Nonlinear Questions in Clamped Plate Models

Authors: HansChristoph Grunau

Publish Date: 2009/08/08

Volume: 77, Issue: 1, Pages: 171-204

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Abstract

The linear clamped plate boundary value problem is a classical model in mechanics The underlying differential equation is elliptic and of fourth order The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order Concerning the clamped plate boundary value problem “linear questions” may be considered as well understood This changes completely as soon as one poses the simplest “nonlinear question” What can be said about positivity preserving Does a plate bend upwards when being pushed upwards It is known that the answer is “no” in general However there are many positivity issues as eg “almost positivity” to be discussedBoundary value problems for the “Willmore equation” are nonlinear counterparts for the linear clamped plate equation The corresponding energy functional involves curvature integrals over the unknown surface The Willmore equation is of interest in mechanics membrane physics and in particular in differential geometry Quite far reaching results were achieved concerning closed surfaces As for boundary value problems by far less is known These will be discussed in symmetric situations

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