For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard differential calculus in order to get second derivatives by a straight forward "repetition of differentiation". Moreover, by using boundary value characerizations for more regular data, a complete boundary integral representation of the second derivative of the objective is possible. Basing on this, one easily obtains that the second derivative contains only normal components for stationary domains, i.e. for domains, satisfying the first order necessary condition for a free optimum. Moreover, the nature of the second derivative is discussed, which is helpful for the investigation of sufficient optimality conditions.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1005,
author = {Karsten Eppler},
title = {Boundary integral representations of second derivatives in shape optimization},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {20},
year = {2000},
pages = {63-78},
zbl = {0961.49024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1005}
}
Karsten Eppler. Boundary integral representations of second derivatives in shape optimization. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 20 (2000) pp. 63-78. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1005/
[000] [1] S. Belov and N. Fujii, Symmetry and sufficient condition of optimality in a domain optimization problem, Control and Cybernetics 26 (1) (1997), 45-56. | Zbl 0871.49033
[001] [2] D. Colton and R. Kress, Integral equation methods in scattering theory, Malabar, Krieger 1992. | Zbl 0522.35001
[002] [3] K. Eppler, Optimal shape design for elliptic equations via BIE-methods, preprint TU Chemnitz, 1998.
[003] [4] K. Eppler, Optimal shape design for elliptic equations via BIE-methods, submitted (1999).
[004] [5] K. Eppler, Fréchet-differentiability and sufficient optimality conditions for shape functionals, in: K.-H. Hoffmann, G. Leugering, F. Tröltzsch (eds.), Optimal control of Partial Differential Equations, ISNM, 133 (1999), 133-143, Basel: Birkhäuser. | Zbl 0935.49019
[005] [6] K. Eppler, Boundary integral representations of second derivatives in shape optimization, dfg-preprint series 'Echtzeit-Optimierung grosser Systeme' preprint 00-1 (2000), http://www.tu-chemnitz.de/eppler
[006] [7] K. Eppler, Second derivatives and sufficient optimality conditions for shape functionals, to appear in Control and Cybernetics (2000). | Zbl 1010.49027
[007] [8] N. Fujii, Second order necessary conditions in a domain optimization problem, Journal of Optimization Theory and Applications 65 (2) (1990), 223-245. | Zbl 0675.49021
[008] [9] N. Fujii, Sufficient conditions for optimality in shape optimization, Control and Cybernetics 23 (3) (1994), 393-406. | Zbl 0822.49029
[009] [10] N.M. Guenther, Die Potentialtheorie und ihre Anwendungen auf Grundaufgaben der mathematischen Physik, Leipzig, Teubner 1957.
[010] [11] W. Hackbusch, Integralgleichungen, Stuttgart, Teubner 1989.
[011] [12] J. Hadamard, Lessons on Calculus of Variations, (in French), Paris, Gauthier-Villards 1910.
[012] [13] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material and Topological Design, Wiley, Chichester 1988. | Zbl 0713.73062
[013] [14] A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems 9 (1993), 81-96. | Zbl 0773.35085
[014] [15] A.M. Khludnev and J. Sokolowski, Modelling and Control in Solid Mechanics, Birkhäuser, Basel 1997. | Zbl 0865.73003
[015] [16] S.G. Michlin, Partielle Differentialgleichungen in der Mathematischen Physik, Akademie-Verlag, Berlin 1978. | Zbl 0397.35001
[016] [17] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer, New York 1983. | Zbl 0496.93029
[017] [18] R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems 10 (1994), 431-447. | Zbl 0805.35157
[018] [19] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Springer, Berlin 1992. | Zbl 0761.73003