09-05
Nonlinear questions in clamped plate models
Preprint series: 09-05, Preprints
- MSC:
- 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
- 35J40 Boundary value problems for higher-order, elliptic equations
- 49Q10 Optimization of the shape other than minimal surfaces, See also {73K40, 90C90}
- 53C42 Immersions (minimal, prescribed curvature, tight, etc.), See also {49Q05, 49Q10, 53A10, 57R40, 57R42}
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 e.g. `almost positivity\' to be discussed. Boundary 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. This survey article reports upon joint work with A. Dall\'Acqua, K. Deckelnick (Magdeburg), S. Fröhlich (Free University of Berlin), F. Gazzola (Milan), F. Robert (Nice), Friedhelm Schieweck (Magdeburg) and G. Sweers (Cologne).
Keywords: clamped plate equation, Willmore surface of revolution, Dirichlet problem, almost positivity, Green function
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