Polyharmonic boundary value problems

1.3 The first eigenvalue 13
1.3.1 The Dirichlet eigenvalue problem
Whenever the biharmonic operator under Dirichlet boundary conditions has a
strictly positive Green’s function, the first eigenvalue Λ2,1 is simple and the corresponding
first eigenfunction is of fixed sign, see Section 3.1.3. Related to the first
eigenvalue is a question posed by Lord Rayleigh in 1894 in his celebrated monograph
[350]. He studied the vibration of (planar) plates and conjectured that among
domains of given area, when the edges are clamped, the form of gravest pitch is
doubtless the circle, see [350, p. 382]. This corresponds to saying that
Λ2,1(B) ≤ Λ2,1(Ω) whenever |Ω| = π (1.21)
for planar domains (n = 2). Szegö [388] assumed that in any domain the first eigenfunction
for the clamped plate has always a fixed sign and proved that this hypothesis
would imply the isoperimetric inequality (1.21). The assumption that the first
eigenfunction is of fixed sign, however, is not true as Duffin pointed out. In [152],
where he explains some counterexamples, he referred to this assumption as Szegö’s
conjecture on the clamped plate. Details of these counterexamples can be found in
[153, 154, 155].
Subsequently, concerning Rayleigh’s conjecture, Mohr [310] showed in 1975
that if among all domains of given area there exists a smooth minimiser for Λ2,1
then the domain is a disk. However, he left open the question of existence. In 1981,
Talenti [392] extended Szegö’s result in two directions. He showed that the statement
remains true under the weaker assumption that the nodal set of the first eigenfunction
ϕ1 of (3.1) is empty or is included in {x ∈ Ω; ∇ϕ1 = 0}. This result holds
in any space dimension n ≥ 2. Moreover, for general domains, instead of (1.21) he
showed that
CnΛ2,1(B) ≤ Λ2,1(Ω) whenever |Ω| = en
where 0.5 < Cn < 1 is a constant depending on the dimension n. These constants
were increased by Ashbaugh-Laugesen [24] who also showed that Cn → 1 as n → ∞.
A complete proof of Rayleigh’s conjecture was finally obtained one century later
than the conjecture itself in a celebrated paper by Nadirashvili [315]. This result was
immediately extended by Ashbaugh-Benguria [22] to the case of domains in R 3 .
More results about the positivity of the first eigenfunction in general domains
and a proof of Rayleigh’s conjecture can be found in Chapter 3.
1.3.2 An eigenvalue problem for a buckled plate
In 1910, Th. von Kármán [403] described the large deflections and stresses produced
in a thin elastic plate subject to compressive forces along its edge by means of a system
of two fourth order elliptic quasilinear equations. For a derivation of this model
from three dimensional elasticity one may also see [174] and references therein. An