Polyharmonic boundary value problems

1.8 Willmore surfaces 23
1.8 Willmore surfaces
At the beginning of this chapter the modeling of thin elastic plates was explained
in some detail. There, curvature expressions were somehow “linearised” in order
to have a purely quadratic behaviour of the leading terms of the energy functionals.
This simplification results in linear Euler-Lagrange equations, which are justified for
small deviations from a horizontal equilibrium shape. As soon as large deflections
occur or a coordinate system is chosen in such a way that the equilibrium shape is
not the x-y-plane, one has to stick to the frame invariant modeling of the bending
energy in terms of differential geometric curvature expressions. When compared
with the “linearised” energy integral (1.5) in Section 1.1, the integral
α + β(H − H0) 2 − γK dω (1.31)
Γ
with suitable constants α,β,γ,H0 may serve as a more realistic model for the bending
and stretching energy of a thin elastic plate, which is described by a twodimensional
manifold Γ ⊂ R3 . Here, H denotes its mean and K its Gaussian curvature.
According to [324], α is related to the surface tension, β and γ are elastic
moduli, while one may think of H0 as some preferred “intrinsic” curvature due to
particular properties of the material under consideration. Physically reasonable assumptions
on the coefficients are α ≥ 0, 0 ≤ γ ≤ β, βγH 2 0 ≤ α(β −γ), which ensure
the functional to be positive definite. For modeling aspects and a thorough explanation
of the meaning of each term we refer again to the survey article [324] by
Nitsche. A discussion of the full model (1.31), however, seems to be out of reach at
the moment, and for this reason one usually confines the investigation to the most
important and dominant term, i.e. the contribution of H2 .
Given a smooth immersed surface Γ , the Willmore functional is defined by
W(Γ ) := H
Γ
2 dω.
Apart from its meaning as a model for the elastic energy of thin shells or biological
membranes, it is also of great geometric interest, see e.g [413, 414]. Furthermore,
it is used in image processing for problems of surface restoration and image inpainting,
see e.g. [105] and references therein. In these applications one is usually
concerned with minima, or more generally with critical points of the Willmore functional.
It is well-known that the corresponding surface Γ has to satisfy the Willmore
equation
∆Γ H + 2H(H 2 − K) = 0 on Γ , (1.32)
where ∆Γ denotes the Laplace-Beltrami operator on Γ with respect to the induced
metric. A solution of (1.32) is called a Willmore surface. An additional difficulty
here arises from the fact that ∆Γ depends on the unknown surface so that the equation
is quasilinear. Moreover, the ellipticity is not uniform which, in the variational
framework, is reflected by the fact that minimising sequences may in general be un-