The Stationary Mean Field Model of Superconductivity: Partial Regularity of the Free Boundary

We prove a partial regularity result for a free boundary problem arising in the mean field theory of superconductivity. To this end we extend a regularity result of Caffarelli and Friedman for the zero-set of functions satisfying a bound Δv C v to functions only satisfying a bound of the form Δv Cv, where v(x) supp0 t 1 v(tx) is the radially maximal function. In addition we extend the unique continuation theorem of Aronszajn and Cordes to cover this case.