Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-11T22:49:01.675Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic properties of unbounded quadrature domains the plane

Published online by Cambridge University Press:  07 January 2015

LAVI KARP
LAVI KARP*
Affiliation:
Department of Mathematics, ORT Braude College, P.O. Box 78, 21982 Karmiel, Israel email: karp@braude.ac.il
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that if Ω is a simply connected quadrature domain (QD) of a distribution with compact support and the point of infinity belongs to the boundary, then the boundary has an asymptotic curve that is a straight line, parabola or infinite ray. In other words, such QDs in the plane are perturbations of null QDs.

Keywords

quadrature domainsasymptotic curveCauchy transformcontact surfacesnull quadrature domainsconformal mappingfree boundaries
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 2 , April 2015 , pp. 175 - 191
Copyright
Copyright © Cambridge University Press 2015 

References

[1]Aharonov, D. & Shapiro, H. S. (1976) Domains on which analytic functions satisfy quadrature identities. J. Anal. Math. 30, 39–73.Google Scholar
[2]Bers, L. (1965) An approximation theorem. J. Anal. Math. 14, 1–4.Google Scholar
[3]Davis, P. (1974) The Schwarz Function and Its Application, Carus Mathematical Monographs, Vol. 14, The Mathematical Association of America, Washington DC, USA.CrossRefGoogle Scholar
[4]Fedorova, N. V. & Tsirulskiy, A. V. (1976) The solvability in finite form of the inverse logarithmic potential problem for contact surface. Phys. Solid Earth 12, 660–665 (translated from Russian).Google Scholar
[5]Gardiner, S. J. & Sjödin, T. (2008) Convexity and the exterior inverse problem of potential theory. Proc. Am. Math. Soc. 136 (5), 1699–1703.Google Scholar
[6]Gustafsson, B. (1983) Quadrature identities and the Schottky double. Acta Appl. Math. 1, 209–240.Google Scholar
[7]Gustafsson, B. (1990) On quadrature domains and on an inverse problem in potential theory. J. Anal. Math. 44, 172–215.CrossRefGoogle Scholar
[8]Hayman, W. K., Karp, L. & Shapiro, H. S. (2000) Newtonian capacity and quasi balayage. Rend. Mat. Appl. 20 (7), 93–129.Google Scholar
[9]Isakov, V. (1993) Inverse Source Problems, Surveys and Monographs, Vol. 34, American Mathematical Society, Providence, RI.Google Scholar
[10]Ivanov, V. K. (1956) On the solvability of the inverse problem of the logarithmic potential in finite terms. Dokl. SSSR Acad. Sci. Math. 104 (4), 598–599 (in Russian).Google Scholar
[11]Karp, L. & Margulis, A. S. (1996) Newtonian potential theory for unbounded sources and applications to free boundary problems. J. Anal. Math. 70, 1–63.Google Scholar
[12]Karp, L. & Shahgholian, H. (2000) Regularity of a free boundary problem near the infinity point. Commun. Partial Differ. Equ. 25 (11–12), 2055–2086.Google Scholar
[13]Lee, S. Y. & Makarov, N. G. (2013) Topology of quadrature domains, arXiv:1307.0487 [math.CV].Google Scholar
[14]Margulis, A. S. (1995) The moving boundary problem of potential theory. Adv. Math. Sci. Appl. 5 (2), 603–629.Google Scholar
[15]Novikov, P. S. (1938) On the inverse problem of potential. Dokl. Akad. Nauk SSSR 18, 165–168 (translated from Russian).Google Scholar
[16]Sakai, M. (1981) Null quadrature domains. J. Anal. Math. 40, 144–154.Google Scholar
[17]Sakai, M. (1982) Quadrature Domains, Lecture Notes in Mathematics, Vol. 934, Springer-Verlag, Berlin–Heidelberg–New York.Google Scholar
[18]Sakai, M. (1991) Regularity of a boundary having a schwarz function. Acta Math. 166, 263–297.CrossRefGoogle Scholar
[19]Sakai, M. (1993) Regularity of boundaries of quadrature domains in two dimensions. SIAM J. Math. Anal. 24, 341–364.Google Scholar
[20]Sakai, M. (2009) Quadrature domains with infinite volume. Complex Anal. Operator Theory 3 (2), 525–549.Google Scholar
[21]Shapiro, H. S. (1987) Unbounded quadrature domains. In: Berenstein, C. (editor), Complex Analysis I, Lecture Notes in Mathematics, Vol. 1275, Springer-Verlag, 1987, pp. 287–331.Google Scholar
[22]Shapiro, H. S. (1992) The Schwarz Function and Its Generalization to Higher Dimensions, Arkansas Lecture Notes in the Mathematical Sciences, Vol. 9, John Wily & Sons Inc., New York.Google Scholar
[23]Strakhov, V. N. (1974a) The inverse logarithmic potential problem for contact surface. Phys. Solid Earth 10, 104–114, (translated from Russian).Google Scholar
[24]Strakhov, V. N. (1974b) The inverse problem of the logarithmic potential for contact surface. Phys. Solid Earth 10, 369–379 (translated from Russian).Google Scholar
[25]Tsirulskiy, A. V. (1963) Some properties of the complex logarithmic potential of a homogeneous region. Bull. (Izv.) Acad. Sci. USSR Geophys. Ser. 7, 653–655, (Russian).Google Scholar
[26]Zalcman, L. (1987) Some inverse problems of potential theory. Integral Geometry, Contemp. Math., Vol. 63, American Mathematical Society, Providence, RI, pp. 337–350.CrossRefGoogle Scholar