by Grunau, H.-Ch..

**Series:** 2009-05, Preprints

- MSC:
- 35J65 Nonlinear boundary value problems for linear elliptic equations
- 35J40 Boundary value problems for higher-order elliptic equations
- 49Q10 Optimization of shapes other than minimal surfaces
- 53C42 Immersions (minimal, prescribed curvature, tight, etc.)

**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