Skip to main content
SpringerLink
Log in

On Null Quadrature Domains

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

The characterization of null quadrature domains in R n, n ≥ 3, has been an open problem throughout the past two and a half decades. A substantial contribution was made by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellipsoids. The first result with unbounded complements appeared in [16], there it is assumed the complement is contained in an infinitely cylinder.

The aim of this paper is to show the relation between null quadrature domains and Newton’s theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a slab and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a slab. In addition, we present a Phragmén-Lindelöf type theorem which we could not find in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Aikawa and M.R. Essèn, Potential Theory: Selected Topics, Lecture Notes in Mathematics 1633, Springer-Velag, Berlin-Heidelberg-New York, 1996.

    MATH  Google Scholar 

  2. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, Second Edition, New York, 1989.

    Book  Google Scholar 

  3. E. DiBenedetto and A. Friedman, Bubble growth in porous media, Indiana Univ. Math. J. 35 (1986), 5730–5606.

    MathSciNet  Google Scholar 

  4. L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Diff. Eq. 5 (1980), 427–448.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. A. Caffarelli, L. Karp and H. Shahgholian, Regularity of a free boundary problem with application to the Pompeiu problem, Ann. Math. 151 (2000), 269–292.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Dover, 1987.

  7. P. Dive, Attraction des ellipsoides homogénes et rèciproque d’um théorème de Newton, Bull Soc. Math. France 59 (1931), 128–140.

    MathSciNet  Google Scholar 

  8. V. M. Entov and P. I. Etingof, Bubble contraction in Hele-Shaw cells, Quart. J. Mech. Appl. Math. 44 (1991) no.4, 507–535.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Essèn, K. Haliste, J. L. Lewis, and D. F. Shea, Harmonic majorization and classical analysis, J. London Math. Soc. (2) 32 (1985) no.3, 506–520.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Friedman and M. Sakai, A characterization of null quadrature domains in ℝn, Indiana Univ. Math. J. 35 (1986), 607–610.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. K. Hayman, Subharmonic Functions, Volume 2, Academic Press, 1989.

  12. S. D. Howison, Bubble growth in porous media and Hele Shaw flow, Proc. Roy. Soc. Edinburgh A102 (1985), 141–148.

    MathSciNet  Google Scholar 

  13. E. Hölder, Über eine potentialtheoretische Eigenschaft der Ellipse, Math. Z. 35 (1932), 632–643.

    Article  MathSciNet  Google Scholar 

  14. L. Karp, Generalized Newtonian potential and its applications, J. Math. Anal. Appl. 174 (1993), 480–497.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Karp, On the Newtonian potential of ellipsoids, Complex Variables 25 (1995), 367–372.

    MathSciNet  Google Scholar 

  16. L. Karp and A. Margulis, Newtonian potential theory for unbounded sources and applications to free boundary problems, J. Analyse Math. 70 (1996), 1–63.

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Karp and H. Shahgholian, Regularity of a free boundary problem near the infinity point, Comm. Partial Differential Equations 25 (2000) no.11-12, 2055-2086.

    Article  MathSciNet  MATH  Google Scholar 

  18. O. D. Kellog, Fundation of Potential Theory, Reprint from the First Edition 1929, Springer-Verlag, Berlin-Heidelberg-New York, 1967.

    Google Scholar 

  19. D. Khavinson, Holomorphic Partial Differential Equations and Classical Potential Theory, Departamento de Analisis Matematico, Universidad de La Laguna, Tenerife, 1996.

    Google Scholar 

  20. V. A. Kondrat’ev and E. M. Landis, Qualitive theory of second order linear partial differential equations, in: Yu. V. Egorov and M.A. Shubin (eds.), Partial Differential Equations III, Springer-Verlag, Berlin 1991.

    Google Scholar 

  21. N. S. Landkof, Fundation of Modern Potential Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

    Book  Google Scholar 

  22. V. G. Maz’ya, On the regularity on the bounday of solutions of elliptic equations, and of a conformal mapping, Dokl. Akad. Nauk SSSR 152 (1963), 1297–1300; English transl.: Sov. Math. Dokl. 4 (1964), 1547–1551.

    MathSciNet  Google Scholar 

  23. W. Nikliborc, Eine Bemerkung über die Volumpotentiale, Math. Z. 35 (1932), 625–631.

    Article  MathSciNet  Google Scholar 

  24. M. Sakai, Null quadrature domains, J. Analyse Math. 40 (1981), 144–154.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Sakai, Regularity of boundaries of quadrature domains in two dimensions, SIAM J. Math. Anal. 24 (1994), 341–364.

    Article  Google Scholar 

  26. H. Shahgholian, Unbounded quadrature domains in Rn (nt ≥3), J. Analyse Math. 57 (1991), 281–291.

    Article  Google Scholar 

  27. H. Shahgholian, On quadrature domains and the Schwarz potential, J. Math. Anal. Appl. 177 (1992), 61–78.

    Article  MathSciNet  Google Scholar 

  28. H. S. Shapiro, A weighted L1 estimate for the Laplace operator, in: Recent Advance in Fourier Analysis atnd its Applications (Il Ciocco 1989), Kluwer Acad Publ., Dordrecht, 1990, 563–577.

    Chapter  Google Scholar 

  29. H. S. Shapiro, Quasi-balayage and a priori estimates for the Laplace operator, Multivariate Approximation (Witten-Bommerholz 1996), 203–230, 231–254, Math. Res. 101, Akademie Verlag, Berlin, 1997.

    Google Scholar 

  30. A. N. Varchenko, P. I. Etingof, Why the Boundary of at Round Drop Becomes a Curve otf Order Four, American Mathematical Society University Lecture Series, Volume 3, Providence, Rhode Island 1992.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, ORT Braude College, P.O. Box 78, 21982, Karmiel, Israel

    Lavi Karp

Authors
  1. Lavi Karp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lavi Karp.

Additional information

Dedicated to Walter K. Hayman on his 80th Birthday

Research partially supported by DFG: 446ISR-112/3/06.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karp, L. On Null Quadrature Domains. Comput. Methods Funct. Theory 8, 57–72 (2008). https://doi.org/10.1007/BF03321670

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03321670

Keywords

2000 MSC

Search

Navigation