# 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 & Continuous Dynamical Systems - S, 2009, 2 (3) : 697-722. doi: 10.3934/dcdss.2009.2.697
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