Abstract

The paper considers the solvability of the first, second, and third boundary value problems, as well as one problem with a directional derivative, in a bounded domain for a scalar improperly elliptic differential equation with complex coefficients. More detailed consideration is given to a model case in which the domain is a unit disk and the equation does not contain lower-order terms. For each of these problems, the classes of boundary data for which there exists a unique solution in the ordinary Sobolev space are characterized. In a typical case, such classes turned out to be the spaces of function with exponentially decreasing Fourier coefficients. These problems have been the subject of several previous publications of the authors, and, in this article, the earlier-obtained results have been collected together and are presented from a unified point of view.

中文翻译：

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圆上一个不正确的椭圆方程的边值问题

摘要

本文考虑了带复数系数的标量不正确椭圆型微分方程在有界域中的一阶，二阶和三阶边值问题的可解性以及一个方向导数的问题。对于模型的情况进行了更详细的考虑，其中的域是单位圆盘，等式不包含低阶项。对于这些问题中的每一个，都对在普通的Sobolev空间中存在唯一解的边界数据进行了分类。在典型情况下，此类类别被证明是傅立叶系数呈指数下降的函数空间。这些问题一直是作者先前发表过的论文的主题，在本文中，