Calculus of Variations and Geometric Measure Theory
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H. Abels - M. G. Mora - S. Müller

Large Time Existence for Thin Vibrating Plates

created by mora on 14 Sep 2009
modified on 09 Jul 2012

[BibTeX]

Published Paper

Inserted: 14 sep 2009
Last Updated: 9 jul 2012

Journal: Comm. Partial Differential Equations
Volume: 36
Pages: 2062-2102
Year: 2011

Abstract:

We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large times under appropriate scaling of the initial values such that the limit system as $h \to 0$ is either the nonlinear von Kármán plate equation or the linear fourth order Germain-Lagrange equation. In the case of the linear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.

Keywords: dimension reduction, nonlinear elasticity, wave equation, plate theory, von Kármán theory, singular perturbation


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