Addendum to: Dacorogna-Moser theorem on the Jacobian determinant equation with control of support

TL;DR

This paper extends the Dacorogna-Moser theorem by demonstrating that solutions to the Jacobian determinant equation can have their support controlled based on initial data, while maintaining optimal regularity, thus generalizing previous results.

Contribution

It provides a generalization of the Dacorogna-Moser theorem, allowing support control of solutions to the Jacobian PDE with preserved regularity, addressing an open problem from the original work.

Findings

Support of solutions can be controlled from initial data.

The regularity of solutions remains optimal.

The result generalizes previous specific cases.

Abstract

In Dacorogna-Moser theorem on the pullback equation $\varphi^* (g)=f$ between two prescribed volume forms (with the same total volume), control of support of the solutions can be obtained from that of the initial data, while keeping optimal regularity. This result answers a problem implicitly raised on page 14 of Dacorogna-Moser's original article ("On a partial differential equation involving the Jacobian determinant", Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire 7 (1990), 1-26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\,\nabla\varphi=f$) given in the author's paper "Dacorogna-Moser theorem on the Jacobian determinant equation with control of support", Discrete Cont. Dyn. Syst. 37 (2017), 4071-4089.

Equations

$$\{\begin{array}{l}{\phi }^{\ast }(g)=f\\ \text{supp}(\phi -\text{id})\subset \Omega \end{array}$$

$$\{\begin{array}{l}{\int }_{\Omega }f={\int }_{\Omega }g\\ \text{supp}(f-g)\subset \Omega \end{array}$$

$$\{\begin{array}{l}{\phi }^{\ast }(g)=f\\ \text{supp}(\phi -\text{id})\subset \Omega .\end{array}$$

$${\int }_{{\Omega }^{\prime }}f={\int }_{\Omega }f-{\int }_{\Omega \setminus {\Omega }^{\prime }}f={\int }_{\Omega }g-{\int }_{\Omega \setminus {\Omega }^{\prime }}g={\int }_{{\Omega }^{\prime }}g$$

$$\{\begin{array}{ll}{\int }_{\Omega }f={\int }_{\Omega }g& \\ f=g& \text{in a neighbourhood of }U\end{array}$$

$$\{\begin{array}{ll}{\phi }^{\ast }(g)=f& \\ \phi =\text{id}& \text{in a neighbourhood of }U.\end{array}$$

$$\{\begin{array}{l}{\int }_{\Omega }f={\int }_{\Omega }g\\ d(\text{supp}(f-g),\partial \Omega )\ge d\end{array}$$

$$\left\{\begin{array}[]{llll}\varphi^{*}(g)=f\\ \varphi=\text{\emph{id}}&in\>V_{d}.\end{array}\right.$$

$$\{\begin{array}{ll}{\int }_{\Omega }g=\text{meas}\Omega & \\ g=\text{det}\nabla \Phi & \text{in a neighbourhood of }U.\end{array}$$

$$\{\begin{array}{ll}\text{det}\nabla \varphi =g& \\ \varphi =\Phi & \text{in a neighbourhood of }U.\end{array}$$

$$h=(g\circ {\Phi }^{-1})\text{det}\nabla {\Phi }^{-1}\in C^{r,\alpha }(\overline{\Omega })$$

$${\int }_{\Omega }h={\int }_{\Omega }g=\text{meas}\Omega$$

$${\int }_{{\Omega }^{\prime }}h=\text{meas}{\Omega }^{\prime }\text{,}\text{supp}(h-1)\subset {\Omega }^{\prime }\text{and}h>0.$$

$$\{\begin{array}{l}\text{det}\nabla \Theta =h{|}_{\overline{{\Omega }^{\prime }}}\\ \text{supp}(\Theta -\text{id})\subset {\Omega }^{\prime }\end{array}$$

$$\varphi =\Theta \circ \Phi$$

Full text

[TABLE]

This result answers a problem implicitly raised on page 14 of Dacorogna-Moser’s original article (“On a partial differential equation involving the Jacobian determinant”, *Ann. Inst. H. Poincaré Anal. Non Linéaire *7 (1990), 1–26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\,\nabla\varphi=f$) given in the author’s paper “Dacorogna-Moser theorem on the Jacobian determinant equation with control of support”, *Discrete Cont. Dyn. Syst. *37 (2017), 4071-4089.

Pedro Teixeira

Centro de Matemática da Universidade do Porto

Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

††2010 Mathematics Subject Classification. Primary 35F30.††Key words and phrases. Volume preserving diffeomorphism, volume correction, volume form, pullback equation, control of support, optimal regularity.

1. Introduction

This short note presents a simple solution to the problem of finding, in the Hölder setting, optimal regularity solutions to the pullback equation $\varphi^{*}(g)=f$ between two prescribed (nondegenerate) volume forms $f,\,g$ which coincide near $\partial\varOmega$ and with the same total volume over a bounded connected open set $\varOmega\subset\mathbb{R}^{n}$, the solution being a diffeomorphism of $\overline{\varOmega}$ onto itself that coincides with the identity near $\partial\varOmega$. This problem has been implicitly raised in Dacorogna and Moser [DM, p.14, iv] and again in [CDK, p.18-19]. In [TE] the solution was given for the particular case of $g\equiv 1$ all over $\overline{\varOmega}$ (prescribed Jacobian PDE, $\text{det}\,\nabla\varphi=f$), but at the time we could not devise the solution to the problem in its full generality. It turns out that the deduction of the general solution from the particular case mentioned above is in fact quite simple, the paper [TE] already containing all the tools needed for that purpose. Here we complete the solution to the problem in question.

2. **Optimal regularity pullback between prescribed volume forms

with control of support**

We recall two definitions introduced in [TE] for brevity of expression, and which actually do not coincide with the more standard definitions of domain and collar.

Definition 1**.**

(Domain). A bounded, connected open set $\varOmega\subset\mathbb{R}^{n}$ is called here a domain. Domains with $C^{r}$ boundary (briefly $C^{r}$ domains), $r\geq 1$ an integer, are defined in the usual way [CDK, p.338]. A domain is smooth if it is $C^{\infty}$.

Definition 2**.**

(Collar of $\overline{\varOmega}$). If $\varOmega\subset\mathbb{R}^{n}$ is a smooth domain, there is a smooth embedding $\zeta:\partial\varOmega\times[0,\infty)\hookrightarrow\overline{\varOmega}$ such that $\zeta(x,0)=x$ (collar embedding). For each $\epsilon>0$ we call $U_{\epsilon}:=\zeta(\partial\varOmega\times[0,\epsilon$]) a (compact)* collar of* $\overline{\varOmega}$.

**Convention. **As usual, volume forms on domains are identified with scalar functions, via the natural correspondence $f=f(x)dx^{1}\wedge\ldots\wedge dx^{n}$.

A simple formulation of the main result is the following (c.f. Theorem 2 below):

Theorem 1**.**

*(Dacorogna-Moser theorem - Case $\text{supp}(f-g)\subset\varOmega)$. *Let $\varOmega\subset\mathbb{R}^{n}$ be a bounded connected open set, $r\geq 0$ an integer and $0<\alpha<1$. Given $f,\,g\in C^{r,\alpha}(\overline{\varOmega})$ with $f\cdot g>0$ in $\overline{\varOmega}$ satisfying:

[TABLE]

there exists $\varphi\in\text{\emph{Diff}}{}^{r+1,\alpha}(\overline{\varOmega},\overline{\varOmega})$ satisfying:

[TABLE]

Proof.

Fix a smooth domain $\varOmega^{\prime}$ such that $\overline{\varOmega^{\prime}}\subset\varOmega$ and $\text{supp}(f-g)\subset\varOmega^{\prime}$ (use e.g. [TE, Lemma 1]). Then

[TABLE]

Fix a collar $U$ of $\overline{\varOmega^{\prime}}$ small enough so that $\text{supp}(f-g)$ is contained in the open set $\overline{\varOmega^{\prime}}\setminus U$. Clearly $f=g$ in a small neighbourhood of $U$ in $\overline{\varOmega^{\prime}}$. Apply Lemma 1 below to $f|_{\overline{\varOmega^{\prime}}}$, $g|_{\overline{\varOmega^{\prime}}}$, $U$ and $\overline{\varOmega^{\prime}}$, thus finding $\varphi\in\text{Diff}{}^{r+1,\alpha}(\overline{\varOmega},\overline{\varOmega})$ such that $\varphi^{*}(g|_{\overline{\varOmega^{\prime}}})=f|_{\overline{\varOmega^{\prime}}}$ and $\text{supp}(\varphi-\text{id})\subset\varOmega^{\prime}$. Now, $\varphi$ extends by id to the whole $\overline{\varOmega}$, in the $C^{r+1,\alpha}$ class, thus providing the desired diffeomorphism (as $\text{supp}(f-g)\subset\varOmega^{\prime}$). ∎

Remark 1*.*

As it is easily seen, in Theorem 1 the hypothesis $f,\,g\in C^{r,\alpha}(\overline{\varOmega})$ can be substantially weakened without compromising the conclusions. It is enough to assume that (a) the restrictions of $f,\,g$ to $\varOmega$ are $C^{r,\alpha}$ and (b) $\int_{\varOmega}f=\int_{\varOmega}g<\infty$. Actually, the hypothesis can be further weakened, for instance, the conclusions still hold if $f$ and $g$ are arbitrary functions on $\overline{\varOmega}$ such that (a’) the restriction of $f$ and $g$ to the closure of some smooth (sub)domain $\varOmega^{\prime}$ satisfying $\overline{\varOmega^{\prime}}\subset\varOmega$ and $\text{supp}(f-g)\subset\varOmega^{\prime}$ are in $C^{r,\alpha}$ and satisfy $f\cdot g>0$ (in $\overline{\varOmega^{\prime}}$), and (b’) $\int_{\varOmega^{\prime}}f=\int_{\varOmega^{\prime}}g$ (note that the hypothesis $\text{supp}(f-g)\subset\varOmega$ guarantees the existence of such $\varOmega^{\prime}$, see e.g. [TE, Lemma 1]). For then, the diffeomorphism of $\overline{\varOmega^{\prime}}$ onto itself solving $\varphi^{*}(g)=f$ in $\overline{\varOmega^{\prime}}$ (obtained through Lemma 1) extends by the identity to the whole $\overline{\varOmega}$ (in the $C^{r+1,\alpha}$ class) and $f=g$ in $\overline{\varOmega}\setminus\varOmega^{\prime}$ by hypothesis, thus everything agrees.

Lemma 1**.**

Let $\varOmega\subset\mathbb{R}^{n}$ be a bounded connected open smooth set and $U$ a collar of $\overline{\varOmega}$. Let $r\geq 0$ be an integer and $0<\alpha<1$. Given $f,\,g\in C^{r,\alpha}(\overline{\varOmega})$ with $f\cdot g>0$ in $\overline{\varOmega}$ satisfying:

[TABLE]

there exists $\varphi\in\text{\emph{Diff}}{}^{r+1,\alpha}(\overline{\varOmega},\overline{\varOmega})$ satisfying:

[TABLE]

Proof.

(Lemma 1). Replace $f,\,g$ respectively by the positive volume forms $|\lambda f|,\,|\lambda g|$ where $\lambda=\text{meas}\,\varOmega/\int_{\varOmega}\,f$, thus getting $\int_{\varOmega}\,f=\text{meas}\,\varOmega=\int_{\varOmega}\,g$ and keeping the coincidence of $f$ and $g$ in a neighbourhood of $U$. Apply in first place [DM, Theorem 1’] to get a solution $\varPhi\in\text{Diff}{}^{r+1,\alpha}(\overline{\varOmega},\overline{\varOmega})$ to $\text{det}\,\nabla\varPhi=f$ and then apply Lemma 2 below to the data $\varPhi$ and $g$ to get a diffeomorphism $\phi$ satisfying the conclusions of that lemma. The desired pullback between the two original volume forms is then given by the diffeomorphism $\varphi=\phi^{-1}\circ\varPhi$. ∎

The next result generalizes [TE, Theorem 7] to the case of the pullback equation between any two (nondegenerate) volume forms with the same total volume (see the introduction to Section 7 in [TE] for the motivation). The proof, relying on Lemma 1, directly follows the pattern of that of [TE, Theorem 7] and presents no difficulty. The statement employs the convention $d(\emptyset,\partial\varOmega):=\text{inradius}\,\varOmega$, introduced and explained in [TE, Section 7] (otherwise, for any two nonvoid subsets of $\mathbb{R}^{n}$, $d(\cdot,*)$ denotes the euclidean distance between them).

Theorem 2**.**

Let $\varOmega\subset\mathbb{R}^{n}$ be a bounded connected open set, $r\geq 0$ an integer and $0<\alpha<1$. For each $0<c\leq R:=\text{\emph{inradius}}\,\varOmega$ there exists a neighbourhood $V_{c}$ of $\partial\varOmega$ in $\overline{\varOmega}$ such that: given any $f,\,g\in C^{r,\alpha}(\overline{\varOmega})$ with $f\cdot g>0$ in $\overline{\varOmega}$ and any $0<d\leq R$ satisfying:

[TABLE]

there exists $\varphi\in\text{\emph{Diff}}{}^{r+1,\alpha}(\overline{\varOmega},\overline{\varOmega})$ satisfying:

[TABLE]

2.1. Concordant solutions to the Jacobian determinant equation for volume

forms agreeing in a collar

Lemma 2**.**

(Existence of concordant solutions).* Let $\varOmega\subset\mathbb{R}^{n}$ be a bounded connected open smooth set and $U$ a collar of $\overline{\varOmega}$. Let $r\geq 0$ be an integer and $0<\alpha<1$. Suppose that $\varPhi\in\text{\emph{Diff}}{}^{r+1,\alpha}(\overline{\varOmega},\overline{\varOmega})$ and $g\in C^{r,\alpha}(\overline{\varOmega})$, $g>0$ in $\overline{\varOmega}$, satisfy:*

[TABLE]

Then, there exists $\phi\in\text{\emph{Diff}}{}^{r+1,\alpha}(\overline{\varOmega},\overline{\varOmega})$ satisfying:

[TABLE]

Proof.

Suppose that $\varPhi$, $g$ and $U$ are as in the statement. The open set $\varOmega^{\prime}=\overline{\varOmega}\setminus\varPhi(U)$ is homeomorphic to $\varOmega$ and thus it is a domain containing $\varPhi(\text{supp}(g-\text{det}\,\nabla\varPhi))$. Let

[TABLE]

Then

[TABLE]

by the change of variables formula, and $h=1$ in a neighbourhood of $\varPhi(U)$, thus

[TABLE]

Apply [TE, Theorem 1] to the data $\varOmega^{\prime}$ and $h|_{\overline{\varOmega^{\prime}}}$, getting $\varTheta\in\text{Diff}{}^{r+1,\alpha}(\overline{\varOmega^{\prime}},\overline{\varOmega^{\prime}})$ satisfying

[TABLE]

Extend $\varTheta$ by id to the whole $\overline{\varOmega}.$ Then

[TABLE]

is the desired diffeomorphism, since

• •

$\phi=\text{id}\circ\varPhi=\varPhi$  in a neighbourhood of $U$

• •

$\text{det}\,\nabla\phi=g$  by definition of $h$.

∎