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 regula rity result of Caffarelli and Friedman for the zero-set of functions satisf ying a bound \Delta v\ less than or equal to C \v\ to functions only satisf ying a bound of the form \Delta v\ less than or equal to C (v) over cap, wh ere (v) over cap(x) := supp(0 less than or equal to t less than or equal to 1)\v(tx)\ is the radially maximal function. In addition we extend the uniq ue continuation theorem of Aronszajn and Cordes to cover this case. (C) 199 9 Academic Press.