Journal of Mathematical Sciences, Vol. 115, No. 6, 2003 BOUNDARY ESTIMATES FOR SOLUTIONS OF THE PARABOLIC FREE BOUNDARY PROBLEM D. E. Apushkinskaya, H. Shahgholian, and N. N. Uraltseva UDC 517.9 Let u and Ω solve the problem H(u)= χΩ, u = |Du| =0 in Q + 1 \ Ω, u =0 on Π ∩ Q1, where Ω is an open set in R n+1 + = {(x, t): x ∈ R n ,t ∈ R 1 ,x1 > 0},n  2, H =Δ - ∂t is the heat operator, χ Ω denotes the characteristic function of Ω, Q 1 is the unit cylinder in R n+1 , Q + 1 = Q1 ∩ R n+1 + , Π= {(x, t): x1 =0}, and the first equation is satisfied in the sense of distributions. We obtain the optimal regularity of the function u, i.e., we show that u ∈ C 1,1 x ∩ C 0,1 t . Bibliography: 6 titles. Dedicated to the memory of A. V. Ivanov The present paper deals with regularity properties of solutions of a certain type of parabolic free boundary problems. Mathematically, the problem is formulated as follows. Let Ω be an open set in R n+1 + and assume that locally there exists a function u such that H(u)= χ Ω in Q + 1 , u = |Du| =0 in Q + 1 \ Ω, u =0 on Π ∩ Q 1 , (1.1) the first equation in (1.1) is understood in the sense of distributions. Results concerning the elliptic problem related to (1.1) can be found in [6]. Also, the recent article [3] should be mentioned, where the parabolic problem (1.1) was considered in the whole unit cylinder, without restrictions on the boundary Π. Paper [2] deals with the elliptic free boundary problem in the unit ball, without conditions on Π. In [6, 3, 2], regularity properties of the free boundary ∂Ω were studied in addition to the optimal regularity of solutions. Notation. Throughout the paper we use the following notation: z =(x,t) is a point in R n+1 , where x =(x 1 ,x ′ )=(x 1 ,x 2 ,...,x n ) ∈ R n , n  2 and t ∈ R 1 ; R n+1 + = {(x,t) ∈ R n+1 : x 1 > 0}; Π= {(x,t) ∈ R n+1 : x 1 =0}; χ Ω denotes the characteristic function of the set Ω (Ω ⊂ R n+1 ); B r (x) is an open ball in R n with center x and radius r; B r = B r (0); Q r (z )= Q r (x,t)= B r (x,t)×]t − r 2 ,t] denotes an open cylinder in R n+1 ; ∂ ′ Q r (x,t) is the parabolic boundary, i.e., the topological boundary minus the top of the cylinder; Q + r (z )= Q + r (x,t)= Q r (x,t) ∩ R n+1 + ; Q r = Q r (0, 0); Q + r = Q + r (0, 0). The indices i and j always vary from 1 to n, whereas the indices τ and μ vary from 2 to n. Repeated indices imply summation, for example, a ij x i x j = ∑ n i,j =1 a ij x i x j . D i denotes the differential operator with respect to x i ; ∂ t = ∂ ∂t ; D =(D 1 ,D ′ )=(D 1 ,D 2 ,...,D n ) denotes the spatial gradient; H =Δ − ∂ t is the heat operator; Λ(u)= {(x,t): u(x,t)= |Du(x,t)| =0} for any C 1 x ∩ C 0 t - function u; Ω(u)= R n+1 \ Λ(u); Γ (u)= ∂Ω(u) ∩ Λ(u) is the free boundary; ‖·‖ p,Ω denotes the norm in L p (Ω), 1 <p  ∞; Published in Zapiski Nauchnykh Seminarov POMI, Vol. 271, 2000, pp. 39–55. Original article submitted October 16, 2000. 2720 1072-3374/03/1156-2720 $25.00 c  2003 Plenum Publishing Corporation