Abstract
In this paper we examine an evolution problem which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal compliance condition and a friction law which are nonmonotone, possibly multivalued and of the subdifferential form while the damping operator is assumed to be coercive and pseudomonotone. We derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish the a priori estimates and the existence of weak solutions by using a surjectivity result.
Supported in part by the State Committee for Scientific Research of the Republic of Poland (KBN) under Research Grants no. 2 P03A 003 25 and 4 T07A 027 26.
Chapter PDF
Similar content being viewed by others
Keywords
References
H.T. Banks, S. Reich, and I.G. Rosen. Estimation of nonlinear damping in second order distributed parameter systems. Control-Theory and Advanced Techn., 6:395–415, 1990.
J. Berkovits and V. Mustonen. Monotonicity methods for nonlinear evolution equations. Nonlinear Analysis, 27:1397–1405, 1996.
O Chau, W. Han, and M. Sofonea. A dynamic frictional contact problem with normal damped response. Acta Appl. Math., 71:159–178, 2002.
F.H. Clarke. Optimization and Nonsmooth Analysis. Wiley-Interscience, New York, 1983.
Z. Denkowski and S. Migórski. Existence of solutions to evolution second order hemivariational inequalities with multivalued damping, submitted.
Z. Denkowski, S. Migórski, and N.S. Papageorgiou. An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.
G. Duvaut and J.L. Lions. Les Inéquations en Mécanique et en Physique. Dunod, Paris, 1972.
L. Gasiński. Hyperbolic Hemivariational Inequalities and their Applications to Optimal Shape Design. PhD thesis, Jagiellonian University, Cracow, Poland, 2000.
L. Gasiński and M. Smołka. An existence theorem for wave-type hyperbolic hemivariational inequalities. Math. Nachr., 242:1–12, 2002.
D. Goeleven, M. Miettinen, and P.D. Panagiotopoulos. Dynamic hemivariational inequalities and their applications. J. Optimiz. Theory and Appl., 103:567–601, 1999.
W. Han and M. Sofonea. Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. AMS and International Press, 2002.
J. Haslinger, M. Miettinen, and P.D. Panagiotopoulos. Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer Academic Publishers, Boston, Dordrecht, London, 1999.
S. Hu and N.S. Papageorgiou. Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, 1997.
J. Jarusek. Dynamic contact problems with given friction for viscoelastic bodies. Czech. Math. J., 46:475–487, 1996.
K. Kuttler. Non-degenerate implicit evolution inclusions. Electronic J. Diff. Equations, 34:1–20, 2000.
K.L. Kuttler and M. Shillor. Set-valued pseudomonotone maps and degenerate evolution inclusions. Comm. Contemp. Math., 1:87–123, 1999.
J.L. Lions. Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Paris, 1969.
S. Migórski. Boundary hemivariational inequalities of hyperbolic type and applications, J. Global Optim., in press, 2004.
S. Migórski. Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Applicable Analysis, in press, 2004.
S. Migórski. Existence and convergence results for evolution hemivariational inequalities. Topological Methods Nonlinear Anal., 16:125–144, 2000.
S. Migórski. Evolution hemivariational inequalities in infinite dimension and their control. Nonlinear Analysis, 47:101–112, 2001.
S. Migórski. Modeling, Analysis and Optimal Control of Systems Governed by Hemivariational Inequalities, pp. 248–279, chapter in the book “Industrial Mathematics and Statistics” dedicated to commemorate the Golden Jubilee of Indian Institute of Technology, Kharagpur, India, 2002, J.C. Misra, ed., Narosa Publishing House. Delhi, 2003.
D. Motreanu and P.D. Panagiotopoulos. Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications. Kluwer Academic Publishers, Boston, Dordrecht, London, 1999.
Z. Naniewicz and P.D. Panagiotopoulos. Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995.
A. Ochal. Optimal Control of Evolution Hemivariational Inequalities. PhD thesis, Jagiellonian University, Cracow, Poland, 2001.
P.D. Panagiotopoulos. Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel, 1985.
P.D. Panagiotopoulos. Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer-Verlag, Berlin, 1993.
P.D. Panagiotopoulos. Hemivariational inequalities and fan-variational inequalities, new applications and results. Atti Sem. Mat. Fis. Univ. Modena, 43:159–191, 1995.
P.D. Panagiotopoulos. Modelling of nonconvex nonsmooth energy problems: dynamic hemivariational inequalities with impact effects. J. Comput. Appl. Math., 63:123–138, 1995.
P.D. Panagiotopoulos and G. Pop. On a type of hyperbolic variational-hemivariational inequalities. J. Applied Anal., 5(1):95–112, 1999.
M. Rochdi, M. Shillor, and M. Sofonea. A quasistatic contact problem with directional friction and damped response. Applicable Analysis, 68:409–422, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 International Federation for Information Processing
About this paper
Cite this paper
Denkowski, Z., Migórski, S. (2005). Existence of Solutions to Evolution Second Order Hemivariational Inequalities with Multivalued Damping. In: Cagnol, J., Zolésio, JP. (eds) System Modeling and Optimization. CSMO 2003. IFIP International Federation for Information Processing, vol 166. Springer, Boston, MA. https://doi.org/10.1007/0-387-23467-5_14
Download citation
DOI: https://doi.org/10.1007/0-387-23467-5_14
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4020-7760-9
Online ISBN: 978-0-387-23467-0
eBook Packages: Computer ScienceComputer Science (R0)