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Minimization arguments in analysis of variational–hemivariational inequalities

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Abstract

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

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References

  1. Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)

    MATH  Google Scholar 

  2. Capatina, A.: Variational Inequalities: Frictional Contact Problems. Advances in Mechanics and Mathematics, vol. 31. Springer, New York (2014)

    MATH  Google Scholar 

  3. Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  Google Scholar 

  4. Eck, C., Jarušek, J., Krbeč, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics, vol. 270. Chapman/CRC Press, New York (2005)

    Book  Google Scholar 

  5. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  6. Han, W.: Numerical analysis of stationary variational–hemivariational inequalities with applications in contact mechanics. Math. Mech. Solids 23, 279–293 (2018)

    Article  MathSciNet  Google Scholar 

  7. Han, W.: Minimization principles for elliptic hemivariational inequalities. Nonlinear Anal. B Real World Appl. 54, 103114 (2020)

    Article  MathSciNet  Google Scholar 

  8. Han, W.: A revisit of elliptic variational–hemivariational inequalities. Numer. Funct. Anal. Optim. 42, 371–395 (2021)

    Article  MathSciNet  Google Scholar 

  9. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics, vol. 30. American Mathematical Society, Providence (2002)

    Book  Google Scholar 

  10. Han, W., Sofonea, M.: Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numer. 28, 175–286 (2019)

    Article  MathSciNet  Google Scholar 

  11. Han, W., Sofonea, M., Barboteu, M.: Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 57, 640–663 (2017)

    Article  MathSciNet  Google Scholar 

  12. Han, W., Sofonea, M., Danan, D.: Numerical analysis of stationary variational–hemivariational inequalities. Numer. Math. 139, 563–592 (2018)

    Article  MathSciNet  Google Scholar 

  13. Han, W., Zeng, S.: On convergence of numerical methods for variational–hemivariational inequalities under minimal solution regularity. Appl. Math. Lett. 93, 105–110 (2019)

    Article  MathSciNet  Google Scholar 

  14. Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications, Kluwer Academic Publishers, Boston (1999)

    Book  Google Scholar 

  15. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)

    Book  Google Scholar 

  16. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)

    Book  Google Scholar 

  17. Migórski, S., Ochal, A., Sofonea, M.: A class of variational–hemivariational inequalities in reflexive Banach spaces. J. Elast. 127, 151–178 (2017)

    Article  MathSciNet  Google Scholar 

  18. Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)

    Article  MathSciNet  Google Scholar 

  19. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York (1995)

    MATH  Google Scholar 

  20. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)

    Book  Google Scholar 

  21. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  Google Scholar 

  22. Sofonea, M., Bollati, J., Tarzia, D.A.: Optimal control of differential quasivariational inequalities with applications in contact mechanics. J. Math. Anal. Appl. 493, 124567 (2021)

    Article  MathSciNet  Google Scholar 

  23. Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  24. Sofonea, M., Migórski, S.: Variational–Hemivariational Inequalities with Applications. Pure and Applied Mathematics, Chapman & Hall/CRC Press, Boca Raton (2018)

    MATH  Google Scholar 

  25. Xiao, Y.B., Sofonea, M.: On the optimal control of variational–hemivariational inequalities. J. Math. Anal. Appl. 475, 364–384 (2019)

    Article  MathSciNet  Google Scholar 

  26. Zeng, B., Liu, Z., Migorski, S.: On convergence of solutions to variational–hemivariational inequalities. Z. Angew. Math. Phys. 69, 87 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research was supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 823731 CONMECH, and by Simons Foundation Collaboration Grants, No. 850737. The authors are grateful to the anonymous referees for their valuable comments and suggestions on the original manuscript.

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  1. Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan, France

    Mircea Sofonea

  2. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Weimin Han

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  1. Mircea Sofonea
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  2. Weimin Han
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Correspondence to Mircea Sofonea.

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Sofonea, M., Han, W. Minimization arguments in analysis of variational–hemivariational inequalities. Z. Angew. Math. Phys. 73, 6 (2022). https://doi.org/10.1007/s00033-021-01638-z

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  • DOI: https://doi.org/10.1007/s00033-021-01638-z

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