Continuous differentiability of the free boundary for weak solutions of the Stefan problem

We announce a result concerning the continuous differentiability of the unknown boundary curve defined by a weak solution of the one-dimensional two-phase Stefan problem. We deal with the following two-phase Stefan problem: to determine u(x, t) for 0<:t<:T, 0<:X<:1 and s(t) for 0^t<±Tsuch that (i) 0<s(t)< 1, s(0)=b; (ii) u^pjUn for 0<t^T9 0<x<s(t) and ut==p2uxx for 0<t^T, s(t)<x<\; (iii) w(0, 0= / i (0>0 and u(l,t)=f2(t)<0 for O^t^T; (iv) u(x, 0)=Y>(JC) for 0 ^ x < l ; (v) u(s(t)91)=0 for 0<st<>T; and (vi) ai(0= —ua(s(t)—09 t)+ux(s(t)+0, t) for 0<t<^T. Here and throughout, ft and a are positive parameters, b e (0, 1), f{ and xp are continuous functions with/i(0)=y>(0),/2(0)=^(1), ¥>(ô)=0, y>(x)>0 for 0<^x<b, y>(x)<0 for b<x£l9 and \ip(x)\£K\b-x\ for O^x^ l . Cannon and Primicerio [3], following the work of Cannon, Douglas and Hill [2] showed that this problem has a unique classical solution (one for which the expressions appearing in (vi) are defined and continuous for 0 < / ^ r ) on condition that the ft and y> are bounded by certain constants which depend on the parameters of the problem. A. Friedman [4], AMS (MOS) subject classifications (1970). Primary 35K60.