ABSTRACT

In this article, we study a certain type of boundary behaviour of positive solutions of the heat equation on the Euclidean upper half-space of ${\mathbb{R}}^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.

中文翻译：

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关于热方程正解的抛物线收敛

摘要

在本文中，我们研究了热方程正解在欧氏上半空间上的某种边界行为 ${\mathbb{[R}}^{ñ+\mathrm{1个}}$。我们证明了边界上某点处热方程正解的抛物线极限的存在等于该点处边界测度的强导数的存在。此外，抛物线极限和强导数相等。