Author | H. Inci, P. Topalov, and T. Kappeler | |

Isbn | 9780821887417 | |

File size | 608.98KB | |

Year | 2013 | |

Pages | 60 | |

Language | English | |

File format | ||

Category | mathematics |

EMOIRS
M
of the
American Mathematical Society
Volume 226 • Number 1062 (third of 5 numbers) • November 2013
On the Regularity of the
Composition of Diffeomorphisms
H. Inci
T. Kappeler
P. Topalov
ISSN 0065-9266 (print)
ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Inci, H., 1982- author.
On the regularity of the composition of diﬀeomorphisms / H. Inci, T. Kappeler, P. Topalov.
pages cm – (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; number 1062)
”November 2013, volume 226, number 1062 (third of 5 numbers).”
Includes bibliographical references.
ISBN 978-0-8218-8741-7 (alk. paper)
1. Diﬀeomorphisms. 2. Riemannian manifolds.
I. Kappeler, Thomas, 1953- author.
II. Topalov, P., 1968- author. III. Title.
QA613.65.I53 2013
2013025511
516.36–dc23
DOI: http://dx.doi.org/10.1090/S0065-9266-2013-00676-4
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10 9 8 7 6 5 4 3 2 1
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Contents
Chapter 1. Introduction
1
Chapter 2. Groups of diﬀeomorphisms on Rn
5
Chapter 3. Diﬀeomorphisms of a closed manifold
31
Chapter 4. Diﬀerentiable structure of H s (M, N )
39
Appendix A
49
Appendix B
51
Bibliography
59
iii
Abstract
For M a closed manifold or the Euclidean space Rn we present a detailed proof
of regularity properties of the composition of H s -regular diﬀeomorphisms of M for
s > 12 dim M + 1.
Received by the editor September 7, 2011, and, in revised form, January 12, 2012.
Article electronically published on March 28, 2013; S 0065-9266(2013)00676-4.
2010 Mathematics Subject Classiﬁcation. Primary 58D17, 35Q31, 76N10.
Key words and phrases. Group of diﬀeomorphisms, regularity of composition, Euler equation.
The ﬁrst author was supported in part by the Swiss National Science Foundation.
The second author was supported in part by the Swiss National Science Foundation.
The third author was supported in part by NSF DMS-0901443.
c
2013
American Mathematical Society
v
CHAPTER 1
Introduction
In this paper we are concerned with groups of diﬀeomorphisms on a smooth
manifold M . Our interest in these groups stems from Arnold’s seminal paper [4]
on hydrodynamics. He suggested that the Euler equation modeling a perfect ﬂuid
on a (oriented) Riemannian manifold M can be reformulated as the equation for
geodesics on the group of volume (and orientation) preserving diﬀeomorphims of
M . In this way properties of solutions of the Euler equation can be expressed in
geometric terms – see [4]. In the sequel, Ebin and Marsden [14], [15] used this
approach to great success to study the initial value problem for the Euler equation
on a compact manifold, possibly with boundary. Later it was observed that other
nonlinear evolution equations such as Burgers equation [6], KdV, or the Camassa
Holm equation [7], [17] can be viewed in a similar way – see [22], [32], as well
as [5], [19], and [23]. In particular, for the study of the solutions of the Camassa
Holm equation, this approach has turned out to be very useful – see e.g. [12], [33].
In addition, following Arnold’s suggestions [4], numerous papers aim at relating the
stability of the ﬂows to the geometry of the groups of diﬀeomorphisms considered
– see e.g. [5].
In various settings, the space of diﬀeomorphisms of a given manifold with prescribed regularity turns out to be a (inﬁnite dimensional) topological group with the
group operation given by the composition – see e.g. [15, p 155] for a quite detailed
historical account. In order for such a group of diﬀeomorphisms to be a Lie group,
the composition and the inverse map have to be C ∞ -smooth. A straightforward
formal computation shows that the diﬀerential of the left translation Lψ : ϕ → ψ ◦ϕ
of a diﬀeomorphism ϕ by a diﬀeomorphism ψ in direction h : M → T M can be
formally computed to be
(dϕ Lψ )(h)(x) = (dϕ(x) ψ)(h(x)), x ∈ M
and hence involves a loss of derivative of ψ. As a consequence, for a space of
diﬀeomorphisms of M to be a Lie group it is necessary that they are C ∞ -smooth
and hence such a group cannot have the structure of a Banach manifold, but only of
a Fr´echet manifold. It is well known that the calculus in Fr´echet manifolds is quite
involved as the classical inverse function theorem does not hold, cf. e.g. [18], [24].
Various aspects of Fr´echet Lie groups of diﬀeomorphisms have been investigated –
see e.g. [18], [31], [34], [35]. In particular, Riemann exponential maps have been
studied in [10], [11], [20], [21].
However, in many situations, one has to consider diﬀeomorphisms of Sobolev
type – see e.g. [12], [13], [14]. In this paper we are concerned with composition
of maps in H s (M ) ≡ H s (M, M ). It seems to be unknown whether, in general, the
composition of two maps in H s (M ) with s an integer satisfying s > n/2 is again in
1
2
1. INTRODUCTION
H s (M ). In all known proofs one needs that one of the maps is a diﬀeomorphism
or, alternatively, is C ∞ -smooth.
First we consider the case where M is the Euclidean space Rn , n ≥ 1. Denote
by Diﬀ1+ (Rn ) the space of orientation preserving C 1 -diﬀeomorphisms of Rn , i.e. the
space of bijective C 1 -maps ϕ : Rn → Rn so that det(dx ϕ) > 0 for any x ∈ Rn and
ϕ−1 : Rn → Rn is a C 1 -map as well. For any integer s with s > n/2 + 1 introduce
D s (Rn ) := {ϕ ∈ Diﬀ1+ (Rn ) | ϕ − id ∈ H s (Rn )}
where H s (Rn ) = H s (Rn , Rn ) and H s (Rn , Rd ) is the Hilbert space
H s (Rn , Rd ) := {f = (f1 , . . . , fd ) | fi ∈ H s (Rn , R), i = 1, . . . , d}
with H s -norm · s given by
f s =
d
fi 2s
1/2
i=1
and H s (Rn , R) is the Hilbert space of elements g ∈ L2 (Rn , R) with the property that
the distributional derivatives ∂ α g, α ∈ Zn≥0 , up to order |α| ≤ s are in L2 (Rn , R).
Its norm is given by
1/2
(1)
g s =
|∂ α g|2 dx
.
|α|≤s
Rn
Here we used multi-index notation, i.e. α = (α1 , . . . , αn ) ∈ Zn≥0 , |α| = ni=1 αi ,
x = (x1 , . . . , xn ), and ∂ α ≡ ∂xα = ∂xα11 · · · ∂xαnn . As s > n/2 + 1 it follows from the
Sobolev embedding theorem that
Ds (Rn ) − id = {ϕ − id | ϕ ∈ Ds (Rn )}
is an open subset of H s (Rn ) – see Corollary 2.1 below. In this way D s (Rn ) becomes
a Hilbert manifold modeled on H s (Rn ). In Section 2 of this paper we present a
detailed proof of the following
Theorem 1.1. For any r ∈ Z≥0 and any integer s with s > n/2 + 1
(2)
μ : H s+r (Rn , Rd ) × D s (Rn ) → H s (Rn , Rd ),
(u, ϕ) → u ◦ ϕ
and
(3)
inv : Ds+r (Rn ) → Ds (Rn ),
ϕ → ϕ−1
are C r -maps.
Remark 1.1. To the best of our knowledge there is no proof of Theorem 1.1
available in the literature. Besides being of interest in itself we will use Theorem
1.1 and its proof to show Theorem 1.2 stated below. Note that the case r = 0 was
considered in [8].
Remark 1.2. The proof for the C r -regularity of the inverse map is valid in a
much more general context: using that Ds (Rn ) is a topological group and that the
composition
Ds+r (Rn ) × D s (Rn ) → Ds (Rn ), (ψ, ϕ) → ψ ◦ ϕ
r
is C -smooth we apply the implicit function theorem to show that the inverse map
D s+r (Rn ) → Ds (Rn ),
ϕ → ϕ−1
H. INCI, T. KAPPELER, and P. TOPALOV
3
is a C r -map as well.
Remark 1.3. By considering lifts to Rn of diﬀeomorphisms of Tn = Rn /Zn , the
same arguments as in the proof of Theorem 1.1 can be used to show corresponding
results for the group Ds (Tn ) of H s -regular diﬀeomorphisms on Tn .
In Section 3 and Section 4 of this paper we discuss various classes of diﬀeomorphisms on a closed1 manifold M . For any integer s with s > n/2 the set H s (M )
of Sobolev maps is deﬁned by using coordinate charts of M . More precisely, let M
be a closed manifold of dimension n and N a C ∞ -manifold of dimension d. We say
that a continuous map f : M → N is an element in H s (M, N ) if for any x ∈ M
there exists a chart χ : U → U ⊆ Rn of M with x ∈ U, and a chart η : V → V ⊆ Rd
of N with f (x) ∈ V, such that f (U) ⊆ V and
η ◦ f ◦ χ−1 : U → V
is an element in the Sobolev space H s (U, Rd ). Here H s (U, Rd ) – similarly deﬁned
as H s (Rn , Rd ) – is the Hilbert space of elements in L2 (U, Rd ) whose distributional
derivatives up to order s are L2 -integrable. In Section 3 we introduce a C ∞ diﬀerentiable structure on the space H s (M, N ) in terms of a speciﬁc cover by open
sets which is especially well suited for proving regularity properties of the composition of mappings as well as other applications presented in subsequent work.
The main property of this cover of H s (M, N ) is that each of its open sets can be
embedded into a ﬁnite cartesian product of Sobolev spaces of H s -maps between
Euclidean spaces.
It turns out that this cover makes H s (M, N ) into a C ∞ -Hilbert manifold – see
Section 4 for details. In addition, we show in Section 4 that the C ∞ -diﬀerentiable
structure for H s (M, N ) deﬁned in this way coincides with the one, introduced by
Ebin and Marsden in [14], [15] and deﬁned in terms of a Riemannian metric on N .
In particular it follows that the standard diﬀerentiable structure does not depend
on the choice of the metric. Now assume in addition that M is oriented. Then,
for any linear isomorphism A : Tx M → Ty M between the tangent spaces of M at
arbitrary points x and y of M , the determinant det(A) has a well deﬁned sign. For
any integer s with s > n2 + 1 deﬁne
⏐
D s (M ) := ϕ ∈ Diﬀ1+ (M )⏐ϕ ∈ H s (M, M )
where Diﬀ1+ (M ) denotes the set of all orientation preserving C 1 smooth diﬀeomorphisms of M . We will show that Ds (M ) is open in H s (M, M ) and hence is a
C ∞ -Hilbert manifold. Elements in D s (M ) are referred to as orientation preserving
H s -diﬀeomorphisms.
In Section 3 we prove the following
Theorem 1.2. Let M be a closed oriented manifold of dimension n, N a C ∞ manifold, and s an integer satisfying s > n/2 + 1. Then for any r ∈ Z≥0 ,
(i)
μ : H s+r (M, N ) × Ds (M ) → H s (M, N ), (f, ϕ) → f ◦ ϕ
and
(ii)
inv : Ds+r (M ) → Ds (M ), ϕ → ϕ−1
are both C r -maps.
1 i.e.,
a compact C ∞ -manifold without boundary
4
1. INTRODUCTION
Remark 1.4. Various versions of Theorem 1.2 can be found in the literature,
however mostly without proofs – see e.g. [13], [14], [16], [34], [35], [36], [37]; cf.
also [30]. A complete, quite involved proof of statement (i) of Theorem 1.2 can be
found in [35], Proposition 3.3 of Chapter 3 and Theorem 2.1 of Chapter 6. Using
the approach sketched above we present an elementary proof of Theorem 1.2. In
particular, our approach allows us to apply elements of the proof of Theorem 1.1 to
show statement (i).
Remark 1.5. Actually Theorem 1.1 and Theorem 1.2 continue to hold if instead
of s being an integer it is an arbitrary real number s > n/2 + 1. In order to keep
the exposition as elementary as possible we prove Theorem 1.1 and Theorem 1.2
as stated in the main body of the paper and discuss the extension to the case where
s > n/2 + 1 is real in Appendix B.
We ﬁnish this introduction by pointing out results on compositions of maps in
function spaces diﬀerent from the ones considered here and some additional literature. In the paper [26], de la Llave and Obaya prove a version of Theorem 1.1 for
H¨
older continuous maps between open sets of Banach spaces. Using the paradifferential calculus of Bony, Taylor [39] studies the continuity of the composition of
maps of low regularity between open sets in Rn – see also [3].
Acknowledgment: We would like to thank Gerard Misiolek and Tudor Ratiu for
very valuable feedback on an earlier version of this paper.
CHAPTER 2
Groups of diﬀeomorphisms on Rn
In this section we present a detailed and elementary proof of Theorem 1.1. First
we prove that the composition map μ is a C r -map (Proposition 2.9) and then, using
this result, we show that the inverse map is a C r -map as well (Proposition 2.13).
To simplify notation we write D s ≡ Ds (Rn ) and H s ≡ H s (Rn ). Throughout this
section, s denotes a nonnegative integer if not stated otherwise.
2.1. Sobolev spaces H s (Rn , R). In this subsection we discuss properties of
the Sobolev spaces H s (Rn , R) needed later. First let us introduce some more notation. For any x, y ∈ Rn denote by x · y the Euclidean inner product, x · y =
n
1/2
. Recall that for
k=1 xk yk , and by |x| the corresponding norm , |x| = (x · x)
s
n
2
s ∈ Z≥0 , H (R , R) consists of all L -integrable functions f : Rn → R with the
property that the distributional derivatives ∂ α f, α ∈ Zn≥0 , up to order |α| ≤ s are
L2 -integrable as well. Then H s (Rn , R), endowed with the norm (1), is a Hilbert
space and for any multi-index α ∈ Zn≥0 with |α| ≤ s, the diﬀerential operator ∂ α is
a bounded linear map,
∂ α : H s (Rn , R) → H s−|α| (Rn , R).
Alternatively, one can characterize the spaces H s (Rn , R) via the Fourier transform.
For any f ∈ L2 (Rn , R) ≡ H 0 (Rn , R), denote by fˆ its Fourier transform
fˆ(ξ) := (2π)−n/2
f (x)e−ix·ξ dx.
Rn
Then fˆ ∈ L2 (Rn , R) and fˆ = f , where f ≡ f 0 denotes the L2 -norm of f .
The formula for the inverse Fourier transform reads
−n/2
fˆ(ξ)eix·ξ dξ.
f (x) = (2π)
Rn
When expressed in terms of the Fourier transform fˆ of f , the operator ∂ α , α ∈ Zn≥0
is the multiplication operator
fˆ → (iξ)α fˆ
where ξ α = ξ1α1 · · · ξnαn and one can show f ∈ L2 (Rn , R) is an element in H s (Rn , R)
α ˆ 2 1/2
,
iﬀ (1 + |ξ|)s fˆ is in L2 (Rn , R) and the H s -norm of f , f s =
|α|≤s ξ f
satisﬁes
(4)
Cs−1 f s ≤ f ∼
s ≤ Cs f s
for some constant Cs ≥ 1 where
∼
(5)
f s :=
Rn
1/2
(1 + |ξ| ) |fˆ(ξ)|2 dξ
2 s
5
.
2. GROUPS OF DIFFEOMORPHISMS ON Rn
6
In this way the Sobolev space H s (Rn , R) can be deﬁned for s ∈ R≥0 arbitrary. See
Appendix B for a study of these spaces.
Using the Fourier transform one gets the following approximation property for
functions in H s (Rn , R).
Lemma 2.1. For any s in Z≥0 , the subspace Cc∞ (Rn , R) of C ∞ functions with
compact support is dense in H s (Rn , R).
Remark 2.1. The proof shows that Lemma 2.1 actually holds for any s real
with s ≥ 0.
Proof. In a ﬁrst step we show that C ∞ (Rn , R) ∩ H s (Rn , R) is dense in
s
H (Rn , R) for any integer s ≥ s. Let χ : R → R be a decreasing C ∞ function
satisfying
χ(t) = 1 ∀t ≤ 1 and χ(t) = 0 ∀t ≥ 2.
For any f ∈ H s (Rn , R) and N ∈ Z≥1 deﬁne
ix·ξ
ˆ
fN (x) = (2π)−n/2
χ |ξ|
dξ.
N f (ξ)e
Rn
ˆ
The support of χ |ξ|
N f (ξ) is contained in the ball {|ξ| ≤ 2N }. Hence fN (x) is in
∞
n
s
n
C (R , R) ∩ H (R , R) for any s ≥ 0. In addition, by the Lebesgue convergence
theorem,
2
ˆ 2
lim
(1 + |ξ|)2s 1 − χ( |ξ|
N ) |f (ξ)| dξ = 0.
N →∞
Rn
In view of (5), we have fN → f in H s (Rn , R). In a second step we show that
Cc∞ (Rn , R) is dense in C ∞ (Rn , R) ∩ H s (Rn , R) for any integer s ≥ 0. We get
the desired approximation of an arbitrary function f ∈ C ∞ (Rn , R) ∩ H s (Rn , R) by
truncation in the x-space. For any N ∈ Z≥1 , let
f˜N (x) = χ |x|
N · f (x).
The support of f˜N is contained in the ball {|x| ≤ 2N } and thus f˜N ∈ Cc∞ (Rn , R).
s
n
To see that f − f˜N = (1 − χ |x|
N )f converges to 0 in H (R , R), note that f (x) −
f˜N (x) = 0 for any x ∈ Rn with |x| ≤ N . Furthermore it is easy to see that
≤ Ms
sup ∂ α 1 − χ |x|
N
x∈Rn
|α|≤s
for some constant Ms > 0 independent on N . Hence for any α ∈ Zn≥0 with |α| ≤ s ,
by Leibniz’ rule,
·
f
(x)
∂ α f − ∂ α f˜N = ∂ α 1 − χ |x|
N
≤
· ∂ γ f .
∂ β 1 − χ |x|
N
β+γ=α
|x|
= 0 for any |x| ≤ N we conclude that
1/2
|x| γ
β
γ 2
∂ 1 − χ N · ∂ f ≤ Ms
|∂ f | dx
Using that 1 − χ
N
|x|≥N
H. INCI, T. KAPPELER, and P. TOPALOV
7
and hence, as f ∈ H s (Rn , R),
lim ∂ α f − ∂ α f˜N = 0.
N →∞
To state regularity properties of elements in H s (Rn , R), introduce for any
r ∈ Z≥0 the space C r (Rn , R) of functions f : Rn → R with continuous partial
derivatives up to order r. Denote by f C r the C r -norm of f ,
f C r = sup sup |∂ α f (x)|.
x∈Rn |α|≤r
By Cbr (Rn , R) we denote the Banach space of functions f in C r (Rn , R) with f C r <
∞ and by C0r (Rn , R) the subspace of functions f in C r (Rn , R) vanishing at inﬁnity.
r
n
These are functions in C (R , R) with the property that for any ε > 0 there exists
M ≥ 1 so that
sup sup |∂ α f (x)| < ε.
|α|≤r |x|≥M
Then
C0r (Rn , R) ⊆ Cbr (Rn , R) ⊆ C r (Rn , R).
By the triangle inequality one sees that C0r (Rn , R) is a closed subspace of Cbr (Rn , R).
The following result is often referred to as Sobolev embedding theorem.
Proposition 2.2. For any r ∈ Z≥0 and any integer s with s > n/2, the
space H s+r (Rn , R) can be embedded into C0r (Rn , R). More precisely H s+r (Rn , R) ⊆
C0r (Rn , R) and there exists Ks,r ≥ 1 so that
f C r ≤ Ks,r f s+r
∀f ∈ H s+r (Rn , R).
Remark 2.2. The proof shows that Proposition 2.2 holds for any real s with
s > n/2.
Proof. As for s > n/2
Rn
(1 + |ξ|2 )−s dξ < ∞
one gets by the Cauchy-Schwarz inequality for any f ∈ Cc∞ (Rn , R) and α ∈ Zn≥0
with |α| ≤ r
sup |∂ α f (x)| ≤ (2π)−n/2
|fˆ(ξ)| |ξ|α dξ
n
x∈Rn
R
1/2
1/2
2 −s
−n/2
2
(1 + |ξ| ) dξ
(2π)
|fˆ(ξ)| (1 + |ξ|2 )s+r dξ
≤
Rn
Rn
(6)
≤ Kr,s f r+s
for some Kr,s > 0. By Lemma 2.1, an arbitrary element f ∈ H s+r (Rn , R) can be
approximated by a sequence (fN )N ≥1 in Cc∞ (Rn , R). As C0r (Rn , R) is a Banach
space, it then follows from (6) that (fN )N ≥1 is a Cauchy sequence in C0r (Rn , R)
which converges to some function f˜ in C0r (Rn , R). In particular, for any compact
subset K ⊆ Rn ,
fN |K → f˜ |K in L2 (K, R).
This shows that f˜ ≡ f a.e. and hence f ∈ C r (Rn , R).
0
As an application of Proposition 2.2 one gets the following
2. GROUPS OF DIFFEOMORPHISMS ON Rn
8
Corollary 2.1. Let s be an integer with s > n/2 + 1. Then the following
statements hold:
(i) For any ϕ ∈ D s , the linear operators dx ϕ, dx ϕ−1 : Rn → Rn are bounded
uniformly in x ∈ Rn∗ . In particular,
inf det dx ϕ > 0.
x∈Rn
(ii) D s − id = {ϕ − id | ϕ ∈ Ds } is an open subset of H s . Hence the map
Ds → H s ,
ϕ → ϕ − id
provides a global chart for D s , giving Ds the structure of a C ∞ -Hilbert
manifold modeled on H s .
(iii) For any ϕ• ∈ D s such that
inf det dx ϕ• > M > 0
x∈Rn
there exist an open neighborhood Uϕ• of ϕ• in Ds and C > 0 such that
for any ϕ in Uϕ• ,
inf det dx ϕ ≥ M and sup dx ϕ−1 < C.†
x∈Rn
x∈Rn
Remark 2.3. The proof shows that Corollary 2.1 holds for any real s with
s > n/2 + 1.
Proof. (i) Introduce
C 1 (Rn ) := ϕ ∈ Diﬀ1+ (Rn ) ϕ − id ∈ C01 (Rn )
where C01 (Rn ) ≡ C01 (Rn , Rn ) is the space of C 1 -maps f : Rn → Rn , vanishing
together with their partial derivatives ∂xi f (1 ≤ i ≤ n) at inﬁnity. By Proposition
2.2, H s continuously embeds into C01 (Rn ) for any integer s with s > n/2 + 1.
In particular, D s → C 1 (Rn ). We now prove that for any ϕ ∈ C 1 (Rn ), dϕ and
dϕ−1 are bounded on Rn . Clearly, for any ϕ ∈ C 1 (Rn ), dϕ is bounded on Rn .
To show that dϕ−1 is bounded as well introduce for any f ∈ C01 (Rn ) the function
F (f ) : Rn → R given by
F (f )(x) := det id + dx f − 1
= det (δi1 + ∂x1 fi )1≤i≤n , . . . , (δin + ∂xn fi )1≤i≤n − 1
where f (x) = f1 (x), . . . , fn (x) . As
lim ∂xk fi (x) = 0 for any
|x|→∞
1 ≤ i, k ≤ n
one has
lim F (f )(x) = 0.
(7)
|x|→∞
It is then straightforward to verify that F is a continuous map,
F : C01 (Rn ) → C00 (Rn , R).
Choose an arbitrary element ϕ in C 1 (Rn ). Then (7) implies that
M1 := infn det(dx ϕ) > 0.
(8)
x∈R
∗ Here
† For
dx ϕ−1 ≡ dx (ϕ−1 ) where ϕ ◦ ϕ−1 = id.
a linear operator A : Rn → Rn , denote by |A| its operator norm, |A| := sup|x|=1 |Ax|
H. INCI, T. KAPPELER, and P. TOPALOV
9
−1
As the diﬀerential of the inverse, dx ϕ−1 = dϕ−1 (x) ϕ
, can be computed in terms
of the cofactors of dϕ−1 (x) ϕ and 1/ det(dϕ−1 (x) ϕ) it follows from (8) that
M2 := sup |dx ϕ−1 | < ∞
(9)
x∈Rn
where |A| denotes the operator norm of a linear operator A : Rn → Rn .
(ii) Using again that D s continuously embeds into C 1 (Rn ) it remains to prove that
C 1 (Rn ) − id is an open subset of C01 (Rn ). Note that the map F introduced above
is continuous. Hence there exists a neighborhood Uϕ of fϕ := ϕ − id in C01 (Rn ) so
that for any f ∈ Uϕ
1
(10)
sup |dx f − dx fϕ | ≤
2M2
x∈Rn
and
M1
sup F f (x) − F fϕ (x) ≤
2
x∈Rn
(11)
with M1 , M2 given as in (8)-(9). We claim that id + f ∈ C 1 (Rn ) for any f ∈ Uϕ .
As ϕ ∈ C 1 (Rn ) was chosen arbitrarily it then would follow that C 1 (Rn ) − id is
open in C01 (Rn ). First note that by (11),
0 < M1 /2 ≤ det(id + dx f ) ∀x ∈ Rn , ∀f ∈ Uϕ .
Hence id + f is a local diﬀeomorphism on Rn and it remains to show that id + f
is 1-1 and onto for any f in Uϕ . Choose f ∈ Uϕ arbitrarily. To see that id + f is
1-1 it suﬃces to prove that ψ := (id + f ) ◦ ϕ−1 is 1-1. Note that
ψ = (id + fϕ + f − fϕ ) ◦ ϕ−1 = id + (f − fϕ ) ◦ ϕ−1 .
For any x, y ∈ Rn , one therefore has
ψ(x) − ψ(y) = x − y + (f − fϕ ) ◦ ϕ−1 (x) − (f − fϕ ) ◦ ϕ−1 (y).
By (9) and (10)
|(f − fϕ ) ◦ ϕ−1 (x) − (f − fϕ ) ◦ ϕ−1 (y)| ≤
≤
1
|ϕ−1 (x) − ϕ−1 (y)|
2M2
1
|x − y|
2
and thus
1
|(x − y) − ψ(x) − ψ(y) | ≤ |x − y| ∀x, y ∈ Rn
2
which implies that ψ is 1-1. To prove that id + f is onto we show that Rf :=
{x + f (x) | x ∈ Rn } is an open and closed subset of Rn . Being nonempty, one then
has Rf = Rn . As id + f is a local diﬀeomorphism on Rn , Rf is open. To see that
it is closed, consider a sequence (xk )k≥1 in Rn so that yk := xk + f (xk ), k ≥ 1,
converges. Denote the limit by y. As lim|x|→∞ f (x) = 0, the sequence f (xk ) k≥1
is bounded, hence xk = yk − f (xk ) is a bounded sequence and therefore admits a
convergent subsequence (xki )i≥1 whose limit is denoted by x. Then
y
=
lim xki + lim f (xki )
i→∞
i→∞
= x + f (x)
i.e. y ∈ Rf . This shows that Rf is closed and ﬁnishes the proof of item (ii). The
proof of (iii) is straightforward and we leave it to the reader.
2. GROUPS OF DIFFEOMORPHISMS ON Rn
10
The following properties of multiplication of functions in Sobolev spaces are
well known – see e.g. [2].
Lemma 2.3. Let s, s be integers with s > n/2 and 0 ≤ s ≤ s. Then there
exists K > 0 so that for any f ∈ H s (Rn , R), g ∈ H s (Rn , R), the product f · g is in
H s (Rn , R) and
f · g s ≤ K f s g s .
(12)
In particular, H s (Rn , R) is an algebra.
Remark 2.4. The proof shows that Lemma 2.3 remains true for any real s and
s with s > n/2 and 0 ≤ s ≤ s.
Proof. First we show that
(1 + |ξ|2 )s /2 f
· g(ξ) = (1 + |ξ|2 )s /2 (fˆ ∗ gˆ)(ξ) ∈ L2 (Rn , R)
where ∗ denotes the convolution
(fˆ ∗ gˆ)(ξ) =
By assumption,
f˜(ξ) := fˆ(ξ) (1 + |ξ|2 )s/2
Rn
fˆ(ξ − η)ˆ
g (η)dη.
g˜(ξ) = gˆ(ξ) (1 + |ξ|2 )s /2
are in L2 (Rn , R). Note that in view of deﬁnition (5), f˜ = f ∼
g = g ∼
s and ˜
s .
It is to show that
|f˜(ξ − η)|
|˜
g (η)|
2 s /2
ξ → (1 + |ξ| )
dη
2 )s/2 (1 + |η|2 )s /2
(1
+
|ξ
−
η|
n
R
and
is square-integrable. We split the domain of integration into two subsets {|η| >
|ξ|/2} and {|η| ≤ |ξ|/2}. Then
|f˜(ξ − η)|
|˜
g (η)|
dη
(1 + |ξ|2 )s /2
2 s/2 (1 + |η|2 )s /2
|η|>|ξ|/2 (1 + |ξ − η| )
|f˜(ξ − η)|
|˜
g (η)|
dη
≤ 2s (1 + |ξ|2 )s /2
2
s/2
(1 + |ξ|2 )s /2
|η|>|ξ|/2 (1 + |ξ − η| )
|f˜(ξ − η)|
s
|˜
g (η)|dη
≤ 2
2 s/2
Rn (1 + |ξ − η| )
g |(ξ).
= 2s |fˆ| ∗ |˜
By Young’s inequality (see e.g. Theorem 1.2.1 in [28]),
|fˆ| ∗ |˜
g
g | ≤ fˆ L1 ˜
and
fˆ L1 ≤
This implies that
1/2
(1 + |ξ| ) |fˆ(ξ)|2 dξ
2 s
Rn
Rn
2 −s
(1 + |ξ| )
1/2
dξ
.
|fˆ| ∗ |˜
g | ≤ C f s g ∼
s .
Similarly, one argues for the integral over the remaining subset. Note that on the
domain {|η| ≤ |ξ|/2} one has
1
(1 + |ξ − η|2 ) ≥ (1 + |η|2 ) and (1 + |ξ − η|2 ) ≥ (1 + |ξ|2 )
4
H. INCI, T. KAPPELER, and P. TOPALOV
and hence
11
(1 + |ξ − η|2 )s/2 ≥ (1 + |η|2 )(s−s )/2 2−s (1 + |ξ|2 )s /2
Hence
(1 + |ξ|2 )s /2
≤ 2s
|η|≤|ξ|/2
|η|≤|ξ|/2
|f˜(ξ − η)|
|f˜(ξ − η)|
|˜
g (η)|
dη
2
s/2
(1 + |ξ − η| ) (1 + |η|2 )s /2
|˜
g (η)|
dη
(1 + |η|2 )s/2
and the L2 -norm of the latter convolution is bounded by
f˜ ˜
g (η)/(1 + |η|2 )s/2 L1 ≤ C f s g ≤ C f s g s
with an appropriate constant C > 0.
The following results concern the chain rule of diﬀerentiation for functions in
H 1 (Rn , R).
Lemma 2.4. Let ϕ ∈ Diﬀ+1 (Rn ) with dϕ and dϕ−1 bounded on all of Rn . Then
the following statements hold:
(i) The right translation by ϕ, f → Rϕ (f ) := f ◦ ϕ is a bounded linear map
on L2 (Rn , R).
(ii) For any f ∈ H 1 (Rn , R), the composition f ◦ ϕ is again in H 1 (Rn , R) and
the diﬀerential d(f ◦ ϕ) is given by the map df ◦ ϕ · dϕ ∈ L2 (Rn , Rn ),
d(f ◦ ϕ) = (df ) ◦ ϕ · dϕ.
(13)
Proof. (i) For any f ∈ L2 (Rn , R), the composition f ◦ ϕ is measurable. As
−1
M1 := infn det(dx ϕ) = sup det dx ϕ−1
>0
x∈R
x∈Rn
one obtains by the transformation formula
1
f ϕ(x) 2 dx ≤
f ϕ(x) 2 det(dx ϕ)dx
M1 Rn
Rn
1
|f (x)|2 dx
=
M1 Rn
and thus f ◦ ϕ ∈ L2 (Rn , R) and the right translation Rϕ is a bounded linear map
on L2 (Rn , R).
(ii) For any f ∈ Cc∞ (Rn , R), f ◦ ϕ ∈ H 1 (Rn , R) and (13) holds by the standard
chain rule of diﬀerentiation. Furthermore for any f ∈ H 1 (Rn , R), df ∈ L2 (Rn , Rn )
and hence by (i), (df ) ◦ ϕ ∈ L2 (Rn , Rn ). As dϕ is continuous and bounded by
assumption it then follows that for any 1 ≤ i ≤ n
n
∂xk f ◦ ϕ · ∂xi ϕk ∈ L2 (Rn , R)
k=1
where ϕk (x) is the k’th component of ϕ(x), ϕ(x) = ϕ1 (x), . . . , ϕn (x) . By Lemma
2.1, f can be approximated by (fN )N ≥1 in Cc∞ (Rn , R). By the chain rule, for any
1 ≤ i ≤ n, one has
∂xi (fN ◦ ϕ) =
n
(∂xk fN ) ◦ ϕ · ∂xi ϕk
k=1
2. GROUPS OF DIFFEOMORPHISMS ON Rn
12
and in view of (i), in L2 ,
n
(14)
(∂xk fN ) ◦ ϕ · ∂xi ϕk −→
N →∞
k=1
n
(∂xk f ) ◦ ϕ · ∂xi ϕk .
k=1
Moreover, for any test function g ∈ Cc∞ (Rn , R),
n
−
∂xi g · fN ◦ ϕdx =
g · ∂xk fN ◦ ϕ · ∂xi ϕk dx.
Rn
k=1
Rn
By taking the limit N →
∞ and using (14), one sees that the distributional den
rivative ∂xi (f ◦ ϕ) equals k=1 (∂xk f ) ◦ ϕ · ∂xi ϕk for any 1 ≤ i ≤ n. Therefore,
1
n
f ◦ ϕ ∈ H (R , R) and d(f ◦ ϕ) = df ◦ ϕ · dϕ as claimed.
The next result concerns the product rule of diﬀerentiation in Sobolev spaces.
To state the result, introduce for any integer s with s > n/2 and ε > 0 the set
Uεs := g ∈ H s (Rn , R) infn 1 + g(x) > ε .
x∈R
By Proposition 2.2,
Uεs
is an open subset of H s (Rn , R) and so is
U s :=
Uεs .
ε>0
Note that U s is closed under multiplication. More precisely, if g ∈ Uεs and
s
h ∈ Uδs , then g + h + gh ∈ Uεδ
. Indeed, by Lemma 2.3, gh ∈ H s (Rn , R), and
hence so is g + h + gh. In addition, 1 + g + h + gh = (1 + g)(1 + h) satisﬁes
s
.
inf x∈Rn (1 + g)(1 + h) > εδ and thus g + h + gh is in Uεδ
Lemma 2.5. Let s, s be integers with s > n/2 and 0 ≤ s ≤ s. Then for any
ε > 0 and K > 0 there exists a constant C ≡ C(ε, K; s, s ) > 0 so that for any
f ∈ H s (Rn , R) and g ∈ Uεs with g s < K, one has f /(1 + g) ∈ H s (Rn , R) and
f /(1 + g) s ≤ C f s .
(15)
Moreover, the map
H s (Rn , R) × U s → H s (Rn , R),
(16)
(f, g) → f /(1 + g)
is continuous.
Remark 2.5. The proof shows that Lemma 2.5 continues to hold for any s real
with s > n/2. The case where in addition s is real is treated in Appendix B.
Proof. We prove the claimed statement by induction with respect to s . For
s = 0, one has for any f in L2 (Rn , R) and g ∈ Uεs
f 1
1 + g ≤ ε f .
Moreover, for any f1 , f2 ∈ L2 (Rn , R), g1 , g2 ∈ Uεs
f1
ε2 1+g
−
1
f2
1+g2
Hence by Proposition 2.2,
f1
−
ε2 1+g
1
f2
1+g2
≤ (f1 − f2 ) + f1 (g2 − g1 ) + (f1 − f2 )g1
≤ 1 + g1 C 0 f1 − f2 + f1 g2 − g1 C 0 .
≤ 1 + Ks,0 g1 s f1 − f2 + Ks,0 f1 g2 − g1 s
H. INCI, T. KAPPELER, and P. TOPALOV
13
and it follows that for any ε > 0
L2 (Rn , R) × Uεs → L2 (Rn , R),
(f, g) → f /(1 + g)
is continuous. As ε > 0 was taken arbitrarily, we see that the map (16) is continuous
as well. Thus the claimed statements are proved in the case s = 0.
Now, assuming that (15) and (16) hold for all 1 ≤ s ≤ k − 1, we will prove
that they hold also for s = k. Take f ∈ H s (Rn , R) and g ∈ Uεs . First, we will
prove that f /(1 + g) ∈ H s (Rn , R) and
f
∂x (f g) − g · ∂xi f
∂xi f
− i
∂xi
(1 ≤ i ≤ n).
=
1+g
1+g
(1 + g)2
Indeed, by Lemma 2.1, there exists (fN )N ≥1 , (gN )N ≥1 ⊆ Cc∞ (Rn , R) so that fN →
f in H s (Rn , R) and gN → g in H s (Rn , R). As Uεs is open in H s (Rn , R) we can
assume that (gN )N ≥1 ⊆ Uεs . By the product rule of diﬀerentiation, one has for any
N ≥ 1, 1 ≤ i ≤ n
fN
∂x (fN gN ) − gN · ∂xi fN
∂xi fN
(17)
∂xi
− i
.
=
1 + gN
1 + gN
(1 + gN )2
As ∂xi fN −→ ∂xi f in H s −1 (Rn , R) it follows by the induction hypothesis that
∂x i f
1+g
∈H
N →∞
s −1
n
(R , R) and
∂xi f
∂xi fN
−→
1 + gN N →∞ 1 + g
(18)
in
H s −1 (Rn , R).
2
(N ≥ 1) and 2g + g 2 are in H s (Rn , R) and
By Lemma 2.3, 2gN + gN
2
2gN + gN
−→ 2g + g 2
(19)
N →∞
As
inf
x∈Rn
2
1 + gN (x) > ε2
and
in
inf
H s (Rn , R).
x∈Rn
2
1 + g(x) > ε2
2
it follows that 2gN + gN
(N ≥ 1) and 2g + g 2 are elements in Uεs2 . By Lemma
2.3, fN · gN (N ≥ 1), f · g are in H s (Rn , R) and fN · gN −→ f · g in H s (Rn , R).
N →∞
Therefore
(20)
∂xi (fN · gN ) → ∂xi (f · g)
in H s −1 (Rn , R).
Similarly, as ∂xi fN −→ ∂xi f in H s −1 (Rn , R) it follows again by Lemma 2.3 that
N →∞
gN · ∂xi fN (N ≥ 1), g · ∂xi f are in H s −1 (Rn , R) and
(21)
gN · ∂xi fN −→ g · ∂xi f
N →∞
in H s −1 (Rn , R).
It follows from (19)-(21), and the induction hypothesis that
(22)
∂xi (f g) − g · ∂xi f
∂xi (fN gN ) − gN · ∂xi fn
−→
2
N →∞
(1 + gN )
(1 + g)2
in
H s −1 (Rn , R).

Author H. Inci, P. Topalov, and T. Kappeler Isbn 9780821887417 File size 608.98KB Year 2013 Pages 60 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare For $M$ a closed manifold or the Euclidean space $mathbb{R}^n$, the authors present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s > frac{1}{2}dim M 1$. Download (608.98KB) Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions Kuznetsovs Trace Formula and the Hecke Eigenvalues of Maass Forms Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three Lectures On Mathematical Logic Volume Iii The Logic Of Arithmetic Spaces of PL Manifolds and Categories of Simple Maps (AM-186 Load more posts