4Elliptic boundary value problems
III Analysis of Partial Differential Equations
4.3 The spectrum of elliptic operators
Let’s recap what we have obtained so far. Given
L
, we have found some
γ
such
that whenever
µ ≥ γ
, there is a unique solution to (
L
+
µ
)
u
=
f
(plus boundary
conditions). In particular,
L
+
µ
has trivial kernel. For
µ ≤ γ
, (
L
+
µ
)
u
= 0 may
or may not have a non-trivial solution, but we know this satisfies the Fredholm
alternative, since L + µ is still an elliptic operator.
Rewriting (
L
+
µ
)
u
= 0 as
Lu
=
−µu
, we are essentially considering eigen-
values of
L
. Of course,
L
is not a bounded linear operator, so our usual spectral
theory does not apply to
L
. However, as always, we know that
L
−1
γ
is compact
for large enough
γ
, and so the spectral theory of compact operators can tell us
something about what the eigenvalues of L look like.
We first recall some elementary definitions. Note that we are explicitly
working with real Hilbert spaces and spectra.
Definition
(Resolvent set)
.
Let
A
:
H → H
be a bounded linear operator.
Then the resolvent set is
ρ(A) = {λ ∈ R : A − λI is bijective}.
Definition (Spectrum). The spectrum of a bounded linear A : H → H is
σ(A) = R \ ρ(A).
Definition
(Point spectrum)
.
We say
η ∈ σ
(
A
) belongs to the point spectrum
of A if
ker(A − ηI) 6= {0}.
If η ∈ σ
p
(A) and w satisfies Aw = ηw, then w is an associated eigenvector.
Our knowledge of the spectrum of
L
will come from known results about the
spectrum of compact operators.
Theorem
(Spectral theorem of compact operators)
.
Let
dim H
=
∞
, and
K : H → H a compact operator. Then
– σ(K) = σ
p
(K) ∪{0}. Note that 0 may or may not be in σ
p
(K).
– σ(K) \ {0} is either finite or is a sequence tending to 0.
– If λ ∈ σ
p
(K), then ker(K − λI) is finite-dimensional.
–
If
K
is self-adjoint, i.e.
K
=
K
†
and
H
is separable, then there exists a
countable orthonormal basis of eigenvectors.
From this, it follows easily that
Theorem (Spectrum of L).
(i)
There exists a countable set Σ
⊆ R
such that there is a non-trivial solution
to Lu = λu iff λ ∈ Σ.
(ii)
If Σ is infinite, then Σ =
{λ
k
}
∞
k=1
, the values of an increasing sequence
with λ
k
→ ∞.
(iii) To each λ ∈ Σ there is an associated finite-dimensional space
E(λ) = {u ∈ H
1
0
(U) | u is a weak solution of (∗) with f = 0}.
We say
λ ∈
Σ is an eigenvalue and
u ∈ E
(
λ
) is the associated eigenfunction.
Proof.
Apply the spectral theorem to compact operator
L
−1
γ
:
L
2
(
U
)
→ L
2
(
U
),
and observe that
L
−1
γ
u = λu ⇐⇒ u = λ(L + γ)u ⇐⇒ Lu =
1 − λγ
λ
u.
Note that L
−1
γ
does not have a zero eigenvalue.
In certain cases, such as Laplace’s equation, our operator is “self-adjoint”,
and more things can be said. As before, we want the “formally” quantifier:
Definition
(Formally self-adjoint)
.
An operator
L
is formally self-adjoint if
L = L
†
. Equivalently, if b
i
≡ 0.
Definition
(Positive operator)
.
We say
L
is positive if there exists
C >
0 such
that
kuk
2
H
1
0
(U)
≤ CB[u, u] for all u ∈ H
1
0
(U).
Theorem.
Suppose
L
is a formally self-adjoint, positive, uniformly elliptic
operator on
U
, an open bounded set with
C
1
boundary. Then we can represent
the eigenvalues of L as
0 < λ
1
≤ λ
2
≤ λ
3
≤ ··· ,
where each eigenvalue appears according to its multiplicity (
dim E
(
λ
)), and there
exists an orthonormal basis
{w
k
}
∞
k=1
of
L
2
(
U
) with
w
k
∈ H
1
0
(
U
) an eigenfunction
of L with eigenvalue λ
k
.
Proof.
Note that positivity implies
c ≥
0. So the inverse
L
−1
:
L
2
(
U
)
→ L
2
(
U
)
exists and is a compact operator. We are done if we can show that
L
−1
is
self-adjoint. This is trivial, since for any f, g, we have
(L
−1
f, g)
L
2
(U)
= B[v, u] = B[u, v] = (L
−1
g, f)
L
2
(U)
.