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Quadrature Domains and Their Two-Phase Counterparts

  • Stephen J. GardinerEmail author
  • Tomas Sjödin
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

This survey describes recent advances on quadrature domains that were made in the context of the ESF Network on Harmonic and Complex Analysis and its Applications (2007–2012). These results concern quadrature domains, and their two-phase counterparts, for harmonic, subharmonic and analytic functions.

Keywords

Analytic function Harmonic function Partial balayage Quadrature domain Subharmonic function 

References

  1. 1.
    D.H. Armitage, S.J. Gardiner, Classical Potential Theory (Springer, London, 2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    H. Brezis, A.C. Ponce, Kato’s inequality when \(\Delta u\) is a measure. C. R. Acad. Sci. Paris Ser. I 338, 599–604 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L.A. Caffarelli, L. Karp, H. Shahgholian, Regularity of a free boundary with application to the Pompeiu problem. Ann. Math. (2) 151, 269–292 (2000)Google Scholar
  4. 4.
    B. Emamizadeh, J.V. Prajapat, H. Shahgholian, A two phase free boundary problem related to quadrature domains. Potential Anal. 34, 119–138 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S.J. Gardiner, T. Sjödin, Quadrature domains for harmonic functions. Bull. Lond. Math. Soc. 39, 586–590 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    S.J. Gardiner, T. Sjödin, Convexity and the exterior inverse problem of potential theory. Proc. Am. Math. Soc. 136, 1699–1703 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    S.J. Gardiner, T. Sjödin, Partial balayage and the exterior inverse problem of potential theory. In: Potential theory and stochastics in Albac, ed. by D. Bakry, et al. (Bucharest, Theta, 2009), pp. 111–123Google Scholar
  8. 8.
    S.J. Gardiner, T. Sjödin, Two-phase quadrature domains. J. Anal. Math. 116, 335–354 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    B. Gustafsson, M. Sakai, H.S. Shapiro, On domains in which harmonic functions satisfy generalized mean value properties. Potential Anal. 7, 467–484 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    B. Gustafsson, M. Sakai, Properties of some balayage operators, with applications to quadrature domains and moving boundary problems. Nonlinear Anal. 22, 1221–1245 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    B. Gustafsson, H.S. Shapiro, What is a quadrature domain? Quadrature domains and their applications, 1–25. Oper. Theory Adv. Appl., vol. 156 (Birkhäuser, Basel, 2005)Google Scholar
  12. 12.
    Ü. Kuran, On the mean-value property of harmonic functions. Bull. Lond. Math. Soc. 4, 311–312 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Roos, Weighted Potential Theory and Partial Balayage. MSc thesis in Mathematics, KTH, 2011Google Scholar
  14. 14.
    E.B. Saff, V. Totik, Logarithmic Potentials with External Fields (Springer, Berlin, 1997)CrossRefzbMATHGoogle Scholar
  15. 15.
    M. Sakai, Regularity of a boundary having a Schwarz function. Acta Math. 166, 263–297 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Sakai, Restriction, localization and microlocalization, Quadrature domains and their applications, 195–205. Oper. Theory Adv. Appl., vol. 156 (Birkhäuser, Basel, 2005)Google Scholar
  17. 17.
    H. Shahgholian, T. Sjödin, Harmonic balls and the two-phase Schwarz function. Complex Var. Elliptic Eqn. 58, 837–852 (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    H. Shahgholian, N. Uraltseva, Regularity properties of a free boundary near contact points with the fixed boundary. Duke Math. J. 116, 1–34 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    H. Shahgholian, N. Uraltseva, G.S. Weiss, The two-phase membrane problem—regularity of the free boundaries in higher dimensions. Int. Math. Res. Not. IMRN 8(Art. ID rnm026), 16 pp (2007)Google Scholar
  20. 20.
    H. Shahgholian, G.S. Weiss, The two-phase membrane problem—an intersection-comparison approach to the regularity at branch points. Adv. Math. 205, 487–503 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    T. Sjödin, Quadrature identities and deformation of quadrature domains. Quadrature domains and Their Applications, 239–255. Oper. Theory Adv. Appl., vol. 156 (Birkhäuser, Basel, 2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity College DublinDublin 4Ireland
  2. 2.Department of MathematicsLinköping UniversityLinköpingSweden

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