Skip to main content
SpringerLink
Log in

Remarks on an overdetermined boundary value problem

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

Abstract

We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragalà et al. (Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006).

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. Alexandrov A.D. (1962). A characteristic property of the spheres. Ann. Mat. Pura Appl. 58: 303–354

    Article  MathSciNet  Google Scholar 

  2. Brock F. and Henrot A. (2002). A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative. Rend. Circ. Mat. Palermo 51: 375–390

    Article  MATH  MathSciNet  Google Scholar 

  3. Degiovanni M., Musesti A. and Squassina M. (2003). On the regularity of solutions in the Pucci-Serrin identity. Calculus Variations 18: 317–334

    Article  MATH  MathSciNet  Google Scholar 

  4. Fragalà I., Gazzola F. and Kawohl B. (2006). Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254: 117–132

    Article  MATH  Google Scholar 

  5. Garofalo N. and Lewis J.L. (1989). A symmetry result related to some overdetermined boundary value problems. Am. J. Math. 111: 9–33

    Article  MATH  MathSciNet  Google Scholar 

  6. Kawohl, B.: Symmetrization - or how to prove symmetry of solutions to partial differential equations. In: Jäger, W., Nečas, J., John, O., Najzar, K., Stara, J. (eds.) Partial Differential Equations, Theory and Numerical Solution. Chapman & Hall CRC Research Notes in Math. 406 London, pp. 214–229 (1999)

  7. Lieberman G. (1988). Boundary regularity for solutions of degenerate elliptic equations. Nonlin. Anal. 12: 1203–1219

    Article  MATH  MathSciNet  Google Scholar 

  8. Payne L.A. and Philippin G.A. (1979). Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature. Nonlin. Anal. 3: 193–211

    Article  MATH  MathSciNet  Google Scholar 

  9. Philippin, G.A.: Applications of the maximum principle to a variety of problems involving elliptic differential equations, In: Schaefer, P.W. Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48 (1988)

  10. Pohožaev S.J. (1965). Eigenfunctions of the equation Δu + λ f(u) = 0. Sov. Math. Doklady 6: 1408–1411

    Google Scholar 

  11. Prajapat J. (1998). Serrin’s result for domains with a corner or cusp. Duke Math. J. 91: 29–31

    Article  MATH  MathSciNet  Google Scholar 

  12. Pucci P. and Serrin J. (1986). A general variational identity. Indiana Univ. Math. J. 35: 681–703

    Article  MATH  MathSciNet  Google Scholar 

  13. Rellich F. (1940). Darstellung der Eigenwerte von Δu + λ u = 0 durch ein Randintegral. Math. Zeitschr. 46: 635–636

    Article  MATH  MathSciNet  Google Scholar 

  14. Serrin J. (1971). A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43: 304–318

    Article  MATH  MathSciNet  Google Scholar 

  15. Sperb R. (1981). Maximum Principles and Applications. Academic,

    MATH  Google Scholar 

  16. Thorbergsson G. (2000). A survey on isoparametric hypersurfaces and their generalizations. In: Dillen, F.J.E. and Verstraelen, L.C.A. (eds) Handbook of differential geometry, vol I., pp 963–995. North-Holland, Amsterdam

    Chapter  Google Scholar 

  17. Weinberger H. (1971). Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal. 43: 319–320

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de Mathématiques et d’Informatique, 33, rue Saint-Leu, 80039, Amiens, France

    A. Farina

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

    B. Kawohl

Authors
  1. A. Farina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Kawohl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Farina.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Farina, A., Kawohl, B. Remarks on an overdetermined boundary value problem. Calc. Var. 31, 351–357 (2008). https://doi.org/10.1007/s00526-007-0115-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-007-0115-8

Mathematics Subject Classification (2000)

Search

Navigation