On some overdetermined free boundary problems

Khatmi Samira, Barkatou Mohammed


This paper deals with some free boundary problems for the Laplacian operator. We first give sufficient conditions of existence of free boundaries. Then combining the maximum principle to the monotonicity of the mean curvature, we will prove a symmetry result in the case where the source term is constant. All the results obtained here can be extended to more general divergence operators.

  • D. Bucur and J.P. Zolesio : N-dimensional shape optimization under capacitary constraints, J. Diff. Eq., 123-2, 1995, 504--522.
  • D. Chenais : Sur une famille de varietes a bord lipschitziennes, application a un probleme d'identification de domaine, Ann. Inst. Fourier, 4-27, 1977, 201--231.
  • E. DiBenditto : $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis, 7, 1983, 827--850.
  • B. Gustafsson and H. Shahgholian : Existence and geometric properties of solutions of a free boundary problem in potential theory, J. fur die Reine und Ang. Math. 473, 1996, 137--179.
  • A. Henrot and M. Pierre : Variation et optimisation de formes, une analyse geometrique, MathÈmatiques and Applications, 48, (2005), SMAI Springer
  • C. Huang and D. Miller : Domain functionals and exit times for Brownian motion. Proc. Amer. Math. Soc. 130(3), 2001, 825--831.
  • K. K. J. Kinateder P. Mcdonald : Hypersurfaces in $R^d$ and the variance of times for Brownian motion. Proc. Amer. Math. Soc. 125(8), 1997, 2453--2462.
  • J. L. Lewis : Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32, 1983, 849--858.
  • G. M. Liberman : Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis, 12, 1988, 1203--1219.
  • F. Murat and J. Simon: Quelques resultats sur le controle par un domaine geometrique, Publ. du labo. d'Anal. Num. Paris VI, 1974, 1--46
  • A. D. Alexandrov: Uniquness theorems for surfaces in large I, II, Amer. Soc. Trans. 31, 1962, 341--388.
  • O. Pironneau : Optimal shape design for elliptic systems, Springer Series in Computational Physics, 1984, Springer, New York.
  • H. Shahgholian : Existence of Quadrature Surfaces for Positive Measures with Finite Support, Potential Analysis, 3, 1994, 245--255.
  • J. Sokolowski et J. P. Zolesio : Introduction to shape optimization : shape sensitity analysis, Springer Series in Computational Mathematics, 10, 1992, Springer, Berlin.
  • P. Tolksdorf (1983) On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8(7), 1983, 773--817.
  • M. Barkatou, D. Seck and I. Ly: An existence result for a quadrature surface free boundary problem, Cent. Eur. Jou. Math. 3(1), 2005, 39--57.
  • M. Barkatou: Some geometric properties for a class of non Lipschitz-domains, New York J. Math. 8, 2002, 189--213. http://nyjm.albany.edu:8000/j/2002/8-13.html
  • D. Bucur and P. Trebeschi : Shape Optimization Problems Governed by Nonlinear State Equations, Proc. Roy. Soc. Edinburgh , 128A, 1998, 945--963

Full Text:


DOI: http://dx.doi.org/10.21914/anziamj.v49i0.168

Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.