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.
