On the asymptotic behavior of variational inequalities set in cylinders

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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

Abstract
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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 and 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.
[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), 133-159. 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), 15-44. 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), 3579-3602. 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), 477-482. 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), 345-370. 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), 21-26. 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), 179-269. 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), 309-332. 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), 97-116. 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), 1-22.
[14]	J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519. 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), 269-290.
[16]	A., Rougirel, Unpublished results.
[17]	R. A. Toupin, Saint-Venant's principle, Arch. Rational Mech. Anal., 18 (1965), 83-96.
[18]	K. Yeressian, "Spatial Asymptotic Behaviour of Elliptic Equations and Variational Inequalities," Ph.D thesis, University of Zurich, 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), 237-250.
[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), 133-159. 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), 15-44. 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), 3579-3602. 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), 477-482. 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), 345-370. 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), 21-26. 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), 179-269. 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), 309-332. 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), 97-116. 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), 1-22.
[14]	J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519. 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), 269-290.
[16]	A., Rougirel, Unpublished results.
[17]	R. A. Toupin, Saint-Venant's principle, Arch. Rational Mech. Anal., 18 (1965), 83-96.
[18]	K. Yeressian, "Spatial Asymptotic Behaviour of Elliptic Equations and Variational Inequalities," Ph.D thesis, University of Zurich, 2010.

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