Functional Analysis and Applications Seminar 

08/03/2002, 15:00 — 16:00 — Room P3.10, Mathematics Building
Vakhtang Kokilashvili, Razmadze Mathematical Institute, Academy of Sciences, Tbilisi, Georgia

The Boundary Value Problems for Analytic and Harmonic Functions

The Dirichlet and Neumann problems for harmonic functions from the Smirnov type classes in domains with arbitrary piecewise smooth boundaries will be discussed. The picture of solvability is described completely; the non-Fredholm cases are exposed; an influence of geometric properties of boundaries on the solvability is revealed; a criterion for the unique solvability of this problem is established for arbitrary boundary values from the Lebesque spaces with exponent greater than one. Similar problems are considered in weighted Smirnov classes of harmonic functions. The weight is an arbitrary power function. In the classes of harmonic functions which are real parts of analytic functions represented in the domains by the Cauchy type integrals we investigated the Dirichlet problem with boundary functions from the weighted Zygmund classes. In all the cases of solvability there are given explicit formulas for the solution in terms of Cauchy type integrals and conformal mapping functions. The proofs are heavily based on the investigation of the linear conjugation problem with oscillating conjugation coefficient, the boundary properties of derivatives of functions which map conformally the unit circle onto a domain with an arbitrary piecewise smooth boundary and two-weighted norm inequalities for singular integrals. The talk is based on joint papers by V.Kokilashvili, V. Paatashvili and Z.Meshveliani.