transactions of the american mathematical society Volume 333, Number 2, October 1992 A FATOU THEOREM FOR THE SOLUTION OF THE HEAT EQUATION AT THE CORNER POINTS OF A CYLINDER KIN MING HUI Abstract. In this paper the author proves existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain D x (0, oo) where D is a bounded smooth domain in Rn with zero lateral values. A unique representation of the strong solution is given in terms of measures p. on D and k on dD. We also show that the strong solution u(x, t) of the heat equation in a cylinder converges a.e. xq e dD x {0} as (x, t) converges to points on 3D x {0} along certain nontangential paths. Introduction The existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain D x (0, T) subject to Dirichlet boundary conditions u\dDx^,T) = 0 where D is a bounded smooth domain in R" have been studied by a large number of researchers. (See [F, LSU, FGS].) In this paper, by following the argument of Dahlberg and Kenig [DK2], I prove the existence and uniqueness of the nonnegative strong solution of the initial Dirichlet problem (IDP) for the heat equation in a cylinder D x (0, co), D £ C°° , with Dirichlet boundary condition «aox(o,oo) = 0 • In fact I show that corresponding to each nonnegative strong solution u(x, t) of IDP, there exists a pair of measures p on D and X on dD such that u(x,t)= [ G(X,t;Q,0)dp(Q) Jd + [ -^-(x,t;Q,0)dX(Q) JdD OHq where G(x, t; Q, s) is the Green function for the heat equation and d/dNQ is the derivative in the direction of the inward normal at Q. I also find that the strong solution u(x, t) of the heat equation in a cylinder converges a.e. xo £ dD x {0} as (x, t) converges to points on dD x {0} along certain nontangential path. In fact I prove that Ñ A1 lim u(x, t) = -j= • -j- a.e. x0 £ dD x {0} (x,/)er>(*o) v47t da _ Í-.0 Received by the editors June 6, 1990. 1980 Mathematics Subject Classification (1985Revision).Primary 35K05,35K15,35K20,35C15, 35D05;Secondary 31B10,31B25. Key words and phrases. Heat equation, initial-Dirichlet problem, Fatou theorem at corner points. ©1992 American Mathematical Society 0002-9947/92 $1.00 + $.25 per page 607 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use