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Journal of Convex Analysis 23 (2016), No. 2, 347--386
Copyright Heldermann Verlag 2016

Thin Elastic Plates Supported over Small Areas. I: Korn's Inequalities and Boundary Layers

Giuseppe Buttazzo
Dept. of Mathematics, UniversitÓ di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

Giuseppe Cardone
Dept. of Engineering, UniversitÓ del Sannio, Corso Garibaldi 107, 82100 Benevento, Italy

Sergey A. Nazarov
Mathematics and Mechanics Faculty, St. Petersburg State University, Universitetsky pr. 28, Stary Peterhof 198504, Russia


A thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $\theta_{h}$ of one base is considered; the diameter of $\theta_{h}$ is of the same order as the plate relative thickness $h\ll 1$. In addition to the standard Kirchhoff model with the Sobolev point condition, a three-dimensional boundary layer is investigated in the vicinity of the support $\theta_{h}$, which with the help of the derived weighted inequality of Korn's type, will provide an error estimate with the bound $ch^{1/2}|\ln h|$. Ignoring this boundary layer effect reduces the precision order down to $|\ln h|^{-1/2}$.

Keywords: Kirchhoff plate, small support zones, asymptotic analysis, boundary layers, weighted Korn inequality.

MSC: 74K20, 74B05

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