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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 384320, 19 pages
http://dx.doi.org/10.1155/2013/384320
Research Article

On the First-Order Shape Derivative of the Kohn-Vogelius Cost Functional of the Bernoulli Problem

1Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, Baguio City 2600, Philippines
2Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz, Austria

Received 18 August 2013; Accepted 1 November 2013

Academic Editor: Sergei V. Pereverzyev

Copyright © 2013 Jerico B. Bacani and Gunther Peichl. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. B. Abda, F. Bouchon, G. Peichl, M. Sayeh, and R. Touzani, “A new formulation for the Bernoulli problem,” in Proceedings of the 5th International Conference on Inverse Problems, Control and Shape Optimization, pp. 1–19, 2010.
  2. L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, American Mathematical Society, Providence, RI, USA, 2005.
  3. J. Crank, Free and Moving Boundary Problems, Oxford University Press, New York, NY, USA, 1984.
  4. M. Flucher and M. Rumpf, “Bernoulli's free-boundary problem, qualitative theory and numerical approximation,” Journal fur die Reine und Angewandte Mathematik, vol. 486, pp. 165–204, 1997. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. A. Friedman, “Free boundary problems in science and technology,” Notices of the AMS, vol. 47, pp. 854–861, 2000. View at Google Scholar
  6. J. I. Toivanen, J. Haslinger, and R. A. E. Mäkinen, “Shape optimization of systems governed by Bernoulli free boundary problems,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 45–48, pp. 3803–3815, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. A. Beurling, “On free boundary problems for the Laplace equation,” in Proceedings of the Seminars on Analytic Functions, pp. 248–263, 1957.
  8. P. Cardaliaguet and R. Tahraoui, “Some uniqueness results for Bernoulli interior free-boundary problems in convex domains,” Electronic Journal of Differential Equations, vol. 2002, pp. 1–16, 2002. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. J. Haslinger, K. Ito, T. Kozubek, K. Kunisch, and G. Peichl, “On the shape derivative for problems of Bernoulli type,” Interfaces and Free Boundaries, vol. 11, no. 2, pp. 317–330, 2009. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. J. Haslinger, T. Kozubek, K. Kunisch, and G. Peichl, “Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type,” Computational Optimization and Applications, vol. 26, no. 3, pp. 231–251, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. K. Eppler and H. Harbrecht, “Shape optimization for free boundary problems-analysis and numerics,” in Constrained Optimization and Optimal Control for Partial Differential Equations, vol. 160, pp. 277–288, 2012. View at Google Scholar
  12. K. Eppler and H. Harbrecht, “Tracking Neumann data for stationary free boundary problems,” SIAM Journal on Control and Optimization, vol. 48, no. 5, pp. 2901–2916, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. K. Ito, K. Kunisch, and G. H. Peichl, “Variational approach to shape derivatives for a class of Bernoulli problems,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 126–149, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. R. Kohn and M. Vogelius, “Determining conductivity by boundary measurements,” Communications on Pure and Applied Mathematics, vol. 37, no. 3, pp. 289–298, 1984. View at Publisher · View at Google Scholar
  15. K. Eppler and H. Harbrecht, “On a Kohn-Vogelius like formulation of free boundary problems,” Computational Optimization and Applications, vol. 52, no. 1, pp. 69–85, 2012. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Delfour and J. Zolesio, Shapes and Geometries, SIAM, Philadelphia, Pa, USA, 2001.
  17. P. Ciarlet, Mathematical Elasticity I, Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 1988.
  18. J. Sokolowski and J. Zolesio, Introduction to Shape Optimization, Springer, Berlin, Germany, 1991.
  19. J. Bacani, Methods of shape optimization in free boundary problems [Ph.D. thesis], Karl-Franzens-Universitaet Graz, Graz, Austria, 2013.
  20. J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization (Theory, Approximation, and Computation), SIAM Advances and Control, Philadelphia, Pa, USA, 2003.
  21. T. Tiihonen, “Shape optimization and trial methods for free boundary problems,” Mathematical Modelling and Numerical Analysis, vol. 31, no. 7, pp. 805–825, 1997. View at Google Scholar · View at Scopus
  22. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing, Marshfield, Mass, USA, 1985.
  23. L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1998.
  24. V. Girault and P. Raviart, Finite Element Methods for Navier-Stoke's Equations (Theory and Algorithms), Springer, Berlin, Germany, 1986.
  25. A. Kufner, O. John, and S. Fucik, Function Spaces, Noordhoff International Publishing, Leyden, The Netherlands, 1977.
  26. M. C. Delfour and J. P. Zolésio, “Anatomy of the shape Hessian,” Annali di Matematica Pura ed Applicata, vol. 159, no. 1, pp. 315–339, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. J. Lamboley and M. Pierre, “Structure of shape derivatives around irregular domains and applications,” Journal of Convex Analysis, vol. 14, no. 4, pp. 807–822, 2007. View at Google Scholar · View at Zentralblatt MATH · View at Scopus