American Institute of Mathematical Sciences

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

On the asymptotic behavior of variational inequalities set in cylinders

Michel Chipot 1, and  Karen Yeressian 2,

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.
Keywords: asymptotic behavior., Variational inequality.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35J86, 35B0.
Citation: Michel Chipot, Karen Yeressian. On the asymptotic behavior of variational inequalities set in cylinders. Discrete & Continuous Dynamical Systems, 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), 237-250.  Google Scholar

[2]

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

[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), 133-159. doi: 10.1016/j.matpur.2008.04.002.  Google Scholar

[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), 15-44. doi: 10.1142/S0219199702000555.  Google Scholar

[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), 3579-3602. doi: 10.1090/S0002-9947-08-04361-4.  Google Scholar

[6]

M. Chipot and Y. Xie, On the asymptotic behaviour of elliptic problems with periodic data, C. R. Math. Acad. Sci. Paris, 339 (2004), 477-482. doi: 10.1016/j.crma.2004.09.007.  Google Scholar

[7]

M. Chipot and Y. Xie, Elliptic problems with periodic data: An asymptotic analysis, J. Math. Pures Appl. (9), 85 (2006), 345-370. doi: 10.1016/j.matpur.2005.07.002.  Google Scholar

[8]

M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci. Paris, 346 (2008), 21-26. doi: 10.1016/j.crma.2007.12.004.  Google Scholar

[9]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle, Adv. in Appl. Mech., 23 (1983), 179-269. doi: 10.1016/S0065-2156(08)70244-8.  Google Scholar

[10]

C. O. Horgan and L. E. Payne, Decay estimates for second-order quasilinear partial differential equations, Adv. in Appl. Math., 5 (1984), 309-332. doi: 10.1016/0196-8858(84)90012-5.  Google Scholar

[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), 97-116. doi: 10.1137/0135008.  Google Scholar

[12]

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

[13]

J. K. Knowles, On Saint-Venant's principle in the two-dimensional linear theory of elasticity, Arch. Rational Mech. Anal., 21 (1966), 1-22.  Google Scholar

[14]

J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302.  Google Scholar

[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), 269-290.  Google Scholar

[16]

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

[17]

R. A. Toupin, Saint-Venant's principle, Arch. Rational Mech. Anal., 18 (1965), 83-96.  Google Scholar

[18]

K. Yeressian, "Spatial Asymptotic Behaviour of Elliptic Equations and Variational Inequalities," Ph.D thesis, University of Zurich, 2010. Google Scholar

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), 237-250.  Google Scholar

[2]

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

[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), 133-159. doi: 10.1016/j.matpur.2008.04.002.  Google Scholar

[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), 15-44. doi: 10.1142/S0219199702000555.  Google Scholar

[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), 3579-3602. doi: 10.1090/S0002-9947-08-04361-4.  Google Scholar

[6]

M. Chipot and Y. Xie, On the asymptotic behaviour of elliptic problems with periodic data, C. R. Math. Acad. Sci. Paris, 339 (2004), 477-482. doi: 10.1016/j.crma.2004.09.007.  Google Scholar

[7]

M. Chipot and Y. Xie, Elliptic problems with periodic data: An asymptotic analysis, J. Math. Pures Appl. (9), 85 (2006), 345-370. doi: 10.1016/j.matpur.2005.07.002.  Google Scholar

[8]

M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci. Paris, 346 (2008), 21-26. doi: 10.1016/j.crma.2007.12.004.  Google Scholar

[9]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle, Adv. in Appl. Mech., 23 (1983), 179-269. doi: 10.1016/S0065-2156(08)70244-8.  Google Scholar

[10]

C. O. Horgan and L. E. Payne, Decay estimates for second-order quasilinear partial differential equations, Adv. in Appl. Math., 5 (1984), 309-332. doi: 10.1016/0196-8858(84)90012-5.  Google Scholar

[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), 97-116. doi: 10.1137/0135008.  Google Scholar

[12]

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

[13]

J. K. Knowles, On Saint-Venant's principle in the two-dimensional linear theory of elasticity, Arch. Rational Mech. Anal., 21 (1966), 1-22.  Google Scholar

[14]

J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302.  Google Scholar

[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), 269-290.  Google Scholar

[16]

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

[17]

R. A. Toupin, Saint-Venant's principle, Arch. Rational Mech. Anal., 18 (1965), 83-96.  Google Scholar

[18]

K. Yeressian, "Spatial Asymptotic Behaviour of Elliptic Equations and Variational Inequalities," Ph.D thesis, University of Zurich, 2010. Google Scholar

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