II. REVISIÓN DE LA LITERATURA
2.7 Teoría del constructivismo y el
This section shows that there exists an orthonormal basis of eigenfunctions ofL. We begin with the following theorem.
Theorem 1.30 ([17], Theorem 7.11, p. 81) Suppose that Ω⊂Rnis bounded,
open and has continuous boundary. Then the embedding
j :W1,2(Ω) →L2(Ω)
is compact.
Theorem 1.31 ([17], Theorem 6.15, p. 67) LetB :W1,2(Ω)×W1,2(Ω)→
R
be a bounded coercive form. Let the operator A:L2(Ω)→L2(Ω) be given by the relation
A={(x, y)∈L2 ×L2 :∃u∈W1,2(Ω) s.t. j(u) =x and
B(u, v) =hy, j(v)i for v ∈W1,2}
has compact resolvent.
Recall that L was given by hLu, vi := B(u, v) and B was a bounded coercive form. We conclude by the above statement that Lhas compact resolvent. This is important for the next theorem. Recall that L is self adjoint, and positive (since the potentialV was nonnegative).
Theorem 1.32 ([17], Theorem 6.17, p. 68) Let A : L2 → L2 be a positive self adjoint operator with compact resolvent. Then there exists an orthonormal basis (ψj)j∈N of L
2(Ω) and a sequence (λ
j)j∈N in [0,∞) with nlim→∞λn = ∞ such that
Aψj =λjψj ∀j ∈N.
By the above theorems, we conclude that the eigenfunctions (ψj)j∈N of L form an orthonormal basis of L2(Ω), with ψj ∈D(L), as long as the boundary of Ω is continuous.
Local Eigenfunctions
In the next sections, we want to compare the above eigenfunctions to localised eigenfunctions, that is we wish to replace Ω with a suitable subset, and use mixed boundary conditions. We wish to remind the reader of 1.11, and remark that we are using mixed boundary conditions on a subset of Ω, so we modify the statements below correspondingly. Let K ⊂ Ω be a compact set, and let U ⊂ Ω\K be a connected component. We remark here thatU may not be open inRd but is open in Ω. Setting Γ =K∩∂U and
V ={u|U :u∈Cc∞(Ω\Γ)}
we restrict Lto VW
1,2(Ω)
. As remarked earlier, this gives Dirichlet conditions on Γ and Neumann conditions on ∂U \Γ =∂U ∩∂Ω.
To simplify the notation, we useW01,2(U) to denote the closure ofC1(Ω) functions which are supported only in U in the W1,2(Ω) norm. We remark that
VW 1,2(Ω) ={u: support(u)⊂U and u∈C∞ c (Ω)} W1,2(Ω) =W01,2(Ω).
where the last line follows becauseC1(Ω) is also dense inW1,2(Ω) (by theorem 1.3). For convenience we take W01,2(Ω) to be the domain for L, that is Lϕ=f weakly on U if it satisfies Z Ω ∇ϕ∇η+V ηdx= Z Ω f ηdx
for all η∈W01,2(U). We remark that when ∂Ω⊂K then we have Γ =∂U V ={u|U :u∈Cc∞(Ω\∂U)}
={u:u∈Cc∞(U)}
recovering the Dirichlet conditions.
Recall from the regularity theory that to ensure H¨older regularity of localised eigenfunctions we require K to satisfy the bi-Lipschitz cone condition. We remind the reader that the bi-Lipschitz cone condition is
Definition 1.33 A set K ⊂Rn, n6= 1 satisfies the bi-Lipschitz cone condition
if there exists r >0 and ε >0 such that for every point x0 ∈∂K there is a map
F :Br(x0)→Rn with F(x0) = 0 satisfying the following bi-Lipschitz bounds
ε|x−y| ≤ |F(x)−F(y)| ≤ 1
ε|x−y|
and such that
F[K]⊃ {x= (x1, x0)∈R×Rn−1 :|x0|< εx1 < ε2}.
While this restriction on K may appear restrictive, we can still pick K with a certain degree of freedom. By this, we mean that for any open set U ⊂ Ω and compact set K ⊂U we can find a compact set K0 which satisfies the bi-Lipschitz cone condition and K ⊂ K0 ⊂ U. We use this result when we approximate eigenfunctions locally.
Lemma 1.34 ([10], Lemma 5.2, p.27) Suppose that K is a compact subset of Ω and let U be an open set in Ω such that K ⊂U. Then there is a compact set
Proof: Let K ⊂Ω be given. Since Ω is compact we can cover Ω with a finite number of closed cubes. For any open setU containingK set
d= inf
x∈Kd(x, ∂U).
For ε = d/4>0, subdivide each closed cube of the covering dyadically so that we have a finite covering by cubes of diameter smaller than ε. Call this cover Cε.
Note that the original covering of Ω may have overlapping cubes, so these dyadic subcubes may overlap. However, Cε is always a covering by finitely many cubes.
We set
K0 :=K∩Cε.
We now note that each cube in Cε satisfies the bi-Lipschitz cone condition, and
since K0 is a finite union of such cubesK0 must also satisfy the bi-Lipschitz cone condition. Finally, K0 ⊂U by our choice of ε.
Chapter 2
Important Identities and Key
Estimates
2.1
Important Identity
Before embarking on outlining the arguments of [10], we first prove a key identity. This identity is used for the fundamental Lemma 2.3.
Lemma 2.1 For v, ϕ∈C1(Ω) we have
∇v· ∇(ϕ2v) = |∇(ϕv)|2− |v|2|∇ϕ|2 (2.1.1)
In particular, if v =u and ϕ=f /u satisfy the requirements then we have
∇u· ∇(f2/u) =|∇f|2− |u|2|∇(f /u)|2 (2.1.2)
Proof: Suppose v, ϕ are continuously differentiable. We may apply the product rule to ∇(ϕ2v). On the left hand side of (2.1.1) we have
∇v· ∇(ϕ2v) =∇v·(ϕv∇ϕ+ϕ∇(ϕv))
=∇v·(ϕv∇ϕ+ϕ(v∇ϕ+ϕ∇v))
= 2ϕv∇v· ∇ϕ+ϕ2∇v· ∇v. (?)
Now we notice that
|∇(ϕv)|2 =∇(ϕv)· ∇(ϕv)
= (ϕ∇v+v∇ϕ)·(φ∇v+v∇ϕ)
=ϕ2∇v· ∇v+ 2ϕv∇v· ∇ϕ+v2∇ϕ· ∇ϕ
a rearrangement gives
2ϕv∇v· ∇ϕ=|∇(ϕv)|2−ϕ2∇v· ∇v−v2∇ϕ· ∇ϕ so if we substitute this into (?) we get
∇v · ∇(ϕ2v) =|∇(ϕv)|2−ϕ2∇v· ∇v−v2∇ϕ· ∇ϕ+ϕ2∇v· ∇v =|∇(ϕv)|2−v2∇ϕ· ∇ϕ
which is (2.1.1). To obtain (2.1.2) simply substitutev =u andϕ=f /u to obtain
∇u· ∇((f /u)2u) =|∇((f /u)u)|2− |u|2|∇(f /u)|2
which is (2.1.2) after cancellation.
Corollary 2.2 For v, ϕ∈W1,2(Ω)∩L∞(Ω) we have
Z Ω ∇v· ∇(ϕ2v) = Z Ω |∇(ϕv)|2− |v|2|∇ϕ|2 (2.2.1)
In particular, if v =u and ϕ=f /u satisfy the requirements then we have
Z Ω ∇u· ∇(f2/u) = Z Ω |∇f|2− |u|2|∇(f /u)|2 (2.2.2)
We recall that the product rule can be generalised for functions v, ψ ∈W1,2(Ω)∩
L∞(Ω) by proposition 1.7. Thus we can generalise both (2.1.1) and (2.1.2) to
v, ψ ∈W1,2(Ω)∩L∞(Ω).