Polyharmonic boundary value problems

2.4 Hilbert space theory 37
Introduction of a suitable Hilbert triple. Divide the boundary operators in (2.33)
into two classes. If m j < m we say that the boundary operator B j(x;D) is stable
while if m j ≥ m we say that it is natural. Assume that there are p stable boundary
operators, with p being an integer between 0 and m. If p = 0 all the boundary operators
are natural, whereas if p = m all boundary operators are stable. In view of
(2.30) the stable operators correspond to indices j ≤ p. Then we define the space
V := {v ∈ H m (Ω); B j(x,D)v = 0 on ∂Ω for j = 1,..., p}. (2.34)
Clearly, if p = 0 we have V = Hm (Ω) while if p = m we have V = Hm 0 (Ω) (provided
the assumption (2.36) below holds). In particular, in the case of Dirichlet boundary
conditions (2.20) we have
V = H m 0 (Ω),
in the case of Navier boundary conditions (2.21) we have
V = H m
ϑ (Ω) := v ∈ H m (Ω); ∆ j v = 0 on ∂Ω for j < m
, (2.35)
2
in the case of Steklov boundary conditions (2.22) we have
V = H 2 ∩ H 1 0 (Ω) = H 2 ϑ (Ω).
In any case, the space V is well-defined since each B j contains trace operators of
maximal order m j < m. Moreover, V is a closed subspace of H m (Ω) which satisfies
H m 0 (Ω) ⊂ V ⊂ Hm (Ω) with continuous embedding. Therefore, V inherits the scalar
product and the Hilbert space structure from H m (Ω). If we put H = L 2 (Ω), then
V ⊂ H ⊂ V ′ forms a Hilbert triple with compact embeddings.
Assumptions on the boundary operators. Assume that
and that the orders of the B j’s satisfy
{B j(x;D)} m j=1 forms a normal system (2.36)
mi + m j = 2m − 1 for all i, j = 1,...,m. (2.37)
This assumption is needed since we are not free to choose the orders of the B j’s. For
every k = 0,...,m − 1 there must be exactly one m j in the set {k,2m − k − 1}.
Let p denote the number of stable boundary operators. In view of (2.30) we know
that these operators are precisely {B j} p
j=1
and, of course, they also form a normal
system of boundary operators. By Proposition 2.12, there exists a family of normal
operators {S j} m j=p+1 such that {B1,...,Bp,Sp+1,...,Sm} forms a Dirichlet system of
order m. We relabel this system and define
{Fj} m j=1 ≡ {B1,...,Bp,Sp+1,...,Sm} (2.38)