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

Hui, Kin Ming

doi:10.1090/S0002-9947-1992-1091707-4

A Fatou theorem for the solution of the heat equation at the corner points of a cylinder
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Trans. Amer. Math. Soc. 333 (1992), 607-642 Request permission

Abstract:

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

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