Nonlinear Questions in Clamped Plate Models

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). 1. Clamped plate models A naive measure for the bending energy of a thin elastic plate under orthogonal load f : Ω → R is given by ∫ Ω ( (∆u) − f u ) dx. Here one should think of the bounded smooth domain Ω ⊂ R as the horizontal equilibrium shape of the plate, while u : Ω → R describes the vertical deflection from its equilibrium. ∆u = ∑n i=1 ∂2 ∂xi u denotes its Laplacian. The corresponding Euler-Lagrange equation is the linear plate equation, which is suitable only for small deflections u. This model however is not even invariant under rotating the x1, x2, u-coordinate system in R. A more realistic measure for the bending energy without external force is given by the Willmore functional ∫ graph [u] (H[u]) dS. Here, H is the mean curvature of graph [u], and dS denotes its surface element. The corresponding Euler-Lagrange equation is the Willmore equation. Still, this model is rather special. For a more realistic modelling like e.g. the Helfrich functional – see e.g. [He] – as well as a lot of background information concerning variational integrals involving curvature terms we refer to the survey article by Nitsche [Nit]. Although being rather special both models give rise to a number of interesting and still largely open mathematical problems which will be outlined in the following subsections. 1991 Mathematics Subject Classification. Primary 35J40; Secondary 35J65, 35B50, 49Q10, 53C42, 34L30.