On a singular integral equation with Weierstrass kernel

Abstract

The singular integral equation

where L ₀₀ is the union of smooth arcs, ζ(t − t ₀) is the Weierstrass ‘ζ-function’, f(t ₀) is given and ϕ(t) is an unknown function, is considered. It is proved that the solution of the equation in a Hölder class always exists. The effective solutions are obtained by means of boundary value problems for sectionally holomorphic doubly quasi-periodic functions in latticed domains.

Keywords:

• singular integrals
• boundary value problems
• doubly quasi-periodic functions

AMS Subject Classifications:

• 30E20
• 30E25
• 45E05

Acknowledgements

The author is grateful to Professors Niko Avazashvili, George Jaiani, Vakhtang Paatashvili, Dazmir Shulaia and Nelli Kilasonia for helpful comments in the early version of this article during the seminars. The author also express thanks to Ms Izo Chichua for helping with the preparation of the electronic version of this article.

Notes

Note

1. To the memory of my teacher Andria Bitsadze.