Annali di Matematica Pura ed Applicata

, Volume 173, Issue 1, pp 127–143 | Cite as

Anatomy of the shape Hessian via lie brackets

  • Dorin Bucur
  • Jean-Paul Zolésio
Article

Abstract

The goal of this paper is to study the anatomy of the shape Hessian for some classes of smooth shape functionals. A structure theorem gives a decomposition of the shape Hessian in three additive bilinear forms acting on the two fields: the first one acting on the normal components at the boundary, the second one being symmetrical and the third one involving a half of the Lie bracket of the pair of fields at which the shape Hessian is computed. Applications to the commutation of the mixed derivatives and the symmetry of the linear operator which appears in the structure theorem are given.

Keywords

Linear Operator Bilinear Form Normal Component Structure Theorem Mixed Derivative 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.Abraham - J.Marsden,Foundations of Mechanics, second edition, Addison Wesley Publ. Co. Inc. (1987).Google Scholar
  2. [2]
    D.Bucur,Contrôle par rapport au domaine dans les EDP, Thèse de Doctorat, Ecole des Mines de Paris (1995).Google Scholar
  3. [3]
    M.Delfour - J.-P.Zolésio,Anatomy of the shape Hessian, Ann. Mat. Pura App.,CLVIII (1989).Google Scholar
  4. [4]
    M. Delfour -J.-P. Zolésio,Velocity method and Lagrangian formulation for the computation of the shape Hessian, SIAM Control and Optimization,29, No. 6 (November 1991), pp. 1414–1442.Google Scholar
  5. [5]
    M. Delfour -J.-P. Zolésio,Shape analysis via oriented distance functions, J. Func. Analysis, No. 1 (July 1994), pp. 129–201.Google Scholar
  6. [6]
    A.Henrot -M. Pierre,About critical points of the energy in the electromagnetic shaping problem, inBoundary Control and Boundary Variation, J.-P.Zolésio (ed.), Lecture Notes in Control and Information Sciences, vol. 178, Springer-Verlag (1991), pp. 238–252.Google Scholar
  7. [7]
    J.Sokolowski - J.-P.Zolésio,Introduction to Shape Optimization, Springer-Verlag (1992).Google Scholar
  8. [8]
    J.-P.Zolésio,Identification de domaines par déformations, PhD thesis, Nice (1979).Google Scholar
  9. [9]
    J.-P.Zolésio,Introduction to Shape Optimization Problems and Free Boundary Problems, Kluwer Academic Publishers, NATO ASI Series, C: Mathematical and Physical Sciences, vol. 380, M.Delfour (ed.) (1990), pp. 397–457.Google Scholar
  10. [10]
    F. Desaint -J.-P. Zolésio,Shape Derivative for the Laplace-Beltrami Equation, Lecture Notes in Pure and Applied Mathematics, vol. 188, Marcel Dekker Inc., New York (1997), pp. 120–145.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1997

Authors and Affiliations

  • Dorin Bucur
    • 1
  • Jean-Paul Zolésio
    • 2
    • 3
  1. 1.CNRS-Equipe de MathématiquesUniversité de Franche-ComtéBesançonFrance
  2. 2.CNRS-INLNFrance
  3. 3.CMA/INRIASophia AntipolisFrance

Personalised recommendations