November  2013, 33(11&12): 4875-4890. doi: 10.3934/dcds.2013.33.4875

On the asymptotic behavior of variational inequalities set in cylinders

1. 

Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich

2. 

Technische Universität Darmstadt, Department of Mathematics, Schlossgartenstr. 7, D-64289 Darmstadt, Germany

Received  September 2011 Revised  March 2012 Published  May 2013

We study the asymptotic behavior of solutions to variational inequalities with pointwise constraint on the value and gradient of the functions as the domain becomes unbounded. First, as a model problem, we consider the case when the constraint is only on the value of the functions. Then we consider the more general case of constraint also on the gradient. At the end we consider the case when there is no force term which corresponds to Saint-Venant principle for linear problems.
Citation: Michel Chipot, Karen Yeressian. On the asymptotic behavior of variational inequalities set in cylinders. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4875-4890. doi: 10.3934/dcds.2013.33.4875
References:
[1]

B. Brighi and S. Guesmia, On elliptic boundary value problems of order 2m in cylindrical domain of large size,, Adv. Math. Sci. Appl., 18 (2008), 237.

[2]

M. Chipot, "l goes to plus infinity,", Birkhäuser, (2002). doi: 10.1007/978-3-0348-8173-9.

[3]

M. Chipot and S. Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction,, J. Math. Pures Appl. (9), 90 (2008), 133. doi: 10.1016/j.matpur.2008.04.002.

[4]

M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded,, Commun. Contemp. Math., 4 (2002), 15. doi: 10.1142/S0219199702000555.

[5]

M. Chipot and A. Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded,, Trans. Amer. Math. Soc., 360 (2008), 3579. doi: 10.1090/S0002-9947-08-04361-4.

[6]

M. Chipot and Y. Xie, On the asymptotic behaviour of elliptic problems with periodic data,, C. R. Math. Acad. Sci. Paris, 339 (2004), 477. doi: 10.1016/j.crma.2004.09.007.

[7]

M. Chipot and Y. Xie, Elliptic problems with periodic data: An asymptotic analysis,, J. Math. Pures Appl. (9), 85 (2006), 345. doi: 10.1016/j.matpur.2005.07.002.

[8]

M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique,, C. R. Math. Acad. Sci. Paris, 346 (2008), 21. doi: 10.1016/j.crma.2007.12.004.

[9]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle,, Adv. in Appl. Mech., 23 (1983), 179. doi: 10.1016/S0065-2156(08)70244-8.

[10]

C. O. Horgan and L. E. Payne, Decay estimates for second-order quasilinear partial differential equations,, Adv. in Appl. Math., 5 (1984), 309. doi: 10.1016/0196-8858(84)90012-5.

[11]

C. O. Horgan and L. T. Wheeler, Spatial decay estimates for the Navier-Stokes equations with application to the problem of entry flow,, SIAM J. Appl. Math., 35 (1978), 97. doi: 10.1137/0135008.

[12]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).

[13]

J. K. Knowles, On Saint-Venant's principle in the two-dimensional linear theory of elasticity,, Arch. Rational Mech. Anal., 21 (1966), 1.

[14]

J. L. Lions and G. Stampacchia, Variational inequalities,, Comm. Pure Appl. Math., 20 (1967), 493. doi: 10.1002/cpa.3160200302.

[15]

O. A. Oleinik and G. A. Yosifian, Boundary value problems for second order elliptic equations in unbounded domains and Saint-Venant's principle,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4 (1977), 269.

[16]

A., Rougirel,, Unpublished results., ().

[17]

R. A. Toupin, Saint-Venant's principle,, Arch. Rational Mech. Anal., 18 (1965), 83.

[18]

K. Yeressian, "Spatial Asymptotic Behaviour of Elliptic Equations and Variational Inequalities,", Ph.D thesis, (2010).

show all references

References:
[1]

B. Brighi and S. Guesmia, On elliptic boundary value problems of order 2m in cylindrical domain of large size,, Adv. Math. Sci. Appl., 18 (2008), 237.

[2]

M. Chipot, "l goes to plus infinity,", Birkhäuser, (2002). doi: 10.1007/978-3-0348-8173-9.

[3]

M. Chipot and S. Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction,, J. Math. Pures Appl. (9), 90 (2008), 133. doi: 10.1016/j.matpur.2008.04.002.

[4]

M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded,, Commun. Contemp. Math., 4 (2002), 15. doi: 10.1142/S0219199702000555.

[5]

M. Chipot and A. Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded,, Trans. Amer. Math. Soc., 360 (2008), 3579. doi: 10.1090/S0002-9947-08-04361-4.

[6]

M. Chipot and Y. Xie, On the asymptotic behaviour of elliptic problems with periodic data,, C. R. Math. Acad. Sci. Paris, 339 (2004), 477. doi: 10.1016/j.crma.2004.09.007.

[7]

M. Chipot and Y. Xie, Elliptic problems with periodic data: An asymptotic analysis,, J. Math. Pures Appl. (9), 85 (2006), 345. doi: 10.1016/j.matpur.2005.07.002.

[8]

M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique,, C. R. Math. Acad. Sci. Paris, 346 (2008), 21. doi: 10.1016/j.crma.2007.12.004.

[9]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle,, Adv. in Appl. Mech., 23 (1983), 179. doi: 10.1016/S0065-2156(08)70244-8.

[10]

C. O. Horgan and L. E. Payne, Decay estimates for second-order quasilinear partial differential equations,, Adv. in Appl. Math., 5 (1984), 309. doi: 10.1016/0196-8858(84)90012-5.

[11]

C. O. Horgan and L. T. Wheeler, Spatial decay estimates for the Navier-Stokes equations with application to the problem of entry flow,, SIAM J. Appl. Math., 35 (1978), 97. doi: 10.1137/0135008.

[12]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).

[13]

J. K. Knowles, On Saint-Venant's principle in the two-dimensional linear theory of elasticity,, Arch. Rational Mech. Anal., 21 (1966), 1.

[14]

J. L. Lions and G. Stampacchia, Variational inequalities,, Comm. Pure Appl. Math., 20 (1967), 493. doi: 10.1002/cpa.3160200302.

[15]

O. A. Oleinik and G. A. Yosifian, Boundary value problems for second order elliptic equations in unbounded domains and Saint-Venant's principle,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4 (1977), 269.

[16]

A., Rougirel,, Unpublished results., ().

[17]

R. A. Toupin, Saint-Venant's principle,, Arch. Rational Mech. Anal., 18 (1965), 83.

[18]

K. Yeressian, "Spatial Asymptotic Behaviour of Elliptic Equations and Variational Inequalities,", Ph.D thesis, (2010).

[1]

Liping Pang, Fanyun Meng, Jinhe Wang. Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-23. doi: 10.3934/jimo.2018116

[2]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437

[3]

S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155

[4]

Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206

[5]

Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721

[6]

Mykhailo Potomkin. Asymptotic behavior of thermoviscoelastic Berger plate. Communications on Pure & Applied Analysis, 2010, 9 (1) : 161-192. doi: 10.3934/cpaa.2010.9.161

[7]

Hunseok Kang. Asymptotic behavior of a discrete turing model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 265-284. doi: 10.3934/dcds.2010.27.265

[8]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165

[9]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621

[10]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183

[11]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[12]

Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545

[13]

Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673

[14]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645

[15]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

[16]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71

[17]

Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331

[18]

G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583

[19]

Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091

[20]

Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]