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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
buttazzo@dm.unipi.it

Giuseppe Cardone
Dept. of Engineering, UniversitÓ del Sannio, Corso Garibaldi 107, 82100 Benevento, Italy
giuseppe.cardone@unisannio.it

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



[Abstract-pdf]

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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