A Fatou Theorem for the Solution of the Heat Equation at the Corner Points of a Cylinder
Kin Ming Hui · Transactions of the American Mathematical Society · 1992
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.