Regularity of Maps between Sobolev Spaces

Regularity of Maps between Sobolev Spaces

Martins Bruveris Martins Bruveris, Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, United Kingdom martins.bruveris@brunel.ac.uk
July 15, 2019
Abstract.

Let be a -map between Sobolev spaces, either on or on a compact manifold. We show that equivariance of under the diffeomorphism group allows to trade regularity of as a nonlinear map for regularity in the image space: for , the map is well-defined and of class . This result is used to study the regularity of the geodesic boundary value problem for Sobolev metrics on the diffeomorphism group and the space of curves.

Key words and phrases:
Diffeomorphism group, Sobolev spaces on manifolds
2010 Mathematics Subject Classification:
Primary 58D15, Secondary 58D05, 58B20
The author was partially supported by the Erwin Schrödinger Institute programme Infinite-Dimensional Riemannian Geometry with Applications to Image Matching and Shape Analysis.
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1. Introduction

The main result of this paper is inspired from and a generalisation of results on the smoothness of geodesics for right-invariant Riemannian metrics on the diffeomorphism group. Riemannian metrics on the diffeomorphism group have been studied since it was recognised in [Arnold1966] that solutions of Euler’s equations for incompressible fluids correspond to geodesics on the group of volume-preserving diffeomorphisms with respect to a right-invariant Riemannian metric. The well-posedness of Euler’s equation was established in [Ebin1970] by showing that the corresponding geodesic spray is a smooth vector field on the group of volume-preserving -diffeomorphisms for and a compact manifold. A smooth geodesic spray on a Hilbert manifold gives rise to a smooth exponential map and because the metric is right-invariant, this exponential maps is -equivariant.

The right-invariance of the exponential map was used in [Ebin1970] to show the following result: if the initial conditions of a geodesic are of class , then so is the whole geodesic. This property implies that smooth () initial conditions for Euler’s equations have smooth solutions. The same property was observed for various other right-invariant Riemannian metrics on the diffeomorphism group as well as for reparametrisation invariant Riemannian metrics on the space of curves.

In this paper we want to prove a general version of this result, both for Sobolev spaces on Euclidean space and for Sobolev spaces on manifolds.

Theorem.

Let , and be a -equivariant -map, i.e. . Then maps into and is a -map.

Here and in the following we assume and . The group is the group of -diffeomorphisms; see Sect. 3.

Previously the strongest statement was that if is a -equivariant -map, then is well-defined. No statement was made about the continuity or differentiability of the resulting map. Next we state the corresponding result for Sobolev spaces on manifolds. Let be a compact manifold and smooth manifolds, all without boundary.

Theorem.

Let , and be a -equivariant -map, i.e. . Then maps into and is a -map.

In a nutshell this results states that given an equivariant map we can trade smoothness of the map to gain spatial smoothness of the image . If is a -map, then it also induces a -map between the spaces .

Corollary.

Let and be a -equivariant -map. Then is a -map.

The same can be also formulated for maps defined on compact manifolds.

Corollary.

Let and be a -equivariant -map. Then is a -map.

In Sect. 5 we apply this theorem to study the regularity of the geodesic boundary value problem for right-invariant Riemannian metrics on the diffeomorphism group. We show that if and are nonconjugate along the geodesic , then the whole geodesic is as smooth as and . In Sect. 6 we show the same result for reparametrization invariant Sobolev metrics on the space of curves.

Note

We will write , if the inequality holds for some constant , that may depend on the parameters and the manifolds involved, but is independent of the functions . The constant may also depend on the auxiliary functions and introduced in the proofs and additional dependencies will be stated in the text.

2. Differentiability in Banach Spaces

For Banach spaces we denote by the space of bounded, symmetric, -linear mappings . Let be open. A function is , if it is Fréchet differentiable and the derivative is continuous.

The following lemma is standard and is stated without proof.

Lemma 2.1.

Let be Banach spaces and a convex, open set. Let . Assume that is a mapping, such that

holds for all . Then and .

The next lemma shows, that if a function is differentiable on a dense subspace of a Banach space and the derivatives can be extended to continuous maps on the bigger space, then the function is differentiable on the bigger space.

Lemma 2.2.

Let be Banach spaces, a dense subspace, with and open. If we can extend and its derivatives to for , then .

Proof.

We have for , and ,

(1)

Since can be extended to and both sides in (1) are continuous on , the identity continues to hold for .

Now we argue inductively: Since and can be extended to , we obtain by Lem. 2.1 that . Now we apply Lem. 2.1 with in place of to conclude that and so . In this way we obtain inductively that . ∎

3. -equivariant Maps

The Sobolev spaces with can be defined in terms of the Fourier transform

and consist of -integrable functions with the property that is -integrable as well. An inner product on is given by

Denote by the group of -diffeomorphisms of , i.e.,

For and there are three equivalent ways of defining the group of Sobolev diffeomorphisms:

If we denote the three sets on the right by , and , then it is not difficult to see the inclusions . The equivalence has first been shown in [Ebin1970b, Sect. 3] for the diffeomorphism group of a compact manifold; a proof for can be found in [Inci2013]. Regarding the inclusion , it is shown in [Palais1959, Cor. 4.3] that if with and , then is a -diffeomorphism.

It follows from the Sobolev embedding theorem, that is an open subset of and thus a Hilbert manifold. Since each has to decay to the identity as , it follows that is orientation preserving. More importantly, is a topological group, but not a Lie group, since left-multiplication and inversion are continuous, but not smooth operations.

We will use the following two results regarding the multiplication and composition of functions in Sobolev spaces.

Lemma 3.1 ([Inci2013, Lem. 2.3]).

Let with and . Then pointwise multiplication

is a bounded bilinear map.

Lemma 3.2 ([Inci2013, Thm. 1.1 and Rem. 1.5]).

Let and . Then

is a -map.

Now we are ready to state and prove the main theorem for .

Theorem 3.3.

Let , and be a -equivariant -map, i.e. . Then maps into and is a -map.

Proof.

The proof is split into several steps.

Step 1. If is , then is .
Let be a uniformly localy finite cover of by open balls, a subordinate smooth partition of unity with uniformly (in ) bounded derivatives, and a basis of . Then an equivalent norm for is given by

(2)

see [Triebel1992, Sect. 7.2.2].

Let be the one-parameter subgroup generated by , i.e., satisfies the ODE . The existence of – nontrivial since is not a Lie group – is shown for example in [Bruveris2014_preprint, Thm. 4.4]. Then by Lem. 3.2 the map

is . Now fix and consider the identity . Both sides of the identity are as maps and by differentiating at we obtain the identity

We can estimate the -norm of the right hand side via the left hand side,

and hence we see using the equivalent -norm (2) that

(3)

This shows that , provided . Regarding continuity we can show in the same manner the estimate

(4)

for , from which the continuity of follows.

Step 2. If is a -map, then is and is .
We will show this together with the explicit estimates

(5)
(6)

which are valid for in a bounded -ball, using induction on .

For this is step 1. Now assume the statement has been shown for and let be . Then and thus also . Since , we obtain by induction that and . We now take lying in a bounded -ball and apply (3) to obtain

From this we obtain

which completes the induction for (6).

The induction assumption (5) applied to the -map shows that for , lying in a bounded -ball,

here we have used the module property of Sobolev spaces. Therefore with

(7)

Since also , by induction . Now we can apply Lem. 2.2 to obtain that and hence by step 1, we have together with the estimate,

Now we combine the induction assumption (5) and the estimates (7) and (6) to obtain

This concludes the induction.

Step 3. If is , then is .
The case is trivial and the case was proven in step 2. Now let . We consider as a map . Then the maps

Thus by step 2 we have

and we have the additional inequality for and in a bounded -ball,

which shows . Thus by Lem. 2.2 we obtain . This concludes the proof. ∎

Remark 3.4.

All the arguments in the proof of the theorem are local in nature, i.e. the one-parameter subgroups are only considered around , the estimates for and are only required to hold for in bounded balls and the statement about differentiability itself is local. Hence the theorem can also be proven for functions defined on open subsets of Sobolev spaces.

Corollary 3.5.

Let , , an open subset and a -equivariant -function, i.e. for , whenever . Then maps into and is a -map.

4. -equivariant Maps

In this section we assume that is a -dimensional compact manifold and are - and -dimensional manifolds respectively, both without boundary.

To define the spaces we require . A continuous map belongs to , if around each point , there exists a chart of and a chart of around , such that . When , then is the Sobolev space of functions on a manifold and the condition is not necessary; see [Aubin1998]. In general is not a vector space, but can be given the structure of a -smooth Hilbert manifold; this was done first in [Eells1966, Palais1968] and a different but compatible differentiable structure is described in [Inci2013].

For the diffeomorphism group can be defined by

and denotes the group of -diffeomorphisms of . The diffeomorphism group is an open subset of and it is a topological group.

We will need the following result on the boundedness of pointwise multiplications and the regularity of the composition map.

Lemma 4.1 ([Inci2013, Lem. 2.13 and Sect. 3]).

Let with and . Then pointwise multiplication

is a bounded bilinear map.

Lemma 4.2 ([Inci2013, Thm. 1.2 and Rem. 1.5]).

Let and . Then

is a -map.

Now we can state the analogue of Thm. 3.3 for Sobolev spaces on manifolds and -equivariant maps.

Theorem 4.3.

Let , and be a -equivariant -map, i.e. . Then maps into and is a -map.

Proof.

Step 1. Reduction to and .
Using Whitney’s embedding theorem we can embed and into Euclidean space. Let be a tubular neighborhood of in and denote by and the inclusion and retraction maps. Similarly we introduce the tubular neighborhood of and the maps and . Then we extend to the map via the following commutative diagram

The extension is again -equivariant, since

We note that and are open subsets of and respectively. If the theorem is proven in the case, when and are the Euclidean space, then together with Rem. 4.4 this shows that is . Now we write

Since composition from the left with -functions are -maps on Sobolev spaces, it follows that is .

For the rest of the proof we will assume that and .

Step 2. If is , then is .
Choose smooth vector fields such that

for all . Then an equivalent norm for is given by

Let , where denotes the Lie group exponential on . Then is a one-parameter subgroup and the map

is . Now fix and consider the identity . Both sides of the identity are as maps and by differentiating at we obtain the identity

We can estimate the -norm of the right hand side via the left hand side,

and hence we see using the -norm from above that

This shows that , provided . Regarding continuity we can show in the same manner the estimate

for , from which the continuity of follows.

The rest of the proof follows in the same way as steps 2 and 3 of the proof of Thm. 3.3. ∎

Remark 4.4.

As in the previous section, all the arguments in the proof of the theorem are local in nature, i.e. the one-parameter subgroups are only considered around and the statement about differentiability itself is local. Hence the theorem continues to hold for functions defined on open subsets of Sobolev spaces and because the local version is used implicitely in the proof we state it below.

Corollary 4.5.

Let , , an open subset and a -equivariant -map, i.e. , whenever . Then maps into and is a -map.

5. Geodesic Boundary Value Problem on the Diffeomorphism Group

In this section we assume that is or a compact manifold without boundary of dimension and with .

The group introduced in the previous sections is a smooth Hilbert manifold and a topological group. Let be a smooth right-invariant metric on , i.e.

for some fixed inner product on and . Note that does not necessarily have to induce the Sobolev topology on . If the geodesic spray of the metric is a smooth vector field on , then it admits an exponential map

defined on an open neighborhood of the zero section. Because the metric is right-invariant, the geodesic spray and the exponential map are -equivariant, i.e.

holds for .

Because is not a Lie group – in particular the inverse map is continuous but not differentiable – not every inner product leads to a smooth right-invariant metric . Similarly, because the topology induced by the metric can be weaker than the manifold topology, not every smooth metric admits an exponential map. Hence the assumption, that is a smooth metric with a smooth exponential map, are nontrivial.

5.1. Metrics with smooth sprays

At this point we should give examples of metrics to which this discussion applies.

Let