On the Theorem of Univalence on the Boundary

Abstract

We give several generalizations of a known theorem from complex analysis, namely the univalence on the boundary theorem. Starting from a purely topological result (Theorems 1 and 11), we obtain univalence conditions for Sobolev mappings.

1. Introduction

The well-known theorem of univalence on the boundary says that if $Q\subset \mathbb{C}$ is open, $D\subset \subset Q$ is a Jordan domain bounded by a rectifiable Jordan curve, $f\in H\left(Q\right)$ and f is injective on $\partial D$, then it results that f is injective on D. Stoilow proved that such a theorem has a purely topological character. He proved in [1] that if $D\subset \mathbb{C}$ is a Jordan domain bounded by a Jordan curve, $f\in C(\overline{D},\mathbb{C})$ is injective on $\partial D$, and f is open and light on D, then it results that f is injective on $\overline{D}$. An old conjecture of Whyburn says that this theorem may be valid for $n\ge 3$. Independently, Avkhadiev [2] and Cristea [3] showed the following:

Theorem 1. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component and let $f\in C(\overline{D},{\mathbb{R}}^n)$ be open, discrete on D and injective on $\partial D$. Then it results that f is injective on $\overline{D}$.

We say that an open, discrete mapping is an interior mapping. The oldest condition of interiority is a result of Titus and Young [4], which says that if $n\ge 2$, $D\subset {\mathbb{R}}^n$ is open and $f\in C^1(D,{\mathbb{R}}^n)$ is a light mapping such that $J_f\left(x\right)\ge 0$ a.e., then f is an interior mapping. The result is improved in Corollary 3 in [5], where it is shown that if $n\ge 2$, $D\subset {\mathbb{R}}^n$ is open, $f\in C(D,{\mathbb{R}}^n)$ is a light mapping, $K\subset D$ is such that ${\mu }_n\left(f\left(K\right)\right)=0$ and f is Fréchet differentiable on $D\setminus K$ and $J_f\left(x\right)\ge 0$ on $D\setminus K$, then it results that f is an interior mapping. Here ${\mu }_n$ is the Lebesgue measure in ${\mathbb{R}}^n$. Also, f is light and sense-preserving, if and only if f is open and discrete (see [6]).

The conditions of openness, sense-preserving nature and condition $\left(N\right)$ are essential in the proof of Theorem 11, while the lightness condition is one of the basic tools in the proof of Theorem 13. The sense-preserving nature condition holds if f has a.e. quasidifferential $L_x$ with det $L_x\ge 0$ a.e. (or if f has a weak differential a.e. and $J_f\left(x\right)\ge 0$ a.e.), and this condition is used in Theorem 12.

We shall prove some univalence on the boundary theorems for open, sense-preserving mappings, or for open mappings $f:D\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ having a.e. a quasidifferential $L_x$ with det $L_x\ge 0$.

It must be mentioned that in his seminal paper in the theory of mathematical models in Nonlinear Elasticity, Ball [7] proved a univalence on the boundary theorem for some special classes of Sobolev mappings.

The theory of elasticity is concerned with deformations $f:D\subset {\mathbb{R}}^n\to V\subset {\mathbb{R}}^n$ of a material body, and hyperelastic materials are the ones that possess and store energy $E:D\times V\times {\mathbb{R}}^{2n}\to \mathbb{R}$. The associated energy of f is given by

$$W\left(f\right)={\displaystyle \underset{D}{\int }}E(x,f\left(x\right),f^{\prime }\left(x\right))dx<\infty$$

and is defined by a suitable class of Sobolev mappings.

By the virtue of the principle of non-penetration of matter, the primary goal is to find injective mappings $f:D\to V$ of smaller energy. Ball [7] proved that the minimizer of the energy is injective, studying the injectivity of this minimizer only on $\partial D$ and in the Sobolev class $W_{loc}^{1,q}(D,V),q>n$.

We shall prove in Theorem 17 that if such a minimizer of the energy is in the Sobolev class $W_{loc}^{1,n}(D,{\mathbb{R}}^n)$, and is open and injective on $\partial D$, then f is injective on $\overline{D}$.

Also, a known class of mappings of finite distortion $f:D\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ which are open and discrete is the class such that $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$ and $K_0(·,f)\in L_{loc}^p\left(D\right)$, where $p>n-1$ if $n\ge 3$, $p=1$ if $n=2$ (see [6]).

The concepts used above will be explained in Section 2.

2. Notations and Definitions

Let $Q\subset \mathbb{C}$ be open. We set $H\left(Q\right)=\{f:Q\to \mathbb{C}$ analytic function}. Let $D\subset {\mathbb{R}}^n$ be open. We set $C(\overline{D},{\mathbb{R}}^n)=\{f:\overline{D}\to {\mathbb{R}}^n|f$ is continuous} and $C^1(D,{\mathbb{R}}^n)=\{f:D\to {\mathbb{R}}^n$ continuously Fréchet differentiable}.

If $D\subset {\mathbb{R}}^n$ is open and bounded, $f\in C(\overline{D},{\mathbb{R}}^n)$ and $p\notin f(\partial D)$, we denote by $d(f,D,p)$ the topological degree of f at p (see [8] for more information about the topological degree). If $Q\subset {\mathbb{R}}^n\setminus f(\partial D)$ is a domain, then $d(f,D,p)=d(f,D,q)$ for every $p,q\in Q$ and we denote this common value by $d(f,D,Q)$.

Let $D\subset {\mathbb{R}}^n$ be open and $f:D\to {\mathbb{R}}^n$. We say that f is open if f carries open sets into open sets, we say that f is discrete if either $f^{-1}\left(y\right)=\varphi$, or $f^{-1}\left(y\right)$ is a discrete set in D and we say that f is a light mapping if $dim$$f^{-1}\left(y\right)\le 0$ for every $y\in {\mathbb{R}}^n$. Here $dim$$f^{-1}\left(y\right)$ is the topological dimension of $f^{-1}\left(y\right)$ if $y\in {\mathbb{R}}^n$. If f is discrete at a point $x\in D$ (i.e., there exists $Q\in \mathcal{V}\left(x\right)$ such that $f^{-1}\left(f\left(x\right)\right)\cap Q=\left\{x\right\}$), then $d(f,U_1,f\left(x\right))=d(f,U_2,f\left(x\right))$ for every $U_1,U_2\in \mathcal{V}\left(x\right)$ such that $U_1\cup U_2\subset Q$, and we denote this common value by $i(f,x)$. If $x\in D$, we set $\mathcal{V}\left(x\right)=\{U\subset {\mathbb{R}}^n$ open $|x\in U\}$. If $p\ge 1$ and $D\subset {\mathbb{R}}^n$ is open, we set $W_{loc}^{1,p}(D,{\mathbb{R}}^n)$, the Sobolev space of all mappings $f:D\to {\mathbb{R}}^n$, which are locally in $L^p$, together with their first order distributional derivatives. A mapping $f:D\to {\mathbb{R}}^n$ is of finite distortion if $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$, $J_f\in L_{loc}^1\left(D\right)$ and there exists $K:D\to [0,\infty ]$${\mu }_n$-measurable and finite a.e. such that $|f^{\prime }{\left(x\right)|}^n\le K\left(x\right)J_f\left(x\right)$ a.e. We set the outer dilatation $K_0(x,f)={\displaystyle \frac{|f^{\prime }{\left(x\right)|}^n}{J_f\left(x\right)}}$ if $J_f\left(x\right)\ne 0$, $K_0(x,f)=1$ if $J_f\left(x\right)=0$ and $x\in D$, and the inner dilatation $K_I(x,f)={\displaystyle \frac{J_f\left(x\right)}{l{\left(f^{\prime }\left(x\right)\right)}^n}}$ if $J_f\left(x\right)\ne 0$, $K_I(x,f)=1$ if $J_f\left(x\right)=0$ and $x\in D$.

If in addition, $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$, we say that f is of finite dilatation.

If $A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)$, we set $l\left(A\right)=\underset{\left|x\right|=1}{inf}\left|A\left(x\right)\right|$.

If $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)\cap C(D,{\mathbb{R}}^n)$, then f has a.e. first partial derivatives and we set $S_f\in \{x\in D|f$ has first partial derivatives at $x\}$.

If $x\in S_f$, we set $f^{\prime }\left(x\right)={({\displaystyle \frac{\partial f_i}{\partial x_j}}\left(x\right))}_{i,j=1,...,n}$, the matrix of the first partial derivative of f at x. If $x=(x_1,...,x_n)\in {\mathbb{R}}^n$, we set $\left|x\right|={({\displaystyle {\displaystyle \sum _{i=1}^n}}{x}_i^2)}^{{\displaystyle \frac{1}{2}}}$.

Let $D\subset {\mathbb{R}}^n$ be open and $f:D\to {\mathbb{R}}^n$. We say that f satisfies condition $\left(N\right)$ if ${\mu }_n\left(f\left(A\right)\right)=0$ if $A\subset D$ and ${\mu }_n\left(A\right)=0$.

Let $D\subset {\mathbb{R}}^n$ be a domain and $f\in C(D,{\mathbb{R}}^n)$. We say that f is sense-preserving if $d(f,G,p)>0$ for every $p\in f\left(G\right)\setminus f(\partial G)$ and for every $G\subset \subset D$. Here, we say that $G\subset \subset D$ if $G\subset D$ is open, $\overline{G}$ compact and $\overline{G}\subset D$. If f is discrete, the sense-preserving nature of the mapping f means that $i(f,x)\ge 1$ for every $x\in D$.

We say that f is non-singular if $Int$$f\left(U\right)\ne \varphi$ for every open $U\subset D$, $U\ne \varphi$. If $y\in {\mathbb{R}}^n$, we set $N(y,f,D)=Card$$f^{-1}\left(y\right)\cap D$.

We denote by $B_f=\{x\in D|f$ is not a local homeomorphism at $x\}$.

Let $D\subset {\mathbb{R}}^n$ be open, $x\in D$, $f\in C(D,{\mathbb{R}}^n)$ and $L_x\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)$. We say that $L_x$ is a quasidifferential of f at x if there exist $r_m\to 0$ such that for every $ϵ>0$ there exists $m_{ϵ}\in \mathbb{N}$ such that $|f\left(z\right)-f\left(x\right)-L_x(z-x)|\le ϵ|z-x|$ if $|z-x|=r_m$ for every $m\ge m_{ϵ}$.

If $p>n-1$ and $f\in W_{loc}^{1,p}(D,{\mathbb{R}}^n)$, then f has a quasidifferential L a.e. in points $x\in D$ (see Theorem 5.21 in [9]). A quasidifferential of f at x may be not unique. Indeed, let $f:\mathbb{R}\to \mathbb{R}$, $f\left(x\right)=xsin{\displaystyle \frac{1}{x}}$ if $x\ne 0$, $f\left(0\right)=0$. Then every $t\in [-1,1]$ is a quasidifferential of f at 0.

Let now $D\subset {\mathbb{R}}^n$ be open, $x\in D$ and $f:D\to {\mathbb{R}}^n$ such that f has first partial derivatives of f at x. If f has in addition a quasidifferential $L_x$ at x, then $L_x\left(e_i\right)={\displaystyle \frac{\partial f}{\partial x_i}}\left(x\right)$, $i=1,...,n$, where $e_1,...,e_n$ is the Euclidean base in ${\mathbb{R}}^n$, and hence $L_x=f^{\prime }\left(x\right)$ and det $L_x=J_f\left(x\right)$. Here $f^{\prime }\left(x\right)$ is the matrix of the first partial derivatives of f at x.

Theorem 2. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ a domain and let $f\in C(D,{\mathbb{R}}^n)$ be open, satisfying condition $\left(N\right)$, such that f has a.e. at points $x\in D$ a quasidifferential $L_x$ such that det $L_x\ge 0$. Then f is sense-preserving.

The following result is proved in [10,11]:

Theorem 3. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ a domain and let $f:D\to {\mathbb{R}}^n$ be a mapping of finite dilatation such that $f^{-1}\left(y\right)$ is either empty or a compact subset of D. Suppose that either $K_I(·,f)\in L_{loc}^1\left(D\right)$, or $K_0(·,f)\in L_{loc}^{n-1}\left(D\right)$. Then f is open and discrete.

The following local inversion theorem is proved in [12]:

Theorem 4. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ open and $K\subset D$ such that $K=\varphi$ if $n=2$, K is at most countable if $n\ge 3$. Let $f\in C(D,{\mathbb{R}}^n)$ be Fréchet differentiable on $D\setminus K$ such that $J_f\left(x\right)\ne 0$ on $D\setminus K$. Then f is a local homeomorphism on D and K may be dense in D.

The following local inversion theorem is proved in [13]:

Theorem 5. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ open, let $x\in D$ and $f\in C(D,{\mathbb{R}}^n)\cap W_{loc}^{1,1}(D,{\mathbb{R}}^n)$ be such that there exist $r,\delta >0$ such that $l\left(A\right)>\delta$ for every $A\in co\left(\{f^{\prime }\left(y\right):y\in (D\setminus S_f)\cap B(x,r)\}\right)$.

Then there exists $V\in \mathcal{V}\left(f\right(x\left)\right)$ such that $f\left|B\right(x,r):B(x,r)\to V$ is a homeomorphism.

Remark 1. 

The following univalence on the border theorem for domains D having multiple components of $\partial D$ is proved in [14]:

Theorem 6. 

Let $n\ge 2$, let $D\subset {\mathbb{R}}^n$ be a bounded domain such that there exist $m\ge 2$, $L_1,...,L_m$ continua such that $Int{L}_i=\varphi$, $\partial D={\displaystyle \bigcup _{i=1}^m}{L}_i$ and ${\mathbb{R}}^n\setminus L_i$ has exactly one bounded component ${\Delta }_i$, $i=1,...,m$. Suppose that $\overline{{\Delta }_i}\subset {\Delta }_1$, $i=2,...,m$, $\overline{{\Delta }_i}\cap {\overline{\Delta }}_j=\varphi$ for $i\ne j$, $i,j=2,...,m$, $D={\Delta }_1\setminus {\displaystyle \bigcup _{i=2}^m}{\overline{\Delta }}_i$ and $d(f,{\Delta }_i,Q_i)=1$ for $i=2,...,m$, where $Q_i$ is the unique component of ${\mathbb{R}}^n\setminus f\left(L_i\right)$, $i=2,...,m$. Let $f\in C(\overline{D},\overline{{\mathbb{R}}^n})$ be open and discrete such that $i(f,x)\ge 1$ for every $x\in D$ and f is injective on $\partial D$. Then f is injective on D.

The condition “$d(f,{\Delta }_i,Q_i)=1$, $i=2,...,m$” cannot be omitted in Theorem 6, as we see from [2]. This implies that for mappings $f\in C(\overline{D},\overline{{\mathbb{R}}^n})$, which are open and discrete on D and injective on $\partial D$, and ${\mathbb{R}}^n\setminus \partial D$ has m components, $m\ge 3$, we may not always obtain the injectivity of the mapping f on the whole domain D. We restrict our research only for domains $D\subset {\mathbb{R}}^n$ such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components.

We used the following results concerning degree theory:

Theorem 7 

(Theorem 2.1 in [8]). Let $f\in C(\overline{D},{\mathbb{R}}^n)$ and $p\notin f(\partial D)$. Then there exists $x\in D$ such that $f\left(x\right)=p$.

Theorem 8 

(Theorem 2.4. in [8]). Let $f,g\in C(\overline{D},{\mathbb{R}}^n)$ such that $f\left(x\right)=g\left(x\right)$ for every $x\in \partial D$ and let $p\notin f(\partial D)$. Then $d(f,D,p)=d(g,D,p)$.

Theorem 9 

(Theorem 2.7 in [8]). Let $f\in C(\overline{D},{\mathbb{R}}^n)$, $p\notin f(\partial D)$ and $f^{-1}\left(p\right)\subset {\displaystyle \bigcup _{i\in \mathbb{N}}}{D}_i$, where ${\left(D_i\right)}_{i\in \mathbb{N}}$ are open disjoint subsets of D. Then $d(f,D,p)={\displaystyle \sum _{i\in \mathbb{N}}}d(f,D_i,p)$.

Theorem 10 

(Theorem 2.10 in [8]). Let $f\in C(\overline{D},{\mathbb{R}}^n$, $g\in C({\mathbb{R}}^n,{\mathbb{R}}^n)$, let $\Delta ={\mathbb{R}}^n\setminus f(\partial D)$ and ${\left({\Delta }_i\right)}_{i\in \mathbb{N}}$ the components of Δ and let $p\notin g\left(f\right(\partial D\left)\right)$. Then $d(g\circ f,D,p)={\displaystyle \sum _{i=1}^{\infty }}d(g({\Delta }_i,p)d(f,D,{\Delta }_i)$.

The strategy used in this paper is to prove a quite topological result in Theorem 11 and also use Theorem 1, to obtain applications under different sufficient conditions (Sobolev mappings, mappings having a.e. a quasidifferential $L_x$ with $det{L}_x\ge 0$ a.e.).

3. The Main Results

Theorem 11. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ be a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component and let $f:\overline{D}\to {\mathbb{R}}^n$ be continuous. Suppose that f satisfies condition $\left(N\right)$ and is sense-preserving on D and injective on $\partial D$. Then ${\mathbb{R}}^n\setminus f(\partial D)$ has exactly one bounded component denoted by Q, $f\left(D\right)\supset Q$ and there exists a set $K\subset Q$ with ${\mu }_n\left(K\right)=0$, such that $Card$$f^{-1}\left(y\right)\cap D=1$ for every $y\in Q\setminus K$. Also, $f^{-1}\left(y\right)$ has a single component for every $y\in Q$. If f is non-singular, then $Int$$f^{-1}\left(K\right)=\varphi$ and f is injective on $D\setminus f^{-1}\left(K\right)$ and if f is open, then $f\left(D\right)=Q$ and f is injective on $\overline{D}$.

Proof. 

Let $g:f(\partial D)\to \partial D$ be the inverse of $f|\partial D:\partial D\to f(\partial D)$ and let $G:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be continuous such that $G\left|f\right(\partial D)=g$. We see from Theorems 2.1.4 and 2.3.1 in [8] that

$$1=d(I{d}_{{\mathbb{R}}^n},D,D)=d(G\circ f,D,D)=d(f,D,Q)d(G,Q,D)$$

and hence

$$d(f,D,Q)=\pm 1$$

(1)

We see from Theorem 2.1.1 in [8] that $f\left(D\right)\supset Q$.

Let $j\in \mathbb{N}$ be fixed. Using Vitali’s theorem, we find a set $E_j$ with ${\mu }_n\left(E_j\right)=0$ and disjoint balls $B_{ij}=B(x_{ij},r_{ij})$ with $0<r_{ij}<{\displaystyle \frac{1}{2^j}}$ for every $i\in \mathbb{N}$, such that $D={\displaystyle \bigcup _{i=1}^{\infty }}{B}_{ij}\cup E_j$. Let $F_j={\displaystyle \bigcup _{i=1}^{\infty }}S(x_{ij},r_{ij})$. Then ${\mu }_n\left(F_j\right)=0$, and since f satisfies condition $\left(N\right)$, we see that ${\mu }_n\left(C_j\right)=0$, where $C_j=f(E_j\cup F_j)$.

Let $y\in Q\setminus C_j$. Then $f^{-1}\left(y\right)\subset {\displaystyle \bigcup _{i=1}^{\infty }}{B}_{ij}$ and there exists $I_j\subset \mathbb{N}$ finite such that $f^{-1}\left(y\right)\subset {\displaystyle \bigcup _{i\in I_j}}{B}_{ij}$. Using Theorems 2.1.3 and 2.2.1 in [8], we find that

$$\pm 1=d(f,D,Q)=d(f,D,y)={\displaystyle \sum _{i\in I_j}}d(f,B_{ij},y).$$

Since f is sense-preserving and $y\in f\left(B_{ij}\right)\setminus f(\partial B_{ij})$ for every $i\in I_j$, we find that $d(f,B_{ij},y)\ge 1$ for every $i\in I_j$. It results that $d(f,D,Q)=1$ and there exists a single ball $B_{i\left(j\right),j}$ such that $f^{-1}\left(y\right)\subset B_{i\left(j\right),j}$ and $diam\left(B_{i\left(j\right),j}\right)\le {\displaystyle \frac{1}{2^j}}$ for every $j\in \mathbb{N}$. Let $K={\displaystyle \bigcup _{j=1}^{\infty }}{C}_j$ and $y\in Q\setminus K$. We find in this way balls $B_{i\left(j\right),j}$ such that $f^{-1}\left(y\right)\subset B_{i\left(j\right),j}$ and $diam\left(B_{i\left(j\right),j}\right)\le {\displaystyle \frac{1}{2^j}}$ for every $j\in \mathbb{N}$ and hence there exists a single point $x\in D\setminus f^{-1}\left(K\right)$ such that $f\left(x\right)=y$.

In the same way we prove that $f^{-1}\left(y\right)$ has a single component for every $y\in Q$.

If f is non-singular, then $Int$$f^{-1}\left(K\right)=\varphi$. Indeed, if there exists an open $U\subset f^{-1}\left(K\right)$, $U\ne \varphi$, then $f\left(U\right)\subset K$, $Int$$f\left(U\right)\ne \varphi$ and ${\mu }_n\left(K\right)=0$ and we reach a contradiction. We proved that if f is non-singular, then $Int$$f^{-1}\left(K\right)=\varphi$ and f is injective on $D\setminus f^{-1}\left(K\right)$.

Suppose now that f is open and let W be the unbounded component of ${\mathbb{R}}^n\setminus f(\partial D)$, and suppose that $W\cap f\left(D\right)\ne \varphi$. Such components W exist since $f(\partial D)$ is a compact set. Since f is open, then $\partial f\left(D\right)\subset f(\partial D)$, and hence $W\cap \partial f\left(D\right)=\varphi$. We have $W=(W\cap f\left(D\right))\cup (W\cap \complement \overline{f\left(D\right)})$, $W\cap f\left(D\right)$ is open and W is connected and hence $f\left(D\right)\supset W$. Since W is unbounded and $f\left(\overline{D}\right)$ is compact, we reach a contradiction. We proved that $W\cap f\left(D\right)=\varphi$ and since f is open, we see that $f\left(D\right)=Q$.

Let us show that f is injective on $\overline{D}$. Indeed, if not, we find points $x_1,x_2\in \overline{D}$, $x_1\ne x_2$ such that $f\left(x_1\right)=f\left(x_2\right)=y$ and we can suppose that $x_1\in D$. Let $U_1\in \mathcal{V}\left(x_1\right)$ be such that $x_2\notin {\overline{U}}_1$ and there exists $V\in \mathcal{V}\left(y\right)$ such that $f\left(U_1\right)=V$. Since f is continuous at $x_2$, we can find $U_2\in \mathcal{V}\left(x_2\right)$ such that ${\overline{U}}_1\cap {\overline{U}}_2=\varphi$ and $f(U_2\cap D)\subset V$. Since $f(U_2\cap D)$ is open and non-empty and ${\mu }_n\left(K\right)=\varphi$, we can find a point $w\in f(U_2\cap D)\setminus K$. Let $z_1,z_2\in D\setminus f^{-1}\left(K\right)$, $z_1\in U_1$, $z_2\in U_2$ be such that $f\left(z_1\right)=f\left(z_2\right)=w$. We reached a contradiction, since we proved that f is injective on $D\setminus f^{-1}\left(K\right)$ and $z_1,z_2$ are distinct points in $D\setminus f^{-1}\left(K\right)$ and $f\left(z_1\right)=f\left(z_2\right)=w$.

We therefore proved that f is injective on $\overline{D}$. □

Remark 2. 

The mapping f is non-singular if either f is light, or if there exists a dense subset $H\subset D$ such that f is open at every point $x\in H$. The last case may hold if for every $x\in H$ there exists a quasidifferential $L_x$ at x such that $det$$L_x\ne 0$ (see Lemma 2.1 in [15]).

Theorem 12. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component. Let $f:\overline{D}\to {\mathbb{R}}^n$ be continuous, satisfying condition $\left(N\right)$ on D and such that f has a.e. at points $x\in D$ a quasidifferential $L_x$ with $det$$L_x\ge 0$ and suppose that f is open on D and injective on $\partial D$. Then f is injective on $\overline{D}$.

Proof. 

We see from Theorem 1 in [16] that f is sense-preserving on D and we apply Theorem 11. □

Theorem 13. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ be a domain, let $K\subset D$ be closed such that $Int$$K=\varphi$ and let $f\in C(D,{\mathbb{R}}^n)$ be light such that $Int$$f\left(K\right)=\varphi$ and f is sense-preserving on $D\setminus K$. Then f is open and discrete.

Proof. 

Let $x\in D$ and $U\in \mathcal{V}\left(x\right)$ be such that $\overline{U}\subset D$, $\overline{U}$ is compact and $f\left(x\right)\notin f(\partial U)$. Let $W\in \mathcal{V}\left(f\right(x\left)\right)$ be such that $W\cap f(\partial U)=\varphi$ and let $Q=U\cap f^{-1}\left(W\right)$. Since f is a light mapping, it is non-singular (see [17]) and let $w\in f\left(Q\right)\setminus f\left(K\right)$. We see that $w\notin f(\partial U)$, $w\in f(U\setminus K)$, $\partial (U\setminus K)\subset \partial U\cup (U\cap K)$ and hence $w\notin \partial f(U\setminus K)\subset f(\partial U)\cup f\left(K\right)$. Let $p\in W$. Then, since f is sense-preserving on $D\setminus K$, we have

$$d(f,U,p)=d(f,U,w)=d(f,U\setminus K,w)\ge 1.$$

(2)

It results from Theorem 2.1.1 in [8] that $d(f,U,W)\ge 1$ and $f\left(U\right)\supset W$ and hence that f is open at x.

Let us show that f is discrete. Let $x\in D$, $U\in \mathcal{V}\left(x\right)$ such that $\overline{U}$ is compact, $\overline{U}\subset D$, $f\left(x\right)\notin f(\partial U)$ and let $m=d(f,U,f(x\left)\right)\ge 1$. Let $W\in \mathcal{V}\left(f\right(x\left)\right)$ be such that $W\cap f(\partial U)=\varphi$, let $Q=U\cap f^{-1}\left(W\right)$ and suppose that we can find distinct points $x_1,...,x_{m+1}$ in Q such that $f\left(x_i\right)=f\left(x\right)$, $i=1,...,m+1$. Let $U_i\in \mathcal{V}\left(x_i\right)$ be such that ${\overline{U}}_i$ is compact, ${\overline{U}}_i\subset Q$, $f\left(x_i\right)\notin f(\partial U_i)$, $i=1,...,m+1$ and ${\overline{U}}_i\cap {\overline{U}}_j=\varphi$ for $i,j=1,...,m+1$, $i\ne j$. Let $V\in \mathcal{V}\left(f\right(x\left)\right)$ be such that $V\cap f(\partial U_i)=\varphi$, $V\subset W$ and, as before, we see that $f\left(U_i\right)\supset V$ and $d(f,U_i,V)\ge 1$ for $i=1,...,m+1$. Let $p\in V\setminus f\left(K\right)$. We see that $p\in f(U_i\setminus K)\setminus \partial f(U_i\setminus K)$, $i=1,...,m+1$ and we have $m=d(f,U,f\left(x\right))=d(f,U,p)=d(f,U\setminus K,p)=d(f,(U\setminus K)\setminus {\displaystyle \bigcup _{i=1}^{m+1}}{\overline{U}}_i,p)+{\displaystyle \sum _{i=1}^{m+1}}d(f,U_i\setminus K,p)\ge {\displaystyle \sum _{i=1}^{m+1}}d(f,U_i,V)\ge m+1$ and we reach a contradiction. We therefore prove that f is open and discrete on D. □

Theorem 14. 

Let $n\ge 2$, let $D\subset {\mathbb{R}}^n$ be a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component and let $K\subset D$ be closed with $Int$$K=\varphi$. Let $f\in C(\overline{D},{\mathbb{R}}^n)$ be light such that $Int$$f\left(K\right)=\varphi$ and suppose that f is open and satisfies condition $\left(N\right)$ on $D\setminus K$. Suppose also that f has a.e. at points $x\in D\setminus K$ a quasidifferential $L_x$ with $det$$L_x\ge 0$ and that f is injective on $\partial D$. Then f is injective on $\overline{D}$.

Proof. 

Using Theorem 1 in [16], we find that f is sense-preserving on $D\setminus K$. We apply now Theorem 13. □

Theorem 15. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component and let $K\subset D$ be such that $D\setminus K$ is connected and $Int$$K=\varphi$. Let $f\in C(\overline{D},{\mathbb{R}}^n)\cap W_{loc}^{1,1}(D,{\mathbb{R}}^n)$ be light such that $Int$$f\left(K\right)=\varphi$ and f is injective on $\partial D$. Suppose that

• For every $x\in D\setminus K$, there exist $r_x,{\delta }_x>0$ such that $l\left(A\right)>{\delta }_x$ for every

$$co\left(\{f^{\prime }\left(y\right):y\in (D\setminus S_f)\cap B(x,r_x)\}\right).$$

(3)

Then, f is injective on $\overline{D}$.

Proof. 

We see from Theorem 3 in [13] that f is a local homeomorphism at every point $x\in D\setminus K$. It results that $B_f\subset K$, and hence $Int$$B_f=\varphi$, $Int$$f\left(B_f\right)=\varphi$ and $D\setminus B_f$ is connected. Since $D\setminus B_f$ is connected, we see that $i(f,·)$ has a constant nonvanishing sign on $D\setminus B_f$ and hence f is sense-preserving on $D\setminus B_f$. We now apply Theorem 13. □

Remark 3. 

Condition (3) holds if f is locally Lipschitz (and hence a.e. Fréchet differentiable) and det $A\ne 0$ for every $A\in \partial f\left(x\right)$ and every $x\in D\setminus K$. Here $\partial f\left(x\right)$ is Clarke’s derivative of f at x, i.e., $\partial f\left(x\right)=\{A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)|$ there exist $x_m\to x$ such that f is Fréchet differentiable at every $x_m$ and $f^{\prime }\left(x_m\right)\to A\}$ (see [18]).

We found in Example 1 in [13] an injective function $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ for which we can apply condition (3), and since $f^{\prime }$ is not locally bounded, we cannot apply Clarke’s local inversion theorem.

Theorem 16. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and let $K\subset D$ be at most countable. Let $f\in C(\overline{D},{\mathbb{R}}^n)\cap W_{loc}^{1,1}(D,{\mathbb{R}}^n)$ be such that f satisfies condition (3) at every point $x\in D\setminus K$ and f is injective on $\partial D$. Then f is injective on $\overline{D}$.

Proof. 

We see from Theorem 3 in [13] that $B_f$ and $f\left(B_f\right)$ are at most countable and hence f is a light mapping. We apply now Theorem 15. □

Theorem 17. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ be a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component and let $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)\cap C(\overline{D},{\mathbb{R}}^n)$ be open on D and such that $J_f\left(x\right)\ge 0$ a.e. Suppose that f is injective on $\partial D$. Then f is injective on $\overline{D}$.

Proof. 

We see from Theorem 5.2.1, page 129 in [9] that f has a.e. a weak differential which coincides with $f^{\prime }\left(x\right)$ and $J_f\left(x\right)\ge 0$ a.e. We also see from [19] that f satisfies condition $\left(N\right)$. We apply now Theorem 12. □

Remark 4. 

Theorem 17 is valid for open mappings of finite dilatation, i.e., for mappings f of finite distortion and such that $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$. We wish to point out that there exist mappings $f\in {\displaystyle \bigcap _{p<n}}{W}_{loc}^{1,p}(D,{\mathbb{R}}^n)$ with $J_f\left(x\right)>0$ a.e. and for which condition $\left(N\right)$ does not hold (see [20]).

Theorem 18. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ be a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component and let $f\in C(\overline{D},{\mathbb{R}}^n)$ be a mapping of finite dilatation such that f is injective on $\partial D$. Suppose that either $K_0(·,f)\in L_{loc}^{n-1}\left(D\right)$, or that $K_I(·,f)\in L_{loc}^1\left(D\right)$. Then f is injective on $\overline{D}$.

Proof. 

We see that $f^{-1}\left(y\right)$ is compact in D for every $y\in f\left(D\right)$. We use now Theorem 1.1 in [10] or Theorem 1 in [11] to show that f is open and discrete. We use now the classical result from Theorem 1 (Avkhadiev–Cristea) to show that f is injective on $\overline{D}$. □

Theorem 19. 

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ be a bounded domain such that ${\mathbb{R}}^n\setminus \partial D$ has exactly two components and D is its bounded component and let $K\subset D$ be such that $K=\varphi$ if $n=2$ and K is at most countable if $n\ge 3$. Let $f\in C(\overline{D},{\mathbb{R}}^n)$ be Fréchet differentiable on $D\setminus K$ with $J_f\left(x\right)\ne 0$ for every $x\in D\setminus K$ and suppose that f is injective on $\partial D$. Then f is injective on $\overline{D}$.

Proof. 

We see from [12] that f is a local homeomorphism on D and we apply Theorem 1 to show that f is injective on $\overline{D}$. □

Remark 5. 

Other univalence on the boundary theorems, given for domains $D\subset {\mathbb{R}}^n$ such that ${\mathbb{R}}^n\setminus \partial D$ may have infinitely many components, may be found in [2,4,14]. Some variants for local homeomorphisms on infinite dimensional Banach spaces may be found in [21,22]. We mention some papers where local homeomorphisms conditions are proven (see [23,24,25,26,27,28,29,30,31,32,33,34,35,36]).

We also mention the conditions of openness and discreteness for the mapping of finite distortion, starting with the book of Reshetniak [37] and the papers [16,17,38,39,40]. Other classes of mappings which generalize quasiregular mappings and are open and discrete are the classes of mappings with direct or inverse Poletsky inequality (see [41,42,43] and the monographs [44,45]).

We mention also the lightness conditions established in the seminal monographs of Stoilow [46] or Whyburn [47] (see also [15]).

These things show that the study of open, discrete mapping has an old and rich tradition, either in a purely topological case, or in an analytic case.

Remark 6. 

Some d-bar Neumann operators are studied in [48] in connection with the work of Ball [7].

4. Conclusions

In his celebrated book [46], Stoilow opened a new chapter in mathematics, namely the topological theory of analytic functions. He showed that some classical theorems from classical complex analysis are in fact topological theorems. His work was continued by Whyburn in [47] on general metric spaces. Our Theorem 1 and Theorem 11 belong to this old chapter in general topology.

We continue the research dedicated to the known theorem of univalence on the boundary. In [6], Ball studied situations in which interpenetration of matter in the interior of a body can be ruled using the deformations of its boundary. He studied this problem for a special class of Sobolev functions. We apply our results for open mappings in the $W^{1,n}$ class and our research may be used both in theoretical and applied mathematics.