Journal of Optimization Theory and Applications

, Volume 20, Issue 1, pp 65–79 | Cite as

On isoperimetric problems for domains with partly known boundaries

  • E. L. Gurvitch
Contributed Papers


In the present paper, we discuss the isoperimetric problems for domains with partly known boundaries, i.e., the problem of determining a domain that minimizes the capacity functional in the class of plain double-connected domains having the same fixed area and outer boundary. The formulas for capacity variations obtained in this paper allows us to formulate necessary conditions.

It is proved that the convexity of the fixed outer boundary implies the convexity of the inner boundary corresponding to an optimal domain. Then, we discuss the case where the fixed part of the boundary is a square.

Further, we consider similar problems with more complicated functionals. We introduce the concept of a minimal function in the class of equimeasurable functions. This concept allows us to unify the approach to all of these problems. At the end, we produce a hypothesis that, if proved, would enable us to characterize the shape of the optimal domains in the isoperimetric problems mentioned above.

Key Words

Isoperimetric problems partly known boundaries convexity equimeasurable functions 


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  1. 1.
    Polya, G., andSzegö, G.,Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, New Jersey, 1951.Google Scholar
  2. 2.
    Payne, L. E.,Isoperimetric Inequalities and Their Applications, SIAM Review, Vol. 9, No. 3, 1967.Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • E. L. Gurvitch
    • 1
  1. 1.A. F. Ioffe Physico-Technical InstituteUSSR Academy of SciencesLeningradUSSR

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