Skip to main content
SpringerLink
Log in

Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

It is known that corners of interior angle less than π/2 in the boundary of a plane domain are initially stationary for Hele–Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.

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.

Similar content being viewed by others

References

  1. Aikawa H.: On the minimal thinness in a Lipschitz domain. Analysis 5, 347–382 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer, London, 2001

  3. Bourgain J.: On the Hausdorff dimension of harmonic measure in higher dimension. Invent. Math. 87, 477–483 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Brezis H., Ponce A.C.: Kato’s inequality when Δu is a measure. C. R. Acad. Sci. Paris Ser. I 338, 599–604 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York, 1984

  6. Friedland S., Hayman W.K.: Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv. 51, 133–161 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gardiner, S.J., Sjödin, T.: Partial balayage and the exterior inverse problem of potential theory. Potential Theory and Stochastics in Albac (Eds. Bakry, D., et al.) Bucharest, Theta, 111–123, 2009

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1983

  10. Gustafsson, B.: Variational inequality formulation of a moving boundary problem in rock mechanics. TRITA-MAT-1984-8, Department of Mathematics, KTH Stockholm, 1984

  11. Gustafsson B., Sakai M.: Properties of some balayage operators, with applications to quadrature domains and moving boundary problems. Nonlinear Anal. 22, 1221–1245 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hansen W., Hueber H.: Singularity of harmonic measure for sub-Laplacians. Bull. Sci. Math. 112(2), 53–64 (1988)

    MATH  MathSciNet  Google Scholar 

  13. Huber A.: Über Wachstumseigenschaften gewisser Klassen von subharmonischen Funktionen. Comment. Math. Helv. 26, 81–116 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  14. King J.R., Lacey A.A., Vázquez J.L.: Persistence of corners in free boundaries in Hele–Shaw flow. Eur. J. Appl. Math. 6, 455–490 (1995)

    MATH  Google Scholar 

  15. Kuran Ü.: On NTA-conical domains. J. Lond. Math. Soc. 40(2), 467–475 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Miyamoto I., Yanagishita M., Yoshida H.: Beurling–Dahlberg–Sjögren type theorems for minimally thin sets in a cone. Can. Math. Bull. 46, 252–264 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Miyamoto I., Yanagishita M., Yoshida H.: On harmonic majorization of the Martin function at infinity in a cone. Czechoslov. Math. J. 55(130), 1041–1054 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sakai M.: Applications of variational inequalities to the existence theorem on quadrature domains. Trans. Am. Math. Soc. 276, 267–279 (1983)

    Article  MATH  Google Scholar 

  19. Sakai M.: Sharp estimates of the distance from a fixed point to the frontier of a Hele–Shaw flow. Potential Anal. 8, 277–302 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sakai, M.: Small modifications of quadrature domains. Mem. Am. Math. Soc. 206(969) (2010)

  21. Widman K.-O.: Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand. 21, 17–37 (1967)

    MATH  MathSciNet  Google Scholar 

Download references

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomas Sjödin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen J. Gardiner.

Additional information

Communicated by G. Dal Maso

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gardiner, S.J., Sjödin, T. Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions. Arch Rational Mech Anal 213, 503–526 (2014). https://doi.org/10.1007/s00205-014-0750-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-014-0750-0

Keywords

Search

Navigation