A Sufficient Condition for the Harmonicity of Functions of Two Variables That Satisfy a Difference Laplace Equation

5th International Conference on "Problems of Mathematical and Theoretical Physics and Mathematical Modelling"

物理学;数学

Telyakovskii, D.S.^1

National Research Nuclear University, MEPhI (Moscow Engineering Physics Institute), 31 Kashirskoe shosse, Moscow

115409, Russia^1

关键词: Absolute values;    Boundness;    Continuity conditions;    Continuous functions;    Harmonicity;    Partial derivatives;    Summability;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/788/1/012059/pdf
DOI  :  10.1088/1742-6596/788/1/012059

来源: IOP

【 摘 要 】

In this paper we weaken the sufficient conditions of harmonicity for functions of two variables. I.I. Privalov had shown that a continuous function that satisfies the Laplace equation in each point of the domain is harmonic. For function of two variables the Privalov's condition on continuity can be weakened. G.P. Tolstov replaced the continuity condition by the boundness condition, later the author had shown that summability is sufficient. At the same time summability condition can not be weakened substantially. In this paper, while we keep the summability condition, we provide the sufficient condition for harmonicity of the functions, that satisfy less restricted condition than the Laplace equation in all points of the domain. We assume that arbitrary close to any point ζ there exists a collection of four nodes for which a difference relation of Schwartz type for the Laplace equation can be made arbitrary small by the absolute value. Nodes are the ends of two mutually perpendicular segments, that intersect at point ζ. We need to impose a certain weakened continuous assumption on function itself, in case of function that satisfy the traditional Laplace condition this continuity condition follows from the existence of the partial derivatives.