A Fatou theorem for the solution of the heat equation at the corner points of a cylinder

Abstract

In this paper the author proves existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain D × ( 0 , ∞ ) D \times (0,\infty ) where D D is a bounded smooth domain in R n {R^n} with zero lateral values. A unique representation of the strong solution is given in terms of measures μ \mu on D D and λ \lambda on ∂ D \partial D . We also show that the strong solution u ( x , t ) u(x,t) of the heat equation in a cylinder converges a.e. x 0 ∈ ∂ D × { 0 } {x_0} \in \partial D \times \{ 0\} as ( x , t ) (x,t) converges to points on ∂ D × { 0 } \partial D \times \{ 0\} along certain nontangential paths.