论文标题
泊松方程的几乎定期解决方案的规律性
Regularity of almost periodic solutions of Poisson's equation
论文作者
论文摘要
本文讨论了$ \ mathbb {r}^n $的泊松方程$-ΔU= f $几乎定期解决方案的一些规律性，其中$ f $几乎是一个周期性的功能。 Sibuya [几乎是Poisson方程的解决方案证明了这一点。 Proc。阿米尔。数学。 Soc。，28：195--198，1971。在这项工作中，我们在分布意义上放宽了通常的界限到界限的假设，我们称为有限的广义函数。有限的广义函数的集合比通常的有限函数集更宽。然后，在假设$ u $是一个有界的广义函数并从分布意义上求解泊松方程后，我们证明该解决方案在通常的意义上，连续且几乎是周期性的。此外，我们表明，解决方案$ \ partial u/ \ partial x_i $，$ i = 1，\ ldots，n $的第一个部分衍生物也是连续的，有限的，几乎是周期性的功能。该技术基于在分布意义上使用Green函数用于解决方案的poisson方程的表示公式。还显示了一些有用的分布特性，可用于研究其他椭圆问题。
This paper discusses some regularity of almost periodic solutions of the Poisson's equation $-Δu = f$ in $\mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poisson's equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poisson's equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poisson's equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $\partial u/ \partial x_i$, $i=1, \ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Green's function for Poisson's equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.
