Archive for Rational Mechanics and Analysis

, Volume 212, Issue 2, pp 415–443 | Cite as

Bernoulli Variational Problem and Beyond

  • Alexander Lorz
  • Peter Markowich
  • Benoît Perthame
Article
  • 271 Downloads

Abstract

The question of ‘cutting the tail’ of the solution of an elliptic equation arises naturally in several contexts and leads to a singular perturbation problem under the form of a strong cut-off. We consider both the PDE with a drift and the symmetric case where a variational problem can be stated. It is known that, in both cases, the same critical scale arises for the size of the singular perturbation. More interesting is that in both cases another critical parameter (of order one) arises that decides when the limiting behaviour is non-degenerate. We study both theoretically and numerically the values of this critical parameter and, in the symmetric case, ask if the variational solution leads to the same value as for the maximal solution of the PDE. Finally we propose a weak formulation of the limiting Bernoulli problem which incorporates both Dirichlet and Neumann boundary condition.

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References

  1. 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)MATHMathSciNetGoogle Scholar
  2. 2.
    Alt H. W., Caffarelli L. A., Friedman A.: Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282, 431–461 (1984)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Berestycki, H., Caffarelli, L. A., Nirenberg, L.: Uniform estimates for regularization of free boundary problems, in Analysis and partial differential equations, vol. 122 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 567–619, 1990Google Scholar
  4. 4.
    Braides, A.: Γ-convergence for beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford 2002Google Scholar
  5. 5.
    Caffarelli, L.A.: A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are C 1, α, Rev. Mat. Iberoamericana, vol. 3, pp. 139–162 1987Google Scholar
  6. 6.
    Caffarelli, L.A.: A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on X. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(1988), 583–602 (1989)Google Scholar
  7. 7.
    Caffarelli L.A.: A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz. Comm. Pure Appl. Math., 42, 55–78 (1989)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Caffarelli L.A., Jerison D., Kenig C. E.: Some new monotonicity theorems with applications to free boundary problems. Ann. Math. 155(2), 369–404 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Caffarelli L.A., Vázquez J.L.: A free-boundary problem for the heat equation arising in flame propagation. Trans. Am. Math. Soc. 347, 411–441 (1995)CrossRefMATHGoogle Scholar
  10. 10.
    Hauswirth L., Hélein F., Pacard F.: On an overdetermined elliptic problem. Pacific J. Math. 250, 319–334 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Lederman C., Wolanski N.: A two phase elliptic singular perturbation problem with a forcing term. J. Math. Pures Appl. 86(9), 552–589 (2006)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Lederman C., Wolanski N.: A local monotonicity formula for an inhomogeneous singular perturbation problem and applications. Ann. Mat. Pura Appl. 187(4), 197–220 (2008)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Lederman C., Wolanski N.: A local monotonicity formula for an inhomogeneous singular perturbation problem and applications. II. Ann. Mat. Pura Appl. 189(4), 25–46 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Mirrahimi S., Barles G., Perthame B., Souganidis P. E.: Singular hamilton-jacobi equation for the tail problem. SIAM J. Math. Anal. 44(6), 4297–4319 (2012)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Moreira D.R.: Least supersolution approach to regularizing free boundary problems. Arch. Ration. Mech. Anal. 191, 97–141 (2009)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Perthame B., Gauduchon M.: Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations. Math. Med. Biol. 27, 195–210 (2010)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Serrin J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)ADSCrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Teixeira E.V.: A variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 633–658 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    www.freefem.org. Accessed 23 Feb 2013

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Lorz
    • 1
    • 2
  • Peter Markowich
    • 3
  • Benoît Perthame
    • 1
    • 2
  1. 1.CNRS UMR 7598, Laboratoire Jacques-Louis LionsUPMC Univ Paris 06Paris Cedex 05France
  2. 2.INRIA-Rocquencourt, EPI BANGParisFrance
  3. 3.CSMSE DivisionKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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