Mappings of a Bounded Dirichlet Integral: The Modulus Method

Abstract

We study the geometric properties of some classes of mappings for which an inverse Poletsky modular inequality holds. In these classes of mappings, we give some extensions of the theorems of Lindelőf and Fatou from the classical complex analysis. We also find some conditions for the existence of injective minimizers for mappings of biconformal energy.

1. Introduction

Throughout this paper, $X,Y$ are locally compact and locally connected metric measure spaces endowed with Borel regular measures $\mu$ and $\nu$, which are finite on compact sets. Also, X is Ahlfors n-regular and $\nu$ is doubling and $D\subset X$ is an open subset of X. We set by d the distance on X and Y. If $f:D\to Y$ is a mapping and $x\in D$, we set

$$J_f(x)=\underset{r\to 0}{lim\; sup}{\displaystyle \frac{\nu (f(B(x,r)))}{\mu (B(x,r))}},$$

and we set

$$l(x,f)=\underset{y\to x}{lim\; inf}{\displaystyle \frac{d(f(y),f(x))}{d(x,y)}}andL(x,f)=\underset{y\to x}{lim\; sup}{\displaystyle \frac{d(f(y),f(x))}{d(y,x)}}.$$

We say that $f:D\to Y$ is of finite conformal energy if $\underset{D}{\int }L{(x,f)}^{n}d\mu <\infty$. If $p>1$ 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. Usually, if $X={\mathbb{R}}^n$, a mapping $f:D\to {\mathbb{R}}^n$ is of finite conformal energy (or of a bounded Dirichlet integral) if $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$ and $\underset{D}{\int }{|f^{\prime }(x)|}^{n}dx<\infty$. If f is a.e. differentiable, then $L(x,f)=|f^{\prime }(x)|$ a.e. and both definitions coincide in this case. The following example shows that our definition is more general, since we find a mapping $f:\Omega \subset {\mathbb{R}}^3\to {\mathbb{R}}^3$ a.e. differentiable such that $\underset{\Omega }{\int }{|f^{\prime }(x,y,z)|}^{3}d{\mu }_3<\infty$ and $f\notin W_{loc}^{1,1}(\Omega ,{\mathbb{R}}^3)$.

We denote by ${\mu }_n$ the Lebesgue measure in ${\mathbb{R}}^n$. If $A\subset {\mathbb{R}}^n$ is Lebesgue measurable and $f:A\to {\mathbb{R}}^n$ is Lebesgue integrable, we set $\underset{A}{\int }fd{\mu }_n=\underset{A}{\int }f(x)dx$.

Example 1.

Let $E\subset [0,1]$ be a Cantor set such that ${\mu }_1(E)>0$ and let $g:[0,1]\to [0,{\mu }_1(\complement E)]$ be given by $g(x)={\displaystyle \underset{0}{\overset{x}{\int }}}{\chi }_{\complement E}(t)dt$ for every $x\in [0,1]$. We see that g is absolutely continuous and strictly increasing and this inverse $h:[0,{\mu }_1(\complement E)]\to [0,1]$ is not absolutely continuous, is a.e. differentiable and $h^{\prime }(t)=1$ a.e. Let $f:B(0,1)\to \mathbb{C}$, $f(z)=z^n$ for every $z\in B(0,1)$, suppose that $n\ge 2$ and let $\Omega =(0,{\mu }_1(\complement E))\times B(0,1)$ and $F:\Omega \to {\mathbb{R}}^3,F(x_1,x_2,x_3)=(h(x_1),f(x_2,x_3))$. Then, ${\mu }_3(B_F)=0$, ${\mu }_3(F(B_F))=0$, $F\notin W_{loc}^{1,1}(\Omega ,{\mathbb{R}}^3)$, $\underset{\Omega }{\int }{|F^{\prime }(x,y,z)|}^{3}d{\mu }_3<\infty$ and F satisfies condition $(N^{-1})$.

Here, if $f:D\to Y$ is a mapping, we set $B_f=\{x\in D|f$ is not a local homeomorphism at $x\}$ and we say that f satisfies condition $(N)$ if $\nu (f(A))=0$ whenever $A\subset X$ is such that $\mu (A)=0$ and we say that f satisfies condition $(N^{-1})$ if $\mu (f^{-1}(A))=0$ whenever $A\subset Y$ is such that $\nu (A)=0$.

Let $D\subset X$. Let $A(D)$ be the set of all nonconstant path families in D and if $\Gamma \in A(D)$, and we set $F(\Gamma )=\{\rho :D\to [0,\infty ]$ Borel functions $|\underset{\gamma }{\int }\rho ds\ge 1$ for every $\gamma \in \Gamma$ locally rectifiable}.

If $p>1$ and $\omega :D\to [0,\infty ]$ is $\mu$ measurable and finite a.e., we define the p-modulus of weight $\omega$ by

$$M_{\omega }^p(\Gamma )=\underset{\rho \in F(\Gamma )}{inf}\underset{X}{\int }\omega (x)\rho {(x)}^{p}d\mu if\Gamma \in A(D).$$

If $F(\Gamma )=\varphi$, we set $M_{\omega }^p(\Gamma )=0$. If $\omega =1$, we set

$$M_p(\Gamma )=\underset{\rho \in F(\Gamma )}{inf}\underset{X}{\int }\rho {(x)}^{p}d\mu if\Gamma \in A(D).$$

One of the basic tools in studying quasiregular mappings is the modular inequality of Poletsky

$$M_n(f(\Gamma )\le K{M}_n(\Gamma )forevery\Gamma \in A(D).$$

(1)

We recommend the monographs [1,2,3,4] to the reader for more information about quasiregular mappings.

Several generalizations of quasiregular mappings were introduced in the last 30 years. The most important is the class of mappings of finite distortion (see monographs [5,6] for further information), and a Poletsky modular inequality holds in this class of mappings (see [7,8] or [9]). A mapping $f:D\to {\mathbb{R}}^n$ is of finite distortion if $n\ge 2$, $D\subset {\mathbb{R}}^n$ is open, $f\in W_{loc}^{1,1}(D,{\mathbb{R}}^n)$, $J_f\in L_{loc}^1(D)$ and there exists $K:D\to [0,\infty ]$ measurable and finite a.e. such that $|f^{\prime }{(x)|}^n\le K(x)J_f(x)$ a.e. and we set the outer dilatation

$$K_0(x,f)={\displaystyle \frac{|f^{\prime }{(x)|}^n}{J_f(x)}}if{J}_f(x)\ne 0,K_0(x,f)=1if{J}_f(x)=0.$$

Here, $f^{\prime }(x)$ is the matrix formed by the partial derivatives of f at x and $J_f(x)=det{f}^{\prime }(x)$ for a.e. $x\in D$ and we use this definition of the Jacobian $J_f$ for mappings of finite distortion.

Martio proposed the study of mappings distinguished by moduli inequalities, first on ${\mathbb{R}}^n$ and then on metric measure spaces. Such mappings satisfy a generalized Poletsky modular inequality of the type

$$M_q(f(\Gamma ))\le M_{\omega }^p(\Gamma )forevery\Gamma \in A(D).$$

(2)

In this class of mappings, Montel- and Picard-type theorems, boundary extension results, equicontinuity results and estimates of the modulus of continuity were given (see [7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]) and we recommend the monographs [22,28] to the reader for further information.

If $f:D\to Y$ is a mapping and $L(x,f,r)=\underset{y\in S(x,r)}{sup}d(f(x),f(y))$, $l(x,f,r)=\underset{y\in S(x,r)}{inf}d(f(x),f(y))$ for $x\in X$ and $r>0$, the linear dilatation $H(·,f):D\to [0,\infty ]$ is defined by $H(x,f)=\underset{r\to 0}{lim\; sup}{\displaystyle \frac{L(x,f,r)}{l(x,f,r)}}$ for every $x\in D$.

A mapping $f:D\to Y$ between general metric measure spaces is quasiregular if it is continuous, open and discrete, and there exists $H\ge 1$ such that $H(x,f)\le H$ for every $x\in D$. This is the metric definition of quasiregularity (see [29,30,31,32,33,34,35,36,37,38,39]). In some particular cases, a Poletsky modular inequality holds for quasiregular mappings defined on general metric measure spaces.

As in the classical case, even on very general metric measure spaces X and Y, a mapping $f:D\to Y$ which is continuous, open and discrete is quasiregular if and only if it is geometrically quasiregular, i.e., $M_n(\Gamma )\le K\underset{Y}{\int }N(y,f,G){\rho }^n(y)d\nu$ for every open set $G\subset \subset D$, every $\Gamma \in A(G)$ and every $\rho \in F(f(\Gamma ))$. This important property inspired the study of some classes of mappings satisfying a generalized inverse Poletsky modular inequality (see [12,28,40]), like

$$M_p(\Gamma )\le {\gamma }_G(M_{{\omega }_G}^q(f(\Gamma )))forevery\Gamma \in A(D)andeveryG\subset \subset D.$$

(3)

Here, $p,q>1$, $f:D\to Y$ is continuous, open and discrete, ${\omega }_G:Y\to [0,\infty ]$ is $\nu$ measurable and finite a.e. and ${\gamma }_G:(0,\infty )\to (0,\infty )$ is increasing and $\underset{t\to 0}{lim}{\gamma }_G(t)=0$ for every $G\subset \subset D$.

The class of mappings satisfying a relation of type (3) is strictly larger than the class of quasiregular mappings, where relation (3) holds for a particular weight ${\omega }_G(y)=N(y,f,G)$ for every $G\subset \subset D$ and every $y\in Y$. This thing is shown by the function $F:\Omega \to {\mathbb{R}}^3$ from Example 1. We see that $F\notin W_{loc}^{1,1}(\Omega ,{\mathbb{R}}^3)$ (and hence F is not quasiregular), and since $\underset{\Omega }{\int }{|F^{\prime }(x)|}^{3}dx<\infty$, we see from Lemma 1 that there exists $\omega \in L^1({\mathbb{R}}^3)$ such that $M_3(\Gamma )\le M_{\omega }^3(F(\Gamma ))$ for every $\Gamma \in A(\Omega )$. In fact, F is not a mapping of finite distortion.

The following class of mappings of finite conformal energy satisfies an inverse Poletsky modular inequality:

Theorem 1.

Let $n\ge 2$, $q>1$, let $f:D\to Y$ be continuous such that $\mu (B_f)=0$, $\nu (f(B_f))=0$, f satisfies condition $(N^{-1})$ and $\underset{D}{\int }L{(z,f)}^{q}d\mu <\infty$. Then, there exists $\omega \in L^1(Y)$ such that $M_q(\Gamma )\le M_{\omega }^q(f(\Gamma ))$ for every $\Gamma \in A(D)$.

Theorem 1 extends Lemma 1 from [17] given for open, discrete mappings between Riemannian n-manifolds. Our result is more general and is valid for non-open mappings, as shown in the example given by $f:B^2\to B^2$, $f(x,y)=(x,y)$ if $y>0$, $f(x,y)=(x-y)$ if $y<0$.

The aim of this paper is to study the geometric properties of such mappings. Let us consider the following class of mappings of finite distortion:

$$f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)suchthat{K}_0(·,f)\in L_{loc}^p(D),wherep>n-1ifn\ge 3,p=1ifn=2.$$

(4)

We see from [41,42] that such mappings $f:D\to {\mathbb{R}}^n$ are open, discrete and satisfy condition $(N^{-1})$ and that ${\mu }_n(B_f)=0$, ${\mu }_n(f(B_f))=0$. Using Theorem 1, we see that for every $G\subset \subset D$ there exists ${\omega }_G\in L^1({\mathbb{R}}^n)$ such that

$$M_n(\Gamma )\le M_{{\omega }_G}^n(f(\Gamma )))forevery\Gamma \in A(G).$$

(5)

Using Hőlder’s inequality, we can replace n by some $n-1<q<n$ in (5). We note that relation (5) is the first inverse Poletsky modular inequality given for an important class of mappings of finite distortion. We will establish, in a future paper, further geometric properties of this class of mappings using the modulus method.

A significant part of mathematical analysis is the study of geometric properties of mappings. In our paper, we establish some geometric properties of mappings of a bounded Dirichlet integral using the fact that such mappings satisfy an inverse Poletsky modular inequality.

First, we study the modulus of continuity of such mappings and we extend in this way Theorem 7.5.2 in [6] given for mappings $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$.

The main objective of this paper is to generalize important theorems from complex analysis and quasiregular mappings in the class of mappings of a bounded Dirichlet integral. These extensions are consistent due to the generality of such mappings and due to the fact that these extensions are established on general metric measure spaces. However, the main spaces we have in mind are Riemannian n-manifolds.

We extend in Theorem 3 the known theorems of Fatou and F. and M. Riesz from complex analysis.

Another important theorem from classical complex analysis is the Lindelöf-type result from Theorem 5 and Theorem 6, which extends some results of Vuorinen (see [4,43,44,45]).

We prove in Theorem 7 a pointwise $\eta$-quasisymmetry property of some homeomorphisms $f:D\to D_f$ between Riemannian n-manifolds satisfying a direct Poletsky modular inequality.

We say that a mapping $f:D\to Y$ is quasimonotone if there exists a constant $L>0$ (the coefficient of quasimonotonicity) such that for every set $K\subset A\subset \subset D$ with K compact and A open, it results that $d(f(K))\le Ld(f(\partial A))$. If $L=1$, we say that f is monotone. If $Y={\mathbb{R}}^n$ and f is open and continuous, then f is monotone.

Let $B^n$ be the unit ball from ${\mathbb{R}}^n$. An important theorem from classical complex analysis is the theorem of Fatou, which says that if $f:B^2\to \mathbb{C}$ is a bounded analytic mapping, then f has a.e. radial limits. Another theorem from classical complex analysis is the theorem of F. and M. Riesz, which says that if $f:B^2\to \mathbb{C}$ is a bounded analytic mapping and $w\in \mathbb{C}$, then, if $E_w=\{y\in S(0,1)|\underset{t\to 1}{lim}f(ty)=w\}$, it results that ${\mathcal{H}}^1(E_w)=0$. Here, if $p\ge 1$, we denote by ${\mathcal{H}}^p$ the p-dimensional Hausdorff measure.

We say that a mapping $f:B^n\to {\mathbb{R}}^n$ has a nontangential limit at a point $x\in \partial B^n$ if for every finite cone $\Gamma$ and ${\Gamma }_1$ with a vertex at x such that $\overline{\Gamma }\backslash \{x\}\subset {\Gamma }_1\subset B^n$, it results that there exists $\underset{\begin{array}{c}y\to x\\ y\in \Gamma \end{array}}{lim}f(y)\in {\mathbb{R}}^n$. Also, the mapping $f:B^n\to \mathbb{R}$ is monotone (in the sense of Lebesgue) if

$$\underset{x\in \overline{G}}{max}f(x)=\underset{x\in \partial G}{max}f(x)and\underset{x\in \overline{G}}{min}f(x)=\underset{x\in \partial G}{min}f(x)foreveryG\subset \subset B^n.$$

Using the modulus method, Manfredi and Villamor showed in [42,46,47] that if $n-1<p\le n$, $f:B^n\to {\mathbb{R}}^n$ has monotone components, $f\in W_{loc}^{1,p}(D,{\mathbb{R}}^n)$ and $\underset{B^n}{\int }{|f^{\prime }(x)|}^{p}dx<\infty$, then f has nontangential limits with the possible exception of a set of zero p-capacity (see [46,47]). Also, Miklyukov showed in [48] that if $f:B^n\to {\mathbb{R}}^n$ is a bounded quasiregular mapping and $\underset{B^n}{\int }{|f^{\prime }(x)|}^{n}dx<\infty$, then f has a.e. nontangential limits.

It is not yet known whether a bounded quasiregular mapping $f:B^n\to {\mathbb{R}}^n$, $n\ge 3$ has at least a radial limit. Using the modulus method, an important result related to the theorems of Fatou and M. and F. Riesz was obtained by Martio and Rickman in [49]. They showed that if $f:B^n\to {\mathbb{R}}^n$ is quasiregular and there exist $C>0$ and $0<\alpha <n-1$ such that $\underset{{\mathbb{R}}^n}{\int }N(y,f,B(0,r))\le C{(1-r)}^{\alpha }$ for every $0<r<1$, then f has a.e. radial limits, and if $w\in {\mathbb{R}}^n$ and $E_w=\{y\in S(0,1)|\underset{t\to 1}{lim}f(ty)=w\}$, they showed that ${\mathcal{H}}^{n-1}(E_w)=0$. Akkinen showed in [50] that if $f:B^n\to {\mathbb{R}}^n$ is a mapping of finite distortion with exponentially integrable outer distortion and there exist $C>0$ and $0<\alpha <n-1$ such that $\underset{B^n}{\int }{J}_f(x)dx<C{(1-r)}^{-\alpha }$ for every $0<r<1$, then f has a.e. radial limits.

We partially extend these results in Theorem 3 for mappings $f:D\to Y$ with Y complete and having doubling measure $\nu$ and such that there exists a homeomorphism $\Phi :{\overline{B}}^n\to \overline{D}$ such that $l(z,\Phi )\ge m>0$ and $L(z,\Phi )<M<\infty$ for every $z\in {\overline{B}}^n$. Due to the generality of homeomorphism $\Phi$ and of spaces X and Y, we obtain a rather general version of the theorems of Fatou and F. and M. Riesz. A version for a particular type of homeomorphism $\Phi$ and for open, discrete mappings $f:D\to Y$ between Riemannian n-manifolds is given in [17].

Let $D\subset {\mathbb{R}}^n$ be open, $q>1$ and $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$. We set

$$K_{I,q}(x,f)={\displaystyle \frac{J_f(x)}{l{(f^{\prime }(x))}^q}}if{J}_f(x)\ne 0,K_{I,q}(x,f)=0otherwise.$$

Here, if $A\in \mathcal{L}({\mathbb{R}}^n,{\mathbb{R}}^n)$, we set $l(A)=\underset{|x|=1}{inf}|A(x)|$. If, in addition, $f:D\to D_f$ is a homeomorphism, then f is a.e. differentiable and satisfies condition $(N)$. Let $Z_f=\{x\in D|f$ is differentiable at x and $J_f(x)=0\}$ and $A_f=\{x\in D|f$ is not differentiable at $x\}$. Using Sard’s lemma from [51], we see that ${\mu }_n(f(Z_f))=0$ and then ${\mu }_n(f(A_f\cup Z_f))=0$. It results that if $g:D_f\to D$ is the inverse of f, then g is a.e. differentiable and $J_g(y)\ne 0$ a.e. Using the change of variable formulae, we see that

$$\underset{D_f}{\int }|g^{\prime }{(y)|}^{q}dy=\underset{D\backslash A_f}{\int }|g^{\prime }{(f(x))|}^q|J_f(x)|dx=\underset{D}{\int }{K}_{I,q}(x,f)dx.$$

It results that $\underset{D}{\int }{K}_{I,q}(x,f)dz<\infty$ if and only if $\underset{D}{\int }|g^{\prime }(y)|dy<\infty$. If, in addition, f satisfies condition $(N^{-1})$, then f is a.e. differentiable and $J_f(x)\ne 0$ a.e.

The chordal distance between two points $a,b\in {\mathbb{R}}^n$ is the number

$q(a,b)={|a-b|(1+|a|}^2{)}^{-{\displaystyle \frac{1}{2}}}{(1+|b|}^2{)}^{-{\displaystyle \frac{1}{2}}}$ if $a,b\in {\mathbb{R}}^n$, $q(a,\infty )={(1+|a|}^2{)}^{-{\displaystyle \frac{1}{2}}}$ if $a\in {\mathbb{R}}^n$.

We shall apply our results to the mappings of finite conformal energy. Such mappings are homeomorphisms $f:D\to D_f$ between two domains from ${\mathbb{R}}^n$ such that $f\in W_{loc}^{1,n}(D,D_f)$ and $\underset{D}{\int }{|f^{\prime }(x)|}^{n}dx<\infty$. We set $E_D(f)=\underset{D}{\int }{|f^{\prime }(x)|}^{n}dx$ the energy of f and $E_{t,D}^q(f)=\underset{D}{\int }(|f^{\prime }(x){|}^n+K_{I,q}(x,f))dx$ the total energy of f and usually we take $q=n$.

If $f^{-1}\in W_{loc}^{1,n}(D_f,D)$ and $\underset{D_f}{\int }{|{(f^{-1})}^{\prime }(y)|}^{n}dy<\infty$, we say that f is of biconformal energy. Such mappings form an important class with connections to mathematical models of Nonlinear Elasticity (see [52,53,54]). The general task in mathematical models of elasticity is to find deformations of the smallest energy $E_D(f)$ or of the smallest total energy $E_{t,D}^q(f)$. These deformations are usually found as weak limits of minimizing sequences. However, in the limiting process, we may lose the injectivity of f, and we refer to this incident as permanent damage in the material. We accept the weak limits of energy-minimizing sequences of homeomorphisms as legitimate deformations and the minimal energy (usually attained) can be strictly smaller then the infimum energy among homeomorphisms.

We are interested to find injective minimizers of the energy $E_M^q(D)$ and of the total energy $E_t^q(D)$ in some classes of deformations.

Let $D\subset {\mathbb{R}}^n$ be a domain, $a,b\in D$, $a\ne b$, $n-1<q\le n$, $r>0$, $H_D^q=\{f\in W_{loc}^{1,n}(D,D_f)$ homeomorphisms such that $f^{-1}\in W_{loc}^{1,q}(D_f,D)$, $E_{t,D}^q(f)<\infty$ and $q(f(a),f(b))\ge r\}$ and let $H_{D,M}^q=\{f\in H_D^q|E_{t,D}^q(f)<M\}$. We set the total energy $E_t^q(D)=\underset{f\in H_D^q}{inf}{E}_{t,D}^q(f)$ and $E_{t,M}^q(D)=\underset{f\in H_{D,M}^q}{inf}{E}_{t,D}^q(f)$ and the energy $E_M^q(D)=\underset{f\in H_{D,M}^q}{inf}{E}_D(f)$.

We apply the preceding results in Theorem 8 and we find some conditions for the existence of injective minimizers for mappings of biconformal energy in connection with the seminal paper of Ball [55].

2. Notations and Definitions

We say that a set $E\subset {\mathbb{R}}^n$ is a cap of a sphere $S(x,t)$ in ${\mathbb{R}}^n$ if $E=H\cap S(x,t)$ where H is an open half-space in ${\mathbb{R}}^n$. We extend now some results from [12] and Theorem 6.4 from [43].

We say that a mapping $f:D\to Y$ is open if it carries open sets into open sets, we say that it is closed if it carries closed sets into closed sets and we say that it is discrete if either $f^{-1}(y)=\varphi$ or $f^{-1}(y)$ is a discrete subset of D. We set $G\subset \subset D$ if G is open, $\overline{G}$ is compact and $\overline{G}\subset D$.

Let $f:D\to Y$ be a mapping. We say that f is $\eta$-quasisymmetric if $\eta :(0,\infty )\to (0,\infty )$ is increasing and $\underset{t\to 0}{lim}\eta (t)=0$ and ${\displaystyle \frac{d(f(y),f(x))}{d(f(z),f(x))}}\le \eta ({\displaystyle \frac{d(y,x)}{d(z,x)}})$ for every distinct point $x,y,z\in D$. Such a continuous, open, discrete mapping is quasiregular, since $H(x,f)\le \eta (1)$ for every $x\in D$. If $X=Y={\mathbb{R}}^n$, every quasiconformal mapping $f:D\to f(D)$ is locally quasisymmetric.

If $\omega :D\to [0,\infty ]$ is $\mu$ measurable and A is a Borel subset of D, we set $\underset{A}{⨏}\omega (z)d\mu =\underset{A}{\int }\omega (z)d\mu /\mu (A)$, and if $x\in D$, we set ${\omega }_x=\underset{r\to 0}{lim\; sup}\underset{B(x,r)}{⨏}\omega (z)d\mu$.

If $E,F\subset \overline{D}$, we set $\Delta (E,F,D)=\{\gamma :[0,1]\to \overline{D}$ path $|\gamma (0)\in E,\gamma (1)\in F$ and $\gamma ((0,1))\subset D\}$. A domain $A\subset {\overline{\mathbb{R}}}^n$ is a ring if $\complement A$ has exactly two components, and if these components of $\complement A$ are $C_0$ and $C_1$, we denote $A=R(C_0,C_1)$ and we set ${\Gamma }_A=\Delta (C_0,C_1,A)$. Given $r>0$, we let ${\Phi }_n(r)$ be the set of all rings $A=R(C_0,C_1)$ such that $C_0$ contains the origin and a point a such that $|a|=1$ and $C_1$ contains ∞ and a point b such that $|b|=r$ and we denote ${\mathcal{H}}_n(r)=inf{M}_n({\Gamma }_A)$ over all rings $A\in {\Phi }_n(r)$. The function ${\mathcal{H}}_n(r):(0,\infty )\to (0,\infty )$ is decreasing and $\underset{r\to 0}{lim}{\mathcal{H}}_n(r)=\infty$, $\underset{r\to \infty }{lim}{\mathcal{H}}_n(r)=0$.

If $A=R(C_0,C_1)$ and $a,b\in C_0$, $c,\infty \in C_1$, then $M_n({\Gamma }_A)\ge {\mathcal{H}}_n({\displaystyle \frac{|c-a|}{|b-a|}})$ (see [3]).

Given $0<r\le 1$, $q>n-1$ and $0<t\le 1$, we let ${\psi }_{n,q}(r,t)$ be the set of all rings $A=R(C_0,C_1)$ such that $q(C_0)\ge r$, $q(C_1)\ge r$, $C_1$ is unbounded, $q(C_0,C_1)\le t$ and we set ${\lambda }_{n,q}(r,t)=inf{M}_q({\Gamma }_A)$ over all rings $A\in {\mathsf{\Psi }}_{n,q}(r,t)$. Then, the function ${\lambda }_{n,q}:(0,1]\times (0,1]\to \mathbb{R}$ is increasing in r and decreasing in t and $\underset{t\to 0}{lim}{\lambda }_{n,q}(r,t)=\infty$ for every fixed $r\in (0,1]$ (see [3,56]).

Here, if $A,B\subset {\overline{\mathbb{R}}}^n$, we set $q(A)$ as the diameter of A corresponding to the chordal distance in ${\overline{\mathbb{R}}}^n$ and we set $q(A,B)$ as the distance between the sets A and B corresponding to the chordal distance in ${\overline{\mathbb{R}}}^n$.

Let $E_1,...,E_m,...$ be a sequence of sets in ${\overline{\mathbb{R}}}^n$. The kernel $Ke{r}_{j\to \infty }{E}_j$ of this sequence is the set of all points in ${\overline{\mathbb{R}}}^n$, which have a neighborhood that is contained in all but a finite number of sets $E_j$, i.e., $Ke{r}_{j\to \infty }{E}_j={\displaystyle \bigcup _{k=1}^{\infty }}Int{\displaystyle \bigcap _{j=k}^{\infty }}{E}_j$.

Let X be a Riemannian n-manifold and $x\in X$. A set $D\subset X$ is a normal neighborhood of x if there exist $r>0$ and a $C^1$ diffeomorphism $\Phi :\overline{B}(0,r)\to \overline{B}(x,r)$ such that $\Phi (0)=x$, $\Phi (S(0,t))=S(x,t)$ for every $0<t\le r$ and $D=B(x,r)$. Here, $B(0,r)$ is a ball from ${\mathbb{R}}^n$ endowed with the Euclidean distance and $B(x,r)$ is the ball in X endowed with the geodesic distance. We see from [57] that if X is a Riemannian n-manifold and $x\in X$, then X has a fundamental system of normal neighborhoods.

Let $\gamma :[a,b]\to X$ be rectifiable. We set $|\gamma |=Im\gamma$ and if $\rho :X\to [0,\infty ]$ is a Borel function, we set $\underset{\gamma }{\int }\rho ds={\displaystyle \underset{0}{\overset{l(\gamma )}{\int }}}\rho ({\gamma }^0(t))dt$, where ${\gamma }^0:[0,l(\gamma )]\to X$ is such that $\gamma ={\gamma }^0\circ s_{\gamma }$ and $s_{\gamma }$ is the length function of $\gamma$. If $\gamma :[a,b]\to X$ is locally rectifiable, we set $\underset{\gamma }{\int }\rho ds=sup\underset{\beta }{\int }\rho ds$, where the supremum is taken over closed rectifiable subpaths of $\gamma$.

If ${\Gamma }_1,{\Gamma }_2\in A(D)$, we say that ${\Gamma }_1>{\Gamma }_2$ if every path ${\gamma }_1\in {\Gamma }_1$ has a subpath ${\gamma }_2\in {\Gamma }_2$. If ${\Gamma }_1>{\Gamma }_2$, $p>1$ and $\omega :D\to [0,\infty ]$ is measurable and finite a.e., then $M_{\omega }^p({\Gamma }_1)\le M_{\omega }^p({\Gamma }_2)$. If $x=(x_1,...,x_n)\in {\mathbb{R}}^n$, we set $|x|={({\displaystyle \sum _{i=1}^n}{x}_i^2)}^{-{\displaystyle \frac{1}{2}}}$.

A metric measure space X is Ahlfors n-regular if there exists a constant $C>0$ such that ${\displaystyle \frac{r^n}{C}}\le \mu (B(x,r))\le C{r}^n$ for every ball $B(x,r)\subset X$.

A domain $D\subset X$ is finitely connected at a boundary point $x\in \partial D$ if there exists ${(U_m)}_{m\in \mathbb{N}}$ a fundamental system of neighborhoods of x such that $U_m\cap D$ has a finite number of components for every $m\in \mathbb{N}$.

We denote by $\overline{Y}$ the Alexandrov compactification of Y.

We set $C_{x,a,b}=B(x,b)\backslash \overline{B}(x,a)$ if $x\in X$ and $0<a<b$ and ${\Gamma }_{x,a,b}=\Delta (\overline{B}(x,a),\complement B(x,b),C_{x,a,b})$ for $x\in X$ and $0<a<b$.

3. Preliminaries

As in Lemma 2.5 in [8], we prove the following:

Lemma 1.

Let $X,Y$ be Ahlfors n-spaces, $p>1$, $D\subset X$, $G\subset Y$ be open and let $f:D\to G$ be a homeomorphism such that there exist $m,M>0$ such that $L(z,f)\le M$ and $l(z,f)>m$ for every $z\in D$. Then, there exists $C>0$ such that

$$M_p(\Gamma )\le C{M}_p(f(\Gamma ))forevery\Gamma \in A(D).$$

(6)

Remark 1.

If $X,Y$ are Riemannian n-manifolds, $D\subset X$, $G\subset Y$ are domains and $f:\overline{D}\to \overline{G}$ is a $C^1$ diffeomorphism, then there exist $m,M>0$ such that $L(z,f)\le M$ and $l(z,f)\ge m$ for every $z\in D$. If $X=Y={\mathbb{R}}^n$, the condition (6) with $p=n$ holds if $f:D\to G$ is quasiconformal.

As in Lemma 2.6 in [8], we prove the following:

Lemma 2.

Let X be a Riemannian n-manifold, let $x\in X$ and let D be a normal neighborhood of x such that there exist $r_x>0$ and a $C^1$ diffeomorphism $\Phi :\overline{B}(0,r_x)\to \overline{D}=\overline{B}(x,r_x)$ and let $0<a<b<r_x$ and $n-1<q\le n$. Then, if $E,F\subset D$ are such that $S(x,t)\cap E\ne \varphi$, $S(x,t)\cap F\ne \varphi$ for every $a<t<b$, it results that

$$M_n(\Delta (E,F,C_{x,a,b}))\ge C(n,\Phi )ln({\displaystyle \frac{b}{a}})ifq=n,$$

$$M_q(\Delta (E,F,C_{x,a,b}))\ge C(n,q,\Phi )(b^{n-q}-a^{n-q})ifn-1<q<n.$$

Lemma 3.

Let $n\ge 2$, let $n-1<q\le n$, let $X,Y$ be Riemannian n-manifolds, let $D\subset X$ a domain and let $f_j:D\to D_j$ be homeomorphisms such that $f_j\to f$ and the family ${(f_j)}_{j\in \mathbb{N}}$ is equicontinuous. Suppose that there exists ${\omega }_j:D\to [0,\infty ]$ μ measurable and finite μ a.e. such that $M_q(f_j(\Gamma ))\le M_{{\omega }_j}^q(\Gamma )$ for every $\Gamma \in A(D)$ and every $j\in \mathbb{N}$ and that for every $x\in D$ there exist $M_x>0$ and $U_x\in V(x)$ such that $\underset{U_x}{\int }{\omega }_j(z)d\mu \le M_x$ for every $j\in \mathbb{N}$. Then, either f is constant on D or $f:D\to f(D)$ is a homeomorphism.

Proof. 

Let us show first that every point $x\in D$ has a neighborhood $U_x$ such that either f is constant on $U_x$ or f is injective on $U_x$. Suppose that this thing is false. Then, there exist $U_x\in \mathcal{V}(x)$ as before and a $C^1$ diffeomorphism $\Phi :{\mathbb{R}}^n\to U_x$ and distinct points $a_1,a_2,a_3\in \Phi (B^n)$ such that $f(a_1)\ne f(a_2)$ and $f(a_2)=f(a_3)$. Since the family ${(f_j)}_{j\in \mathbb{N}}$ is equicontinuous, we can also suppose that there exists $V_x\in \mathcal{V}(f(x))$ such that $f_j(U_x)\subset V_x$ for every $j\in \mathbb{N}$ and that there exists a $C^1$ diffeomorphism $\psi :B^n\to V_x$. Let $x_k\in B^n$ be such that $\Phi (x_k)=a_k$ for $k=1,2,3.$ Let $J_0$ be an arc joining $x_1$ with $x_2$ in $B^n$ and let $J_1$ be an arc joining $x_3$ with some point $x_4\in S(0,1)$ and such that $J_0\cap J_1=\varphi$. Let $A=R(J_0,J_1\cup \complement B^n)$ and let $F_j={\psi }^{-1}\circ f_j\circ \Phi$ for every $j\in \mathbb{N}$. Then, $F_j:{\mathbb{R}}^n\to F_j({\mathbb{R}}^n)$ is a homeomorphism, and $F_j({\Gamma }_A)$ is a ring $A_j=(C_{0j},C_{1j})$, where $C_{0j}=F_j(J_0)$ and $C_{1j}=F_j(J_1)\cup \complement F_j(B^n)$ for every $j\in \mathbb{N}$. Since $f(a_1)\ne f(a_2)$, there exists $j_0\in \mathbb{N}$ and $0<\rho <q(\complement B^n)$ such that $q(F_j(J_0))>\rho$ for every $j\ge j_0$ and we can take $j_0=1$. Let $d(\Phi (J_0),\Phi (J_1\cup \complement B^n))=\delta >0$. Since $F_j(B^n)\subset B^n$ for $j\in \mathbb{N}$, we see that $\complement F_j(B^n)\supset \complement B^n$ and then $q(C_{1j})\ge q(\complement B^n)>\rho >0$ if $j\in \mathbb{N}$.

Let $r_j=min\{q(C_{0j}),q(C_{1j})\}$ and $t_j=q(C_{0j},C_{1j})$ for every $j\in \mathbb{N}$. Then, $r_j\ge \rho$ for every $j\in \mathbb{N}$ and

$${\lambda }_{n,q}(\rho ,t_j)\le {\lambda }_{n,q}(r_j,t_j)\le M_q(F_j({\Gamma }_A))=M_q({\psi }^{-1}\circ f_j\circ \Phi ({\Gamma }_A))\le$$

$$\le C{M}_q(f_j(\Phi ({\Gamma }_A)))\le C{M}_{{\omega }_j}^q(\Phi ({\Gamma }_A))\le {\displaystyle \frac{C}{{\delta }^q}}\underset{U_x}{\int }{\omega }_j(z)d\mu \le {\displaystyle \frac{C{M}_x}{{\delta }^q}}$$

for every $j\in \mathbb{N}$. Since ${\lambda }_{n,q}(\rho ,t_j)\to \infty$ if $j\to \infty$, we reached a contradiction. We therefore proved that for every point $x\in D$ there exists $U_x\in \mathcal{V}(x)$ such that either f is constant on $U_x$ or f is injective on $U_x$.

Let $E_1=\{x\in D|$ there exists $U_x\in \mathcal{V}(x)$ such that f is constant on $U_x\}$ and $E_2=\{x\in D|$ there exists $U_x\in \mathcal{V}(x)$ such that f is injective on $U_x\}$. Then, $D=E_1\cup E_2$, $E_1$ and $E_2$ are open and we proved that $E_1\cap E_2=\varphi$.

Suppose that f is not injective on D and let $x_1,x_2\in D$, $x_1\ne x_2$ be such that $f(x_1)=f(x_2)$ and let $\delta =d(x_1,x_2)>0$. Let $U_1=B(x_1,{\displaystyle \frac{\delta }{4}})$, $U_2=B(x_2,{\displaystyle \frac{\delta }{4}})$. Then, ${\overline{U}}_1\cap {\overline{U}}_2=\varphi$, and since $f_j$ is a homeomorphism, we see that $f_j({\overline{U}}_1)$ and $f_j({\overline{U}}_2)$ are compact disjoint sets. Let $0<ϵ<{\displaystyle \frac{\delta }{4}}$ and let $d_j:[0,1]\to Y$ be a geodesic joining $f_j(x_1)$ with $f_j(x_2)$ for every $j\in \mathbb{N}$. Then, $f_j(x_2)\notin f_j({\overline{U}}_1)$ and $d_j(0)=f_j(x_1)$, $d_j(1)=f_j(x_2)$. Let $t_j\in (0,1)$ be such that $d_j(t_j)\in \partial f_j(B(x_1,ϵ))=f_j(S(x_1,ϵ))$ for every $j\in \mathbb{N}$ and let $x_j\in S(x_1,ϵ)$ be such that $d_j(t_j)=f_j(x_j)$ for every $j\in \mathbb{N}$ and can suppose that $x_j\to x_{ϵ}\in S(x_1,ϵ)$ and then $f_j(x_j)\to f(x_{ϵ})$. On the other side, $d(f_j(x_j),f_j(x_2))=d(d_j(t_j),d_j(1))<d(d_j(0),d_j(1))=d(f_j(x_1),f_j(x_2))\to 0$ and hence $f(x_{ϵ})=f(x_2)=f(x_1)$. We found in every sphere $S(x_1,ϵ)$ points $x_{ϵ}$ such that $f(x_1)=f(x_{ϵ})$ and hence $x_1\notin E_2$. It results that $x_1\in E_1$ and hence $D=E_1$.

We proved that either f is injective on D or f is constant on D. □

Lemma 4.

Let $q>1$, $D\subset {\mathbb{R}}^n$ be open and $f_m\in W_{loc}^{1,q}(D,{\mathbb{R}}^n)$ such that $f_m\to f$ uniformly on the compact subsets of D. Then, $f_m\to f$ weakly in $W_{loc}^{1,q}(D,{\mathbb{R}}^n)$ and $\underset{D}{\int }|f^{\prime }{(x)|}^{q}dx\le \underset{m\to \infty }{lim\; inf}\underset{D}{\int }{|f_m^{\prime }(x)|}^{q}dx$.

Proof. 

Since $f_m\to f$ uniformly on the compact subsets of D, it results that also $f\in W_{loc}^{1,q}(D,{\mathbb{R}}^n)$ and $\underset{F}{\int }{|f_m^{\prime }(x)-f^{\prime }(x)|}^{q}dx\to 0$ if $m\to \infty$ for every fixed compact set $F\subset D$. We can now suppose that $I=\underset{m\to \infty }{lim\; inf}\underset{D}{\int }{|f_m^{\prime }(x)|}^{q}dx<\infty$ and also that $\underset{D}{\int }{|f^{\prime }(x)|}^{q}dx>0$. Let $F\subset \subset D$ be such that $\underset{F}{\int }{|f^{\prime }(x)|}^{q}dx>0$ and let $ϵ>0$. Let ${\delta }_{ϵ}>0$ be such that $\underset{A}{\int }{|f^{\prime }(x)|}^{q}dx<{\displaystyle \frac{ϵ}{3}}$ if $A\subset F$ and ${\mu }_n(A)<{\delta }_{ϵ}$ and let $m_{ϵ}\in \mathbb{N}$ be such that $\underset{D}{\int }{|f_m^{\prime }(x)|}^{q}dx\le I+{\displaystyle \frac{ϵ}{3}}$ and $\underset{F}{\int }{|f_m^{\prime }(x)-f^{\prime }(x)|}^{q}dx\le {\left({\displaystyle \frac{ϵ}{3q(\underset{F}{\int }|f^{\prime }(x){{|}^{q}dx)}^{\frac{q-1}{q}}}}\right)}^q$ if $m\ge m_{ϵ}$. Using Riesz’s theorem and taking a subsequence if necessary, we can suppose that $|f_m^{\prime }(x)|\to |f^{\prime }(x)|$ a.e. Using Egorov’s theorem, we find $F_{ϵ}\subset F$ measurable such that ${\mu }_n(F\backslash F_{ϵ})<{\delta }_{ϵ}$ and $|f_m^{\prime }(x)|\to |f^{\prime }(x)|$ uniformly on $F_{ϵ}$.

Let $n_{ϵ}\ge m_{ϵ}$ be such that $|f^{\prime }(x)|-|f_m^{\prime }(x)|<ϵ$ for every $x\in F_{ϵ}$ and every $m\ge n_{ϵ}$. Let $A_{ϵ}(x)=<q{|f^{\prime }(x)|}^{q-2}{f}^{\prime }(x),f_m^{\prime }(x)-f^{\prime }(x)>$ if $x\in F_{ϵ}$ and $m\ge n_{ϵ}$. We have

$$|f_m^{\prime }{(x)|}^q-|f^{\prime }{(x)|}^q\ge qmin\{|f_m^{\prime }(x)|,|f^{\prime }{(x)|\}}^{q-1}(|f_m^{\prime }(x)|-|f^{\prime }(x)|)\ge$$

$$\ge q|f^{\prime }{(x)|}^{q-1}{\displaystyle \frac{1}{|f^{\prime }(x)|}}<f^{\prime }(x),f_m^{\prime }(x)-f^{\prime }(x)>=A_{ϵ}(x)$$

if $x\in F_{ϵ}$ and $m\ge n_{ϵ}$. Using Hőlder’s inequality, we have

$$\underset{F_{ϵ}}{\int }|A_{ϵ}(x)|dx\le q{\left(\underset{F_{ϵ}}{\int }{|f^{\prime }(x)|}^{q}dx\right)}^{{\displaystyle \frac{q-1}{q}}}{\left(\underset{F_{ϵ}}{\int }{|f_m^{\prime }(x)-f^{\prime }(x)|}^{q}dx\right)}^{{\displaystyle \frac{1}{q}}}\le$$

$$\le q{\left(\underset{F}{\int }{|f^{\prime }(x)|}^{q}dx\right)}^{{\displaystyle \frac{q-1}{q}}}{\left(\underset{F}{\int }{|f_m^{\prime }(x)-f^{\prime }(x)|}^{q}dx\right)}^{{\displaystyle \frac{1}{q}}}\le {\displaystyle \frac{ϵ}{3}}$$

if $m\ge n_{ϵ}$. Then, $\underset{F}{\int }|f^{\prime }{(x)|}^{q}dx=\underset{F_{ϵ}}{\int }|f^{\prime }{(x)|}^{q}dx+\underset{F\backslash F_{ϵ}}{\int }|f^{\prime }{(x)}^{q}dx\le \underset{F_{ϵ}}{\int }|f_m^{\prime }{(x)|}^{q}dx+\underset{F_{ϵ}}{\int }|A_{ϵ}(x)|dx+\underset{F\backslash F_{ϵ}}{\int }{|f^{\prime }(x)|}^{q}dx\le I+{\displaystyle \frac{ϵ}{3}}+{\displaystyle \frac{ϵ}{3}}+{\displaystyle \frac{ϵ}{3}}=I+ϵ$ if $m\ge n_{ϵ}$.

Letting $ϵ\to 0$ and $F\nearrow D$, the theorem is proved. □

Let X be a metric space, $A\subset X$ and $k,r>0$. We set ${\mathcal{H}}_r^k(A)=inf{\displaystyle \sum _{i=1}^{\infty }}d{(B_i)}^k$, where the infimum is taken over all balls $B_i$ such that $A\subset {\displaystyle \bigcup _{i=1}^{\infty }}{B}_i$ and $d(B_i)<r$ for every $i\in \mathbb{N}$.

We set ${\mathcal{H}}^k(A)=\underset{r\to 0}{lim}{\mathcal{H}}_r^k(A)$.

Lemma 5.

Let $X,Y$ be Ahlfors n-regular, let $U\subset X$, let $V\subset Y$ be open and let $\Phi :\overline{U}\to \overline{V}$ be a homeomorphism such that there exist $m,M>0$ such that $l(z,\Phi )>m$ and $L(z,\Phi )<M$ for every $z\in \overline{U}$. Then, if $A\subset \overline{U}$ is such that ${\mathcal{H}}^k(A)=0$, it results that ${\mathcal{H}}^k(\Phi (A))=0$ and the mapping $\Phi |U:U\to V$ satisfies condition $(N)$ and condition $(N^{-1})$.

Proof. 

Let $A\subset \overline{U}$ be such that ${\mathcal{H}}^k(A)=0$ and let $r>0$, $ϵ>0$. We can find a covering $A\subset {\displaystyle \bigcup _{i=1}^{\infty }}B(x_i,r_i)$ such that $d(\Phi (x),\Phi (x_i))\le Md(x,x_i)$ for every $x\in A\cap B(x_i,r_i)$, $r_i<r$ for every $i\in \mathbb{N}$ and ${\displaystyle \sum _{i=1}^{\infty }}{(2r_i)}^k<ϵ$. Then, $\Phi (A)\subset {\displaystyle \bigcup _{i=1}^{\infty }}\Phi (B(x_i,r_i)\cap A)\subset {\displaystyle \bigcup _{i=1}^{\infty }}B(\Phi (x_i),M{r}_i)$ and ${\displaystyle \sum _{i=1}^{\infty }}{(2M{r}_i)}^k\le M^k{\displaystyle \sum _{i=1}^{\infty }}{(2r_i)}^k\le M^kϵ$. It results that ${\mathcal{H}}_{Mr}^k(\Phi (A))\le M^kϵ$ and letting $ϵ\to 0$ and then $r\to 0$ we find that ${\mathcal{H}}^k(\Phi (A))=0$.

Let now $A\subset U$ be such that $\mu (A)=0$ and let $ϵ>0$. Since $\mu$ is Borel regular, we find $Q\subset U$ open such that $A\subset Q\subset U$ and $\mu (Q)<ϵ$. We see that $\Phi (B(x,r))\subset B(\Phi (x),Mr)$ and $B(x,r)\subset {\Phi }^{-1}(B(\Phi (x),Mr))$ for every $x\in A$ and $r>0$ small enough. Let $C_1,C_2>0$ be such that ${\displaystyle \frac{r^n}{C_1}}\le \mu (B(x,r))\le C_1{r}^n$ for every ball $B(x,r)\subset X$ and ${\displaystyle \frac{r^n}{C_2}}\le \nu (B(y,r))\le C_2{r}^n$ for every ball $B(y,r)\subset Y$. We cover the set A with balls $B(x,r_x)$ such that $B(x,r_x)\subset Q$, $B(\Phi (x),5M{r}_x)\subset \Phi (Q)$ and then $\Phi (A)\subset \Phi (\bigcup _{x\in A}B(x,r_x))\subset \bigcup _{x\in A}B(\Phi (x),M{r}_x)$.

Using the covering lemma from page 30 in [35], we find disjoint balls $B(\Phi (x_i),M{r}_i)$, $i\in \mathbb{N}$ such that $\bigcup _{x\in A}B(\Phi (x),M{r}_x)\subset {\displaystyle \bigcup _{i=1}^{\infty }}B(\varphi (x_i),5M{r}_i)$. Then, $\Phi (A)\subset {\displaystyle \bigcup _{i=1}^{\infty }}B(\Phi (x_i),5M{r}_i)$ and hence $\nu (\Phi (A))\le \nu ({\displaystyle \bigcup _{i=1}^{\infty }}B(\Phi (x_i),5M{r}_i))\le {\displaystyle \sum _{i=1}^{\infty }}\nu (B(\Phi (x_i),5M{r}_i))\le C_2{(5M)}^n{\displaystyle \sum _{i=1}^{\infty }}{r}_i^n$. We see that also the sets ${\Phi }^{-1}(B(\Phi (x_i),M{r}_i))$, $i\in \mathbb{N}$ are disjoint, and since $B(x_i,r_i)\subset {\Phi }^{-1}(B(\Phi (x_i),M{r}_i))$ for every $i\in \mathbb{N}$, it results that the balls $B(x_i,r_i)$, $i\in \mathbb{N}$ are disjoint subsets of Q and $\mu ({\displaystyle \bigcup _{i=1}^{\infty }}B(x_i,r_i))\le \mu (Q)\le ϵ$. We find that

$${\displaystyle \frac{1}{C_1}}{\displaystyle \sum _{i=1}^{\infty }}{r}_i^n\le {\displaystyle \sum _{i=1}^{\infty }}\mu (B(x_i,r_i))=\mu ({\displaystyle \bigcup _{i=1}^{\infty }}B(x_i,r_i))\le \mu (Q)\le ϵ.$$

We have $\nu (\Phi (A))\le C_2{(5M)}^n{\displaystyle \sum _{i=1}^{\infty }}{r}_i^n\le C_1{C}_2{(5M)}^nϵ$.

Letting $ϵ\to 0$, we see that $\nu (\Phi (A))=0$ and we proved that $\Phi |U:U\to V$ satisfies condition $(N)$. In the same way, we prove that $\Phi |U:U\to V$ satisfies condition $(N^{-1})$. □

Lemma 6.

Let $p>1$, let X be a metric measure space, let $x\in X$ and let $\omega :X\to [0,\infty ]$ be μ measurable and finite μ a.e. such that ${\omega }_x<\infty$. Then, there exists $r_x>0$ such that $M_{\omega }^p({\Gamma }_{x,a,b})\le {\displaystyle \frac{C{e}^p{\omega }_x{\displaystyle \sum _{k=1}^{\infty }}\frac{1}{k^p}}{{(lnln(\frac{be}{a}))}^n}}$ for every $0<a<b<be<r_x$.

Proof. 

We follow the ideas from Lemma 2.2 in [16]. □

A measure $\mu$ in a metric space X is called doubling if the balls have a finite and positive measure and there exists a constant C such that $\mu (2B)\le C\mu (B)$ for all balls $B\subset X$.

Lemma 7.

Let $(X,\mu )$ be a metric measure space and let μ be doubling. Let $D\subset \subset X$ and let $f:\overline{D}\to {\overline{D}}_f\subset Y$ be a homeomorphism satisfying condition $(N)$ and let $g:Y\to [0,\infty ]$ be a Borel function. Then, $\underset{A}{\int }g(f(x))J_f(x)d\mu =\underset{f(A)}{\int }g(y)d\nu$ for every Borel set $A\subset D$.

Proof. 

Let ${\nu }_f(A)=\nu (f(A))$ for every Borel set $A\subset D$. Then, ${\nu }_f$ is a finite Borel measure, and since f satisfies condition $(N)$, we see that ${\nu }_f\le \mu$. Using Radon–Nikodym’s theorem, we find $h\in L^1(f(D))$ such that ${\nu }_f(A)=\underset{A}{\int }h(x)d\mu$ for every Borel set $A\subset D$. Using Lebesgue’s differentiation theorem (see Theorem 1.8 in [32]), we see that in every Lebesgue point x of h we have

$$J_f(x)=\underset{r\to 0}{lim\; sup}{\displaystyle \frac{\nu (f(B(x,r)))}{\mu (B(x,r))}}=\underset{r\to 0}{lim\; sup}{\displaystyle \frac{{\nu }_f(B(x,r))}{\mu (B(x,r))}}=\underset{r\to 0}{lim\; sup}\underset{B(x,r)}{⨏}h(z)d\mu =h(x).$$

Then, $\nu (f(A))={\nu }_f(A)=\underset{A}{\int }h(x)d\mu =\underset{A}{\int }{J}_f(x)d\mu$ for every Borel set $A\subset D$. Using now standard arguments, we see that $\underset{A}{\int }g(f(x))J_f(x)d\mu =\underset{f(A)}{\int }g(y)d\nu$ for every Borel set $A\subset D$. □

4. The Main Results

Proof of Theorem 1. 

Let $D_k\subset \subset D\backslash B_f$ be such that ${\overline{D}}_k\subset D_{k+1}$ for every $k\in \mathbb{N}$ and $D\backslash B_f={\displaystyle \bigcup _{k=1}^{\infty }}{D}_k$. Let $k\in \mathbb{N}$ be fixed. Since $\mu (B_f)=0$, f is open on $D_k\backslash B_f$ and every ball $B(x,r)$ in Y with $r>0$ has a positive measure and $\nu (f(B_f))=0$, we see that $f(D_k)\backslash f(B_f)\ne \varphi$. Let $y\in f(D_k)\backslash f(B_f)$. Since f is a local homeomorphism on ${\overline{D}}_k$ and ${\overline{D}}_k$ is compact, we see that $f^{-1}(y)\cap {\overline{D}}_k$ is finite and we can find $V\in \mathcal{V}(y)$ and open sets $U_i\subset \subset D\backslash B_f$ such that $f^{-1}(V)\cap {\overline{D}}_k\subset {\displaystyle \bigcup _{i=1}^m}{U}_i\subset D_{k+1}$ and $f|U_i:U_i\to V$ is a homeomorphism for every $i=1,...,m$. Since $\nu$ is doubling, we can use Vitali’s covering theorem (see Theorem 1.6 in [32]) and we find $H_k\subset Y$ with $\nu (H_k)=0$, disjoint balls $V_i$ such that $f(D_k)\backslash H_k={\displaystyle \bigcup _{i=1}^{\infty }}{V}_i$ and disjoint sets $U_{ij}$ such that $f|U_{ij}:U_{ij}\to V_i$ is a homeomorphism, $D_k\backslash f^{-1}(H_k)\subset {\displaystyle \bigcup _{i=1}^{\infty }}{\displaystyle \bigcup _{j=1}^{j(i)}}{U}_{ij}$, and if we denote by $g_{ij}:V_i\to U_{ij}$, $g_{ij}={(f|U_{ij})}^{-1}$ for $i\in \mathbb{N}$, $j=1,...,j(i)$, we can also suppose that $J_{g_{ij}}(y)<\infty$ for every $y\in V_i\backslash H_k$ and for $i\in \mathbb{N}$ and $j=1,...,j(i)$. Since f satisfies condition $(N^{-1})$, we see that $\mu (f^{-1}(H_k))=0$.

Let ${\omega }_k:Y\to [0,\infty ]$, ${\omega }_k(y)={\displaystyle \sum _{i=1}^{\infty }}{\chi }_{V_i}(y){\displaystyle \sum _{j=1}^{j(i)}}{J}_{g_{ij}}(y)L{(g_{ij}(y),f)}^q$ if $y\in f(D_k)\backslash H_k$, ${\omega }_k(y)=0$ otherwise. Using Lemma 7, we have $\underset{f(D_k)}{\int }{\omega }_k(y)d\nu ={\displaystyle \sum _{i=1}^{\infty }}\underset{V_i}{\int }{\omega }_k(y)d\nu ={\displaystyle \sum _{i=1}^{\infty }}\underset{V_i}{\int }{\displaystyle \sum _{j=1}^{j(i)}}{J}_{g_{ij}(y)}L{(g_{ij}(y),f)}^{q}d\nu ={\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^{j(i)}}\underset{U_{ij}}{\int }L{(z,f)}^{q}d\mu \le \underset{D_k}{\int }L{(z,f)}^{q}d\mu$ and we find that

$$\underset{f(D_k)}{\int }{\omega }_k(y)d\nu \le \underset{D_k}{\int }L{(z,f)}^{q}d\mu$$

(7)

If $\Gamma \in A(D)$ is such that $F(\Gamma )=\varphi$, then $M_q(\Gamma )=0$ and the theorem is proved. We can suppose that $F(\Gamma )\ne \varphi$.

Let $\Gamma \in A(D)$ and $\Delta =\{\gamma \in \Gamma |\gamma$ is locally rectifiable and $f\circ {\gamma }^0$ is absolutely continuous}. Using Fuglede’s theorem from [15], we see that $M_q(\Gamma )=M_q(\Delta )$. Let $\eta \in F(f(\Gamma ))$ and $\rho :X\to [0,\infty ]$ be defined by $\rho (z)=\eta (f(z))L(z,f)$ if $z\in D$, $\rho (z)=0$ otherwise. We see from [15] that $\rho \in F(\Delta )$, and using Lemma 7, we have:

$$\underset{D_k}{\int }\rho {(z)}^{q}d\mu =\underset{D_k}{\int }\eta {(f(z))}^{q}L{(z,f)}^{q}d\mu \le {\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^{j(i)}}\underset{U_{ij}}{\int }\eta {(f(z))}^{q}L{(z,f)}^{q}d\mu =$$

$$={\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^{j(i)}}\underset{V_i}{\int }\eta {(f(g_{ij}(y)))}^{q}L{(g_{ij}(y),f)}^q{J}_{g_{ij}}(y)d\nu =$$

$$={\displaystyle \sum _{i=1}^{\infty }}\underset{V_i}{\int }\eta {(y)}^q{\chi }_{V_i}(y){\displaystyle \sum _{j=1}^{j(i)}}L{(g_{ij}(y),f)}^q{J}_{g_{ij}}(y)d\nu =$$

$$=\underset{Y}{\int }\eta {(y)}^q{\omega }_k(y)d\nu .$$

We proved that

$$\underset{D_k}{\int }\rho {(z)}^{q}d\mu \le \underset{Y}{\int }\eta {(y)}^q{\omega }_k(y)d\nu$$

(8)

Let $\omega :Y\to [0,\infty ]$, $\omega (y)=\sum _{x\in f^{-1}(y)}{J}_{g_x}(y)L{(g_x(y),f)}^q$ if $y\in f(D)\backslash f(B_f)$, $\omega (y)=0$ otherwise. Here, if $x\in f^{-1}(y)$, we set by $g_x$ a local inverse of f such that $g_x(y)=x$ and we see that ${\omega }_k=\omega |D_k$$\nu$ a.e. and ${\omega }_k\nearrow \omega$$\nu$ a.e. Using the fact that $\mu (B_f)=0$, $\nu (f(B_f))=0$ and letting $k\to \infty$ in (7) and (8), we obtain that

$$\underset{f(D)}{\int }\omega (y)d\nu \le \underset{D}{\int }L{(z,f)}^{q}d\mu <\infty and{M}_q(\Gamma )=M_q(\Delta )\le \underset{D}{\int }\rho {(z)}^{q}d\mu \le \underset{Y}{\int }\eta {(y)}^q\omega (y)d\nu .$$

Since $\eta \in F(f(\Gamma ))$ was arbitrarily chosen, we find that $M_q(\Gamma )\le M_{\omega }^q(f(\Gamma ))$ for every $\Gamma \in A(D)$ and that $\omega \in L^1(Y)$. □

Remark 2.

Let us take in Theorem 1 the measure ${\nu }_{\omega }$ given by ${\nu }_{\omega }(A)=\underset{A}{\int }\omega (x)d\mu$ for every Borel set $A\subset Y$. Since $\omega \in L^1(Y)$, we see that ${\nu }_{\omega }\le \mu$, and if $\eta :Y\to [0,\infty ]$ is a Borel function and $A\subset Y$ is a Borel set, then

$$\underset{A}{\int }\omega (x)\eta {(x)}^{q}d\mu =\underset{A}{\int }\eta {(x)}^{q}d{\nu }_{\omega }.$$

Since $M_q(\Gamma )\le M_{\omega }^q(f(\Gamma ))$ for every $\Gamma \in A(D)$, it results that

$$M_q(\Gamma )\le M_q(f(\Gamma ))forevery\Gamma \in A(D).$$

(9)

Here, we take the measure μ on X and the measure ${\nu }_{\omega }$ on Y.

Remark 3.

The idea in the proof of Theorem 1 comes from the old methods in the book of Väisälä [3].

Here, if $D,G$ are domains in ${\mathbb{R}}^n$ and $f:D\to G$ is a homeomorphism satisfying condition $(N)$, the Jacobian is defined by

$$J_f(x)=\underset{r\to 0}{lim}{\displaystyle \frac{{\mu }_n(f(B(x,r)))}{{\mu }_n(B(x,r))}}$$

which exists a.e. and $J_f$ is measurable and

$${\mu }_n(f(A))=\underset{A}{\int }{J}_f(x)dx$$

for every Borel set $A\subset D$ (see page 85 in [3]). Using standard arguments, we see that

$$\underset{A}{\int }g(f(x))J_f(x)dx=\underset{f(A)}{\int }g(y)dy$$

for every positive Borel function g and every Borel set $A\subset D$.

Suppose now that $D,G$ are domains in ${\mathbb{R}}^n$, let $q>1$ and let $f:D\to G$ be a homeomorphism such that its inverse $g:G\to D$ satisfies condition $(N)$ and $\underset{D}{\int }L{(x,f)}^{q}dx<\infty$. Let $\omega :{\mathbb{R}}^n\to [0,\infty ]$, $\omega (y)=J_g(y)L{(g(y),f)}^q$ if $y\in G$, $\omega (y)=0$ otherwise.

We see that $\underset{{\mathbb{R}}^n}{\int }\omega (y)dy=\underset{G}{\int }{J}_g(y)L{(g(y),f)}^{q}dy=\underset{D}{\int }L{(x,f)}^{q}dx<\infty$ and hence $\omega \in L^1({\mathbb{R}}^n)$.

Let $\Gamma \in A(D)$, $\Delta =\{\gamma \in \Gamma |f\circ {\gamma }^0$ is absolutely continuous} and let $\eta \in F(f(\Gamma ))$. Let $\rho :{\mathbb{R}}^n\to [0,\infty ]$, $\rho (x)=\eta (f(x))L(x,f)$ if $x\in D$, $\rho (x)=0$ otherwise and let $\gamma \in \Delta$. Then, using Theorem 5.3 in [3], we have $1\le \underset{f\circ \gamma }{\int }\eta ds\le \underset{\gamma }{\int }\eta (f(x))L(x,f)ds=\underset{\gamma }{\int }\rho ds$ and hence $\rho \in F(\Delta )$.

We have

$$M_q(\Gamma )=M_q(\Delta )\le \underset{D}{\int }\rho {(x)}^{q}dx=\underset{D}{\int }\eta {(f(x))}^{q}L{(x,f)}^{q}dx=$$

$$=\underset{G}{\int }\eta {(f(g(y)))}^{q}L{(g(y),f)}^q{J}_g(y)dy=\underset{G}{\int }\eta {(y)}^q\omega (y)dy.$$

Since $\eta \in F(f(\Gamma ))$ was arbitrarily chosen, we see that

$$M_q(\Gamma )\le M_{\omega }^q(f(\Gamma ))$$

for every $\Gamma \in A(D)$.

Now, we see from Example 1 and Theorem 1 that we can take the homeomorphism $f:D\to G$ such that $f\notin W_{loc}^{1,1}(D,G)$.

Also, the modular inequality (9) implies that

$$M_q(\Gamma )\le M_q(f(\Gamma ))forevery\Gamma \in A(D)$$

(10)

and we take the measure ${\mu }_n$ on D and the measure ${({\mu }_n)}_{\omega }$ on G.

Theorem 2.

Let $1\le n-1<q\le n$, let $X,Y$ be Riemannian n-manifolds, let $f:D\to Y$ be continuous, open and discrete, satisfying condition $(N^{-1})$ and such that $\mu (B_f)=0$, $\nu (f(B_f))=0$ and $\underset{D}{\int }L{(z,f)}^{q}d\mu <\infty$. Then, there exists $r_x>0$ such that

$$d{(f(y),f(x))}^n\le {\displaystyle \frac{C_1}{ln(\frac{d(x,z)}{d(x,y)})}}\underset{B(x,d(x,z))}{\int }L{(w,f)}^{n}d\mu and0<d(x,y)<d(x,z)<r_x.$$

(11)

$$d{(f(y),f(x))}^q\le {\displaystyle \frac{C_2}{d{(x,y)}^{n-q}}}\underset{B(x,2d(x,y))}{\int }L{(w,f)}^{q}d\mu ifn-1<q<nand2d(x,y)<r_x.$$

(12)

If $X={\mathbb{R}}^n$, we can take $r_x=d(x,\partial D)$. The constant $C_1$ depends only on n and x, and the constant $C_2$ depends on $n,q$ and x. If f is quasimonotone of constant L instead of being open and discrete, then $C_1$ depends on $L,n$ and x and the constant $C_2$ depends on $L,q,n$ and x.

Proof of Theorem 2. 

We see from Theorem 1 that there exists $\omega \in L^1(Y)$ such that $M_q(\Gamma )\le M_{\omega }^q(f(\Gamma ))$ for every $\Gamma \in A(D)$ and that $\underset{f(B)}{\int }\omega (y)d\nu \le \underset{B}{\int }L{(z,f)}^{q}d\mu$ for every ball $B\subset D$. Let $\Phi :\overline{B}(0,r_x)\to \overline{B}(x,r_x)$ be a $C^1$ diffeomorphism such that $\overline{B}(x,r_x)\subset D$, $\Phi (S(0,t))=S(x,t)$ for every $0<t<r_x$ and let $y,z\in X$ be such that $0<d(x,y)<d(x,z)<r_x$. Suppose that f is quasimonotone. We can find for every $d(x,y)<t<d(x,z)$ some points ${\alpha }_t,{\beta }_t\in S(x,t)$ such that $df(\overline{B}(x,y))\le Ld(f({\alpha }_t),f({\beta }_t))$. Let $E=\{{\alpha }_t\}$, $F=\{{\beta }_t\}$ for $d(x,y)<t<d(x,z)$ and let $\Gamma =\Delta (E,F,C_{x,d(x,y),d(x,z)})$. Then, $\eta ={\displaystyle \frac{L}{d(f(x),f(y))}}{\chi }_{f(B(x,d(x,z)))}\in F(f(\Gamma ))$. Here, L is the coefficient of quasimonotonicity of f.

If $q=n$, we see from Lemma 2 that

$$C(x,n)ln({\displaystyle \frac{d(x,z)}{d(x,y)}})\le M_n(\Gamma )\le M_{\omega }^n(f(\Gamma ))\le {\displaystyle \frac{L^n}{d{(f(x),f(y))}^n}}\underset{f(B(x,d(x,z)))}{\int }\omega (w)d\nu \le$$

$$\le {\displaystyle \frac{L^n}{d{(f(x),f(y))}^n}}\underset{B(x,d(x,z))}{\int }L{(w,f)}^{n}d\mu .$$

If $n-1<q<n$ and $2d(x,y)=d(x,z)$, we see from Lemma 2 that $$\begin{gathered} C(x,n,q)(2^{n-q}- \\ 1)d{(x,y)}^{n-q}=C(x,n,q)(d{(x,z)}^{n-q}-d{(x,y)}^{n-q})\le M_q(\Gamma )\le M_{\omega }^q(f(\Gamma ))\le \underset{Y}{\int }\omega (w)\eta {(w)}^{q}d\nu \le \end{gathered}$$ ${\displaystyle \frac{L^q}{d{(f(x),f(y))}^q}}\underset{f(B(x,d(x,z)))}{\int }\omega (w)d\nu \le {\displaystyle \frac{L^q}{d{(f(x),f(y))}^q}}\underset{B(x,d(x,z))}{\int }L{(w,f)}^{q}d\mu$.

Suppose now that f is open and discrete. In this case, we take the normal coordinates $\Phi :\overline{B}(0,r_x)\to \overline{B}(x,r_x)$ such that $f(\overline{\Phi }(x,r_x))\subset V_x$, where $V_x\in \mathcal{V}(f(x))$ and there exists a $C^1$ diffeomorphism $\psi :B^n\to V_x$ such that $\psi (0)=f(x)$. We can suppose that there exists $L_x>0$ such that ${\displaystyle \frac{d(a,b)}{L_x}}\le d({\psi }^{-1}(a),{\psi }^{-1}(b))\le L_{x}d(a,b)$ for every $a,b\in V_x$. Let $F={\psi }^{-1}\circ f:B(x,r_x)\to {\mathbb{R}}^n$. Then, F is well defined, is continuous, open, discrete and hence monotone and $L(w,F)\le L_{x}L(w,f)$ for every $w\in B(x,r_x)$.

If $q=n$, using the preceding result, we have

$${\displaystyle \frac{d{(f(y),f(x))}^n}{L_x^n}}\le d{(F(y),F(x))}^n\le {\displaystyle \frac{C(n,x)}{ln(\frac{d(x,z)}{d(x,y)})}}\underset{B(x,d(x,z))}{\int }L{(w,F)}^{n}dw\le$$

$$\le {\displaystyle \frac{C(n,x)L_x^n}{ln(\frac{d(x,z)}{d(x,y)})}}\underset{B(x,d(x,z))}{\int }L{(w,f)}^{n}d\mu .$$

Also, if $n-1<q<n$, we have

$${\displaystyle \frac{d{(f(y),f(x))}^q}{L_x^q}}\le d{(F(y),F(x))}^q\le {\displaystyle \frac{C(n,q,x)}{d{(x,y)}^{n-q}}}\underset{B(x,2d(x,y))}{\int }L{(w,F)}^{q}dw\le$$

$$\le {\displaystyle \frac{C(n,q,x)L_x^q}{d{(x,y)}^{n-q}}}\underset{B(x,2d(x,y))}{\int }L{(w,f)}^{q}d\mu .$$

□

Also, the family W from Remark 5 is obviously equicontinuous on D.

Remark 4.

Relation (11) is proved in Theorem 7.5.2 in [6] for mappings $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$ of finite dilatation. However, as Example 1 shows, relation (11) may be valid even for mappings $f\notin W_{loc}^{1,1}(D,{\mathbb{R}}^n)$.

Remark 5.

If $W=\{f:D\to Y|f$ is continuous, has the coefficient of quasimonotonicity L and satisfies condition $(N^{-1})$, $\mu (B_f)=0$, $\nu (f(B_f))=0$ and there exists a constant $M<\infty$ such that $\underset{D}{\int }L{(x,f)}^{n}d\mu \le M\}$, then the family W is equicontinuous on D.

Theorem 3.

Let Y be complete, let $D\subset X$ be a domain, let $x\in D$ be such that there exists a homeomorphism $\Phi :{\overline{B}}^n\to \overline{D}$ such that $\Phi (0)=x$ and there exist $m,M>0$ such that $l(z,\Phi )>m$, $L(z,\Phi )<M$ for every $z\in {\overline{B}}^n$. Let us denote by ${\gamma }_y:[0,1]\to Y$ the radial path joining x and y in D given by ${\gamma }_y(t)=\Phi (t{\Phi }^{-1}(y))$ for every $t\in [0,1]$ and every $y\in \partial D=\Phi (S(0,1))$. Let $f:D\to Y$ be continuous such that $\mu (B_f)=0$, $\nu (f(B_f))=0$, f satisfies condition $(N^{-1})$ and there exists $q>1$ such that $\underset{D}{\int }L{(z,f)}^{q}d\mu <\infty$. Then, there exists $\underset{t\to 1}{lim}f({\gamma }_y(t))\in Y$ for ${\mathcal{H}}^{n-1}$ a.e. $y\in \partial D$, and if $E_w=\{y\in \partial D|\underset{t\to 1}{lim}f({\gamma }_y(t))=w\}$, it results that ${\mathcal{H}}^{n-1}(E_w)=0$ for every $w\in Y$.

Proof of Theorem 3.  

We use the ideas from Theorem 1 in [17] and we apply Theorem 1 instead of Lemma 1 in [17]. □

Theorem 4.

Let $D\subset {\mathbb{R}}^n$ be a domain, let $x\in D$ be such that there exists $\Phi :{\overline{B}}^n\to \overline{D}$ a $C^1$ diffeomorphism such that $\Phi (0)=x$ and let us denote by ${\gamma }_y:[0,1]\to {\mathbb{R}}^n$ the radial path joining x and y in D given by ${\gamma }_y(t)=\Phi (t{\Phi }^{-1}(y))$ for every $t\in [0,1]$ and every $y\in \partial D=\Phi (S(0,1))$. Let $f:D\to {\mathbb{R}}^n$ be a mapping of finite distortion satisfying relation (4) and such that $\underset{D}{\int }{|f^{\prime }(z)|}^{n}dz<\infty$. Then, there exists $\underset{t\to 1}{lim}f({\gamma }_y(t))\in {\mathbb{R}}^n$ for ${\mathcal{H}}^{n-1}$ a.e. $y\in \partial D$, and if $E_w=\{y\in \partial D|\underset{t\to 1}{lim}f({\gamma }_y(t))=w\}$, it results that ${\mathcal{H}}^{n-1}(E_w)=0$ for every $w\in {\mathbb{R}}^n$.

Proof of Theorem 4.  

The mapping f satisfies the conditions from Theorem 1 and we apply Theorem 3. □

Theorem 5.

Let $n\ge 2$, $r>0$, $D_1\subset B(0,r)\subset {\mathbb{R}}^n$ be a domain such that $D_1\cap S(0,t)$ is a cap of a sphere in ${\mathbb{R}}^n$ for every $0<t\le r$ and let $D_2$ be a domain such that ${\overline{D}}_2\backslash \{0\}\subset D_1$, $0\in {\overline{D}}_2\cap {\overline{D}}_1$ and there exists $0<a<1$ such that $B(y,a|y|)\subset D_1$ for every $y\in D_2$.

Let $D\subset X$ and $x\in \partial D$ and suppose that there exists a homeomorphism $\Phi :\overline{B}(0,r)\to {\overline{U}}_r$, with $U_r\in \mathcal{V}(x)$ such that $\Phi (0)=x$ and there exist $m,M>0$ such that $l(z,\Phi )>m$, $L(z,\Phi )<M$ for every $z\in \overline{B}(0,r)$ and let $G_1=\Phi (D_1)$ be such that $G_1\backslash \{x\}\subset D\cap U_r$ and let $G_2=\Phi (D_2)$.

Let $f:D\to Y$ be quasimonotone and continuous, satisfying condition $(N^{-1})$, such that $\mu (B_f)=0$, $\nu (f(B_f))=0$ and $\underset{D\cap U_r}{\int }L{(z,f)}^{n}d\mu <\infty$ and suppose that there exists a path $\gamma :[0,1)\to G_1$ such that $\underset{t\to 1}{lim}\gamma (t)=x$ and $\underset{t\to 1}{lim}f(\gamma (t))=c\in Y$. Then, $\underset{\begin{array}{c}y\to x\\ y\in G_2\end{array}}{lim}f(y)=c$.

Proof of Theorem 5.  

Let $\delta >0$ be such that $\Phi (D_1\cap B(0,\delta ))\subset G_1\cap U_r\subset D\cap U_r$. Let $0<\alpha <\beta <\delta$ and let us take points ${\alpha }_t,{\beta }_t\in D_1\cap S(0,t)$ for every $\alpha <t<\beta$ and let ${\Gamma }_t=\Delta ({\alpha }_t,{\beta }_t,D_1\cap S(0,t))$ for $t\in (\alpha ,\beta )$ and let $\Gamma =\bigcup _{t\in (\alpha ,\beta )}{\Gamma }_t$. Let $\rho \in F(\Gamma )$ and we can suppose that $\rho =0$ on $C_{0,\alpha ,\beta }\backslash D_1$. We see from Theorem 10.12 in [3] that there exists a constant $C(n)$ depending only on n such that $M_{S(0,t)}^n({\Gamma }_t)\ge {\displaystyle \frac{C(n)}{t}}$ for every $\alpha <t<\beta$. Here, $M_{S(0,t)}^n({\Gamma }_t)=\underset{\eta \in F({\Gamma }_t)}{inf}\underset{S(0,t)}{\int }{\eta }^n(z)d_{S(0,t)}$ for every $\alpha <t<\beta$. Also, $\rho |S(0,t)\in F({\Gamma }_t)$ for every $t\in (\alpha ,\beta )$. We have

$$C(n)ln({\displaystyle \frac{\beta }{\alpha }})=C(n){\displaystyle \underset{\alpha }{\overset{\beta }{\int }}}{\displaystyle \frac{dt}{t}}\le {\displaystyle \underset{\alpha }{\overset{\beta }{\int }}}(M_{S(0,t)}^n({\Gamma }_t))dt\le {\displaystyle \underset{\alpha }{\overset{\beta }{\int }}}(\underset{S(0,t)}{\int }{\rho }^n(z)d_{S(0,t)})dt=\underset{D_1\cap C_{0,\alpha ,\beta }}{\int }{\rho }^n(z)dz$$

and hence

$$C(n)ln({\displaystyle \frac{\beta }{\alpha }})\le M_n(\Gamma )$$

(13)

Let ${\gamma }_1={\Phi }^{-1}\circ \gamma :[0,1)\to D_1$. Then, ${\gamma }_1$ is a path, ${\gamma }_1(0)\in D_1$ and $\underset{t\to 1}{lim}{\gamma }_1(t)=0$. Let $F=f\circ \Phi :D_1\cap B(0,\delta )\to Y$. Using Lemma 5, we see that F is quasimonotone, continuous and $\underset{t\to 1}{lim}F({\gamma }_1(t))=c$, ${\mu }_n(B_F)=0$, $\nu (F(B_F))=0$, F satisfies condition $(N^{-1})$ and $\underset{D_1\cap B(0,\delta )}{\int }L{(z,F)}^{n}dz<\infty$. We see from Theorem 1 that there exists $\omega \in L^1(Y)$ such that $M_n(\Gamma )\le M_{\omega }^n(F(\Gamma ))$ for every $\Gamma \in A(D_1\cap B(0,\delta ))$ and $\underset{F(A)}{\int }\omega (y)d\nu \le \underset{A}{\int }L{(z,F)}^{n}dz$ for every open set $A\subset D_1\cap B(0,\delta )$. We show that $\underset{\begin{array}{c}y\to 0\\ y\in D_2\end{array}}{lim}F(y)=c$. Indeed, otherwise we find $ϵ>0$ and points $y_k\to 0$, $y_k\in D_2\cap B(0,{\displaystyle \frac{\delta }{2}})$ such that $d(F(y_k),c)\ge 3ϵ$ for every $k\in \mathbb{N}$, $Im{\gamma }_1\cap S(0,t)\cap D_1\ne \varphi$ and we can suppose that $d(F({\gamma }_1(t)),c)\le ϵ$ for every $0<t<\delta$. We also suppose that $B(y_k,2a|y_k|)\subset D_1\cap B(0,\delta )$ for every $k\in \mathbb{N}$. Let $k\in \mathbb{N}$. Since F is quasimonotone, we can find $L>0$ and ${\alpha }_t,{\beta }_t\in D_1\cap S(y_k,t)$ such that $d(F(y),F(y_k))\le Ld(F({\alpha }_t),F({\beta }_t))$ for every $a|y_k|<t<2a|y_k|$ and every $y\in B(y_k,a|y_k|)$. Let ${\eta }_k={\displaystyle \frac{L}{d(F(y),F(y_k))}}{\chi }_{F(B(y_k,2a|y_k|))}$ and let us take $\Gamma =\bigcup _{a|y_k|<t<2a|y_k|}\Delta ({\alpha }_t,{\beta }_t,S(y_k,t)\cap D_1)$.

Then, ${\eta }_k\in F(F(\Gamma ))$, and as before, we have

$$C(n)ln2=C(n)ln({\displaystyle \frac{2a|y_k|}{a|y_k|}})\le M_n(\Gamma )\le M_{\omega }^n(F(\Gamma ))\le \underset{Y}{\int }{\eta }_k{(z)}^n\omega (z)d\nu \le$$

$$\le {\displaystyle \frac{L^n}{d{(F(y),F(y_k))}^n}}\underset{F(B(y_k,2a|y_k|))}{\int }\omega (z)d\nu \le {\displaystyle \frac{L^n}{d{(F(y),F(y_k))}^n}}\underset{B(y_k,2a|y_k|)}{\int }L{(z,F)}^{n}dz.$$

We showed that $d{(F(y),F(y_k))}^n\le {\displaystyle \frac{L^n}{C(n)ln2}}\underset{B(y_k,2a|y_k|)}{\int }L{(z,F)}^{n}dz\to 0$ if $k\to \infty$.

We showed that there exists $k_{ϵ}\in \mathbb{N}$ such that

$$d(F(y),F(y_k))\le ϵforeveryy\in B(y_k,2a|y_k|)andeveryk\ge k_{ϵ}$$

(14)

and we can take $k_{ϵ}=1$. Then, it results that

$$d(F(y),c)\ge 2ϵforeveryy\in B(y_k,2a|y_k|)andeveryk\ge k_{ϵ}.$$

(15)

Let $k\in \mathbb{N}$. We see that $S(0,t)\cap B(y_k,a|y_k|)\ne \varphi$ for every $t\in ((1-a)|y_k|,(1+a)|y_k|)$, and since $Im{\gamma }_1\cap D_1\cap S(0,t)\ne \varphi$ for every $t\in (0,\delta )$, we can find points ${\alpha }_t\in Im{\gamma }_1\cap D_1\cap S(0,t)$, ${\beta }_t\in D_1\cap S(0,t)\cap B(y_k,a|y_k|)$ for every $t\in ((1-a)|y_k|,(1+a)|y_k|)$. Let $\alpha =(1-a)|y_k|$ and $\beta =(1+a)|y_k|$ and let us take in relation (13) the path family $\Gamma =\bigcup _{t\in (\alpha ,\beta )}\Delta ({\alpha }_t,{\beta }_t,D_1\cap S(0,t))$.

Since $d(F({\alpha }_t),F({\beta }_t))\ge ϵ$ for every $\alpha <t<\beta$, we see that ${\rho }_k={\displaystyle \frac{1}{ϵ}}{\chi }_{F(B(0,\delta )\cap D_1)}\in F(F(\Gamma ))$.

We have

$$C(n)ln({\displaystyle \frac{1+a}{1-a}})\le M_n(\Gamma )\le M_{\omega }^n(F(\Gamma ))\le \underset{Y}{\int }{\rho }_k{(z)}^n\omega (z)d\nu \le$$

$$\le {\displaystyle \frac{1}{{ϵ}^n}}\underset{F(B(0,\delta )\cap D_1)}{\int }\omega (z)d\nu \le {\displaystyle \frac{1}{{ϵ}^n}}\underset{B(0,\delta )\cap D_1}{\int }L{(z,F)}^{n}dz\to 0if\delta \to 0.$$

We reached a contradiction and we therefore proved that $\underset{\begin{array}{c}y\to 0\\ y\in D_2\end{array}}{lim}F(y)=c$ and hence $\underset{\begin{array}{c}y\to x\\ y\in G_2\end{array}}{lim}f(y)=c$. □

Our theorem is a Lindelőf-type theorem for mappings of finite conformal energy. This theorem was inspired by the work of M. Vuorinen concerning nontangential limits of quasiregular mappings and angular limits of the mappings having a bounded Dirichlet integral and defined on open sets from ${\mathbb{R}}^n$ (see [4,43,44,45]). As in the corresponding theorem from classical complex analysis or from the theory of quasiregular mappings (see [43]), $D_1$ may be a cone centered at 0, direction d and angle $\phi \in (0,{\displaystyle \frac{\pi }{2}})$ and $D_2$ a cone centered at 0, direction d and angle $0<\psi <\phi$. In our case, $D_1$ and $D_2$ may be a kind of spirallike domain and hence can be more general than cones in ${\mathbb{R}}^n$. The preceding theorem may be a theorem concerning angular limits over domains $G_2$ with $x\in \partial G_2$, which are images of a spirallike domain $D_2\subset {\mathbb{R}}^n$ with $0\in \partial D_2$ by a normal parametrization $\Phi$ of some manifold X. Due to the generality of domains $G_1$ and $G_2$, our theorem is an extension of Lindelőf’s theorem even in ${\mathbb{R}}^n$. We immediately prove the following:

Theorem 6.

Let $n\ge 2$, $r>0$, $D_1\subset B(0,r)\subset {\mathbb{R}}^n$ be a domain such that $D_1\cap S(0,t)$ is a cap of a sphere for every $0<t\le r$ and let $D_2$ be a domain such that ${\overline{D}}_2\backslash \{0\}\subset D_1$, $0\in {\overline{D}}_2\cap {\overline{D}}_1$ and there exists $0<a<1$ such that $B(y,a|y|)\subset D_1$ for every $y\in D_2$. Let $D\subset {\mathbb{R}}^n$ be open such that $0\in \partial D$, $D_1\backslash \{0\}\subset D$ and let $f:D\to {\mathbb{R}}^n$ be a mapping of finite distortion satisfying relation (4) such that $\underset{D\cap B(0,r)}{\int }L{(z,f)}^{n}dz<\infty$ and suppose that there exists a path $\gamma :[0,1)\to D_1$ such that $\underset{t\to 1}{lim}\gamma (t)=0$ and $\underset{t\to 1}{lim}f(\gamma (t))=c\in {\mathbb{R}}^n$. Then, $\underset{\begin{array}{c}y\to 0\\ y\in D_2\end{array}}{lim}f(y)=c$.

We remind that Miklyukov showed in [48] that if $f:B^n\to {\mathbb{R}}^n$ is a bounded quasiregular mapping and $\underset{B^n}{\int }{|f^{\prime }(x)|}^{n}dx<\infty$, then f has a.e. nontangential limits. Theorem 4 and Theorem 6 generalized this result in the class of mappings of finite distortion satisfying relation (4).

Corollary 1.

Let $n\ge 2$, let X be a Riemannian n-manifold, let $G\subset X$ be open and $x\in \partial G$ be an isolated point of $\partial G$. Let $f:G\to Y$ be continuous and quasimonotone, satisfying condition $(N^{-1})$ such that $\mu (B_f)=0$, $\nu (f(B_f))=0$ and $\underset{G}{\int }L{(z,f)}^{n}d\mu <\infty$. Suppose that there exists a path $\gamma :[0,1)\to G$ such that $\underset{t\to 1}{lim}\gamma (t)=x$ and $\underset{t\to 1}{lim}f(\gamma (t))=c$. Then, $\underset{y\to x}{lim}f(y)=c$.

Remark 6.

Usually in Theorem 5, we take X a Riemannian n-manifold and the mapping $\Phi :\overline{B}(0,r_x)\to {\overline{U}}_r$ a $C^1$ diffeomorphism. Also, in Theorem 3, we can take X a Riemannian n-manifold and $\Phi :{\overline{B}}^n\to \overline{D}$ a $C^1$ diffeomorphism. If $D\subset {\mathbb{R}}^n$ is a domain and $\Phi :{\overline{B}}^n\to \overline{D}$ is a homeomorphism such that there exist $m,M>0$ such that $l(z,\Phi )>m$, $L(z,\Phi )<M$ for every $z\in {\overline{B}}^n$ and $f:D\to {\mathbb{R}}^n$ is a mapping of biconformal energy, then there exists $\underset{t\to 1}{lim}f(\Phi (t{\Phi }^{-1}(y)))$ for ${\mathcal{H}}^{n-1}$ a.e. $y\in \partial D$.

Theorem 7.

Let $n\ge 2$, let $X,Y$ be Riemannian n-manifolds, let $D\subset X$ be a domain, let $\omega :D\to [0,\infty ]$ be μ measurable and finite a.e. and let $f:D\to D_f$ be a homeomorphism such that $M_n(f(\Gamma ))\le M_{\omega }^n(\Gamma )$ for every $\Gamma \in A(D)$. Let $x\in D$ be such that ${\omega }_x<\infty$. Then, there exist constants $C_1,C_2,C_3,C_4$ and a function ${\eta }_x:(0,\infty )\to (0,\infty )$ such that $\underset{t\to 0}{lim}{\eta }_x(t)=0$, ${\eta }_x$ is increasing on each interval $(0,1)$ and $(1,\infty )$,

$${\eta }_x(t)=\left\{\begin{array}{c}(C_1/{\mathcal{H}}_n^{-1}({\displaystyle \frac{C_2{\omega }_x}{{(lnln(\frac{e}{t}))}^n}})),if0<t<1\hfill \\ C_{4}exp(C_3{\omega }_x{(2+t)}^n),ift\ge 1.\hfill \end{array}\right.$$

and there exists $r_x>0$ such that

$${\displaystyle \frac{d(f(z),f(x))}{d(f(y),f(x))}}\le {\eta }_x({\displaystyle \frac{d(x,z)}{d(x,y)}})ify,z\in B(x,r_x),x\ne y,x\ne z.$$

(16)

Proof of Theorem 7.  

Let $r_x>0$ and $\Phi :\overline{B}(0,r_x)\to \overline{B}(x,r_x)$ be a $C^1$ diffeomorphism such that $\Phi (0)=x$ and $\Phi (S(0,t))=S(x,t)$ for every $0<t<r_x$ and let $V_x\in \mathcal{V}(f(x))$ be such that there exists $\psi :B(0,1)\to V_x$ a $C^1$ diffeomorphism such that $\psi (0)=f(x)$ and $f(\overline{B}(x,r_x))\subset V_x$. Let $y,z\in B(x,r_x)$ be such that $0<d(x,y)<d(x,z)<{\displaystyle \frac{r_x}{e}}$, and let $a,b\in B(0,r_x)$ be such that $\Phi (a)=y$ and $\Phi (b)=z$. Let $F:B(0,1)\to B(0,1)$, $F={\psi }^{-1}\circ f\circ \Phi$. Then, $F:B(0,1)\to F(B(0,1))$ is a homeomorphism between open sets from ${\mathbb{R}}^n$, $F(0)=0$ and let $\Gamma ={\Gamma }_{0,|a|,|b|}$. Then, $F(\Gamma )$ is a ring $R(C_0,C_1)$ and $F(0),F(a)\in C_0$, $F(b),\infty \in C_1$. Let L be such that ${\displaystyle \frac{1}{L}}d(\alpha ,\beta )\le d({\psi }^{-1}(\beta ),{\psi }^{-1}(\alpha ))\le Ld(\alpha ,\beta )$ for every $\alpha ,\beta \in V_x$. Using Lemma 1 and Lemma 7, we have the following:

$${\mathcal{H}}_n(L^2{\displaystyle \frac{d(f(z),f(x))}{d(f(y),f(x))}})={\mathcal{H}}_n({\displaystyle \frac{L^{2}d(f\circ \Phi (b),f\circ \Phi (0))}{d(f\circ \Phi (a),f\circ \Phi (0))}})\le {\mathcal{H}}_n({\displaystyle \frac{d(F(b),F(0))}{d(F(a),F(0))}})\le$$

$$\le M_n(F(\Gamma ))\le C{M}_n(f\circ \Phi (\Gamma ))\le C{M}_{\omega }^n(\Phi (\Gamma ))=C{M}_{\omega }^n({\Gamma }_{x,d(x,y),d(x,z)})\le {\displaystyle \frac{C{\omega }_x}{{(lnln(\frac{ed(x,z)}{d(x,y)}))}^n}}$$

if $0<d(x,y)<d(x,z)<{\displaystyle \frac{r_x}{e}}$. We proved that

$${\displaystyle \frac{d(f(z),f(x))}{d(f(y),f(x))}}\ge {\displaystyle \frac{1}{L^2}}{\mathcal{H}}_n^{-1}({\displaystyle \frac{C{\omega }_x}{{(lnln(\frac{ed(x,z)}{d(x,y)}))}^n}})$$

if $0<d(x,y)<d(x,z)<{\displaystyle \frac{r_x}{e}}$.

Suppose now that ${\displaystyle \frac{1}{L_1}}d(\alpha ,\beta )\le d(\Phi (\alpha ),\Phi (\beta ))\le L_{1}d(\alpha ,\beta )$ if $\alpha ,\beta \in \overline{B}(0,r_x)$. Let $y,z\in B(x,r_x)$ be such that $0<d(x,y)\le d(x,z)\le 2d(x,y)+d(x,z)<r_x$ and let $a,b\in B(0,r_x)$ be such that $\Phi (a)=y$, $\Phi (b)=z$, and as before, let $F={\psi }^{-1}\circ f\circ \Phi$. Let $w\in S(0,|a|+|b|)$ be such that $L(0,F,|a|+|b|)=d(F(0),F(w))$ and let $B\in S(0,2|a|+|b|)$ be such that $L(0,F,2|a|+|b|)=d(F(0),F(B))$. Let P be the plane determined by the point $0,a,w$ and let C be the circle $P\cap S(a,|a|)$. Let E be the tangency point of C of a ray emerging from 0 to w and let $\gamma$ be the shortest arc in the circle C joining 0 with E and let $E_1=[E,w]\cup |\gamma |$. Let d be the line perpendicular on the plane P in the point a and let $M=d\cap S(0,2|a|+|b|)$. Let $E_2=[a,M]\cup S(0,2|a|+|b|)$. Then, $d(E_1,E_2)\ge |a|$ and $d(F(0),F(a))\le L(0,F,|a|)\le L(0,F,|a|+|b|)=d(F(0),F(w))\le L(0,F,2|a|+|b|)=d(F(0),F(B))$. Let $\Delta =\Delta (E_1,E_2,B(0,2|a|+|b|))$. Then, ${\displaystyle \frac{1}{|a|}}{\chi }_{B(0,2|a|+|b|)}\in F(\Delta )$. Let ${\Gamma }^{\prime }=\Delta (F(E_1),F(E_2),C_{0,d(F(0),F(a)),d(F(0),F(w))})$.

We see that ${\Gamma }^{\prime }>F(\Delta )$, that $f(E_1)$ is a path joining $F(0)$ with $F(w)$ and that $F(E_2)$ contains a subpath joining $F(a)$ with $F(B)$, and since $d(F(0),F(w))\le d(F(0),F(B))$, we see from (13) that $M_n({\Gamma }^{\prime })\ge C(n)ln({\displaystyle \frac{d(F(0),F(w))}{d(F(0),F(a))}})$. We see that $d(\Phi (E_1),\Phi (E_2))\ge {\displaystyle \frac{d(E_1,E_2)}{L_1}}\ge {\displaystyle \frac{d(x,y)}{L_1}}$ and hence $\rho ={\displaystyle \frac{L_1}{d(x,y)}}{\chi }_{(B(x,2d(x,y)+d(x,z)))}\in F(\Phi (\Delta ))$. We have the following:

$$C(n)ln({\displaystyle \frac{d(f(z),f(x))}{L^{2}d(f(y),f(x))}})=C(n)ln({\displaystyle \frac{d(f(\Phi (b))),d(f(\Phi (0)))}{L^{2}d(f(\Phi (a)),f(\Phi (0)))}})\le$$

$$C(n)ln({\displaystyle \frac{d(F(b),F(0))}{d(F(a),F(0))}})\le C(n)ln({\displaystyle \frac{d(F(w),F(0))}{d(F(a),F(0))}})\le M_n({\Gamma }^{\prime })\le$$

$$\le M_n(F(\Delta ))=M_n({\mathsf{\Psi }}^{-1}\circ f\circ \Phi (\Delta ))\le C{M}_n(f\circ \Phi (\Delta ))\le$$

$$C{M}_{\omega }^n(\Phi (\Delta ))\le C\underset{X}{\int }{\rho }^n(z)\omega (z)d\mu \le {\displaystyle \frac{L_1^{n}C}{d{(x,y)}^n}}\underset{B(x,2d(x,y)+d(x,z))}{\int }\omega (u)du\le$$

$$\le (if{r}_{x}issmallenough)\le {\displaystyle \frac{C{L}_1^n{(2d(x,y)+d(x,z))}^n}{d{(x,y)}^n}}\underset{B(x,2d(x,y)+d(x,z))}{⨏}\omega (u)du\le$$

$$\le C{L}_1^n{\omega }_x{(2+{\displaystyle \frac{d(x,z)}{d(x,y)}})}^{n}if0<d(x,y)\le d(x,z)<2d(x,y)+d(x,z)<r_x.$$

We proved that

$${\displaystyle \frac{d(f(z),f(x))}{d(f(y),f(x))}}\le L^{2}exp{({\displaystyle \frac{C{L}_1^n{\omega }_x}{C(n)}}(2+{\displaystyle \frac{d(x,z)}{d(x,y)}}))}^n$$

if $0<d(x,y)\le d(x,z)<2d(x,y)+2d(x,z)<r_x$. □

Corollary 2.

Let $f:D\to D_f$ be a homeomorphism satisfying condition $(N)$ and such that $\underset{Y}{\int }L{(y,g)}^{n}d\nu <\infty$, where $g:D_f\to D$ is the inverse of f. Then, relation (16) is satisfied for a.e. $x\in D$ and $H(x,f)<\infty$ a.e.

Proof of Corollary 2. 

We see from Theorem 1 that there exists $\omega \in L^1(X)$ such that $M_n({\Gamma }^{\prime })\le M_{\omega }^n(g(\Gamma ))$ for every ${\Gamma }^{\prime }\in A(D_f)$ and this implies that $M_n(f(\Gamma ))\le M_{\omega }^n(\Gamma )$ for every $\Gamma \in A(D)$. Since $\omega \in L^1(X)$, we see that ${\omega }_x<\infty$$\mu$ a.e. in D and we apply Theorem 7. Also, the proof of Corollary 3 follows immediately from Corollary 2. □

Corollary 3.

Let $f:D\to D_f$ be a homeomorphism between two domains in ${\mathbb{R}}^n$ such that $f\in W_{loc}^{1,n}(D,D_f)$ and $f^{-1}\in W_{loc}^{1,n}(D_f,D)$. Then, relation (16) holds for a.e. $x\in D$.

Theorem 8.

Let $n\ge 2$, $D\subset {\mathbb{R}}^n$ a domain, $a,b\in D$, $a\ne b$, $n-1<q\le n$ and $r>0$ be such that $H_D^q\ne \varphi$. Then, there exists an injective minimizer $f\in H_D^q$ for the total energy $E_t^q(D)$ such that $E_{t,D}^q(f)=E_t^q(D)$. Also, if $H_{D,M}^q\ne \varphi$, there exist injective minimizers $f_1,f_2\in H_{D,M}^q$ such that $E_M^q(D)=E_D(f_1)$ and $E_{t,M}^q(D)=E_{t,D}^q(f_2)$ and $f_1$ and $f_2$ may be different.

Proof of Theorem 8. 

Let $I=E_t^q(D)<\infty$ and let $f_j\in H_D^q$ be such that $I<E_{t,D}^q(f_j)<I+{\displaystyle \frac{1}{j}}$ for every $j\in \mathbb{N}$. We see from Corollary 7.5.1 in [6] that the family ${(f_j)}_{j\in \mathbb{N}}$ is equicontinuous and, eventually taking a subsequence, we find $f:D\to {\mathbb{R}}^n$ such that $f_j\to f$ uniformly on the compact subsets D. We see from Lemma 4 that $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$ and that $\underset{D}{\int }|f^{\prime }{(x)|}^{n}dx\le \underset{j\to \infty }{lim\; inf}\underset{D}{\int }{|f_j^{\prime }(x)|}^{n}dx=I$. Let $g_j:D_{f_j}\to D$ be the inverse of $f_j:D\to D_{f_j}$ for every $j\in \mathbb{N}$. Then, $g_j$ is a.e. differentiable, satisfies condition $(N^{-1})$ and $\underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy=\underset{D}{\int }{K}_{I,q}(x,f_j)\le E_{t,D}^q(f_j)<\infty$ for every $j\in \mathbb{N}$. We see from Theorem 1 that there exists ${\omega }_j\in L^1({\mathbb{R}}^n)$ such that $M_q({\Gamma }^{\prime })\le M_{{\omega }_j}^q(g_j({\Gamma }^{\prime }))$ for every ${\Gamma }^{\prime }\in A(D_{f_j})$ and every $j\in \mathbb{N}$ and hence $M_q(f_j(\Gamma ))\le M_{{\omega }_j}^q(\Gamma )$ for every $\Gamma \in A(D)$ and every $j\in \mathbb{N}$. We see that

$$\underset{D}{\int }{\omega }_j(z)dz=\underset{g_j(f_j(D))}{\int }{\omega }_j(z)dz\le \underset{f_j(D)}{\int }L{(y,g_j)}^{q}dy=\underset{f_j(D)}{\int }{|g_j^{\prime }(y)|}^{q}dy=$$

$$\underset{D}{\int }{K}_{I,q}(x,f_j)dx\le E_{t,D}^q(f_j)<I+1$$

for every $j\in \mathbb{N}$.

We see from Lemma 3 that either f is constant on D or there exists a domain $G\subset {\mathbb{R}}^n$ such that $f:D\to G$ is a homeomorphism and $f_j\to f$ uniformly on the compact subsets of D. Since $q(f_j(a),f_j(b))\ge r$ for every $j\in \mathbb{N}$, we see that $q(f(a),f(b))\ge r>0$ and hence there exists a domain $G\subset {\mathbb{R}}^n$ such that $f:D\to G$ is a homeomorphism. Using Lemma 1 in [13], we find that G is a component of $Ke{r}_{j\to \infty }{D}_{f_j}$ and let $g:G\to D$ be the inverse of $f:D\to G$. Let now K and U be domains such that $K\subset \subset G$ and $U\subset \subset D$ and $K\subset f(U)$. Then, $K\subset f(U)\subset Ke{r}_{j\to \infty }{f}_j(U)$. Let us show that there exists $j_0\in \mathbb{N}$ such that $g_j(K)\subset \overline{U}$ for $j\ge j_0$. Indeed, otherwise we find infinitely many points $a_j\in K$ such that ${\alpha }_j=g_j(a_j)\notin \overline{U}$. Then, $f_j({\alpha }_j)=a_j\in K\subset f(U)\subset Ke{r}_{j\to \infty }{f}_j(U)$; and for j great enough, we see that $f_j({\alpha }_j)\subset f_j(U)$ and hence we find ${\beta }_j\in U$ such that $f_j({\alpha }_j)=f_j({\beta }_j)$; and since ${\alpha }_j\ne {\beta }_j$, this contradicts the injectivity of the mapping $f_j$. We proved that there exists $j_0\in \mathbb{N}$ such that $g_j(K)\subset \overline{U}$ for every $j\ge j_0$ and we can suppose that $K\subset f(U)\subset D_{f_j}$ for every $j\ge j_0$. We can take $j_0=1$ and then the mappings $g_j|K:K\to D$ are well defined for every $j\in \mathbb{N}$. The mappings $g_j$ satisfy condition $(N^{-1})$ and $\underset{K}{\int }|g_j^{\prime }{(y)|}^{q}dy\le \underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy=\underset{D}{\int }{K}_{I,q}(x,f_j)dx\le E_{t,D}^q(f_j)<I+1$ for every $j\in \mathbb{N}$ and we see from Theorem 2 that the family $W={(g_j|K)}_{j\in \mathbb{N}}$ is equicontinuous. In fact, the family $W_1={(g_j|K_1)}_{j\in \mathbb{N}}$ is equicontinuous even if $K\subset \subset K_1$ and $K_1\subset f(U)$. We can suppose that there exists $h:\overline{K}\to {\mathbb{R}}^n$ such that $g_j\to h$ uniformly on $\overline{K}$. We see that $h(\overline{K})\subset \overline{U}\subset D$ and let us show that $g_j\to g$ uniformly on $\overline{K}$. Indeed, otherwise we can find $ϵ>0$ and $z_j,z\in \overline{K}$ such that $z_j\to z$ and $|g_j(z_j)-g(z_j)|>ϵ$ for every $j\in \mathbb{N}$. Now, $w_j=g_j(z_j)\to h(z)=w\in \overline{U}\subset D$ and let $x_j=g(z_j)$ for every $j\in \mathbb{N}$. We can suppose that $x_j\to x\in g(\overline{K})\subset D$, and since the family ${(f_j)}_{j\in \mathbb{N}}$ is equicontinuous at w, we find that $f_j(w_j)\to f(w)$. Then, $f_j(w_j)=f_j(g_j(z_j))=z_j=f(x_j)\to f(x)$; and letting $j\to \infty$, we find that $f(w)=f(x)$; and since f is injective, it results that $x=w$. On the other side, $|x_j-w_j|=|g(z_j)-g_j(z_j)|>ϵ$ for every $j\in \mathbb{N}$ and then $|x-w|\ge ϵ$ and we reached a contradiction.

We proved that $g_j\to g$ uniformly on K and hence $g_j\to g$ in $W_{loc}^{1,q}(K,{\mathbb{R}}^n)$. As before, using Lemma 4, we have

$$\underset{K}{\int }|g^{\prime }{(y)|}^{q}dy\le \underset{j\to \infty }{lim\; inf}\underset{K}{\int }|g_j^{\prime }{(y)|}^{q}dy\le \underset{j\to \infty }{lim\; inf}\underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy$$

and letting $K\nearrow G$, we find that $g\in W_{loc}^{1,q}(G,{\mathbb{R}}^n)$ and

$$\underset{G}{\int }|g^{\prime }{(y)|}^{q}dy\le \underset{j\to \infty }{lim\; inf}\underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy.$$

We have

$$E_{t,D}^q(f)=\underset{D}{\int }(|f^{\prime }{(x)|}^n+K_{I,q}(x,f))dx=\underset{D}{\int }|f^{\prime }{(x)|}^{n}dx+\underset{G}{\int }{|g^{\prime }(y)|}^{q}dy\le$$

$$\le \underset{j\to \infty }{lim\; inf}\underset{D}{\int }|f_j^{\prime }{(x)|}^{n}dx+\underset{j\to \infty }{lim\; inf}\underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy=$$

$$=\underset{j\to \infty }{lim\; inf}\underset{D}{\int }{|f_j^{\prime }(x)|}^{n}dx+\underset{j\to \infty }{lim\; inf}\underset{D}{\int }{K}_{I,q}(x,f_j)dx\le$$

$$\le \underset{j\to \infty }{lim\; inf}\underset{D}{\int }(|f_j^{\prime }(x){|}^n+K_{I,q}(x,f_j))dx=\underset{j\to \infty }{lim\; inf}{E}_{t,D}^q(f_j)=I.$$

We proved that $f\in H_D^q$ and that $E_{t,D}^q(f)=E_t^q(D)$.

Suppose now that $H_{D,M}^q\ne \varphi$. Then, $I=E_M^q(D)\le E_{t,M}^q(D)<M$ and let $f_j\in H_{D,M}^q$ be such that $I<E_D(f_j)<I+{\displaystyle \frac{1}{j}}$ for every $j\in \mathbb{N}$. As before, taking eventually a subsequence, we can suppose that there exists $f\in W_{loc}^{1,n}(D,{\mathbb{R}}^n)$ such that $f_j\to f$ uniformly on the compact subsets of D and $\underset{D}{\int }|f^{\prime }{(x)|}^{n}dx\le \underset{j\to \infty }{lim\; inf}\underset{D}{\int }{|f_j^{\prime }(x)|}^{n}dx$. Let $g_j:D_{f_j}\to D$ be the inverse of $f_j:D\to D_{f_j}$ for every $j\in \mathbb{N}$. Then, $g_j$ satisfies condition $(N^{-1})$, is a.e. differentiable and $\underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy=\underset{D}{\int }{K}_{I,q}(x,f_j)dx<M$ for every $j\in \mathbb{N}$. Using Theorem 1, we find ${\omega }_j\in L^1({\mathbb{R}}^n)$ such that $M_q({\Gamma }^{\prime })\le M_{{\omega }_j}^q(g_j({\Gamma }^{\prime }))$ for every ${\Gamma }^{\prime }\in A(D_{f_j})$ and every $j\in \mathbb{N}$ and hence $M_q(f_j(\Gamma ))\le M_{{\omega }_j}^q(\Gamma )$ for every $\Gamma \in A(D)$ and every $j\in \mathbb{N}$. Then, $\underset{D}{\int }{\omega }_j(z)dz=\underset{g_j(f_j(D))}{\int }{\omega }_j(z)dz\le \underset{D_{f_j}}{\int }L{(z,g_j)}^{q}dz=\underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy=\underset{D}{\int }{K}_{I,q}(x,f_j)dx\le E_{t,D}^q(f_j)<M$ for every $j\in \mathbb{N}$.

Using Lemma 3, we find a domain $G\subset {\mathbb{R}}^n$ such that $f:D\to G$ is a homeomorphism, and if $g:G\to D$ is the inverse of $f:D\to G$, we prove as before that $\underset{G}{\int }|g^{\prime }{(y)|}^{q}dy\le \underset{j\to \infty }{lim\; inf}\underset{D_{f_j}}{\int }{|g_j^{\prime }(y)|}^{q}dy$ and hence that $E_{t,D}^q(f)\le \underset{j\to \infty }{lim}{E}_{t,D}^q(f_j)<M$. We proved that $f\in H_{D,M}^q$ and that $E_M^q(D)=E_D(f)$. □

5. Conclusions

Vuorinen (see [4,43,44,45]) was the first who remarked that the mappings of a bounded Dirichlet integral satisfy a modular inequality. He used this method to prove some geometric properties of such mappings. We develop his method and, by showing that such mappings satisfy an inverse Poletsky modular inequality, we can obtain strong generalizations of some classical theorems from complex analysis. We open the systematic study of geometric properties of some classes of mappings of finite distortion using the modulus method. The conditions for finding injective minimizers may be useful both in theoretical and applied mathematics.