Skip to main content

Advertisement

SpringerLink
Log in

An isoperimetric inequality related to a Bernoulli problem

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

Given a bounded domain Ω we look at the minimal parameter Λ(Ω) for which a Bernoulli free boundary value problem for the p-Laplacian has a solution minimising an energy functional. We show that amongst all domains of equal volume Λ(Ω) is minimal for the ball. Moreover, we show that the inequality is sharp with essentially only the ball minimising Λ(Ω). This resolves a problem related to a question asked in Flucher et al. (Reine Angew Math 486:165–204, 1997).

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. Alt H.W., Caffarelli L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)

    MATH  MathSciNet  Google Scholar 

  2. Beurling, A.: On free-boundary value problems for the Laplace equation. Seminars on Analytic Functions (Princeton 1957), vol. I, pp. 248–263. Institute for Advanced Study, Princeton (1958)

  3. Brothers J.E., Ziemer W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)

    MATH  MathSciNet  Google Scholar 

  4. Danielli D., Petrosyan A.: A minimum problem with free boundary for a degenerate quasilinear operator. Calc. Var. Partial Differ. Equ. 23, 97–124 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Flucher M., Rumpf M.: Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486, 165–204 (1997)

    MATH  MathSciNet  Google Scholar 

  6. Frehse J.: Capacity methods in the theory of partial differential equations. Jahresber. Deutsch. Math. -Verein. 84, 1–44 (1982)

    MATH  MathSciNet  Google Scholar 

  7. Friedrichs K.: Über ein Minimumproblem für Potentialströmungen mit freiem Rande. Math. Ann. 109, 60–82 (1934)

    Article  MathSciNet  Google Scholar 

  8. Heinonen J., Kilpeläinen T., Martio O.: Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs, Clarendon Press, New York (1993)

    MATH  Google Scholar 

  9. Henrot A., Shahgholian H.: Existence of classical solutions to a free boundary problem for the p-Laplace operator. II. The interior convex case. Indiana Univ. Math. J. 49, 311–323 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kawohl B.: Rearrangements and convexity of level sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer-Verlag, Berlin (1985)

    Google Scholar 

  11. Ly I., Seck D.: Isoperimetric inequality for an interior free boundary problem with p-Laplacian operator. Electron. J. Differ. Equ. 2004(109), 1–12 (2004)

    MathSciNet  Google Scholar 

  12. Martínez S., Wolanski N.: A minimum problem with free boundary in Orlicz spaces. Adv. Math. 218, 1914– (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976). doi:10.1007/BF02418013

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, 2006, Australia

    Daniel Daners

  2. Mathematisches Institut, Universität zu Köln, 50923, Köln, Germany

    Bernd Kawohl

Authors
  1. Daniel Daners
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernd Kawohl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Daners.

Additional information

Communicated by L. Caffarelli.

Dedicated to Giorgio Talenti on the occasion of his 70th birthday

This research is part of the ESF program “Global and geometric aspects of nonlinear partial differential equations (GLOBAL)”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Daners, D., Kawohl, B. An isoperimetric inequality related to a Bernoulli problem. Calc. Var. 39, 547–555 (2010). https://doi.org/10.1007/s00526-010-0324-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-010-0324-4

Mathematics Subject Classification (2000)

Search

Navigation