Glasnik Matematicki, Vol. 41, No.2 (2006), 283-297.

ISOPERIMETRIC INEQUALITIES FOR AN ELECTROSTATIC PROBLEM

L. Boukrim and T. Mekkaoui

Abstract.   We study the problem of the (p-)capacity c_p of a multiconnected configuration Ω = (G \ E) \ (∪ H_i) when ∂G and ∂E have given potentials. Here Ω represents a nonhomogeneous medium and the H_i, which separate the different connected components of Ω, represent perfect conductors. By comparison with a similar configuration with spherical symmetry, we give isoperimetric inequalities for c_p and the unknown potentials on H_i.

2000 Mathematics Subject Classification.   26D10, 35J65.

Key words and phrases.   Isoperimetric inequality, rearrangement, nonlinear elliptic PDE's.

DOI: 10.3336/gm.41.2.11

References:

1. A. Alvino and G. Trombetti, Isoperimetric inequalities connected with torsion problem and capacity, Boll. Un. Mat. Ital. B (6), 4 (1985), 773-787.
   MathSciNet

2. C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, Mass.-London, 1980.
   MathSciNet

3. L. Boukrim, Inégalités isopérimétriques pour un problème d'électrostatique, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 435-438.
   MathSciNet

4. L. Boukrim, Inégalités isopérimétriques pour certains problèmes de conductivité dans desmilieux non homogènes, Doctoral Thesis, Université Paris-Sud, Orsay, France, 1994.

5. J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries I. Elliptic equations, Pitman, Boston, 1985.
   MathSciNet

6. V. Ferone, Symmetrization results in electrostatic problems, Ricerche Mat. 37 (1988), 359-370.
   MathSciNet

7. J. Mossino, Inégalités Isopérimétriques et Applications en Physique, Hermann, Paris, 1984.
   MathSciNet

8. J. Mossino and R. Temam, Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math. J. 48 (1981), 475-495.
   MathSciNet     CrossRef

9. M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl. (4) 80 (1968), 1-122.
   MathSciNet

10. G. Pólya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quart. Appl. Math. 6 (1948), 267-277.
    MathSciNet

11. G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.
    MathSciNet

12. J. M. Rakotoson, Some properties of the relative rearrangement, J. Math. Anal. Appl. 135 (1988), 488-500.
    MathSciNet     CrossRef

13. J. M. Rakotoson and R. Temam, Relative rearrangement in quasilinear elliptic variational inequalities, Indiana Univ. Math. J 36 (1987), 757-810.
    MathSciNet     CrossRef