Skip to main content
SpringerLink
Log in

A Duality Approach in Some Boundary Value Problems

  • Chapter
  • First Online:
Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

Part of the book series: Springer INdAM Series ((SINDAMS,volume 22))

  • 632 Accesses

Abstract

We describe several results from the literature concerning approximation procedures for variational boundary value problems, via duality techniques. Applications in shape optimization are also indicated. Some properties are quite unexpected and this is an argument that the present duality approach may be of interest in a large class of problems.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arnăutu, V., Langmach, H., Sprekels, J., Tiba, D.: On the approximation and optimization of plates. Numer. Funct. Anal. Optim. 21(3–4), 337–354 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barbu, V., Precupanu, T.: Convexity and optimization in Banach spaces. Sijthoff & Noerdhoff, Alphen aan de Rijn (1978)

    MATH  Google Scholar 

  3. Beirao da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual elements methods. Math. Models Methods Appl. Sci. 23, 114–214 (2013)

    Google Scholar 

  4. Berg, J., Nordstrom, J.: Duality based boundary treatment for the Euler and Navier- Stokes equations. In: AIAA Aerospace Sciences - Fluid Sciences Event, pp. 1–19 (2013). http://dx.doi.org/10.2514/6.2013-2959

  5. Chenais, D., Paumier, J.-C.: On the locking phenomenon for a class of elliptic problems. Numer. Math. 67, 427–440 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chenais, D., Zerner, M.: Numerical methods for elliptic boundary value problems with singular dependence on a small parameter, necessary conditions. Comput. Methods Appl. Mech. Eng. 115(1–2), 145–163 (1994)

    Article  Google Scholar 

  7. Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  8. De Szoeke, R.A., Samelson, R.M.: The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. Oceanogr. 32, 2194–2203 (2002)

    Article  MathSciNet  Google Scholar 

  9. Gao, D.Y.: Nonlinear elastic beam theory with application in contact problems and variational approaches. Mech. Res. Commun. 23, 11–17 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ghoussoub, N.: Self-Dual Partial Differential Systems and Their Variational Principles. Springer, New York (2008)

    MATH  Google Scholar 

  11. Gomes, D.A.: Duality principles for fully nonlinear elliptic equations. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 61, pp. 125–136. Springer, Berlin (2005)

    Google Scholar 

  12. Harvey, F.R., Lawson, H.B. Jr.: Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. J. Differ. Geom. 88, 395–482 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ignat, A., Sprekels, J., Tiba, D.: Analysis and Optimization of nonsmooth arches. SICON 40(4), 1107–1133 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Krell, S., Manzini, G.: The discrete duality finite volume method for the stokes equations on 3-d polyhedral meshes. SIAM J. Numer. Anal. 50(2), 808–837 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Machalova, J., Netuka, H.: Solution to contact problems for nonlinear Gao beam and obstacle. J. Appl. Math. 2015, 12 pp. (2015). art. 420649 http://dx.doi.org/10.1155/2015/420649

  16. Merluşcă, D.: A duality algorithm for the obstacle problem. Ann. Acad. Rom. Sci. Ser. Math. Appl. 5(1–2), 209–215 (2013)

    MATH  MathSciNet  Google Scholar 

  17. Merluşcă, D.: A duality-type method for the obstacle problem. An. Şt. Univ. “Ovidius”, Constanţa 21(3), 181–195 (2013)

    Google Scholar 

  18. Merluşcă, D.: A duality-type method for the fourth order obstacle problem. U.P.B. Sci. Bull., Ser. A 76(2), 147–158 (2014)

    Google Scholar 

  19. Neittaanmäki, P., Sprekels, J., Tiba, D.: Optimization of Elliptic Systems. Theory and Applications. Springer, New York (2006)

    MATH  Google Scholar 

  20. Nguyen, V.P., Rabczak, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sprekels, J., Tiba, D.: A duality approach in the optimization of beams and plates. SICON 37(2), 486–501 (1998/1999)

    Google Scholar 

  22. Sprekels, J., Tiba, D.: Optimal design of mechanical structures. In: Imanuvilov, O., Leugering, G., Triggiani, R., et al. (eds.) Control Theory of Partial Differential Equations. Pure and Applied Mathematics, vol. 242, pp, 259–271. Chapman and Hall/CRC, Boca Raton (2005)

    Google Scholar 

  23. Sprekels, J., Tiba, D.: Extensions of the control variational method. Control Cybern. 40(4), 1099–1108 (2011)

    MATH  MathSciNet  Google Scholar 

  24. Tiba, D.: A duality approximation of some nonlinear PDE’s. Ann. Acad. Rom. Sci. Ser. Math. Appl. 8(1), 68–77 (2016)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by CNCS Romania under Grant 211/2011.

Author information

Authors and Affiliations

  1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702, Bucharest, Romania

    Dan Tiba

Authors
  1. Dan Tiba
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Tiba, D. (2017). A Duality Approach in Some Boundary Value Problems. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_20

Download citation

Publish with us

Policies and ethics

Search

Navigation