A nonlinear history-dependent boundary value problem
Authors:
Mircea Sofonea and Aissa Benseghir
Journal:
Quart. Appl. Math. 75 (2017), 181-199
MSC (2010):
Primary 35A01, 35Q74, 35R03, 47J20, 49J40
DOI:
https://doi.org/10.1090/qam/1456
Published electronically:
August 29, 2016
MathSciNet review:
3580100
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider a nonlinear boundary value problem with unilateral constraints in a two-dimensional rectangle. We derive a variational formulation of the problem which is in the form of a history-dependent variational inequality. Then, we establish the existence of a unique weak solution to the problem. We also prove two convergence results. The first one provides the continuous dependence of the solution with respect to the unilateral constraint. The second one shows the convergence of the solution of the penalized problem to the solution of the original problem, as the penalization parameter converges to zero.
References
- Claudio Baiocchi and António Capelo, Variational and quasivariational inequalities, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. Applications to free boundary problems; Translated from the Italian by Lakshmi Jayakar. MR 745619
- Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005
- Roland Glowinski, Jacques-Louis Lions, and Raymond Trémolières, Numerical analysis of variational inequalities, Studies in Mathematics and its Applications, vol. 8, North-Holland Publishing Co., Amsterdam-New York, 1981. Translated from the French. MR 635927
- Weimin Han and B. Daya Reddy, Plasticity, Interdisciplinary Applied Mathematics, vol. 9, Springer-Verlag, New York, 1999. Mathematical theory and numerical analysis. MR 1681061
- Weimin Han, Stanisław Migórski, and Mircea Sofonea (eds.), Advances in variational and hemivariational inequalities, Advances in Mechanics and Mathematics, vol. 33, Springer, Cham, 2015. Theory, numerical analysis, and applications. MR 3380521
- Weimin Han and Mircea Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1935666
- I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek, Solution of variational inequalities in mechanics, Applied Mathematical Sciences, vol. 66, Springer-Verlag, New York, 1988. Translated from the Slovak by J. Jarník. MR 952855
- Kamran Kazmi, Mikael Barboteu, Weimin Han, and Mircea Sofonea, Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 3, 919–942. MR 3264340, DOI https://doi.org/10.1051/m2an/2013127
- N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258
- David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Classics in Applied Mathematics, vol. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1980 original. MR 1786735
- Stanislaw Migorski, Meir Shillor, and Mircea Sofonea, Editorial [Special section: Contact mechanics], Nonlinear Anal. Real World Appl. 22 (2015), 435–436. MR 3280844, DOI https://doi.org/10.1016/j.nonrwa.2014.10.005
- Jindřich Nečas and Ivan Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Studies in Applied Mechanics, vol. 3, Elsevier Scientific Publishing Co., Amsterdam-New York, 1980. MR 600655
- P. D. Panagiotopoulos, Inequality problems in mechanics and applications, Birkhäuser Boston, Inc., Boston, MA, 1985. Convex and nonconvex energy functions. MR 896909
- M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Physics 655, Springer, Berlin, 2004.
- M. Sofonea and K. Bartosz, A dynamic contact model for viscoelastic plates, submitted for publication in Quarterly Journal of Mechanics and Applied Mathematics, 2015.
- Mircea Sofonea and Andaluzia Matei, History-dependent quasi-variational inequalities arising in contact mechanics, European J. Appl. Math. 22 (2011), no. 5, 471–491. MR 2834015, DOI https://doi.org/10.1017/S0956792511000192
- M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series 398, Cambridge University Press, Cambridge, 2012.
- M. Sofonea and Y. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Applicable Analysis, DOI 10.1080/00036811.2015.1093623, published on line: October 7, 2015.
References
- Claudio Baiocchi and António Capelo, Variational and quasivariational inequalities, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. Applications to free boundary problems; Translated from the Italian by Lakshmi Jayakar. MR 745619
- Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005
- Roland Glowinski, Jacques-Louis Lions, and Raymond Trémolières, Numerical analysis of variational inequalities, Studies in Mathematics and its Applications, vol. 8, North-Holland Publishing Co., Amsterdam-New York, 1981. Translated from the French. MR 635927
- Weimin Han and B. Daya Reddy, Plasticity, Interdisciplinary Applied Mathematics, vol. 9, Springer-Verlag, New York, 1999. Mathematical theory and numerical analysis. MR 1681061
- Advances in variational and hemivariational inequalities, Advances in Mechanics and Mathematics, vol. 33, Springer, Cham, 2015. Theory, numerical analysis, and applications; Edited by Weimin Han, Stanisław Migórski and Mircea Sofonea. MR 3380521
- Weimin Han and Mircea Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1935666
- I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek, Solution of variational inequalities in mechanics, Applied Mathematical Sciences, vol. 66, Springer-Verlag, New York, 1988. Translated from the Slovak by J. Jarník. MR 952855
- Kamran Kazmi, Mikael Barboteu, Weimin Han, and Mircea Sofonea, Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 3, 919–942. MR 3264340, DOI https://doi.org/10.1051/m2an/2013127
- N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258
- David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Classics in Applied Mathematics, vol. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1980 original. MR 1786735
- Stanislaw Migorski, Meir Shillor, and Mircea Sofonea, Editorial [Special section: Contact mechanics], Nonlinear Anal. Real World Appl. 22 (2015), 435–436. MR 3280844, DOI https://doi.org/10.1016/j.nonrwa.2014.10.005
- Jindřich Nečas and Ivan Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Studies in Applied Mechanics, vol. 3, Elsevier Scientific Publishing Co., Amsterdam-New York, 1980. MR 600655
- P. D. Panagiotopoulos, Inequality problems in mechanics and applications, Birkhäuser Boston, Inc., Boston, MA, 1985. Convex and nonconvex energy functions. MR 896909
- M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Physics 655, Springer, Berlin, 2004.
- M. Sofonea and K. Bartosz, A dynamic contact model for viscoelastic plates, submitted for publication in Quarterly Journal of Mechanics and Applied Mathematics, 2015.
- Mircea Sofonea and Andaluzia Matei, History-dependent quasi-variational inequalities arising in contact mechanics, European J. Appl. Math. 22 (2011), no. 5, 471–491. MR 2834015, DOI https://doi.org/10.1017/S0956792511000192
- M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series 398, Cambridge University Press, Cambridge, 2012.
- M. Sofonea and Y. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Applicable Analysis, DOI 10.1080/00036811.2015.1093623, published on line: October 7, 2015.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35A01,
35Q74,
35R03,
47J20,
49J40
Retrieve articles in all journals
with MSC (2010):
35A01,
35Q74,
35R03,
47J20,
49J40
Additional Information
Mircea Sofonea
Affiliation:
Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan Cedex, France
Email:
sofonea@univ-perp.fr
Aissa Benseghir
Affiliation:
Université Ferhat Abbas Sétif 1, El Bez Sétif 19000, Algérie
Email:
aissa5919@yahoo.fr
Keywords:
Nonlinear problem,
unilateral constraint,
history-dependent variational inequality,
weak solution,
penalty method,
convergence results
Received by editor(s):
March 3, 2016
Published electronically:
August 29, 2016
Article copyright:
© Copyright 2016
Brown University
