Jump boundary-value problem on a contour with elongate singularities
The problem under consideration is the following: to find two analytic functions knowing their difference on a given contour. This problem is well known and has numerous applications. In the case of sufficiently smooth contours, it is investigated in the classical books devoted to the theory of linear conjugation for analytic functions and the theory of singular integral equations. In the case of contours from wider classes, the investigation needs essential additional efforts. The paper under review continues the work of the authors [Russ. Math. 59, No. 2, 57--61 (2015); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2015, No. 2, 70--75 (2015; Zbl 1327.30043)]. Let \(\Gamma\) be the image of an injective mapping \(\varphi: (0,1)\to \mathbb C\). If the set \(\overline{\Gamma}\setminus\Gamma\) has more than two points, then \(\Gamma\) is called a contour with elongate singularities. For such contours, under some additional assumptions and for some class of functions, solvability and uniqueness of the problem is proved
scientific article; zbMATH DE number 6838071
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