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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. (dambrine@bretagne.ens-cachan.fr)
Michel Pierre
Affiliation:
Antenne de Bretagne de l'ENS Cachan, Institut de Recherche Mathématique de Rennes, Campus de Ker Lann, 35170 Bruz, France. (pierre@bretagne.ens-cachan.fr)
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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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