An existence result for a quadrature surface free boundary problem

Mohammed Barkatou; Diaraf Seck; Idrissa Ly

Open Mathematics (2005)

  • Volume: 3, Issue: 1, page 39-57
  • ISSN: 2391-5455

Abstract

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The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.

How to cite

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Mohammed Barkatou, Diaraf Seck, and Idrissa Ly. "An existence result for a quadrature surface free boundary problem." Open Mathematics 3.1 (2005): 39-57. <http://eudml.org/doc/268803>.

@article{MohammedBarkatou2005,
abstract = {The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.},
author = {Mohammed Barkatou, Diaraf Seck, Idrissa Ly},
journal = {Open Mathematics},
keywords = {35J05; 35A35},
language = {eng},
number = {1},
pages = {39-57},
title = {An existence result for a quadrature surface free boundary problem},
url = {http://eudml.org/doc/268803},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Mohammed Barkatou
AU - Diaraf Seck
AU - Idrissa Ly
TI - An existence result for a quadrature surface free boundary problem
JO - Open Mathematics
PY - 2005
VL - 3
IS - 1
SP - 39
EP - 57
AB - The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.
LA - eng
KW - 35J05; 35A35
UR - http://eudml.org/doc/268803
ER -

References

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  1. [1] H.W. Alt and L.A. Caffarelli: “Existence and regularity for a minimum problem with free boundary”, J. Reine angew. Math., Vol. 325, (1981), pp. 105–144. Zbl0449.35105
  2. [2] M. Barkatou, D. Seck and I. Ly: “An existence result for a free boundary problem for the p-Laplace operator”, submitted. Zbl1207.35106
  3. [3] M. Barkatou: “Some geometric properties for a class of non Lipschitz-domains”, New York J. of Math., Vol. 8, (2002), pp. 189–213. Zbl1066.35028
  4. [4] M. Barkatou: “Existence of quadrature surfaces for a uniform density supported by a segment”, submitted. Zbl1160.35380
  5. [5] J.A. Bello, E. Fernandez-Cara, J. Lemoine and J. Simon: “On drag differentiability for Lipschitz domains”, Control of Part. Diff. Eq. and Appl., Lec. Notes In Pure and Applied Math. Series, Vol. 174, (1995), Dekker, New York. Zbl0860.35093
  6. [6] A. Beurling: “On free-boundary problems for the Laplace equation”, Sem. Anal. Funct., Inst. Adv. Study Princeton, Vol. 1, (1957), pp. 248–263. 
  7. [7] D. Bucur and P. Trebeschi: “Shape Optimization Problems Governed by Nonlinear State Equations”, Proc. Roy. Sc. Edinburgh, Vol. 128 A (1998), pp. 945–963. Zbl0918.49030
  8. [8] D. Bucur and J.P. Zolesio: “N-dimensional shape optimization under capacitary constraints”, J. Diff. Eq., Vol. 123(2), (1995), pp. 504–522. http://dx.doi.org/10.1006/jdeq.1995.1171 Zbl0847.49029
  9. [9] G. Buttazzo, V. Ferone and B. Kawhol: “Minimum problems over sets of concave functions and related questions”, Math. Nachr., (1995), pp. 71–89. Zbl0835.49001
  10. [10] T. Carleman: “Über ein Minimumproblem der mathematischen Physik”,Math. Z.,Vol. 1, (1918),pp.208–212. http://dx.doi.org/10.1007/BF01203612 Zbl46.0765.02
  11. [11] D. Chenais, “On the existence of a solution in a domain identification problem”, J. Math. Anal. Appl., Vol. 52, (1975), pp. 189–289. http://dx.doi.org/10.1016/0022-247X(75)90091-8 
  12. [12] R. Dautray and J.L. Lions: Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 1, 2, Masson, Paris, 1984 
  13. [13] E. DiBendetto: “ C 1+∞ local regularity of weak solutions of degenerate elliptic equations”, Nonlinear Analysis., Vol. 7, (1983), pp. 827–850. http://dx.doi.org/10.1016/0362-546X(83)90061-5 
  14. [14] K. Friedrichs: “Über ein Minimumproblem für Potentialströmungen mit freiem Rand”.Math. Ann.,Vol. 109, (1934),pp.208–212. 
  15. [15] G. Gidas, Wei-Ming Ni and L. Nirenberg: “Symmetry and related properties via the maximum principle”,Comm. Math. Phys., Vol. 68, (1979),pp.209–300. http://dx.doi.org/10.1007/BF01221125 Zbl0425.35020
  16. [16] B. Gustafsson and H. Shahgholian: “Existence and geometric properties of solutions of a free boundary problem in potential theory”, J. Reine angew. Math., Vol. 473, (1996), pp. 137–179. Zbl0846.31005
  17. [17] A. Henrot: “Subsolutions and supersolutions in a free boundary problem”, Arkiv för Math., Vol. 32(1), (1994), pp. 79–98. Zbl0809.35172
  18. [18] H. Hosseinzadeh and H. Shahgholian: “Some qualitative aspects of a free boundary problem for the p-Laplacian”, Ann. Acad. Scient. Fenn. Math., Vol. 24, (1999), pp. 109–121. Zbl0914.35149
  19. [19] D. Gilbarg and N.S. Trudinger: Elliptic partial equations of second order, Springer-Verlag, 1983. Zbl0562.35001
  20. [20] M.V. Keldyš: “On the solvability and the stability of the Dirichlet problem”, Amer. Math. Soc. Trans., Vol. 51 (2), (1966), pp. 1–73. 
  21. [21] J.L. Lewis: “Regularity of the derivatives of solutions to certain degenerate elliptic equations”, Indiana Univ. Math. J., Vol. 32, (1983), pp. 849–858. http://dx.doi.org/10.1512/iumj.1983.32.32058 Zbl0554.35048
  22. [22] G.M. Lieberman: “Boundary regularity for solutions of degenerate elliptic equations”, Nonlinear Analysis., Vol. 12, (1988), pp. 1203–1219. http://dx.doi.org/10.1016/0362-546X(88)90053-3 
  23. [23] V. Mikhailov: Équation aux dérivées partielles, Mir, Moscow, 1980. 
  24. [24] F. Murat and J. Simon: “Quelques résultats sur le contrôle par un domaine géométrique”, Publ. du labo. d’Anal. Num., Paris VI, (1974), pp. 1–46. 
  25. [25] O. Pironneau: Optimal shape design for elliptic systems, Springer series in Computational Physics, Springer, New York, 1984. 
  26. [26] J. Serrin: “A symmetry problem in potential theory”,Arch. Rat. Mech. Anal.,Vol. 43, (1971),pp.304–318. http://dx.doi.org/10.1007/BF00250468 Zbl0222.31007
  27. [27] H. Shahgholian: “Quadrature surfaces as free boundaries”, Arkiv för Math., Vol. 32(2), (1994), pp. 475–492. Zbl0827.31004
  28. [28] J. Sokolowski and J.P. Zolesio: Introduction to shape optimization: shape sensitity analysis, Springer Series in Computational Mathematics, Vol. 10, Springer, Berlin, 1992. 
  29. [29] P. Tolksdorf: “On the Dirichlet problem for quasilinear equations in domains with conical boundary points”, Comm. Partial Differential Equations, Vol. 8(7), (1983), pp. 773–817. Zbl0515.35024

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