The p-Dirichlet-to-Neumann operator on open sets
James Kennedy
Institute of Applied Analysis
University of Ulm
Operator semigroups meet complex analysis,
harmonic analysis and mathematical physics
Herrnhut, 3 June, 2013
Joint work with Daniel Hauer (University of Sydney)
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
The Dirichlet-to-Neumann operator
Start with the homogeneous Dirichlet problem
(
−∆u = 0 in Ω
u=ϕ
on ∂Ω
(1)
where Ω ⊂ RN , N ≥ 2, is an open set (maybe also bounded,
connected, smooth) and ϕ is a function defined on ∂Ω
Heuristically, we define the Dirichlet-to-Neumann operator D
by Dϕ = ∂u
∂ν , the outer normal derivative of u on ∂Ω.
Also known as the ‘voltage-to-current’ operator: ϕ is the
voltage on the boundary inducing a potential u in Ω (Ohm’s
law), current across the boundary is ∂u
∂ν
Related to Calderón’s inverse problem
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
Other applications
Can be used to prove properties of the Laplacian with
generalised Wentzell boundary conditions
∆u + β
∂u
+ γu = 0
∂ν
on ∂Ω
Eigenvalues are the Steklov eigenvalues
Can also be used to give information on the Dirichlet and
Neumann Laplacians, e.g., to prove Friedlander’s inequality
between Dirichlet and Neumann eigenvalues.
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
Realising the DtN operator
D can be realised as a well-defined operator
D : L2 (∂Ω) → L2 (∂Ω), also on other Lp (∂Ω) spaces, also on
C (∂Ω).
E.g.: if P : H 1/2 (∂Ω) → H 1 (Ω), Pϕ = u is the (weak)
solution operator associated with (1), then D is associated
with a form a : H 1/2 (∂Ω) × H 1/2 (∂Ω) → C
Z
a(ϕ, ψ) =
∇(Pϕ) · ∇(Pψ) dx.
Ω
This form is sesquilinear, bounded and L2 (∂Ω)-elliptic, and
H 1/2 (∂Ω) embeds compactly in L2 (∂Ω). Hence D generates a
compact, analytic C0 -semigroup on L2 (∂Ω).
If ∂Ω is smooth, D can also be realised as a pseudodifferential
operator on ∂Ω.
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
DtN operators and Calderón’s problem
For γ ∈ C 1 (Ω) we can consider
(
− div(γ(x)∇u(x)) = 0 in Ω
u=ϕ
on ∂Ω
and Dγ ϕ = γ ∂u
∂ν .
Calderón’s problem: determine γ from Dγ
More generally: A is a second-order elliptic operator,
∂u
DA ϕ = ∂ν
(the co-normal derivative of u with respect to A),
A
where Au = 0 in Ω and u = ϕ on ∂Ω.
Why should A be linear?
Natural question: can one do this when A = ∆p , the
p-Laplacian?
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
The p-Dirichlet-to-Neumann operator
Let 1 < p < ∞ and Ω ⊂ Rd bounded, Lipschitz.
For ϕ ∈ L2 (∂Ω) we define the (weak) W 1,p -solution u of
(
−∆p u := −div(|∇u|p−2 ∇u) = 0 in Ω
u=ϕ
on ∂Ω
(2)
as the unique solution u ∈ W 1,p (Ω) of
Z
|∇u|p−2 ∇u · ∇v dx = 0
Ω
Cc∞ (Ω)
for all v ∈
which satisfies u|∂Ω = ϕ in L2 (∂Ω). In this case
we write u = Pp ϕ.
For u ∈ W 1,p (Ω), if there exists a function ψ ∈ L2 (∂Ω) such that
Z
Z
Z
p−2
|∇| ∇u · ∇v dx =
ψv dσ −
∆p uv dx
Ω
C ∞ (Ω),
for all v ∈
derivative of u.
∂Ω
then we write ψ =
James Kennedy
Ω
|∇u|p−2 ∂u
∂ν ,
the p-normal
The p-Dirichlet-to-Neumann operator on open sets
The p-Dirichlet-to-Neumann operator
We define the p-Dirichlet-to-Neumann operator Dp on L2 (∂Ω) as
having operator domain
D(Dp ) := {ϕ ∈ L2 (∂Ω) : there exists a
W 1,p -solution u of (2) with |∇u|p−2
∂u
∈ L2 (∂Ω)}
∂ν
and being given by
Dp ϕ = |∇u|p−2
James Kennedy
∂u
.
∂ν
The p-Dirichlet-to-Neumann operator on open sets
The p-Dirichlet-to-Neumann operator
Theorem (Hauer-K., 2013)
−Dp generates a strongly continuous semigroup (e −tDp )t≥0 of
contractions on L2 (∂Ω) having the regularising effect. That is, for
every ϕ0 ∈ L2 (∂Ω) there is a unique function ϕ(t) := e −tDp ϕ with
ϕ(0) = ϕ0 satisfying
(i) ϕ ∈ C ([0, ∞); L2 (∂Ω)) such that Pp ϕ ∈ C ((0, ∞); W 1,p (Ω));
(ii) ϕ ∈ W 1,∞ (δ, ∞; L2 (∂Ω)) for all δ > 0, and ϕ is
right-differentiable at every t > 0, with
dϕ
(t) + Dp ϕ(t) = 0.
dt +
This semigroup is order preserving on L2 (∂Ω) and L∞ -accretive,
and extrapolates to an order-preserving semigroup of contractions
on Lq (∂Ω) for 1 ≤ q ≤ ∞, which is strongly continuous if q < ∞.
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
Idea of proof
First prove existence and uniqueness of the solution Pp ϕ of
(2) for each ϕ ∈ L2 (∂Ω). This is equivalent to the existence
of a unique minimiser of the problem
Z
1,p
p
|∇v | dx : v − Φ ∈ W0 (Ω)
inf
Ω
for any Φ ∈ W 1,p (Ω) with Φ|∂Ω = ϕ.
Define a functional E : L2 (∂Ω) → R ∪ {∞} by
 Z
1
|∇Pp ϕ|p dx if ∃v ∈ W 1,p (Ω): v |∂Ω = ϕ,
p
E(ϕ) =
Ω

∞
otherwise.
E is well-defined, convex, densely defined and lower
semicontinuous.
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
Idea of proof
The subgradient ∂E of E is single-valued and coincides with
Dp .
The existence of a semigroup of contractions e −tDp on
L2 (∂Ω) now follows from standard nonlinear semigroup theory.
Moreover: the map t 7→ E(e −tDp ϕ0 ) is locally absolutely
continuous on (0, ∞ for each initial value ϕ0 ∈ L2 (∂Ω), and
satisfies the energy equality
2
Z t2 d −sD
p
+ E(e −t2 Dp ϕ0 ) = E(e −t1 Dp ϕ0 ).
e
ϕ
0
ds
t1
2
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets
Further properties and generalisations
Stability of the semigroup on Lq , q ∈ [1, ∞): for every
ϕ0 ∈ L2 (∂Ω) theR limit limt→∞ e −tDp ϕ0 exists in L2 (∂Ω) and
equals σ(∂Ω)−1 ∂Ω ϕdσ · χ∂Ω (x) for σ-almost all x ∈ ∂Ω.
Semigroup on C (∂Ω).
Eigenvalue properties.
Generalises trivially to operators of the form
div(γ(x)|∇u|p−2 ∇u) in place of ∆p u.
Theory on general open sets: Arendt and ter Elst developed a
linear theory of Dirichlet-to-Neumann operators on (almost)
arbitrary open sets using variational techniques plus a new
generation result for non-closable forms.
James Kennedy
The p-Dirichlet-to-Neumann operator on open sets