Geodesic
Boundary integral representations of second derivatives in shape optimization
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 20 (2000) no. 1, pp. 63-78

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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.
Keywords: optimal shape design, fundamental solution, boundary integral equation, second-order derivatives, optimality conditions 
Eppler, Karsten. Boundary integral representations of second derivatives in shape optimization. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 20 (2000) no. 1, pp. 63-78. http://geodesic.mathdoc.fr/item/DMDICO_2000_20_1_a4/
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