The Riemann and Dirichlet Problems with Data from the Grand Lebesgue Spaces

Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)

Abstract

In Section 1, we present a solution of the following boundary value problem: find an analytic function Φon the plane cut along a closed piecewisesmooth curve Γ which is represented by a Cauchy type integral with a density from the Grand Lebesgue Space \( L^{p)},\theta(\Gamma)(1 < p < \infty, 0 < \theta < \infty) \) and whose boundary values satisfy the conjugacy condition
$$ \Phi^{+}(t)=G(t)\Phi^{-}(t)+g(t),\quad t \in \Gamma $$
Here G and g are functions defined on Γ such that G is a piecewise continuous function, \( G(t)\neq 0 \) and \( g \in L^{{p}),\theta}(\Gamma) \) The conditions for the problem to be solvable are established and the solutions are constructed in explicit form.

In Section 2, the Dirichlet problem for harmonic functions, real parts of Cauchy type integrals with densities from weighted generalized Grand Lebesgue Spaces is studied when boundary data belong to the same space.

Keywords

Boundary value problem analytic function Cauchy type integral the Grand Lebesgue Space piecewise smooth boundary piecewise continuous coefficient Dirichlet problem Lyapunov contour. 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.A. Razmadze Mathematical Institute I. Javakhishvili Tbilisi State UniversityTbilisiGeorgia

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