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  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:
has a solution with on and .
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.
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):
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:
- , 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:
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 .
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:
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 ).
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
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 .