The Dirichlet-to-Neumann operator on exterior domains
Tom ter Elst
University of Auckland
Joint work with Wolfgang Arendt (Ulm)
28-4-2017
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Outline
1 Introduction
2 Sobolev spaces
3 DtN operators on exterior domains
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Introduction
The operator
Let Ω ⊂ Rd be bounded, connected, Lipschitz boundary.
Let Γ = ∂Ω.
The Dirichlet-to-Neumann operator AΩ is the self-adjoint operator on
L2 (Γ) such that for all ϕ, ψ ∈ L2 (Γ) one has

∆u = 0 weakly on Ω


ϕ ∈ D(AΩ ) and AΩ ϕ = ψ ⇔ ∃u∈H 1 (Ω)  ϕ = Tr u
 ∂u
=ψ
∂ν
The Dirichlet-to-Neumann operator is the voltage-to-current map.
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Introduction
Form method
Define a : H 1 (Ω) × H 1 (Ω) → C by
Z
∇u · ∇v.
a(u, v) =
Ω
Then
µ kuk2H 1 (Ω) ≤ Re a(u, u) + ω kTr uk2L2 (Γ)
for all u ∈ H 1 (Ω) for suitable µ, ω > 0.
Lemma. Let ϕ, ψ ∈ L2 (Γ). TFAE.
•
ϕ ∈ D(AΩ ) and AΩ ϕ = ψ.
•
There exists a u ∈ H 1 (Ω) such that Tr u = ϕ and
a(u, v) = (ψ, Tr v)L2 (Γ)
for all v ∈
H 1 (Ω).
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Introduction
Proof
Proof. ⇐. Suppose there exists a u ∈ H 1 (Ω) such that Tr u = ϕ and
Z
∇u · ∇v = a(u, v) = (ψ, Tr v)L2 (Γ) for all v ∈ H 1 (Ω).
Ω
First, if v ∈ Cc∞ (Ω), then
Z
Z
∇u · ∇v = a(u, v) =
ψ Tr v = 0.
Ω
Γ
So ∆u = 0 as distribution.
Secondly, if v ∈ H 1 (Ω), then
Z
Z
Z
∇u · ∇v + (∆u) v = a(u, v) =
ψ Tr v.
Ω
So by Green
∂u
∂ν
Ω
Γ
= ψ.
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Introduction
Aim
Exterior domain Ω = Rd \ Ω0 , where Ω0 is bounded open with Lipschitz
boundary. Assume Ω is connected.
Boundary
Γ = ∂Ω = ∂Ω0 .
Wish to define a Dirichlet-to-Neumann operator on Ω.
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Introduction
First attempt
Define a : H 1 (Ω) × H 1 (Ω) → C by
Z
∇u · ∇v.
a(u, v) =
Ω
Wish to consider operator A associated with (a, Tr ), that is
ϕ ∈ D(A) and Aϕ = ψ
if and only if
there exists a u ∈ H 1 (Ω) such that Tr u = ϕ and
a(u, v) = (ψ, Tr v)L2 (Γ)
for all v ∈
H 1 (Ω).
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Introduction
First attempt
Define a : H 1 (Ω) × H 1 (Ω) → C by
Z
∇u · ∇v.
a(u, v) =
Ω
Wish to consider operator A associated with (a, Tr ), that is
ϕ ∈ D(A) and Aϕ = ψ
if and only if
there exists a u ∈ H 1 (Ω) such that Tr u = ϕ and
a(u, v) = (ψ, Tr v)L2 (Γ)
for all v ∈
H 1 (Ω).
Bad luck! There are no µ, ω > 0 such that
µ kuk2H 1 (Ω) ≤ Re a(u, u) + ω kTr uk2L2 (Γ)
for all u ∈ H 1 (Ω).
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Sobolev spaces
Notation
For large R > 0 define ΩR = Ω ∩ BR .
Suppose d ≥ 3.
Then W 1,2 (Rd ) ⊂ Lp (Rd ), where
Note p ∈ (2, ∞).
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1
p
=
1
2
− d1 .
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Sobolev spaces
Sobolev space I
Define
Z
1
W (Ω) = {u ∈ Hloc
(Ω) :
|∇u|2 < ∞}.
Ω
Theorem (Lu–Ou). If u ∈ W (Ω), then
1
hui = lim
R→∞ |ΩR |
Z
u exists.
ΩR
There exists a c > 0 such that for all u ∈ W (Ω)
Z
2
2
ku − huikLp (Ω) ≤ c
|∇u|2
Ω
Norm
kukW (Ω) =
Z
|∇u|2 + |hui|2
1/2
.
Ω
Then W (Ω) is a Hilbert space.
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Sobolev spaces
Sobolev space II
Let W D (Ω) be the closure of the space {u|Ω : u ∈ Cc∞ (Rd )} in W (Ω).
Theorem (Lu–Ou). The space W D (Ω) has codimension 1 in W (Ω).
Moreover,
W D (Ω) = W (Ω) ∩ Lp (Ω) = {u ∈ W (Ω) : hui = 0}.
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Sobolev spaces
Weak normal derivative
Let u ∈ W (Ω) and ψ ∈ L2 (Γ).
We say that u has normal derivative ψ on Γ if
Z
Z
Z
∇u · ∇v + (∆u) v =
ψ Tr v
Ω
Ω
(1)
Γ
for all v ∈ {w|Ω : w ∈ Cc∞ (Rd )}. Notation ∂ν u = ψ.
Consequently, (1) is valid for all v ∈ W D (Ω).
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DtN operators on exterior domains
DtN operator with Dirichlet boundary conditions at infinity
Theorem. There exists a self-adjoint operator AD on L2 (Γ) such that the
graph is given as follows. Let ϕ, ψ ∈ L2 (Γ). Then
ϕ ∈ D(AD ) and AD ϕ = ψ
if and only if
∆u = 0 weakly on Ω

such that  Tr u = ϕ

there exists a function u ∈
W D (Ω)
∂ν u = ψ
Proof. Define
aD :
W D (Ω)
×
W D (Ω)
→ C by
Z
∇u · ∇v.
a(u, v) =
Ω
Then AD is the operator associated with (aD , Tr ).
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DtN operators on exterior domains
DtN operator with Neumann boundary conditions at infinity
Define a : W (Ω) × W (Ω) → C by
Z
∇u · ∇v.
a(u, v) =
Ω
Then there are µ, ω > 0 such that
µ kuk2W (Ω) ≤ Re a(u, u) + ω kTr uk2L2 (Γ)
for all u ∈ W (Ω).
Let A be the (self-adjoint) operator associated with (a, Tr ), that is
ϕ ∈ D(A) and Aϕ = ψ
if and only if
there exists a u ∈ W (Ω) such that Tr u = ϕ and
a(u, v) = (ψ, Tr v)L2 (Γ)
for all v ∈ W (Ω).
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DtN operators on exterior domains
DtN operator with Neumann boundary conditions at infinity
Part II
Theorem. Let ϕ, ψ ∈ L2 (Γ). TFAE.
•
ϕ ∈ D(A) and Aϕ = ψ.
•
∆u = 0 weakly on Ω

 Tr u = ϕ
There exists a u ∈ W (Ω) such that 
 ∂ u=ψ
 ν
R
Γ ψ = 0.

Recall a(u, v) =
R
Ω ∇u · ∇v
Tom ter Elst (University of Auckland)
and a(u, v) = (ψ, Tr v)L2 (Γ) for all v ∈ W (Ω).
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DtN operators on exterior domains
Semigroups
Let S D and S be the semigroups generated by −AD and −A.
Then S and S D are real, positive and submarkovinan.
If t > 0 and ϕ ∈ L2 (Γ) with ϕ ≥ 0, then StD ϕ ≤ St ϕ.
The semigroups S and S D are both irreducible.
Remark. We do not assume that Γ is connected, only Ω is connected.
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DtN operators on exterior domains
Approximation, Dirichlet version
For all large R > 0 there exists a self-adjoint operator AD
R on L2 (Γ) such
that the graph is given as follows. Let ϕ, ψ ∈ L2 (Γ). Then
D
ϕ ∈ D(AD
R ) and AR ϕ = ψ
if and only if
∆u = 0 weakly on Ω

 Tr u = ϕ
there exists a function u ∈ W (Ω) such that 
 u|
 Ω\ΩR = 0

∂ν u = ψ
Theorem. Let λ ∈ [0, ∞) and q ∈ [1, ∞). Then
−1
lim (λ I + AD
= (λ I + AD )−1
R)
R→∞
in L(Lq (Γ)).
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DtN operators on exterior domains
Approximation, Neumann version
For all large R > 0 there exists a self-adjoint operator AR on L2 (Γ) such
that the graph is given as follows. Let ϕ, ψ ∈ L2 (Γ). Then
ϕ ∈ D(AR ) and AR ϕ = ψ
if and only if
∆u = 0 weakly on ΩR

 Tr u = ϕ on Γ
there exists a function u ∈ W (ΩR ) such that 
 ∂ u = 0 on ∂B
 ν
R

∂ν u = ψ on Γ
Theorem. Let λ ∈ (0, ∞). Then
lim (λ I + AR )−1 = (λ I + A)−1
R→∞
in L(L2 (Γ)).
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DtN operators on exterior domains
Relation A and AD
Proposition. 1Γ ∈ D(AD ).
Theorem. D(AD ) = D(A) and
AD ϕ = Aϕ +
for all ϕ ∈ D(A), where β =
R
1
(ϕ, AD 1Γ )L2 (Γ) AD 1Γ
β
ΓA
D1 .
Γ
Corollary.
The operator A − AD is bounded.
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DtN operators on exterior domains
Three DtN operators
Recall Ω = Rd \ Ω0 , where Ω0 is bounded open with Lipschitz boundary.
So we also have the DtN operator AΩ0 on Ω0 .
Example. Suppose that Ω0 is the unit ball. Then
AD = AΩ0 + (d − 2) I.
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DtN operators on exterior domains
References
W. Arendt and A.F.M. ter Elst,
The Dirichlet-to-Neumann operator on exterior domains.
Potential Anal. 43 (2015), 313–340.
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