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2009-05

Nonlinear questions in clamped plate models

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