The Dirichlet-to-Neumann map as a
Pseudodifferential Operator
Universität Paderborn
Fakultät für Elektrotechnik, Informatik und Mathematik Institut für Mathematik
Jan Möllers
Matrikelnummer: 6310491
Hiermit versichere ich, die vorliegende Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie die Zitate deutlich kenntlich gemacht zu haben.
Zuerst möchte ich mich bei Herrn Prof. Dr. Hansen ganz herzlich für die Vergabe des Diplomthemas und seine ausgezeichnete Betreuung bedanken. Seine Art und Weise, schwierige Sachverhalte auf den Kern zu reduzieren, half mir sowohl bei der Bearbeitung des Diplomthemas, als auch in seinen Vorlesungen erheblich weiter. Für ein offenes Ohr bei Fragen und Problemen danke ich außerdem Herrn Prof. Dr. Hilgert.
Mein Dank geht auch an die Studienstifung des deutschen Volkes, die mein Studium drei Jahre lang finanziert hat und mir die Teilnahme am Naturwissenschaftlichen Kolleg ermöglichte.
Zuletzt danke ich noch meiner Familie und meinen Freunden für die Unterstützung, die ich während meines Studiums erfahren habe.
Introduction 1
1 The Dirichlet Problem 3
1.1 Distributions in Ω⊆Rn . . . . 3
1.2 Distributions on Manifolds . . . 7
1.3 The Dirichlet Problem . . . 13
2 Partial Hypoellipticity and the Dirichlet-to-Neumann map 15 2.1 Partial Regularity and Péetre’s Theorem . . . 16
2.2 Vector Valued Distributions . . . 19
2.3 The Proof of Péetre’s Theorem . . . 24
2.4 The Dirichlet-to-Neumann map . . . 25
3 Pseudodifferential Operators 29 3.1 Amplitudes and Operators . . . 29
3.2 Transpose and Action on Distributions . . . 33
3.3 Properly Supported Operators and Composition . . . 36
3.4 The Symbol Isomorphism and Asymptotic Expansions . . . 41
3.5 Elliptic Operators . . . 45
4 Tangential Operators 49 4.1 Tangential Amplitudes and Tangential Operators . . . 50
4.2 Properly Supported Tangential Operators and Composition . . . 55
4.3 The Symbol Isomorphism and Asymptotic Expansions . . . 60
4.4 Elliptic Tangential Operators . . . 67
5 The Heat Equation 69 5.1 The Transport Equation for Tangential Symbols . . . 70
6 The Dirichlet-to-Neumann map as a Pseudodifferential Operator 77 6.1 A Factorization of ∆g . . . 77 6.2 The Dirichlet-to-Neumann map on the unit circle . . . 83
A Functional Analysis 85
A.1 Transposing Operators . . . 85 A.2 Properties of Function and Distribution spaces . . . 86 A.3 The Schwartz Kernel Theorem . . . 87
B Ck-Curves and Weak Integrals 89
B.1 Ck-Curves . . . 89 B.2 Weak Integrals . . . 90
C Boundary Normal Coordinates 93
C.1 The Proof of Theorem 2.4.1 . . . 93
D Properly Supported Functions 97
D.1 Construction of Properly Supported Functions . . . 97
The Dirichlet problem plays an important role in many branches of mathematics and physics. IfM is a Riemannian manifold with boundary ∂M, e.g. M = Ω for a
smoothly bounded domain Ω ⊆ Rn, then for given boundary data f the Dirichlet problem consists in finding solutions u of
∆gu= 0 in M◦, u|∂M =f,
(0.1) where ∆g denotes the Laplace-Beltrami operator on M. This problem naturally turns
up if M describes a solid body, its conductivity determining the metric g, and an
electric potential f is applied to the surface ∂M. The solution u of (0.1) then has an
interpretation as the electric potential in the interior of M, depending of course on
the conductivity given in terms of the metric. The current flux (∂νu)|∂M through the surface is described by the normal derivative ∂ν of the inner potential u restricted to the boundary.
The inverse problem consists in determining the conductivity g of the solid body M by applying different potentials f to the surface and measuring the current flux
(∂νu)|∂M through the surface. This is interesting since (∂νu)|∂M can be obtained without “looking into M”. (For further study of the inverse problem see [LU89].)
This boundary data is described by the Dirichlet-to-Neumann operator Λg which maps the Dirichlet data f (in this case the applied potential) to the Neumann data Λgf = (∂νu)|∂M (here the current flux measured at the surface). In this thesis we will first define the Dirichlet-to-Neumann map for a compact connected Riemannian manifold M in the Sobolev setting, i.e. as an operator
Λg :H1/2(∂M)−→H−1/2(∂M).
The definition involves two important results: First we have to formulate and solve the Dirichlet problem in the Sobolev setting. For given data f ∈H1/2(∂M) on the
boundary there is a unique solution u∈H1(M) to the Dirichlet problem (0.1).
Sec-ondly it is necessary to use a theorem of Péetre on partial hypoellipticity to conclude that the derivative∂νu actually has a well-defined restriction Λgf := (∂νu)|∂M to the
boundary ∂M.
Our main result is the following theorem which shows that locally the Dirichlet-to-Neumann operator is given as sum of a first order pseudodifferential operator and a smoothing operator.
Theorem 6.1.3 ([LU89, Proposition 1.2]). Let M be a compact connected Rie-mannian manifold with boundary ∂M, n = dimM. For any coordinate chart
κ :∂M ⊇Ω−→Ωκ ⊆Rn−1 there is a properly supported pseudodifferential
opera-tor A∈Ψ1(Ωκ) such that for every f ∈H1/2(∂M) we have
κ∗(Λgf)−A(κ∗f)∈ C∞(Ωκ).
A is elliptic and has a coordinate invariant positively1-homogeneous principal symbol
a ∈ C∞(T∗Ωκ\0) =C∞(Ωκ×( Rn−1\0)) given by a(x,ξ) = v u u u t n−1 X j,k=1 gjk(x)ξjξk ∀(x,ξ)∈T∗Ωκ\0.
We follow the proof in [LU89] which uses a microlocal factorization of the Laplace-Beltrami operator ∆g near the boundary. For this we need the calculus of tangential (pseudodifferential) operators which will be developed in detail. Moreover we have to apply a regularity statement for the heat equation which can also be obtained using tangential operators.
This thesis is structured as follows. In chapter 1 we will introduce some notations for distributions on open sets Ω⊆Rn and on manifolds as well as state the Dirichlet problem and quote a solution theorem. Chapter 2 treats Péetre’s Theorem on Partial Hypoellipticity and sketches the proof such that at the end of the chapter we are able to define the Dirichlet-to-Neumann map. In chapter 3 the calculus of pseudodifferential operators is developed and generalized to tangential operators in chapter 4. The heat equation will be discussed in chapter 5 where we give the construction of a parametrix. Finally in chapter 6 we prove the main result and give a short example.
The Dirichlet Problem
1.1 Distributions in Ω
⊆
R
nLet us recall the basic notions of distribution theory. We make no claim to be complete since the purpose of this section merely lies in fixing some notations.
Function spaces. Let Ω⊆Rn be open. Then C∞(Ω) denotes the space of smooth complex valued functions in Ω. Its locally convex topology is generated by the seminorms
kϕkK,m := sup |α|≤mx∈Ksup
|∂αϕ(x)|
for ϕ∈ C∞(Ω) and m ≥ 0, K ⊂⊂ Ω compact, where the supremum is taken over all multiindices α∈Nn
0 with |α|=α1+. . .+αn≤m. Equipped with this topology
C∞(Ω) becomes a Fréchet space.
The subspaceC∞(Ω)⊆ C∞(Ω) contains all smooth functions in Ω whose derivatives have continuous extensions to Ω. It carries an own Fréchet space structure defined by the seminorms k−kK,m for m≥0, K ⊂⊂Ω compact. Functions ϕ∈ C∞(Ω) are called smooth up to the boundary.
This idea can be generalized to smooth functions defined on so called locally convex subsets of Rn with dense interior. Since we do not need this in full generality we merely introduce two special cases. If X = Ω×[0,T) or X = Ω×[0,T] for some open
set Ω⊆Rnand T >0 then C∞(X) is the space of all smooth functions in the interior
X◦ whose derivatives have continuous extensions to X. The Fréchet space topology
of C∞(X) is given by the seminormsk−k
K,m with m≥0 and K ⊂⊂X compact. Another subspace of C∞(Ω) is C∞
c (Ω) which consists of all smooth functions on Ω whose support (taken in Ω) is compact. It carries the inductive limit topology of all subspaces C∞
c (K) = {ϕ∈ C∞(Ω) : supp ϕ⊆ K} for K ⊂⊂ Ω compact. Cc∞(K) becomes a Fréchet space if equipped with the topology induced by C∞(Ω). If we
choose an exhaustion of Ω by compact sets Kj ⊂⊂Ω, i.e. Kj ⊆Kj◦+1 and
S
jKj = Ω, then the subspaces C∞
c (Kj) form a sequence of definition for C ∞
c (Ω) and henceC ∞ c (Ω) is an LF space.
Though typical seminorms on C∞
c (Ω) look quite uncomfortable the space has some nice properties which make life easier.
Firstly linear maps C∞
c (Ω)−→E into any locally convex topological vector space E are continuous if and only if all restrictions C∞
c (K)−→E for compact K ⊂⊂Ω are continuous (cf. Proposition A.2.1(1)). To check continuity of maps into C∞
c (Ω) it is often useful to know that for any seminorm p onC∞
c (Ω) and any compact subset
K ⊂⊂Ω there is by Lemma A.2.2 an integerm ≥0 and a constant C >0 such that p(ϕ)≤CkϕkK,m ∀ϕ∈ Cc∞(K).
To work with topologies of bounded convergence later we also use the fact that bounded subsets of C∞
c (Ω) are already contained and bounded in Cc∞(K) for some
K ⊂⊂Ω compact (see Proposition A.2.1(2)).
For two open sets Ω1 ⊆Rn, Ω2 ⊆Rm there is a canonical algebraic isomorphism
C∞(Ω1×Ω2)
∼
=
−→ C∞(Ω1,C∞(Ω2)), ϕ7−→(x7→ϕ(x,−)),
where the space on the right hand side is defined in appendix B. (In fact this even defines a topological isomorphism. For details see e.g. [Trè06, Theorem 40.1 and Corollary 1].) It restricts to an isomorphism
Cc∞(Ω1×Ω2)
∼
=
−→ Cc∞(Ω1,Cc∞(Ω2)).
We will also need this for Ω1 = [0,T], T >0. Then similarly
C∞(Ω2×[0,T])
∼
=
−→ C∞([0,T],C∞(Ω2)).
Distribution spaces. The strong dual D0(Ω) := (C∞ c (Ω))
0
b is called the space of distributions in Ω. We use the duality bracket hu,ϕi for u(ϕ) if u ∈ D0(Ω) and
ϕ ∈ C∞
c (Ω). One can canonically embed the space L1`oc(Ω) of locally integrable functions on Ω into D0(Ω) by putting
hu,ϕi:=
Z
u(x)ϕ(x)dx ∀ϕ∈ Cc∞(Ω),
for u ∈ L1`oc(Ω). The transpose of the inclusion map C∞
c (Ω) ,−→ C
∞(Ω) gives a continuous injection ofE0(Ω) := (C∞(Ω))0
b into D
0(Ω). If viewed as subspace of D0(Ω),
E0(Ω) consists precisely of those distributions with compact support. C∞
c (Ω) is dense in E0(Ω) and D0(Ω) and hence continuous operators on these space are uniquely determined by their values on Cc∞(Ω).
The Schwartz kernel theorem A.3.1 establishes a bijection between continuous linear operatorsA:C∞
c (Ω) −→ D
0(Ω) and distributions K ∈ D0(Ω×Ω) by
hAϕ,ψi=hK,ψ⊗ϕi ∀ϕ,ψ∈ Cc∞(Ω),
where (ψ⊗ϕ)(x,y) :=ψ(x)·ϕ(y) for every (x,y)∈Ω×Ω. K is called kernel of A.
Transposing the inclusion map C∞
c (Ω1) ,−→ Cc∞(Ω) for an open subset Ω1 ⊆ Ω
gives the restriction D0(Ω)−→ D0(Ω
1), u7→u|Ω1. This turns D
0(Ω) into a sheaf of vector spaces (cf. [Hör03, Theorem 2.2.4]): If (Ωα)α∈A is any open cover of Ω then for every system (uα)α∈A of distributions uα ∈ D0(Ωα) such that
uα|Ωα∩Ωβ =uβ|Ωα∩Ωβ ∀α,β ∈ A
there is a unique distribution u∈ D0(Ω) such that
u|Ωα =uα ∀α ∈ A.
Weak differentiation of distributions and multiplication by C∞ functions is defined in the usual manner via duality. Together this gives the action of differential operators onD0(Ω). A general differential operator will be of the form
P = X
|α|≤m
aα(x)Dα
with coefficients aα ∈ C∞(Ω). Here Dα =Dα11. . . Dαnn with
Dj := 1 i∂j = 1 i ∂ ∂xj.
Of particular importance will be the Laplacian ∆ =∂12+. . .+∂2n=−D12+. . .+Dn2.
For u∈ D0(Ω) thesingular support singsuppu will denote the complement (in Ω) of the largest open set in which u isC∞.
The angular bracket. Forξ ∈Rn
hξi:=1 +|ξ|21/2 =1 +ξ12+. . .+ξ2n1/2
is called the angular bracket. It will be used to estimate growth and decay of functions. For |ξ| ≥1 we have the following basic inequality
which shows thathξidescribes polynomial growth of order 1 and hencehξispolynomial growth of order s ∈R. This also gives
Z
hξisdξ < ∞ if and only if s <−n.
The advantage of the brackethξi in contrast to the simple modulus|ξ| is that it is
smooth up to ξ = 0 and everywhere≥1.
Fors ∈Rand ξ,η∈Rn we have Péetre’s inequality (cf. [SR91, Lemma 1.18])
hηis≤2|s|hη−ξi|s|hξis. (1.2)
Finally the action of the Laplacian on trigonometric functions can be expressed using the angular bracket:
(1−∆x)[e±ix·ξ] =hξi2e±ix·ξ ∀x,ξ ∈Rn.
The Fourier transform. For f ∈ L1(
Rn) its Fourier transform fb ∈ C(Rn) is
defined by b f(ξ) = Z e−ix·ξf(x)dx. It restricts to an isomorphism S(Rn) −→ S(
Rn) of the Schwartz space S(Rn) of rapidly decreasing smooth functions and extends to an isomorphismS0(
Rn)−→ S0(Rn) between tempered distributions via duality. Using the modified Lebesgue measure
¯
dξ = (2π)−ndξ Fourier’s inversion formula can be stated as follows (cf. [Hör03,
Theorem 7.1.5]): If ϕ∈ C∞ c (Ω) (or ϕ∈ S(Rn)) thenϕb ∈ S(R n) and ϕ(x) = (2π)−nϕbb(−x) = Z Z ei(x−y)·ξϕ(y)dydξ¯ ∀x∈Rn. (1.3)
Sobolev spaces. For s ∈ R the Sobolev space Hs(
Rn) consists of all tempered distributionsu∈ S0(
Rn) such that the Fourier transformubis locally square-integrable
and kuk2 s := Z hξi2s| b u(ξ)|2dξ <¯ ∞.
The norm k−ks provides Hs(Rn) with a Hilbert space structure with the obvious inner product. There is a canonical isomorphism H−s(Rn)∼= (Hs(
Rn))0.
If Ω⊆Rn is open let H`oc(Ω) be the space of all distributionss u∈ D0(Ω) such that
χ·u∈ E0(Ω)⊆ S0(
Rn) is an element ofHs(Rn) for everyχ∈ Cc∞(Ω). H`oc(Ω) becomess a Fréchet space if given the locally convex topology generated by the seminorms
If s= k is a nonnegative integer Hk(
Rn) turns out to be the space of all functions
u ∈ L2(
Rn) such that Dαu ∈ L2(Rn) for every |α| ≤ k and in this case the norm
k−kk is equivalent to the norm k−k(k) defined by
kuk2 (k)= X |α|≤k kDαuk2 L2( Rn).
This motivates the definition of Sobolev spaces on general domains Ω ⊂Rn. For a non-negative integerkthe Sobolev spaceHk(Ω) consists of all functionsu∈L2(Ω) ⊆ D0(Ω) such that Dαu∈L2(Ω) for every |α| ≤k. The norm k−k(k) which turns Hk(Ω) into
a Hilbert space is given by
kuk2 (k)= X |α|≤k kDαuk2 L2(Ω). Also put H−k(Ω) :=Hk 0(Ω) 0 whereHk
0(Ω) denotes the closure of C
∞
c (Ω) inHk(Ω). Then for an arbitrary integer k ∈ Z multiplication a· : Hk(Ω) −→ Hk(Ω) by
a∈ C∞(Ω) and differentiation ∂
j :Hk(Ω)−→Hk−1(Ω) for 1≤j ≤n are defined in the usual way. Combined this shows that differential operators with coefficients in
C∞(Ω) also act between the spacesHk(Ω).
1.2 Distributions on Manifolds
Manifolds. All manifolds we consider will be smooth. By a closed manifold we mean a compact one without boundary. But first let us consider Riemannian mani-foldsM which do not need to be compact. (This also includes open coordinate charts
of a compact Riemannian manifold as well as the interior of a Riemannian manifold with boundary.)
First let M be without boundary. Coordinate charts are of the form κ: Ω −→Ωκ
with Ω ⊆ M open and Ωκ ⊆ Rn open, n = dimM. In such a coordinate chart we denote the metric by
κ∗g = n
X
j,k=1
gjk·dxj ⊗dxk
with gjk ∈ C∞(Ωκ). (gjk)jk is a real valued n×n matrix which is non-degenerate at every point. Letg = det (gjk)jk ∈ C∞(Ωκ) denote the everywhere positive determinant of (gjk)jk and
gjk
jk the inverse matrix of (gjk)jk, g
jk ∈ C∞(Ωκ).
To state the Dirichlet problem we also need the notion of a Riemannian manifold
M with boundary ∂M. Here M = M denotes the manifold and M◦ := M \∂M
coordinate charts of the form κ: Ω−→Ωκ with Ω⊆M open and Ωκ = Ω0κ×[0,T),
Ω0 κ ⊆R
n−1 open, T > 0, n= dimM. By definition the coefficients of the metric in
local coordinates are smooth up to the boundary, i.e. gjk,gjk,g ∈ C∞(Ω0κ ×[0,T)). The matrix (gjk)jk is also non-degenerate up to the boundary.
Note that if M is a compact Riemannian manifold with boundary then M◦ is
a boundaryless Riemannian manifold (in general noncompact) and ∂M a closed
Riemannian manifold, each equipped with the induced metric.
For any Riemannian manifold, with or without boundary, we have the canonical Riemannian measure µg on M which is locally in a coordinate chart κ: Ω −→Ωκ given by Z Ω ϕ dµg = Z Ωκ ϕ(κ−1(x))·g1/2(x)dx ∀ϕ∈ Cc(Ω),
dx denoting the Lebesgue measure in Ωκ. This invariantly defines the spaces Lp(M), 1≤p≤ ∞, and L1
`oc(M).
Functions on Manifolds. In this part we only need to discuss functions on Rie-mannian manifolds M without boundary.
If κ: Ω−→Ωκ is a coordinate chart we have canonical maps
κ∗ :C∞(Ω) −→ C∞(Ωκ), ϕ7→ϕ◦κ−1,
κ∗ :C∞(Ωκ)−→ C∞(Ω), ϕ7→ϕ◦κ
called push-forward and pull-back, respectively. Obviouslyκ∗ and κ∗ are inverse to each other and also act between C∞
c (Ω) and C ∞ c (Ωκ).
The spaceC∞(M) of smooth functions on M is provided with the initial topology of the mapsC∞(M)−→ C∞(Ωκ), ϕ7→κ
∗(ϕ|Ω) forκ: Ω −→κ(Ω) a coordinate chart.
Thanks toM being second countableC∞(M) is a Fréchet space. As in the Euclidean case we equip C∞
c (M), the space of compactly supported smooth functions onM, with the LF topology which comes from the inductive system
C∞
c (K) = {ϕ∈ C
∞(M) : suppϕ⊆K} for K ⊂⊂M compact where C∞
c (K) is given the Fréchet topology induced by C∞(M). If M is compact we haveC∞
c (M) =C ∞(M) topologically.
Now differential operators act continuously on C∞(M) and C∞
c (M). If P is a differential operator and κ: Ω−→Ωκ a coordinate chart denote by Pκ the operator
onC∞
c (Ωκ) given by
Pκϕ:=κ∗P(κ∗ϕ) ∀ϕ∈ Cc∞(Ωκ).
By definition Pκ is a differential operator with coefficients in C∞(Ωκ) and hence extends toD0(Ωκ).
To extend P to distributions on M we need the notion of adjoint operators. Using
the Riemannian measure it can easily be seen that for every differential operator P
onM there is a unique differential operator P∗ on M of the same order such that
Z M P ϕ·ψ dµg = Z M ϕ·P∗ψ dµg ∀ϕ,ψ∈ Cc∞(M).
P∗ is called the formal adjoint ofP with respect to the Riemannian structure. The
Fundamental Lemma of Calculus of Variations shows thatP∗∗ =P.
An important example is the Laplace-Beltrami operator ∆g. It is a differential operator of second order on M and locally given by
∆κ g = 1 g1/2 n X j,k=1 ∂j g1/2gjk∂k (1.4) (see e.g. [Jos95, Equation (2.1.12)]). ∆g is formally selfadjoint, i.e. ∆∗
g = ∆g.
Distributions on Manifolds. IfM is boundaryless, define the space of distributions
D0(M) as the strong dual of C∞
c (M). In particular this explains the space D
0(M◦) of distributions in the interior M◦ of some manifold M with boundary.
The Riemannian measure µg can be used to identify L1`oc(M) with a subspace of
D0(M) via
hf,ϕi:=
Z
M
f·ϕ dµg ∀ϕ∈ Cc∞(M).
for f ∈L1`oc(M). In particular we have L2(M)⊆ D0(M).
As in the euclidean case we define the action of differential operators P onM on
distributions via duality. For u∈ D0(M) the distributionP u∈ D0(M) is given by
hP u,ϕi:=hu,P∗ϕi ∀ϕ∈ C∞(M),
where P∗ is the formal adjoint of P. By definition of the adjointP∗ this is consistent
with the action of P onC∞(M).
The restriction D0(M)−→ D0(Ω), u7→u|
Ω to an open subset Ω⊆M is defined in
map C∞
c (Ω) ,−→ C ∞ c (M).
We can also extend push-forward and pull-back to distributions. If κ:M ⊇Ω−→
Ωκ ⊆Rn is a coordinate chart we define push-forward and pull-back by
κ∗ :D0(Ω)−→ D0(Ωκ), hκ∗u,ϕi:= D u,κ∗ϕ·g−1/2E ∀ϕ∈ Cc∞(Ωκ), u∈ D0(Ω), κ∗ :D0(Ωκ)−→ D0(Ω), hκ∗u,ϕi:=Du,κ∗ϕ·g1/2 E ∀ϕ∈ Cc∞(Ω), u∈ D0(Ωκ).
The factors g−1/2 andg1/2 assure that these operations agree with those on functions
defined previously. As for functions κ∗ and κ∗ are inverse to each other and if P is a differential operator P the identity Pκu=κ∗(P(κ∗u)) extends to distributions
u∈ D0(Ωκ). We will often write κ
∗u forκ∗(u|Ω) if u∈ D0(M).
After these preparations we can now define the Sobolev spaces Hs(M) for s∈ R and M a closed manifold as subspaces of D0(M). The definition is due to [Tay96, Chapter 4].
Definition 1.2.1. Let M be a closed manifold. For s ∈ R the Sobolev space
Hs(M) is the space of all distributions u ∈ D0(M) such that for every coordinate chart κ:M ⊇Ω−→Ωκ ⊆Rn we have
κ∗u∈H`oc(Ωκ)s .
Hs(M) is equipped with the initial topology of the mappings
Hs(M)−→H`oc(Ωκ)s , u7→κ∗u.
In fact by the invariance property of Sobolev spaces it suffices to work with a finite atlas of M. Forming inner products in coordinates and using a partition of unity it
is easy to show that Hs(M) actually carries a Hilbert space structure. Defining a canonical inner product though requires some additional work.)
Nevertheless, for a nonnegative integer k we can describe Hk(M) in terms of weak L2 derivatives.
Lemma 1.2.2. Let M be a closed Riemannian manifold. For k ∈ N0 the space
Hk(M) consists of all functions u∈L2(M) such that for every differential operator
P of order ≤k we have P u∈L2(M).
Sketch of the Proof. “⊆”: Letu∈Hk(M) and P be a differential operator of
order ≤k. We claim that P u∈L2(M)⊆ D0(M).
Using a partition of unity subordinate to some finite atlas this can be reduced to the case that u ∈ Hk(M) with supp u ⊆ Ω for κ : Ω −→ Ωκ ⊆
Rn some coordinate chart, Ω⊆M open.
Choose χ∈ C∞
c (Ω) with χ≡1 on supp u. Then by definition
κ∗u=κ∗(χ·u)∈Hk(Ωκ). and hence
κ∗(P u) = Pκ(κ∗u)∈L2(Ωκ).
Since supp P u⊆supp u⊂⊂Ω is compact this gives P u∈L2(M) .
In particular this holds for P = Id and hence u∈L2(M).
“⊇”: Letu∈L2(M) such thatP u∈L2(M) for every differential operatorP of order
≤k. For every coordinate chart κ: Ω−→Ωκ ⊆Rn and every cutoff function
χ∈ C∞
c (Ω) we have to show thatκ∗(χu)∈Hk(Ωκ). For any multiindexα ∈Nn
0 with |α| ≤k there is a differential operator P on
M of order ≤k such that Pκ =∂α on the compact set supp χ. Then
∂α[κ∗(χu)] =Pκ(κ∗(χu)) =κ∗(P(χu)).
An easy calculation shows that P(χu) = (χP∗)∗u, so P(χu) ∈ L2(M) since
(χP∗)∗is also a differential operator of order≤k. But suppP(χu)⊆suppχ⊂⊂
Ω is compact and hence alsoκ∗(P(χu))∈L2(Ωκ). This shows that
κ∗(χu)∈Hk(Ωκ).
This description suggests how to define Sobolev spacesHk(M) for compact
Rieman-nian manifoldsM with boundary∂M in terms of weakL2 derivatives. Note that since ∂M is a null set for the Riemannian measure we have L2(M) =L2(M◦)⊆ D0(M◦) and the action of differential operators on D0(M◦) is defined.
Definition 1.2.3. Let M be a compact Riemannian manifold with boundary ∂M.
For k ∈N0 the Sobolev space Hk(M) is the space of all functionsu∈L2(M) such
that for any differential operatorP onM of order≤k with coefficients smooth up to
the boundary we have
P u∈L2(M).
Hk(M) carries the initial topology with respect to the maps Hk(M)−→L2(M), u7→P u
for P a differential operator on M of order ≤k with coefficients smooth up to the
boundary.
In fact it suffices to check this for a finite numberP1, . . . ,Pr of differential operators which generate the left C∞(M) module of differential operators on M of order ≤k. This gives rise to an inner product
(u,v)7→
r
X
j=1
(Pju|Pjv)L2(M)
for u,v ∈Hk(M) and turns Hk(M) into a Hilbert space. To work in local coordinates we need the following Lemma.
Lemma 1.2.4. For any coordinate chart κ : Ω −→ Ωκ, Ω ⊆ M, Ωκ ⊆ Rn open, which is relatively compact in another coordinate chart we have κ∗u ∈ Hk(Ωκ) for every u∈Hk(M) and the map
Hk(M)−→Hk(Ωκ), u7→κ∗u, (1.5)
is a linear continuous operator.
Note that in particular this allows coordinate charts of the typeκ: Ω−→Ωκ×(0,T)
which extend to coordinate charts onto Ω0
κ ×[0,T +ε) near the boundary, Ωκ ⊆Ω 0 κ,
ε >0.
Proof. We have to show that for given α ∈ Nn0 the distribution κ∗u ∈ D
0(Ωκ) is already contained in Hk(Ωκ). Thanks to Ω being relatively compact in another coordinate chart we can find a differential operator P on M with coefficients smooth
up to the boundary such that Pκ =∂α. By definition we have P u ∈L2(M). Now g−1/2 is bounded on Ωκ by some constant C >0 since Ω is compact and contained in
some bigger coordinate chart. We then have
Z Ωκ |(∂ακ∗u) (x)|2dx ≤C Z Ωκ |κ∗(P u) (x)|2·g1/2(x)dx =C Z Ω |P u|2dµg ≤CkP uk2L2(M).
1.3 The Dirichlet Problem
The classical Dirichlet problem can be stated as follows. Let M be a compact
Riemannian manifold with boundary ∂M. For given boundary data f ∈ C∞(∂M) one is interested in solutions u∈ C∞(M) of
∆gu= 0 in M◦ u|∂M =f
In order to give a formulation of the Dirichlet problem in the Sobolev setting we need to translate both conditions to the distribution case. For the first one this has already been done. If u∈H1(M) then ∆gu∈ D0(M◦) is defined by the action of the second order differential operator ∆g on distributions in the interior. Thus we have to specify what we mean by the restriction u|∂M for u∈H1(M).
Proposition 1.3.1 ([Tay96, Proposition 4.5 in Chapter 4]). Fork ≥1the restriction mapC∞(M)−→ C∞(∂M), ϕ7→ϕ|∂M has a unique continuous extension as a bounded linear operator
Hk(M)−→Hk−1/2(∂M), u7→u|∂M.
In fact this Proposition even holds for k > 1/2, but since we just need the case k = 1 we did not define Sobolev spaces of fractional order on manifolds with boundary.
We are now able to formulate the Dirichlet problem in the Sobolev setting and quote a solution theorem. For uniqueness we also require M to be connected. (In
fact it would of course suffice to require every connected component of M to have
nonempty boundary.)
Theorem 1.3.2 ([Tay96, Proposition 1.7 in Chapter 5]). Let M be a compact con-nected Riemannian manifold with nonempty boundary ∂M, k ≥1. Then there is a continuous linear operator
PI :Hk−1/2(∂M)−→Hk(M),
such that for every f ∈Hk−1/2(∂M) the unique solution u∈Hk(M) to the Dirichlet problem
∆gu= 0 in M◦, u|∂M =f,
(1.6) is given by u= PI f.
Partial Hypoellipticity and the Dirichlet-to-Neumann map
At the end of this chapter we will be able to define the Dirichlet-to-Neumann map
Λg :H1/2(∂M)−→ D0(∂M). For f ∈H1/2(∂M) we want Λgf to be the restriction to the boundary of the outer normal derivative∂νu of u∈H1(M) where u:= PI f denotes the unique solution to the Dirichlet problem (1.6).
The first part, stating and solving the Dirichlet problem, has already been estab-lished in the previous chapter. What we lack at this point is an adequate definition of the restriction to the boundary.
Now one might say that with Proposition 1.3.1 we already stated a result which defines a restriction in the Sobolev setting. Unfortunately this does not apply to our situation. In fact, if we choose a smooth vectorfield X onM which equals the outer
unit normal vector on∂M then Xdefines a first order differential operator onM and
hence by definitionXu∈H0(M) =L2(M). However, a priori there is no well-defined
restriction of an L2 function to the boundary ∂M, so we have to use some extra
information, namely that u is a solution of ∆gu= 0.
In this chapter we will prove a theorem of Péetre which gives that a function
u∈H1(M) which is in addition a solution of ∆gu= 0 is, locally near the boundary,
smooth in the normal variable up to the boundary and hence its normal derivative has a restriction to the boundary.
In section 2.1 we will state Péetre’s Theorem on partial hypoellipticity and explain the notion of partially smooth distributions. Section 2.2 explains some facts about vector valued distributions which are required for the proof of Péetre’s Theorem in section 2.3. And finally in 2.4 we will be able to define the Dirichlet-to-Neumann map.
2.1 Partial Regularity and Péetre’s Theorem
To state Péetre’s Theorem we have to specify what we mean by saying that a distribution u ∈ D0(Ω ×(0,T)) is smooth in the t-variable t ∈ (0,T) up to the boundary.
Proposition 2.1.1. Let Ω⊆Rn−1 be open and T >0. Then the map
ι :C∞([0,T],D0(Ω))−→ D0(Ω×(0,T)) given by hι(u),ϕi:= Z T 0 hu(t),ϕ(−,t)idt ∀ϕ∈ Cc∞(Ω×(0,T)) (2.1)
for u∈ C∞([0,T],D0(Ω)) is a linear continuous injection.
Proof. Since linearity is obvious it remains to show that ι is defined, continuous
and injective. In what follows we use the identification
Cc∞(Ω×(0,T))−→ C∼= c∞((0,T),Cc∞(Ω)), ϕ7→(t7→ϕ(−,t)).
(1) Using that D0(Ω) carries the topology of bounded convergence one easily shows that for u∈ C∞([0,T],D0(Ω)) and ϕ∈ C∞
c (Ω×(0,T))∼=C ∞ c ((0,T),C ∞ c (Ω)) the map [0,T]−→K, t7→ hu(t),ϕ(t)i
is smooth with derivative
d dthu(t),ϕ(t)i= * du dt(t),ϕ(t) + + * u(t),dϕ dt(t) + . (2.2)
This shows that the integral in (2.1) exists. To see that (2.1) in fact de-fines a distribution ι(u) it is sufficient to show that ι(u) maps bounded sets B ⊆ C∞
c (Ω×(0,T)) to bounded sets in K (cf. [Trè06, Corollary to Proposition 14.8]). This will follow from part (2).
(2) Let B ⊆ C∞
c (Ω×(0,T)) be a bounded subset, so B ⊆ C ∞
c (K1×K2) bounded
for some compact subsets K1 ⊂⊂Ω, K2 ⊂⊂(0,T) (see Proposition A.2.1(2)).
Thus
e
is bounded and we have sup ϕ∈B |hι(u),ϕi| ≤sup ϕ∈B Z T 0 |hu(t),ϕ(t)i|dt ≤T · sup t∈[0,T] sup ψ∈Be |hu(t),ψi|<∞. (2.3)
This shows that the bounded setB ⊆ C∞
c (Ω×(0,T)) gets mapped to a bounded set by ι(u) : C∞
c (Ω×(0,T)) −→K and hence ι(u) ∈ D0(Ω×(0,T)). But now the left hand side in (2.3) is a typical continuous seminorm onD0(Ω×(0,T)) at
ι(u) and the right hand side is a continuous seminorm onC∞([0,T],D0(Ω)) at u which gives continuity of ι.
(3) If 06= u∈ C∞([0,T],D0(Ω)) then u(t) 6= 0 for every t in some relatively open set in [0,T]. In this case it is not hard to construct ϕ∈ C∞
c ((0,T),Cc∞(Ω)) such that hι(u),ϕi= Z T 0 hu(t),ϕ(t)idt6= 0, hence ι(u)6= 0.
Thus we can identify C∞([0,T],D0(Ω)) with a subspace of D0(Ω×(0,T)). Distribu-tions u∈ C∞([0,T],D0(Ω)) are called smooth in t up to the boundary.
To actually work with C∞([0,T],D0(Ω)) as a subspace of D0(Ω×(0,T)) we need some basic facts.
Lemma 2.1.2. Let Ω⊂Rn−1 be open and T >0.
(1) ι preserves derivatives in t-direction, i.e. for u∈ C∞([0,T],D0(Ω)) we have
ι du dt ! =∂tι(u). (2) If κ: Ω −→Ω0 is a diffeomorphism, Ω0 ⊆ Rn−1 open, and u∈ C∞([0,T],D0(Ω)) then (κ×id)∗(ι(u))∈ C∞([0,T],D0(Ω0)) and
(κ×id)∗(ι(u)) =ι(κ∗◦u) (2.4)
where κ∗ is viewed as continuous linear operator D0(Ω)−→ D0(Ω0).
By part (1) it is clear that for distributions u ∈ C∞([0,T],D0(Ω)) the derivative
∂tu∈ C∞([0,T],D0(Ω)) has a restriction ∂tu(−,0)∈ D0(Ω) to t= 0.
parts: * ι du dt ! ,ϕ + = Z T 0 * du dt(t),ϕ(−,t) + dt (2.2) = hu(t),ϕ(−,t)i T 0 − Z T 0 hu(t),∂tϕ(−,t)idt = 0− hι(u),∂tϕi = h∂tι(u),ϕi.
(2) By the linear chain rule B.1.2 we have κ∗◦u∈ C∞([0,T],D0(Ω0)) since
κ∗ :D0(Ω)−→ D0(Ω0) is a linear continuous operator. Hence it remains to show (2.4). Again for ϕ∈ C∞ c (Ω0×(0,T))∼=Cc∞((0,T),Cc∞(Ω0)) we compute hι(κ∗◦u),ϕi= Z T 0 hκ∗u(t),ϕ(−,t)idt =Z T 0 hu(t),(κ×id)∗ϕ(−,t)idt =hι(u),(κ×id)∗ϕi =h(κ×id)∗(ι(u)),ϕi.
To state Péetre’s Theorem we use the notation
HPm−1(Ω×(0,T)) := ker(P :Hm−1(Ω×(0,T))−→H−1(Ω×(0,T)))
for a differential operator P of orderm in Ω×(0,T). Note thatHPm−1(Ω×(0,T)) is
a closed subspace of Hm−1(Ω×(0,T)) and hence a Hilbert space itself.
Theorem 2.1.3 (Péetre). Let Ω⊂Rn−1 be open andT > 0. Suppose
P = X
|α|≤m
aαDα
is a differential operator of order m∈N0 in Ω×(0,T) with aα ∈ C∞(Ω×[0,T])and the coefficient of Dm
n being equal to 1. If u∈Hm−1(Ω×(0,T))is a solution ofP u= 0 then u∈ C∞([0,T],D0(Ω)) and the inclusion
HPm−1(Ω×(0,T)),−→ C∞([0,T],D0(Ω)) (2.5)
is a continuous linear operator.
Note that we require the coefficients ofP to be smooth up to the boundary.
2.2 Vector Valued Distributions
In this section we briefly introduce the basic notions of vector valued distributions which are needed for the proof of Theorem 2.1.3. More details can be found e.g. in [Wlo82, Chapters 24 and 25]. These concepts will not be used in other chapters, only for the proof of Péetre’s Theorem. The reader could as well skip sections 2.2 and 2.3 and continue with the construction of the Dirichlet-to-Neumann map in section 2.4. Let H be any separable Hilbert space and (a,b)⊆R an open interval, −∞ ≤a < b≤ ∞. The space
D0((a,b),H) :=Lb(Cc∞((a,b)),H)
is called space of distributions with values in H. Weak derivatives are defined in
the obvious way:
du dt(ϕ) :=−u dϕ dt ! for ϕ∈ C∞ c ((a,b)) and u∈ D 0((a,b),H).
Now for p = 1,2 let Lp((a,b),H) be the space of equivalence classes of functions
u: (a,b)−→H such thatku(t)kpH is measurable and
Z b
a
ku(t)kpHdt <∞.
L2((a,b),H) becomes a Hilbert space if equipped with the norm k−k
0,H given by kuk2 0,H := Z b a ku(t)k2 Hdt <∞ ∀u∈L 2((a,b),H).
We can canonically embed C∞
c ((a,b),H) into L2((a,b),H).
In [Wlo82, Satz 24.8] it is shown that for functions u∈L1((a,b),H) an integral
Z b a
u(t)dt∈H
called theBochner integral can be defined by approximating with step functions. (This construction is quite similar to the construction of the Lebesgue integral. For
details see [Wlo82, Chapter 24].) The integral allows us to identify L2((a,b),H) with
a subspace of D0((a,b),H) via
u(ϕ) :=
Z b
a
for ϕ∈ C∞
c ((a,b)) andu∈L2((a,b),H) and the embedding
L2((a,b),H),−→ D0((a,b),H) (2.6)
is a continuous linear operator.
As in the usual theory of Sobolev spaces we put
Hk((a,b),H) := ( u∈L2((a,b),H) : d ju dtj ∈L 2((a,b),H)∀j = 0, . . . ,k )
for k ∈N0. The norm k−kk,H given by
kuk2k,H := m X j=0 dju dtj 2 0,H =Xk j=0 Z b a dju dtj (t) 2 H dt
provides Hk((a,b),H) with a Hilbert space structure.
We will further need a version of the Sobolev Embedding Theorem for the spaces
Hk((a,b),H). To give a proof of it let us use the following two classical results:
Theorem 2.2.1. (1) Let−∞ ≤a < b ≤ ∞. There is a linear continuous extension operator Hk((a,b),H)−→Hk((−∞,∞),H) for every k∈
N0.
(2) C∞
c ((−∞,∞),H) is dense in Hk((−∞,∞),H) for every k ∈N0.
Proof. Proofs for slightly different spaces can be found in [LM72, Theorems 2.1
and 2.2]. The same arguments carry through in this case.
Theorem 2.2.2 (Sobolev Embedding Theorem). Let −∞< a < b < ∞. For any
k ∈N0 we have Hk+1((a,b),H)⊆ Ck([a,b],H) with linear continuous embedding.
Proof. We only need to show that the restriction
Cc∞((−∞,∞),H)−→ Ck([a,b],H)
has a continuous extension R to Hk+1((−∞,∞),H). Then the desired embedding
can be factorized into
Hk+1((a,b),H) ⊆ // ESSSSSSS)) S S S S S S S Ck([a,b],H) Hk+1((−∞,∞),H) R 5 5 l l l l l l l l l l l l l l
where E denotes an extension operator (cf. Theorem 2.2.2(1)). (It is obvious that
this does not depend on the chosen extension map.) So fix χ ∈ C∞(
u∈ C∞ c ((−∞,∞),H) and 0≤j ≤k we have dju dtj(t) = Z t t−1 d dt " χ(τ −t)d ju dtj(τ) # dτ ≤ Z t t−1 dχ dt(τ −t) dju dtj(τ) dτ + Z t t−1 χ(τ −t)d j+1u dtj+1(τ) dτ ≤ Z 0 −1 dχ dt(τ) 2 dτ 1/2 Z ∞ −∞ dju dtj (τ) 2 dτ 1/2 +Z 0 −1 |χ(τ)|2 dτ 1/2 Z ∞ −∞ dj+1u dtj+1(τ) 2 dτ 1/2 .
This shows continuity of the restrictionC∞
c ((−∞,∞),H)−→ Ck([a,b],H) in the norm ofHk+1((−∞,∞),H), so, thanks to the density result in Theorem 2.2.2(2), it extends
as a linear continuous operator to Hk+1((−∞,∞),H) −→ Ck([a,b],H). (That the extension is actually the restriction follows by an easy approximation argument.)
We will apply the theory of vector valued distributions to the case whereH =H`(Ω),
` ∈ Z, for some open set Ω ⊂ Rn−1 and (a,b) = (0,T) and denote the norm of
Hk((0,T),H`(Ω)) byk−kk,`. The following embedding allows us to consider the space
Hk((0,T),H`(Ω)) as a space of distributions in Ω×(0,T): Hk((0,T),H`(Ω))⊆L2((0,T),H`(Ω))(2.6)⊆ D0((0,T),H`(Ω)) =L(Cc∞((0,T)),H`(Ω))⊆L(Cc∞((0,T)),D0(Ω)) ∼ =D0(Ω×(0,T)), (2.7) where we have used the Schwartz kernel theorem A.3.1 for the last isomorphism.
On the other hand, to consider certain distributions in Ω×(0,T) as elements of Hk((0,T),H`(Ω)) we need the following Lemma.
Lemma 2.2.3. The map
J :Hk(Ω×(0,T))−→Hk((0,T),L2(Ω)) (2.8)
given by
J u(t) := u(−,t) (2.9)
for almost every t ∈(0,T) is a linear continuous injection.
Sketch of the Proof. Basic measure theory applies to check that (2.9) defines
derivatives dj(J u)
dtj are given byJ
∂jtuand hence are measurable maps (0,T)−→L2(Ω)
by the same argument as above.
Using Fubini’s Theorem we now have the estimate
kJ uk2 k,0 = k X j=0 Z T 0 dj(J u) dtj (t) 2 L2(Ω) dt =Xk j=0 Z T 0 ∂ j tu(−,t) 2 L2(Ω) dt =Xk j=0 Z T 0 Z Ω |∂tju(x,t)|2dx dt =Xk j=0 Z Ω×(0,T) |∂tju(x,t)|2d(x,t) ≤ X |α|≤k Z Ω×(0,T) |∂αu(x,t)|2d(x,t) = kuk2 (k)<∞.
This gives J u∈Hk((0,T),L2(Ω)) and continuity of (2.8).
Now the advantage of this splitting of variables is that one can talk about partial regularity. If u∈Hk(Ω×(0,T)) then we only know that derivatives in every direction exist up to order k ∈ N0. We are not able to express that u admits higher order
derivatives e.g. in the t-variable t∈(0,T).
Whereas by viewingu as a function in the space Hk((0,T),L2(Ω)) we can later state
the Sobolev version of Péetre’s theorem on partial hypoellipticity. This will provide that for a certain class of differential operators P a solutionu∈Hm−1((0,T),L2(Ω))
of P u= 0 is also for arbitrary k∈N an element of the spaceHk((0,T),H−`(Ω)) with
` ∈N depending onk. This basically means that we can improve regularity ofu in
the t-variable in (0,T) if we take a loss of regularity in the x-variable in Ω.
To prove this version of Péetre’s theorem we need to understand how differential operators act on the spaces Hk((0,T),H`(Ω)). A technique to define general operators on these spaces is provided by the following abstract Lemma.
Lemma 2.2.4. A map a ∈ C∞([0,T],L(H`1(Ω),H`2(Ω))), `
1,`2 ∈ Z, induces for
every k ∈N0 a linear continuous operator
A:Hk((0,T),H`1(Ω))−→Hk((0,T),H`2(Ω))
by (Au)(t) := a(t)u(t) for u∈Hk((0,T),H`1(Ω)).
Sketch of the Proof. Approximating with step functions one first shows that
Then the following estimate gives Au∈L2((0,T),H`2(Ω)): kAuk20,`2 = Z T 0 ka(t)u(t)k2(`2)dt ≤ Z T 0 ka(t)k2 L(H`1,H`2)· ku(t)k2(` 1)dt ≤ sup t∈[0,T] ka(t)k2L(H`1,H`2)· kuk20,` 1.
Now weak derivatives of Au can be calculated using the product rule Au dt = dA dt u+A du dt where dA dt is given by da dt ∈ C
∞([0,T],L(H`1(Ω),H`2(Ω))). The previous argument
applied to the derivatives gives Au ∈ Hk((0,T),H`2(Ω)) and estimates for the L2
norms of derivatives of Au. Hence A is defined and continuous.
Proposition 2.2.5. Let Ω⊂Rn−1 be open, T > 0, m
1,m2 ∈N. Suppose
P = X
|α|≤m1,αn≤m2
aαDα
is a differential operator of order m = max(m1,m2) in Ω×(0,T) with
aα ∈ C∞(Ω×[0,T]). Then P induces a bounded linear operator
P :Hk((0,T),H`(Ω))−→Hk−m2((0,T),H`−m1(Ω)) (2.10)
for every k ≥m2 and ` ∈Z. Under the embedding (2.7) this action coincides with
the action of P on D0(Ω×(0,T)).
Proof. (1) Clearly ∂n is defined as (weak) differentiation
∂n:Hk((0,T),H`(Ω)) −→Hk−1((0,T),H`(Ω)).
(2) Forj = 1, . . . ,n−1 we have ∂j ∈L(H`(Ω),H`−1(Ω)) which can also be viewed as the constant function ∂j ∈ C∞([0,T],L(H`(Ω),H`−1(Ω))). Thus by Lemma 2.2.4 we have
∂j :Hk((0,T),H`(Ω))−→Hk((0,T),H`−1(Ω)).
(3) First observe that the map λ : C∞(Ω) −→ L(H`(Ω)) mapping a function
b ∈ C∞(Ω) to the multiplication operator λ(b) := b· : H`(Ω) −→ H`(Ω) is a bounded linear operator itself. Then the multiplication on Ω× (0,T) by
a function a ∈ C∞(Ω×[0,T]) ∼= C∞([0,T],C∞(Ω)) is expressed by λ◦ a ∈
2.2.4 we obtain
a·:Hk((0,T),H`(Ω)) −→Hk((0,T),H`(Ω)).
Putting these arguments together gives the desired mapping property.
That (2.10) is actually the restriction of P :D0(Ω×(0,T))−→ D0(Ω×(0,T)) follows from a close examination of the embedding (2.7).
2.3 The Proof of Péetre’s Theorem
Let us first prove a Sobolev version of Péetre’s Theorem by induction. The theorem stated in section 2.1 then follows easily using the Sobolev Embedding Theorem 2.2.2. Proposition 2.3.1. Let Ω⊂Rn−1 be open, T >0, m∈
N. Suppose
P = X
|α|≤m
aαDα
is a differential operator of order m in Ω×(0,T) with aα ∈ C∞(Ω×[0,T]) and the coefficient of Dm
n being equal to 1. If u∈H
m−1((0,T),L2(Ω)) is a solution of
P u= 0 (2.11)
then for every k ∈N0 we have u∈Hm−1+k((0,T),H−km(Ω)) and there is a constant
Ck >0 such that
kukm−1+k,−km ≤Ckkukm−1,0
for every u∈Hm−1((0,T),L2(Ω)) with P u= 0.
Proof. We only prove the case k= 1. The rest follows easily by induction.
Let u∈Hm−1((0,T),L2(Ω)) withP u= 0. Equation (2.11) can by assumption on the
coefficient of Dmn be written as Dmnu=− X
|α|≤m,αn≤m−1
aαDαu. (2.12)
(A priory this equation does not make sense in L2((0,T),L2(Ω)), but in
D0((0,T),H−m(Ω)) where weak derivatives of arbitrary order exist.) The right hand side of (2.12) defines a differential operator Qsatisfying the conditions of Proposition
2.2.5 with m1 =m, m2 =m−1. Hence Q induces a continuous linear operator
But this shows that
Dnmu=−Qu∈L2((0,T),H−m(Ω)).
Since u∈Hm−1((0,T),L2(Ω))⊆Hm−1((0,T),H−m(Ω)) this yields u∈Hm((0,T),H−m(Ω)).
Existence of the constant C1 then follows from continuity of Q.
Now let us finally prove Péetre’s Theorem as stated in section 2.1.
Proof of Theorem 2.1.3. Using (2.8) we first split variables of
u∈Hm−1(Ω×(0,T)) and identify u with the vector valued distribution
u∈Hm−1((0,T),L2(Ω)). Proposition 2.3.1 shows that for arbitraryk ∈
N0 there is
s∈N with u∈Hk+1((0,T),H−s(Ω)).
But now by the Sobolev embedding theorem 2.2.2
Hk+1((0,T),H−s(Ω))⊆ Ck((0,T),H−s(Ω))
.
Composed with the embedding Ck([0,T],H−s(Ω)),−→ Ck([0,T],D0(Ω)) (see Lemma B.1.2) this gives u∈ Ck([0,T],D0(Ω)). Since this holds for arbitrary k ∈
N0 we finally
haveu∈ C∞([0,T],D0(Ω)).
To check continuity of the inclusion (2.5) observe that C∞([0,T],D0(Ω)) carries the initial topology with respect to the embeddings
C∞([0,T],D0(Ω)),−→ Ck([0,T],D0(Ω)).
Hence it remains to check that all maps HPm−1(Ω×(0,T)),−→ Ck([0,T],D0(Ω)) for
k ∈N0 are continuous which is evident since the inclusion maps in Proposition 2.3.1
and Theorem 2.2.2 are continuous.
2.4 The Dirichlet-to-Neumann map
Let M be a compact connected Riemannian manifold with boundary ∂M. We will
now construct the Dirichlet-to-Neumann Map Λg as a continuous linear operator
Λg :H1/2(∂M)−→ D0(∂M).
For f ∈ H1/2(∂M) let u = PI f ∈ H1(M) be the unique solution of the Dirichlet
problem
∆gu= 0 in M◦, u|∂M =f