Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary

  • We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.

Download full text files

Export metadata

Additional Services

Share in Twitter Search Google Scholar Statistics
Author:Serguey Grudsky, Nikolai Tarkhanov
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Potsdam (1(2012)10)
Document Type:Preprint
Year of Completion:2012
Publishing Institution:Universität Potsdam
Release Date:2012/01/18
Tag:boundary value problems; nonsmooth curves; singular integral equations
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC Classification:30-XX FUNCTIONS OF A COMPLEX VARIABLE (For analysis on manifolds, see 58-XX) / 30Exx Miscellaneous topics of analysis in the complex domain / 30E25 Boundary value problems [See also 45Exx]
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Qxx Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] / 35Q15 Riemann-Hilbert problems [See also 30E25, 31A25, 31B20]
45-XX INTEGRAL EQUATIONS / 45Exx Singular integral equations [See also 30E20, 30E25, 44A15, 44A35] / 45E05 Integral equations with kernels of Cauchy type [See also 35J15]
Collections:Universität Potsdam / Schriftenreihen / Preprints des Instituts für Mathematik der Universität Potsdam, ISSN 2193-6943 / 2012
Licence (German):License LogoKeine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Notes extern:RVK-Notation: SI 990 , SK 540