Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity

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December 2011, 31(4): 1293-1305. doi: 10.3934/dcds.2011.31.1293

Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity

Youri V. Egorov ^1, and Evariste Sanchez-Palencia ^2,

1.	IMT, Université Paul Sabatier, 118, route de Narbonne, Toulouse, 31062, France
2.	Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie 4, place Jussieu, Paris, 75252, France

Received June 2010 Revised October 2010 Published September 2011

Abstract
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Some problems of elasticity and shell theory are considered. The common feature of these problems is the presence of a small parameter $\varepsilon$. If $\varepsilon>0$ the corresponding equations are elliptic and the boundary conditions satisfy the Shapiro - Lopatinsky condition. When $\varepsilon=0$, this condition is violated and the problem can be non-solvable in the distribution spaces. The rather difficult passing to the limit is studied using the related Cauchy problem for elliptic equations. This approach allows to show that the most important is the transition zone where the frequencies $|\xi|\asymp \log (\varepsilon^{-1})$.

Keywords: Two-dimensional elasticity, Cauchy problem., Elliptic boundary value problems.

Mathematics Subject Classification: Primary: 35J25; Secondary: 74B0.

Citation: Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293

References:

[1]	S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I., Comm. Pure. Applied Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.
[2]	F. Béchet, O. Millet and E. Sanchez-Palencia, Singular perturbations generating complexification phenomena in elliptic shells, Comput. Mech., 43 (2008), 207-221.
[3]	R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations," Interscience Publishers, New York-London, 1962.
[4]	Yu. V. Egorov and V. A. Kondratév, The oblique derivative problem, Matem. sbornik (N.S.), 78 (1969), 148-176.
[5]	Yu. V. Egorov, N. Meunier and E. Sanchez-Palencia, Rigorous and heuristic treatment of certain sensitive singular perturbations, Journal Math. Pures et Appliques (9), 88 (2007), 123-147. doi: 10.1016/j.matpur.2007.04.010.
[6]	Yu. V. Egorov, N. Meunier and E. Sanchez-Palencia, "Rigorous and Heuristic Treatment of Sensitive Singular Perturbations Arising in Elliptic Shells," Around the research of V. Maz'ya, II, Int. Math. Ser. (N.Y.), 12, Springer, New York, (2010), 159-202.
[7]	Yu. V. Egorov and M. A. Shubin, "Foundations of the Classical Theory of Partial Differential Equations," Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 1998.
[8]	J. Hadamard, "Lectures on Cauchy's Problem for Linear Partial Differential Equations," Dover, 1952.
[9]	P. R. Popivanov and D. K. Palagachev, "The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations," Mathematical Research, 93, Akademie Verlag, Berlin, 1997.
[10]	L. Schwartz, "Théorie des Distributions," Hermann, Paris, 1961.
[11]	M. E. Taylor, "Pseudodifferential Operators," Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981.

show all references

References:

[1]	S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I., Comm. Pure. Applied Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.
[2]	F. Béchet, O. Millet and E. Sanchez-Palencia, Singular perturbations generating complexification phenomena in elliptic shells, Comput. Mech., 43 (2008), 207-221.
[3]	R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations," Interscience Publishers, New York-London, 1962.
[4]	Yu. V. Egorov and V. A. Kondratév, The oblique derivative problem, Matem. sbornik (N.S.), 78 (1969), 148-176.
[5]	Yu. V. Egorov, N. Meunier and E. Sanchez-Palencia, Rigorous and heuristic treatment of certain sensitive singular perturbations, Journal Math. Pures et Appliques (9), 88 (2007), 123-147. doi: 10.1016/j.matpur.2007.04.010.
[6]	Yu. V. Egorov, N. Meunier and E. Sanchez-Palencia, "Rigorous and Heuristic Treatment of Sensitive Singular Perturbations Arising in Elliptic Shells," Around the research of V. Maz'ya, II, Int. Math. Ser. (N.Y.), 12, Springer, New York, (2010), 159-202.
[7]	Yu. V. Egorov and M. A. Shubin, "Foundations of the Classical Theory of Partial Differential Equations," Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 1998.
[8]	J. Hadamard, "Lectures on Cauchy's Problem for Linear Partial Differential Equations," Dover, 1952.
[9]	P. R. Popivanov and D. K. Palagachev, "The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations," Mathematical Research, 93, Akademie Verlag, Berlin, 1997.
[10]	L. Schwartz, "Théorie des Distributions," Hermann, Paris, 1961.
[11]	M. E. Taylor, "Pseudodifferential Operators," Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981.

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