Equilibrium points of a singular cooperative system with free boundary

We study maps minimising the energy $$ \int_{D} (|\nabla {\mathbf u}|^2+2|{\mathbf u}|)\ dx, $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ \Delta {\mathbf u}=\frac{{\mathbf u}}{|{\mathbf u}|}\chi_{\left\lbrace |{\mathbf u}|>0\right\rbrace }, \qquad {\mathbf u} = (u_1, \cdots , u_m) \ . $$ 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)