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On a Kohn-Vogelius like formulation of free boundary problems

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Abstract

The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L 2-distance of the gradients of the state functions. We compute the corresponding shape gradient and Hessian. By the investigation of sufficient second order conditions we prove algebraic ill-posedness of the present formulation. Our theoretical findings are supported by numerical experiments.

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Authors and Affiliations

  1. Institut für Numerische Mathematik, Technische Universität Dresden, Zellescher Weg 12–14, 01062, Dresden, Germany

    Karsten Eppler

  2. Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Helmut Harbrecht

Authors
  1. Karsten Eppler
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  2. Helmut Harbrecht
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Correspondence to Helmut Harbrecht.

Additional information

This work was supported by the DFG Priority Program “Optimierung mit partiellen Differentialgleichungen” (PDE constrained optimization).

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Eppler, K., Harbrecht, H. On a Kohn-Vogelius like formulation of free boundary problems. Comput Optim Appl 52, 69–85 (2012). https://doi.org/10.1007/s10589-010-9345-3

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