An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball

An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhäuser, Basel, 1997 ) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101–117, 1996 ). Although, the problem in one dimension has a solution (cf. Kreĭn in AMS Transl. Ser. 2(1):163–187, 1955 ) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185–220, 1989 ), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328–337, 1983 ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball

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Publisher
Springer-Verlag
Copyright
Copyright © 2009 by Springer Science+Business Media, LLC
Subject
Mathematics; Numerical and Computational Methods ; Mathematical Methods in Physics; Mathematical and Computational Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-008-9061-x
Publisher site
See Article on Publisher Site

Abstract

The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhäuser, Basel, 1997 ) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101–117, 1996 ). Although, the problem in one dimension has a solution (cf. Kreĭn in AMS Transl. Ser. 2(1):163–187, 1955 ) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185–220, 1989 ), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328–337, 1983 ).

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2009

References

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