# American Institute of Mathematical Sciences

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

 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

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 $|\xi|\asymp \log (\varepsilon^{-1})$.
Citation: Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293
##### References:

show all references

##### References:
 [1] Felix Sadyrbaev, Inara Yermachenko. Multiple solutions of nonlinear boundary value problems for two-dimensional differential systems. Conference Publications, 2009, 2009 (Special) : 659-668. doi: 10.3934/proc.2009.2009.659 [2] Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems & Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709 [3] Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065 [4] Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713 [5] Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057 [6] Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292 [7] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 [8] K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624 [9] Paolo Tilli. Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length. Networks & Heterogeneous Media, 2012, 7 (1) : 127-136. doi: 10.3934/nhm.2012.7.127 [10] Simone Creo, Maria Rosaria Lancia, Alexander Nazarov, Paola Vernole. On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 57-64. doi: 10.3934/dcdss.2019004 [11] Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431 [12] Al-hassem Nayam. Constant in two-dimensional $p$-compliance-network problem. Networks & Heterogeneous Media, 2014, 9 (1) : 161-168. doi: 10.3934/nhm.2014.9.161 [13] Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011 [14] Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347 [15] Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819 [16] Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271 [17] Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059 [18] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [19] Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399 [20] Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

2018 Impact Factor: 1.143