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This article is cited in 4 scientific papers (total in 4 papers) The boundary behavior of S. K. Vodopyanova, A. O. Molchanovab a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia b University of Vienna, Vienna, Austria
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2023, Volume 87, Issue 4, DOI: https://doi.org/10.4213/im9376 
Bibliographic databases:
    IntroductionIn this section, we briefly survey the articles dealing with the boundary behaviour of mappings in quasiconformal analysis. Consider two domains $D,D' \subset \mathbb R^2$ bounded by Jordan curves and a conformal mapping $f\colon D \to D'$. The classical result, established independently by Carathéodory [1] and Osgood and Taylor [2], asserts that $f$ extends to the boundary, giving a homeomorphism $\overline{f}\colon \overline{D} \to \overline{D'}$. The Jordan condition for the boundary is necessary, which is easy to see in the example of a slit disc. Nevertheless, a homeomorphic extension is possible for some generalized boundary accounting for the geometry of the domain. This construction, introduced by Carathéodory [1] and called the prime end boundary, initiated intensive applications of the geometric approach to study the boundary behaviour of mappings. Carathéodory’s prime end theory received developments on the plane $\mathbb{R}^2$ (see [3], [4]) and in the space $\mathbb{R}^n$ for $n> 2$ (see [5], [6]), in studying Dirichlet problems for elliptic equations [7], and in the theory of dynamical systems (see [8], [9]). For more detailed surveys of the available results and literature, see [10]–[13]. A natural development of these questions is to study the boundary behaviour of quasiconformal mappings in space. It requires a more refined analysis of the geometric properties of domains. Indeed, in the higher-dimensional case there exist a Jordan domain and a quasiconformal mapping admitting no homeomorphic extension to the boundary of this domain [14]. In some questions it turned out helpful to describe the geometric properties of domains using the concept of modulus of a curve family [15]. With that, a simple classification of boundary points was introduced: for instance, the properties of the boundary to be quasiconformally flat or quasiconformally accessible in [16], [17], or properties $P1$ and $P2$ of [18]. This approach became widely used in the last decade to study the geometric properties of mappings. Let us mention only some articles concerning the boundary correspondence of quasiconformal mappings [19], [20], $Q$-homeomorphisms, see the book [21] and the articles [13], [22] (a more detailed discussion appears in § 4), as well as the mappings satisfying generalized modular inequalities [23]. An alternative functional-geometric approach to study the boundary behaviour of quasiconformal mappings is based on the relation between the Euclidean geometry of the domain and the functional space $L^1_n$ via the concept of the variational capacity of a condenser. This approach was founded in [24]–[26] and applied also to studying mappings which are not quasiconformal [27]. As [17] shows, the functional-geometric approach can be interpreted in the language of moduli of curve families. The three main approaches to the boundary behaviour of mappings, using prime ends, geometric description, and functional-geometric definition, form an hierarchy, as each of them adequately describes the boundary behaviour of certain classes of mappings. This article studies the problem of boundary correspondence for $\mathcal Q_{p,q}$-homeomorphisms, whose fundamental properties were established in [29]–[34]. To this end, we complete the domains in special capacity metrics on the image and the preimage, associated with the geometry of a suitable Sobolev class. The elements adjoined to the domain in the completion of the corresponding metric space constitute an improper boundary, which we call the capacity boundary $H_{\rho}$. In § 2 the study of the boundary behaviour of the homeomorphism $f\in\mathcal Q_{p,q}$ defined in § 1 consists in: (1) continuing $f$ to the capacity boundary $H_{\rho}$, with the main result stated as Theorem 2.22; (2) establishing a connection between the elements of the capacity boundary and the points of the Euclidean boundary of the domain, see Theorem 2.37 and Corollaries 2.38 and 2.39. In § 3, we compare the approaches stated in the languages of moduli and capacity. In § 4 we contrast the conclusions of this article with the main results of other approaches to the problem of boundary behaviour of mappings. Some applications of our results are given in § 5. This article naturally enters the line of publications [28]–[36], preceded by the results of [37]–[39] and the articles cited in the bibliographies in [28]–[34] and arising on the crossroads of the theory of Sobolev function spaces [40], [41] and geometric theory of functions [18], [42]–[48]. Some results of this series of articles have found applications in nonlinear elasticity, see [49]. § 1. Classes of $\mathcal Q_{p,q}$-homeomorphismsIn what follows $D$ and $D'$ stand for domains (open connected sets) in $\mathbb{R}^n$. The norm $|x|_p$ of a vector $x=(x_1,x_2,\dots,x_n)\in\mathbb R^n$ is defined as $|x|_p=\bigl(\sum_{k=1}^n|x_k|^p\bigr)^{1/p}$ for $p\in[1,\infty)$ and $|x|_\infty=\max_{k=1,\dots,n}|x_k|$. A ball in the norm $|x|_2$ is a Euclidean ball, while in the norm $|x|_\infty$ it is a Euclidean cube. 1.1. Definitions of Sobolev spaces and the capacity of condensersFor the general theory of Sobolev spaces, the reader is referred to [40], [41]. We recall that a function $u\colon D\to\mathbb R$ is of Sobolev class $L^1_{p}(D)$ if $u\in L_{1,\mathrm{loc}}(D)$, meaning that $u\in L_1(U)$ for every domain $U$ compactly embedded into $D$, written $U\Subset D$, and it has the generalized derivatives $\partial u/dx_j\in L_{1,\mathrm{loc}}(D)$ for every $j=1,\dots,n$ and finite seminorm
$$
\begin{equation*}
\|u\mid L^1_{p}(D)\|=\biggl(\int_{D}|\nabla u(y)|^p\,dy\biggr)^{1/p},\qquad 1\leqslant p\leqslant \infty,
\end{equation*}
\notag
$$
where $\nabla u(y)=(\partial u/dx_1,\partial u/dx_2,\dots,\partial u/dx_n)$ is the generalized gradient of $u$. A mapping $\varphi=(\varphi_1,\dots, \varphi_n)\colon D \to \mathbb R^n$ belongs to the Sobolev class $W^1_{p,\mathrm{loc}}(D; \mathbb{R}^n)$ whenever $\varphi_j(x) \in L_{p,\mathrm{loc}}(D)$ and $\partial\varphi_j/dx_i\in L_{p,\mathrm{loc}}(D)$ for all $j,\,i=1,\dots,n$. We say that a mapping $\varphi\colon D\to \mathbb R^n$ of Sobolev class $W^1_{1,\mathrm{loc}}(D;\mathbb{R}^n)$ is a mapping with finite distortion whenever
$$
\begin{equation}
D\varphi(x)=0\text{ almost everywhere (a.e.) on the set }Z=\{x\in D\colon \det D\varphi (x)=0\}.
\end{equation}
\tag{1.1}
$$
(Meaning $\det D\varphi (x)=0$ at all points of $Z$ except for a set of Lebesgue measure zero.) Here, and henceforth, $D\varphi (x)=(\partial\varphi_j(x)/\partial x_i)_{i,j=1}^{n}$ stands for the Jacobi matrix of the mapping $\varphi$ at $x\in D$, while $|D\varphi (x)|$, for its Euclidean operator norm, and $\det D\varphi (x)$, for its determinant, the Jacobian. A locally integrable function $\omega\colon D'\to\mathbb R$ is called a weight whenever $0\,{<}\,\omega(y)\,{<}\,\infty$ for a.e. $y\in D'$. A function $u\colon D'\to\mathbb R$ belongs to the weighted Sobolev class $L^1_{p}(D';\omega)$, with $p\in[1,\infty)$, if $u \in L_{1,\mathrm{loc}}(D')$ and $\partial u/\partial y_j \in L_{p}(D';\omega)$ for every $j=1,\dots,n$. The seminorm of a function $u\in L^1_{p}(D';\omega)$ is then defined as
$$
\begin{equation}
\|u\mid L^1_{p}(D';\omega)\|=\biggl(\int_{D'}|\nabla u(y)|^p\omega(y)\,dy\biggr)^{1/p}.
\end{equation}
\tag{1.2}
$$
In the case $\omega\equiv 1$, instead of $L^1_{p}(D';1)$ we write simply $L^1_{p}(D')$. Henceforth, the symbol $\operatorname{Lip}_{\mathrm{loc}}(D')$ stands for the space of locally Lipschitz functions on $D'$. It is obvious that
$$
\begin{equation*}
\operatorname{Lip}_{\mathrm{loc}}(D')=W^1_{\infty,\mathrm{loc}}(D')\cap C(D'),
\end{equation*}
\notag
$$
where $W^1_{\infty,\mathrm{loc}}(D')$ is the space of locally bounded measurable functions on $D'$ with locally bounded generalized derivative. We say that a homeomorphism $\varphi \colon D \to D'$ induces the bounded composition operator
$$
\begin{equation*}
\varphi^* \colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D), \qquad 1\leqslant q \leqslant p < \infty,
\end{equation*}
\notag
$$
acting as $D\ni x\mapsto(\varphi^*u)(x)=u(\varphi(x))$, whenever for some constant $K_{q,p}<\infty$ the inequality
$$
\begin{equation*}
\|\varphi^*u\mid L^1_q(D)\|\leqslant K_{q,p}\|u\mid L^1_p(D';\omega)\|
\end{equation*}
\notag
$$
holds for every function $u\in L^1_p(D')\cap \operatorname{Lip}_{\mathrm{loc}}(D')$. 1.2. Condensers and their capacity in Sobolev spacesA condenser in a domain $D\subset \mathbb{R}^n$ is a pair $\mathcal E=(F_1,F_0)$ of connected compact sets (continua) $F_1$, $F_0\subset D$. For a continuum $F\subset U$, where $U\Subset D$ is an open connected compactly embedded set, we denote the condenser $\mathcal E=(F,\partial U)$ by $\mathcal E=(F,U)$. A condenser $\mathcal E=(F, U)$ is called annular whenever the complement in $\mathbb R^n$ to the open set $U\setminus F$ consists of two closed sets each of which is connected: the bounded connected component is the continuum $F$, and the unbounded component is $\mathbb R^n\setminus U$. A condenser $\mathcal E=(F, U)$ in $\mathbb R^n$ is called spherical whenever $U=B(x,R)=\{y\in\mathbb R^n \colon |y-x|_2< R\}$ and $F=\overline{B(x,r)}=\{y\in\mathbb R^n \colon |y-x|_2 \leqslant r\}$, where $r<R$, and cubical whenever $U=Q(x,R)=\{y\in\mathbb R^n \colon |y-x|_\infty< R\}$ and $F=\overline{Q(x,r)}=\{y\in\mathbb R^n \colon |y-x|_\infty\leqslant r\}$, respectively. Definition 1.1. A function $u\colon D\to\mathbb R$ of the class $W^1_{1,\mathrm{loc}}(D)$ is called admissible for a condenser $\mathcal E=(F_1,F_0)\subset D$ whenever (1) $u$ is continuous, (2) $u\equiv 1$ on $F_1$, and (3) $u \equiv 0$ on $F_0$. We denote the collection of admissible functions for a condenser $\mathcal E=(F_1,F_0)$ by $\mathcal A(\mathcal E)$. The capacity of a condenser $\mathcal E=(F_1,F_0)$ in the space $L^1_q(D)$ with $q\in[1,\infty)$ is defined as
$$
\begin{equation}
\operatorname{cap}\bigl(\mathcal E; L^1_q(D)\bigr)=\inf_{u}\|u\mid L^1_{q}(D)\|^q,
\end{equation}
\tag{1.3}
$$
where the infimum is taken over all admissible functions $u\in \mathcal A(\mathcal E)\cap L^1_{q}(D)$ for the condenser $\mathcal E=(F_1,F_0)\subset D$. Let us now define the weighted capacity of a condenser $\mathcal E=(F_1,F_0)\subset D'$ in the space $L^1_p(D';\omega)$ by analogy with (1.3): where the infimum is over all admissible functions $u \in\mathcal A(\mathcal E)\cap \operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ for the condenser $\mathcal E=(F_1,F_0)$.See the books [41], [44], which present the properties of capacity in Sobolev spaces. For more details on the properties of weighted capacity (for a special class of admissible weights), see [50], Chap. 2. The definition of capacity yields the following property. Property 1.2 (subordination principle). Consider two condensers $\mathcal E'\,{=}\,(F'_1,F'_0)$ and $\mathcal E=(F_1,F_0)$ in a domain $D'$ with the plates of the first condenser included in those of the second one, $F'_1\subset F_1$ and $F'_0\subset F_0$. Then 1.3. A quasi-additive set function and its propertiesDenote by ${\mathcal O}(D)$ a system of open sets in $D$ with the following properties: (1) $D\in{\mathcal O}(D)$ and if the closure of an open ball $B$ (cube $Q$) lies in $D$, then $B\in{\mathcal O}(D)$ ($Q\in{\mathcal O}(D)$); (2) if $U_1,\dots,U_k\in{\mathcal O}(D)$ is a disjoint system of open sets, then $\bigcup_{i=1}^kU_i\in{\mathcal O}(D)$, where $k\in \mathbb N$ is an arbitrary number. The choice of a ball or cube in this definition depends on the choice of a system of elementary sets with respect to which the set function is differentiated, see (1.6). Definition 1.3. A mapping $\Phi\colon {\mathcal O}(D)\to[0,\infty]$ is called a quasi-additive set function if (1) for every point $x\in D$ there exists a number $\delta(x)\in(0,\infty)$ such that $\overline{B(x,\delta(x))}\subset D$ and $0<\Phi(B(x,\delta))<\infty$ for all $\delta\in(0, \delta(x))$, and the ball in this condition can be replaced with a cube; (2) every finite tuple $\{U_i\in{\mathcal O}(D)\}$, for $i=1,\dots,l$, of disjoint open sets with
$$
\begin{equation}
\bigcup_{i=1}^lU_i \subset U,\quad \text{where }U\in{\mathcal O}(D), \text{ satisfies } \sum_{i=1}^{l}\Phi(U_i)\leqslant \Phi(U).
\end{equation}
\tag{1.4}
$$
If every finite tuple $\{U_i\in{\mathcal O}(D)\}$ of pairwise disjoint open sets satisfies
$$
\begin{equation}
\sum_{i=1}^{n}\Phi(U_i)=\Phi\biggl(\bigcup_{i=1}^{n} U_i \biggr),
\end{equation}
\tag{1.5}
$$
then this set function is called finitely additive, while if (1.5) holds for every countable tuple $\{U_i\in{\mathcal O}(D)\}$ of disjoint open sets, then this set function is called countably additive. The function $\Phi$ is monotone whenever $\Phi(U_1)\leqslant \Phi(U_2)$ as soon as $U_1\subset U_2 \subset D$ with $U_1,U_2\in{\mathcal O}(D)$. Every quasi-additive set function is obviously monotone. A quasi-additive set function $\Phi\colon {\mathcal O}(D)\to[0,\infty]$ is called a bounded quasi-additive set function whenever $D\in {\mathcal O}(D)$ and $\Phi(D)<\infty$. It is known (see [51]–[53] for instance) that every quasi-additive set function $\Phi$ defined on some system ${\mathcal O}(D')$ of open subsets of a domain $D'$ is differentiable in the following sense: for a.e. point $y\in D'$ there exists the finite derivative1:
$$
\begin{equation}
\lim_{\delta\to 0,\, y\in B_\delta}\frac{\Phi(B_\delta)}{\mathcal{H}^n(B_{\delta})}=\Phi'(y);
\end{equation}
\tag{1.6}
$$
and, for every open set $U\in \mathcal O(D')$, 1.4. Definition of the class of $\mathcal Q_{p,q}(D',\omega;D)$-homeomorphisms and their propertiesDenote by $\mathcal O_{\mathrm c}(D')$ the minimal system of open sets in $D'$, which contains: (1) $D'$; (2) every open cube $Q$ whenever $\overline Q\subset D'$; (3) the complement $Q_2\setminus \overline Q_1$ whenever $Q_1\subset Q_2$ are two cubes with a common center and $\overline Q_2\subset D'$. In Definition 1.4 and Theorem 1.6, we consider the mapping $\Phi\colon \mathcal O_{\mathrm c}(D')\to[0,\infty)$ as the bounded quasi-additive set function. Definition 1.4 [31]. Given two domains $D$, $D'\subset\mathbb R^n$, for $n\geqslant2$, we say that a homeomorphism $f\colon D'\to D$ is of class2 $\mathcal{CRQ}_{p, q}(D',\omega;D)$, where $1< q\leqslant p<\infty$ for $n\geqslant3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$, while $\omega\in L_{1,\mathrm{loc}}(D')$ is a weight function, if there exist (1) a constant $K_p>0$ for $q=p$ or (2) a bounded quasi-additive function $\Psi_{p,q}$ defined on the system $\mathcal O_{\mathrm c}(D')$ of open sets in $D'$ for $q<p$ such that for every cubical condenser $\mathcal E=(\overline{Q(x,r)}, Q(x,R))\subset D'$ with $0<r<R$ with the image $f(\mathcal E)=(f(\overline{Q(x,r)}), f(Q(x,R))\subset D$ we have
$$
\begin{equation}
\begin{cases} \operatorname{cap}^{1/p}\bigl(f(\mathcal E); L^1_p(D)\bigr) \leqslant K_p\operatorname{cap}^{1/p}\bigl(\mathcal E; L^1_p(D';\omega)\bigr), & q=p, \\ \operatorname{cap}^{1/q}\bigl(f(\mathcal E); L^1_q(D)\bigr) \leqslant \Psi_{p,q}(Q(x,R)\setminus \overline{Q(x,r)})^{1/\sigma} \operatorname{cap}^{1/p}\bigl(\mathcal E; L^1_p(D';\omega)\bigr), &q<p, \end{cases}
\end{equation}
\tag{1.8}
$$
where $1/\sigma=1/q-1/p$. Definition 1.5 (see [31], [32]). Let $D$ and $D'$ be open sets in $\mathbb R^n$ with $n\geqslant 2$, $1< q\leqslant p<\infty$ for $n\geqslant3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$, and $\omega\in L_{1,\mathrm{loc}}(D')$ be a weight function. We say that a homeomorphism $\varphi \colon D\to D'$ belongs to the class $\mathcal{Q}_{p, q}(D',\omega;D)$, whenever each condenser $\mathcal E=(F_1,F_0)$ in $D'$ with the preimage $\varphi^{-1}(\mathcal E)=(\varphi^{-1}(F_1),\varphi^{-1}(F_0))$ in $D$ satisfies It is easy to see that if $\varphi\in\mathcal{Q}_{p, q}(D',\omega;D)$, then $f=\varphi^{-1}\in\mathcal{CRQ}_{p, q}(D',\omega;D)$. The following Theorem 1.6 gives an analytic description of the mappings with inverses of class $\mathcal{CRQ}_{p, q}(D',\omega;D)$. Theorem 1.6 (see [33], Theorem 1). A homeomorphism $f\colon D' \to D$ belongs to the class $\mathcal{CRQ}_{p, q}(D',\omega;D)$ with $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ if and only if the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ enjoys one of the following properties: (1) the composition operator $\varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D)$, with $1< q \leqslant p<\infty$, is bounded; (2) the homeomorphism $\varphi\colon D\to D'$ is of class $\mathcal{Q}_{p, q}(D',\omega;D)$ in the sense of Definition 1.5, with some bounded quasi-additive set function $\widetilde\Psi$ defined on open subsets of $D'$; (3) a homeomorphism $\varphi \colon D \to D'$
(4) if $n=2$, then claims (1)–(3) also hold in the case $1=q \leqslant p<\infty$. Note that Theorem 1.6 is a consequence of [29], Theorem 1, [30], and [31], [32], Theorem 1, see details in [33], Theorem 1. The smallest quantities $K_p$ and $\widetilde K_p$ (quasiadditive functions $\Psi$ and $\widetilde\Psi$) in (1.8), (1.9) satisfy
$$
\begin{equation}
\text{for }q=p \quad \|\varphi^*\| =\|K^{1,\omega}_{p,p}(\,{\cdot}\,)\mid L_\infty(D)\|= K_p=\widetilde K_p
\end{equation}
\tag{1.11}
$$
$$
\begin{equation}
\bigl(\text{for }q<p\quad \|\varphi^*_W\|^\sigma =\|K^{1,\omega}_{q,p}(\,{\cdot}\,)\mid L_\sigma(\varphi^{-1}(W))\|^\sigma= \Psi(W)=\widetilde\Psi(W)\bigr)
\end{equation}
\tag{1.12}
$$
for an open set $W\subset D'$, where $\|\varphi^*_W\|$ is the norm of the restriction
$$
\begin{equation*}
\varphi_W\colon {L}^1_p(W;\omega) \cap \mathring{\mathrm{Lip}}_{\mathrm{loc}}(W)\to L^1_q(D);
\end{equation*}
\notag
$$
here, $\mathring{\mathrm{Lip}}_{\mathrm{loc}}(W)$ stands for the space of locally Lipschitz functions vanishing on the boundary of $W$, see [34], Theorem 4. Let us formulate the following corollary of Theorem 1.6. Corollary 1.7. A homeomorphism $f\colon D\to D'$ is of class $\mathcal{CRQ}_{p, q}(D',\omega;D)$ with $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$ if and only if $\varphi=f^{-1}$ is also of class $\mathcal{Q}_{p, q}(D',\omega;D)$. Therefore, from now on, we use only $\mathcal{Q}_{p, q}(D',\omega;D)$ to refer to both classes $\mathcal{CRQ}_{p, q}(D',\omega;D)$ and $\mathcal{Q}_{p, q}(D',\omega;D)$. The differential properties of mappings of the classes $\mathcal{Q}_{p, q}(D',\omega;D)$ are established in [30] and [31], Theorem 2. Remark 1.8. The homeomorphisms $\varphi\colon D \to D'$ with $f=\varphi^{-1}\in \mathcal{Q}_{p, q}(D',\omega;D)$ in the cases (1) $q=p=n$ and $\omega\equiv1$ coincide with quasiconformal mappings [18], [42]–[45]; (2) $1<q=p<\infty$ and $\omega\equiv1$ were studied in [28]; (3) $1<q<p<\infty$ and $\omega\equiv1$ were studied in [28], [37]–[39]. Let us extract from Theorem 1.6 and Corollary 1.7 the following two examples of $Q_{p,q}$-homeomorphisms. Example 1.9 (see [29], [32]). If a homeomorphism $\varphi\colon D \to D'$ induces a bounded composition operator $\varphi^*\colon {L}^1_p(D';\omega) \cap \operatorname{Lip}_{\mathrm{loc}}(D')\to L^1_q(D)$, with $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$, then the inverse homeomorphism $f=\varphi^{-1}\colon D'\to D$ is of class $\mathcal{Q}_{p, q}(D',\omega;D)$. Example 1.10 (see [29], [32]). Consider a homeomorphism $\varphi \colon D \to D'$ of the Sobolev class $W^1_{q, \mathrm{loc}}(D)$ with finite distortion (1.1) and the operator function distortion (1.10) of class $L_{\sigma}(D)$, where $1/\sigma=1/q-1/p$ for $1\leqslant q<p<\infty$ and $\sigma=\infty$ for $q=p$. If $1<q\leqslant p<\infty$ for $n\geqslant 3$ and $1\leqslant q\leqslant p<\infty$ for $n=2$, then the inverse homeomorphism $f=\varphi^{-1}\colon D'\to D$ is of class $\mathcal{Q}_{p, q}(D',\omega;D)$. In addition to Examples 1.9 and 1.10, other classes of mappings in the family $\mathcal Q_{p,q}(D',\omega;D)$ were considered in [31]. Let us present some of them. Example 1.11 (see [31], Example 3)). Consider a homeomorphism $\varphi\colon D\to D'$ of Sobolev class $W^1_{p,\mathrm{loc}}(D)$, where $1<p<\infty$ for $n\geqslant3$ and $1\leqslant p<\infty$ for $n=2$, with finite distortion. The inverse homeomorphism $f=\varphi^{-1}\colon D'\to D$ is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ and the weight function Example 1.13 (see [31], Example 4). For $n-1< s<\infty$, consider a homeomorphism $f \colon D' \to D$ of open domains $D',D\subset \mathbb{R}^n$, where $n\geqslant 2$, such that (1) $f\in W^1_{n-1, \mathrm{loc}}(D')$; (2) the mapping $f$ has finite distortion; (3) the outer distortion function
$$
\begin{equation}
D'\ni y \mapsto K^{1,1}_{n-1,s}(y,f) = \begin{cases} \dfrac{|Df(y)|}{|{\det Df (y)}|^{1/s}} &\text{if } \det Df (y)\neq 0, \\ 0 &\text{if }\det Df (y) = 0 \end{cases}
\end{equation}
\tag{1.14}
$$
lies in $L_{\sigma}(D)$, where $\sigma=(n-1)p$ with $p=s/(s-(n-1))$. Then the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ has the properties (4) $\varphi\in W^1_{p, \mathrm{loc}}(D)$, $p=s/(s-(n-1))$; (5) $\varphi$ has finite distortion; while the homeomorphism $f\colon D'\to D$ (6) is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ and the weight function $\omega\in L_{1,\mathrm{loc}}(D')$ defined as where $Z'=\{y\in D'\colon Df(y)=0\}$.Say that a mapping $f\in W^1_{1, \mathrm{loc}}(D')$ has finite codistortion if the adjoint matrix $\operatorname{adj} Df(y)$ of the differential equals $0$ a.e. on the zero set of the Jacobian Example 1.14 (see [31], Example 5). For $n-1< s<\infty$, consider a homeomorphism $f \colon D' \to D$ of domains $D',D\subset \mathbb{R}^n$, with $n\geqslant 2$, such that (1) $f\in W^1_{n-1, \mathrm{loc}}(D')$; (2) the mapping $f$ has finite codistortion; (3) the inner distortion function
$$
\begin{equation}
D'\ni y \mapsto \mathcal K^{1,1}_{n-1,s}(y,f) = \begin{cases} \dfrac{|{\operatorname{adj} Df(y)}|}{|{\det Df (y)}|^{(n-1)/s}} &\text{if }\det Df (y)\neq 0, \\ 0 &\text{if }\det Df (y) = 0 \end{cases}
\end{equation}
\tag{1.16}
$$
belongs to $L_{p}(D')$, where $p=s/(s-(n-1))$ and $n-1<s<\infty$. Then the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ has the properties (4) $\varphi\in W^1_{p, \mathrm{loc}}(D)$ and $p=s/(s-(n-1))$; (5) $\varphi$ has finite distortion; and the homeomorphism $f\colon D'\to D$ (6) is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ and the weight function (1.15); (7) has finite distortion for $n-1< s<n+1/(n-2)$. Example 1.15 (see [35], Definition 11, Theorem 34). A homeomorphism $f\colon D'\,{\to}\, D$ is called a homeomorphism with inner bounded $\theta$-weighted $(s,r)$-distortion, or of class $\mathcal{ID}(D';s,r;\theta,1)$, where $n-1< s\leqslant r<\infty$, whenever: (1) $f\in W^1_{n-1,\mathrm{loc}}(D')$; (2) the mapping $f$ has finite codistortion; (3) the function of local $\theta$-weight $(s,r)$-distortion
$$
\begin{equation}
D' \ni x\mapsto \mathcal K_{s,r}^{\theta,1}(x,f) =\begin{cases} \dfrac{\theta^{(n-1)/s}(x)|{\operatorname{adj}} D f(x)|}{|{\det D f(x)}|^{(n-1)/r}} &\text{if } \det D f(x)\ne0, \\ 0 &\text{otherwise} \end{cases}
\end{equation}
\tag{1.17}
$$
belongs to $L_{\varrho}(\Omega)$, where $\varrho$ can be found from the condition $1/\varrho = (n-1)/s-(n- 1)/r$, and $\varrho= \infty$ for $s=r$. Hence, under the condition $n-1< s\leqslant r<\infty$ and the local summability of the function $\omega(x)=\theta^{-(n-1)/(s-(n-1))}(x)$, the homeomorphism $f\colon D'\to D$ belongs to $\mathcal Q_{p,q}(D',\omega;D)$, where $q=r/(r-(n-1))$ and $p=s/(s-(n-1))$, for $1<q\leqslant p<\infty$. Furthermore, the factors on the right-hand side of (1.8) are equal to $K_p=\|\mathcal K_{r,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\infty}(\Omega)\|$ for $q=p$ and where $1/\sigma=1/q-1/p=1/\varrho$.Example 1.16 (see [36], Definition 3, Theorem 19). A homeomorphism $f\colon D'\to D$ is of class $\mathcal{OD}(D';s,r;\theta,1)$, with $n-1< s\leqslant r<\infty$, and is called a mapping with outer bounded $\theta$-weighted $(s,r)$-distortion, whenever: (1) $f\in W^1_{n-1,\mathrm{loc}}(D')$; (2) the mapping $f$ has finite distortion; (3) the function of local $\theta$-weighted $(s,r)$-distortion
$$
\begin{equation*}
D' \ni x\mapsto K_{s,r}^{\theta,1}(x,f)= \begin{cases} \dfrac{\theta^{1/s}(x)|D f(x)|}{|{\det D f(x)}|^{1/r}} &\text{if }\det D f(x)\ne0, \\ 0 &\text{otherwise} \end{cases}
\end{equation*}
\notag
$$
belongs to $L_{\rho}(D')$, where $\rho$ can be found from the conditions $1/\rho = 1/s-1/r$ and $\rho = \infty$ for $s=r$. Hence, under the condition $n-1< s\leqslant r<\infty$ and the local summability of $\omega(x)=\theta^{-(n-1)/(s-(n-1))}(x)$, the homeomorphism $f\colon D'\to D$ belongs to $\mathcal Q_{p,q}(D',\omega;D)$, where $q=r/(r-(n-1))$ and $p=s/(s-(n-1))$ with $1<q\leqslant p<\infty$. The factors on the right-hand side of (1.8) are equal to $K_p=\|K_{r,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\infty}(D')\|^{n-1}$ for $q=p$ and for $q<p$, where $1/\sigma=1/q-1/p=(n-1)/\varrho$.It is shown in [36], Theorem 8, that the inclusion
$$
\begin{equation*}
\mathcal{OD}(D';s,r;\theta,1) \subset \mathcal{ID}(D';s,r;\theta,1)
\end{equation*}
\notag
$$
holds under the condition $n-1< s\leqslant r<\infty$. Moreover, for every homeomorphism $f\colon D'\to D$ of class $\mathcal{OD}(D';s,r;\theta,1)$, with $n-1< s\leqslant r<\infty$, we have
$$
\begin{equation*}
\|\mathcal K_{s,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\sigma}(D')\|\leqslant \| K_{s,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_\rho(D')\|^{n-1},
\end{equation*}
\notag
$$
where the numbers $\rho$ and $\sigma$ are defined in Examples 1.15 and 1.16. More examples of $\mathcal{OD}(D'; s, r; \theta, 1)$-homeomorphisms in $\mathbb R^2$ can be found in [54]. § 2. Behaviour of mappings with respect to the capacity metricWe fix two domains $D, D'\subset \mathbb R^n$, a locally integrable weight function $\omega\colon D'\to\mathbb R$ on $D'$, and a mapping $f\in\mathcal Q_{p,q}(D',\omega;D)$ with $n-1<q\leqslant p<\infty$. Recall that Corollary 1.7 guarantees that $f$ satisfies (1.9) for every condenser $\mathcal E=(F_1,F_0)$ in $D'$. We also fix some continuum $F_0\subset D'$ with non-empty interior such that the open set $D'\setminus F_0$ is connected. 2.1. Capacity metric functions in domains for the homeomorphisms of class $\mathcal Q_{p,q}(D',\omega;D)$ for $n-1<q\leqslant p\leqslant n$Observe that in the case $n-1<q\leqslant n$ the left-hand side of (1.9) is non-zero as long as the continuum $f(F_1)$ is distinct from a point. Indeed, we have the following proposition. Lemma 2.1. In a domain $D\subset \mathbb R^n$, let $B_0\Subset D$ and $B_1\Subset D$ be two balls satisfying $\overline{B_0}\cap\overline{B_1}=\varnothing$. Then, for $n-1<q\leqslant n$, a fixed continuum $T_0\subset B_0$, and an arbitrary continuum $T_1\subset \overline{B_1}$, the relation Proof. Let us present the scheme of the proof of Lemma 2.1.
Necessity. By [48], Lemma 3, there is a John domain $\Omega\in J(\alpha,\beta)$ (see [48], Definition 8) compactly embedded into $D$, with some positive parameters $\alpha$ and $\beta$ depending on $D$ and the balls $B_0$ and $B_1$, which includes the closures of both balls. On the domain $\Omega$ under the conditions $1\leqslant q < n$ and $q\leqslant q^*\leqslant nq/(n-q)$ we have the following Poincaré inequality [55], Theorems 4 and 9:
$$
\begin{equation}
\|u - c_u\mid L_{q^*}(\Omega)\| \leqslant C_\Omega\biggl(\frac{\alpha}{\beta}\biggr)^{n}(\operatorname{diam}\Omega)^{1-n/q+n/q^*} \|\nabla u \mid L_q(\Omega)\|,
\end{equation}
\tag{2.2}
$$
where $c_u$ and $C_\Omega$ are constants, with $C_\Omega>0$ independent of $u$, $\alpha$, and $\beta$. By (2.1) there exists a sequence of continua $T_{1,k}\subset B_1$ and admissible functions $u_k\in C(\Omega)\cap L^1_q(\Omega)$ for the capacity $\operatorname{cap}((T_{1,k}, T_0); L^1_q(\Omega))$ such that
$$
\begin{equation}
u_k\vert_{T_{1,k}}= 1,\quad u_k\vert_{T_0}= 0,\quad 0\leqslant u_k\leqslant 1\quad \text{and} \quad \|\nabla u_k \mid L_q(\Omega)\|\to 0\quad \text{as}\quad k\to\infty.
\end{equation}
\tag{2.3}
$$
Now inequality (2.2) implies that $\|u_k - c_{u_k}\mid L_{q^*}(\Omega)\|\to 0$ as $k\to\infty$. Note that the sequence of numbers $\{c_{u_k}\}$ is bounded. Indeed, if $\{c_{u_k}\}$ is not bounded, then, since $0\leqslant u_k\leqslant 1$, the left-hand side of (2.2) is also not bounded, which contradicts the right convergence in (2.3). Therefore, we may assume that $c_{u_k}$ converges to some number $c_0$, and up to subsequence $u_{k} - c_{u_{k}}\to 0$ for a.e. $x\in \Omega$ as $k\to\infty$. Hence, $u_{k} \to c_0$ for a.e. $x\in \Omega$ as $k\to\infty$, and due to $u_{k}|_{B_0}\equiv0$ we deduce $c_0=0$. In addition, $\Omega$ is a bounded domain, and the Lebesgue dominated convergence theorem shows that
$$
\begin{equation}
\|u_l \mid L_{q}(\Omega)\|\to 0\quad\text{as}\quad l\to\infty.
\end{equation}
\tag{2.4}
$$
From (2.3) and (2.4) we infer that $\|u_l \mid W^1_{q}(\Omega)\|\to 0$ as $l\to\infty$. We can extend the restrictions $u_l|_{B_1}$ to the functions $\widetilde u_l\in W^1_{q}(\mathbb R^n)$ so that the extension operator is bounded. Therefore,
$$
\begin{equation*}
\|\widetilde u_l \mid W^1_{q}(\mathbb R^n)\|\to 0\quad \text{as}\quad l\to\infty.
\end{equation*}
\notag
$$
We obtain then that the capacity of the continua $T_{1,l}$ in the space $W^1_{q}(\mathbb R^n)$ of Bessel potentials is positive and tends to $0$ as $l\to\infty$. For $n-1< q < n$ the latter is possible only if $\operatorname{diam} T_{1,l}\to 0$ as $l\to\infty$; see the details in [56], [41].
The case $q =n$ reduces to the previous one using Hölder’s inequality. Sufficiency. By Property 1.2 we have
$$
\begin{equation*}
\operatorname{cap}\bigl((T_1,T_0); L^1_q(D)\bigr) \leqslant \operatorname{cap}\bigl((T_1,\overline{B_0}); L^1_q(D)\bigr),
\end{equation*}
\notag
$$
and so, it suffices to prove that $\operatorname{cap}((T_1,\overline{B_0}); L^1_q(D))\to0$ as $\operatorname{diam} T_1\to 0$.
We put $R=\operatorname{dist}(B_0,B_1)$ and suppose that the continuum $T_1$ satisfies $r_{T_1}<R$. Then we may assume that every admissible function for the condenser $(\overline{B(x,r_{T_1})}, B(x,R))$ is also admissible for the condenser $(T_1,T_0)$, and so
$$
\begin{equation*}
\operatorname{cap}\bigl((T_1,T_0); L^1_q(D)\bigr)\leqslant \operatorname{cap}\bigl(\bigl(\overline{B(x,r_{T_1})},B(x,R)\bigr); L^1_q(B(x,R))\bigr).
\end{equation*}
\notag
$$
From Example 2.7 below for $\alpha=0$, we conclude
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{cap}\bigl(\bigl(\overline{B(0,r)},B(0,R)\bigr);L^1_q(B(0,R))\bigr) \\ &\qquad= \begin{cases} \sigma_{n-1}\biggl(\dfrac{n-q}{n-1}\biggr)^{q-1} (r^{(q-n)/(q-1)}-R^{(q-n)/(q-1)})^{1-q} &\text{for }q<n, \\ \sigma_{n-1}\biggl(\ln\dfrac{R}r\biggr)^{1-n} &\text{for }q=n, \end{cases} \end{aligned}
\end{equation*}
\notag
$$
where $r\in(0,R)$, while $\sigma_{n-1}$ is the measure of the unit $(n-1)$-dimensional sphere in the space $\mathbb R^n$. Thus,
$$
\begin{equation*}
\operatorname{cap}\bigl(\bigl(\overline{B(x,r_{T_1})},B(x,R)\bigr); L^1_q(B(x,R)\bigr) \to0 \quad \text{as}\quad r_{T_1}\to 0,
\end{equation*}
\notag
$$
and the proof of Lemma 2.1 is complete. Corollary 2.2. For $n-1\,{<}\,q\,{\leqslant}\, n$, the existence of a mapping $f\,{\in}\,\mathcal Q_{p,q}(D',\omega;D)$ is ensured by the condition for an arbitrary condenser $\mathcal E=(\gamma,F_0)$, where $\gamma\colon[a,b]\to D'\setminus F_0$ is an arbitrary closed curve with distinct endpoints $x=\gamma(a)$ and $y=\gamma(b)$. Proof. Since the continuum $F_0\subset D'$ has non-empty interior, there exists a closed ball $\overline{B_0'}\subset F_0$ and a closed ball $\overline{B_1'}\subset D'$ centered on $\gamma$ such that $\overline{B_1'}\cap \overline{B_0'}=\varnothing$. Consider the condenser $\mathcal E=(\gamma\cap\overline{B_1'}, \overline{B_0'})$. By (1.8), it suffices to show that
$$
\begin{equation}
\operatorname{cap}\bigl(f(\mathcal E); L^1_q(D)\bigr)\ne0.
\end{equation}
\tag{2.6}
$$
The latter follows from Lemma 2.1. Indeed, there are closed disjoint balls $\overline{B_0''}\subset f(\overline{B_0'})$ and $\overline{B_1''}\subset f(\overline{B_1'})$ whose intersection $\gamma\cap\overline{B_1''}$ is a nondegenerate continuum. Then, Lemma 2.1 and (1.8) yield
$$
\begin{equation*}
0 \ne \operatorname{cap}\bigl(\bigl(\gamma\cap\overline{B_1'}, \overline{B_0'}\bigr);L^1_q(D)\bigr)\leqslant \operatorname{cap}\bigl(f(\mathcal E); L^1_q(D)\bigr).
\end{equation*}
\notag
$$
This justifies Corollary 2.2. With (2.5) we can define a metric function similar to the one introduced in [25], [26], Chap. 5, in the unweighted case. Definition 2.3. The capacity $(\omega,p)$-metric function between two distinct points $x,y \in D' \setminus F_0$ with respect to $F_0$ is defined as where the infimum is over all curves $\overline{xy}$ in $D'\setminus F_0$ with endpoints $x,y\in D'\setminus F_0$. By analogy, we define the capacity $q$-metric function $\rho_{q,f(F_0)}(a,b)$ between two points $a,b \in D \setminus f(F_0)$ with respect to the continuum $f(F_0)$ in the image $D'$: Proposition 2.4. If a homeomorphism $f\colon D' \to D$ belongs to $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$, then the capacity metric functions satisfy Proof. Take $\mathcal E=(\overline{xy},F_0)$ in $D'$, then from (1.9) it follows that
$$
\begin{equation*}
\begin{aligned} \, \rho_{q,f(F_0)}(f(x),f(y))& \leqslant\operatorname{cap}^{1/q}\bigl((f(\overline{xy}),f(F_0)); L^1_q(D)\bigr) \\ &\leqslant \Psi_{p,q}(D'\setminus F_0)^{1/\sigma} \operatorname{cap}^{1/p}\bigl((\overline{xy},F_0); L^1_p(D';\omega)\bigr) \end{aligned}
\end{equation*}
\notag
$$
provided that $q<p$. Passing to the infimum over all curves $\overline{xy}\subset D'\setminus F_0$ with endpoints $x$ and $y$, we arrive at the second inequality in (2.9).
The case $q=p$ is similar. The proposition is proved. Proposition 2.5. In the case $n-1<q\leqslant p\leqslant n$, for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$, the capacity $(\omega,p)$-metric function $\rho^{\omega}_{p,F_0}(x,y)$ enjoys the properties (1) $\rho^{\omega}_{p,F_0}(x,y)=\rho^{\omega}_{p,F_0}(y,x)$ for all points $x,y\in D' \setminus F_0$; (2) $\rho^{\omega}_{p,F_0}(x,z)\leqslant \rho^{\omega}_{p,F_0}(x,y)+\rho^{\omega}_{p,F_0}(y,z)$ for all points $x,y,z\in D' \setminus F_0$. Proof. Property (1) is obvious.
To verify the second property, consider the case $x\ne z$, $x\ne y$, $y\ne z$; otherwise property (2) obviously holds. We fix $\varepsilon>0$ and some curves $\overline{xy}$ and $\overline{yz}$ with endpoints $x$, $y$ and $y$, $z$, respectively, such that
$$
\begin{equation}
\operatorname{cap}^{1/p}\bigl((\overline{xy}, F_0); L^1_p(D';\omega)\bigr) < \rho^{\omega}_{p,F_0}(x,y)+\frac{\varepsilon}{4},
\end{equation}
\tag{2.10}
$$
$$
\begin{equation}
\operatorname{cap}^{1/p}\bigl((\overline{yz}, F_0); L^1_p(D';\omega)\bigr) < \rho^{\omega}_{p,F_0}(y,z)+\frac{\varepsilon}{4}.
\end{equation}
\tag{2.11}
$$
Take two functions $u_1$ and $u_2$ admissible for the capacities $\operatorname{cap}((\overline{xy}, F_0); L^1_p(D';\omega))$ and $\operatorname{cap}((\overline{yz}, F_0); L^1_p(D';\omega))$ such that
$$
\begin{equation}
\biggl(\int_{D'}|\nabla u_1|^p(y)\omega(y)\,dy\biggr)^{1/p} < \operatorname{cap}^{1/p}\bigl((\overline{xy}, F_0); L^1_p(D';\omega)\bigr)+\frac{\varepsilon}{4},
\end{equation}
\tag{2.12}
$$
$$
\begin{equation}
\biggl(\int_{D'}|\nabla u_2|^p(y)\omega(y)\,dy\biggr)^{1/p} < \operatorname{cap}^{1/p}\bigl((\overline{yz}, F_0); L^1_p(D';\omega)\bigr)+\frac{\varepsilon}{4}.
\end{equation}
\tag{2.13}
$$
It is easy to see that $u_1+u_2$ is admissible for the capacity $\operatorname{cap}^{1/p}((\overline{xy}\cup\overline{yz}, F_0); L^1_p(D';\omega))$. Hence, from (2.10)–(2.13), we obtain
$$
\begin{equation*}
\begin{aligned} \, \rho^{\omega}_{p,F_0}(x,z) &\leqslant \operatorname{cap}^{1/p}\bigl((\overline{xy}\cup\overline{yz}, F_0); L^1_p(D';\omega)\bigr) \leqslant \biggl(\int_{D'}|\nabla (u_1+u_2)|^p(y)\omega(y)\,dy\biggr)^{1/p} \\ &\leqslant \biggl(\int_{D'}|\nabla u_1|^p(y)\omega(y)\,dy\biggr)^{1/p}+ \biggl(\int_{D'}|\nabla u_2|^p(y)\omega(y)\,dy\biggr)^{1/p} \\ &<\rho^{\omega}_{p,F_0}(x,y)+\rho^{\omega}_{p,F_0}(y,z)+\varepsilon. \end{aligned}
\end{equation*}
\notag
$$
Since $\varepsilon>0$ is arbitrarily, the triangle inequality is verified, proving the proposition. Recall that the metric function $\rho^{\omega}_{p,F_0}$ is defined in (2.7) for distinct points $x\ne y$ of the open set $D' \setminus F_0$. If $x=y\in D' \setminus F_0$, we put
$$
\begin{equation}
\rho^{\omega}_{p,F_0}(x,x)=\operatorname{cap}^{1/p}\bigl((\{x\}, F_0); L^1_p(D';\omega)\bigr).
\end{equation}
\tag{2.14}
$$
For the capacity metric function $\rho^{\omega}_{p,F_0}$ to be a metric, we must ensure that for every point $x\in D'\setminus F_0$. Proposition 2.6. Given $x\in D'\setminus F_0$, condition (2.15) holds if and only if Proof. Since the condenser $(\{x\}, F_0)$ is a part of the condenser $(\overline{B(x,r)}, F_0)$, Property 1.2 yields
$$
\begin{equation*}
\rho^{\omega}_{p,F_0}(x,x)\leqslant \lim_{r\to0}\operatorname{cap}^{1/p} \bigl(\bigl(\overline{B(x,r)}, F_0\bigr); L^1_p(D';\omega)\bigr).
\end{equation*}
\notag
$$
Granted (2.16), this implies (2.15).
Suppose now that (2.15) holds: $\rho^{\omega}_{p,F_0}(x,x)=\operatorname{cap}^{1/p}((\{x\}, F_0); L^1_p(D';\omega))=0$. By the definition of capacity, for every $\varepsilon\in(0,1/2)$, there exists a function $u_\varepsilon\in \operatorname{Lip}_{\mathrm{loc}}(D')$ such that $u_\varepsilon(y)\in [0,1]$ for all $y\in D'$, while $u_\varepsilon\vert_{F_0}=0$, $u_\varepsilon(x)=1$, and
$$
\begin{equation}
\int_{D'}|\nabla u_\varepsilon |^p(y)\omega(y)\,dy<\varepsilon.
\end{equation}
\tag{2.17}
$$
Since $x$ is an interior point of $\{y\in D'\colon u_\varepsilon(y)>1-\varepsilon\}$, we have $B(x,r_0)\subset \{y\in D'\colon u_\varepsilon(y)>1-\varepsilon\}$ for some ball $B(x,r_0)$. Consequently, the function
$$
\begin{equation*}
\frac{\min(u_\varepsilon(y),1-\varepsilon)}{1-\varepsilon}
\end{equation*}
\notag
$$
is admissible for the capacity of the condenser $(\overline{B(x,r)}, F_0)$ provided that $r\in (0,r_0)$. Therefore,
$$
\begin{equation*}
\begin{aligned} \, \operatorname{cap}\bigl(\bigl(\overline{B(x,r)}, F_0\bigr); L^1_p(D';\omega)\bigr) &\leqslant \frac1{(1-\varepsilon)^p}\int_{D'} \bigl|\nabla\bigl(\min(u_\varepsilon(y),1-\varepsilon)\bigr)\bigr|^p\omega(y)\,dy \\ &\leqslant \frac1{(1-\varepsilon)^p}\int_{D'} |\nabla u_\varepsilon|^p(y)\omega(y)\,dy\leqslant \frac{\varepsilon}{(1-\varepsilon)^p}< 2^p\varepsilon \end{aligned}
\end{equation*}
\notag
$$
by (2.17). Since $\varepsilon\in(0,1/2)$ is arbitrary, (2.16) is justified.
This completes the proof of Proposition 2.6. Observe that (2.16) always holds in the case $q \leqslant p \leqslant n$ and $\omega\equiv1$. In the case of a non-trivial weight function condition (2.15) need not hold, see Examples 2.7 and 2.8. Example 2.7 (see [50], Example 2.22). Consider the domain $D' = B(0,2)$ with the weight $\omega(x) = |x|^{\alpha}$, where $\alpha> -n$, and $p>1$. The capacity of the condenser $\mathcal E=(\overline{B(0,r)}, B(0,1))$ with $0<r<1$ in the space $L_p(D';\omega)$, where the weight function $\omega$ belongs to the special class of weight functions called admissible in [50], is In the following example, we will construct a weight function for which condition (2.15) is violated on a countable dense subset of $D'$. Example 2.8. Consider an arbitrary bounded domain $D' \subset\mathbb R^n$, a continuum $F_0$, and a number $\alpha$ satisfying $p-n-\alpha>0$. With each point $x_i$ of some countable dense subset of $D'$, we associate the function Example 2.9. Consider a bounded domain $D' \subset\mathbb R^n$, a point $x\in D'$, a continuum $F_0 \subset D' \setminus B(x, e^{-1})$, and a weight $\omega\colon D' \to [1,\infty)$ with $\omega \in \mathrm{BMO}(D')$. For $0<r<e^{-2}$, define the function Examples 2.7–2.9 show that condition (2.15) depends on the properties of the weight function $\omega$. We let $d(x,y)$ denote the Euclidean distance between two points $x,y\,{\in}\,\mathbb R^n$. Proposition 2.10. Consider a homeomorphism $f\colon D' \to D$ belonging to the class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$. (1) If $y\in D'\setminus F_0$ and $\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as $m\to\infty$ in the domain $D'\setminus F_0$, then
(2) Provided with (2.15) at $y\in D'\setminus F_0$, the convergence $d(z_m,y)\to0$ as $m\to\infty$ implies the convergence $\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ with respect to the capacity $(\omega,p)$-metric function $\rho^{\omega}_{p,F_0}$ in the domain $D'\setminus F_0$. Proof. (1) By Definition 2.3, for each $m\in \mathbb N$, there exists a continuous curve $\gamma_m\colon [0,1]\to D'\setminus F_0$ with endpoints $z_m=\gamma_m(0)$, $y=\gamma_m(1)\in D'\setminus F_0$ such that
$$
\begin{equation}
\operatorname{cap}^{1/p}\bigl((\overline{\gamma_m}, F_0); L^1_p(D';\omega)\bigr) \leqslant 2\rho^{\omega}_{p,F_0}(z_m,y),
\end{equation}
\tag{2.18}
$$
where $\overline{\gamma_m}=\gamma_m([0,1])$ stands for the image of the curve $\gamma_m\colon [0,1]\to D'\setminus F_0$. Using the inequality
$$
\begin{equation*}
\operatorname{cap}^{1/p}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr) \leqslant \operatorname{cap}^{1/p}\bigl((\overline{z_my}, F_0); L^1_p(D';\omega)\bigr),
\end{equation*}
\notag
$$
valid for all $m\in \mathbb N$, from (2.18) and the condition $\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as $m\to\infty$ in the domain $D'\setminus F_0$, we infer that
$$
\begin{equation*}
\operatorname{cap}^{1/p}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr)=0.
\end{equation*}
\notag
$$
Furthermore, from (2.9) and the condition $\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as $m\to\infty$ we find that $\rho_{q,f(F_0)}(f(z_m),f(y))\to 0$ as $m\to\infty$. By Lemma 2.1, the latter is possible if and only if $f(z_m)\to f(y)$ as $m\to\infty$. Hence $z_m\to y$ as $m\to\infty$.
(2) Assume that condition (2.15) holds at $y\in D'\setminus F_0$ and $d(z_m,y)\to0$ as $m\to\infty$ for some sequence $z_m\in D'\setminus F_0$. On assuming condition (2.15), Proposition 2.6 implies that
$$
\begin{equation}
\lim_{r\to0} \operatorname{cap}^{1/p}\bigl(\bigl(\overline{B(y,r)}, F_0\bigr); L^1_p(D';\omega)\bigr)=0.
\end{equation}
\tag{2.19}
$$
For $z_m\in B(y,r)$, from the properties of capacity, we infer that
$$
\begin{equation*}
\rho^{\omega}_{p,F_0}(z_m,y)\leqslant \operatorname{cap}^{1/p}\bigl(\bigl(\overline{B(y,r)}, F_0\bigr); L^1_p(D';\omega)\bigr),
\end{equation*}
\notag
$$
and hence $\rho^{\omega}_{p,F_0}(z_m,y)\to 0$ as $m\to \infty$.
This proves Proposition 2.10. Given a set $B\subset\mathbb R^n$, we let $\operatorname{dist}(y,B):=\inf_{z\in B}d(y,z)$ denote the distance from a point $y\in\mathbb R^n$ to $B$, where $d(\,{\cdot}\,,{\cdot}\,)$ is the Euclidean distance. The following proposition generalizes Proposition 2.10. Proposition 2.11. Consider a homeomorphism $f\colon D' \to D$ belonging to the class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$. If $\{y_l\in D'\setminus F_0\}$, for $l\in \mathbb N$, is a fundamental sequence with respect to the metric function $\rho^{\omega}_{p,F_0}$, while $y$ is one of its partial limits in the topology of the extended space $\overline{\mathbb R^n}$, then the following claims hold: (1) if $y\in D'\setminus F_0$, then $d(y_l,y)\to 0$ as $l\to \infty$; (2) if $y\in F_0$, then $d(y_l,y)\to 0$ as $l\to \infty$; (3) if $y\in \partial D'$ and $\{y_{l}\in D'\}$ is bounded, then $\operatorname{dist}(y_l, \partial D')\to 0$ as $l\to \infty$; (4) if $\{y\}=\overline{\mathbb R^n}\setminus \mathbb R^n$, either $y_l\to y$ as $l\to \infty$ in the topology of $\overline{\mathbb R^n}$, or $\varliminf_{\,l\to \infty}d(y_l,0)\,{<}\,\infty$ and $\lim_{k\to \infty}\operatorname{dist}(y_{l_k}, \partial D')=0$ for every subsequence $\{y_{l_k}\in D'\}$ bounded in $\mathbb R^n$. Proof. Let us prove the claims of Proposition 2.11 one by one.
(1) Take a fundamental sequence $\{y_l\in D'\setminus F_0\}$, for $l\in \mathbb N$, with respect to the metric function $\rho^{\omega}_{p,F_0}$ and its subsequence $\{y_{l_k}\in D'\setminus F_0\}$, for $k\in \mathbb N$, converging in the topology of the Euclidean space $\mathbb R^n$ to some point $y\in D'\setminus F_0$ as $k\to \infty$. By (2.9), the sequence $\{f(y_l)\in D\setminus f(F_0)\}$, $l\in \mathbb N$, is also fundamental with respect to $\rho_{q,f(F_0)}$. In addition, since $f$ is continuous at $y\in D'\setminus F_0$, we have the convergence $f(y_{l_k})\to f(y)$ as $k\to \infty$. Lemma 2.1 implies the convergence $\rho_{q,f(F_0)}(f(y_{l_k}),f(y))\to 0$ as $k\to \infty$. Since the sequence $\{f(y_l)\in D\setminus f(F_0)\}$, for $l\in \mathbb N$, is fundamental with respect to the metric function $\rho_{q,f(F_0)}$, we see that $\rho_{q,f(F_0)}(f(y_{l}),f(y))\to 0$ as $l\to \infty$. Moreover, $f(y_{l})\to f(y)$ as $l\to \infty$, again by Lemma 2.1. Since $f^{-1}$ is continuous at $f(y)$, we infer that $y_{l}\to y$ as $l\to \infty$. (2) Take a fundamental sequence $\{y_l\in D'\setminus F_0\}$, $l\in \mathbb N$, with respect to the metric function $\rho^{\omega}_{p,F_0}$ and its subsequence $\{y_{l_k}\in D'\setminus F_0\}$, $k\in \mathbb N$, converging in the topology of the Euclidean space $\mathbb R^n$ to some point $y\in F_0$ as $k\to\infty$. The second claim will be justified once we verify that the stated properties contradict the existence of a subsequence $\{y_{l_j}\}$, $j\in \mathbb N$, such that $d(y_{l_j},y)\geqslant 1/\beta$ for all $j\in \mathbb N$, where $\beta>1$ is some number. Indeed, if such a subsequence exists, then for all $j\in \mathbb N$, where $\beta'>1$ is some number, whose existence is ensured by the locally uniform continuity of the homeomorphism $f$. On the other hand, the sequence $\{f(y_l)\in D\setminus f(F_0)\}$, for $l\in \mathbb N$, is fundamental with respect to the metric function $\rho_{q,f(F_0)}$. Applying the subordination principle, see Property 1.2, we infer that this sequence is also fundamental with respect to the metric function $\rho_{q,K}$ for an arbitrary compact set $K\subset \operatorname{int}f(F_0)$ with non-empty interior. By Lemma 2.1, the sequence $f(y_{l})$ converges to $f(y)\notin K$ as $l\to \infty$. The latter contradicts (2.20).(3) Take a partial limit $y=\lim_{j\to \infty} y_{l_j}\in \partial D'$ and assume on the contrary that there exists a subsequence $\{y_{l_k}\}$, for $k\in \mathbb N$, such that $\operatorname{dist}(y_{l_k}, \partial D')\geqslant \beta_0>0$ for all $k\in \mathbb N$, where $\beta_0$ is some number. By the latter property, since $\{y_{l}\}$ is bounded, we may assume that the subsequence $\{y_{l_k}\}$ converges to some $z\in D'$. Consequently, the hypotheses of the first claim are fulfilled, and so $y_l\to z$ as $l\to\infty$, which contradicts the property $\lim_{j\to \infty} y_{l_j}=y \in \partial D'$. (4) If under the condition $\{y\}=\overline{\mathbb R^n}\setminus \mathbb R^n$ we have $\varliminf_{\,l\to \infty}d(y_l,0)=\infty$, then $y_l\to y$ as $l\to \infty$ in the topology of $\overline{\mathbb R^n}$. Assume that if $\varliminf_{\,l\to \infty}d(y_l,0)<\infty$, then $\varlimsup_{k\to \infty}\operatorname{dist}(y_{l_k}, \partial D')> 0$ for some bounded subsequence $\{y_{l_k}\}$, for $k\in \mathbb N$. Then some subsequence $y_{l_{k_j}}\to z\in D'$ as $j\to\infty$. The first claim yields $y_{l}\to z\in D'$ as $l\to\infty$, which contradicts the hypotheses of claim (4). Proposition 2.11 is proved. Remark 2.12. Below, we consider the fundamental sequences with respect to the metric function $\rho^{\omega}_{p,F_0}$ which satisfy just one of claims (1), (3), and (4) of Proposition 2.11. 2.2. Capacity metric and completion of the domainDefinition 2.13. Let $D'_{\rho,p}$ be the collection of points $\{y\in D'\setminus F_0\}$ with the capacity metric function $\rho^{\omega}_{p,F_0}$. Definition 2.14. Two fundamental sequences $\{y_{l}\in D'_{\rho,p}\}$ and $\{z_{l}\in D'_{\rho,p}\}$, $l\in \mathbb N$, with respect to the capacity metric function $\rho^{\omega}_{p,F_0}$ are called equivalent whenever $\rho^{\omega}_{p,F_0}(y_l,z_l)\to 0$ as $l\to \infty$. Let us define a new metric space $({\widetilde D}'_{\rho,p}, \widetilde\rho^{\,\omega}_{p,F_0})$: (1) its elements are the classes of equivalent fundamental sequences, and (2) the distance between two elements $X, Y\in {\widetilde D}'_{\rho,p}$ equals
$$
\begin{equation}
\widetilde\rho^{\,\omega}_{p,F_0}(X,Y)=\lim_{l\to\infty}\rho^{\omega}_{p,F_0}(x_l,y_l),
\end{equation}
\tag{2.21}
$$
where $\{x_l\}$ and $\{y_l\}$ are fundamental sequences in $X$ and $Y$, respectively. Assume henceforth that the metric space $({\widetilde D}'_{\rho,p}, \widetilde\rho^{\,\omega}_{p,F_0})$ is non-empty. By analogy with the Hausdorff completion theorem, see [57], Chap. 2, § 6, and [58], § 21.3, for instance, we can prove the following result. Proposition 2.15. The following claims hold: (1) the metric function (2.21) is independent of the choice of fundamental sequences $\{x_l\}$ in the class $X$ and $\{y_l\}$ in the class $Y$; (2) the metric function (2.21) in Definition 2.14 satisfies on ${\widetilde D}'_{\rho,p}$ the axioms of a metric space; (3) the space $({\widetilde D}'_{\rho,p}, \widetilde\rho^{\,\omega}_{p,F_0})$ includes a subset isometric to the metric space with the metric $\rho^{\omega}_{p,F_0}$. Proof. Recall how we identify the points of $\{y\in D'\setminus F_0\mid \rho^{\omega}_{p,F_0}(y,y)=0\}$ with the metric $\rho^{\omega}_{p,F_0}$ and those of some subset in $({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$.
With a point $y\in {D}'_{\rho,p}$ we associate the equivalence class $i(y)\in {\widetilde D}'_{\rho,p}$ containing the constant sequence $\{y,y,\dots, y,\dots\}$. It is obvious that
$$
\begin{equation*}
\widetilde\rho^{\,\omega}_{p,F_0}(i(x),i(y))=\rho^{\omega}_{p,F_0}(x,y),
\end{equation*}
\notag
$$
so that the embedding is an isometry. Proposition 2.15 is proved. Definition 2.16. Refer to the metric space $({D}'_{\rho,p}, \rho^{\omega}_{p,F_0})$ to the subset $\{y\in D'\setminus F_0\mid \rho^{\omega}_{p,F_0}(y,y)=0\}$ with the metric $\rho^{\omega}_{p,F_0}$. Proposition 2.17. Consider a homeomorphism $f\colon D' \to D$ from $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant2$ for $n=2$. We fix an equivalence class $h\in {\widetilde D}'_{\rho,p}$ and take an arbitrary fundamental sequence $\{y_l\}$ in this class. Then the following behaviour of $\{y_l\}$ is possible: (1) (a) $y_l\to y\in D'\setminus F_0$ as $l\to\infty$ in the Euclidean metric and the limit $y$ is unique, meaning that it is independent of the choice of sequence in $h$; (b) $y_l\to y\in F_0$ as $l\to\infty$ in the Euclidean metric and the limit $y$ is unique; (2) otherwise, depending on the choice of fundamental sequence in $h$, the following cases are possible: (a) $\varlimsup_{l\to \infty}d(y_l,0)<\infty$ and then $\operatorname{dist}(y_l, \partial D')\to 0$ as $l\to \infty$; (b) $\varlimsup_{l\to \infty}d(y_l,0)=\infty$ and $\varliminf_{\,l\to \infty}d(y_l)<\infty$, and then
$$
\begin{equation*}
\lim_{l\to \infty}\operatorname{dist}(y_{l_k}, \partial D')=0
\end{equation*}
\notag
$$
for every bounded subsequence $\{y_{l_k}\in D'\}$ of $\mathbb R^n$; (c) $\lim_{l\to \infty}d(y_l,0)=\infty$. Proof. The fundamental sequence $\{y_l\}$ of class $h\in {\widetilde D}'_{\rho,p}$ bounded in $\mathbb R^n$ satisfies the hypotheses of Proposition 2.11, and so its claims (1)–(4) can hold for it. It remains to verify that the same claims hold for every bounded sequence $\{z_l\}$ of the class $h\in {\widetilde D}'_{\rho,p}$.
Indeed, the sequence $y_1,z_1,y_2,z_2,\dots,y_n,z_n,\dots$ is fundamental with respect to the metric function $\rho^{\omega}_{p,F_0}$, bounded in $\mathbb R^n$, and has an accumulation point $y$, which lies either in $D'$ or in $\partial D'$. In the first case by claim (1) of Proposition 2.11 some subsequence of the sequence converges to $y\in D'$. Hence, both sequences (2.22) and $\{z_l\}$ converge to $y$ as $l\to\infty$. In the second case, no subsequence $\{z_{l_k}\}$ of the sequence $\{z_l\}$ can converge to any point $z\in D'$, because similar arguments would yield the impossible coincidence $y=z$. So, if the sequence $\{z_l\}$ is bounded, then claim (3) of Proposition 2.11 shows that $\operatorname{dist}(z_l, \partial D')\to 0$ as $l\to \infty$.If some sequence $\{y_l\}$ of class $h\in {\widetilde D}'_{\rho,p}$ is not bounded, then we should apply claim (4) of Proposition 2.11 to justify claims (2)(b) and (2)(c) of Proposition 2.17. Now we take another fundamental sequence $\{z_l\}$, $l\in \mathbb N$, in the same class $h\in {\widetilde D}'_{\rho,p}$. Applying Proposition 2.11 to it, we conclude that $z_l$ cannot converge to any point $z\in D'$, as otherwise $y_l$ would also converge to $z\in D'$ as $l\to \infty$. Thus, for the sequence $z_l$, only claims (3) or (4) of Proposition 2.11 can hold, which proves Proposition 2.17. The following example shows that each of the possibilities (a), (b) and (c) of part 2 of Proposition 2.17 can be realized in various sequences of the same class. Example 2.18 (ridge domain). In [18], [26], and [45], one can find an example of a simply-connected domain with non-trivial boundary elements, although the domain is locally connected at all boundary points of the Euclidean boundary. For $q=p=n=3$ and $\omega \equiv 1$, consider the ridge domain With the new notation and concepts, we can interpret Proposition 2.4 as follows. Theorem 2.19 (extension of $\mathcal Q_{p,q}$-homeomorphisms). Consider a homeomorphism $f\colon D' \to D$ of the class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$. Then (1) the mapping $f\colon D' \to D$ induces the Lipschitz mapping
$$
\begin{equation*}
f\colon \bigl({D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}\bigr)\to \bigl({D}_{\rho,q},\widetilde\rho_{q,f(F_0)}\bigr)
\end{equation*}
\notag
$$
of metric spaces, with the estimate for metric distances
$$
\begin{equation}
\begin{cases} \widetilde\rho_{p,f(F_0)}\bigl(f(x),f(y)\bigr) \leqslant K_p\widetilde\rho^{\,\omega}_{p,F_0}(x,y) &\textit{if } q=p, \\ \widetilde\rho_{q,f(F_0)}\bigl(f(x),f(y)\bigr) \leqslant \Psi_{p,q}(D'\setminus F_0)^{1/\sigma} \widetilde\rho^{\,\omega}_{p,F_0}(x,y) &\textit{if } q<p, \end{cases}
\end{equation}
\tag{2.23}
$$
for all points $x,y\in {D}'_{\rho,p}$, where $1/\sigma=1/q-1/p$; (2) the mapping $f\colon D' \to D$ induces the Lipschitz mapping
$$
\begin{equation*}
\widetilde f\colon \bigl({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}\bigr) \to \bigl({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)}\bigr)
\end{equation*}
\notag
$$
of the “completed” metric spaces: with each element $X\in( {\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ associate the element $\widetilde f(X)\in ({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ containing the fundamental sequence $\{f(x_l)\}$, where $\{x_l\}\in X$, with the estimate for metric distances
$$
\begin{equation}
\begin{cases} \widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr) \leqslant K_p\widetilde\rho^{\,\omega}_{p,F_0}(X,Y) &\textit{if } q=p, \\ \widetilde\rho_{q,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr) \leqslant \Psi_{p,q}(D'\setminus F_0)^{1/\sigma} \widetilde\rho^{\,\omega}_{p,F_0}(X,Y) &\textit{if } q<p, \end{cases}
\end{equation}
\tag{2.24}
$$
for $x,y\in {\widetilde D}'_{\rho,p}$. Proof. Claim (1) and (2.23) follow directly from Proposition 2.4, while (2.24) is secured by Definition (2.21) of the metric distance between the elements of “completed” spaces. Indeed, if a sequence $\{x_l\}$ belongs to $X\in {\widetilde D}'_{\rho,p}$, then by (2.23) the sequence $\{f(x_l)\}$ is fundamental with respect to the metric function $\widetilde\rho_{q,f(F_0)}$. We call the class of equivalent sequences containing $\{f(x_l)\}$ the image of the class $X$, and denote the resulting mapping by $\widetilde f$. Deducing that
$$
\begin{equation*}
\widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr) =\lim_{l\to\infty}\widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(x_l),\widetilde f(y_l)\bigr)
\end{equation*}
\notag
$$
and using Definition (2.21), as well as (2.23), we obtain the claim. Therefore, Proposition 2.19 determines the extended mapping $\widetilde f$. Definition 2.20. Let $f\colon D'{\to}\, D$ be a homeomorphism of the class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$. Denote by $\widetilde f\colon ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})\to({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ the extension of $f$ to the “completed” metric spaces: to each $X\in({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ we associate $\widetilde f(X)\in ({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ containing the fundamental sequence $\{f(x_l)\}$. 2.3. Capacity boundary. Boundary correspondence of mappingsBy Proposition 2.17, in the topology of the extended space $\mathbb R^n$, the limit points of the fundamental sequence $\{y_l\}$, $l\in \mathbb N$, of some class $h\in {\widetilde D}'_{\rho,p}$ can be (1a) the points $y\in D'\setminus F_0$: in this case, $y_l\to y\in D'\setminus F_0$ as $l\to\infty$ in the Euclidean metric; (1b) the points $y\in F_0$: in this case, $y_l\to y\in F_0$ as $l\to\infty$ in the Euclidean metric. Otherwise, depending on the choice of fundamental sequence $\{y_l\}$, $l\in \mathbb N$, of class $h$, the possible variants are (2a) the points $y\in \partial D'$; (2b) the point $y=\infty$. Clearly, in case (1a) we can identify the class $h\in {\widetilde D}'_{\rho,p}$ with some point $y\in D'\setminus F_0$, while in case (1b), with some point $y\in F_0$. With this observation at hand, we define the concept of the capacity boundary. By claim (3) of Proposition 2.15, the points of the metric space $({D}'_{\rho,p},\rho^{\omega}_{p,F_0})$ are identified with those in some subset of $({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ so that the embedding is an isometry. Henceforth, we identify ${D}'_{\rho,p}$ with the image $i({D}'_{\rho,p})$ in ${\widetilde D}'_{\rho,p}$.Definition 2.21. The complement is called the capacity boundary of $D'$ (respectively, $D$). The metric on the boundary is induced from the ambient space. The capacity boundary elements of the domain $D'$ or $D$ are the points of the capacity boundary $H^{\omega}_{\rho,p}(D')$ or $H_{\rho,q}(D)$. Theorem 2.22 (boundary correspondence of $\mathcal Q_{p,q}$-homeomorphisms). Let $f\colon D' \to D$ be a homeomorphism of the class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$. Then the restriction $\widetilde f\mid_{H^\omega_{\rho,p}(D')}$ is a Lipschitz mapping of capacity boundaries. Proof. Let $\widetilde f\colon ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})\to({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$ be the mapping from Theorem 2.19. Then the restriction $\widetilde f\mid_{H^\omega_{\rho,p}(D')}$ is the Lipschitz mapping To prove the claim, it remains to verify that the image of this mapping lies in $(H_{\rho,q}(D), \widetilde\rho_{q,f(F_0)})$.
Assume on the contrary that there exists a boundary element $h\,{\in}\, (H^\omega_{\rho,p}(D'), \widetilde\rho^{\,\omega}_{p,F_0})$ such that $\widetilde f(h)= y\in (D,\widetilde\rho_{q,f(F_0)})$. Then there exists a sequence $\{x_l\}\in h$, where $h\in ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$, such that $f(x_l)\to y$ in the metric space $(D_{\rho,q},\widetilde\rho_{q,f(F_0)})$. By Proposition 2.10, the sequence $f(x_l)$ converges to $y\in D$ in the Euclidean metric as well. Therefore, $f^{-1}( f(x_l))=x_l$ converges to $\varphi(y)\in D'$ in $\mathbb{R}^n$. Proposition 2.17 shows that every sequence $\{z_l\}\in h$ converges to $\varphi(y)\in D'$ in the Euclidean metric, and so in the metric space $({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ as well, see Proposition 2.10, which obviously contradicts the initial assumption. Theorem is proved. 2.4. Support of a boundary elementIn this section, we fix an arbitrary number $p$ satisfying $n-1< p\leqslant n$ for $n\geqslant 3$ and $1\leqslant p\leqslant 2$ for $n=2$. Definition 2.23. Given a domain $D'$ in $\mathbb R^n$, the support $\mathcal{S}_h$ of a boundary element $h \in H^\omega_{\rho,p}(D')$ is the set of all accumulation points in the topology of the extended space $\overline{\mathbb R^n}$ of all fundamental sequences with respect to the capacity metric lying in the equivalence class defining $h$. Remark 2.24. Proposition 2.17 and Definition 2.21 show that no accumulation point of a sequence in $h \in H^\omega_{\rho,p}(D')$ fundamental with respect to the capacity metric belongs to $D'$. Therefore, Proposition 2.25. If $D'$ is a domain in $\mathbb R^n$, then (1) the support $\mathcal{S}_h$ of a boundary element $h \in H^\omega_{\rho,p}(D')$ coincides with the intersection $\bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'}$,
$$
\begin{equation}
\mathcal{S}_h= \bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'},
\end{equation}
\tag{2.27}
$$
where the closure is taken in the topology of the extended space $\overline{\mathbb R^n}$; (2) if $\rho^{\omega}_{p,F_0}(h_1,h_2)=0$ for two boundary elements $h_1,h_2 \in H^\omega_{\rho,p}(D')$, then $\mathcal{S}_{h_1}= \mathcal{S}_{h_2}$. Proof. The proof is in three steps.
(1) We fix a boundary element $h \in H^\omega_{\rho,p}(D')$. Let us verify the inclusion
$$
\begin{equation}
\mathcal{S}_h \subset \bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'}.
\end{equation}
\tag{2.28}
$$
By the definition of a boundary element $h \in H^\omega_{\rho,p}(D')$, there exists a fundamental sequence $\{y_l\}\in h$ with respect to the $(\omega,p)$-metric function with $\rho^{\omega}_{p,F_0}(y_l,h)\to 0$ as $l\to\infty$. For the sequence $\{y_l\in D'_{\rho,p}\}$ and its subsequences, only the behaviour described in Proposition 2.17 is possible:
(a) $y_l\to y\in D'\setminus F_0$ or $y_l\to y\in F_0$ as $l\to\infty$ in the Euclidean metric, and the limit $y$ is unique, meaning independent of the choice of sequence in $h$; (b) $\varlimsup_{l\to \infty}d(y_l,0)<\infty$ and then $\operatorname{dist}(y_l, \partial D')\to 0$ as $l\to \infty$; (c) $\varlimsup_{l\to \infty}d(y_l,0)=\infty$ and $\varliminf_{\,l\to \infty}d(y_l,0)<\infty$, and then
$$
\begin{equation*}
\lim_{l\to \infty}\operatorname{dist}(y_{l_k}, \partial D')=0
\end{equation*}
\notag
$$
for every subsequence $\{y_{l_k}\in D'\}$ bounded in $\mathbb R^n$;
(d) if $d(y_l,0)\to\infty$, then $\infty\in \mathcal{S}_h$. Definition 2.21 excludes case (a). In cases (b)–(d), we have In these cases, for every $\varepsilon > 0$ the elements of the sequence $\{y_l\in D'\}$ starting with some index $l_0$ lie in $B_{\rho}(h,\varepsilon)\cap D'$ for all $l\geqslant l_0$. Thus the accumulation points of $\{y_l\in D'\}$ lie in the closure $\overline{B_{\rho}(h,\varepsilon) \cap D'}$ in the topology of the extended space $\overline{\mathbb R^n}$. Since we choose the fundamental sequence $\{y_l\}\in h$ for the boundary element $h$ arbitrarily, it follows that $\mathcal{S}_h \subset\overline{B_{\rho}(h,\varepsilon) \cap D'}$. The inclusion (2.28) is established as $\varepsilon > 0$ is arbitrary.(2) In the case $\rho^{\omega}_{p,F_0}(h_1,h_2)=0$, the equivalence classes of fundamental sequences for the boundary elements $h_1$ and $h_2$ coincide. Hence, we conclude that the supports of $h_1$ and $h_2$ coincide. (3) To justify (2.27), it remains to verify the reverse inclusion to (2.28):
$$
\begin{equation}
\bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'} \subset \mathcal{S}_h.
\end{equation}
\tag{2.29}
$$
Indeed, if $x\,{\in} \bigcap_{\varepsilon > 0} \overline{B_{\rho}(h,\varepsilon) \cap D'}$, then, for each $l\in\mathbb N$, there exists $x_l\in B_{\rho}(h,1/l) \cap D'$ such that simultaneously $\rho^{\omega}_{p,F_0}(x_l,h)\to 0$ as $l\to\infty$ and (using Proposition 2.17 and extracting a subsequence if necessary) $x_l\to x$ in the topology of the extended space $\overline{\mathbb R^n}$. Therefore, the fundamental sequence $\{x_l\}$ with respect to the capacity metric determines a boundary element, which coincides with $h$. Thus, $x\in \mathcal{S}_h$ and (2.29) is established. The inclusions (2.28) and (2.29) are equivalent to (2.27). This proves the proposition. Proposition 2.26. The support $\mathcal{S}_h$ of each boundary element $h\in H^\omega_{\rho,p}(D')$ is connected in the topology of the space $\overline{\mathbb R^n}$. Proof. Assume on the contrary that, for some boundary element $h\in H^\omega_{\rho,p}(D')$, there are two disjoint open sets $V,W\subset\overline{\mathbb R^n}$ with $\mathcal{S}_h\subset V\cup W$, while $\mathcal{S}_h\cap V\ne\varnothing$ and $\mathcal{S}_h\cap W\ne\varnothing$. Take two points $x\in\mathcal{S}_h\cap V$ and $y\in\mathcal{S}_h\cap W$ and fundamental sequences $\{x_m\},\{y_m\}\in h$ with respect to the capacity metric such that $x_m\to x$ and $y_m\to y$ as $m\to\infty$. There is a curve $\gamma_m\subset D'$ with endpoints $x_m$ and $y_m$ such that $\operatorname{cap}((\gamma_m, F_0); L^1_p(D';\omega))\to 0$ as $m\to\infty$. For all big enough $m$, starting with some there exists a point $z_m\in \gamma_m$ satisfying $z_m\notin V\cup W$. We emphasize that the sequence $\{z_m\}$, fundamental with respect to the capacity metric, belongs to the equivalence class $h$. Extracting a subsequence, we may assume that $z_m\to z_0$, where $z_0\in \overline{D'}\setminus (V\cup W)$; here, the closure is taken in the topology of the extended space $\overline{\mathbb R^n}$. Since $z_0\notin \mathcal{S}_h$, we arrive at a contradiction with the definition of the support of a boundary element. Proposition 2.26 is proved. Proposition 2.27. Consider the support $\mathcal{S}_h$ of $h\in H^\omega_{\rho,p}(D')$. For every sequence $\{x_m\}\in h$ we have the convergence $x_m\to \mathcal{S}_h$ as $m \to \infty$ in the topology of the extended space $\overline{\mathbb R^n}$. Proof. Proposition 2.25 excludes the possibility that $\mathcal{S}_h\cap D'\ne\varnothing$.
Suppose that $\mathcal{S}_h$ is bounded in $\mathbb R^n$ and $\mathcal{S}_h\subset \partial D'$. Suppose that there exists a subsequence $\{x_{m_k}\in D'\}$, for $k\in \mathbb N$, of some fundamental sequence $\{x_m\}\in h$ such that $d(x_{m_k},\mathcal{S}_h) \geqslant\alpha> 0$ for all $k\in \mathbb N$, where $\alpha$ is some constant. Then the sequence $\{x_{m}\}$ has an accumulation point at some positive distance from $\mathcal{S}_h$. This point must lie in the support of the boundary element $h$, which contradicts the connectedness of $\mathcal{S}_h$. However, if the support $\mathcal{S}_h$ is unbounded and the sequence $x_m$ does not converge to $\mathcal{S}_h$ in the topology of the extended space $\overline{\mathbb R^n}$, then $\varliminf_{\,m\to\infty}x_m<\infty$. Consequently, there exists a finite accumulation point at some positive distance from $\mathcal{S}_h$. As in the previous case, we arrive at a contradiction with the connectedness of $\mathcal{S}_h$. Proposition 2.27 is proved. Proposition 2.28 (criterion for singleton support). Given a boundary element $h\in H^\omega_{\rho,p}(D')$ of the domain $D'$, the support $\mathcal{S}_h$ amounts to a single point if and only if for all fundamental sequences $\{x_m\},\{y_m\}\in h$ with respect to the capacity metric there exist curves $\overline{x_m y_{m}}$, for $m\in \mathbb N$, with $\operatorname{diam}(\overline{x_m y_{m}}) \to 0$ as $m\to \infty$. Proof. Necessity. Suppose that $\mathcal{S}_h=\{x_0\}$. Assume on the contrary that there exist fundamental sequences $\{x_m\}$ and $\{y_m\}$ of class $h$ with respect to the capacity metric converging to $x_0$, curves $\gamma_{m} = \overline{x_m y_{m}}$ with
$$
\begin{equation}
\operatorname{cap}^{1/p}\bigl((\gamma_{m}, F_0); L^1_p(D';\omega)\bigr)\to 0 \quad\text{as}\quad m\to \infty,
\end{equation}
\tag{2.30}
$$
and a number $\alpha>0$ such that
$$
\begin{equation*}
\operatorname{diam} \gamma_{m}\geqslant \alpha > 4d(x_m, y_{m})\quad\text{for all}\quad m\in\mathbb N
\end{equation*}
\notag
$$
because $x_m\to x_0$ and $y_m\to x_0$ as $m\to\infty$. Then, for each $m\in\mathbb N$, there exists a point $z_{m} \in \gamma_{m}$ such that, on the one hand,
$$
\begin{equation}
d(x_m,z_{m})> \frac{\alpha}{4}, \qquad d(y_{m},z_{m})> \frac{\alpha}{4}
\end{equation}
\tag{2.31}
$$
and on the other hand, (2.7) and (2.30) yield $\rho^{\omega}_{p,F_0} (z_{m}, x_{m}) \to 0$ as $m\to \infty$. Hence, we infer that the sequence $\{z_{m}\}$, for $m\in \mathbb N$, is fundamental with respect to the capacity metric and belongs to the boundary element $h$. On the other hand, there exists a subsequence $\{z_{m_i}\}$, for $i\in \mathbb N$, converging to some point $z_0$; moreover, (2.31) implies that $z_0 \neq x_0$. Since $z_0\in \mathcal{S}_h$ by the definition of support, we arrive at a contradiction with its being a singleton.
Sufficiency. Assume on the contrary that there are two sequences $\{x_m\}$, $\{y_m\} \in h$ fundamental with respect to the capacity metric and converging to distinct points $x$ and $y$ of the support $\mathcal{S}_h$. By the hypotheses, there exist curves $\gamma_m = \overline{x_m y_m}$ such that $\operatorname{diam} \gamma_m \to 0$ as $m \to \infty$. In particular, $\operatorname{diam} \gamma_m\,{\geqslant}\, d(x_m, y_m)\,{\to}\, d(x, y)\,{>}\,0$ as $m \to \infty$, which, evidently, contradicts the property $\operatorname{diam} \gamma_m \to 0$ as $m \to \infty$ inferred from the assumption, proving the proposition. 2.5. Continuous extension of mappings of class $\mathcal{Q}_{p, q}(D',\omega;D)$ to the Euclidean boundaryIn this section, we fix arbitrary numbers $q$ and $p$ satisfying $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$. In what follows, we define domains $\mu$-connected at boundary points. Definition 2.29 (connectedness properties [16], [18]). (1) A domain $D'$ is called locally connected at $x \in \partial D'$ if for every neighbourhood $U$ of $x$ there is a neighbourhood $V \subset U$ of this point such that $V \cap D'$ is connected. (2) An unbounded domain $D'$ is called locally connected at $\infty$ if for every neighbourhood $U$ of $\infty$ there is a neighbourhood $V \subset U$ of this point such that $V \cap D'$ is connected. (3) A domain $D'$ is called locally $\mu$-connected at $x \in \partial D'$, where $\mu\in\mathbb N$, if for every neighbourhood $U$ of $x$ there is a neighbourhood $V \subset U$ of this point such that $V\cap D'$ consists of $\mu$ connected components, each of which is locally connected at $x$. Observe that a domain $D'$ locally $1$-connected at $x \in \partial D'$ is precisely the domain $D'$ locally connected at $x \in \partial D'$. (4) An unbounded domain $D'$ is called locally $\mu$-connected at $\infty$, where $\mu\in\mathbb N$, if for every neighbourhood $U$ of $\infty$ there is a neighbourhood $V \subset U$ of this point such that $V \cap D'$ consists of $\mu$ connected components, each of which is locally connected at $\infty$. In the case $\mu=1$ we obtain the domain $D'$ locally connected at $\infty$. (5) A domain $D'$ is called finitely connected at $x \in \partial D'$ or $x=\infty$ whenever it is $\mu$-connected at $x$ for some $\mu\in\mathbb N$. The following example demonstrates the appearance of domains which are multiply connected at boundary points. Example 2.30 (slit ball). Let $D' = B(0,1) \setminus (\{0\}\times[0,1)^{n-1})$. It is not difficult to see that $D'$ is locally $2$-connected at each point $x\in \{0\}\times(0,1)^{n-1}$. If $\omega= 1$ is the trivial weight and $p=n$, then condition (2.37) is met for every point $x\in \{0\}\times(0,1)^{n-1}$, and $x$ lies in the support of two distinct boundary elements $h_{+},h_{-} \in H_{\rho,n}(D')$. Let us present the methods of [16], Theorem 1.10, for describing connectedness alternative to Definition 2.29 and useful below. Proposition 2.31. Given a domain $D'\in \mathbb R^n$ and its boundary point $x \in \partial D'$, the following statements are equivalent: (1) $D'$ is locally $\mu$-connected at $x$; (2) for every neighbourhood $U$ of $x$ there exists a neighbourhood $V \subset U$ of this point such that $V \cap D'$ consists of $\mu$ connected components, the boundary of each of which contains $x$; (3) $\mu$ is the smallest integer for which the following condition holds: given $\mu + 1$ sequences $\{x_ {1, k}\}, \dots, \{x_ {\mu+1, k}\}$ of points in $D'$ converging to $x$, if $V$ is some neighbourhood of $x$, then there exists a connected component of $V \cap D'$ including subsequences of two distinct sequences. To obtain similar properties at $\infty$, we should use the stereographic projection to map the domain $D'$ onto the unit sphere in $\mathbb R^{n+1}$ with the point $\infty$ going into the north pole, on which the property of local $\mu$-connectedness at $\infty$ can be stated by analogy with the above. Example 2.32. On the plane $\mathbb R^2$, consider the complement The domain $D'$ is obviously $4$-connected at $0$: each intersection $B(0,r)\cap D'$, for $r\in (0,2)$, consists of $4$ connected components. We denote them by $V_1$ and $V_3$ if $x_1\cdot x_2>0$ and by $V_2$ and $V_4$ otherwise. It is natural to define the weighted capacity of the condenser $\mathcal E=(\{0\},F_0)\subset D'$ in the space $L^1_p(D';\sigma)$ with respect to the connected component $V_i$ as
$$
\begin{equation}
\operatorname{cap}\bigl((\{0\}, F_0); L^1_p(V_i,D';\omega)\bigr) =\inf_{u}\|u\mid L^1_{p}(D';\omega)\|^p,
\end{equation}
\tag{2.32}
$$
where the infimum is over all functions $u\in\operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ such that $u|_{B(0,r)\cap V_i}\equiv1$ for some $r>0$, depending on $u$, and $u|_{F_0}\equiv0$. On account of Example 2.7, the capacity of the point $0$ with respect to $V_1$ and $V_3$ is positive, and with respect to $V_2$ and $V_4$ it vanishes. This example motivates the following definition. Definition 2.33. Suppose that a domain $D'$ is locally $\mu$-connected at a boundary point $x\in \partial D'$ and denote by $V_1,V_2,\dots,V_\mu$ the distinct connected components of $B(x,r)\cap D'$, where $r\in(0,r_0)$ for sufficiently small $r_0>0$, whose boundaries contain $x$. Next, we define the weighted capacity of the condenser $\mathcal E=(\{x\},F_0)\subset D'$ in the space $L^1_p(D';\omega)$ with respect to the connected component $V_i$ by where the infimum is over all functions $u\in\operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ such that $u|_{B(x,r)\cap V_i}\equiv1$ for some $r\in (0,r_0)$, depending on $u$, and $u|_{F_0}\equiv 0$. If $\mu=1$, then instead of notation (2.33) we will simply write
$$
\begin{equation*}
\operatorname{cap}\bigl((\{x\}, F_0); L^1_p(D';\omega)\bigr).
\end{equation*}
\notag
$$
In the case $x=\infty$, the lower bound in (2.33) is taken over all functions $u\in\operatorname{Lip}_{\mathrm{loc}}(D')\cap L^1_{p}(D';\omega)$ such that $u|_{(\mathbb R^n\setminus B(x,r))\cap V_i}\equiv1$ for some $r>0$, depending on $u$, and $u|_{F_0}\equiv0$, and denoted by
$$
\begin{equation}
\operatorname{cap}\bigl((\{\infty\}, F_0); L^1_p(V_i,D';\omega)\bigr).
\end{equation}
\tag{2.34}
$$
A boundary point $x\in \partial D'$ is called a point of zero capacity with respect to the connected component $V_i$ whenever
$$
\begin{equation}
\operatorname{cap}\bigl((\{x\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0.
\end{equation}
\tag{2.35}
$$
If condition (2.35) is independent of the choice of continuum $F_0$, we simply write Proposition 2.28 yields the following corollary. Corollary 2.34. The following claims hold. (1) If the domain $D'$ is locally connected at $x_0$ and the condition
$$
\begin{equation}
\operatorname{cap}\bigl((\{x_0\}, F_0); L^1_p(D';\omega)\bigr)=0
\end{equation}
\tag{2.37}
$$
holds at $x_0$, then the boundary elements $h_1$ and $h_2\in H^\omega_{\rho,p}(D')$ of the domain $D'$ whose supports $\mathcal{S}_{h_1}$ and $\mathcal{S}_{h_2}$ meet at $x_0$ cannot be distinct: $h_1=h_2$. (2) Suppose that the domain $D'$ is locally $\mu$-connected at $x_0$, and that at $x_0$ condition (2.35) holds for all $i=1,\dots,\mu$. Then the boundary elements $h_1,h_2,\dots,h_\mu,h_{\mu+1} \in H^\omega_{\rho,p}(D')$ of $D'$ whose supports $\mathcal{S}_{h_1},\mathcal{S}_{h_2},\dots,\mathcal{S}_{h_\mu},\mathcal{S}_{h_{\mu+1}}$ share the point $x_0$ cannot be distinct: at least two of them coincide. Proof. (1) Suppose that the supports $\mathcal{S}_{h_1}$ and $\mathcal{S}_{h_2}$ of two boundary elements $h_1,h_2\in H^\omega_{\rho,p}(D')$ of $D'$ meet at $x_0$. Take two arbitrary sequences $\{x_k\}\in h_1$ and $\{y_k\}\in h_2$ fundamental with respect to the metric $\rho^{\omega}_{p,F_0}$ such that $x_k\to x_0$ and $y_k\to x_0$ as $k\to\infty$. Since $D'$ is locally connected at $x_0$, we can connect $x_k$ and $y_k$ with curves $\gamma_k = \overline{x_k y_k}$ such that $\operatorname{diam}\gamma_k \to 0$ as $k\to\infty$. Since $D'$ is locally connected at $x$, condition (2.37) also yields Hence, we see that the sequence $\{x_k\}$ and $\{y_k\}$ are equivalent, which implies $h_1\,{=}\,h_2$.
(2) Assume that the supports $\mathcal{S}_{h_1},\mathcal{S}_{h_2},\dots,\mathcal{S}_{h_{\mu+1}}$ of some boundary elements $h_1,h_2,\dots,h_{\mu+1}\,{\in}\, H^\omega_{\rho,p}(D')$, for $\mu\in \mathbb N$, of $D'$ meet at $x_0$. Take an arbitrary fundamental sequence $\{x_{ik}\}\in h_i$ with respect to the metric $\rho^{\omega}_{p,F_0}$ such that $x_{ik}\to x_0$ as $k\to\infty$, for $i=1,\dots,\mu+1$. By claim (3) of Proposition 2.31, since $D'$ is locally $\mu$-connected at $x_0$, there exists a connected component $V_{i_0}$, for $1\leqslant i_0\leqslant \mu_0$, of the intersection $B(x_0,r)\cap D'$ containing subsequences, for instance, $x_{1k_j}$ and $x_{2l_j}$, for $j\in \mathbb N$, of two distinct sequences $x_{1k}$ and $x_{2k}$, for $k\in \mathbb N$. Since the connected component $V_{i_0}$ is locally connected at $x_0$ and the hypotheses of claim 1 hold, which yields $h_1=h_2$. This proves the corollary.Definition 2.35 (associated support and connected components). Consider some boundary element $h \in H^\omega_{\rho,p}(D')$ whose support $\mathcal{S}_h$ contains $x\in \partial D'$ such that the domain $D'$ is $\mu$-connected at $x$, while $\{y_{m}\}$ is a fundamental sequence with respect to the metric $\rho^{\omega}_{p,F_0}$ belonging to the boundary element $h$ and converging to $x$ in the topology of $\overline{\mathbb R^n}$. Since $D'$ is $\mu$-connected at $x$, there exists at least one connected component $V_i$ of the intersection $B(x,r)\cap D'$, where $r>0$ is a sufficiently small number, which contains some subsequence $\{y_{m_k}\}$, for $k\in\mathbb N$. In this case, say that the support $\mathcal{S}_h$ of the boundary element $h$ and the connected component $V_i$ are associated with each other at $x\in \mathcal{S}_h$. Proposition 2.36. The following claims hold. (1) If $D'$ is a locally $\mu$-connected domain at $x$, the support $\mathcal{S}_h$ of some boundary element $h \in H^\omega_{\rho,p}(D')$ contains $x\in \partial D'$ and is associated with the connected component $V_i$ at $x$, while the weighted capacity of $x$ with respect to the connected component $V_i$ vanishes,
$$
\begin{equation*}
\operatorname{cap}\bigl((\{x\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0,
\end{equation*}
\notag
$$
then, for every sequence $\{x_m \in V_i\cap D'\}$ of points, $d(x_m,x) \to 0$ implies that $\{x_m\}\in h$ and
$$
\begin{equation}
\rho_{q,f(F_0)} \bigl(f(x_m), \widetilde f(h)\bigr)\to 0\quad \textit{as}\quad m \to \infty.
\end{equation}
\tag{2.38}
$$
(2) If $D'$ is a locally $\mu$-connected domain at $\infty$, the support $\mathcal{S}_h$ of some boundary element $h \in H^\omega_{\rho,p}(D')$ contains $\infty$ and is associated with the connected component $V_i$ at $\infty$, while the weighted capacity of the point $\infty$ with respect to some connected component $V_i$ vanishes,
$$
\begin{equation*}
\operatorname{cap}\bigl((\{\infty\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0,
\end{equation*}
\notag
$$
then, for every sequence $\{ x_m \in V_i\cap D'\}$ of points, $d(x_m,0) \to \infty$ implies that $\{x_m\}\,{\in}\, h$ and (2.38) holds. Proof. (1) Choose $x\,{\in}\, \partial D'$ and a sequence $\{x_m{\in}\, V_i\,{\cap}\, D'\}$ such that $d(x_m,x)\,{\to}\, 0$ as $m \to \infty$. Since $V_i\cap D'$ is locally connected at $x$, see claim (2) of Proposition 2.31, we infer the existence of curves $\overline{x_mx_{m+k}}$ with endpoints $x_m$ and $x_{m+k}$, for $k\geqslant1$, such that $\operatorname{diam} \overline{x_mx_{m+k}} \to 0$ as $m,k\to \infty$. Since $\operatorname{cap}((\{x\}, F_0); L^1_p(V_i,D';\omega))=0$, Definition 2.33 yields $\rho^{\omega}_{p,F_0} (x_m, x_{m+k})\to 0$ as $m,k\to \infty$. Thus, on the one hand the sequence $\{x_{m}\}$ is fundamental with respect to the metric $\rho^{\omega}_{p,F_0}$, and on the other, $d(x_m,x) \to 0$ as $m\to \infty$.
Now we take an arbitrary sequence $\{y_{m}\in V_i\cap D'\}$, for $m\in\mathbb N$, fundamental with respect to the metric $\rho^{\omega}_{p,F_0}$, belonging to some boundary element $h$, and converging $x$ in the Euclidean metric. We claim that each fundamental sequence $\{x_{m}\}$ with respect to the metric $\rho^{\omega}_{p,F_0}$ satisfies
$$
\begin{equation}
\rho^{\omega}_{p,F_0} (x_m, y_m)\to 0 \quad\text{as} \quad m\to \infty.
\end{equation}
\tag{2.39}
$$
As in the previous argument, we conclude that $\rho^{\omega}_{p,F_0} (x_m, y_m)\to 0$ as $m\to \infty$. Thus, property (2.39) and property $\{x_m\}\in h$ together with it are justified.
Applying (2.9), we deduce (2.38): indeed, the sequences $\{f(x_m)\}$ and $\{f(y_m)\}$ are equivalent with respect to the capacity metric function $\rho_{q,f(F_0)}$ in the domain $D$. Hence, $\{f(x_m)\}\in \widetilde f(h)$ and $\rho_{q,f(F_0)} (f(x_m), \widetilde f(h))\to 0$ as $m \to \infty$. (2) The second claim can be justified similarly. Proposition 2.36 is proved. Theorem 2.37 (boundary behaviour of homeomorphisms). Consider a homeomorphism $f \colon D'\to D$ of class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1<q\leqslant p\leqslant n$ for $n\geqslant 3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$, as well as a weight function $\omega\in L_{1,\mathrm{loc}}(D')$. Suppose that the domain $D'$ (1) is locally $\mu$-connected at some boundary point $y\in\partial D'$, (2) the support $\mathcal{S}_{h}$ of some boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$, (3) $\operatorname{cap}((\{y\}, F_0); L^1_p(V_i,D';\omega))=0$, where $V_i$ is the connected component associated with the support $\mathcal{S}_{h}$ at $y$. Then the boundary behaviour of the mapping $f \colon D'\to D$ at $x\in\partial D'$ is in the topology of the extended space $\mathbb R^n$. Proof. We take a sequence $\{y_m\in V_i\cap D'\}$ converging to $y\in \partial D'$ as $m\to\infty$. Proposition 2.36 shows that $\rho^{\omega}_{q,f(F_0)} (f(y_m),\widetilde f(h))\to 0$ as $m \to \infty$. In addition, by Proposition 2.27 the sequence $\{f(y_m)\}$ converges to the support $\mathcal{S}_{\widetilde f( h)}$ in the topology of the extended space $\mathbb R^n$. The proof of Theorem 2.37 is complete. Corollary 2.38 (continuous extension to boundary points). Consider a homeomorphism $f \colon D'\to D$ of the class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1< q\leqslant p\leqslant n$ for $n\geqslant3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$, as well as a weight function $\omega\in L_{1,\mathrm{loc}}(D')$. Suppose also that (1) the domain $D'$ is locally $\mu$-connected at some boundary point $y\in\partial D'$; (2) the support $\mathcal{S}_{h}$ of a boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$; (3) $\operatorname{cap}((\{y\}, F_0); L^1_p(V_i,D';\omega))=0$, where $V_i$ is the connected component associated with the support $\mathcal{S}_{h}$ at $y$; (4) the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ amounts to a singleton: $\mathcal{S}_{\widetilde f(h)}=\{x\}\in \partial D$. Then the mapping $f \colon D'\to D$ extends by continuity to $y\in\partial D'$ and Proof. We take a sequence $\{y_m\in V_i\cap D'\}$ converging to $y\in \partial D'$ as $m\to\infty$. Theorem 2.37 shows that
$$
\begin{equation*}
f(z)\to \mathcal{S}_{\widetilde f(h)}\quad \text{as}\quad z\to y,\quad z\in V_i\cap D'
\end{equation*}
\notag
$$
in the topology of the extended space $\mathbb R^n$. Since by the assumption the support $\mathcal{S}_{\widetilde f( h)}$ of the boundary element $\widetilde f(h)$ is a singleton, $\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$, the above implies that the sequence $\{f(y_m)\}$ converges to $x\in \partial D$. The proof of Corollary 2.38 is complete. From Corollary 2.38 we have Corollary 2.39 (continuous extension to the Euclidean boundary). Consider a homeomorphism $f \colon D'\to D$ of the class $\mathcal{Q}_{p, q}(D',\omega;D)$, where $n-1< q\leqslant p\leqslant n$ for $n\geqslant3$ and $1\leqslant q\leqslant p\leqslant 2$ for $n=2$, as well as a weight function $\omega\in L_{1,\mathrm{loc}}(D')$. The following claims hold: (1) if $D'$ is locally connected at $y\in\partial D'$ and $\operatorname{cap}((\{y\}, F_0); L^1_p(D';\omega))=0$, then $y$ lies in the support $\mathcal{S}_{h}$ of some boundary element $h\in H^\omega_{\rho,p}(D')$; (2) if the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton, $\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$, then the mapping $f\colon D'\to D$ extends by continuity to $y\in\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$, and Proof. All hypotheses of Proposition 2.36 are obviously met, and so $y$ lies in some boundary element $h\in H^\omega_{\rho,p}(D')$. The argument above and the hypotheses of the corollary ensure the fulfillment of the conditions of Corollary 2.38 for $\mu=1$. It shows that the mapping $f \colon D'\to D$ extends by continuity to $y\in\mathcal{S}_{h}$, and the limit equals (2.40). Corollary 2.39 is proved. Example 2.40 (a domain with non-trivial boundary elements). Let us consider $D=(0,1)^2\setminus\bigcup_{k\in\mathbb{N}}I_k\subset \mathbb{R}^2$, where $I_k=[1/2,1) \times \{1/2^k\}$ determine the cuts. It is not difficult to see that $I=[1/2,1)\times\{0\}$ is the support of a boundary element for $p=2$ and $\omega\equiv1$. Example 2.41. For the domain from Example 2.18, the edge of the ridge is indeed the support of a boundary element. Remark 2.42. For the weight $\omega$ and the domain $D'$ such that the collection $H^\omega_{\rho,p}(D')$ of boundary elements is independent of the choice of the continuum $F_0$, the support $\mathcal{S}_h$ of an arbitrary boundary element $h\in H^\omega_{\rho,p}(D')$ is independent of the choice of $F_0$, and consequently, all statements of this section are absolute. § 3. Moduli of curve families and homeomorphisms of class $\mathcal Q_{p,q}(D',\omega)$Consider a domain $D'$ in $\mathbb{R}^{n}$, where $n \geqslant 2$, a weight function $\omega\colon D' \to (0, \infty)$ of class $L_{1,\mathrm{loc}}$, and a family $\Gamma$ of (continuous) curves or paths $\gamma\colon[a,b]\to D'$. Recall that, given a curve family $\Gamma$ in $D'$ and a real number $p\geqslant 1$, the weighted $p$-modulus of $\Gamma$ is defined as
$$
\begin{equation*}
\operatorname{mod}^{\omega}_{p}(\Gamma)=\inf_\rho \int_{D'} \rho^{p}(x)\omega(x)\, dx,
\end{equation*}
\notag
$$
where the infimum is over all nonnegative Borel functions $\rho\colon D' \to [0, \infty]$ with for all (locally) rectifiable curves $\gamma \in \Gamma$. In the case of trivial weight $\omega \equiv 1$ we write $\operatorname{mod}_{p}(\Gamma)$ instead of $\operatorname{mod}^1_{p}(\Gamma)$. Recall that the integral in (3.1) for a rectifiable curve $\gamma\colon[a,b]\to D'$ is defined as
$$
\begin{equation*}
\int_{\gamma} \rho\, ds = \int_{0}^{l(\gamma)} \rho(\widetilde{\gamma}(t))\, dt,
\end{equation*}
\notag
$$
where $l(\gamma)$ is the length of $\gamma\colon[a,b]\to D'$, while $\widetilde{\gamma}\colon[0,l(\gamma)]\to D'$ is its natural parametrization, that is, the unique continuous mapping with $\gamma=\widetilde{\gamma}\circ S_{\gamma}$, where $S_{\gamma}\colon[a,b]\to[0,l(\gamma)]$ is the length function, defined at $t\in [a,b]$ as $S_{\gamma}(t)=l(\gamma\vert_{[a,t]})$. If $\gamma$ is only a locally rectifiable curve, then we put
$$
\begin{equation*}
\int_{\gamma} \rho\,ds = \sup \int_{\gamma'} \rho\,ds
\end{equation*}
\notag
$$
with the least upper bound taken over all rectifiable subcurves $\gamma'\colon [a', b'] \to D'$ of $\gamma$, where $[a', b']\subset(a,b)$ and $\gamma'= \gamma_{[a', b']}$. The functions $\rho$ satisfying (3.1) are called admissible functions, or metrics, for the family $\Gamma$. An equivalent description of the mappings of classes $\mathcal Q_{p,q}(D',\omega;D)$ is obtained in [33] in the modular language: to this end, we should replace capacity in the definition of $\mathcal Q_{p,q}(D',\omega;D)$ by the modulus of the curve family whose endpoints lie on the plates of the condenser. Remark 3.1. It is observed in [32], § 4.4, that in the case $q=p=n$ ($n-1<q=p< n$) the class of homeomorphisms $\mathcal Q_{n,n}(D',\omega;D)$ ($\mathcal Q_{p,p}(D',\omega;D)$) is included into the class of $\omega$-homeomorphisms ($(p, \omega)$-homeomorphisms)4 [21] (and [59]), defined via a controlled variation of the modulus of the curve family. We will verify that, actually, the class $\mathcal Q_{n,n}(D',\omega;D)$ coincides with the family of $\omega$-homeomorphisms of [21], § 4.1. Consider two domains $D'$ and $D$ in $\mathbb{R}^{n}$, where $n \geqslant 2$, and a function $\omega\colon D' \to [1, \infty)$ of class $L_{1,\mathrm{loc}}$. Recall that a homeomorphism $f\colon D' \to D$ is called an $\omega$-homeomorphism whenever
$$
\begin{equation}
\operatorname{mod}_n(f \Gamma) \leqslant \int_{D'} \omega(x) \cdot \rho^{n}(x)\,dx
\end{equation}
\tag{3.2}
$$
for each family $\Gamma$ of paths in $D'$ and every admissible function $\rho$ for $\Gamma$. By [33], Theorem 19, the homeomorphisms satisfying (3.2) coincide with the homeomorphisms $f\colon D'\to D$ of class $\mathcal Q_{n,n}(D',\omega; D)$. Some properties of the homeomorphisms of class $\mathcal Q_{p,q}(D',\omega)$ were studied in [27] (for $n-1<q<p=n$, the value $\Psi_{q,n}(U)$ instead of $\Psi_{q,n}(U\setminus F)$, and $\omega\equiv 1$), [21], [60]–[64] (all for $q=p=n$ and $\omega=Q$), [65], [66] (for $1<q=p<n$ and $\omega=Q$), and many others. In all articles mentioned except [27] the distortion of the geometry of condensers is stated in the language of moduli of curve families, which in a series of cases is a more restrictive characteristic than capacity as far as meaningful applications are concerned. § 4. Geometry the boundaryIn this section, we consider geometric concepts and the main results of other approaches to the boundary behaviour problem. Definition 4.1. The boundary $\partial D'$ of a domain $D'$ is called $(p,\omega)$-weakly flat at $x_0 \in \partial D'$, where $p > 1$, if for every neighbourhood $U$ of $x_0$ and every number $\lambda > 0$, there is a neighbourhood $V \subset U$ of $x_{0}$ such that for all continua5 $F_0$ and $F_1$ in $D'$, intersecting $\partial U$ and $\partial V$, the capacity of the condenser $\mathcal E=(F_1,F_0)$ satisfies $\operatorname{cap} (\mathcal E; L_p(D',\omega)) \geqslant \lambda$. The boundary $\partial D'$ is called $(p,\omega)$-weakly flat whenever it is $(p,\omega)$-weakly flat at each of its points. A point $x_0 \in \partial D'$ is called $(p,\omega)$-strongly accessible, where $p > 1$, if for every neighbourhood $U$ of $x_0$, there exist a neighbourhood $V \subset U$ of this point, a compact set $F_0 \subset D'$, and a number $\delta > 0$, such that for all continua $F_1$ in $D'$ intersecting $\partial U$ and $\partial V$ the capacity of the condenser $\mathcal E=(F_1,F_0)$ is bounded from below: $\operatorname{cap} (\mathcal E; L_p(D',\omega)) \geqslant \delta$. The boundary $\partial D'$ is called $(p,\omega)$-strongly accessible whenever each of its points is $(p,\omega)$-strongly accessible. In the unweighted case, for $p = n$, the properties of the boundary to be weakly flat and strongly accessible are introduced in [21], § 3.8, in terms of moduli of curve families. These conditions generalize properties $P1$ and $P2$ of [18], § 17, and the properties of the boundary to be quasiconformally flat and quasiconformally accessible [16]. The case of arbitrary $p >n-1$ is considered, for instance, in [67]. Proposition 4.2. Suppose that $1\leqslant p <\infty$. If a domain $D'\subset \mathbb{R}^n$, where $n\geqslant 2$, has $(p,\omega)$-weakly flat boundary and $\omega\in L_{1,\mathrm{loc}}(D')$, then (1) the boundary $\partial D'$ is $(p,\omega)$-strongly accessible; (2) $D'$ is locally connected at the boundary points. Proof. The proof follows is similar to that of Proposition 3.1 and Lemma 3.15 of [21] with obvious adjustments. Remark 4.3. In the unweighted case the modulus and capacity coincide [68]–[70], and hence the properties of the boundary to be weakly flat and strongly accessible of [21] precisely coincide with the case of trivial weight and $p = n$ in Definition 4.1 of $(n,1)$-weakly flat and $(n,1)$-strongly accessible boundary. Moreover, a point $x_0\in \partial D'$ is $(n,1)$-strongly accessible whenever it is quasiconformally accessible [16], Definition 1.7: given a neighbourhood $U$ of $x_0$, there are a continuum $F_0 \subset D'$ and a number $\delta > 0$ such that $\operatorname{cap} ((F_1,F_0); L^1_p(D',\omega)) \geqslant \delta$ for all connected sets $F_1$ in $D'$ satisfying $x_0\in\overline F_1$ and $F_1 \cap \partial U \neq \varnothing$. Note the following connection between the singleton support of a boundary element and the above conditions on the geometry of the boundary. Proposition 4.4. Given a weight $\omega$ and a domain $D'$ satisfying Remark 2.42, let $h\in H^\omega_{\rho,p}(D')$ a boundary element and let a point $x_0\in S_h$ be $(p,\omega)$-strongly accessible in the sense of Definition 4.1. Then $\mathcal{S}_h = \{x_0\}$. Proof. Assume on the contrary that $x_0$ is $(p,\omega)$-strongly accessible and there exists a point $y_0 \in \mathcal{S}_h$ with $d(x_0,y_0)\geqslant \alpha>0$. By the definition of the support of a boundary element, there exist fundamental sequences $\{x_m \,{\in}\, D'_{\rho,p}\}$ and $\{y_m \,{\in}\, D'_{\rho,p}\}$ with respect to the metric $\rho^{\omega}_{p,F_0}$ such that $x_m \to x_0$ and $y_m\to y_0$ in the topology of the extended Euclidean space. We fix a neighbourhood $V \subset U = B(x_0,\alpha/3)$ of $x_0$, a compact set $F_0 \subset D'$, and a number $\delta > 0$ according to Definition 4.1. Find a number $m_0$ such that $x_m \in V$ and $y_m \in B(y_0,\alpha/3)$ for all $m\geqslant m_0$. It is obvious that for $m\geqslant m_0$ every curve $\overline{x_m y_m}$ crosses $\partial V$ and $\partial U$, and so, since the image of the curve is a continuum, the definition of strong accessibility yields $\operatorname{cap}((\overline{x_m y_m},F_0); L_p(D',\omega)) \geqslant \delta$.
By the definition of the capacity metric (2.7), among the mentioned continua with endpoints $x_m \in V$ and $y_m \in B(y_0,\alpha/3)$. there is $\gamma_m = \overline{x_m y_m}$ such that
$$
\begin{equation}
\rho^{\omega}_{p,F_0}(x_m, y_m) \geqslant \operatorname{cap}\bigl((\gamma_m,F_0); L_p(D',\omega)\bigr) - \frac{\delta}{2^m} \geqslant \delta\biggl(1-\frac{1}{2^m}\biggr).
\end{equation}
\tag{4.1}
$$
On the other hand, $x_0$, $y_0 \in \mathcal{S}_h$ implies that the sequences $\{x_m \in D'_{\rho,p}\}$ and $\{y_m \in D'_{\rho,p}\}$ are equivalent. Therefore, $\rho^{\omega}_{p,F_0}(x_m, y_m) \to 0$, which contradicts (4.1). Proposition 4.4 is proved. Corollary 4.5 of Theorem 2.19 ([25], [26], Chap. 5, Theorem 1.3, [17], Theorem 10.4). Consider two domains $D$ and $D'$ in $\mathbb{R}^n$, where $n \geqslant 2$. Every quasiconformal mapping $f\colon D' \to D$ admits a homeomorphic extension to the capacity boundary Proof. By Definition 1.4, the quasiconformal mapping belongs to $\mathcal{Q}_{n,n}(D',1;D)$. The claim follows directly from Theorem 2.22. Corollary 4.6 of Theorem 2.38. Consider two domains $D$ and $D'$ in $\mathbb{R}^n$, where $n \geqslant 2$, and a homeomorphism $f\colon D' \to D$ satisfying one of the following conditions: (1) $f$ is quasiconformal, $D'$ is locally connected on the boundary, and $\partial D$ is quasiconformally accessible [16], Theorem 2.4. (2) $f\in\mathcal Q_{n,n}(D',\omega;D)$, in particular, $f$ is an $\omega$-homeomorphism in the sense of Remark 3.1, for6 $\omega\in \mathrm{BMO}(\overline{D'})$, $D'$ is locally connected on the boundary, and $\partial D$ is $(n,1)$-strongly accessible [21], Lemma 5.3. Then $f$ admits a continuous extension $\overline f \colon \overline{D'} \to \overline{D}$ to the boundary. Proof. Verify that the hypotheses of Corollary 2.39 hold in both cases, and so $f\colon D' \to D$ extends by continuity to the closure $\overline{D'}$.
In case (1), for every point $x\in \overline{D'}$ we have $\operatorname{cap}((\{x\}, F_0); L^1_n(D'))=0$. Since every quasiconformal mapping is of class $\mathcal{Q}_{n,n}(D',1;D)$, it remains to verify that if $x\in\mathcal{S}_{h}$ and $h\in H_{\rho,n}(D')$, then the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton, where $\widetilde f$ is the extension of $f$ of Theorem 2.19. The latter follows from the quasiconformal accessibility of $\partial D$, Proposition 4.4, and Remark 4.3. The possibility of extending the mapping $f$ by continuity to $\partial D'$ follows from Corollary 2.39. In case (2), we observe first that $\operatorname{cap}((\{x\}, F_0); L^1_p(D';\omega))\,{=}\,0$ for every boundary point $x\in \partial D'$ (see Example 2.9), and this property is local. Hence, it is independent of the continuum $F_0$. Moreover, by Remark 3.1, the $\omega$-homeomorphism $f$ belongs to $\mathcal{Q}_{n,n}(D',\omega;D)$. As above, Proposition 4.4 shows that the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton, and Corollary 2.39 guarantees the required result. Corollary 4.6 is proved. Remark 4.7. In the planar case, $n=2$, the capacity boundary $H_{\rho,2}$ with respect to the Sobolev class $L^1_2$ is homeomorphic to the boundary of prime ends, see [71], for instance. In the space $\mathbb{R}^n$, where $n\geqslant 3$, it is known that for the domains quasiconformally equivalent to a domain with locally quasiconformal boundary, called regular domains, the completion in the prime ends topology is equivalent to the completion in the modular [17] and capacity [26] metrics. Example 4.8. Consider the domain $D' = [0,1]^3 \subset \mathbb{R}^3$, the weight $\omega(y) = y_1^{\beta}$ with $\beta>-3$, and the ridge domain from Example 2.18: As far as the authors are aware, this example cannot be handled in the framework of other articles concerning boundary correspondence. For instance, [13], [22] require that the boundary of the domain $D$ be $(n,1)$-strongly accessible. In the case of $D$ under consideration, the ridge is neither $(n,1)$-weakly flat nor $(n,1)$-strongly accessible for $\alpha>2$. Indeed, [16], Example 5.5, shows that the points on the ridge are quasiconformally accessible if and only if $1<\alpha<2$ and are not quasiconformally flat for any $\alpha>1$. In addition, it is not difficult to verify that necessary conditions for the ridge to be quasiconformally flat and quasiconformally accessible are also necessary for the ridge to be $(n,1)$-weakly flat and $(n,1)$-strongly accessible, see [16], Theorems 5.3, 5.4. § 5. ApplicationsIn this section, we apply the results on boundary behaviour to the homeomorphisms of certain classes $\mathcal Q_{p,q}(D',\omega;D)$ considered in the examples of this article. 5.1. The homeomorphism of Example 1.13The following mapping is considered in [31]. For $n-1< s<\infty$, consider a homeomorphism $f \colon D' \to D$ of open domains $D'$, $D\subset \mathbb{R}^n$, where $n\geqslant 2$, such that (1) $f\in W^1_{n-1, \mathrm{loc}}(D')$; (2) the mapping $f$ has finite distortion; (3) the outer distortion function
$$
\begin{equation}
D'\ni y \mapsto K^{1,1}_{n-1,s}(y,f) = \begin{cases} \dfrac{|Df(y)|}{|{\det Df (y)}|^{1/s}} &\text{if }\det Df (y)\neq 0, \\ 0 &\text{if }\det Df (y) = 0 \end{cases}
\end{equation}
\tag{5.1}
$$
belongs to $L_{\sigma}(D)$, where $\sigma=(n-1)p$ and $p=s/(s-(n-1))$. According to Theorem 4 in [28], the inverse homeomorphism $\varphi=f^{-1}\colon D\to D'$ has the following properties: (4) $\varphi\in W^1_{p, \mathrm{loc}}(D)$, $p=s/(s-(n-1))$; (5) $\varphi$ has finite distortion. The original homeomorphism $f\colon D'\to D$ has the following properties: (6) it is of class $\mathcal Q_{p,p}(D',\omega;D)$ with the constant $K_p=1$ [31], Corollary 26, and the weight function $\omega\in L_{1,\mathrm{loc}}(D')$ defined as
$$
\begin{equation}
\omega(y)= \begin{cases} \dfrac{|{\operatorname{adj} Df(y)}|^{p}}{|{\det Df (y)}|^{p-1}} &\text{if }y\in D'\setminus Z', \\ 1 &\text{otherwise}, \end{cases}
\end{equation}
\tag{5.2}
$$
see [31], formula (37), where $Z'=\{y\in D'\colon Df(y)=0\}$; (7) if $p>n-1$ (which corresponds to $s<n+1/(n-2)$), then the composition operator
$$
\begin{equation*}
f^*\colon L^1_{p'}(D)\cap \operatorname{Lip}_{\mathrm{loc}}(D)\to L^1_{p'}(D';\theta)
\end{equation*}
\notag
$$
is bounded, where $p'=p/(p-(n-1))$ and $\theta(y)=\omega^{-(n-1)/(p-(n-1))}(y)$. Proposition 5.1. The results of this article concerning the boundary behaviour of homeomorphisms, namely, Theorems 2.19 and 2.37, Corollaries 2.38 and 2.39, are applicable to the mapping $f$ of § 5.1. Explicitly, for $n\leqslant s < n+1/(n-2)$ the homeomorphism $f$ introduced above has the following properties: (1) the mapping $f$ induces a Lipschitz mapping $f\colon ({D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}) \to({D}_{\rho,p},\widetilde\rho_{p,f(F_0)})$ of metric spaces: $\widetilde\rho_{p,f(F_0)}(f(x),f(y))\leqslant \widetilde\rho^{\,\omega}_{p,F_0}(x,y)$ for all points $x,y\in {D}'_{\rho,p}$; (2) the mapping $f$ induces a Lipschitz mapping $\widetilde f\colon ({\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0}) \to({\widetilde D}_{\rho,p},\widetilde\rho_{p,f(F_0)})$ of “completed” metric spaces: to $X\in( {\widetilde D}'_{\rho,p},\widetilde\rho^{\,\omega}_{p,F_0})$ associate $\widetilde f(X)\in ({\widetilde D}_{\rho,q},\widetilde\rho_{q,f(F_0)})$, which contains the fundamental sequence $\{f(x_l)\}$, where $\{x_l\}\in X$:
$$
\begin{equation*}
\widetilde\rho_{p,f(F_0)}\bigl(\widetilde f(X),\widetilde f(Y)\bigr)\leqslant \widetilde\rho^{\,\omega}_{p,F_0}(X,Y)
\end{equation*}
\notag
$$
for $X,Y\in {\widetilde D}'_{\rho,p}$; (3) the restriction $\widetilde f\mid_{H^\omega_{\rho,p}(D')}\colon (H^\omega_{\rho,p}(D'), \widetilde\rho^{\,\omega}_{p,F_0})\to (H_{\rho,p}(D), \widetilde\rho_{p,f(F_0)})$ is a Lipschitz mapping of capacity boundaries; (4) if the domain $D'$ is locally $\mu$-connected at a boundary point $y\in\partial D'$, the support $\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$, and
$$
\begin{equation*}
\operatorname{cap}\bigl((\{y\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0,
\end{equation*}
\notag
$$
where $V_i$ is the connected component associated with $\mathcal{S}_{h}$ at $y$, then $f(z)\to \mathcal{S}_{\widetilde f(h)}$ as $z\to y$ with $z\in V_i\cap D'$ in the topology of the extended space $\mathbb R^n$; (5) if the domain $D'$ is locally $\mu$-connected at a boundary point $y\in\partial D'$, the support $\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ contains $y$ and
$$
\begin{equation*}
\operatorname{cap}\bigl((\{y\}, F_0); L^1_p(V_i,D';\omega)\bigr)=0,
\end{equation*}
\notag
$$
where $V_i$ is the connected component associated with $\mathcal{S}_{h}$ at $y$ and $\mathcal{S}_{\widetilde f(h)}=\{x\}\in \partial D$, then the mapping $f \colon D'\to D$ extends by continuity to $y\in\partial D'$ and (6) if the domain $D'$ is locally connected at $y\in\partial D'$ and
$$
\begin{equation*}
\operatorname{cap}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr)=0,
\end{equation*}
\notag
$$
then $y$ lies in the support $\mathcal{S}_{h}$ of some boundary element $h\in H^\omega_{\rho,p}(D')$; (7) if $\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$, then the mapping $f \colon D'\to D$ extends by continuity to $y\in\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ and Let us compare the above example with the mapping of [72], which considers a $W^{1}_{1, \mathrm{loc}}$-homeomorphism $f\colon D' \to D$ with finite distortion, whose outer distortion function
$$
\begin{equation}
K^{1,1}_{n,n}(y,f) = \begin{cases} \dfrac{|Df(y)|}{|{\det Df (y)}|^{1/n}} &\text{if }\det Df (y)\neq 0, \\ 0 &\text{if } \det Df (y)=0 \end{cases}
\end{equation}
\tag{5.3}
$$
belongs to $L_{(n-1)n, \mathrm{loc}}(D')$. Verify that this mapping is a particular case for $s=n$ of the scale mapping considered above: $f\in W^1_{n-1, \mathrm{loc}}(D')$ with the distortion function (5.1). To this end, we have to show that the $W^{1}_{1, \mathrm{loc}}$-homeomorphism $f\colon D' \to D$ is of class $f\in W^1_{n-1, \mathrm{loc}}(D')$. To verify the last property, observe that $f$ induces the composition operator
$$
\begin{equation*}
f^*\colon L^1_{n}(D)\cap \operatorname{Lip}_{\mathrm{loc}}(D) \to L^1_{n-1, \mathrm{loc}}(D')
\end{equation*}
\notag
$$
in the sense that $u\circ f \in L^1_{n-1, \mathrm{loc}}(D')$ for every function $u\in L^1_{n}(D)\cap \operatorname{Lip}_{\mathrm{loc}}(D)$. Indeed, consider a compactly embedded domain $U\Subset D'$. Take $u\in {L}^1_n(f(U)) \cap \operatorname{Lip}_{\mathrm{loc}}(f(U))$. The composition $u\circ f$ clearly lies in $\mathrm{ACL}(U)$. Let us show that the derivatives of the composition are integrable. We can find the derivative of the composition as
$$
\begin{equation*}
\frac{\partial (u\circ f)}{\partial y_i}(y)= \sum_{j=1}^n\frac{\partial u}{\partial x_j}(f(y))\, \frac{\partial f_j}{\partial y_i}(y)
\end{equation*}
\notag
$$
provided that $f(y)$ is a point of differentiability of $u$ and $\partial (u\circ f)(y)/\partial y_i=0$ otherwise because in this case, $y\in Z'$ and $Df(y)=0$ a.e. Since the distortion function (5.3) is of class $L_{(n-1)n}(U)$, we have
$$
\begin{equation}
\qquad\leqslant \int_{U\setminus (Z'\cup \Sigma')}|\nabla u(f(y))|^{n-1}\det Df(y)^{(n-1)/n} \cdot\frac{|Df(y)|^{n-1}}{\det Df(y)^{(n-1)/n}} \,dy
\end{equation}
\tag{5.4}
$$
$$
\begin{equation}
\qquad\leqslant \biggl(\int_{U\setminus (Z'\cup \Sigma')}|\nabla u(f(y))|^n\det Df(y)\,dy\biggr)^{(n-1)/n} \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad \times \biggl(\int_{U\setminus (Z'\cup\Sigma')} \biggl(\frac{|Df(y)|}{|{\det Df(y)}|^{1/n}}\biggr)^{(n-1)n} \,dy\biggr)^{1/n}
\end{equation}
\tag{5.5}
$$
$$
\begin{equation}
\qquad=\|K^{1,1}_{n,n}(\,{\cdot}\,,f)\mid L_{(n-1)n}(U)\|^{n-1} \biggl(\int_{f(U)}|\nabla u(x)|^n\,dx\biggr)^{(n-1)/n}. \nonumber
\end{equation}
\notag
$$
To change from (5.4) to (5.5), we use Hölder’s inequality with the summability exponents $n/(n-1)$ and $n$. Furthermore, we note that $f(U)$ is a bounded open set, so that the coordinate function $u_j(x)\mapsto x_j$ lies in $L^1_n(f(U))$. By (5.4), (5.5) the composition $(u_j\circ f)(y)=f_j(y)$ for $y\in D'$ is of class $f_j\in L^1_{n-1, \mathrm{loc}}(D')$, for $j=1,\dots,n$, while the mapping $f\colon D'\to D$ is of class $W^1_{n-1, \mathrm{loc}}(D')$. Therefore, the mapping of [72] satisfies all hypotheses of Example 1.13 with $s=n$, and thus, the claim of Proposition 5.1 holds for it. 5.2. The homeomorphism of Example 1.16Consider the mapping of Example 1.16 in the case that it is a homeomorphism. Then we have some homeomorphism $f\colon D'\to D$ of class $\mathcal{OD}(D';s,r;\theta,1)$, where $n-1< s\leqslant r<\infty$, with outer bounded $\theta$-weighted $(s,r)$-distortion, meaning that (1) $f \in W^1_{n-1,\mathrm{loc}}(D')$; (2) $f$ has finite distortion; (3) the distortion function
$$
\begin{equation*}
D' \ni x\mapsto K_{s,r}^{\theta,1}(x,f)= \begin{cases} \dfrac{\theta^{1/s}(x)|D f(x)|}{|{\det Df(x)}|^{1/r}} &\text{if } \det Df(x)\ne0, \\ 0 &\text{otherwise} \end{cases}
\end{equation*}
\notag
$$
is of class $L_{\rho}(D')$, where $\rho$ is found from the condition $1/\rho = 1/s-1/r$ and $\rho = \infty$ for $s=r$. Proposition 5.2. On assuming that $\omega(x)=\theta^{-(n-1)/(s-(n-1))}(x)$ is locally integrable, the homeomorphism $f\colon D'\to D$ of class $\mathcal{OD}(D';s,r;\theta,1)$, where $n\leqslant s \leqslant r < n+ 1/(n+2)$, belongs to the family $\mathcal Q_{p,q}(D',\omega;D)$, where $q=r/(r-(n-1))$ and $p=s/(s-(n-1))$ with $n-1<q\leqslant p\leqslant n$. Furthermore, the factors in the right-hand side of (1.8) are equal to $K_p=\|K_{r,r}^{\theta,1}(\,{\cdot}\,,f)\mid L_{\infty}(D')\|^{n-1}$ for $q=p$ and where $1/\sigma=1/q-1/p=(n-1)/\varrho$. Therefore, Theorems 2.22 and 2.37 concerning boundary behaviour and their Corollaries 2.38 and 2.39 apply to the mapping $f\colon D'\to D$. In particular, applying Corollary 2.39, we obtain the following proposition. Proposition 5.3. Under the hypotheses of Proposition 5.2, assume that (1) the domain $D'$ is locally connected at every point $y\in\partial D'$ and
$$
\begin{equation*}
\operatorname{cap}\bigl((\{y\}, F_0); L^1_p(D';\omega)\bigr)=0,
\end{equation*}
\notag
$$
(2) the support $\mathcal{S}_{\widetilde f(h)}$ of the boundary element $\widetilde f(h)$ is a singleton: $\mathcal{S}_{\widetilde f( h)}=\{x\}\in \partial D$, where $h\in H^\omega_{\rho,p}(D')$ is the boundary element containing $\{y\}$. Then we obtain an extension by continuity of the homeomorphism $f \colon D'\to D$ at the point $y$ of the support $\mathcal{S}_{h}$ of the boundary element $h\in H^\omega_{\rho,p}(D')$ such that A similar result is obtained in [67], Theorem 2, under stronger restrictions: $f\in W^1_{s,\mathrm{loc}}(D')$, and so $n-1< s$, condition (1) holds, but instead of condition (2) it is assumed that the points $x\in\partial D$ are $q$-strongly accessible for $q=r /(r- (n- 1))$. Recall that under this condition the support $\mathcal{S}_{h}$ of $x\in h$ is a singleton, see Proposition 4.4. Therefore, the fulfillment of the hypotheses of Theorem [67], Theorem 2, ensures that conditions (1) and (2) above hold. Then, there exists a continuous extension of the mapping $f \colon D'\to D$ to the Euclidean boundary. Proposition 5.4. Assume the hypotheses of Proposition 5.2. If the domain $D'$ is locally connected at the boundary, while the boundary $\partial D$ is $q$-weakly flat for $q=r /(r-(n-1))$, then the mapping $f^{-1}$ admits a continuous extension $\widetilde{f}^{-1}\colon\overline{D} \to \overline{\mathbb{R}^{n}}$. Proof. Assume on the contrary that the mapping $f^{-1}$ has no limit at some point $x_{0} \in \partial D$. Then there exist two distinct points $y_1,y_2\in \partial D'$ and two sequences $\{x_{1,k} \in D\}$, $\{x_{2,k} \in D\}$ such that Choose two balls $B_i=B(y_i, r_i)$, for $i=1,2$, satisfying $\overline{B}_1 \cap \overline{B}_2=\varnothing$. Since the domain $D'$ is locally connected at the boundary, for the ball $B_i$ there is a connected component of $B_i \cap D'$ which includes $U_i=B(y_i, \widetilde{r}_i) \cap D'$ for some $\widetilde{r}_i \in(0, r_i)$, for $i=1,2$.
Consider a positive number $h<\operatorname{dist}(B_1, B_2)$. By the subordination principle, Property 1.2, the piecewise linear function $u$ defined as
$$
\begin{equation*}
u(y)= \begin{cases} 1 &\text{for } y \in B(y_1, r_1) \cap D', \\ 0 & \text{for } y \in \mathbb{R}^{n} \setminus(B(y_1, r_1+h) \cap D') \end{cases}
\end{equation*}
\notag
$$
is admissible for the condenser $E'=(F_1',F_2')$ for every continuum $F_i'\Subset B_i \cap D'$. Let $P$ be a number such that $P > C \|u \mid L^1_p (D', \omega)\|$, where $C$ is the constant in (1.9).
By construction, $x_{0} \in \overline{f(U_1)} \cap \overline{f(U_2)}$. Suppose that $V$ is a neighbourhood of $x_{0}$ so small that Since $\partial D$ is $q$-weakly flat, for some neighbourhood $W \subset V$ of $x_{0}$ and some continuum $F_i {\subset}\, f(U_i)$, for $i\,{=}\,1,2$, intersecting $\partial V$ and $\partial W$, we have
$$
\begin{equation*}
\operatorname{cap}^{1/q} ((F_1, F_2); L_q(D))\,{\geqslant}\, P.
\end{equation*}
\notag
$$
Let $F_i'$ be such that $F_i'=f(F_i)$. Then the relations
$$
\begin{equation*}
\begin{aligned} \, P &\leqslant \operatorname{cap}^{1/q}\bigl((F_1,F_2); L^1_q(D)\bigr) = \operatorname{cap}^{1/q}\bigl(f^{-1}(E'); L^1_q(D)\bigr) \\ &\leqslant C \operatorname{cap}^{1/p} (E'; L^1_p(D',\omega)) < P \end{aligned}
\end{equation*}
\notag
$$
lead to a contradiction. Proposition 5.4 is proved. Some results similar to Propositions 5.2–5.4 were obtained in [67], Theorem 1, under stronger restrictions: $f\in W^1_{s,\mathrm{loc}}(D')$ and $s>n-1$.
Citation in format AMSBIB
\Bibitem{VodMol23} |
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