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Inverse Coefficient Problem for Elliptic Hemivariational Inequality

  • S. Migórski
  • A. Ochal
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 50)

Abstract

In this paper we consider the problem of identification of a discontinuous coefficient in elliptic hemivariational inequality. First we prove an existence theorem for an inverse problem and we establish the boundary homogenization result for the direct problem. Then we study the asymptotic behavior of the set of solutions to the inverse problem. We show that the solution set to the inverse problem for homogenized hemivariational inequality has the upper semicontinuity property with respect to the solution set of the original identification problem.

Keywords

Inverse Problem Direct Problem Hemivariational Inequality Semicontinuity Property Lipschitz Continuous Boundary 
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References

  1. Aubin, J.-P., and Cellina, A. (1984). Differential Inclusions, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo.MATHGoogle Scholar
  2. Banks, H.T., and Ito, K. (1988). A unified framework for approximation Min-verse problems for distributed parameter systems, Control-Theory and Advanced Technology, 4: 73–90.MathSciNetGoogle Scholar
  3. Banks, H.T., and Rebnord, D.A. (1991). Estimation of material parameters for grid structures, Journal of Mathematical Systems, Estimation and Control, 1: 107–130.MathSciNetGoogle Scholar
  4. Chang, K.C. (1981). Variational methods for nondifferentiable functionals and applications to partial differential equations, Journal of Mathematical Analysis and Applications, 80: 102–129.MathSciNetMATHCrossRefGoogle Scholar
  5. Clarke, F.H. (1983). Optimization and Nonsmooth Analysis, Wiley - Inter-science, New York.MATHGoogle Scholar
  6. Damlamian, A., and Ta-tsien, Li. (1987). Boundary homogenization for elliptic problems, Journal de Mathematiques Pures etAppliquees, 66: 351–361.MathSciNetMATHGoogle Scholar
  7. Denkowski, Z., and Migórski, S. (1998). Optimal shape design problems for a class of systems described by hemivariational inequalities, Journal of Global Optimization, 12: 37–59.MATHCrossRefGoogle Scholar
  8. Giusti, E. (1984). Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, Basel, Stuttgart.MATHGoogle Scholar
  9. Gutman, S. (1990). Identification of discontinuous parameters in flow equations, SIAM Journal of Control Optimization, 28: 1049–1060.MATHCrossRefGoogle Scholar
  10. Haslinger, J., and Panagiotopoulos, P.D. (1995). Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Analysis, Theory, Methods, and Applications, 24: 105–119.MathSciNetMATHGoogle Scholar
  11. Miettinen, M., and Haslinger, J. (1992). Approximation of optimal control problems of hemivariational inequalities, Numerical Functional Analysis and Optimization, 13: 43–68.MathSciNetMATHCrossRefGoogle Scholar
  12. Migórski, S. (1993). Stability of Parameter Identification Problems with Applications to Nonlinear Evolution Systems, Dynamics Systems and Applications, 2: 387–404.MATHGoogle Scholar
  13. Migórski, S. (1996). Boundary Homogenization Technique for Estimation of Coefficients in Elliptic Equations, Proceedings of 2nd International Conference on Inverse Problems in Engineering: Theory and Practice, Port aux Rocs, Le Croisic, France, June 9–14,1996, D. Delaunay et al., Eds., 375–384, The American Society of Mechanical Engineers.Google Scholar
  14. Migórski, S. (1998). Identification of nonlinear heat transfer laws in problems modeled by hemivariational inequalities, Proceedings of International Symposium on Inverse Problems in Engineering Mechanics1998 (ISIP’98), Nagano, Japan, March 24–27, 1998, M. Tanaka and G. S. Dulikravich, Eds., Elsevier Science B.V., 27–37.Google Scholar
  15. Migórski, S. (1999). Sensitivity Analysis of Inverse Problems with Applications to Nonlinear Systems, Dynamic Systems and Applications, 8 (1): 73–89.MathSciNetMATHGoogle Scholar
  16. Migórski, S., and Ochal, A. (2000). Optimal control of parabolic hemivariational inequalities, J. Global Optimization, to appear.Google Scholar
  17. Naniewicz, Z., and Panagiopopoulos, P.D. (1995). Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong.Google Scholar
  18. Ochal, A. (2000). Domain Identification Problem for Elliptic Hemivariational Inequalities, Topological Methods in Nonlinear Analysis, Submitted.Google Scholar
  19. Panagiotopoulos, P.D. (1985). Inequality Problems in Mechanics and Applica-tions. Convex and Nonconvex Energy Functions, Birkhäuser, Basel.Google Scholar
  20. Panagiotopoulos, P.D. (1993). Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin.MATHGoogle Scholar
  21. Yongji, T. (1986). An inverse problem for nonlocal elliptic BVP and resistivity identification, Lecture Notes in Mathematics, 1306: 149–159.Google Scholar
  22. Zeidler, E. (1990). Nonlinear Functional Analysis and Applications II A/B, Springer, New York.Google Scholar
  23. Zhikov, V.V., Kozlov, S.M., and Oleinik, O.A. (1994). Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin.MATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • S. Migórski
    • 1
  • A. Ochal
    • 1
  1. 1.Institute of Computer ScienceJagiellonian UniversityCracowPoland

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