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About stability of equilibrium shapes

Published online by Cambridge University Press:  15 April 2002

Marc Dambrine  and
Michel Pierre
Marc Dambrine
Affiliation:
Antenne de Bretagne de l'ENS Cachan, Institut de Recherche Mathématique de Rennes, Campus de Ker Lann, 35170 Bruz, France. ([email protected])
Michel Pierre
Affiliation:
Antenne de Bretagne de l'ENS Cachan, Institut de Recherche Mathématique de Rennes, Campus de Ker Lann, 35170 Bruz, France. ([email protected])
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Abstract

We discuss the stability of "critical" or "equilibrium" shapes ofa shape-dependent energy functional. We analyze a problem arising whenlooking at the positivity of the second derivative in order to provethat a critical shape is an optimal shape. Indeed, often whenpositivity -or coercivity- holds, it does for a weaker norm than thenorm for which the functional is twice differentiable and localoptimality cannot be a priori deduced. We solve this problem for aparticular but significant example. We prove "weak-coercivity" ofthe second derivative uniformly in a "strong" neighborhood of theequilibrium shape.

Keywords

Shape optimisationstability of critical shapeweak coercivityarea-preserving transformations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 811 - 834
Copyright
© EDP Sciences, SMAI, 2000

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References

Brancher, J.-P., Etay, J. and Séro-Guillaume, O., Formage d'une lame métallique liquide. Calculs et expériences. J. Mec. Theor. Appl. 2 (1983) 977-989.
D. Bucur and J.-P. Zolésio, Anatomy of the Shape Hessian Via Lie Brackets. Ann. Mat. Pura Appl. (IV) CLXXIII (1997) 127-143.
Crouzeix, M., Variational approach of a magnetic shaping problem. Eur. J. Mech. B Fluids 10 (1991) 527-536.
M. Dambrine, Hessiennes de formes et stabilité de formes critiques. Ph.D. thesis, Université de Rennes 1, France (2000).
R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 2. Masson, Paris (1985).
Delfour, M. and Zolésio, J.-P., Velocity Method and Lagrangian Formulation for the Computation of the Shape Hessian. SIAM Control Optim. 29 (1991) 1414-1442. CrossRef
J. Descloux, On the two dimensional magnetic shaping problem without surface tension. Report, Analysis and numerical analysis, 07.90, École Polytechnique Fédérale de Lausanne (1990).
Descloux, J., Stability of the solutions of the bidimensional magnetic shaping problem in absence of surface tension. Eur. J. Mech. B Fluids 10 (1991) 513-526.
Descloux, J., A stability result for the magnetic shaping problem. Z. Angew. Math. Phys. 45 (1994) 543-555. CrossRef
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 2nd edn (1983).
A. Henrot and M. Pierre, Stability in shaping problems (to appear).
A. Henrot and M. Pierre, About existence of a free boundary in electromagnetic shaping, in Recent advances in nonlinear elliptic and parabolic problems (Nancy, 1988), Longman Sci. Tech., Harlow (1989) 283-293.
Henrot, A. and Pierre, M., Un problème inverse en formage des métaux liquides. RAIRO Modél. Math. Anal. Numér. 23 (1989) 155-177. CrossRef
A. Henrot and M. Pierre, About critical points of the energy in the electromagnetic shaping problem, in Boundary Control and Boundary variations, Springer-Verlag, 178 (1991) 238-252.
Henrot, A. and Pierre, M., About existence of equilibria in electromagnetic casting. Quart. Appl. Math. 49 (1991) 563-575. CrossRef
F. Murat and J. Simon, Sur le contrôle par un domaine géométrique. Rapport du L.A. 189, Université Paris VI, France (1976).
A. Novruzi, Contribution en Optimisation de Formes et Applications. Ph.D. thesis, Université Henri Poincaré, Nancy (1996).
A. Novruzi and M. Pierre, Structure of Shape Derivatives. Prépublication IRMAR, n° 00-07, Rennes (2000).
Séro-Guillaume, O. and Bernardin, D., Note on a Hamiltonian formalism for the flow of a magnetic fluid with a free surface. J. Fluid Mech. 181 (1987) 381-386.
Simon, J., Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649-687. CrossRef
J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization. Springer-Verlag, Berlin (1992).