# American Institute of Mathematical Sciences

September  2009, 2(3): 697-722. doi: 10.3934/dcdss.2009.2.697

## On shallow shell equations

 1 Key Laboratory of Control and Systems, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

Received  April 2008 Revised  October 2008 Published  June 2009

We present explicit formulas for the shallow shell model consisting of a couple of a wave equation and a plate equation, where the middle surface is viewed as a natural manifold with the induced metric from the classical Euclidean space of three dimensions. The Green formula for the shallow is given by the displacement field which expresses the relationship between the interior and the boundary. Next, the ellipticity of the strain energy for the shallow shell is studied under some curvature assumptions on the middle surface. Finally, the motion equations for shallow shells are obtained in terms of the displacement field as an unknown. The new ingredients in these formulas are that they take a form which is not described by a coordinate patch to provide the shell theory with the modern geometry.
Citation: Peng-Fei Yao. On shallow shell equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 697-722. doi: 10.3934/dcdss.2009.2.697
 [1] Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267 [2] Francisco Brito, Maria Luiza Leite, Vicente de Souza Neto. Liouville's formula under the viewpoint of minimal surfaces. Communications on Pure and Applied Analysis, 2004, 3 (1) : 41-51. doi: 10.3934/cpaa.2004.3.41 [3] Marius Mitrea. On Bojarski's index formula for nonsmooth interfaces. Electronic Research Announcements, 1999, 5: 40-46. [4] Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421 [5] Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195 [6] Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533 [7] Julien Dambrine, Morgan Pierre. Continuity with respect to the speed for optimal ship forms based on Michell's formula. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021049 [8] Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control and Related Fields, 2022, 12 (1) : 115-146. doi: 10.3934/mcrf.2021004 [9] Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763 [10] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [11] Jon Johnsen. Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators. Electronic Research Archive, 2019, 27: 20-36. doi: 10.3934/era.2019008 [12] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [13] Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607 [14] Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Optimality of (s, S) policies with nonlinear processes. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 161-185. doi: 10.3934/dcdsb.2017008 [15] Serena Dipierro, Alessio Figalli, Giampiero Palatucci, Enrico Valdinoci. Asymptotics of the $s$-perimeter as $s\searrow 0$. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2777-2790. doi: 10.3934/dcds.2013.33.2777 [16] Uri Shapira. On a generalization of Littlewood's conjecture. Journal of Modern Dynamics, 2009, 3 (3) : 457-477. doi: 10.3934/jmd.2009.3.457 [17] David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319 [18] John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367 [19] Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics and Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011 [20] Azniv Kasparian, Ivan Marinov. Duursma's reduced polynomial. Advances in Mathematics of Communications, 2017, 11 (4) : 647-669. doi: 10.3934/amc.2017048

2021 Impact Factor: 1.865