$W^{s,\frac{n}{s}}$-maps with positive distributional Jacobians

TL;DR

This paper extends the continuity and degree estimate results for maps with positive Jacobians from classical Sobolev spaces to fractional Sobolev spaces, broadening the understanding of their regularity and topological properties.

Contribution

It proves that maps in fractional Sobolev spaces with positive distributional Jacobians are continuous, generalizing classical results to fractional settings.

Findings

Maps in $W^{s,rac{n}{s}}$ with positive Jacobian are continuous for $s \\geq \frac{n}{n+1}$.

Degree estimates for sense-preserving maps extend to fractional Sobolev spaces.

Positive Jacobian implies regularity properties in fractional Sobolev spaces.

Abstract

We extend the well-known result that any $f \in W^{1,n}(\Omega,\mathbb{R}^n)$, $\Omega \subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(\Omega)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also obtain extensions to fractional Sobolev spaces $W^{s,\frac{n}{s}}$ of the degree estimates known for $W^{1,n}$-maps with positive or non-negative Jacobian, such as the sense-preserving property.

Equations

$$\mathrm{Jac}(f):=\det (Df)>0\text{a.e. in Ω},$$

$$f(x):=(\tilde{f}(x),0,\dots ,0).$$

$$[f{]}_{W^{s,p}(\Omega )}:={\left({\int }_{\Omega }{\int }_{\Omega }\frac{|f(x)-f(y){|}^p}{|x-y{|}^{n+sp}}dxdy\right)}^{\frac{1}{p}}<\infty .$$

$$\mathrm{Jac}(f)[\phi ]:=\underset{k\to \infty }{\lim }{\int }_{\Omega }\det (D{f}_k){\phi }_k$$

$$\mathrm{Jac}(f)[\phi ]\ge 0.$$

$$\mathrm{Jac}(f)[\phi ]=0\text{implies that φ≡0}.$$

$${\rm curl\,}(f)[\varphi]=-\int_{\Omega}f_{2}\,\partial_{1}\varphi-f_{1}\,\partial_{2}\varphi=0\quad\mbox{for all $\varphi\in C_{c}^{\infty}(\Omega)$},$$

$$\mathrm{deg}(f,B(x,r),p)\le \mathrm{deg}(f,B(x,R),p).$$

$$\mathrm{deg}(f,B(x,R),p)\ge 0.$$

$$R\le \Lambda \mathrm{diam}(f(\partial B(r))),$$

$$\{x\in B(r):f(x)\notin B(q,R)\}\text{is a null set}.$$

$${\int }_{\Omega }\det (Df)\phi ={\int }_{{\mathbb{R}}^n}\det (Df)\phi .$$

$$\left|{\int }_{{\mathbb{R}}^n}\det (Df)\phi \right|$$

$$\le {\left({\int }_{{\mathbb{R}}_{+}^{n+1}}|t^{1-\frac{s}{n}-s}DF{|}^{\frac{n}{s}}\right)}^s{\left({\int }_{{\mathbb{R}}_{+}^{n+1}}|t^{1-(1-s)-(1-s)n}D\Phi {|}^{\frac{1}{1-s}}\right)}^{1-s}.$$

$${\left({\int }_{{\mathbb{R}}_{+}^{n+1}}|t^{1-\frac{s}{n}-s}DF{|}^{\frac{n}{s}}\right)}^s\approx [f{]}_{W^{s,\frac{n}{s}}(\Omega )}^n$$

$${\left({\int }_{{\mathbb{R}}_{+}^{n+1}}|t^{1-(1-s)-(1-s)n}D\Phi {|}^{\frac{1}{1-s}}\right)}^{1-s}\approx [\phi {]}_{W^{(1-s)n,1-s}(\Omega )}.$$

$${\int }_{\Omega }\det (Df)\phi \precsim [f{]}_{W^{s,\frac{n}{s}}(\Omega )}^n[\phi {]}_{W^{(1-s)n,1-s}(\Omega )}.$$

$${\left({\int }_0^R[f{]}_{W^{s,p}(\partial B(x_0,r))}^{p}dr\right)}^{\frac{1}{p}}\precsim [f{]}_{W^{s,p}(B(x_0,R))}.$$

$${\Vert (x_{n+1}{)}^{1-\frac{1}{p}-s}DF\Vert }_{L^p({\mathbb{R}}_{+}^{n+1})}\approx [f{]}_{W^{s,p}({\mathbb{R}}^n)}<\infty .$$

$${\Vert (x_{n+1}{)}^{1-\frac{1}{p}-s}DF\Vert }_{L^p(\partial B(r)\times (0,\infty ))}<\infty .$$

$${\int }_0^R[f{]}_{W^{s,p}(\partial B(x_0,r))}^{p}dr\precsim {\int }_0^R{\int }_{\partial B(x_0,r)\times (0,\infty )}|(x_{n+1}{)}^{1-\frac{1}{p}-s}DF{|}^p={\int }_{B(x_0,R)\times (0,\infty )}|(x_{n+1}{)}^{1-\frac{1}{p}-s}DF{|}^p.$$

$$h:=\{\begin{array}{ll}f& \text{in Ω}\\ g& \text{in RnΩ}\end{array}$$

$$[h]_{W^{s,p}(\mathbb{R}^{n})}\leq C(\Omega)\,\Big{(}[f]_{W^{s,p}(\Omega)}+[g]_{W^{s,p}(\mathbb{R}^{n}\setminus\Omega)}\Big{)}.$$

$$h:=\{\begin{array}{ll}f& \text{in Ω}\\ 0& \text{in RnΩ}\end{array}$$

$$h:=\{\begin{array}{ll}g& \text{in }{\mathbb{R}}^n\backslash B,\\ f& \text{in }B.\end{array}$$

$$h_k(x):=h\left(\frac{k+1}{k}(x)\right)$$

$$\deg(f,B_{r},p):=\text{Brouwer degree of $\bigg{(}\psi:=\frac{f-p}{|f-p|}:\partial B(r)\to{\mathbb{S}}^{n-1}\bigg{)}$}.$$

$$\omega =\sum _{j=1}^n(-1{)}^{j-1}{x}^{j}d{x}^1\wedge \dots \wedge d{x}^{j-1}\wedge d{x}^{j+1}\wedge \dots \wedge d{x}^n.$$

$$\mathrm{deg}(f,B(r),p)=\mathrm{deg}(f_{\epsilon },B(r),p)={\int }_{\partial B(r)}{\psi }_{\epsilon }^{\ast }(\omega ).$$

$$\mathrm{deg}(f,B(r),p)={\int }_{B(r)}{\psi }_{\epsilon }^{\ast }(d\omega )=C{\int }_{B(r)}\det (D{\psi }_{\epsilon }).$$

$$c:=\min \left\{\mathrm{dist}(f(\partial B(r)),p),\mathrm{dist}(f(\partial B(R)),p)\right\}$$

Full text

Along the way we also obtain extensions to fractional Sobolev spaces $W^{s,\frac{n}{s}}$ of the degree estimates known for $W^{1,n}$-maps with positive or non-negative Jacobian, such as the sense-preserving property.

1. 1 Introduction
2. 2 Fractional Sobolev spaces
3. 3 Degree of maps with signed Jacobians: Proof of Propositions 1.7, 1.8, 1.9, 1.10
4. 4 Continuity of maps with positive Jacobian: Proof of Theorem 1.5, 1.6
5. A Two functions

1. Introduction

The following well-known theorem was first proven by Goldšteĭn and Vodopyanov [8]; see also [19, 5, 9] and the recent extension to manifolds in [7]:

Theorem 1.1**.**

Let $\Omega\subset\mathbb{R}^{n}$ be an open set. If $f\in W^{1,n}(\Omega,\mathbb{R}^{n})$ and

[TABLE]

then $f$ is continuous.

The strict inequality $\operatorname{Jac}(f)>0$ is necessary as the following counterexample shows:

Example 1.2**.**

Let $B$ denote the unit ball in $\mathbb{R}^{n}$. Let $\tilde{f}\in W^{1,n}(B,\mathbb{R})$ be discontinuous, e.g. $\tilde{f}(x):=\log\log\frac{2}{|x|}$. Set

[TABLE]

Clearly $f\in W^{1,n}(B,\mathbb{R}^{n})$ and $\operatorname{Jac}(f)=\det(Df)\equiv 0$. However, $f$ is still discontinuous.

The aim of this note is to give a reasonable extension to Theorem 1.1 to fractional Sobolev spaces $W^{s,p}(\Omega,\mathbb{R}^{n})$, $s\in(0,1)$. These are the spaces of maps $f\in L^{p}(\Omega,\mathbb{R}^{n})$ with finite $W^{s,p}$-Gagliardo semi-norm

[TABLE]

Clearly, for a pointwise definition of the Jacobian of $f$ to make sense, $f$ should be almost everywhere differentiable; however, as a distributional operator, the Jacobian also exists for maps in fractional Sobolev spaces $W^{s,p}$, where $s<1$ is large enough. For the sake of presentation we restrict our attention to the critical scaling, that is to the Sobolev spaces $W^{s,\frac{n}{s}}$, $s\in(0,1)$. The space $W^{s,p}_{0}(\Omega)$ denotes, as usual, the closure of $C_{c}^{\infty}(\Omega)$-functions in the $W^{s,p}$-norm.

Lemma 1.3**.**

Let $\Omega\subset\mathbb{R}^{n}$ be open with smooth boundary, $n\geq 2$. For111The case $s=1$ is also true (with $W^{0,\infty}$ replaced by $BMO$): it is the famous theorem by Coifman-Lions-Meyer-Semmes [3]. $s\in(\frac{n-1}{n},1)$ and $f\in W^{s,\frac{n}{s}}(\Omega)$ the Jacobian operator extends to a bounded linear operator on $W^{(1-s)n,\frac{1}{1-s}}_{0}(\Omega)$ in the following sense. The operator

[TABLE]

is well-defined for any $f_{k}\in C^{\infty}(\overline{\Omega})$ which is a smooth approximation of $f\in W^{s,\frac{n}{s}}(\Omega)$ and any $\varphi_{k}\in C_{c}^{\infty}(\Omega)$ which is a smooth approximation of $\varphi$ in $W^{(1-s)n,\frac{1}{1-s}}(\Omega)$.

We recall a proof of Lemma 1.3 in Section 2.

We will restrict our attention to the case $s\geq\frac{n}{n+1}$. This threshold appears in several situations on degree-type estimates in fractional Sobolev spaces; see, e.g., [6, 16]. It is exactly the case when (up to the boundary data) a map $f\in W^{s,\frac{n}{s}}$ can serve as a testfunction for its own Jacobian $\operatorname{Jac}(f)$. Lemma 1.3 warrants the following definition for a distributional Jacobian.

Definition 1.4**.**

Assume $s\geq\frac{n}{n+1}$ and $\Omega\subset\mathbb{R}^{n}$ is a smooth, bounded domain. Let $f\in W^{s,\frac{n}{s}}(\Omega,\mathbb{R}^{n})$.

• •

We say $\operatorname{Jac}(f)\geq 0$ in $\Omega$ if for any $\varphi\in W_{0}^{s,\frac{n}{s}}(\Omega)$, $\varphi\geq 0$ a.e., there holds

[TABLE]

• •

We say $\operatorname{Jac}(f)>0$ if $\operatorname{Jac}(f)\geq 0$ and for any $\varphi\in W_{0}^{s,\frac{n}{s}}(\Omega)$, $\varphi\geq 0$ a.e.,

[TABLE]

Our main result is the following version of Theorem 1.1 for fractional Sobolev spaces $W^{s,\frac{n}{s}}$.

Theorem 1.5**.**

Let $f\in W^{s,\frac{n}{s}}(\Omega,\mathbb{R}^{n})$, $s\geq\frac{n}{n+1}$, for some open and bounded set $\Omega\subset\mathbb{R}^{n}$ with smooth boundary.

If $\operatorname{Jac}(f)>0$ then $f$ is continuous.

By the counterexample, Example 1.2, there is no hope of getting Theorem 1.5 under merely the assumption $\operatorname{Jac}(f)\geq 0$. However, as it is used for the planar Monge-Ampère equation, a curl-free condition is a remedy – similar properties are known, e.g. for $W^{1,n}$-maps, see [14, Lemma 2.1.], or $C^{0,\alpha}$, $\alpha>\frac{2}{3}$, see [13]. Namely we have

Theorem 1.6**.**

Let $f=(f_{1},f_{2})\in W^{s,\frac{2}{s}}(\Omega,\mathbb{R}^{2})$ for some open and bounded set $\Omega\subset\mathbb{R}^{2}$ with smooth boundary and for some $s\geq\frac{2}{3}$. If $\operatorname{Jac}(f)\geq 0$ and if ${\rm curl\,}(f)=0$ in distributional sense, i.e. if

[TABLE]

then $f$ is continuous.

Along the way of proving Theorem 1.5 and Theorem 1.6 we obtain degree estimates for maps with signed Jacobian which are of independent interest.

If $f\in W^{s,\frac{n}{s}}(\Omega)$ then for any $x_{0}\in\Omega$ we have $f\in W^{s,\frac{n}{s}}(\partial B(x_{0},r))\hookrightarrow C^{0,\frac{s}{n}}(\partial B(x_{0},r))$ for almost every $0<r<{\rm dist\,}(x_{0},\partial\Omega)$, by means of Sobolev embedding and Fubini’s theorem, Lemma 2.2. In particular, for any $p\in\mathbb{R}^{n}\backslash f(\partial B(x,r))$ the degree $\deg(f,B(x,r),p)$ is well-defined as the Brouwer degree of the map $\frac{f-p}{|f-p|}:\partial B_{r}(x)\to{\mathbb{S}}^{n-1}$ for almost every $r$, cf.  [4].

We first observe that $f$ with non-negative Jacobian is monotone in the following sense:

Proposition 1.7**.**

Let $f\in W^{s,\frac{n}{s}}(\Omega,\mathbb{R}^{n})$ for $\Omega\subset\mathbb{R}^{n}$ open and $s\geq\frac{n}{n+1}$. Let $B(x,r)\subset B(x,R)\subset\Omega$ and assume that $f$ (in the trace sense) restricted to $\partial B(x,r)$ and $\partial B(x,R)$ is continuous.

If $\operatorname{Jac}(f)\geq 0$ in $\Omega$, then for any $p\not\in(f(\partial B(x,r))\cup f(\partial B(x,R)))$ we have

[TABLE]

We also have

Proposition 1.8**.**

Let $f\in W^{s,\frac{n}{s}}(\Omega,\mathbb{R}^{n})$ for $\Omega\subset\mathbb{R}^{n}$ open and $s\geq\frac{n}{n+1}$. Let $B(x,R)\subset\Omega$ and assume that $f$ is continuous on $\partial B(x,R)$.

If $\operatorname{Jac}(f)\geq 0$, then for any $p\not\in f(\partial B(x,R))$ we have

[TABLE]

Next, we obtain that if $f$ is continuous and the Jacobian of $f$ is positive then $f$ is sense-preserving:

Proposition 1.9**.**

Let $f\in W^{s,\frac{n}{s}}\cap C^{0}(\Omega,\mathbb{R}^{n})$, $\Omega\subset\mathbb{R}^{n}$ open, $s\geq\frac{n}{n+1}$.

If $\operatorname{Jac}(f)>0$ in $\Omega$ then for any ball $B(r)\subset\Omega$ if $p\in f(B(r))\backslash f(\partial B(r))$ then $\deg(f,B(r),p)\geq 1$.

If the Jacobian is positive, the image of a ball $f(B(r))$ has an essential diameter comparable to the diameter of $f(\partial B(r))$. This will be the main ingredient towards the proof of Theorem 1.5.

Proposition 1.10**.**

There exists some $\Lambda>0$ depending only on the dimension such that the following holds. Let $f\in W^{s,\frac{n}{s}}(\Omega,\mathbb{R}^{n})$, $\Omega\subset\mathbb{R}^{n}$ open, and $s\geq\frac{n}{n+1}$. Assume that $\operatorname{Jac}(f)>0$ in $\Omega$. For any $B(r)\subset\subset\Omega$ such that f\Big{|}_{\partial B(r)} is continuous, we can find a ball $B(q,R)\subset\mathbb{R}^{n}$ with

[TABLE]

and

[TABLE]

The number $2R$ may be viewed as the “essential diameter” of $f(B(r))$.

The remainder of this paper is organized as follows. In Section 2 we refer to some needed results for Sobolev spaces. In Section 3 we prove the degree estimates for maps with signed Jacobians, namely Propositions 1.7, 1.8, 1.9, 1.10. In Section 4 we prove Theorem 1.5 and Theorem 1.6.

Acknowledgments. This work has been done during SL’s stay as a CRM–ISM postdoctoral fellow at Centre de Recherches Mathématiques, Université de Montréal and Institut des Sciences Mathématiques. SL thanks these institutes for their hospitality. AS acknowledges funding by the Simons foundation, grant no 579261.

The authors would like to thank P. Hajłasz for helpful discussions; in particular he told us about Example 1.2.

2. Fractional Sobolev spaces

Lemma 1.3 was (essentially) proven in [17] as an extension of the ground-breaking paper [3], which showed that Jacobians of $W^{1,n}$-maps can be tested with BMO-maps. The proof in [17] uses Littlewood-Paley theory and paraproducts. In [2] Brezis and Nguyen gave a simpler and more elegant proof of this result for $s=\frac{n}{n+1}$. We present here the following slight adaptation of their argument due to [12].

We restrict our attention to the a priori estimates, from which the claim follows easily due to multi-linearity.

Proof of Lemma 1.3 (a priori estimates).

Let $\varphi\in C_{c}^{\infty}(\Omega)$ and $f\in C^{\infty}(\overline{\Omega})$. $\Omega$ is an extension domain, [11, 20], so we may assume that $f\in W^{s,\frac{n}{s}}(\mathbb{R}^{n})\cap C^{1}(\mathbb{R}^{n})$. Then

[TABLE]

Extend $f$ and $\varphi$ harmonically to $\mathbb{R}^{n+1}_{+}$, say to $F$ and $\Phi$ respectively. We write $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}_{+}=\mathbb{R}^{n+1}_{+}$. By Stokes’ theorem and Hölder’s inequality,

[TABLE]

If $s\in(\frac{n-1}{n},1]$, then $(1-s)n\in(0,1)$. Then, by trace estimates, see e.g. [12, Proposition 10.2], we have

[TABLE]

and

[TABLE]

Here we also used the fact that $[f]_{W^{s,\frac{n}{s}}(\mathbb{R}^{n})}\precsim[f]_{W^{s,\frac{n}{s}}(\Omega)}$. This is because $\Omega$ is an extension domain; see [11, 20]. We conclude, because we have shown

[TABLE]

∎

The ensuing result on trace operators will be useful for the subsequent developments. For detailed treatments we refer to [15, §2.4.2, Theorem 1], [1, Theorem 7.43, Remark 7.45] and [18, Lemma 36.1].

Lemma 2.1** (Trace Theorem).**

Let $\Omega\subset\mathbb{R}^{n}$ be either bounded or the complement of a bounded set, with smooth boundary. If $s\in(0,1)$, $p\in(1,\infty)$ with $s-\frac{1}{p}>0$, then the trace operator on T=\Big{|}_{\partial\Omega} is a bounded, linear, surjective operator from $W^{s,p}(\Omega)$ to $W^{s-\frac{1}{p},p}(\partial\Omega)$. The harmonic extension is a bounded linear right-inverse of $T$.

The following is well-known for Sobolev functions in $W^{1,p}$ (it is essentially Fubini’s theorem):

Lemma 2.2** (Restriction theorem).**

For $\Omega$ a smooth, bounded domain let $f\in W^{s,p}(\Omega)$. Fix $x_{0}\in\Omega$. There exists a representative of $f$ such that for $\mathcal{L}^{1}$-almost every $r\in(0,{\rm dist\,}(x_{0},\partial\Omega))$ we have $f\in W^{s,p}(\partial B(x_{0},r))$.

Moreover, for $\Omega=B(x_{0},R)$ we have

[TABLE]

Proof.

As $\Omega$ is an extension domain, see [11, 20], we may assume that $\Omega=\mathbb{R}^{n}$ and $f\in W^{s,p}(\mathbb{R}^{n})$ with $f\equiv 0$ outside a compact set. Denote by $F:\mathbb{R}^{n+1}_{+}\to\mathbb{R}$ the harmonic extension of $f$, and w.l.o.g. set $x_{0}=0$. Then (see [12, Proposition 10.2])

[TABLE]

By Fubini’s theorem, for $\mathcal{L}^{1}$-almost every $r>0$,

[TABLE]

This implies that $f\in W^{s,p}(\partial B(r))$ for almost every $r>0$.

The last claim also follows from Fubini’s theorem in $\mathbb{R}^{n+1}_{+}$:

[TABLE]

∎

Lemma 2.3**.**

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary. For $s\in(0,1)$, $p\in(1,\infty)$ such that $s-\frac{1}{p}>0$:

1. (1)

If $f\in W^{s,p}(\Omega)$ and $g\in W^{s,p}(\mathbb{R}^{n}\backslash\Omega)$ with $f=g$ on $\partial\Omega$ in the trace sense. Then

[TABLE]

belongs to the Sobolev space and

[TABLE] 2. (2)

In particular, if $f\in W^{s,p}(\Omega)$ satisfies $f=0$ on $\partial\Omega$ in the trace sense, then $f\in W^{s,p}_{0}(\Omega)$ and that

[TABLE]

belongs to $W^{s,p}(\mathbb{R}^{n})$.

Lemma 2.4**.**

Let $B(R)$ be a ball in $\mathbb{R}^{n}$. Let $f\in W^{s,\frac{n}{s}}(B(R))$ for some $s\in(0,1)$ and f\Big{|}_{\partial B(R)}\in C^{0}(\partial B(R)). Then there exists an approximation $f_{k}\in C^{\infty}_{c}(\mathbb{R}^{n})$ converging to $f$ in $W^{s,\frac{n}{s}}(B(R))$ and $f_{k}\rightrightarrows f$ uniformly on $\partial B(R)$.

Proof.

W.l.o.g. $B(R)=B:=B(0,1)$.

By the trace theorem, Lemma 2.1, $f\in W^{s-\frac{n}{s},\frac{n}{s}}(\partial B)$. Let $g$ be the harmonic extension of $f$ to $\mathbb{R}^{n}\backslash B$. Then $g\in W^{s,\frac{n}{s}}(\mathbb{R}^{n}\backslash B)$, again by Lemma 2.1. Also, since $f$ is continuous on $\partial B$, $g$ is also continuous. Set

[TABLE]

By Lemma 2.3, $h\in W^{s,\frac{n}{s}}(\mathbb{R}^{n})$ and $h$ is locally uniformly continuous on $\mathbb{R}^{n}\backslash B$. This last fact implies that

[TABLE]

converges uniformly to $h$ on $\partial B$ as $k\to\infty$, and also in $W^{s,\frac{n}{s}}_{loc}(\mathbb{R}^{n})$.

Now let us consider the standard mollification $f_{\varepsilon}:=h_{k}\ast\eta_{\varepsilon}$. For $\varepsilon$ small enough in comparison with $\frac{1}{k}$, $f_{\varepsilon}$ converges uniformly on $\partial B$ to $f$ and in $W^{s,\frac{n}{s}}(B))$. This completes the proof. ∎

3. Degree of maps with signed Jacobians: Proof of Propositions 1.7, 1.8, 1.9, 1.10

Here and hereafter, without further specifications, a null set is understood with respect to the Lebesgue measure $\mathcal{L}^{n}$.

For continuous $f\in C^{0}(\partial B(r),\mathbb{R}^{n})$ for some given ball $B(r)\subset\mathbb{R}^{n}$ and some point $p\in\mathbb{R}^{n}\backslash f(\partial B_{r})$, the degree of $f(B(r))$ around this point $p$ is simply the number of times that $f(\partial B(r))$ winds around $p$, i.e.,

[TABLE]

We can approximate $f$ by smooth functions $f_{\varepsilon}:\partial B(r)\to{\mathbb{S}}^{n}$ which are uniformly close to $f$. Moreover, the Brouwer degree of $\psi=\frac{f-p}{|f-p|}$ is the same as that of $\psi_{\varepsilon}=\frac{f_{\varepsilon}-p}{|f_{\varepsilon}-p|}$ for $\varepsilon$ small enough, since maps that are uniformly close to each other have the same Brouwer degree.

For the smooth functions $\psi_{\varepsilon}$ we can compute the Brouwer degree from an integral formula: denote by $\omega\in C^{\infty}(\bigwedge\nolimits^{n-1}\mathbb{R}^{n})$ the standard volume form on ${\mathbb{S}}^{n-1}$:

[TABLE]

Then, for all $\varepsilon$ small enough,

[TABLE]

If we extend $\psi_{\varepsilon}$ from a map $\partial B(r)\to{\mathbb{S}}^{n-1}$ to a map $\psi_{\varepsilon}:B(r)\to\mathbb{R}^{n+1}$, then from Stokes’ theorem we may obtain:

[TABLE]

In the last equation we used the fact that $d\omega=C\,dx^{1}\wedge\ldots\wedge dx^{n}$. Most of our arguments below are based on choosing a suitable extension of $\psi_{\varepsilon}$.

3.1. Monotonicity for non-negative Jacobian: Proof of Proposition 1.7, Proposition 1.8

We only give the proof of Proposition 1.7, the proof of Proposition 1.8 is almost verbatim (it is the “$r=0$” case).

Proof of Proposition 1.7.

Recall that $f\in C^{0}(\partial B(r))\cap C^{0}(\partial B(R))$ and that $p\in\mathbb{R}^{n}\backslash\left(f(\partial B(r))\cup f(\partial B(R))\right)$. We set

[TABLE]

For this $c>0$ let us take $d=d_{c}\in C^{1,1}([0,\infty),(0,\infty))$ as in Lemma A.1.

Let $f_{\varepsilon}$ be the approximation in Lemma 2.4 and set (for $\varepsilon\ll 1$)

[TABLE]

Then, by Stokes’ theorem, we have

[TABLE]

Below we assume $p=0$ for simplicity of notation. Observe that

[TABLE]

so we have $D\psi_{\varepsilon}=W(f_{\varepsilon})Df_{\varepsilon}$, where

[TABLE]

From the properties of $d$ (see Lemma A.1), in particular, since $d^{\prime}(|v|)=0$ whenever $|v|$ is small and $|d^{\prime}(|v|)|\approx|v|^{-2}$ whenever $|v|$ is large, we have

1. (1)

$\sup_{v\in\mathbb{R}^{n}}|W(v)|<\infty$. 2. (2)

$W\in{\rm Lip\,}(\mathbb{R}^{n})$. 3. (3)

$\det(W(v))\geq 0$ for all $v\in\mathbb{R}^{n}$.

Indeed, if $v=0$, then $d^{\prime}(|v|)=0$ and hence $\det(W(v))=\det(d(0)I)\geq 0$. Assume now $v\neq 0$. Since $W(v)$ is symmetric, it suffices to show that the eigenvalues are nonnegative. Observe that $v/|v|$ and any orthonormal basis of $v^{\perp}$ are the eigenvectors of $W(v)$.

In the former case, we compute

[TABLE]

That is, the eigenvalue for the eigenvector $v/|v|$ is non-negative.

In the latter case, given any $o\in v^{\perp}$ with $|o|=1$, one has

[TABLE]

So the eigenvalue for any eigenvector $o$ perpendicular to $v$ is $d(|v|)\geq 0$.

Therefore, all the eigenvalues of $W(v)$ are non-negative, thus $\det(W(v))\geq 0$. 4. (4)

$\det(W(v))=0$ whenever $|v|>\frac{c}{2}$.

Indeed, this is because

[TABLE]

where, for $|v|>\frac{c}{2}$, we have

[TABLE]

This shows that $W(v)$ is non-invertible, so $\det(W(v))=0$.

Now, since $W$ is Lipschitz (and globally bounded), $\det(W(f_{\varepsilon}))$ lies uniformly in $W^{s,\frac{n}{s}}(B(R)\backslash B(r))$ and converges strongly in $W^{s,\frac{n}{s}}(B(R)\backslash B(r))$ to $\det(W(f))$.

On the other hand, for any fixed small enough $\varepsilon>0$ there exists a neighborhood of $\partial(B(R)\backslash B(r))$ where $|f_{\varepsilon}-p|>\frac{3c}{4}$ – this holds since $|f_{\varepsilon}-p|>\frac{4c}{5}$ on $\partial(B(R)\backslash B(r))$ for all $\varepsilon$ small enough, due to uniform convergence.

That is, for any small $\varepsilon>0$ there exists a neighborhood around $\partial(B(R)\backslash B(r))$ where $\det(W(f_{\varepsilon}))\equiv 0$.

This implies that $\det(W(f_{\varepsilon}))$ and $\det(W(f))$ all lie in $W^{s,\frac{n}{s}}_{0}(B(R)\backslash B(r))$. By virtue of Lemma 2.3, we can extend these functions by zero to all of $\Omega$, and they belong consequently to $W^{s,\frac{n}{s}}_{0}(\Omega)$.

That is, we have shown that

[TABLE]

The right-hand side is nonnegative by assumption, and Proposition 1.7 is proven. ∎

3.2. Positive Jacobian implies sense-preserving: Proof of Proposition 1.9

Proof of Proposition 1.9.

The proof is very similar to that of Proposition 1.7. With the notation used therein, we have

[TABLE]

where again

[TABLE]

for $d$ taken from Lemma A.1 with $c:=\frac{1}{2}{\rm dist\,}(f(\partial B(r)),p)$.

As before, we have $\det(W(f))\geq 0$. The assumption $\operatorname{Jac}(f)>0$ implies $\deg(f,B(r),p)\geq 0$. It remains to show that if $\deg(f,B(r),p)=0$ then $p\not\in f(B(r))$, for the claim that $\deg(f,B(r),p)\geq 1$ in $f(B(r))$ shall follow immediately.

So, assume that $p\not\in f(\partial B(r))$ and $\deg(f,B(r),p)=0$. From Definition 1.4 we see that $\operatorname{Jac}(f)>0$ readily implies $\det(W(f))\equiv 0$. That is, one of the eigenvalues of $W(f)$ is zero. As computed in the proof of Proposition 1.7, the eigenvalues of $W(v)$ are

[TABLE]

Since $d(|v|)\neq 0$ for all $v$, $\det(W(f(x)))\equiv 0$ implies that necessarily

[TABLE]

By the properties of $d$ (see Lemma A.1), we deduce that $\inf_{B(r)}|f(x)-p|>0$. Thus $p\not\in f(B(r))$ as claimed. ∎

3.3. Comparability of diameters: Proof of Proposition 1.10

The proof below is an adaptation from the argument in [19, 7]. Modifications are necessary due to the fact that we do not have a pointwise Jacobian.

Proof.

Since $f$ is continuous on $\partial B(r)$, we can find a large ball $B(q,\rho)$ of radius $\rho:={\rm diam\,}f(\partial B(r))$ such that $f(\partial B(r))\subset B(q,\rho)$.

Take $\pi=\pi_{\lambda}$ from Lemma A.2 for $\lambda:=10\rho$.

Let $f_{\varepsilon}$ be the smooth approximation of $f$ from Lemma 2.4. For all small enough $\varepsilon>0$ we have $f_{\varepsilon}(\partial B(r))\subset B(q,2\rho)$.

In particular if we set

[TABLE]

then $g_{\varepsilon}=f_{\varepsilon}$ on $\partial B(r)$. Consequently (by an integration by parts argument it is easy to see that the integral of the Jacobian of a map on a ball only depends on the boundary value of that map, [10, Lemma 4.7.2]),

[TABLE]

Computing $Dg_{\varepsilon}$ similar as in the proof of Proposition 1.7, setting

[TABLE]

we obtain

[TABLE]

As in the proof of Proposition 1.7, the map $1-\det(W(f_{\varepsilon}-q))$ belongs to $W^{s,\frac{n}{s}}(B(r))$ and converges strongly in that space to $1-\det(W(f-q))$.

Moreover, as in the proof of Proposition 1.7, we can compute

[TABLE]

and by the properties of $\pi$, see Lemma A.2,

[TABLE]

Moreover, since $\pi(|v|)\equiv 1$ for $|v|\leq 10\rho$, we have

[TABLE]

That is,

[TABLE]

By Lemma 2.1 and Lemma 2.3 we can thus again extend $1-\det(W(f_{\varepsilon}-q))$ and $1-\det(W(f-q))$ by zero to a $W^{s,\frac{n}{s}}_{0}(\Omega)$-function. Thus, we conclude that

[TABLE]

Since by assumption $\operatorname{Jac}(f)>0$ and $1-\det(W(f-q))\geq 0$ a.e., we may infer (see Definition 1.4) that

[TABLE]

That is,

[TABLE]

But by the properties of $\pi$, see Lemma A.2, this implies

[TABLE]

Therefore,

[TABLE]

∎

4. Continuity of maps with positive Jacobian: Proof of Theorem 1.5, 1.6

The proof of Theorem 1.5 crucially relies on the diameter estimates of Proposition 1.10. Once we have this, we adapt the argument in [19] to fractional Sobolev spaces in a more or less straightforward fashion, namely Theorem 1.5 is a corollary of the following statement.

Proposition 4.1**.**

Let $\Omega\subset\mathbb{R}^{n}$. Assume that $f\in W^{s,\frac{n}{s}}(\Omega,\mathbb{R}^{n})$, $s\in(0,1)$ satisfies the following: for any $x_{0}\in\Omega$ and $\mathcal{L}^{1}$-almost all radii $0<r<\rho<{\rm dist\,}(x_{0},\Omega)$, there holds

[TABLE]

Then $f$ is continuous. Moreover, for any ball $B\Subset\Omega$, $s>0$, and $x,y\in B$, we have

[TABLE]

Observe that an easy extension of Proposition 4.1 holds for $W^{s,p}$-maps whenever $s-\frac{n-1}{p}>0$.

Proof of Theorem 1.5.

Fix $x_{0}\in\Omega$ and let $R:={\rm dist\,}(x_{0},\partial\Omega)$. W.l.o.g. $x_{0}=0$. Let $0<r<\rho<R$ such that $f$ is continuous on $\partial B(r)$ and $\partial B(\rho)$. By Lemma 2.2 we know this happens for $\mathcal{L}^{1}$-a.e. $r$ and $\rho$.

By Proposition 1.10, for almost any $0<r<\rho<R$ we have the monotonicity

[TABLE]

Indeed, by Lemma 2.2, for almost any $0<r<\rho<R$ the map $f$ is continuous on $\partial B(r)$ and $\partial B(\rho)$. Thus

[TABLE]

where ${\rm diam\,}(f(B(\rho))$ is understood as

[TABLE]

The trace f\Big{|}_{\partial B(r)} is $\mathcal{H}^{n-1}$-a.e. attained by sequences of f\Big{|}_{\partial B(\tilde{r})} as $\tilde{r}\to r$. If $A$ is as in the definition of ${\rm diam\,}$ above then for $\mathcal{L}^{1}$-almost every $\tilde{r}$ we have $\mathcal{H}^{n-1}(A\cap\partial B(\tilde{r}))=\mathcal{H}^{n-1}(\partial B(\tilde{r}))$. So, we find a sequence $r_{i}\to r$ with $\partial\mathcal{H}^{n-1}(A\cap\partial B(r_{i}))=\mathcal{H}^{n-1}(\partial B(r_{i}))$ and f\Big{|}_{\partial B(r_{i})} converging $\mathcal{H}^{n-1}$-a.e. to f\Big{|}_{\partial B(r)}. Thus, whenever $\rho>r$,

[TABLE]

This establishes (4.1).

Now, we may deduce from (4.1) that, for almost any $0<r<\rho<R$,

[TABLE]

From here one concludes the continuity property with Proposition 4.1. ∎

Proof of Proposition 4.1.

By Sobolev embedding for $s>0$,

[TABLE]

Thus we find

[TABLE]

In view of Lemma 2.2 we obtain

[TABLE]

This readily implies the claim. ∎

Theorem 1.6 also follows from Proposition 4.1, and additionally the following distortion argument.

Lemma 4.2**.**

Let $f\in W^{s,\frac{2}{s}}(\Omega,\mathbb{R}^{2})$, $\Omega\subset\mathbb{R}^{2}$ open, $s\geq\frac{2}{3}$. Assume that $\operatorname{Jac}(f)\geq 0$ and ${\rm curl\,}(f):=\partial_{1}f^{2}-\partial_{2}f^{1}=0$ in distributional sense.

For $\delta\in\mathbb{R}\backslash\{0\}$ set $f_{\delta}(x_{1},x_{2}):=f(x_{1},x_{2})+\delta(-x_{2},x_{1})^{T}$. Then $\operatorname{Jac}(f_{\delta})>0$.

Proof.

Let $f_{\varepsilon}$ be an approximation of $f$ in $W^{s,\frac{2}{s}}(\Omega)$. We have

[TABLE]

That is, for any $\varphi\in C_{c}^{\infty}(\Omega)$,

[TABLE]

Integrating by parts and the pointwise a.e. convergence of $f_{\varepsilon}$ to $f$ implies

[TABLE]

Thus,

[TABLE]

In particular if $\varphi\geq 0$ and $\operatorname{Jac}(f_{\delta})[\varphi]=0$ we have $\varphi\equiv 0$, i.e. $\operatorname{Jac}(f_{\delta})>0$. ∎

Proof of Theorem 1.6.

By Lemma 4.2 and Theorem 1.5 we have $f_{\delta}$ is continuous, and indeed we have a estimate on the modulus of continuity of $f_{\delta}$ by Proposition 4.1. This estimate is uniform in $\delta$, and by Arzela-Ascoli we conclude that $f=\lim_{\delta\to 0}f_{\delta}$ still enjoys the same continuity estimate. ∎

Appendix A Two functions

Lemma A.1**.**

For any $c>0$ there exists $d=d_{c}\in C^{1,1}([0,\infty),(0,\infty))$ such that

[TABLE]

Proof.

Observe that if $d(t)$ satisfies the above assumptions for $c=2$, then $d_{c}(t):=c^{-1}d(t/c)$ satisfies the assumptions for generic $c>0$. So, w.l.o.g. $c=2$.

Set

[TABLE]

Observe that $\lim_{t\to 1^{+}}d(t)=\lim_{t\to 1^{-}}d(t)=1$, and that $\lim_{t\to 1/2^{+}}d(t)=\lim_{t\to 1/2^{-}}d(t)=\frac{5}{4}$. Also,

[TABLE]

In particular, $\lim_{t\to 1^{+}}d^{\prime}(t)=\lim_{t\to 1^{-}}d^{\prime}(t)=-1$ and $\lim_{t\to 1/2^{+}}d^{\prime}(t)=\lim_{t\to 1/2^{-}}d^{\prime}(t)=0$. That is, $d\in C^{1,1}([0,\infty))$. The only thing left to check is that

[TABLE]

is non-negative. But this is immediate. ∎

Lemma A.2**.**

For any $n\in{\mathbb{N}}$ and any $\lambda>0$ there exists $\pi\in C^{1,1}([0,\infty),(0,\infty))$ with the following properties:

[TABLE]

Proof.

Setting $\pi_{\lambda}(t):=\pi(t/\lambda)$ we can reduce to the case $\lambda=1$, which we shall now consider.

Set $r(t):=t^{n}\pi(t)^{n}$. Then the differential inequalities become

[TABLE]

Clearly we can find a $C^{1,1}$-function $r$ that satisfies these conditions, e.g.

[TABLE]

where $a(t)$ is any smooth non-decreasing function such that $a(1)=a^{\prime}(1)=0$, $a(2)=1$, and $a^{\prime}(2)=\frac{1}{2}$.

Then $p(t):=t^{-1}\sqrt[n]{r(t)}$ is bounded, has derivatives bounded, and satisfies all the other assumptions as well. ∎