A few days ago I crossed by chance the book “The pullback equation for differential forms” of G. Csató, B. Dacorogna and O. Kneuss. Very roughly speaking, this book concerns the existence and regularity of solutions to the partial differential equation (PDE)
where and
are given
–forms and
is an unknown map.
As it turns out, this is a non-linear (for ) homogenous (of degree
in the derivatives of
) of first-order system of
PDEs on
.
The first time I got interested in this (pullback) PDE was some years ago because of its connection with the question of constructing “nice” perturbations of volume-preserving dynamical systems (see this paper here [and its corrigendum here]).
More concretely, suppose that you are studying the dynamical features of volume-preserving diffeomorphism and you want to test whether
has some given robust property
, e.g.,
is robustly transitive (i.e., all volume-preserving diffeomorphisms
obtained from small perturbations of
have some dense orbit). Then, you can use the pullback equation to contradict this robust property for
along the following lines.
Assume that you found some region of the phase space where
is “strangely” close to a local volume-preserving dynamical system
violating the property
(e.g.,
is not transitive because it leaves invariant some open subset
). Hence, you could show that
doesn’t satisfy
in a robust way if you can “glue”
and
in a conservative way to obtain a volume-preserving diffeomorphism
behaving as
inside
and behaving as
outside some neighborhood of
. Since the local dynamics
violates
, we have that the perturbation
of
also violates
, so that the robust property
is not verified by
.
Here, the pullback equation for volume forms says that you can glue and
in a volume-preserving way once you glued them in some non-volume-preserving way: indeed, let
be a possibly non-volume preserving diffeomorphism gluing
and
; since the pullback equation for volume forms
and
translates into the Jacobian determinant PDE
where
is a density function (corresponding to
), by letting
be a solution of the Jacobian determinant equation with
, we can “correct” the “defect” of
to preserve volume by replacing
by the volume-preserving map
.
Historically speaking, the Jacobian determinant equation (i.e., the pullback equation for volume forms) was discussed in details in this paper of B. Dacorogna and J. Moser (from 1990), and some of the results in this paper were recently used to perform “nice” perturbations/regularizations of volume-preserving dynamical systems (see, e.g., this paper here of A. Avila and the references therein).
An interesting technical point about the paper of B. Dacorogna and J. Moser is that they provide two results about the Jacobian determinant equation (stated below only for domains in for sake of simplicity of the exposition):
- (a) given a positive density
(where
and
) on the closure
of an open bounded smooth connected domain
with total mass
, there exists a
diffeomorphism
such that
on
and
on
;
- (b) given a positive density
(where
,
), on the closure
of an open bounded smooth connected domain
with total mass
, there exists a
diffeomorphism
such that
and
; furthermore, if
(i.e., the density
equals
near the boundary
), then
can be chosen so that
(i.e.,
is the identity diffeomorphism near
).
In a nutshell, the result in item (a) is proven in (Section 2 of) Dacorogna-Moser’s paper via global methods based on elliptic regularity of the Laplacian operator, while the result in item (b) is proven in (Section 3 of) Dacorogna-Moser’s paper via local methods based on reduction to ordinary differential equations (ODE’s).
As the attentive reader noticed, there is a tradeoff in the results in items (a) and (b): together, these results say that, if you want to prescribe a density while controlling the behavior of the diffeomorphism near the boundary , then you can do it via item (b) at the cost that you will get no gain of regularity in the sense that the density and the diffeomorphism have the same regularity
; on the other hand, if you want a gain of regularity, then you can do it via item (a) at the cost of giving up on the control of the diffeomorphism near
(the best you can ensure is that the boundary is pointwise fixed but you don’t know that the diffeomorphism is the identity near
).
Remark 1 In some sense, the loss of control on the behavior of the diffeomorphism in item (a) is essentially due to the fact that Dacorogna and Moser construct their diffeomorphisms
using the solutions of the elliptic PDE
(with Neumann boundary condition say) because
and
is the linearized equation associated to the Jacobian determinant PDE. Indeed, since the solutions of
are obtained by convolution of
and the Poisson kernel, the fact that
near
doesn’t imply that
near
(in other words, the convolution is a global operation and thus the local behavior of
is not sufficient to control the local behavior of
). In particular, given that the behavior of the diffeomorphisms
of Dacorogna-Moser near
are driven by the behavior of
(or rather
) near
, we see that the gain regularity in item (a) above comes with the cost of loss of control of
.
For the applications of Dacorogna-Moser’s theorems in Dynamical Systems so far (e.g., Avila’s theorem on the regularization of volume-preserving diffeomorphisms), the control of is more relevant than the gain of regularity and this explains why item (b) is more often used in dynamical contexts than item (a).
Given this scenario, it is natural to ask whether one can have the best from both worlds (or items), i.e., gain of regularity and control of support of diffeomorphisms with prescribed Jacobian determinant.
Of course, as it was pointed out by B. Dacorogna and J. Moser themselves in their 1990 paper (see item (iv) at page 14 of their article), one needs a new ingredient to attack this question. So, after stumbling upon the book of Csató, Dacorogna and Kneuss from 2012, I thought that there might be news on this question since the last time I have looked at it.
However, I saw some “bad” news by page 18-19 of Csató-Dacorogna-Kneuss book, where they say: “In Section 10.5 (cf. Theorem 10.11), we present a different approach proposed by Dacorogna and Moser [33] to solve our problem. This method is constructive and does not use the regularity of elliptic differential operators; in this sense, it is more elementary. The drawback is that it does not provide any gain of regularity, which is the strong point of the above theorem. However, the advantage is that it is much more flexible. For example, if we assume in (1.2) [a variant of Jacobian determinant equation] that
then we will be able to find such that
This type of result, unreachable by the method of elliptic partial differential equations, will turn out to be crucial in Chapter 11.”
Nevertheless, the book of Csató-Dacorogna-Kneuss brought also some good news: there were some progress on the question of gain of regularity in Poincaré lemma, that is, the question of writing a given closed cohomologically trivial form as an exact form (i.e., the differential of another form). After reading about this advance on regularity in Poincaré lemma, I noticed that this is precisely what one needs to modify Dacorogna-Moser’s arguments in order to get gain of regularity and control of the support thanks to a “correction of support argument” that I first read in this paper of Avila here.
In summary, it is possible to gain regularity and control the support in the Jacobian determinant equation by combining the arguments of Dacorogna-Moser and the Poincaré lemma in Csató-Dacorogna-Kneuss book. Evidently, this result per se is not publication-quality (it is just a combination of important results by other authors), but I thought that it could be a nice idea to leave some trace of this fact in this blog in case someone needs this information in the future.
So, the main goal of today’s post is the existence of solutions of the Jacobian determinant equation whose support are controlled and with a gain of regularity.
For sake of simplicity of the exposition, we will simplify our setting by considering the following particular situation (appearing naturally in some dynamical applications of the Jacobian determinant equation): where
denotes the open ball of radius
centered at the origin
and
is a positive (density) function of class
for some
and
such that
on the compact neighborhood
of
. In this context, we will show that:
Proposition 1 Given
, there exists a constant
such that, if
, the Jacobian determinant equation
has a solution
with
on
and
.
Remark 2 As we will see, even though Proposition 1 implies item (a), a theorem of Dacorogna-Moser, the proof of Proposition 1 uses the results of Dacorogna and Moser.
As we already mentioned, the proof of this proposition is a modification of the arguments of Dacorogna-Moser using the Poincaré lemma of Csató-Dacorogna-Kneuss. For this reason, we will divide this post into two sections: the next one serves to recall some preliminary results (mostly consisting of modifications of some results in Section 2 of Dacorogna-Moser’s paper) and in the short final section we will complete the proof of Proposition 1.
1. Preliminaries
Following Dacorogna-Moser paper, let us consider and let us study first the linearized equation associated the Jacobian determinant equation:
In this direction, Dacorogna-Moser note that so that, if we impose
and Neumann boundary condition
(where
is the outward unit normal to
), then, by elliptic regularity (and Schauder’s estimates), the Laplace equation
has a unique solution
such that
, and, a fortiori, we get a solution of the linearized equation
with
. After this, they modify
by adding some convenient term with zero divergence to “adjust” the boundary values of
to obtain a solution
of the linearized equation such that
for all
(cf. pages 7 and 8 of Dacorogna-Moser’s paper). Furthermore, they take extra care in the choice of this extra term so that it depends linearly on
. In this way, they prove the following theorem (cf. Theorem 2 of Dacorogna-Moser’s paper):
Theorem 2 (Dacorogna-Moser) There exists a bounded linear operator
associating to each
with
a solution
of the linearized equation
such that
for each
.
For our later purposes, we will need a version of this theorem where there is a control of the support of in addition to the gain of regularity. For this sake, we will need the following version of Poincaré’s lemma with gain of regularity (cf. Theorems 1.23 and 8.4 in Csató-Dacorogna-Kneuss book):
Theorem 3 (Csató-Dacorogna-Kneuss) Let
be a
-form on
of class
such that
(i.e.,
is closed) and
for all “harmonic
-forms”
, i.e., all
with
(vanishing exterior derivative),
(vanishing interior derivative) and
(vanishing normal component [with
denoting the interior product]), where
denotes the inner product between
-forms. Then, there exists a
–
of class
such that
and
.
Remark 3 In fact, the proof of this result in Csató-Dacorogna-Kneuss book relies on the methods of Dacorogna-Moser and it is possible to check that the form
constructed above depends linearly on
, i.e.,
where
is a linear operator. We will use this fact in a few moments.
At this point, we are ready to prove a version of Theorem 2 where we get a gain of regularity and control of the support of the solution of the linearized equation of the Jacobian determinant PDE:
Proposition 4 There exists a bounded linear operator
associating to each
with
and
, i.e.,
on
a solution
of the linearized equation
such that
(i.e.,
for each
).
Proof: We will use an argument inspired of the proof of Theorem 3 in Avila’s paper.
By Dacorogna-Moser’s theorem 2, we have a solution of the equation
with
and
for all
.
Consider the restriction of to
. By assumption,
vanishes on
, so that
on
.
Now, we recall that there is a duality between vector fields and -forms given by
. Under this duality, the condition
on
becomes
on
, that is,
is a closed
-form on the compact neighborhood
of
. Since
for each
, we see that
for each
. By the classical Poincaré lemma, since
retracts to
, we get that
is exact on
, i.e.,
for some
-form
on
.
By plugging this information in the integration by parts formula (for instance, see Theorem 1.11 in Csató-Dacorogna-Kneuss book)
we deduce that the -form
on
satisfies all the assumptions of Theorem 3 (global Poincaré lemma with gain of regularity of Csató-Dacorogna-Kneuss).
Therefore, we can write on
where
is the
-form on
depending linearly on
mentioned in Remark 3.
Now, we extend to a
-form
on
using some (bounded, linear) extension operator
(such as Whitney’s extension operator) and we consider the exact
-form
of class
on
. By definition,
coincides with
on
, and thus, by duality, we obtain from
a vector field
on
coinciding with
on
. Furthermore, since
is exact, we have that
on
and hence
on
.
In particular, it follows that is a
vector field obtained from
by a bounded linear operator
such that
and on the compact neighborhood
of
, as desired.
Once we know how to solve the linearized equation of the Jacobian determinant PDE with gain of regularity and control of the support of the solution, we can follow Dacorogna-Moser to start the discussion of the solutions of the Jacobian determinant PDE itself. In this direction, we need the following consequence of the proof of Lemma 3 in Dacorogna-Moser’s paper:
Lemma 5 Let
be a positive
density on
with total mass
and
on the compact neighborhood
of
. Then, there exists
a
diffeomorphism of
such that
,
on
and
.
Proof: Let and consider
be the solution of the linearized equation provided by Proposition 4. A short computation (detailed in pages 9 and 10 of Dacorogna-Moser’s paper) reveals that the diffeomorphism
obtained as the time-one map of the solution
of the ODE
with initial data satisfies the conclusions of the lemma.
Remark 4 The idea in the previous lemma of obtaining the diffeomorphism
via a deformation of the identity using an adequate ODE (related to the linearized equation) goes back to the work of J. Moser and it is known in the literature as “Moser’s trick”.
For our purposes, the “drawback” of Lemma 5 is the absence of gain of regularity on (in the sense that
belongs to the same class of differentiability of
). In order to circumvent this technical difficulty, we will need the following variant of Lemma 4 of Dacorogna-Moser’s paper ensuring that one has a gain of regularity and control of the support of the solution of the Jacobian determinant equation under a smallness assumption on
.
Lemma 6 Fix
. Then, there are constants
and
such that, if
and
, then there exists a
-diffeomorphism
of
with
and
.
Proof: Following Dacorogna-Moser’s paper, we will find as the fixed point of some contraction in a Banach space. More precisely, let us consider the Banach spaces:
and
where is our preferred compact neighborhood of
.
By Proposition 4, we have a bounded linear operator such that
for each .
Now, given a
-matrix, let us put
. By letting
, we see that the Jacobian determinant equation
(with on
).
In other words, if we set , then the desired lemma follows once we can find a fixed point
Before searching for a fixed point, let us observe that (2) is well-defined in the sense that . In fact, given
, the identity
follows from the fact that
(as on
and
; cf. pages 11 and 12 in Dacorogna-Moser’s paper), and the fact that
equals to
on
follows from the facts that
on
and
on
(as
).
At this stage, we will solve (2) by showing that is a contraction in the ball
for small enough. Here, we select
a constant controlling the norm of the bounded linear operation
, i.e.,
Next, using that is a sum of monomials of degree
on the entries of
, we can also select a constant
such that, if
, then
In this setting, a short calculation (detailed in page 12 of Dacorogna-Moser’s paper) reveals that if
then, by putting , one has
and
for all . This shows that
is a contraction on
and thus it admits a (unique) fixed point in
.
2. Proof of Proposition 1
The proof of Proposition 1 follows the lines of Dacorogna-Moser’s proof of Theorem 1′ in their paper. More concretely, given a positive density function with total mass
and
on
, let us fix
and let us consider the constant
provided by Lemma 6. By the density of
function among
functions in the
-norm, we can select a positive density function
such that
and on
. By Lemma 6, we have a
diffeomorphism
of
with
for all ,
on
, and
.
Now, let us observe that is a positive density of class
and total mass
whose -norm is bounded by a constant multiple of the
-norm of
and
on
(as
on
and
on
). By Lemma 5, we have a
-diffeomorphism
of
such that
on ,
on
and
.
At this point, the proof of Proposition 1 is complete: is a
diffeomorphism of
such that
on ,
on
and
.
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