Globally diffeomorphic $\sigma$--harmonic mappings

TL;DR

This paper establishes conditions under which a two-dimensional divergence-structure elliptic equation solution mapping is a global diffeomorphism, ensuring invertibility and smoothness.

Contribution

It provides necessary and sufficient boundary conditions for the global diffeomorphism property of $\sigma$-harmonic mappings in two dimensions.

Findings

Characterization of boundary conditions for diffeomorphism

Conditions ensuring invertibility of $\sigma$-harmonic mappings

Theoretical framework for global diffeomorphism in elliptic PDEs

Abstract

Given a two--dimensional mapping $U$ whose components solve a divergence structure elliptic equation, we give necessary and sufficient conditions on the boundary so that $U$ is a global diffeomorphism.

Equations

$$\begin{array}[]{ccrllll}\hbox{$\hskip 11.38092pt\sigma(x)\xi\cdot\xi\geq K^{-1}|\xi|^{2}$, for every $\xi\in\mathbb{R}^{2}\ ,x\in B$\,,}\\ \hbox{$\sigma^{-1}(x)\xi\cdot\xi\geq K^{-1}|\xi|^{2}$, for every $\xi\in\mathbb{R}^{2}\ ,x\in B$\,,}\par\end{array}$$

$$\left\{\begin{array}[]{lll}{\rm div}(\sigma\nabla u^{i})=0,&\hbox{in}&B,\\ u^{i}=\varphi^{i},&\hbox{on}&\partial B\ ,i=1,2\ .\end{array}\right.$$

$$\begin{array}[]{lll}\det DU>0&\hbox{everywhere on}&\partial B.\end{array}$$

$$\begin{array}[]{lll}\det DU>0&\hbox{everywhere on}&\partial B\ ,\end{array}$$

$$\mathrm{div}(\sigma \nabla u)=0$$

$$\mathrm{div}(A\nabla u)+b\cdot \nabla u=0,$$

$$\hat{\sigma }=\frac{1}{2}(\sigma +{\sigma }^T),\check{\sigma }=\frac{1}{2}(\sigma -{\sigma }^T),$$

$$0=\mathrm{div}(\sigma \nabla u)=\mathrm{div}(\hat{\sigma }\nabla u)+{\partial }_{x_i}{\check{\sigma }}_{ij}{\partial }_{x_j}u,$$

$$0=\gamma \mathrm{div}(A\nabla u)+{\partial }_{x_i}(\gamma {\delta }_{ij}+{\check{\sigma }}_{ij}){\partial }_{x_j}u,$$

$$u_{\alpha }=\cos \alpha u^1+\sin \alpha u^2,\alpha \in \mathbb{R},$$

$$M_{\alpha }=\text{number of critical points of uα in B, counted with their multiplicities}.$$

$$M_{\alpha }=\frac{1}{2\pi }{\int }_{\partial B}\mathrm{d}\arg ({\partial }_z{u}_{\alpha }),$$

$$M_{\alpha }=\frac{1}{2\pi }{\int }_{\partial B}\mathrm{d}\arg (\xi ).$$

$$\xi =\frac{1}{|\nabla u_1|}J\nabla u_1.$$

$$J=\left(\begin{array}[]{ccc}0&-1\\ 1&0\end{array}\right)\,,$$

$$\nabla u_{\alpha }\cdot \xi =\frac{\sin \alpha }{|\nabla u_1|}\nabla u_2\cdot J\nabla u_1=\frac{\sin \alpha }{|\nabla u_1|}\det DU$$

$$\mathrm{div}(\sigma \nabla u)=0$$

$$\nabla v=J\sigma \nabla u,$$

$$f=u+iv,$$

$$\begin{array}[]{ll}f_{\bar{z}}=\mu f_{z}+\nu\overline{f_{z}}&\hbox{in $B$}\ ,\end{array}$$

$$\begin{array}[]{llll}\mu=\frac{\sigma_{22}-\sigma_{11}-i(\sigma_{12}+\sigma_{21})}{1+{\rm Tr\,}\sigma+\det\sigma}&\ ,&\nu=\frac{1-\det\sigma+i(\sigma_{12}-\sigma_{21})}{1+{\rm Tr\,}\sigma+\det\sigma}\ ,\end{array}$$

$$|\mu |+|\nu |\le k<1,$$

$$f=F\circ \chi ,$$

$$\begin{array}[]{ll}\chi_{\bar{z}}=\widetilde{\mu}\chi_{z}&\hbox{in $B$}\ ,\end{array}$$

$$\tilde{\mu }=\mu +\frac{\overline{f_z}}{f_z}\nu ,$$

$${\omega }_{\bar{z}}=\tilde{\mu }{\omega }_z.$$

$$\mathrm{div}(\tilde{\sigma }\nabla w)=0,\text{ in }B\setminus \overline{B_{\rho }}$$

$$\widetilde{\sigma}=\left(\begin{array}[]{ccc}\displaystyle{\frac{|1-\widetilde{\mu}|^{2}}{1-|\widetilde{\mu}|^{2}}}&\displaystyle{-\frac{2\Im m(\widetilde{\mu})}{1-|\widetilde{\mu}|^{2}}}\\ \\ \displaystyle{-\frac{2\Im m(\widetilde{\mu})}{1-|\widetilde{\mu}|^{2}}}&\displaystyle{\frac{|1+\widetilde{\mu}|^{2}}{1-|\widetilde{\mu}|^{2}}}\end{array}\right)\,,$$

$$\frac{d\Phi }{d\vartheta }\ne 0,\text{for every}\vartheta \in [0,2\pi ],$$

$$\mathrm{WN}(\gamma )=\frac{1}{2\pi }{\int }_0^{2\pi }\mathrm{d}\arg \left(\frac{d\Phi }{d\vartheta }\right).$$

Full text

1. Introduction

Let $B=\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\}$ denote the unit disk. We denote by $\sigma=\sigma(x)$, $x\in B$, a possibly non–symmetric matrix having measurable entries and satisfying the ellipticity conditions

[TABLE]

for a given constant $K\geq 1$.

Given a diffeomorphism $\Phi=(\varphi^{1},\varphi^{2})$ from the unit circle $\partial B$ onto a simple closed curve $\gamma\subseteq\mathbb{R}^{2}$, we denote by $D$ the bounded domain such that $\partial D=\gamma$. With no loss of generality, we may assume that $\Phi$ is orientation preserving.

Let us consider the mapping $U=(u^{1},u^{2})\in W^{1,2}(B;\mathbb{R}^{2})\cap C(\overline{B};\mathbb{R}^{2})$ whose components are the solutions to the following Dirichlet problems

[TABLE]

Loosely speaking, the question that we intend to address here is:

Under which conditions can we assure that $U$ is an invertible mapping between $B$ and $D$ (or $\overline{B}$ and $\overline{D}$)?

The classical starting point for this issue is the celebrated Radò–Kneser–Choquet Theorem [20, 17, 12, 14] which asserts that assuming $\sigma=I$, the identity matrix, (that is: $u^{1},u^{2}$ are harmonic) if $D$ is convex then $U$ is a homeomorphism. Generalizations to equations with variable coefficients have been obtained in [8, 3] and to certain nonlinear systems in [9, 7, 18]. Counterxamples [12, 4] show that if $D$ is not convex then the invertibility of $U$ may fail.

In [4] the present authors investigated, in the case of harmonic mappings, which additional conditions are needed for invertibility in the case of a possibly non–convex target $D$. In particular, in [4, Theorem 1.3] it is proven that, assuming $\sigma=I$, $U$ is a diffeomorphism if and only if $\det DU>0$ everywhere on $\partial B$. An improvement to this result, still in the harmonic case, is due to Kalaj [15].

Here we intend to treat the case of equations with variable coefficients. The main result in this note is the following:

Theorem 1.1**.**

Assume that the entries of $\sigma$ satisfy $\sigma_{ij}\in C^{\alpha}(\overline{B})$ for some $\alpha\in(0,1)$ and for every $i,j=1,2$. Assume, in addition, that $U\in C^{1}(\overline{B};\mathbb{R}^{2})$.

The mapping $U$ is a diffeomorphism of $\overline{B}$ onto $\overline{D}$ if and only if

[TABLE]

It is evident that, if $U$ is a diffeomorphism on $\overline{B}$, then $\det DU\neq 0$ on $\partial B$. Thus, from now on, we shall focus on the reverse implication only.

In Section 2 we begin by proving Theorem 2.1, that is, a version of Theorem 1.1 which requires stronger regularity on $\sigma$ and on $\Phi$.

Section 3 contains the completion of the proof of Theorem 1.1, let us mention that, as an intermediate step, we also prove Theorem 3.4, which treats the case when the Dirichlet data $\Phi$ is merely a homeomorphism, extending to the case of variable coefficients the result proved in [4, Theorem 1.7] for the case of $\sigma=I$.

In the final Section 4 we sketch the arguments for an improvement, Theorem 4.2, to Theorem 1.1, in analogy with [4, Theorem 5.2].

2. A smoother case

Theorem 2.1**.**

In addition to the hypotheses of Theorem 1.1, let us assume that the entries of $\sigma$ satisfy $\sigma_{ij}\in C^{0,1}(\overline{B})$and that $\Phi=(\varphi^{1},\varphi^{2})\in C^{1,\alpha}(\partial B,\mathbb{R}^{2})$ for some $\alpha\in(0,1)$. If

[TABLE]

then the mapping $U$ is a diffeomorphism of $\overline{B}$ onto $\overline{D}$.

We observe that, assuming that $\sigma_{ij}$ are Lipschitz continuous in $\overline{B}$, it is a straightforward matter to rewrite the equation

[TABLE]

in the form

[TABLE]

where $b=(b^{1},b^{2})$ is in $L^{\infty}$ and $A$ is a uniformly elliptic symmetric matrix in the sense of (1.1), with Lipschitz entries, and it satisfies ${\rm det}A=1$ everywhere.

The calculation is as follows. Denote

[TABLE]

where $(\cdot)^{T}$ denotes the transposition. Writing the equation in weak form and using smooth test functions, we obtain

[TABLE]

next we pose $\gamma=\sqrt{{\rm det}\widehat{\sigma}}$ and $A=\frac{1}{\gamma}\widehat{\sigma}$ and we compute

[TABLE]

hence $b^{j}=\frac{1}{\gamma}\partial_{x_{i}}(\gamma\delta_{ij}+\widecheck{\sigma}_{ij})$.

We recall that local weak solutions $u$ to (2.2) are indeed $C^{1,\alpha}$, their critical points are isolated and have finite integral multiplicity. This theory has been developed by R. Magnanini and the first named author [1]. As a consequence of such a theory, we can state the following auxiliary result. Let us start with some notation.

We denote

[TABLE]

where $u^{1},u^{2}$ are the components of the mapping $U$ appearing in Theorem 1.1. Next we define

[TABLE]

Note that, in view of (1.3), $M_{\alpha}$ is finite for all $\alpha$.

Proposition 2.2**.**

Under the assumptions of Theorem 2.1, we have

[TABLE]

moreover $M_{\alpha}=M$ is constant with respect to $\alpha$.

Here $\partial_{z}$ denotes the usual complex derivative, where it is understood $z=x_{1}+ix_{2}$.

Proof.

Formula (2.5) is a manifestation of the argument principle. A proof, with some changes in notation, can be found in [1, Proof of Theorem 2.1]. Also, a special case of Theorem 2.1 in [1], tell us that if $\xi$ is a $C^{1}$ unitary vector field on $\partial B$ such that $\nabla u_{\alpha}\cdot\xi>0$ everywhere on $\partial B$ then we have

[TABLE]

Let us denote

[TABLE]

where the matrix $J$ represents the counterclockwise $90^{\circ}$ rotation

[TABLE]

and we compute

[TABLE]

which is positive for all $\alpha\in(0,\pi)$. Hence $M_{\alpha}$ is constant for all $\alpha\in(0,\pi)$, by continuity the same is true for all $\alpha\in[0,\pi]$. The proof is complete, by noticing that $u_{\alpha+\pi}=-u_{\alpha}$. ∎

Our next goal being to prove that $M=M_{\alpha}=0$, we return to the equation in pure divergence form. Denoting $u=u_{\alpha}$ for any fixed $\alpha$, we have that the equation

[TABLE]

holds in $B$. It is well–known that there exists $v\in W^{1,2}(B)$, called the stream function of $u$ such that

[TABLE]

where, again, the matrix $J$ denotes the counterclockwise $90^{\circ}$ rotation (2.8), see, for instance, [2]. Denoting

[TABLE]

it is well–known that $f$ solves the Beltrami type equation

[TABLE]

where, the so called complex dilatations $\mu,\nu$ are given by

[TABLE]

and satisfy the following ellipticity condition

[TABLE]

where the constant $k$ only depends on $K$, see [5, Proposition 1.8] and the notation ${\rm Tr\,}A$ is used for the trace of a square matrix $A$.

Furthermore, it is also well–known, Bers and Nirenberg [10], Bojarski [11], that a $W^{1,2}$ solution to (2.11) fulfills the so–called Stoilow representation

[TABLE]

where $F$ is holomorphic and $\chi$ is a quasiconformal homeomorphism, which can be chosen to map $B$ into itself. Moreover, $\chi$ solves the Beltrami equation

[TABLE]

where $\widetilde{\mu}$ is defined almost everywhere by

[TABLE]

Note that, under the present assumptions, $\mu,\nu$ are Lipschitz continuous in $\overline{B}$ and $f$ is in $C^{1,\alpha}(\overline{B},\mathbb{C})$.

From now on, for simplicity, we denote by $B_{\rho}$ be the disk of radius $\rho>0$ concentric to $B$.

In view of (1.3), there exists $0<\rho<1$ such that $\partial_{z}f\neq 0$ on $\overline{B}\setminus B_{\rho}$. As a consequence, $\widetilde{\mu}$ is $C^{\alpha}$ on $\overline{B}\setminus B_{\rho}$, and the following Lemma holds.

Lemma 2.3**.**

Under the assumptions of Theorem 2.1, there exists $0<\rho<1$ such that the mapping $\chi$, appearing in (2.14), belongs to $C^{1,\alpha}$, for some $0<\alpha<1$, when restricted to $\overline{B}\setminus B_{\rho}$.

Proof.

For $\rho$ sufficiently close to $1$ we may represent $\chi=\exp\left(\omega\right)$ in the annulus $\overline{B}\setminus B_{\rho}$. Also, for every determination of $\omega$, we have

[TABLE]

Now, posing $w=\Re e(\omega)=\log|\chi|$, it is well–known that we have

[TABLE]

where $\widetilde{\sigma}$ is given by

[TABLE]

and satisfies uniform ellipticity conditions of the form (1.1), see, for instance, [5]. Moreover, $\widetilde{\sigma}$ has Hölder continuous entries in $\overline{B}\setminus B_{\rho}$. Now, since, trivially, $w=0$ on $\partial B$, then, by standard regularity at the boundary, $w$ is $C^{1,\alpha}$ near $\partial B$. Such a regularity extends to $\omega$, and then to $\chi$, because (2.16) can be rewritten as $\nabla\Im m(\omega)=J\widetilde{\sigma}\nabla w$. ∎

Next we recall the following classical notion, see for instance [21].

Definition 2.4**.**

Given a closed curve $\gamma$, parametrized by $\Phi\in C^{1}(\left[0,2\pi\right];\mathbb{R}^{2})$ and such that

[TABLE]

we define the winding number of $\gamma$ as the following integer

[TABLE]

Proposition 2.5**.**

Under the previously stated assumptions

[TABLE]

with $M$ as in Proposition 2.2.

Proof.

With no loss of generality, we may assume $\chi(1)=1$.

We have that, for every $\vartheta\in\mathbb{R}$,

[TABLE]

where

[TABLE]

hence $\varphi$ is a strictly increasing function from $[0,2\pi]$ into itself, with $C^{1,\alpha}$ regularity. Consequently

[TABLE]

For the second integral we trivially have

[TABLE]

whereas, by the argument principle, the integral

[TABLE]

equals the number of zeroes of $F^{\prime}$ when counted with their multiplicities, which coincides with the number of critical points of $u$, again counted with their multiplicities, that is, $M$. This is a consequence of the notions of geometrical critical points and geometric index introduced in [2, Definition 2.4], which in the present circumstances, coincide with the usual concepts of critical points and multiplicity, respectively.∎

Next we compute:

Proposition 2.6**.**

[TABLE]

Proof.

We may fix $\alpha=0$, that is, $u=u^{1}$, and let $v^{1}$ be its stream function. For every $t\in[0,1]$ let us consider $U_{t}=(u^{1},(1-t)v^{1}+tu^{2})$. Trivially

[TABLE]

We compute

[TABLE]

consequently

[TABLE]

never vanishes. By homotopic invariance of the winding number, [21, Theorem 1], the thesis follows. ∎

Proof of Theorem 2.1.

Combining Propositions 2.2, 2.5 and 2.6 we deduce that, for all $\alpha$, $\nabla u_{\alpha}$ nowhere vanishes. Hence $\det DU>0$ everywhere in $\overline{B}$. Hence it is a local diffeomorphism which is one–to–one on the boundary, by the Monodromy Theorem, see for instance [16, p.175], the thesis follows. ∎

3. Proof of Theorem 1.1

We start by removing the hypothesis of Lipschitz continuity on $\sigma$, and obtain an intermediate weaker result.

Lemma 3.1**.**

In addition to the hypotheses of Theorem 1.1, let us assume $\Phi=(\varphi^{1},\varphi^{2})\in C^{1,\alpha}(\partial B,\mathbb{R}^{2})$, for some $\alpha\in(0,1)$. Then $U$ is locally a homeomorphism in ${B}$.

Proof.

Let $\sigma_{\varepsilon}$ be a family of $C^{\infty}$ mollifications of $\sigma$, which satisfy ellipticity and Hölder regularity uniformly with respect to $\varepsilon$. Let $U_{\varepsilon}$ be the solution to

[TABLE]

By regularity theory, $U_{\varepsilon}\in C^{1,\alpha}(\overline{B},\mathbb{R}^{2})$ uniformly with respect to $\varepsilon$, hence, by the Ascoli–Arzelà Theorem, $U_{\varepsilon_{n}}\to U$ in $C^{1}(\overline{B},\mathbb{R}^{2})$ for some sequence $\varepsilon_{n}\to 0$. Therefore, for $n$ large enough

[TABLE]

thus, by Theorem 2.1, $U_{\varepsilon_{n}}$ is a diffeomorphism of $\overline{B}$ onto $\overline{D}$. In particular the number $(M_{\varepsilon_{n}})_{\alpha}$, associated to $U_{\varepsilon_{n}}$ according to the definition (2.4), equals zero for all $\alpha$ and for $n$ large enough. In view of the stability of the geometric index, established in [2, Proposition 2.6], we have that $u_{\alpha}=\cos\alpha\,u^{1}+\sin\alpha\,u^{2}$ has no (geometrical) critical point in $B$ for any $\alpha$. We may invoke now [3, Theorem 3] to obtain that $U$ is locally a homeomorphism in $B$. ∎

We now recall a variant to the celebrated H. Lewy’s Theorem [19], recently obtained in [6, Theorem 1.1]. Here $\Omega\subset\mathbb{R}^{2}$ is any open set.

Theorem 3.2**.**

Assume that the entries of $\sigma$ satisfy $\sigma_{ij}\in C^{\alpha}_{loc}(\Omega)$ for some $\alpha\in(0,1)$ and for every $i,j=1,2$. Let $U=(u^{1},u^{2})\in W^{1,2}_{loc}(\Omega,\mathbb{R}^{2})$ be such that

[TABLE]

weakly in $\Omega$. If $U$ is locally a homeomorphism, then it is, locally, a diffeomorphism, that is

[TABLE]

The following Theorem mimics an analogous result obtained for harmonic mappings in [4]. We recall the following definition.

Definition 3.3**.**

Given $P\in\overline{B}$, a mapping $U\in C(\overline{B};\mathbb{R}^{2})$ is a local homeomorphism at $P$ if there exists a neighborhood $G$ of $P$ such that $U$ is one–to–one on $G\cap\overline{B}$.

Theorem 3.4**.**

Let $\Phi:\partial B\to\gamma\subset\mathbb{R}^{2}$ be a homeomorphism onto a simple closed curve $\gamma$. Let $D$ be the bounded domain such that $\partial D=\gamma$. Let $U\in W^{1,2}_{\rm loc}(B;\mathbb{R}^{2})\cap C(\overline{B};\mathbb{R}^{2})$ be the solution to (1.2). Assume that the entries of $\sigma$ satisfy $\sigma_{ij}\in C^{\alpha}_{loc}({B})$ for some $\alpha\in(0,1)$ and for every $i,j=1,2$. If, for every $P\in\partial B$, the mapping $U$ is a local homeomorphism at $P$, then it is a homeomorphism of $\overline{B}$ onto $\overline{D}$ and it is a diffeomorphism of $B$ onto $D$ .

We first need the following Lemma, which is adapted from [4, Lemma 4.1]. Let us recall that $B_{\rho}$ denotes the disk of radius $\rho>0$ concentric to $B$.

Lemma 3.5**.**

Assume $\Phi:\partial B\to\gamma\subset\mathbb{R}^{2}$ is a homeomorphism onto a simple closed curve $\gamma$. Let $U\in W^{1,2}_{loc}(B;\mathbb{R}^{2})\cap C(\overline{B};\mathbb{R}^{2})$ be the solution to (1.2). Assume that the entries of $\sigma$ satisfy $\sigma_{ij}\in C^{\alpha}_{loc}({B})$ for some $\alpha\in(0,1)$ and for every $i,j=1,2$. If, in addition, for every $P\in\partial B$ the mapping $U$ is a local homeomorphism near $P$, then there exists $\rho\in(0,1)$ such that $U$ is a diffeomorphism of $B\setminus\overline{B_{\rho}}$ onto U\Big{(}B\setminus\overline{B_{\rho}}\Big{)}.

Proof.

For every $P\in\partial B$ let

[TABLE]

the function $s(P)$ is positive valued and lower semicontinuous hence, by the compactness of $\partial B$, there exists $\delta>0$ such that $s(P)>2\delta$ for all $P\in\partial B$. Again by compactness, there exist finitely many points $P_{1},\ldots,P_{K}\in\partial B$ such that

[TABLE]

and $U$ is one–to–one on $B_{2\delta}(P_{k})\cap\overline{B}$ for every $k$. Note that there exists $\rho_{0}\in(0,1)$ such that

[TABLE]

Let $P,Q$ be two distinct points in $\overline{B}\setminus B_{\rho_{0}}$. If $|P-Q|<\delta$, then there exists $k=1,\ldots,K$ such that $P,Q\in B_{2\delta}(P_{k})$ and, hence, $U(P)\neq U(Q)$. Assume now $|P-Q|\geq\delta$. Let

[TABLE]

We have $|P-P^{\prime}|<1-\rho,\,|Q-Q^{\prime}|<1-\rho,$ and thus

[TABLE]

Choosing $\rho_{1}$, $\rho_{0}\leq\rho_{1}<1$ such that $(1-\rho_{1})<\frac{\delta}{4}$, we have $|P^{\prime}-Q^{\prime}|>\frac{\delta}{2}.$ Now we use the fact that $P^{\prime}$ and $Q^{\prime}$ belong to $\partial B$ and $\Phi$ is one–to–one to deduce that there exists $c>0$ such that

[TABLE]

Recall that $U$ is uniformly continuous on $\overline{B}$. Denoting by $\omega$ its modulus of continuity, we have

[TABLE]

[TABLE]

Choosing $\rho$, $\rho_{1}\leq\rho<1$, such that 1-\rho<\omega^{-1}\big{(}\frac{c}{4}\big{)} we obtain

[TABLE]

which implies the injectivity of $U$ in $\overline{B}\setminus B_{\rho}$. Consequently, by Theorem 3.2, $\det DU\neq 0$ in $B\setminus\overline{B_{\rho}}$ and the thesis follows. ∎

Proof of Theorem 3.4.

In view of the already quoted Monodromy Theorem, it suffices to show that $\det DU\neq 0$ everywhere in $B$.

For every $r\in(0,1)$, let us write $\Phi^{r}:\partial B_{r}\to\mathbb{R}^{2}$ to denote the application given by

[TABLE]

It is obvious, by interior regularity of $U$, that $\Phi^{r}$ belongs to $C^{1,\alpha}$. On the other hand, by Lemma 3.5, there exists $\rho\in(0,1)$ such that for every $r\in(\rho,1)$ the mapping $\Phi^{r}:\partial B_{r}\to\gamma_{r}\subset\mathbb{R}^{2}$ is a diffeomorphism of $\partial B_{r}$ onto a simple closed curve $\gamma_{r}$. Now, when restricted to $\overline{B_{r}}$, $U$ solves (1.2) with $\Phi$ replaced by $\Phi^{r}$, and $B$ by $B_{r}$ . Then, up to a rescaling of coordinates, Lemma 3.1 is applicable and we obtain, in combination with Theorem 3.2,

[TABLE]

Finally, by Lemma 3.5 we have $\det DU\neq 0$ in $B\setminus\overline{B_{\rho}(0)}$ so that $\det DU\neq 0$ everywhere in $B$. ∎

We now conclude the proof of the main Theorem 1.1.

Proof of Theorem 1.1.

Having assumed $\det DU>0$ on $\partial B$, by continuity, one can find $0<\rho<1$, sufficiently close to $1$ such that $\det DU>0$ on $\overline{B}\setminus B_{\rho}$. By Theorem 3.4, we have that $U$ is a global homeomorphism, and that $\det DU>0$ in $B$. Consequently, $\det DU>0$ on all of $\overline{B}$ and the thesis follows. ∎

4. An improvement

In accordance with [4], we prove a variation of Theorem 1.1. First, we recall the following:

Definition 4.1**.**

Given a Jordan domain $D$, let us denote by ${\rm co}(D)$ its convex hull. We define the convex part of $\partial D$ as the closed set $\gamma_{c}=\partial D\cap\partial({\rm co}(D))$. Consequently we define the non–convex part of $\partial D$ as the open subset $\gamma_{nc}=\partial D\setminus\partial({\rm co}(D))$.

Theorem 4.2**.**

Under the assumptions of Theorem 1.1, if

[TABLE]

where $\gamma_{nc}$ is the set introduced in Definition 4.1 above, then the mapping $U$ is a diffeomorphism of $\overline{B}$ onto $\overline{D}$.

It is worth noticing that, if $D$ is convex, then the condition (4.1) is void, which agrees with the known adaptations [8, 3] of the well–known Radó–Kneser–Choquet [17] to the equation (1.2).

Proof.

The proof follows the same line of [4, Theorem 5.2], the only change is that the classical Zaremba–Hopf Lemma for harmonic functions must be replaced by its appropriate adaptation to divergence structure equations with Hölder coefficients, which is due to Finn and Gilbarg [13]. We omit the details. ∎

Acknowledgements G.A. was supported by Università degli Studi di Trieste FRA 2016, V.N. was supported by Fondi di Ateneo Sapienza “Metodi di Analisi Reale e Armonica per problemi stazionari ed evolutivi”.