# About stability of equilibrium shapes

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 4, page 811-834
- ISSN: 0764-583X

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topDambrine, Marc, and Pierre, Michel. "About stability of equilibrium shapes." ESAIM: Mathematical Modelling and Numerical Analysis 34.4 (2010): 811-834. <http://eudml.org/doc/197610>.

@article{Dambrine2010,

abstract = {
We discuss the stability of "critical" or "equilibrium" shapes of
a shape-dependent energy functional. We analyze a problem arising when
looking at the positivity of the second derivative in order to prove
that a critical shape is an optimal shape. Indeed, often when
positivity -or coercivity- holds, it does for a weaker norm than the
norm for which the functional is twice differentiable and local
optimality cannot be a priori deduced. We solve this problem for a
particular but significant example. We prove "weak-coercivity" of
the second derivative uniformly in a "strong" neighborhood of the
equilibrium shape.
},

author = {Dambrine, Marc, Pierre, Michel},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Shape optimisation; stability of critical shape; weak coercivity;
area-preserving transformations.; shape-dependent energy functional; critical shape; optimal shape},

language = {eng},

month = {3},

number = {4},

pages = {811-834},

publisher = {EDP Sciences},

title = {About stability of equilibrium shapes},

url = {http://eudml.org/doc/197610},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Dambrine, Marc

AU - Pierre, Michel

TI - About stability of equilibrium shapes

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 4

SP - 811

EP - 834

AB -
We discuss the stability of "critical" or "equilibrium" shapes of
a shape-dependent energy functional. We analyze a problem arising when
looking at the positivity of the second derivative in order to prove
that a critical shape is an optimal shape. Indeed, often when
positivity -or coercivity- holds, it does for a weaker norm than the
norm for which the functional is twice differentiable and local
optimality cannot be a priori deduced. We solve this problem for a
particular but significant example. We prove "weak-coercivity" of
the second derivative uniformly in a "strong" neighborhood of the
equilibrium shape.

LA - eng

KW - Shape optimisation; stability of critical shape; weak coercivity;
area-preserving transformations.; shape-dependent energy functional; critical shape; optimal shape

UR - http://eudml.org/doc/197610

ER -

## References

top- J.-P. Brancher, J. Etay and O. Séro-Guillaume, 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.
- M. Crouzeix, Variational approach of a magnetic shaping problem. Eur. J. Mech. B Fluids10 (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).
- M. Delfour and J.-P. Zolésio, Velocity Method and Lagrangian Formulation for the Computation of the Shape Hessian. SIAM Control Optim.29 (1991) 1414-1442.
- 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).
- J. Descloux, Stability of the solutions of the bidimensional magnetic shaping problem in absence of surface tension. Eur. J. Mech. B Fluids10 (1991) 513-526.
- J. Descloux, A stability result for the magnetic shaping problem. Z. Angew. Math. Phys.45 (1994) 543-555.
- 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.
- A. Henrot and M. Pierre, Un problème inverse en formage des métaux liquides. RAIRO Modél. Math. Anal. Numér.23 (1989) 155-177.
- 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.
- A. Henrot and M. Pierre, About existence of equilibria in electromagnetic casting. Quart. Appl. Math.49 (1991) 563-575.
- 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).
- O. Séro-Guillaume and D. Bernardin, Note on a Hamiltonian formalism for the flow of a magnetic fluid with a free surface. J. Fluid Mech.181 (1987) 381-386.
- J. Simon, Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim.2 (1980) 649-687.
- J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization. Springer-Verlag, Berlin (1992).

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