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On stability analysis in shape optimization: critical shapes for Neumann problem. (English) Zbl 1126.49034

Summary: The stability issue of critical shapes for shape optimization problems with the state function given by a solution to the Neumann problem for the Laplace equation is considered. To this end, the properties of the shape Hessian evaluated at critical shapes are analysed. First, it is proved that the stability cannot be expected for the model problem. Then, the new estimates for the shape Hessian are derived in order to overcome the classical two norms-discrepancy well known in control problems, K. Malanowski [Diss. Math. 394, 51 p. (2001; Zbl 1017.49027)]. In the context of shape optimization, the situation is similar compared to control problems, actually, the shape Hessian can be coercive only in the norm strictly weaker with respect to the norm of the second order differentiability of the shape functional. In addition, it is shown that an appropriate regularization makes possible the stability of critical shapes.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49K40 Sensitivity, stability, well-posedness

Citations:

Zbl 1017.49027
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