Skip to main content
SpringerLink
Log in

On the Optimal Insulation of Conductors

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We coat a conductor with an insulator and equate the effectiveness of this procedure with the rate at which the body dissipates heat when immersed in an ice bath. In the limit, as the thickness and conductivity of the insulator approach zero, the dissipation rate approaches the first eigenvalue of a Robin problem with a coefficient determined by the shape of the insulator. Fixing the mean of the shape function, we search for the shape with the least associated Robin eigenvalue. We offer exact solutions for balls; for general domains, we establish existence and necessary conditions and report on the results of a numerical method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Friedman, A., Reinforcement of the Principal Eigenvalue of an Elliptic Operator, Archive for Rational Mechanics and Analysis, Vol. 73, pp. 1–17, 1980.

    Google Scholar 

  2. Gilbarg, D., and Trudinger, N., Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer, Berlin, Germany, 1983.

    Google Scholar 

  3. Buttazzo, G., Thin Insulating Layers: The Optimization Point of View, Material Instabilities in Continuum Mechanics and Related Problems, Edited by J. M. Ball, Oxford University Press, Oxford, England, pp. 11–19, 1988.

    Google Scholar 

  4. Dacorogna, B., Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals, Lecture Notes in Mathematics, Springer, Berlin, Germany, Vol. 922, 1982.

    Google Scholar 

  5. Clarke, F., Optimization and Nonsmooth Analysis, SIAM, Philadelphia, Pennsylvania, 1990.

    Google Scholar 

  6. Cea, J., and Malanowski, K., An Example of a Max-Min Problem in Partial Differential Equations, SIAM Journal of Control and Optimization, Vol. 8, pp. 305–316, 1970.

    Google Scholar 

  7. Cox, S. J., and Uhlig, P. X., Where Best to Hold a Drum Fast, SIAM Journal on Optimization, 1999.

  8. Cox, S. J., The Generalized Gradient at a Multiple Eigenvalue, Journal of Functional Analysis, Vol. 33, pp. 30–40, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, Houston, Texas

    S. J. Cox (Associate Professor)

  2. Mathematisches Institut, Universität Köln, Köln, Germany

    B. Kawohl (Professor)

  3. Mathematics Department, St. Mary's University, San Antonio, Texas

    P. X. Uhlig (Assistant Professor)

Authors
  1. S. J. Cox
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Kawohl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. X. Uhlig
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cox, S.J., Kawohl, B. & Uhlig, P.X. On the Optimal Insulation of Conductors. Journal of Optimization Theory and Applications 100, 253–263 (1999). https://doi.org/10.1023/A:1021773901158

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021773901158

Search

Navigation