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Quadrature Domains and Their Two-Phase Counterparts

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Harmonic and Complex Analysis and its Applications

Part of the book series: Trends in Mathematics ((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.

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References

  1. D.H. Armitage, S.J. Gardiner, Classical Potential Theory (Springer, London, 2001)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. B. Emamizadeh, J.V. Prajapat, H. Shahgholian, A two phase free boundary problem related to quadrature domains. Potential Anal. 34, 119–138 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. S.J. Gardiner, T. Sjödin, Quadrature domains for harmonic functions. Bull. Lond. Math. Soc. 39, 586–590 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. S.J. Gardiner, T. Sjödin, Convexity and the exterior inverse problem of potential theory. Proc. Am. Math. Soc. 136, 1699–1703 (2008)

    Article  MATH  Google Scholar 

  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. 111–123

    Google Scholar 

  8. S.J. Gardiner, T. Sjödin, Two-phase quadrature domains. J. Anal. Math. 116, 335–354 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Ü. Kuran, On the mean-value property of harmonic functions. Bull. Lond. Math. Soc. 4, 311–312 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Roos, Weighted Potential Theory and Partial Balayage. MSc thesis in Mathematics, KTH, 2011

    Google Scholar 

  14. E.B. Saff, V. Totik, Logarithmic Potentials with External Fields (Springer, Berlin, 1997)

    Book  MATH  Google Scholar 

  15. M. Sakai, Regularity of a boundary having a Schwarz function. Acta Math. 166, 263–297 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  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. H. Shahgholian, T. Sjödin, Harmonic balls and the two-phase Schwarz function. Complex Var. Elliptic Eqn. 58, 837–852 (2013)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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 

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

    Stephen J. Gardiner

  2. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Tomas Sjödin

Authors
  1. Stephen J. Gardiner
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  2. Tomas Sjödin
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Corresponding author

Correspondence to Stephen J. Gardiner .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Alexander Vasil'ev

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Gardiner, S.J., Sjödin, T. (2014). Quadrature Domains and Their Two-Phase Counterparts. In: Vasil'ev, A. (eds) Harmonic and Complex Analysis and its Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-01806-5_5

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