On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems

On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems We consider a two-phase problem in thermal conductivity: inclusions filled with a material of conductivity σ 1 are layered in a body of conductivity σ 2 . We address the shape sensitivity of the first eigenvalue associated with Dirichlet boundary conditions when both the boundaries of the inclusions and the body can be modified. We prove a differentiability result and provide the expressions of the first and second order derivatives. We apply the results to the optimal design of an insulated body. We prove the stability of the optimal design thanks to a second order analysis. We also continue the study of an extremal eigenvalue problem for a two-phase conductor in a ball initiated by Conca et al. (Appl. Math. Optim. 60(2):173–184, 2009 ) and pursued in Conca et al. (CANUM 2008, ESAIM Proc., vol. 27, pp. 311–321, EDP Sci., Les Ulis, 2009 ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems

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Publisher
Springer-Verlag
Copyright
Copyright © 2011 by Springer Science+Business Media, LLC
Subject
Mathematics; Numerical and Computational Physics; Mathematical Methods in Physics; Theoretical, 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/s00245-010-9111-z
Publisher site
See Article on Publisher Site

Abstract

We consider a two-phase problem in thermal conductivity: inclusions filled with a material of conductivity σ 1 are layered in a body of conductivity σ 2 . We address the shape sensitivity of the first eigenvalue associated with Dirichlet boundary conditions when both the boundaries of the inclusions and the body can be modified. We prove a differentiability result and provide the expressions of the first and second order derivatives. We apply the results to the optimal design of an insulated body. We prove the stability of the optimal design thanks to a second order analysis. We also continue the study of an extremal eigenvalue problem for a two-phase conductor in a ball initiated by Conca et al. (Appl. Math. Optim. 60(2):173–184, 2009 ) and pursued in Conca et al. (CANUM 2008, ESAIM Proc., vol. 27, pp. 311–321, EDP Sci., Les Ulis, 2009 ).

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2011

References

  • Lipschitz continuity of state functions in some optimal shaping
    Briançon, T.; Hayouni, M.; Pierre, M.

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