Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Optimal shape design for systems governed by variational inequalities, part 1: Existence theory for the elliptic case


Some general existence results for optimal shape design problems for systems governed by elliptic variational inequalities are established by the mapping method and variational convergence theory. Then, an existence theorem is given for the optimal shape for an electrochemical machining problem, in which the cost functional is not lower semicontinuous, by extending the general results to this case. Furthermore, this problem is approximated by a set of optimal shape design problems which have more smooth cost functionals and are easier to handle computationally.

This is a preview of subscription content, log in to check access.


  1. 1.

    Pironneau, J.,Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, New York, 1984.

  2. 2.

    Haug, E. J., andCea, J., Editors,Optimization of Distributed-Parameter Structures, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, 1980.

  3. 3.

    Murat, F., andSimon, J.,Optimal Control with Respect to the Domains, Thesis, University of Paris 6, 1977 (in French).

  4. 4.

    Murat, F., andSimon, J.,Study on Optimal Shape Design Problem, Optimization Techniques, Edited by J. Cea, Springer-Verlag, Berlin, Germany, Vol. 2, pp. 54–62, 1976.

  5. 5.

    Fujii, N.,Necessary Conditions for Domain Optimization Problems in Elliptic Boundary-Value Problems, SIAM Journal on Control and Optimization, Vol. 24, pp. 348–359, 1986.

  6. 6.

    Fujii, N.,Existence of an Optimal Domain in a Domain Optimization Problem, System Modelling and Optimization, Edited by M. Iri and R. Yajima, Springer-Verlag, Berlin, Germany, pp. 251–258, 1988.

  7. 7.

    Fujii, N.,Lower Semicontinuity in Domain Optimization Problems, Journal of Optimization Theory and Applications, Vol. 59, pp. 407–422, 1988.

  8. 8.

    Butt, R.,Optimal Shape Design for Variational Inequalities, Thesis, University of Leeds, 1988.

  9. 9.

    Delfour, M. C., andZolesio, J. P.,Shape Sensitivity Analysis via a Penalization Method, Annali di Mathematica Pura ed Applicata, Vol. 151, pp. 179–212, 1988.

  10. 10.

    Haslinger, J., andNeittaanmaki, P.,On the Existence of Optimal Shapes in Contact Problems: Perfectly Plastic Bodies, Computational Mechanics, Vol. 1, pp. 293–299, 1986.

  11. 11.

    Haslinger, J., andNettaanmaki, P.,On the Design for the Optimal Covering of an Obstacle, Boundary Control and Boundary Variations, Edited by J. P. Zolesio, Springer-Verlag, Berlin, Germany, pp. 192–211, 1988.

  12. 12.

    Haslinger, J., andNettaanmaki, P.,Finite-Element Approximation for Optimal Shape Design, John Wiley, Chichester, England, 1988.

  13. 13.

    Sokolowski, J., andZolesio, J. P.,Shape Sensitivity Analysis for Variational Inequalities, System Modeling and Optimization, Edited by R. F. Drenick and F. Kozin, Springer-Verlag, New York, New York, pp. 400–407, 1982.

  14. 14.

    Sokolowski, J., andZolesio, J. P.,Shape Sensitivity Analysis for Unilateral Problems, Publication Mathématique No. 67, Université de Nice, 1985.

  15. 15.

    Serrin, J.,On the Definition and Properties of Variational Integrals, Transactions of the American Mathematical Society, Vol. 101, pp. 139–167, 1961.

  16. 16.

    Lions, J. L.,Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod and Gauthier-Villars, Paris, France, 1969.

  17. 17.

    Barbu, V.,Optimal Control of Variational Inequalities, Pitman, London, England, 1984.

  18. 18.

    Attouch, H.,Variational Convergence for Functions and Operators, Pitman, London, England, 1984.

  19. 19.

    McGeough, J. A.,On the Derivation of the Quasi-steady Model in Electrochemical Machining, Journal of the Institute of Mathematics and Its Applications, Vol. 13, pp. 12–21, 1978.

  20. 20.

    McGeough, J. A.,Principles of Electrochemical Machining, Chapman and Hall, London, England, 1979.

  21. 21.

    Lacey, A. A., andSchillor, M.,Electrochemical and Electrodischarge Machining with a Threshold Current, IMA Journal of Applied Mathematics, Vol. 39, pp. 121–142, 1987.

  22. 22.

    McGeough, J. A.,Unsolved Moving Boundary Problems in Electrochemical Machining, Free Boundary Problems, Edited by A. Bossavit, Pitman, London, England, Vol. 3, pp. 120–127, 1985.

  23. 23.

    Elliott, C. M.,On a Variational Inequality Formulation of an Electrochemical Machining and Its Approximation by Finite Element Methods, Journal of the Institute of Mathematics and Its Applications, Vol. 25, pp. 121–131, 1980.

  24. 24.

    Saguez, C.,Contrôle Optimal d'un Système Gouverné par une Inéquation Variationnelle Parabolique, Comptes Rendus, Serie A, Vol. 287, pp. 957–959, 1978.

  25. 25.

    Lions, J. L.,The Work of G. Stampacchia in Variational Inequalities, Variational Inequalities and Complementarity Problems, Edited by C. Cottle and J. L. Lions, John Wiley, New York, New York, pp. 1–24, 1980.

  26. 26.

    Friedman, A.,Variational Principles and Free-Boundary Problems, John Wiley, New York, New York, 1982.

Download references

Author information

Additional information

The authors with to express their sincere thanks to the reviewers for supplying additional references and for their valuable comments, which made the paper more readable.

Communicated by E. J. Haug

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liu, W.B., Rubio, J.E. Optimal shape design for systems governed by variational inequalities, part 1: Existence theory for the elliptic case. J Optim Theory Appl 69, 351–371 (1991).

Download citation

Key Words

  • Optimal shape design
  • variational convergence theory
  • elliptic variational inequalities
  • existence theorems