Polyharmonic boundary value problems

1.3 The first eigenvalue 15
also [244, Proposition 4.4]. Therefore, for a complete proof of (1.24), “only” Item
2 is missing!
1.3.3 A Steklov eigenvalue problem
Usually, eigenvalue problems arise when one studies oscillation modes in the respective
time dependent problem in order to have a physically well motivated theory
and representation of solutions.
However, in what follows, a most natural motivation for considering a further
eigenvalue problem comes from a seemingly quite different mathematical question.
We explain how L2-estimates for the Dirichlet problem for harmonic functions link
with the Steklov eigenvalue problem for biharmonic functions.
Let Ω ⊂ Rn be a bounded smooth domain and consider the problem
∆u = 0 in Ω,
(1.25)
u = g on ∂Ω,
where g ∈ L 2 (∂Ω). It is well-known that (1.25) admits a unique solution u ∈
H 1/2 (Ω) ⊂ L 2 (Ω), see e.g. [275, Remarque 7.2, p. 202] and also [237, 238] for
an extension to nonsmooth domains. One is then interested in a priori estimates,
namely in determining the sharp constant CΩ such that
u L 2 (Ω) ≤ CΩ g L 2 (∂Ω) .
By Fichera’s principle of duality [170] (see also Section 3.3.2) one sees that CΩ
coincides with the inverse of the first Steklov eigenvalue δ1 = δ1(Ω), namely the
smallest constant a such that the problem
∆ 2u = 0 in Ω,
(1.26)
u = ∆u − auν = 0 on ∂Ω,
admits a nontrivial solution. Notice that the “true” eigenvalue problem for the hinged
plate equation should include the curvature in the second boundary condition, see
(1.8). The map Ω ↦→ δ1(Ω) has several surprising properties which we establish
in Section 3.3.2. By rescaling, one sees that δ1(kΩ) = k −1 δ1(Ω) for any bounded
domain Ω and any k > 0 so that δ1(kΩ) → 0 as k → ∞. One is then led to seek
domains which minimise δ1 under suitable constraints, the most natural one being
the volume constraint. Smith [373] stated that, analogously to the Faber-Krahn result
[162, 253, 254], the minimiser for δ1 should exist and be a ball, at least for planar
domains. But, as noticed by Kuttler and Sigillito, the argument in [373] contains a
gap. In the “Note added in proof” in [374, p. 111], Smith writes:
Although the result is probably true, a correct proof has not yet been found.