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Optimal Control of the Obstacle Problem in a Perforated Domain

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Abstract

We study the problem of optimally controlling the solution of the obstacle problem in a domain perforated by small periodically distributed holes. The solution is controlled by the choice of a perforated obstacle which is to be chosen in such a fashion that the solution is close to a given profile and the obstacle is not too irregular. We prove existence, uniqueness and stability of an optimal obstacle and derive necessary and sufficient conditions for optimality. When the number of holes increase indefinitely we determine the limit of the sequence of optimal obstacles and solutions. This limit depends strongly on the rate at which the size of the holes shrink.

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References

  1. Ammari, H., Kang, H.: Polarization and Moment Tensors. Applied Mathematical Sciences, vol. 162. Springer, New York (2007). With applications to inverse problems and effective medium theory

    MATH  Google Scholar 

  2. Adams, D.R., Lenhart, S.M., Yong, J.: Optimal control of the obstacle for an elliptic variational inequality. Appl. Math. Optim. 38(2), 121–140 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Attouch, H., Picard, C.: Variational inequalities with varying obstacles: the general form of the limit problem. J. Funct. Anal. 50(3), 329–386 (1983)

    Article  MathSciNet  Google Scholar 

  4. Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4(4–5), 383–402 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carbone, L., Colombini, F.: On convergence of functionals with unilateral constraints. J. Math. Pures Appl. (9) 59(4), 465–500 (1980)

    MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Lee, K.-a.: Viscosity method for homogenization of highly oscillating obstacles. Indiana Univ. Math. J. 57(4), 1715–1741 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cioranescu, D., Murat, F.: A strange term coming from nowhere. In: Topics in the Mathematical Modelling of Composite Materials, pp. 45–93 (1997)

    Chapter  Google Scholar 

  8. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  9. Helms, L.L.: Introduction to Potential Theory. Pure and Applied Mathematics, vol. XXII. Wiley-Interscience, New York (1969)

    MATH  Google Scholar 

  10. Haslinger, J., Makinen, R.A.E.: Introduction to Shape Optimization. Advances in Design and Control, vol. 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2003). Theory, approximation, and computation

    Book  MATH  Google Scholar 

  11. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Classics in Applied Mathematics, vol. 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000). Reprint of the 1980 original

    Book  MATH  Google Scholar 

  12. Kesavan, S., Saint Jean Paulin, J.: Optimal control on perforated domains. J. Math. Anal. Appl. 229(2), 563–586 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rajesh, M.: Convergence of some energies for the Dirichlet problem in perforated domains. Rend. Mat. Appl. (7) 21(1–4), 259–274 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27–59 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Saint Jean Paulin, J., Zoubairi, H.: Optimal control and “strange term” for a Stokes problem in perforated domains. Port. Math. 59(2), 161–178 (2002)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden

    Martin H. Strömqvist

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  1. Martin H. Strömqvist
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Correspondence to Martin H. Strömqvist.

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Strömqvist, M.H. Optimal Control of the Obstacle Problem in a Perforated Domain. Appl Math Optim 66, 239–255 (2012). https://doi.org/10.1007/s00245-012-9170-4

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