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.

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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