# Variational approach to shape derivatives

Kazufumi Ito; Karl Kunisch; Gunther H. Peichl

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 14, Issue: 3, page 517-539
- ISSN: 1292-8119

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topIto, Kazufumi, Kunisch, Karl, and Peichl, Gunther H.. "Variational approach to shape derivatives." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2008): 517-539. <http://eudml.org/doc/250272>.

@article{Ito2008,

abstract = {
A general framework for calculating shape derivatives for
optimization problems with partial differential equations as
constraints is presented. The proposed technique allows to obtain
the shape derivative of the cost without the necessity to involve
the shape derivative of the state variable. In fact, the state
variable is only required to be Lipschitz continuous with respect
to the geometry perturbations. Applications to inverse interface
problems, and shape optimization for elliptic systems and the
Navier-Stokes equations are given.
},

author = {Ito, Kazufumi, Kunisch, Karl, Peichl, Gunther H.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Shape derivative; shape derivative},

language = {eng},

month = {2},

number = {3},

pages = {517-539},

publisher = {EDP Sciences},

title = {Variational approach to shape derivatives},

url = {http://eudml.org/doc/250272},

volume = {14},

year = {2008},

}

TY - JOUR

AU - Ito, Kazufumi

AU - Kunisch, Karl

AU - Peichl, Gunther H.

TI - Variational approach to shape derivatives

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/2//

PB - EDP Sciences

VL - 14

IS - 3

SP - 517

EP - 539

AB -
A general framework for calculating shape derivatives for
optimization problems with partial differential equations as
constraints is presented. The proposed technique allows to obtain
the shape derivative of the cost without the necessity to involve
the shape derivative of the state variable. In fact, the state
variable is only required to be Lipschitz continuous with respect
to the geometry perturbations. Applications to inverse interface
problems, and shape optimization for elliptic systems and the
Navier-Stokes equations are given.

LA - eng

KW - Shape derivative; shape derivative

UR - http://eudml.org/doc/250272

ER -

## References

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- J. Haslinger and P. Neittaanmaki, Introduction to shape optimization. SIAM, Philadelphia (2003).
- K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl.314 (2006) 126–149.
- F. Murat and J. Simon, Sur le contrôle par un domaine géometrique. Rapport 76015, Université Pierre et Marie Curie, Paris (1976).
- J. Sokolowski and J.P. Zolesio, Introduction to shape optimization. Springer, Berlin (1991).
- R. Temam, Navier Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1979).
- J.T. Wloka, B. Rowley and B. Lawruk, Boundary value problems for elliptic systems. Cambridge Press (1995).
- J.P. Zolesio, The material derivative (or speed method) for shape optimization, in Optimization of Distributed Parameter Structures, Vol. II, E. Haug and J. Cea Eds., Sijthoff & Noordhoff (1981).

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