On Boundary Value Problems for an Improperly Elliptic Equation in a Circle

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.