An existence result for a quadrature surface free boundary problem. (English) Zbl 1207.35106

Barkatou, Mohammed; Seck, Diaraf; Ly, Idrissa

Cent. Eur. J. Math. 3, No. 1, 39-57 (2005).

Summary: The aim of this paper is to present two different approaches 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.

MSC:

35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35A35	Theoretical approximation in context of PDEs

Keywords:

Dirichlet problem; Neumann problem; quadrature surface; maximum principle, shape optimization; optimality conditions

References:

[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. · Zbl 0449.35105
[2]	M. Barkatou, D. Seck and I. Ly: “An existence result for a free boundary problem for the p-Laplace operator”, submitted. · Zbl 1207.35106
[3]	M. Barkatou: “Some geometric properties for a class of non Lipschitz-domains”, New York J. of Math., Vol. 8, (2002), pp. 189-213. · Zbl 1066.35028
[4]	M. Barkatou: “Existence of quadrature surfaces for a uniform density supported by a segment”, submitted. · Zbl 1160.35380
[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. · Zbl 0860.35093
[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]	D. Bucur and P. Trebeschi: “Shape Optimization Problems Governed by Nonlinear State Equations”, Proc. Roy. Sc. Edinburgh, Vol. 128 A (1998), pp. 945-963. · Zbl 0918.49030
[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 · Zbl 0847.49029
[9]	G. Buttazzo, V. Ferone and B. Kawhol: “Minimum problems over sets of concave functions and related questions”, Math. Nachr., (1995), pp. 71-89. · Zbl 0835.49001
[10]	T. Carleman: “Über ein Minimumproblem der mathematischen Physik”,Math. Z.,Vol. 1, (1918),pp.208-212. http://dx.doi.org/10.1007/BF01203612 · JFM 46.0765.02
[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]	R. Dautray and J.L. Lions: Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 1, 2, Masson, Paris, 1984
[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]	K. Friedrichs: “Über ein Minimumproblem für Potentialströmungen mit freiem Rand”.Math. Ann.,Vol. 109, (1934),pp.208-212.
[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 · Zbl 0425.35020
[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. · Zbl 0846.31005
[17]	A. Henrot: “Subsolutions and supersolutions in a free boundary problem”, Arkiv för Math., Vol. 32(1), (1994), pp. 79-98. · Zbl 0809.35172
[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. · Zbl 0914.35149
[19]	D. Gilbarg and N.S. Trudinger: Elliptic partial equations of second order, Springer-Verlag, 1983. · Zbl 0562.35001
[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]	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 · Zbl 0554.35048
[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]	V. Mikhailov: Équation aux dérivées partielles, Mir, Moscow, 1980.
[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]	O. Pironneau: Optimal shape design for elliptic systems, Springer series in Computational Physics, Springer, New York, 1984.
[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 · Zbl 0222.31007
[27]	H. Shahgholian: “Quadrature surfaces as free boundaries”, Arkiv för Math., Vol. 32(2), (1994), pp. 475-492. · Zbl 0827.31004
[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]	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. · Zbl 0515.35024