Fractional order Sobolev spaces Thesis Matematikus MSc

Készítette:

Témavezet®:

Gerencsér Máté Izsák Ferenc adjunktus

Alkalmazott Analízis és Számításmatematikai Tanszék

Eötvös Loránd Tudományegyetem Természettudományi Kar Budapest, 2012

CONTENTS

1

Contents

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

. . . . . . . . . . . . . . . . . . . .

5

3.1.

Real interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.2.

Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3.3.

Triebel-Lizorkin spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1. Introduction

2. Preliminaries

3. Classical scales of function spaces

. . . . . . . . . . . . . . . . . . . . . . .

13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

4. The fractionalization of

Hcurl

4.2.

s The scale Hcurl .s The scale Hcurl

4.3.

Fractionalization of the

operator . . . . . . . . . . . . . . . . . .

19

4.4.

Non-positive indices . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

4.1.

curl

2

1. INTRODUCTION

1. Introduction The term fractional order Sobolev space might sound like a precise mathematical concept but in fact it is not. There are several methods to ll in the gaps between the traditional Sobolev spaces of integer order and in some cases the function spaces obtained are equivalent, while in other cases they are not.

Dierent approaches

focus on generalizing dierent properties of the Sobolev spaces and each has its own advantage. These generalizations can be interesting and useful both theoretically and in the applications as well. The aim of the thesis is to give an overview of these ideas and apply these techniques to the a non-classical Sobolev space

Hcurl .

The thesis is structured as follows. Chapter 2 summarizes the usual notions used in the following.

Chapter 3 describes the dierent scales of function spaces that

are usually referred to as fractional order Sobolev spaces, based on [1], [2], [5]. Chapter 4 examines the space of

L2

functions whose

fractionalization problems regarding this space.

curl

is also in

L2 ,

and some

The classical results of the topic

follows [4], the rest of the chapter is partially based on [3], partially my own work. I would like to thank my supervisor, Izsák Ferenc, for his help throughout the making of this thesis.

3

2. Preliminaries We will use the following notions throughout the thesis.

Ω ⊂ Rn . A bounded domain is called Lipschitz, if boundary ∂Ω there exists a neighborhood U of x such that

A domain is an open subset for every point

U ∩ ∂Ω

x

of its

is a graph of a Lipschitz-continous function.

For a normed space

Cg(u) (cf (u) ≤ g(u))

X

by default

||·||X

denotes its norm. The relation

denotes that there exists a xed

inequality holds for all

u

C > 0 (c > 0)

f (u) ≤

for which the

from a given space that is always clear from the context.

This constant can change from line to line but is always independent from

u.

The

f (u) ∼ g(u) denotes that cg(u) ≤ f (u) ≤ Cg(u). Two norms ||·|| and ||·||0 0 on a normed space U are said to be equivalent if ||u|| ∼ ||u|| . The space D(Ω) consists of the compactly supoorted innitely many times dierentiable functions with the topology dened by the convergence: φn → φ if and only if there is a compact set K ⊂ Ω with supp(φn ), supp(φ) ⊂ K and for all multiindex α ∂ α φn → ∂ α φ uniformly. Its dual, D0 is the space of the continuous linear functionals, or, the distributions, with the weak topology. For every locally integrable f correR Pn sponds a distribution with the eect φ → j=1 fj φj . For a normed function space V in which D is imbedded, we dene its dual to be the subset of distributions which extends uniquely to V . n α β The Schwartz space of functions u on R with sup |∂ u(x)x | < ∞ for all α, β 0 multiindices is denoted by S . The elements of S , the dual of S are called tempered 0 distributions. The Fourier transformation operator, which can be dened on S , is denoted by F , and the notation u ˆ = F(u) is also used. On S , we dene F(φ)(ξ) = R (2π)−n/2 exp(i < x, ξ >)dx and we extend it to S 0 by Fu(φ) = u(Fφ). n p Given a domain Ω ⊂ R we use the notation L (Ω) for the space of functions u R p with ||u||Lp (Ω) = |u|p < ∞ (1 ≤ p < ∞). When it does not cause confusion, we Ω use the abbreviation ||·||Lp (Ω) = ||·||p . The standard Sobolev space of the functions p k,p with their α-th partial derivatives in L (Ω) for all |α| ≤ k is denoted with W (Ω), P α accompanied with the norm ||u||W k,p (Ω) = |α|≤k ||∂ u||p (k ∈ N, 1 ≤ p < ∞). Here 0 we use the derivatives in the distributional sense, i.e. for u ∈ D , φ ∈ D , we dene ∂ alpha u(φ) = (−1)|α| u(∂ α (φ)). We distinguish the special case H k (Ω) = W k,2 (Ω). We p p n also use the abbreviation L = L (R ), and similarly with the Sobolev spaces. We relation

4

2. PRELIMINARIES

also use this convention for the spaces introduced later.

X and Y their direct sum is denoted by X ⊕ Y and 2 consists of the formal sums x + y where x ∈ X, y ∈ Y , and ||x + y||X⊕Y = (||x||X + ||y||2Y )1/2 . Note that changing 2 to any 1 ≤ p < ∞ we obtain equivalent Banach spaces. If X and Y are Hilbert spaces, the norm can also be characterized by the identities ||x + 0||X⊕Y = ||x||X , ||0 + y||X⊕Y = ||y||Y , and the orthogonality of the components. Given a closed subspace U in a Hilbert space PU denotes the projection operator to U . Given two Banach spaces

5

3. Classical scales of function spaces This section aims to cover most of the possible denitions of fractional order Sobolev spaces that can be found in the literature and describe their relations to each other.

To avoid confusion, we will omit the term fractional order Sobolev

space and use other common names for these spaces instead. We will formulate the dierent but equivalent denitions in forms of theorems. For the detailed proofs we refer to [1], [2], [5].

3.1. Real interpolation

X0 X0 ∩ X1 = 6 {0}

Given two Banach spaces space

X

with a

and

X1 ,

both continously imbedded in a Banach

- such a pair is called an interpolation couple -,

interpolation methods provide ways to construct intermediate spaces between them. In many cases, including the ones we will deal with,

X1 .

X0

is continuously imbedded in

The two main dierent methods are the real and complex interpolation but we

will now only go into details with the real method. The intersection

X 0 ∩ X1

and the algebraic sum

X0 + X 1

are themselves Banach

spaces with the norms

||u||X0 ∩X1 = max{||u||X0 , ||u||X1 }, ||u||X0 +X1 = inf{||x0 ||X0 + ||x1 ||X1 |u = x0 + x1 , x0 ∈ X0 , x1 ∈ X1 }. The intersection is continuously imbedded in

Xj

and

Xj

is continuously imbedded

j = 0, 1. In general, we say that a Banach space U is intermediate between X0 and X1 if X0 ∩ X1 is continuously imbedded in U and U is continuously imbedded in X0 + X1 . When X0 ⊂ X1 , this equals the intuitive requirement that and intermediate space has to be larger than X0 and smaller than X1 . For any given x ∈ X0 + X1 ⊂ X , t ≥ 0, 1 ≤ p < ∞ dene in the algebraic sum for

Kp (t, x) = inf{||x0 ||pX0 + ||tx1 ||pX1 |x = x0 + x1 , x0 ∈ X0 , x1 ∈ X1 }1/p . The usual approach takes

p = 1,

but it will be clear that all choice of

p

result in

6

3. CLASSICAL SCALES OF FUNCTION SPACES

equivalent Banach spaces. Since in the case of Hilbert spaces the choice turn out to be the suitable one, we introduce the abbreviation

1 ≤ q < ∞ and 0 < θ < 1. x ∈ X0 + X1 for which

The interpolation space

||x||q(X0 ,X1 )θ,q

(X0 , X1 )θ,q

∞

Z

[t−θ K(t, x)]q

= 0

K = K2 .

p=2

will

Now take

consists of the vectors

dt < ∞. t

Let (X0 , X1 ) be an interpolation couple, 0 < θ < 1, and 1 ≤ q < ∞. Then (X0 , X1 )θ,q is a Banach space and its norm satises Theorem 3.1

||u||(X ,X ) √ 1 √ ||u||X0 +X1 ≤ R ∞ −θ 0 1 θ,q q dt ≤ 2 ||u||X0 ∩X1 , [t min{1, t}] t 2 0

therefore (X0 , X1 )θ,q is an intermediate space between X0 and X1 . The extremal cases for

q = ∞ and/or θ = 0, 1 can also be dened, the correspond-

ing theorems are often trivial, but to avoid technical diculties we will not deal with these cases. Chopping the dening integral to integrals between

2j

and

2j+1

one can show the

following discretization theorem.

Theorem 3.2

Let (X0 , X1 ) be an interpolation couple, 0 < θ < 1, and 1 ≤ q < ∞.

Then for all x ∈ X0 + X1

||x||q(X0 ,X1 )θ,q

∼

∞ X

2−jqθ (K(2j , x))q .

j=−∞

Let (X0 , X1 ) be an interpolation couple, 0 < θ < 1, and 1 ≤ q < ∞. Then X0 ∩ X1 is dense in (X0 , X1 )θ,q . Theorem 3.3

Let us introduce a functional similar to

K,

this time on

X0 ∩ X1 :

J(t, u) = max{||u||X0 , ||tu||X1 }. This functional can be used to dene interpolation in another way resulting in equivalent spaces, but more importantly, it can be used to formulate the Reiteration

3.1. Real interpolation

7

X

Theorem. First, we dene an intermediate space

to be in the class

H(θ, X0 , X1 ),

if

K(t, u) ≤ Ctθ ||u||X ||u||X ≤ Ct−θ J(t, u) Lemma 3.4

for all for all

u ∈ X,

and

u ∈ X0 ∩ X1 .

Let (X0 , X1 ) be an interpolation couple, 0 < θ < 1, and 1 ≤ q < ∞.

Then (X0 , X1 )θ,q ∈ H(θ, X0 , X1 ).

Let (X0 , X1 ) be an interpolation couple, 0 < λ < 1, 1 ≤ q < ∞, 0 ≤ θ0 < θ1 ≤ 1, and Y0 , Y1 intermediate spaces between X0 and X1 such that Yj ∈ H(θj , X0 , X1 ), j = 0, 1. Let θ = (1 − λ)θ0 + λθ1 .Then Theorem 3.5 (Reiteration Theorem)

(Y0 , Y1 )λ,q = (X0 , X1 )θ,q . The immediate consequence - and the reason for the name of the theorem - is that with the previous notations

((X0 , X1 )θ0 ,q0 , (X0 , X1 )θ1 ,q1 )λ,q = (X0 , X1 )θ,q where

1 ≤ q0 , q1 < ∞

are arbitrary.

It is also an important property of the interpolation that its eect on the dual spaces can be expressed easily.

Let (X0 , X1 ) be an interpolation couple, 0 < θ < 1, and 1 < q < ∞ and assume that X0 ∩ X1 is dense in both X0 and X1 . Dene q 0 by 1/q + 1/q 0 = 1. Then (X1∗ , X0∗ ) is also an interpolation couple and

Theorem 3.6 (Duality Theorem)

(X1∗ , X0∗ )θ,q = (X0 , X1 )∗1−θ,q0 . It is worth to note another simple identity in which this kind of change of parameters appear:

(X0 , X1 )θ,q = (X1 , X0 )1−θ,q . When interpolating between two Hilbert spaces, it is natural to expect that the result is a Hilbert space as well. By checking the paralelogram identity the following theorem provides a sucient condition.

8

3. CLASSICAL SCALES OF FUNCTION SPACES

Let (X0 , X1 ) be an interpolation couple consisting of two Hilbert spaces and 0 < θ < 1. Then (X0 , X1 )θ,2 is also a Hilbert space.

Theorem 3.7

3.2. Besov spaces The scale of Besov spaces is obtained by using the real interpolation method to create intermediate spaces between Sobolev spaces. We have to note that in general the classical Sobolev spaces are not closed under interpolation, i.e. does not equal to

Lemma 3.8

Then

W k,p (Ω) usually

(Lp (Ω), W m,p (Ω))k/m,q .

Let Ω be a Lipschitz domain. Let 0 < k < m be integers and 1 ≤ p < ∞. W k,p (Ω) ∈ H(k/m, Lp (Ω), W m,p (Ω)).

Let 0 < s < ∞, 1 ≤ p ≤ ∞, and 1 ≤ q < ∞. Let m be the smallest integer larger than s. Then the Besov space B s,p,q (Ω) is dened by Definition 3.9

B s,p,q (Ω) = (Lp (Ω), W m,p (Ω))s/m,q . The Reiteration Theorem and the previous Lemma show us that in fact

B s,p,q (Ω) = (W k,p (Ω), W m,p (Ω))θ,q for any

k 0 such that s − 1/p > 0. Then the trace operator is continuous from B s,p,q (Rn ) to B s−1/p,p,q (Rn−1 ).

Theorem 3.11

Repeating taking traces gives imbedding theorems to spaces on large

k.

Rk

for suciently

These theorems extend to traces on suciently smooth surfaces of suciently

high dimension as well.

In case there exists a suitable extension operator from

these theorems also extend to functions in

B

s,p,q

Ω,

(Ω).

It is sometimes useful to examine what more well-known spaces includes the Besov space in question. This motivates the imbedding theorems similar to the following one.

Let 1 < p < ∞, 1 ≤ q < ∞, s > 0, such that sp > n. Then B s,p,q is imbedded in the space of continuous and bounded functions.

Theorem 3.12

The norms of the Besov spaces on

have a more intristic equivalent expressed

h ∈ Rn and a function u ∈ L(R3 ) dene uh to be the mapping x to u(x − h). Let ∆h u = u − uh , ωp (u, h) = ||∆h u||p , (m) m and for positive integers m let ωp (u, h) = ||(∆h ) u||p .

with the

L

p

Rn

-modulus of continuity. First, for a point

Let 1 < p < ∞, 1 ≤ q < ∞, m > s > 0 with m being an integer and u ∈ L (R ). Then u ∈ B s,p,q (Rn ) if and only if

Theorem 3.13 p

n

Z



Rn

|h|−s ωp(m) (u, h)

q dh < ∞. |h|n

Moreover, the q-th root of the expression above is equivalent to ||·||Bs,p,q . For non-integers

s = bsc + {s},

bsc ∈ N and {s} ∈ (0, 1), bsc-th order derivatives:

where

intristic norm can be dened using the

a closely related

Let 1 < p < ∞, 1 ≤ q < ∞, s > 0, and u ∈ W bsc,p (Rn ). Then u ∈ B s,p,q (Rn ) if and only if Theorem 3.14

X Z |α|=bsc

Rn

Z

α

α

p

|∂ u(x) − ∂ u(x − h)| Rn

q/p

|h|−{s}q

dh < ∞. |h|n

10

3. CLASSICAL SCALES OF FUNCTION SPACES

There is another equivalent intristic norm that will show some kind of relation between the Besov spaces and the Triebel-Lizorkin spaces dened in the next subsec-

Φ be an even function on the real line with the properties Φ(t) = 1 for |t| ≤ 1, Φ(t) = 0 for |t| ≥ 2, and |Φ(t)| ≤ 1 for all t. For each integer i i+1 let φi (t) = Φ(t/2 ) − Φ(t/2i ) and for any ξ ∈ Rn let ψi (ξ) = φi (|ξ|). Finally, dene −1 the operator Ti u = F (ξ → ψi (ξ)ˆ u(ξ)). One can regard the functions Ti u as dyadic parts of u with nearly disjoint frequencies. tion. To this end, rst let

Theorem 3.15

The norm ||u||Bs,p,q (Rn ) is equivalent to "

∞ Z X

(1 + 2sj )p |Tj u(x)|p dx

q/p #1/q .

Rn

j=−∞

3.3. Triebel-Lizorkin spaces

Let 0 < s < ∞, 1 ≤ p < ∞, and 1 ≤ q < ∞. Then the TriebelLizorkin space F (Ω) is dened by Definition 3.16

s,p,q

F s,p,q (Rn ) = {u : ||u||F s,p,q (Rn )

Clearly

F s,p,p = B s,p,p ,

 Z  =

Rn

∞ X

!p/q (1 + 2sj )q |Tj u(x)|q

1/q dx

< ∞}.

j=−∞

and therefore the Slobodeckij spaces are included in this

scale as well. Another important special case is obtained by generalizing the wellknown fact that for any

|ξ|2 )k/2 uˆ(ξ) is changing k to

k

integer

u ∈ W k,p

if and only if the function

the Fourier transform of a function from any

s≥0

Lp .

ξ → (1 +

The spaces we get by

in this property are often referred to as the Bessel potential

spaces.

Let 0 < s < ∞ and 1 ≤ p < ∞. Then u ∈ F s,p,2 if and only if the function ξ → (1 + |ξ|2 )s/2 uˆ(ξ) is the Fourier transform of a function from Lp . Theorem 3.17

(1+|ξ|2 )s/2 can be bounded from above and below by constant times 1+|ξ|s , −1 the previous condition can be rephrased that require u and ξ → F |ξ|s uˆ(ξ) to be p in L . The latter expression is also known as the fractional Laplacian of u, and so Since

the Bessel potential spaces can be obtained by generalizing the denition of Sobolev

3.3. Triebel-Lizorkin spaces

11

spaces by the derivatives of the function.

The fractional Laplacian is one of the

possible generalizations of the dierentiation operator (see [7], Chapter 2), it simply fractionalizes the positive operator that

F

m,p,2

=W

m,p

for integers

−∆.

It also follows from the previous Theorem

m.

It also turns out that the Bessel potential spaces can be obtained from the classical Sobolev spaces the same way as the Besov spaces if we use the complex interpolation method instead of the real interpolation.

Let (X0 , X1 ) be an interpolation couple. Let A denote the collection of bounded analytic functions f from the strip {θ + iτ |0 < θ < 1} to X0 + X1 that extend continously to the boundary with the property f (j + τ ) ∈ Xj and ||f (j + iτ )||Xj → 0 as |τ | → ∞, for j = 0, 1. For 0 < θ < 1 the complex interpolation space between X0 and X1 is Definition 3.18

[X0 , X1 ]θ = {u ∈ X0 + X1 |∃f ∈ A : f (θ) = u}

with the norm ||u||[X0 ,X1 ]θ = inf{max{sup ||f (iτ )||X0 , sup ||f (1 + iτ )||X1 }|f (θ) = u}. τ

τ

Although we omit the detailed description of this method, it is worth to note that it has similar properties to the real method. For example the analogues of the Reiteration Theorem and the Duality Theorem hold.

We also have the following

identities that show the connection between the two methods

(X0 , X1 )θ,1 ⊂ [X0 , X1 ]θ ⊂ (X0 , X1 )θ,∞ , ([X0 , X1 ]θ0 , [X0 , X1 ]θ1 )λ,q = (X0 , X1 )(1−λ)θ0 +λθ1 ,q , [(X0 , X1 )θ0 ,q0 , (X0 , X1 )θ1 ,q1 ]λ = (X0 , X1 )(1−λ)θ0 +λθ1 ,q , where

q

is dened by

1/q = (1 − λ)/q0 + λ/q1 .

Let 0 < s < ∞, 0 < θ < 1, m ≤ k integers with (1 − θ)m + θk = s, and 1 ≤ p < ∞. Then Theorem 3.19

F s,p,2 = [W m,p , W k,p ]θ .

12

3. CLASSICAL SCALES OF FUNCTION SPACES

Similar imbedding theorems holds for the Triebel-Lizorkin spaces as for the Besov spaces. For example, analogously to the one mentioned in the previous subsection, the following theorem holds.

Let 1 < p < ∞, 1 ≤ q < ∞, s > 0, such that sp > n. Then F s,p,q is imbedded in the space of continuous and bounded functions.

Theorem 3.20

Finally, we summarize the relations between the spaces introduced:

In the special case case

m

F s,p,q ⊂ B s,p,q

if

q ≤ p,

B s,p,q ⊂ F s,p,q

if

p ≤ q,

B s,p,q0 ⊂ B s,p,q1

if

q0 ≤ q 1 ,

F s,p,q0 ⊂ F s,p,q1

if

q0 ≤ q1 .

p = q = 2, B s,2,2 = F s,2,2

with the special cases

F m,2,2 = H m

in

is integer, and furthermore we have

B s,2,2 = (B s0 ,2,2 , B s1 ,2,2 )θ,2 = [B s0 ,2,2 , B s1 ,2,2 ]θ for

0 < s0 < s < s1 < ∞ Remark.

with

(1 − θ)s0 + θs1 = s.

Some of the denitions and the theorems were stated only in the case

n

Ω = R . These properties can be extended to more general domains via the use of an extension operator. In [6] it is shown that there exists an extension operator E such s,p,q that it simultaneously and boundedly extends functions in B (Ω) to B s,p,q (Rn ) s,p,q and F (Ω) to F s,p,q (Rn ) if the domain Ω is nice enough, for example, if it is a s,p,q Lipschitz domain. Here F (Ω) is dened as restrictions of functions on F s,p,q (Rn ). s,p,q The spaces B (Ω) were already dened, but from the existence of this extension operator it follows that they can also be dened through restriction.

13

Hcurl

4. The fractionalization of

In this section we deal with the possibilities of the fractionalization of the nonstandard Sobolev space

Hcurl = {u ∈ (L2 )3 |curl(u) ∈ (L2 )3 }.

important role in the theory of Maxwell equations.

This space plays an

Even the extension to integer

orders is not evident, we present two approaches. First recall some properties of the

 curl(u) =

curl

operator. It is dened by

∂u3 ∂u2 ∂u1 ∂u3 ∂u2 ∂u1 − , − , − ∂2 ∂3 ∂3 ∂1 ∂1 ∂2



where derivatives are understood in the distributional sense. Its Fourier counterpart is the vectorial product with the longitudinal direction:

F(curl(u))(ξ) = i(ξ × uˆ(ξ)). The same formula holds if we replace

j ∈ N.

Here the operator

j

ξ×

curl

by

(curl)j , ξ×

by

ξ×j ,

and

i

by

ij

for any

is dened by

ξ ×j f (ξ) = ξ × · · · ξ× f (ξ) | {z } j

and similarly for

(curl)j .

The space

Hcurl

can be decomposed

Hcurl = Ker(curl) ⊕ Ker(curl)⊥ where the second component is compactly imbedded in

(L2 )3 .

This is known as the

Helmhlotz decomposition. The Fourier transform of a function in the kernel of the

curl from

points in the longitudinal direction almost everywhere. Conversely, if a function

(L2 )3

has a Fourier transform that points in the longitudinal direction, its

is zero. Therefore it is easy to see that if

u ∈ Ker(curl)⊥ ,

curl

then its Fourier transform

has to point perpendicular to the longitudinal direction. It is known that the sequence called the De Rahm-diagram

grad

curl

div

(H 1 )3 −→ Hcurl −→ Hdiv −→ (L2 )3

(1)

14

4. THE FRACTIONALIZATION OF

HCU RL

has the property that the image of each operator is included in the kernel of the next one, where

Hdiv = {u ∈ (L2 )3 |div(u) = Similarly, if our initial space is

∂u1 ∂u2 ∂u3 + + ∈ L2 }. ∂1 ∂2 ∂3

(L2 (Ω))3 ,

we can dene

Ω is bounded and has Σ1 , Σ2 , . . . ΣL such that

domain. Assume that connected surfaces

• Σl

Hcurl (Ω), Hdiv (Ω)

the property that there exist

for any

L

open

is an open part of a smooth surface,

• ∂Σl ⊂ ∂Ω, • Σl ∩ Σm = ∅ •

if

For any point

l 6= m,

x ∈ ∂Ω

there is an integer

rx ∈ {1, 2}

ρx > 0 center x

and a

0 < ρ < ρx the intersection of Ω with the ball with has rx connected components, each one being a Lipschitz

all

such that for and radius

ρ

domain.

Under these conditions the diagram (1) also has the property that the image of each operator is a closed subspace of nite codimension in the kernel of the next operator [4].

4.1. The scale

s Hcurl

If we consider the

curl

operator as a kind of dierentiation, it is natural to intro-

duce the following spaces

n Hcurl = {u ∈ (L2 )3 |curlj (u) ∈ (L2 )3 n Hcurl

for

j = 1, 2, . . . n}.

P 2 ||u||2H n = nj=0 ||curlj (u)||2 . curl s > 0 by dening

is a Hilbert space with the norm

this denition for any non-integer

dse

s Hcurl = ((L2 )3 , Hcurl )s/dse,2

where

dse

is the smallest integer larger than

We can extend

(2)

s.

Before moving on to examine these spaces further, we establish some simple properties of the real interpolation.

4.1. The scale

s Hcurl

15

Let X, Y, V, W be Banach spaces, 1 ≤ q < ∞, and 0 < θ < 1.

Proposition 4.1

Then

(X ⊕ V, Y ⊕ W )θ,q = (X, Y )θ,q ⊕ (V, W )θ,q

with equivalent norms. In case q = 2, the same holds for Hilbert spaces. Proof.

The equality of the sums simply follows from considering

1 √ (t ||x||X + ||y||Y + t ||v||V + ||w||W ) ≤ t ||x + v||X⊕V + ||y + w||Y ⊕W ≤ 2 ≤ t ||x||X + ||y||Y + t ||v||V + ||w||W . After taking inmums, multiplying by

dt respect to the measure on t In case

q =2

perpendicular to

[0, ∞),

tθ ,

taking

q -th

powers, and integrating with

we get the inequalities required.

it remains to show that for any

z + u ∈ (X ⊕ V, Y ⊕ W )θ,q , z

is

u.

4 < z, u >= ||z + u||2(X⊕V,Y ⊕W )θ,q − ||z − u||2(X⊕V,Y ⊕W )θ,q = Z =

∞

t−2θ (K(t, z + u))2 − (K(t, z − u))2

0 Note that the inmum dening

K


dt . t

can be decomposed:

K(t, z + u) = inf{||t(x + v)||2X⊕V + ||y + w||2Y ⊕W |z + u = (x + v) + (y + w)}1/2 = = inf{||tx||2X + ||y||2Y |z = x + y}1/2 + inf{||tv||2V + ||w||2W |u = v + w}. K(t, z − u)

has exactly same form, except that in the second inmum

−1. This does not aect the u)) − (K(t, z − u)) = 0 and < z, u >= 0. multiplied by

v

w are K(t, z +

and

value of the inmum, therefore

Let U, V, W be subspaces of a Hilbert space X , and suppose that W is closed, 1 ≤ q < ∞, and 0 < θ < 1. Then

Proposition 4.2

(U, V )θ,q ∩ W = (U ∩ W, V ∩ W )θ,q

with equivalent norms.

16

4. THE FRACTIONALIZATION OF

Proof.

HCU RL

w ∈ W . For every decomposition w = u + v we can take w = (PU ∩W u + PV ∩W v) + (PU ∩W ⊥ u + PV ∩W ⊥ v) instead. The second term is in W ⊥ by defnition, and ⊥ it is also a dierence of two elements of W , therefore it is in W ∩ W and thus it is Set

zero. The norm of the components can only decrease with this modication, therefore

KU ∩W,V ∩W (t, w) ≤ KU,V (t, w), while the other inequality is trivial. The of the two K functionals then implies the equivalence of the norms.

equivalence

Let U, V be Banach spaces with U compactly imbedded in V , 1 ≤ q < ∞, and 0 < θ < 1. Then (U, V )θ,q is also compactly imbedded in V . Proposition 4.3

Proof.

Let

{zn }

be a sequence in the unit ball of

n version of interpolation and K1 there are uj and

∈U

and

(U, V )θ,q . Using the discrete vjn ∈ V such that unj + vjn = zn

∞ h X iq 2−jθ (2j unj U + vjn V ) ≤ 2 ||zn ||(U,V )θ,q ≤ 2. j=−∞

We have

n uj ≤ 2j(θ−1)+1 , U

therefore

{unj } ⊂ V

has a Cauchy subsequence for all

j

mk and with the diagonal argument we get a subsequence {zmk } such that {uj } ⊂ V jθ is Cauchy for all j . Fix  > 0 and choose j so that 2 < . Then we have vjn V < ml k 2 and therefore if N is chosen so that for mk , ml > N um < , then j − uj V

||zmk − zml ||V < 5. This statement is true in a more general way, but this weaker version easily implies a stronger one. Note that from the Schauder theorem it follows that if compactly imbedded in

V

then

V∗

is compactly imbedded in

U ∗.

In case

U

and

are reexive spaces we can apply the previous proposition to the dual spaces. the Duality Theorem we obtain that

U

is compactly imbedded in

the Reiteration Theorem we can conclude that

(U, V )θ2 ,q for 0 < θ1 < θ2 < 1. n Let H0,curl denote the orthogonal

(U, V )θ1 ,q

(U, V )θ,q

U

is

V

By

and by

is compactly imbedded in

complement of Ker(curl) in

n Hcurl .

the inner product, and therefore the orthogonality, is the same in every

Note that

n Hcurl

if one

of the functions is in Ker(curl). Proposition 4.1 shows that it suces to interpolate between these spaces. It turns out that they are subspaces of the traditional Sobolev spaces.

4.2. The scale

.s Hcurl

17

Proposition 4.4 n (H n )3 ∩ Ker(curl)⊥ = H0,curl

with equivalent norms. Proof.

Let

u∈

⊥ Ker(curl) . For the Fourier transform of such a function

ξ⊥ˆ u(ξ)

holds almost everywhere, therefore

||u||H n

0,curl

n n X X j ξ → ξ ×j uˆ(ξ) = ∼ curl (u) 2 = 2 j=0

j=0

n n X X j ξ → |ξ|j |ˆ = ξ → |ξ × uˆ(ξ)| 2 = u(ξ)| 2 ∼ ξ → (1 + |ξ|2 )n/2 |ˆ u(ξ)| 2 = j=0

j=0

= ξ → (1 + |ξ|2 )n/2 uˆ(ξ) 2 ∼ ||u||(H n )3 .

Using the results from the interpolation of Sobolev spaces we can write the following

Corollary 4.5 s Hcurl = Ker(curl) ⊕ ((B s,2,2 )3 ∩ Ker(curl)⊥ )

(3)

as Hilbert spaces, where by Proposition 4.3 the second component is compactly imbedded in (L2 )3 . This is a slightly generalized form of the Helmholtz decomposition. The consequence of this decomposition is that these spaces "behave well" under interpolation, i.e.

s1 s2 s (Hcurl , Hcurl )θ,2 = Hcurl for any

0 ≤ s1 < s < s 2

4.2. The scale

with

(1 − θ)s1 + θs2 = s.

.s Hcurl

There is another common way to extend the denition of

Hcurl

to integer indices,

found in e.g [4]. In this case we are not interested in the higher order

curls

of the

18

4. THE FRACTIONALIZATION OF

functions, but instead the smoothness of the function and the

curl

HCU RL

of the function is

.n simultaneously prescribed. Dene Hcurl by .n = {u ∈ (H n )3 |curl(u) ∈ (H n )3 }. Hcurl

These are also Hilbert spaces with the norm

||u||2H .n = ||u||2(H n )3 +||curl(u)||2(H n )3 . curl

It is possible to derive a similar decomposition to (3) for these spaces. Obviously, if

u ∈ Ker(curl)

then

Proposition 4.6

(L2 )3 . Then

.n u ∈ Hcurl

if and only if

u ∈ (H n )3 .

Let Ker(curl)⊥ denote the orthogonal complement of Ker(curl) in .n Hcurl ∩ Ker(curl)⊥ = (H n+1 )3 ∩ Ker(curl)⊥

with equivalent norms. Proof.

Similarly to the proof of Proposition 4.4, consider a function

for which therefore

ξ⊥ˆ u(ξ)

u ∈ Ker(curl)⊥

holds. Using again the equivalent Sobolev norm and the

unitarity of the Fourier transform we can write

||u||H .n ∼ ξ → (1 + |ξ|2 )n/2 |ˆ u(ξ)| 2 + ξ → (1 + |ξ|2 )n/2 |ξ × uˆ(ξ)| 2 = curl

= ξ → (1 + |ξ|2 )n/2 |ˆ u(ξ)| 2 + ξ → |ξ|(1 + |ξ|2 )n/2 |ˆ u(ξ)| 2 ∼ ∼ ξ → (1 + |ξ|2 )(n+1)/2 |ˆ u(ξ)| 2 ∼ ||u||(H n+1 )3 .

u ∈ Ker(curl) ∩ (H n )3 and v ∈ Ker(curl)⊥ ∩ (H n )3 , then they are n 3 respect to the (H ) inner product as well. Indeed, taking partial

Observe that if orthogonal with

derivatives from the functions does not change the direction of the Fourier transform of the function, and thus

F(∂ α u)

points perpendicular to the longitudinal direction. Therefore

0

F(∂ α v) < F(∂ α u), F(∂ α v) >=

points in the longitudinal direction, and

pointwise. We thus get the decomposition

.n = ((H n )3 ∩ Ker(curl)) ⊕ ((H n+1 )3 ∩ Ker(curl)⊥ ) Hcurl

4.3. Fractionalization of the

curl

operator

19

as Hilbert spaces and we can again conclude that

.m .n .s (Hcurl , Hcurl )θ,2 = Hcurl

0 ≤ m < s < n integers with (1 − θ)m + θn = s. We can also extend this scale to any non-integer s by using the above interpolation equation as a denition. By the for any

same interpolation properties as we used in (3), we then obtain the decomposition, now for all

s ≥ 0, .s Hcurl = ((B s,2,2 )3 ∩ Ker(curl)) ⊕ ((B s+1,2,2 )3 ∩ Ker(curl)⊥ ).

4.3. Fractionalization of the

curl

operator

Similarly to the fractional versions of the Laplacian operator, it is possible to fractionalize the

curl

as well through fractionalizing the appropriate operator acting

on the Fourier transform. A natural expectation for these operators to form a semi-

curln operator dened ⊥ earlier. Let u ∈ Ker(curl) ∩ S and examine how the curl operator changes u ˆ at a ⊥ point ξ . Both u ˆ(ξ) and ξ × uˆ(ξ)/i falls in the subspace ξ . In fact ξ × uˆ(ξ)/i = Aˆ u(ξ), group, and also we wish that for integer indices we get back the

where

A = Aξ = |ξ|Rπ/2 = |ξ|

cos(π/2) − sin(π/2) sin(π/2) cos(π/2)

!

ξ ⊥ . Similarly, ξ ×n uˆ(ξ)/in = An uˆ(ξ). The s-th power s s by A = |ξ| Rsπ/2 for any s ≥ 0 and this leads to the

is a linear operator acting on of

A

can be easily dened

denition

curls (u) = F −1 (ξ → is Asξ uˆ(ξ)). This denition can be extended to any integrable, so is

u ∈

⊥ Ker(curl)

∩ (L2 )3 :

since

uˆ

is locally

ξ → is Asξ uˆ(ξ) and therefore we can take its inverse Fourier transform.

In fact this denition extends to any tempered distribution having locally integrable

curls (curlt (u)) = curls+t (u) and we dened curln so that coincides with the n-th power of the curl operator if n is an integer. 2 3 s If u ∈ Ker(curl) ∩ (L ) then we can simply take curl (u) = 0 for s > 0. Then we s 2 3 s t can extend curl to (L ) linearly. It is clear that the properties curl (curl (u)) = Fourier-transform. It is easy to see that

20

4. THE FRACTIONALIZATION OF

curls+t (u)

and

curln = (curl)n

still hold. This extension therefore satises our ex-

pectations and also, it is closely related to the scale of spaces

Proposition 4.7

HCU RL

s Hcurl .

s if and only if curls (u) ∈ (L2 )3 . Let u ∈ (L2 )3 . Then u ∈ Hcurl

This shows that this fractionalization of the

curl could also be used to give another

s equivalent denition of Hcurl quite analogously to the traditional Sobolev spaces.

Proof.

Clearly it is enough to show the claim for

(3), u0 ∈ (B

composition

s,2,2 3

) = (F

s,2,2 3

)

u0 = PKer(curl)⊥ u.

By the de-

, and we can use the equivalent norm of

Proposition 3.17. Therefore,

||u0 ||(L2 )3 + ||u0 ||H s

curl

∼ ||ξ → uˆ0 (ξ)||2 + ξ → (1 + |ξ|2 )s/2 uˆ0 (ξ) 2 ∼

∼ ||ξ → uˆ0 (ξ)||2 + ||ξ → ||ξ|s uˆ0 (ξ)|||2 = = ||ξ → uˆ0 (ξ)||2 + ξ → |is |ξ|s Rsπ/2 uˆ0 (ξ)| 2 = ||u0 ||2 + ||curls (u0 )||2 . Note that this calculation shows that, in fact, if we introduce the norm

||curls (u)||22 )1/2

on

s Hcurl ,

(||u||22 +

we obtain an equivalent Hilbert space.

The analogue of the diagram (1) can be written as

grad

curls

div

s −→ Hdiv −→ (L2 )3 (H 1 )3 −→ Hcurl

for any

s > 0,

and it preserves the property that the image of each operator is

included in the kernel of the next one. Indeed, the kernel of

2 3

curls

is the same for

u ∈ (L )

whose Fourier transform

points in the longitudinal direction almost everywhere.

On the other hand, the

any

s > 0,

kernel of

namely it consists of the functions

div

consists of the functions

2 3

u ∈ (L )

whose Fourier transform points in

perpendicular to longitudinal direction. By denition, all functions of form has this property. inclusion Im(curl

s

However, using the fractional

) ⊂ Ker(div)

curl

curls u

it is easy to show that the

is not of nite codimension.

Set 0 < T ≤ ∞ and let the family of linear operators {C s }s∈(0,T ) acting on the spaces {Vs }s∈(0,T ) such that C s C t is dened and equals to C s+t for all choice of s, t > 0 with s + t < T . Suppose furthermore that Ker(C s ) = Ker(C t ) for Proposition 4.8

4.3. Fractionalization of the

curl

operator

21

all s, t > 0. Then Im(C t ) ⊂ Im(C s ) for all 0 < s < t < T and the inclusions are either strict for every choice of s and t, or are trivial for every choice. C t u = C s (C t−s u). For the second part, it s s is enough to check the actions of C on Vs /Ker(C ), thus we can assume that the t s operators are injectives. Suppose that the inclusion Im(C ) ⊂ Im(C ) is srtict for a xed 0 < s < t. For any r > q > t x m such that (t − s)/m ≤ r − q . At least for s+j(t−s)/m one j ∈ {1, 2, . . . , m} one of the inclusions Im(C ) ⊂ Im(C s+(j−1)(t−s)/m ) is s+(j−1)(t−s)/m strict, that is, there is a v such that v = C u, but there is no such w that v = C s+j(t−s)/m w. Consequently there is no such w that v = C s+(j−1)(t−s)/m+(r−q) w 0 q−(s+(j−1)(t−s)/m) either. Thus v = C v equals to C q u, but doesnt equal to C r w for any w, and thus Im(C r ) ⊂ Im(C q ). We can conclude that there exists an s0 such that the inclusion of images is trivial if both indices are smaller than s0 , and strict if both s 2s s s indices are larger than s0 . On the other hand, if Im(C ) = Im(C ) = Im(C C ) for s s ms a xed s, then C maps Im(C ) onto itself, therefore Im(C ) = Im(C ns ) for any n, m integers. Hence s0 can be only 0 or T , which proves the statement. Proof.

The rst part is trivial from

It now suces to nd an

u ∈ Im(curl)\ Im(curl2 ), and then the Proposition yields

that all the intermediate inclusions in

Im(curl

s

n

) ⊂ Im(curls/2 ) ⊂ · · · ⊂ Im(curls/2 ) ⊂ · · · ⊂ Ker(div)

are strict, therefore the codimension of Im(curl

s

) in Ker(div) is innite. Consider −1 the function v = F (ξ → χ|ξ| + < curl∗s (u), v2 > for all

(4)

s 2 n 3 u ∈ Hcurl where the latter inner products are understood in the (L (R )) sense,

24

4. THE FRACTIONALIZATION OF

and

vj = v|Qj

for

j = 1, 2.

HCU RL

Tf for the regular distribution for R P3 Tf (φ) = j=1 fj φj . Consider the

Let us use the notation

any given locally integrable function

f.

That is,

distribution

T = Tv1 + curls (Tv2 ). We now show that

Tv1 (φ) =< φ, v1 >,

T

s Hcurl

is extended to

by

L.

(5)

Taking

φ ∈ S,

we clearly have

and

s

s

curl (Tv2 )(φ) = F(curl (Tv2 )(F

−1

3 X (Rsπ/2 vˆ2 )j (ξ)φˆj (−ξ) = i |ξ|

Z

s

(φ)) =

s

j=1

Z =

s

s

T

i |ξ| vˆ2 (ξ)

T ˆ Rsπ/2 φ(−ξ)

Z =

s

s

i |ξ|

3 X

T ˆ j (−ξ) = (ˆ v2 )j (ξ)(Rsπ/2 φ)

j=1

= F(v2 )(F −1 (curl∗s (φ)) =< curl∗s (φ), v2 > . Comparing this with (4), we get

T (φ) = L(φ),

so

L

indeed extends

T.

T is of the form (5), we need to show that it has a unique s 2 3 s extension to Hcurl . Since S is dense in (L ) , which is dense in Hcurl , for any xed ∞ s we can nd a sequence {φn }n=1 ⊂ S converging to it. We can write u ∈ Hcurl On the other hand, if

|T (φk ) − T (φl )| ≤ | < φk − φl , v1 > | + | < curl∗s (φk − φl ), v2 > | ≤ ≤ ||φk − φl ||2 ||v1 ||2 + ||curls (φk − φl )||2 ||v2 ||2 ≤ ||φk − φl ||H s

curl

||v||L2 (V ) .

{T (φn )} is a Cauchy sequence in C and so converges to a limit that we can denote by L(u) since it is clear that we obtain the same limit to any other sequence {ψn } ⊂ S converging to u. The functional L is linear and also bounded, since Thus

|L(u)| = lim |T (φn )| ≤ lim ||φn ||H s n→∞

n→∞

We can now extend the scale

s Hcurl

curl

||v||L2 (V ) = ||u||H s

curl

||v||L2 (V ) .

to negative indices in a natural way, while also

ensuring the expectations for duality. Let

A

denote the subspace of

S0

with distri-

butions whose Fourier transform is locally integrable and its longitudinal component is in

(L2 )3 .

For any

s∈R

set

As = A ∩ {u ∈ S 0 |curls (u) ∈ (L2 )3 }.

4.4. Non-positive indices

Proposition 4.10

25

The scale s Hcurl

 (L2 )3 ∩ A s = (L2 )3 + A

s

if s ≥ 0 if s ≤ 0

−s s )0 = Hcurl coincides with the initial denition (2) for s ≥ 0 and has the property (Hcurl .

Proof.

The rst claim is simply a rephrasing of Proposition 4.7. The only dierence

is the intersection with for

s ≥ 0.

A

s ⊂ (L2 )3 ⊂ A Hcurl the case s ≤ 0 follows

which does not change our space since

The second part is Proposition 4.9 for

from the reexivity of Hilbert spaces.

s ≥ 0,

while

26

REFERENCES

References [1] R. A. Adams, J. J. F. Fournier,

Sobolev Spaces, Second Edition,

Academic Press,

Amsterdam (2003) [2] J. Bergh, J. Löfström,

Interpolation Spaces, an Introduction,

Springer-Verlag,

Berlin-New York (1976) [3] N. Engheta, Fractional curl operator in electromagnetics,

Lett., 17: [4] P. Monk,

Microw. Opt. Technol.

86-91 (1998)

Finite Element Methods for Maxwell's Equations,

Clarendon Press,

Oxford (2003)

Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Dierential Equations, de Gruyter, Berlin (1996)

[5] T. Runst, W. Sickel,

[6] V. S. Rychkov On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains,

J. London Math. Soc. (2) 60, 237-257

(1999) [7] Q. Yang Novel analytic and numerical methods for solving fractional dynamical systems, PhD Thesis, Queensland University of Technology (2010)