^2+2|\mathbf{u}|)dx,$$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$\mathrm{\Delta }\mathbf{u}=\frac{\mathbf{u}}{|\mathbf{u}|}{\chi }_{\{|\mathbf{u}|>0\}},\mathbf{u}=(u_1,\cdots ,u_m).$$ Our results here concern regularity of the solution as well as that of the free boundary. The main ingredient consists in an epiperimetric inequality.

(The result is a joint work with J. Andersson, H. Shahgholian and N. Uraltseva, accepted for publication in Advances in Mathematics)