# Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower
semicontinuity of energy

Anna Doležalová  Department of Mathematical Analysis, Charles University,
Sokolovská 83, 186 00 Prague 8, Czech Republic<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Stanislav Hencl  and  Anastasia Molchanova
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,
A-1090 Vienna, Austria<EMAIL_ADDRESS>
###### Abstract.

Let $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be bounded domains and
let $f_{m}\colon\Omega\to\Omega^{\prime}$ be a sequence of homeomorphisms with
positive Jacobians $J_{f_{m}}>0$ a.e. and prescribed Dirichlet boundary data.
Let all $f_{m}$ satisfy the Lusin $\mathrm{(N)}$ condition and
$\sup_{m}\int_{\Omega}(|Df_{m}|^{n-1}+A(|\operatorname{cof}Df_{m}|)+\varphi(J_{f}))<\infty$,
where $A$ and $\varphi$ are positive convex functions. Let $f$ be a weak limit
of $f_{m}$ in $W^{1,n-1}$. Provided certain growth behaviour of $A$ and
$\varphi$, we show that $f$ satisfies the $\mathrm{(INV)}$ condition of Conti
and De Lellis, the Lusin $\mathrm{(N)}$ condition, and polyconvex energies are
lower semicontinuous.

###### Key words and phrases:

limits of Sobolev homeomorphisms, invertibility

###### 2000 Mathematics Subject Classification:

46E35

The first two authors were supported by the grant GAČR P201/21-01976S. The
first author is a Ph.D. student in the University Centre for Mathematical
Modelling, Applied Analysis and Computational Mathematics and was supported by
the Charles University, project GA UK No. 480120. The third author was
supported by the European Unions Horizon 2020 research and innovation
programme under the Marie Składowska-Curie grant agreement No 847693.

##  1\. Introduction

In this paper, we study classes of mappings that might serve as classes of
deformations in Continuum Mechanics models. Let $\Omega\subset\mathbb{R}^{n}$
be a domain, i.e., a non-empty connected open set, and let
$f\colon\Omega\to\mathbb{R}^{n}$ be a mapping with $J_{f}>0$ a.e. Following
the pioneering papers of Ball [2] and Ciarlet and Nečas [11] we ask if our
mapping is in some sense injective as the physical “non-interpenetration of
the matter” indicates that a deformation should be one-to-one. We continue our
study from [17] and suggest studying the class of weak limits of Sobolev
homeomorphism. We show that under natural assumptions on energy functional
these limits are also invertible a.e. and that the energy functional is weakly
lower semicontinuous which makes it a suitable class for variational approach.

Concerning invertibility we use the $\mathrm{(INV)}$ condition which was
introduced for $W^{1,p}$-mappings, $p>n-1$, by Müller and Spector [33] (see
also e.g. [4, 24, 25, 34, 36, 37, 38]). Informally speaking, the
$\mathrm{(INV)}$ condition means that the ball $B(x,r)$ is mapped inside the
image of the sphere $f(S(a,r))$ and the complement
$\Omega\setminus\overline{B(x,r)}$ is mapped outside $f(S(a,r))$ (see
Preliminaries for the formal definition). From [33] we know that mappings in
this class with $J_{f}>0$ a.e. are one-to-one a.e. and that this class is
weakly closed. Moreover, any mapping in this class has many desirable
properties, it maps disjoint balls into essentially disjoint balls,
$\operatorname{deg}(f,B,\cdot)\in\\{0,1\\}$ for a.e. ball $B$, under an
additional assumption its distributional Jacobian equals to the absolutely
continuous part of $J_{f}$ plus a countable sum of positive multiples of Dirac
measures (these corresponds to created cavities) and so on.

In all results in the previous paragraph the authors assume that $f\in
W^{1,p}(\Omega,\mathbb{R}^{n})$ for some $p>n-1$. However, in some real models
for $n=3$ one often works with integrands containing the classical Dirichlet
term $|Df|^{2}$ and thus this assumption is too strong. Therefore, for $n=3$,
Conti and De Lellis [12] introduced the concept of $\mathrm{(INV)}$ condition
also for $W^{1,2}\cap L^{\infty}$ (see also [5] and [6] for some recent work)
and studied Neohookean functionals of the type

$\int_{\Omega}\left(|Df(x)|^{2}+\varphi(J_{f}(x))\right)\,dx,$

where $\varphi$ is convex, $\lim_{t\to 0+}\varphi(t)=\infty$ and
$\lim_{t\to\infty}\frac{\varphi(t)}{t}=\infty$. They proved that mappings in
the $\mathrm{(INV)}$ class that satisfy $J_{f}>0$ a.e. have nice properties
like mappings in [33], but this class is not weakly closed and hence cannot be
used in variational models easily. To fix this problem we add an additional
term to the energy functional and we work only with the class of weak limits
of homeomorphisms.

Let us note that homeomorphisms clearly satisfy the $\mathrm{(INV)}$ condition
and so do their weak limits in $W^{1,p}$, $p>n-1$ (see [33, Lemma 3.3]).
Unfortunately, this is not true anymore in the limiting case of limit of
$W^{1,n-1}$ homeomorphisms as shown by Conti and De Lellis [12] (see also
Bouchala, Hencl and Molchanova [7]). Let us also note that the class of weak
limits of Sobolev homeomorphisms was recently characterized in the planar case
by Iwaniec and Onninen [28, 29] and De Philippis and Pratelli [16]. Our paper
contributes to the study of this class in higher dimensions $n\geq 3$.

In our paper, we study the energy functional

$\mathcal{F}(f)=\int_{\Omega}\left(|Df(x)|^{n-1}+A(|\operatorname{cof}Df(x)|)+\varphi(J_{f}(x))\right)\,dx$

where

(1.1) $\varphi\text{ is a positive convex function on }(0,\infty)\text{ with
}\lim_{t\to 0^{+}}\varphi(t)=\infty,\ $ (1.2)
$\lim_{t\to\infty}\frac{\varphi(t)}{t}=\infty$

and

(1.3) $A\text{ is a positive convex function on }(0,\infty)\text{ with
}\lim_{t\to\infty}\frac{A(t)}{t}=\infty.$

We further assume that our homeomorphisms have the same Dirichlet boundary
data and that they satisfy the Lusin $\mathrm{(N)}$ condition, i.e. that for
every $E\subset\Omega$ with $|E|=0$ we have $|f(E)|=0$. Our main result is the
following:

######  Theorem 1.1.

Let $n\geq 3$ and $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be bounded
domains. Let $\varphi$ and $A$ satisfy (1.1) and (1.3). Let $f_{m}\in
W^{1,n-1}(\Omega,\mathbb{R}^{n})$, $m=0,1,2\dots$, be a sequence of
homeomorphisms of $\overline{\Omega}$ onto $\overline{\Omega^{\prime}}$ with
$J_{f_{m}}>0$ a.e., such that $f_{m}$ satisfies the Lusin $\mathrm{(N)}$
condition and

(1.4) $\sup_{m}\mathcal{F}(f_{m})<\infty.$

Assume further that $f_{m}=f_{0}$ on $\partial\Omega$ for all
$m\in\mathbb{N}$. Let $f$ be a weak limit of $f_{m}$ in
$W^{1,n-1}(\Omega,\mathbb{R}^{n})$, then $f$ satisfies the $\mathrm{(INV)}$
condition.

Moreover, under the additional assumption (1.2) our $f$ satisfies the Lusin
$\mathrm{(N)}$ condition and we have lower semicontinuity of energy

(1.5) $\mathcal{F}(f)\leq\liminf_{m\to\infty}\mathcal{F}(f_{m}).$

Assuming further that $|\partial\Omega^{\prime}|=0$ we have

$\text{ for a.e.\ }x\in\Omega\text{ we have }h(f(x))=x\text{ and for a.e.\
}y\in\Omega^{\prime}\text{ we have }f(h(y))=y,$

where $h$ is a weak limit of (some subsequence of) $f_{k}^{-1}$ in
$W^{1,1}(\Omega^{\prime},\mathbb{R}^{n})$.

Let us comment on our assumptions. Each homeomorphism
$f_{m}\colon\Omega\to\Omega^{\prime}$, $f_{m}\in
W^{1,n-1}(\Omega,\mathbb{R}^{n})$ with $J_{f_{m}}>0$ a.e. satisfies
$f_{m}^{-1}\in W^{1,1}(\Omega^{\prime},\mathbb{R}^{n})$ (see [13]). Moreover
(see e.g. [13] or [18]), we have

$\int_{\Omega^{\prime}}|Df_{m}^{-1}(y)|\,dy\leq\int_{\Omega}|Df_{m}(x)|^{n-1}\,dx$

and hence (1.4) implies that there is a subsequence of $f_{m}^{-1}$ which
converges weak-$*$ to some $h\in BV(\Omega^{\prime},\mathbb{R}^{n})$. Using

(1.6) $\sup_{m}\int_{\Omega}A(|\operatorname{cof}Df_{m}(x)|)\,dx<\infty$

we get that $Df_{m}^{-1}$ are equiintegrable (see Theorem 2.5 below) and hence
(up to a subsequence) $f_{m}^{-1}$ converge to $h\in
W^{1,1}(\Omega^{\prime},\mathbb{R}^{n})$ weakly in
$W^{1,1}(\Omega^{\prime},\mathbb{R}^{n})$. This assumption (1.6) is also
crucial in our proof of the $\mathrm{(INV)}$ condition as it implies that
image $f_{m}(A)$ of small set $A\subset\partial B(c,r)$ is “uniformly” small
in $m$ and therefore cannot enclose some big set that would like to escape
from $f(\partial B(c,r))$ violating the $\mathrm{(INV)}$ condition.

The condition

$\sup_{m}\int_{\Omega}\varphi(J_{f_{m}}(x))\,dx<\infty\text{ with }\lim_{t\to
0+}\varphi(t)=\infty\
\Bigl{(}\text{resp.}\lim_{t\to\infty}\frac{\varphi(t)}{t}=\infty\Bigr{)}$

implies that small sets have uniformly small preimages (resp. small sets have
uniformly small images) and these conditions are quite standard in the theory.
Moreover, we need to assume that $f_{m}$ maps null sets to null sets (by the
Lusin $\mathrm{(N)}$ condition), which is again natural as our deformation
cannot create a new material from “nothing”. Let us note that this is crucial
for the lower semicontinuity of our functional. In Lemma 4.5 below we
construct a series of homeomorphisms that do not satisfy the Lusin
$\mathrm{(N)}$ condition as they map some null set to a set of positive
measure $a$, though they satisfy all other assumptions, converge weakly to
$f(x)=x$ and

$\int_{(0,1)^{n}}J_{f_{m}}(x)\,dx=1-a<1=\int_{(0,1)^{n}}J_{f}(x)\,dx$

and hence lower semicontinuity fails at least for some polyconvex functionals.
Similarly, if we omit the condition (1.2), we can construct a counterexample
to semicontinuity of some functional if all $f_{m}$ satisfy $\mathrm{(N)}$ but
$f$ does not. The lower semicontinuity of functionals below the natural
$W^{1,n}$ energy has attracted a lot of attention in the past and we refer the
reader e.g. to Ball and Murat [3], Malý [31], Dal Maso and Sbordone [15] and
Celada and Dal Maso [9] for further information.

Let us note one disadvantage of our approach. In the previous models [12],
[33] it was possible to model also the cavitation, i.e., the creation of small
holes. Unfortunately, this is not possible for us as the condition (1.2)
together with (1.4) tells us that $f_{m}$ cannot map small sets onto big sets.
However, this is exactly what is needed to be done by our approximating
homeomorphisms around the point where the cavity is created by $f$. On the
other hand, the condition (1.2) is crucial for us in order to prove the Lusin
$\mathrm{(N)}$ condition for $f$ and this condition is the key for the proof
and the validity of the lower semicontinuity of our functional.

Let us briefly comment on the structure of this paper. We recall the
definition of the degree and of the $\mathrm{(INV)}$ condition in the
Preliminaries and we prove the equiintegrability of $Df_{m}^{-1}$ there. Our
proof of $\mathrm{(INV)}$ condition for $f$ uses some techniques and results
that we have developed in our previous paper [17] on this topic. We recall
some of those in the Preliminaries and then we give a detailed proof of the
$\mathrm{(INV)}$ condition using some of those techniques in Section 3. In
Sections 4.1–4.3 we use the $\mathrm{(INV)}$ condition to prove that $f$
satisfies the $\mathrm{(N)}$ condition and that $h$ ($W^{1,1}$ weak limit of
$f_{m}^{-1}$) is the “a.e. inverse” of $f$. Then we use the $\mathrm{(N)}$
condition to prove the lower semicontinuity of our polyconvex functional in
Section 4.2 and we show some counterexamples to lower semicontinuity without
our assumption (1.2) in Section 4.4. Finally, we return to the result of [17]
where we have shown $\mathrm{(INV)}$ under different assumptions and we show
that the lower semicontinuity of energy is valid also there if we additionally
assume that $f_{m}$ satisfy $\mathrm{(N)}$ and that we have (1.2). In the last
Section 5 we give a quick application of our result in Calculus of Variations.

##  2\. Preliminaries

###  2.1. Change of variables estimates

Let $\Omega\subset\mathbb{R}^{n}$ be open, $A\subset\Omega$ be measurable and
let $g\in W_{{\rm loc}}^{1,1}(\Omega;\mathbb{R}^{n})$ be one-to-one. Without
any additional assumption we have (see e.g. [26, Theorem A.35] for
$\eta=\chi_{g(A)}$)

(2.1) $\int_{A}|J_{g}(x)|\,dx\leq|g(A)|.$

Moreover, for general $g$ satisfying the Lusin $\mathrm{(N)}$ condition we
have (see e.g. [26, Theorem A.35] for $\eta=\chi_{g(A)}$)

(2.2) $\int_{A}|J_{g}(x)|\,dx=\int_{\mathbb{R}^{n}}N(g,A,y)\,dy,$

where $N(g,y,A)$ is defined as a number of preimages of $y$ under $g$ in $A$.

Analogous change of variables formula holds also for mappings
$h\colon\Omega\to\mathbb{R}^{n}$, $\Omega\subset\mathbb{R}^{n-1}$. For
Lipschitz $h$ we have (see e.g. [20, Theorem 3.2.3])

(2.3)
$\int_{A}J_{n-1}h(x)\;dx=\int_{\mathbb{R}^{n}}N(f,A,y)\;d\mathcal{H}^{n-1}(y),$

where $A\subset\Omega$ is measurable, $N(f,A,y)$ denotes the number of
preimages $f^{-1}(y)$ in a set $A$ and $J_{n-1}h$ is the $(n-1)$-dimensional
Jacobian of $h$, i.e. it consists of sizes of $(n-1)\times(n-1)$
subdeterminants. We know that each $h\in W^{1,1}(\Omega,\mathbb{R}^{n})$ is
approximatively differentiable a.e. (see e.g. [1, Theorem 3.83]) and for each
approximately differentiable function we can exhaust $\Omega$ up to a set of
measure zero by sets so that the restriction of $h$ is Lipschitz continuous on
those sets (see [20, Theorem 3.1.8 and Theorem 3.1.4]). It follows that (2.3)
holds for Sobolev mapping $h\in W^{1,1}(\Omega,\mathbb{R}^{n})$ if we know
that for every $E\subset\Omega$ with $\mathcal{H}^{n-1}(E)=0$ we have
$\mathcal{H}^{n-1}(h(E))=0$. In general the area formula (2.3) holds for
Sobolev $h$ only up to a set of $(n-1)$-dimensional measure zero
$E\subset\Omega$ (see also [22, Chapter 3, Section 1.5, Theorem 1 and
Corollary 2]).

######  Lemma 2.1.

Given $C_{1}<\infty$ and $\varphi$ satisfying (1.1), there exist monotone
functions $\Phi$, $\Psi\colon(0,\infty)\to(0,\infty)$ with

$\lim_{s\to 0^{+}}\Phi(s)=0\text{ and }\lim_{s\to 0^{+}}\Psi(s)=0$

such that: Let $g\in W^{1,1}(\Omega,\mathbb{R}^{n})$ be a one-to-one mapping
with $\int_{\Omega}\varphi(J_{g})\leq C_{1}$. Then for each measurable set
$A\subset\Omega$ we have

(2.4) $\Phi(|A|)\leq|g(A)|.$

If we moreover assume that the Lusin $\mathrm{(N)}$ condition holds for $g$
and that (1.2) holds, then also

(2.5) $|g(A)|\leq\Psi(|A|).$

###### Proof.

The proof of (2.4) can be found in the proof of [17, Lemma 2.9] (we omit here
the assumption on $\|g\|_{L^{\infty}}$ as it is not necessary). The proof of
(2.5) follows from De la Vallee Pousin theorem [30, Theorem B.103] applied to
$|J_{g}|$ and the fact that the Lusin $\mathrm{(N)}$ condition implies an
equality in (2.1). Note that we can assume that both $\Phi$ and $\Psi$ are
monotonous. ∎

The following lemma was shown in [17, Lemma 2.8].

######  Lemma 2.2.

Let $\Omega\subset\mathbb{R}^{n}$ be an open set of finite measure and $f\in
W_{{\rm loc}}^{1,1}(\Omega;\mathbb{R}^{n})$ satisfy $J_{f}\neq 0$ a.e. Then
for every $\varepsilon>0$ there is $\delta>0$ such that for every measurable
set $F\subset\mathbb{R}^{n}$ we have

$|F|<\delta\implies|f^{-1}(F)|<\varepsilon.$

In order to apply the previous lemma we use the following observation.

######  Lemma 2.3.

Let $\Omega\subset\mathbb{R}^{n}$ be open, and let $f_{k}\in
W^{1,1}(\Omega,\mathbb{R}^{n})$ be a sequence of homeomorphisms with
$J_{f_{k}}>0$ a.e. such that $f_{k}\to f\in W^{1,1}(\Omega,\mathbb{R}^{n})$
pointwise a.e. Assume further that

$\sup_{k}\int_{\Omega}\varphi(J_{f_{k}}(x))\,dx<\infty,$

where $\varphi$ satisfies (1.1). Then $J_{f}\neq 0$ a.e.

###### Proof.

Assume by contradiction that

$E:=\\{x\in\Omega:\ J_{f}(x)=0\\}\text{ satisfies }|E|>0.$

As usual we find a set $E_{0}\subset E$ with $|E_{0}|=|E|$ such that the
$\mathrm{(N)}$ condition holds on $E_{0}$ for $f$ (see e.g. [26, proof of
Theorem A.35]). Moreover, we assume that $f_{k}(x)\to f(x)$ for every $x\in
E_{0}$. By (2.2) we obtain

$|f(E_{0})|=0.$

We find an open set $G\subset\Omega$ such that

$f(E_{0})\subset G\text{ and
}|G|<\tfrac{1}{2}\Phi\bigl{(}\tfrac{1}{2}|E_{0}|\bigr{)},$

where $\Phi$ comes from Lemma 2.1. Since $f_{k}(x)\to f(x)$ we can find
$k_{0}(x)$ such that for every $k\geq k_{0}(x)$ we have $f_{k}(x)\in G$. It
follows that

$E_{0}=\bigcup_{k_{0}=1}^{\infty}E_{k_{0}},\text{ where }E_{k_{0}}=\\{x\in
E_{0}:\ f_{k}(x)\in G\text{ for every }k\geq k_{0}\\}.$

These sets are nested and hence we can fix $k_{0}$ such that
$|E_{k_{0}}|>\tfrac{1}{2}|E_{0}|$. It follows that

$f_{k_{0}}(E_{k_{0}})\subset G\text{ with
}|E_{k_{0}}|>\tfrac{1}{2}|E_{0}|\text{ and
}|G|<\tfrac{1}{2}\Phi\bigl{(}\tfrac{1}{2}|E_{0}|\bigr{)}$

which contradicts (2.4).

∎

######  Theorem 2.4.

Let $B(c,R)\subset\mathbb{R}^{n}$ and let $g\in
W^{1,n-1}(B(c,R),\mathbb{R}^{n})$ be a homeomorphism. Then for a.e.
$r\in(0,R)$ we know that $g\in W^{1,n-1}(\partial B(c,r),\mathbb{R}^{n})$ and
that $g$ satisfies the Lusin $\mathrm{(N)}$ condition on the sphere $\partial
B(c,r)$, i.e.,

$\text{ for every }E\subset\partial B(c,r)\text{ with
}\mathcal{H}^{n-1}(E)=0\text{ we have }\mathcal{H}^{n-1}(g(E))=0.$

Moreover, for such $r$ and every relatively open set $E\subset\partial B(c,r)$
we have

(2.6) $\mathcal{H}^{n-1}(g(E))\leq
C(r)\int_{E}|\operatorname{cof}Dg|\,d\mathcal{H}^{n-1}.$

###### Proof.

The fact that $g\in W^{1,n-1}(\partial B(c,r),\mathbb{R}^{n})$ for a.e. $r$ is
standard and follows e.g. by using the ACL condition (on circles and not
lines). The part about the validity of Lusin $\mathrm{(N)}$ condition on a.e.
sphere follows from [13, Lemma 4.1].

Let us have a homeomorphism $h\in W^{1,1}(\mathbb{R}^{n-1},\mathbb{R}^{n})$
which satisfies the Lusin $\mathrm{(N)}$ condition. Then the area formula
(2.3) implies that for every measurable set $E\subset\mathbb{R}^{n-1}$ we have

$\mathcal{H}^{n-1}(h(E))=\int_{E}|J_{n-1}h(x)|\,dx,$

where $J_{n-1}h$ is the $(n-1)$-dimensional Jacobian, i.e. it consists of all
$(n-1)\times(n-1)$ subdeterminants. To obtain the wanted estimate we simply do
a bilipschitz change of variables (locally) from round $\partial B(c,r)$ to
flat $\mathbb{R}^{n-1}$ and the result for $h$ implies our estimate (2.6) for
$g$. Of course the constant in the bilipschitz change of variables might
depend on $r$ so our constant in (2.6) could depend on $r$. ∎

###  2.2. Equiintegrability of $Df^{-1}_{m}$

The following theorem tells us that our mappings $f_{m}$ from Theorem 1.1 have
equiintegrable $Df^{-1}_{m}$. It follows that up to a subsequence $f^{-1}_{m}$
converge weakly to some $h\in W^{1,1}(\Omega^{\prime},\mathbb{R}^{n})$, see
[30, Theorem B.103] and [14, Lemma 1.2 in Chapter 2.1].

######  Theorem 2.5.

Let $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be domains. Let functions
$\varphi$ and $A$ satisfy (1.1) and (1.3). Then there is a continuous monotone
function $B$ with $\lim_{t\to\infty}\frac{B(t)}{t}=\infty$ such that:
Let $f_{m}\in W^{1,n-1}(\Omega,\Omega^{\prime})$ be homeomorphisms with
$J_{f_{m}}(x)>0$ a.e., $J_{f_{m}^{-1}}(y)>0$ a.e. and

$\sup_{m}\int_{\Omega}\left(|Df_{m}(x)|^{n-1}+A(|\operatorname{cof}Df_{m}(x)|)+\varphi(J_{f_{m}}(x))\right)\,dx<\infty.$

Then

$\sup_{m}\int_{\Omega^{\prime}}B\bigl{(}|Df_{m}^{-1}(y)|\bigr{)}\,dy<\infty.$

###### Proof.

Let us write $A(t)=ta(t)$, $B(t)=tb(t)$ and assume that $b$ is a suitable
function such that $b(t)\leq a(t)$,

(2.7) $b(st)\leq b(s)+b(t),$

and $B$ is continuous and monotone with superlinear growth. We give a detailed
construction of such $b$ below. By differentiation of $f_{m}^{-1}(f_{m}(x))=x$
we obtain

$Df_{m}^{-1}(f_{m}(x))Df_{m}(x)=I\text{ and
}J_{f_{m}^{-1}}(f_{m}(x))J_{f_{m}}(x)=1$

for a.e. $x$ (see [21, Lemma 2.1]). Using the previous line, (2.7), (2.1) and
$A\cdot\operatorname{cof}A=\operatorname{det}A\cdot I$ we have

$\displaystyle\int_{\Omega^{\prime}}$ $\displaystyle
B(|Df_{m}^{-1}(y)|)\,dy=\int_{\Omega^{\prime}}|Df_{m}^{-1}(y)|b(|Df_{m}^{-1}(y)|)\frac{J_{f_{m}^{-1}}(y)}{J_{f_{m}^{-1}}(y)}\,dy$
$\displaystyle\leq\int_{\Omega}|Df_{m}^{-1}(f_{m}(x))|\
b(|Df_{m}^{-1}(f_{m}(x))|)\frac{1}{J_{f_{m}^{-1}}(f_{m}(x))}\,dx$
$\displaystyle=\int_{\Omega}|(Df_{m}(x))^{-1}|\
b(|(Df_{m}(x))^{-1}|)J_{f_{m}(x)}\,dx$
$\displaystyle=\int_{\Omega}|\operatorname{cof}Df_{m}(x)|\
b\left(|\operatorname{cof}Df_{m}(x)|\frac{1}{J_{f_{m}}(x)}\right)\,dx$
$\displaystyle\leq\int_{\Omega}|\operatorname{cof}Df_{m}(x)|\
b(|\operatorname{cof}Df_{m}(x)|)\,dx+\int_{\Omega}|\operatorname{cof}Df_{m}(x)|\
b\left(\frac{1}{J_{f_{m}}(x)}\right)\,dx.$

From $B(t)=tb(t)\leq A(t)$ we obtain that the first term is uniformly bounded.
By the Young inequality, we estimate the second term

$\int_{\Omega}|\operatorname{cof}Df_{m}(x)|\
b\left(\frac{1}{J_{f_{m}}(x)}\right)\,dx\leq\int_{\Omega}A(|\operatorname{cof}Df_{m}(x)|)\,dx+\int_{\Omega}A^{\prime}\left(b\left(\frac{1}{J_{f_{m}}(x)}\right)\right)\,dx$

where $A^{\prime}$ is the fixed conjugate function to our convex function $A$
(see [23, Chapter 2.4]). If we ask also for

$b(t)\leq(A^{\prime})^{-1}(\varphi(\tfrac{1}{t}))$

for large $t$, we have

$A^{\prime}\left(b\left(\frac{1}{J_{f_{m}}(x)}\right)\right)\leq\varphi(J_{f_{m}}(x))+C$

for every $t$ and we are finished.

Now we find such function $b$. We define auxiliary functions $\overline{\psi}$
and $\overline{b}$ and take $\psi$ and $b$ which are smaller than their
counterparts and monotone continuous. Let us set

$\overline{\psi}(t)=\frac{a(t)}{\log(t)}$

for $t>1$. It is continuous and from the continuity and positivity of $a$ on
$(0,\infty)$ we know that $\lim_{t\to 1+}\overline{\psi}(t)=\infty$. Therefore
we can define

$\psi(t)=\begin{cases}1,&1\leq t<t_{0}=\min\\{s>1:\overline{\psi}(s)=1\\},\\\
\min\\{\overline{\psi}(s),s\in[t_{0},t]\\},&t_{0}\leq t,\end{cases}$

which is a positive continuous nonincreasing function less or equal to
$\overline{\psi}$.

Define

$\overline{b}(t)=\begin{cases}0,&0<t\leq 1,\\\
\psi(t)\log(t),&1<t<\infty.\end{cases}$

Since $\psi$ is continuous nonincreasing bounded on $(1,\infty)$,
$\overline{b}$ is also continuous and (2.7) holds for $s,t\geq 1$ (since
$\psi$ is nonincreasing and $\log$ satisfies (2.7)) and $s,t<1$ (since
$\overline{b}(st)=0$). Moreover we have $\overline{b}(t)\leq a(t)$. Now we
check that

$\lim_{t\to\infty}\overline{b}(t)=\lim_{t\to\infty}\psi(t)\log(t)=\infty.$

Either $\lim_{t\to\infty}\psi(t)>0$ and the statement holds, or
$\lim_{t\to\infty}\psi(t)=0$. In the later case, we can find a sequence
$t_{0}<t_{1}<t_{2}\dots$ such that

$t_{k}=\min\\{s>t_{0}:\overline{\psi}(s)=1/k\\}$

and $t_{k}\to\infty$ (since $\overline{\psi}$ is positive). Then on
$[t_{0},t_{k}]$ we have $\psi(t)\geq
1/k=\overline{\psi}(t_{k})=a(t_{k})/\log(t_{k})$ and consequentially

$\displaystyle\liminf_{t\to\infty}\overline{b}(t)$
$\displaystyle=\liminf_{k\to\infty}\min_{t\in[t_{k},t_{k+1}]}\overline{b}(t)=\liminf_{k\to\infty}\min_{t\in[t_{k},t_{k+1}]}\psi(t)\log(t)\geq\liminf_{k\to\infty}\psi(t_{k+1})\log(t_{k})$
$\displaystyle\geq\liminf_{k\to\infty}\frac{\log(t_{k})}{k+1}\geq\frac{1}{2}\liminf_{k\to\infty}\frac{\log(t_{k})}{k}=\frac{1}{2}\liminf_{k\to\infty}a(t_{k})=\infty.$

Now we want to resolve (2.7) for $s<1$, $t\geq 1$. If $st\leq 1$, it is clear.
In the other case, we need $\overline{b}(st)\leq\overline{b}(t)$ which we do
not have for $\overline{b}$ in general as it does not have to be monotonous.
Therefore we define

$b(t)=\inf_{s\in[t,\infty)}\overline{b}(s).$

That function is clearly monotone, continuous, smaller than $\overline{b}$ and
tends to $\infty$. For $s,t<1$ (2.7) is still trivial. For $s<1$, $t\geq 1$ it
follows from monotonicity. For $s,t\geq 1$ we find
$s_{0}=\max\\{r:\overline{b}(r)=b(s)\\}$ and $t_{0}$ analogously. Obviously
from the definition of $b$ we have $s\leq s_{0}$, $t\leq t_{0}$. Then we have

$b(st)\leq
b(s_{0}t_{0})\leq\overline{b}(s_{0}t_{0})\leq\overline{b}(s_{0})+\overline{b}(t_{0})=b(s)+b(t).$

Now we know that $B(t)=tb(t)$ is continuous nonnegative non-decreasing with
$B(0)=0$ and $\lim_{t\to\infty}\frac{B(t)}{t}=\infty$.

The last step is to show that we can ask

$b(t)\leq(A^{\prime})^{-1}(\varphi(\tfrac{1}{t}))$

for large $t$. We can use a similar procedure as before (replacing $a$ by
$\min\\{a,(A^{\prime})^{-1}\\}$), since
$\lim_{t\to\infty}(A^{\prime})^{-1}(t)=\infty$ ($A^{\prime}$ is convex,
negative in $0$ and positive for large values, so going to $\infty$ — and so
does its inverse, too).

∎

###  2.3. Degree for continuous mappings

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set. Given a continuous
map $f\colon\overline{\Omega}\to\mathbb{R}^{n}$ and
$y\in\mathbb{R}^{n}\setminus f(\partial\Omega)$, we can define the topological
degree as

$\operatorname{deg}(f,\Omega,y)=\sum_{\Omega\cap
f^{-1}(y)}\operatorname{sgn}(J_{f}(x))$

if $f$ is smooth in $\Omega$ and $J_{f}(x)\neq 0$ for each $x\in\Omega\cap
f^{-1}(y)$. By uniform approximation, this definition can be extended to an
arbitrary continuous mapping $f\colon\overline{\Omega}\to\mathbb{R}^{n}$. Note
that the degree depends only on values of $f$ on $\partial\Omega$.

If $f\colon\overline{\Omega}\to\mathbb{R}^{n}$ is a homeomorphism, then either
$\operatorname{deg}(f,\Omega,y)=1$ for all $y\in f(\Omega)$ ($f$ is sense
preserving), or $\operatorname{deg}(f,\Omega,y)=-1$ for all $y\in f(\Omega)$
($f$ is sense reversing). If, in addition, $f\in
W^{1,n-1}(\Omega,\mathbb{R}^{n})$, then this topological orientation
corresponds to the sign of the Jacobian. More precisely, we have

######  Proposition 2.6 ([27]).

Let $f\in W^{1,n-1}(\Omega,\mathbb{R}^{n})$ be a homeomorphism on
$\overline{\Omega}$ with $J_{f}>0$ a.e. Then

$\operatorname{deg}(f,\Omega,y)=1,\qquad y\in f(\Omega).$

###  2.4. Degree for $W^{1,n-1}\cap L^{\infty}$ mappings

Let $B$ be a ball, $f\in W^{1,n-1}(\partial B,\mathbb{R}^{n})\cap C(\partial
B,\mathbb{R}^{n})$, $|f(\partial B)|=0$, and $\mathbf{u}\in
C^{1}(\mathbb{R}^{n},\mathbb{R}^{n})$, then (see [33, Proposition 2.1])

(2.8)
$\int_{\mathbb{R}^{n}}\operatorname{deg}(f,B,y)\operatorname{div}\mathbf{u}(y)\,dy=\int_{\partial
B}(\mathbf{u}\circ f)\cdot(\Lambda_{n-1}D_{\tau}f)\nu\,d\mathcal{H}^{n-1},$

where $D_{\tau}f$ denotes the tangential gradient and $\Lambda_{n-1}D_{\tau}f$
is the restriction of $\operatorname{cof}Df$ to the corresponding subspace
(see [17] for details).

Let $\mathcal{M}(\mathbb{R}^{n})=C_{0}(\mathbb{R}^{n})^{*}$ be the space of
all signed Radon measures on $\mathbb{R}^{n}$. By (2.8) we see that
$\operatorname{deg}(f,B,\cdot)\in BV(\mathbb{R}^{n})$ and

(2.9) $\|D\operatorname{deg}(f,B,\cdot)\|_{\mathcal{M}(\mathbb{R}^{n})}\leq
C\|\Lambda_{n-1}D_{\tau}f\|_{L^{1}(\partial B)}\leq
C\|D_{\tau}f\|_{L^{n-1}(\partial B)}^{n-1}.$

Following [12] (see also [8]) we need a more general version of the degree
which works for mappings in $W^{1,n-1}\cap L^{\infty}$ that are not
necessarily continuous. Although only the three-dimensional case is discussed
in [12], the arguments pass in the general case as well. The definition is in
fact based on (2.8).

######  Definition 2.7.

Let $B\subset\mathbb{R}^{n}$ be a ball and let $f\in W^{1,n-1}(\partial
B,\mathbb{R}^{n})\cap L^{\infty}(\partial B,\mathbb{R}^{n})$. Then we define
$\operatorname{Deg}(f,B,\cdot)$ as the distribution satisfying

(2.10)
$\int_{\mathbb{R}^{n}}\operatorname{Deg}(f,B,y)\psi(y)\,dy=\int_{\partial
B}(\mathbf{u}\circ f)\cdot(\Lambda_{n-1}D_{\tau}f)\nu\,d\mathcal{H}^{n-1}$

for every test function $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$ and every
$C^{\infty}$ vector field $\mathbf{u}$ on $\mathbb{R}^{n}$ satisfying
$\operatorname{div}\mathbf{u}=\psi$.

As in [12] (see also [17]) it can be verified that the right-hand side does
not depend on the way $\psi$ is expressed as $\operatorname{div}\mathbf{u}$
and that the distribution $\operatorname{Deg}(f,B,\cdot)$ can be represented
as a $BV$ function.

Assume that $f$, $g\in W^{1,n-1}(\partial B,\mathbb{R}^{n})\cap
L^{\infty}(\partial B,\mathbb{R}^{n})$. As in [17, (2.5)] we obtain the
following version of some “weak isoperimetric inequality”

(2.11) $\displaystyle\bigl{|}\bigl{\\{}y\in\mathbb{R}^{n}:$
$\displaystyle\operatorname{Deg}(f,B,y)\neq\operatorname{Deg}(g,B,y)\bigr{\\}}\bigr{|}^{\frac{n-1}{n}}\leq$
$\displaystyle\leq C\int_{\partial B\cap\\{f\neq
g\\}}\left(|D_{\tau}f(x)|^{n-1}+|D_{\tau}g(x)|^{n-1}\right)d\mathcal{H}^{n-1}(x).$

We need also the classical isoperimetric inequality (see e.g. [19, Theorem 2
in section 5.6.2 and Theorem 2 in section 5.7.3]). Let
$E\subset\mathbb{R}^{n}$ be and open set with finite perimeter. Then

(2.12) $|E|^{1-\frac{1}{n}}\leq C\mathcal{H}^{n-1}(\partial E).$

######  Remark 2.8.

Let $B$ be a ball and $f\in W^{1,n-1}(\partial B,\mathbb{R}^{n})\cap
C(\overline{B},\mathbb{R}^{n})$. If $|f(\partial B)|=0$, then
$\operatorname{Deg}(f,B,y)=\operatorname{deg}(f,B,y)$ for a.e.
$y\in\mathbb{R}^{n}$. We use different symbols to distinguish and emphasize
that $\operatorname{deg}$ is defined pointwise on $\mathbb{R}^{n}\setminus
f(\partial B)$, whereas $\operatorname{Deg}$ is determined only up to a set of
measure zero.

###  2.5. $\mathrm{(INV)}$ condition

Analogously to [12] (see also [33]) we define the $\mathrm{(INV)}$ class.

Let $A\subset\Omega\subset\mathbb{R}^{n}$. We say that $x\in\mathbb{R}^{n}$ is
a point of density one (or just point of density) of a set $A$ if

$\lim_{r\to 0+}\frac{|B(x,r)\cap A|}{|B(x,r)|}=1.$

It is well-known that a.e. $x\in A$ is a point of density of $A$.

######  Definition 2.9 (geometrical image).

Let $\Omega\subset\mathbb{R}^{n}$ be open, $f\colon\Omega\to\mathbb{R}^{n}$ be
a function which is approximately differentiable almost everywhere. Given a
set $A\subset\Omega$ we call the geometrical image of $A$ through $f$ the set
given by $f\left(\Omega_{d}\cap A\right)$, where $\Omega_{d}$ denotes the set
where $f$ is approximatively differentiable. Further on, we denote this
geometrical image by $f(A)$ (since $f$ is nevertheless defined only up to a
set of measure zero).

######  Definition 2.10 (topological image).

Let $B\subset\mathbb{R}^{n}$ be a ball and let $f\in W^{1,n-1}(\partial
B,\mathbb{R}^{n})\cap L^{\infty}(\partial B,\mathbb{R}^{n})$. We define the
topological image of $B$ under $f$, $\operatorname{im}_{T}(f,B)$, as the set
of all points of density one of the set $\\{y\in\mathbb{R}^{n}:\
\operatorname{Deg}(f,B,y)\neq 0\\}$.

######  Definition 2.11 ($\mathrm{(INV)}$ condition).

Let $f\in W^{1,n-1}(\Omega,\mathbb{R}^{n})\cap
L^{\infty}(\Omega,\mathbb{R}^{n})$. We say that $f$ satisfies $\mathrm{(INV)}$
in a ball $B\subset\subset\Omega$ if

  1. (i)

its trace on $\partial B$ is in $W^{1,n-1}(\partial B,\mathbb{R}^{n})\cap
L^{\infty}(\partial B,\mathbb{R}^{n})$;

  2. (ii)

$f(x)\in\operatorname{im}_{T}(f,B)$ for a.e. $x\in B$;

  3. (iii)

$f(x)\notin\operatorname{im}_{T}(f,B)$ for a.e. $x\in\Omega\setminus B$.

We say that $f$ satisfies $\mathrm{(INV)}$ if for every $a\in\Omega$ there is
$r_{a}>0$ such that for $\mathcal{H}^{1}$-a.e. $r\in(0,r_{a})$ it satisfies
$\mathrm{(INV)}$ in $B(a,r)$.

######  Remark 2.12.

If $f$, in addition, satisfies $J_{f}>0$ a.e., then preimages of sets of
measure zero are of measure zero and thus we can characterize the
$\mathrm{(INV)}$ condition in a simpler way. Namely, such a mapping satisfies
the $\mathrm{(INV)}$ condition in the ball $B\subset\subset\Omega$ if and only
if

  1. (i)

its trace on $\partial B$ is in $W^{1,n-1}(\partial B,\mathbb{R}^{n})\cap
L^{\infty}(\partial B,\mathbb{R}^{n})$;

  2. (ii)

$\operatorname{Deg}(f,B,f(x))\neq 0$ for a.e. $x\in B$;

  3. (iii)

$\operatorname{Deg}(f,B,f(x))=0$ for a.e. $x\in\Omega\setminus B$.

######  Definition 2.13.

Let $f\in W^{1,n-1}(\Omega,\mathbb{R}^{n})\cap
L^{\infty}(\Omega,\mathbb{R}^{n})$. The distributional Jacobian of $f$ is the
distribution defined by setting

$\operatorname{Det}Df(\varphi):=-\int_{\Omega}f_{1}(x)J(\varphi,f_{2},...,f_{n})(x)\,dx\qquad\text{
for all }\varphi\in C^{\infty}_{C}(\Omega),$

where $J(\varphi,f_{2},...,f_{n})$ is the classical Jacobian defined as the
determinant of the Jacobi matrix $Dg$ of $g=(\varphi,f_{2},\ldots,f_{n})$.

We need the following lemmata from [12]. They are stated there only for $n=3$
but it is clear from the proofs that everything works also in higher
dimensions.

######  Lemma 2.14 (Lemma 3.8, [12]).

Let $f\in W^{1,n-1}(\Omega,\mathbb{R}^{n})\cap
L^{\infty}(\Omega,\mathbb{R}^{n})$, with $J_{f}\neq 0$ on $\Omega_{d}$, and
choose $B\subset\Omega$ such that $f$ satisfies $\mathrm{(INV)}$ on $B$. Then
$f(B)\subset\operatorname{im}_{T}(f,B)$, and $f(\mathbb{R}^{n}\backslash
B)\subset\mathbb{R}^{n}\setminus\operatorname{im}_{T}(f,B)$.

######  Lemma 2.15 (Lemma 4.3, [12]).

Let $f\in W^{1,n-1}(\Omega,\mathbb{R}^{n})\cap
L^{\infty}(\Omega,\mathbb{R}^{n})$. Suppose that condition $\mathrm{(INV)}$
holds for $f$, and that $J_{f}>0$ a.e. Then,

  1. (i)

$\operatorname{Det}Df\geq 0$, hence it is a Radon measure;

  2. (ii)

the absolutely continuous part of $\operatorname{Det}Df$ with respect to
$\mathcal{L}^{n}$ has density $J_{f}$;

  3. (iii)

for every $a\in\Omega$ and for a.e. $r\in(0,r_{a})$,

(2.13) $\operatorname{Det}Df(B(a,r))=|\operatorname{im}_{T}(f,B(a,r))|.$

###  2.6. Minimizers of the tangential Dirichlet integral

In our main proof, we have a sphere $\partial B$ in $\mathbb{R}^{n}$ and on
this sphere we have a small $(n-2)$-dimensional circle which is a boundary of
an open spherical cap $S\subset\partial B$. Our map $f$ is in $W^{1,n-1}$,
therefore we can choose the sets so that $f$ is continuous on the
$(n-2)$-dimensional circle $\overline{S}\setminus S$. Our mapping $f$ can have
a big oscillation on $S$ so we need to replace it with a reasonable mapping.
We do this by choosing a minimizer of the tangential Dirichlet energy over
this cap $S$ which has the same value on the circle $\overline{S}\setminus S$.
In fact, we need this even for more general shapes than spheres and circles.

We say that a relatively open set $S\subset\partial B$ satisfies the exterior
ball condition if for each $z\in\overline{S}\setminus S$ there exists a ball
$B(z^{\prime},r)$ with $z^{\prime}\in\partial B$ such that $z\in\partial
B(z^{\prime},r)$ and $B(z^{\prime},r)\cap S=\varnothing$. The following
Theorem was shown in [17, Theorem 2.10]:

######  Theorem 2.16.

Let $B\subset\mathbb{R}^{n}$ be a ball. Let $S\subset\partial B$ be a
connected relatively open subset of $\partial B$. Let $T$ be the relative
boundary of $S$ with respect to $\partial B$. Suppose that
$\operatorname{diam}S<\frac{r}{4n}$ and that $S$ satisfies the exterior ball
condition. Let $f=(f^{1},\dots,f^{n})\in W^{1,n-1}(\partial B,\mathbb{R}^{n})$
be continuous on $T$. Then there exists a unique function
$h=(h^{1},\dots,h^{n})\in C(\overline{S},\mathbb{R}^{n})\cap
W^{1,n-1}(S,\mathbb{R}^{n})$ such that each coordinate $h^{i}$ minimizes
$\int_{S}|D_{\tau}u|^{n-1}\,d\mathcal{H}^{n-1}$ among all functions $u\in
f^{i}+W_{0}^{1,n-1}(S,\mathbb{R}^{n})$. We have $h=f$ on $T$, the function $h$
satisfies the estimate

(2.14) $\operatorname{diam}h(\overline{S})\leq\sqrt{n}\operatorname{diam}f(T)$

and we have $\mathcal{L}^{n}(h(S))=0$. Moreover, let $f_{m}$ be continuous and
converge to $f$ uniformly on $T$, then $h_{m}$ converge to $h$ uniformly on
$S$, where $h_{m}$ are minimizers corresponding to boundary values $f_{m}$.

###### Proof.

Everything except $\mathcal{L}^{n}(h(S))=0$ was shown already in [17, Theorem
2.10].

It is standard that the change of variable formula holds for Sobolev mappings
up to a null set (see (2.3) and the paragraph after) and hence using $h\in
W^{1,n-1}$ there is $N\subset S$ with $\mathcal{H}^{n-1}(N)=0$ such that

$\mathcal{H}^{n-1}\bigl{(}h(S\setminus N)\bigr{)}<\infty\text{ and hence
}\mathcal{L}^{n}\bigl{(}h(S\setminus N)\bigr{)}=0.$

We claim that $h$ is pseudomonotone, i.e. there is $C>0$ such that for each
spherical cap $A\subset S$ we have

$\operatorname{diam}h(\overline{A})\leq C\operatorname{diam}f(\partial A)$

(here $\partial A$ is the relative $(n-2)$-dimensional boundary with respect
to $\partial B$). This fact follows from (2.14), i.e., we consider the
corresponding minimizer on $A$ with respect to boundary data $h|_{\partial
A}$. By the uniqueness of this minimizer we obtain that $h|_{A}$ is this
minimizer and (2.14) holds for $\overline{A}$ and $\partial A$ (instead of
$\overline{S}$ and $T$) gives us the pseudomonotonicity. Let us now consider a
mapping $g:=P\circ h\colon S\to\mathbb{R}^{n-1}$, where $P$ is the projection
to the hyperplane $\\{x_{1}=0\\}$. It is easy to see that $g\in W^{1,n-1}$ and
that $g$ is continuous and pseudomonotone. By the result of Malý and Martio
[32, Theorem A] and $\mathcal{H}^{n-1}(N)=0$ we obtain that

$\mathcal{H}^{n-1}\bigl{(}g(N)\bigr{)}=0\text{ and hence
}\mathcal{L}^{n}\bigl{(}h(N)\bigr{)}=0.$

∎

##  3\. Proof of Theorem 1.1: $\mathrm{(INV)}$ condition

###### Proof of Theorem 1.1: $\mathrm{(INV)}$ condition.

Assume on the contrary that $f$ does not satisfy the $\mathrm{(INV)}$
condition. Then we can find a center $c$ such that for
$\mathcal{H}^{1}$-positively many radii $r>0$ our $f$ maps either something
from $B(c,r)$ outside of $\operatorname{im}_{T}(f,B(c,r))$ or something
outside of $B(c,r)$ inside of $\operatorname{im}_{T}(f,B(c,r))$. We treat the
first case in detail and at the end we briefly explain the analogous second
case.

Step 1. Outline of the proof: We assume that $f$ does not satisfy
$\mathrm{(INV)}$ and hence there is $c\in\Omega$ such that the set

(3.1) $\left\\{r:\ B(c,r)\subset\Omega,\ \exists V_{r}\subset B(c,r)\text{
with }|V_{r}|>0\text{ and }\operatorname{Deg}(f,B(c,r),f(x))=0\text{ for all
}x\in V_{r}\right\\}$

has positive (one-dimensional) measure.

Let us now briefly explain the idea of the proof. We know that $f_{m}$
converge to $f$ weakly in $W^{1,n-1}$ and thus up to a subsequence strongly in
$L^{n-1}$ and a.e. We can thus imagine that $f_{m}$ is really close to $f$
both on some fixed $\partial B(c,r)$ and on $V_{r}$. The situation is
illustrated in Fig. 1.

a $S(c,r)$$V_{r}$$f_{m}(S(c,r))$$f(V_{r})$$f(S(c,r))$

Figure 1. Behaviour of mappings $f$ (in black) and $f_{m}$ (in green) on
$S(c,r)$ and $V_{r}$.

That means $f(V_{r})$ is outside of $\operatorname{im}_{T}(f,B(c,r))$ but
$f_{m}(V_{r})$ lies inside $f_{m}(\partial B(c,r))$ since $f_{m}$ is a
homeomorphism. It follows that $f_{m}(\partial B(c,r))$ is close to
$f(\partial B(c,r))$ in most of the places but it makes a “bubble” (or several
bubbles) around $f_{m}(V_{r})$ which is really close to $f(V_{r})$.

Firstly, we define those bubbles and then we show that the number of bubbles
that contain a big part of $f_{m}(V_{r})$ is uniformly bounded and hence in
one of them we have a big portion of the volume of $f_{m}(V_{r})$ (and that
$|f_{m}(V_{r})|\geq C$ using (1.1) and (1.4)). Since $f_{m}(V_{r})$ is quite
big (in one of the big bubbles) we obtain that the boundary of the bubble has
big $\mathcal{H}^{n-1}$ measure. However, this boundary is essentially image
of some very small set (as small as we wish for $m$ big enough) on $\partial
B$ under $f_{m}$ and using (2.6) we obtain that the integral of
$|\operatorname{cof}Df_{m}|$ over this small set is big. This contradicts the
equiintegrability of $|\operatorname{cof}Df_{m}|$ which results in
$\sup_{m}\int_{\partial
B(c,r)}A(|\operatorname{cof}Df_{m}|)\,d\mathcal{H}^{n-1}<\infty$.

Step 2. Replacement of $f$ on $\partial B(c,r)$ with continuous $g$ that has
similar degree:

We need to apply plenty of techniques and results developed in [17]. For the
convenience of the reader, we include most of the details in the current
proof.

We fix $B(c,R)\subset\subset\Omega$. Since $f_{m}$ converge weakly in
$W^{1,n-1}$, which is compactly embedded into $L^{(n-1)^{*}}$, and so into
$L^{n-1}$, we obtain that $f_{m}$ converge to $f$ in $L^{n-1}(B(c,R))$. Up to
a subsequence we can thus assume that $f_{m}\to f$ pointwise a.e. and by Lemma
2.3 we obtain $J_{f}\neq 0$ a.e. We fix $r\in(0,R)$ (and pass to a subsequence
if necessary, see e.g. [33, Lemma 2.9]) such that

$f_{m}\to f\text{ weakly in }W^{1,n-1}(\partial B(c,r),\mathbb{R}^{n})\text{
and }\mathcal{H}^{n-1}\text{-a.e.~{}on }\partial B(c,r)$

and such that there exists a constant $C_{2}$ so that

(3.2) $\int_{\partial
B(c,r)}(|D_{\tau}f|^{n-1}+|D_{\tau}f_{m}|^{n-1})\,d\mathcal{H}^{n-1}<C_{2}\text{
for all }m\in\mathbb{N}.$

Moreover, using Theorem 2.4 we can assume that all $f_{m}\in
W^{1,n-1}(\partial B(c,r),\mathbb{R}^{n})$ satisfy the $\mathrm{(N)}$
condition on $\partial B(c,r)$. Analogously to the proof of [33, Lemma 2.9] we
use the Fatou Lemma and (1.4) to deduce

$\int_{0}^{R}\liminf_{m\to\infty}\int_{\partial
B(c,r)}A(|\operatorname{cof}Df_{m}|)\,d\mathcal{H}^{n-1}\,d\varrho\leq\liminf_{m\to\infty}\int_{0}^{R}\int_{\partial
B(c,r)}A(|\operatorname{cof}Df_{m}|)\,d\mathcal{H}^{n-1}\,d\varrho<\infty.$

Choosing further $r$ so that the $\liminf$ on the lefthand side is finite and
thus (passing again to a subsequence) we have

(3.3) $\int_{\partial
B(c,r)}A\bigl{(}|\operatorname{cof}Df_{m}|\bigr{)}\,d\mathcal{H}^{n-1}<C_{2}\text{
for all }m\in\mathbb{N}.$

We set $B:=B(c,r)$ and choose $\varepsilon>0$ small enough with the exact
value to be specified later. Find
$\rho\in(0,\,\min\\{\frac{1}{16n}r,\frac{\varepsilon}{2}\\})$ such that for
each $z\in\partial B$ we have

(3.4) $\int_{\partial B\cap
B(z,2\rho)}|D_{\tau}f|^{n-1}\,d\mathcal{H}^{n-1}<\varepsilon^{n-1}$

(we can do that since the integral over the whole $\partial B$ is finite). For
each fixed $z\in\partial B$ we find $\rho_{z}\in(\rho,2\rho)$ such that

(3.5) $\rho\int_{\partial B\cap\partial
B(z,\rho_{z})}|D_{\tau}f|^{n-1}\,d\mathcal{H}^{n-2}<\varepsilon^{n-1},$

which is possible because the length of $(\rho,2\rho)$ is $\rho$, combined
with (3.4). Moreover, we can also choose $\rho_{z}$ such that $f_{m}\to f$
occurs $\mathcal{H}^{n-2}$-a.e. on $\partial B\cap\partial B(z,\rho_{z})$ and
that

$\liminf_{m\to\infty}\|f_{m}\|_{W^{1,n-1}(\partial B\cap\partial
B(z,\rho_{z}),\mathbb{R}^{n})}<\infty.$

It follows that up to a subsequence (depending on $z$ and $\rho_{z}$, see e.g.
[33, Lemma 2.9])

(3.6) $f_{m}\to f\text{ weakly in }W^{1,n-1}\text{ and also uniformly on
}\partial B\cap\partial B(z,\rho_{z}).$

Note that on the $(n-2)$-dimensional space $\partial B\cap\partial
B(z,\rho_{z})$ we have embedding into Hölder functions
$W^{1,n-1}\hookrightarrow C^{0,1-\frac{n-2}{n-1}}$, thus $f$ is continuous
there and we have the estimate

(3.7) $\operatorname{diam}f(\partial B\cap\partial B(z,\rho_{z}))\leq
C(\rho_{z})^{1-\frac{n-2}{n-1}}\left(\int_{\partial B\cap\partial
B(z,\rho_{z})}|D_{\tau}f|^{n-1}\,d\mathcal{H}^{n-2}\right)^{\frac{1}{n-1}}\leq
C_{3}\varepsilon.$

Using the Vitali type covering, we find $B_{j}=B(z_{j},\rho_{j})$ such that
$\rho_{j}=\rho_{z_{j}}$,

$\partial B\subset\bigcup_{j}B(z_{j},\rho_{j})$

and the balls $B(z_{j},\frac{1}{5}\rho_{j})$ are pairwise disjoint. Here
$j=1,\dots,j_{\max}$. Furthermore, the balls in the Vitali covering theorem
are chosen inductively so we can also assume using (3.6) that for a
subsequence (chosen in a diagonal argument)

(3.8) $f_{m}\to f\text{ weakly in }W^{1,n-1}\text{ and uniformly on }\partial
B\cap\partial B(z_{j},\rho_{j})\text{ for each }j.$

Given $j$, denote

$S_{j}=\partial B\cap B_{j}\setminus\bigcup_{l<j}\overline{B_{l}}.$

Note that $S_{j}$ satisfies the exterior ball condition of Subsection 2.6. Let
$T_{j}$ denote the relative boundary of $S_{j}$ with respect to $\partial B$.

For each $j$ we define $h_{j}$ on $S_{j}$ such that $h_{j}$ minimizes
coordinate-wise the tangential $(n-1)$-Dirichlet integral among functions with
boundary data $f$ on $T_{j}$ (see Theorem 2.16). We define $h_{j}=f$ on
$\partial B\setminus S_{j}$. Also we define the function $g$ on $\partial B$
as $g=h_{j}$ on each $\overline{S_{j}}$. Set (see Fig. 2)

$\displaystyle F$
$\displaystyle=\\{y\in\Omega^{\prime}\colon\operatorname{Deg}(f,B,y)\neq\operatorname{deg}(g,B,y)\\},$
$\displaystyle F_{j}$
$\displaystyle=\\{y\in\Omega^{\prime}\colon\operatorname{Deg}(f,B,y)\neq\operatorname{Deg}(h_{j},B,y)\\}.$

Let us recall that by Theorem 2.16 we have $\mathcal{L}^{n}(g(S_{j}))=0$ and
hence Remark 2.8 gives us $\operatorname{deg}g=\operatorname{Deg}g$.

$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$g(\partial
B)$$f(\partial B)$$F_{j}$ Figure 2. Behaviour of mappings $f$ (in black) and
$g$ (red) on $\partial B$ in 2D representation. $T_{j}$ corresponds to points
on $f(\partial B)$ (of course in $\mathbb{R}^{n}$ they are
$(n-2)$-dimensional), $g$ is represented by dashed lines connecting these
points (of course these are minimizers of $(n-1)$-energy in higher dimensions
and not lines) and the gray set $F_{j}$ is created “between” $g(S_{j})$ and
$f(S_{j})$.

It is not difficult to find out that

$y\in\bigcup_{j}F_{j}\quad\text{for a.e.\ }y\in F$

(this can be viewed e.g. by using (2.8)). Now, by (2.11), (3.4), and the
minimizing property $\int_{S_{j}}|D_{\tau}h_{j}|^{n-1}\,d\mathcal{H}^{n-1}\leq
C\int_{S_{j}}|D_{\tau}f|^{n-1}\,d\mathcal{H}^{n-1}$ we have

(3.9) $\displaystyle\sum_{j}|F_{j}|$ $\displaystyle\leq
C\sum_{j}\Big{(}\int_{S_{j}}(|D_{\tau}f|^{n-1}+|D_{\tau}h_{j}|^{n-1})\,d\mathcal{H}^{n-1}\Big{)}^{\frac{n}{n-1}}$
$\displaystyle\leq
C\sum_{j}\Big{(}\int_{S_{j}}|D_{\tau}f|^{n-1}\,d\mathcal{H}^{n-1}\Big{)}^{\frac{n}{n-1}}$
$\displaystyle\leq
C\varepsilon\sum_{j}\int_{S_{j}}|D_{\tau}f|^{n-1}\,d\mathcal{H}^{n-1}\leq
CC_{2}\varepsilon.$

It follows that $f$ and $g$ have the same degree up to a very small set. It is
more convenient for us to work with $g$ since for this continuous mapping on
$\partial B$ we can use the classical degree $\operatorname{deg}$ and not
$\operatorname{Deg}$ as for $f$.

Step 3. Replacement of $f_{m}$ on $\partial B(c,r)$ with continuous $g_{m}$
that is close to $g$:

From Theorem 2.16 we know that $\mathcal{L}^{n}(h_{j}(S_{j}))=0$ for each $j$
and thus $|g(\partial B)|=0$. It follows that we can find a compact set
$H\subset\Omega^{\prime}\setminus g(\partial B)$ such that

(3.10) $\Omega^{\prime}\setminus H<\Phi(\tfrac{1}{10}|V_{r}|),$

where $\Phi$ comes from Lemma 2.1. For each $m\in\mathbb{N}$ and
$j\in\\{1,\dots,j_{\max}\\}$ let $g_{m,j}$ be defined in $S_{j}$ as the
coordinate-wise minimizer of the $(n-1)$-Dirichlet integral among functions
with boundary data $f_{m}$ on $T_{j}$. We define $g_{m,j}$ as $f_{m}$ on
$\partial B\setminus S_{j}$. We also define $g_{m}$ on $\partial B$ as
$g_{m,j}$ on each $\overline{S_{j}}$.

Since $f_{m}\to f=g$ uniformly on $T_{j}$ by (3.8), we have $g_{m}\to g$
uniformly on $\partial B$ using Theorem 2.16. Hence we find $m\in\mathbb{N}$
such that $g_{m}(\partial B)$ does not intersect $H$ and

(3.11)
$\operatorname{deg}(g_{m},B,\cdot)=\operatorname{deg}(g,B,\cdot)\quad\text{in
}H.$

Also, we require

(3.12) $|f_{m}-f|=|g_{m}-g|<\varepsilon\quad\text{on all }T_{j}.$

With the help of (2.14) and (3.7) (which holds also for $T_{j}$) this implies
that

$|g_{m}-g|<C\varepsilon\text{ on }\partial B.$

Similarly as in Fig. 2 (but using $f_{m}$ instead of $f$), we define

(3.13) $\displaystyle E$
$\displaystyle=\\{y\in\Omega^{\prime}\colon\operatorname{deg}(f_{m},B,y)=1\neq\operatorname{deg}(g_{m},B,y)\\},$
$\displaystyle E_{j}$
$\displaystyle=\\{y\in\Omega^{\prime}\colon\operatorname{deg}(f_{m},B,y)=1\neq\operatorname{deg}(g_{m,j},B,y)\\},$

see Fig. 3.

$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$g_{m}(\partial
B)$$f_{m}(\partial B)$$E_{j}$ Figure 3. Behaviour of mappings $f_{m}$ (in
black) and $g_{m}$ (red) on $\partial B$ in 2D representation.

Let us note that these sets $E_{j}$ are exactly those bubbles discussed in the
first step (see Fig. 1). Then

$y\in\bigcup_{j}E_{j}\quad\text{for a.e.\ }y\in E.$

Using (2.11) and the minimizing property
$\int_{S_{j}}|D_{\tau}g_{m,j}|^{n-1}\,d\mathcal{H}^{n-1}\leq\int_{S_{j}}|D_{\tau}f_{m}|^{n-1}\,d\mathcal{H}^{n-1}$,
we obtain

(3.14) $|E_{j}|^{1-\frac{1}{n}}\leq
C\int_{S_{j}}|D_{\tau}f_{m}|^{n-1}\,d\mathcal{H}^{n-1}.$

Step 4. Not that many big bubbles where $f_{m}$ and $g_{m}$ have different
degree:

Choose $a>0$ and set

$\displaystyle
J^{+}=\\{j\colon\int_{S_{j}}|D_{\tau}f_{m}|^{n-1}\,d\mathcal{H}^{n-1}>a\\},$
$\displaystyle
J^{-}=\\{j\colon\int_{S_{j}}|D_{\tau}f_{m}|^{n-1}\,d\mathcal{H}^{n-1}\leq
a\\}.$

Note that (3.14) implies that $|E_{j}|$ are small for $j\in J^{-}$. Hence
using (3.2)

(3.15) $\displaystyle\sum_{j\in J^{-}}|E_{j}|$ $\displaystyle\leq C\sum_{j\in
J^{-}}\Big{(}\int_{S_{j}}|D_{\tau}f_{m}|^{n-1}\,d\mathcal{H}^{n-1}\Big{)}^{\frac{n}{n-1}}$
$\displaystyle\leq Ca^{\frac{1}{n-1}}\sum_{j\in
J^{-}}\int_{S_{j}}|D_{\tau}f_{m}|^{n-1}\,d\mathcal{H}^{n-1}$
$\displaystyle\leq Ca^{\frac{1}{n-1}}\int_{\partial
B}|D_{\tau}f_{m}|^{n-1}\,d\mathcal{H}^{n-1}\leq C_{4}a^{\frac{1}{n-1}},$

where $C_{4}=CC_{2}$. We fix $a$ such that

(3.16) $C_{4}a^{\frac{1}{n-1}}\leq\Phi\left(\tfrac{1}{10}|V_{r}|\right).$

We set

$W=f_{m}^{-1}\Big{(}\bigcup_{j\in J^{-}}E_{j}\Big{)}.$

and using Lemma 2.1, (3.15) and (3.16) we obtain

$\Phi(|W|)\leq|f_{m}(W)|=\Bigl{|}\bigcup_{j\in J^{-}}E_{j}\Bigr{|}\leq
C_{4}a^{\frac{1}{n-1}}\leq\Phi\left(\tfrac{1}{10}|V_{r}|\right).$

From the monotonicity of $\Phi$ we get

(3.17) $|W|\leq\tfrac{1}{10}|V_{r}|.$

From (3.2) we have

(3.18) $\\#J^{+}\leq M:=\frac{C_{2}}{a}.$

It follows that we have only boundedly many $E_{j}$, $j\in J^{+}$, where the
size of the bubble $|E_{j}|$ could be big. This bound depends on $|V_{r}|$ and
$\Phi$ (and hence $\varphi$) and $C_{2}$ but it does not depend on $m$ nor on
$\varepsilon$. We could have plenty of other small bubbles $E_{j}$, $j\in
J^{-}$, but (3.17) implies that the union of their preimages is really small.

Step 5. Big part of $f_{m}(V_{r})$ lies in big bubbles:

Set

$Y=\\{x\in V_{r}\setminus W\colon\operatorname{deg}(g_{m},B,f_{m}(x))=0\\}.$

With the help of (3.11) we have

$\displaystyle V_{r}\setminus Y\subset$ $\displaystyle W\cup\\{x\in
V_{r}\colon\operatorname{Deg}(f,B,f(x))\neq 0\\}\cup$
$\displaystyle\cup\\{x\in V_{r}\colon\operatorname{deg}(g_{m},B,f_{m}(x))\neq
0,\ \operatorname{Deg}(f,B,f(x))=0\\}$ $\displaystyle\subset$ $\displaystyle
W\cup\\{x\in V_{r}\colon\operatorname{Deg}(f,B,f(x))\neq 0\\}\cup$
$\displaystyle\cup\\{x\in
V_{r}\colon\operatorname{deg}(g,B,f(x))\neq\operatorname{Deg}(f,B,f(x))\\}\cup$
$\displaystyle\cup\\{x\in
V_{r}\colon\operatorname{deg}(g,B,f_{m}(x))\neq\operatorname{deg}(g,B,f(x))\\}\cup\\{x\in
V_{r}\colon f_{m}(x)\notin H\\}.$

From (3.17) and (3.1) we obtain

$\bigl{|}W\cup\\{x\in V_{r}\colon\operatorname{Deg}(f,B,f(x))\neq
0\\}\bigr{|}\leq\frac{1}{10}|V_{r}|$

as the second set is empty. Using (3.9) we have

$\bigl{|}\bigl{\\{}y\in\Omega^{\prime}\colon\operatorname{deg}(g,B,y)\neq\operatorname{Deg}(f,B,y)\bigr{\\}}\bigr{|}=|F|\leq
CC_{2}\varepsilon.$

Using Lemmata 2.2 and 2.3 we obtain that (for $\varepsilon$ small enough)

$\bigl{|}\bigl{\\{}x\in
V_{r}\colon\operatorname{deg}(g,B,f(x))\neq\operatorname{Deg}(f,B,f(x))\bigr{\\}}\bigr{|}\leq\frac{1}{10}|V_{r}|.$

Since the sets $\\{y\colon\operatorname{deg}(g,B,y)=0\\}$ and
$\\{y\colon\operatorname{deg}(g,B,y)=1\\}$ are open and $f_{m}\to f$ a.e., we
can take $m$ so large that

(3.19) $\bigl{|}\bigl{\\{}x\in
V_{r}\colon\operatorname{deg}(g,B,f_{m}(x))\neq\operatorname{deg}(g,B,f(x))\bigr{\\}}\bigr{|}<\frac{1}{10}|V_{r}|.$

Finally using (3.10) and (2.4) (as in (3.17)) we obtain

$|\\{x\in V_{r}\colon f_{m}(x)\notin H\\}|\leq\frac{1}{10}|V_{r}|$

and all these inequalities together give us

(3.20) $|V_{r}\setminus Y|\leq\frac{1}{2}|V_{r}|.$

It follows that for many points $x\in V_{r}$ we have

$\operatorname{deg}(g_{m},B,f_{m}(x))=0,$

but

$\operatorname{deg}(f_{m},B,f_{m}(x))=1$

since $f_{m}$ is a homeomorphism and $x\in V_{r}\subset B$. Therefore

(3.21) $\left|\\{x\in V_{r}:\ f_{m}(x)\in E_{j}\text{ for some }j\in
J^{+}\\}\right|\geq\frac{1}{2}|V_{r}|.$

Step 6. Integral $\int_{S_{j}}|\operatorname{cof}Df_{m}|$ is big on a small
set $S_{j}$ for some $j\in J^{+}$:

Using (3.18) and (3.21) we fix $j\in J^{+}$ such that for

$U:=\\{x\in V_{r}:\ f_{m}(x)\in E_{j}\\}\text{ we have
}\bigl{|}U\bigr{|}\geq\frac{1}{2\\#J^{+}}|V_{r}|\geq C|V_{r}|.$

From Lemma 2.1 (2.4) we obtain that

(3.22) $|f_{m}(U)|\geq 2\delta,$

where $\delta$ is constant which does not depend on $m$ or $\varepsilon$.

From the definition of $E_{j}$ (see (3.13) and Fig. 3) we obtain that $E_{j}$
is an open set and

$\partial E_{j}\subset f_{m}(S_{j})\cup g_{m}(S_{j}).$

We know that (see (3.7), (3.12) and (2.14))

$\operatorname{diam}(g_{m}(\overline{S_{j}}))\leq C\varepsilon$

and thus we can find a ball $B_{0}$ of radius $C\varepsilon$ such that
$g_{m}(\overline{S_{j}})\subset B_{0}$. Now the set

$\tilde{E}_{j}:=E_{j}\setminus\overline{B_{0}}$

is open,

$\partial\tilde{E}_{j}\subset f_{m}(S_{j})\cup\partial B_{0}.$

and using (3.22) we obtain that

$|\tilde{E}_{j}|\geq 2\delta-C\varepsilon^{n}>\delta$

once $\varepsilon$ is small enough.

It is not difficult to show that the set $\tilde{E}_{j}$ has finite perimeter,
and therefore we can use the isoperimetric inequality (2.12) and Theorem 2.4
to get

$\displaystyle\delta^{1-\frac{1}{n}}$
$\displaystyle\leq|\tilde{E}_{j}|^{1-\frac{1}{n}}\leq
C\mathcal{H}^{n-1}(\partial\tilde{E}_{j})\leq
C\bigl{(}\mathcal{H}^{n-1}(f_{m}(S_{j}))+\mathcal{H}^{n-1}(\partial
B_{0})\bigr{)}$ $\displaystyle\leq
C(r)\Bigl{(}\int_{S_{j}}|\operatorname{cof}Df_{m}|\,d\mathcal{H}^{n-1}+C_{0}\varepsilon^{n-1}\Bigr{)}.$

It follows that for $\varepsilon$ sufficiently small we get

$\frac{1}{2}\delta^{1-\frac{1}{n}}\leq
C(r)\int_{S_{j}}|\operatorname{cof}Df_{m}|\,d\mathcal{H}^{n-1}$

and this estimate on $S_{j}$ (with $\operatorname{diam}S_{j}\leq\varepsilon$)
clearly contradicts the uniform integrability of $|\operatorname{cof}Df_{m}|$
given by (3.3).

Step 7. Something from outside of $B(c,r)$ goes inside
$\operatorname{im}_{T}(f,B(c,r))$:

This case works analogously. We can find a ball $B(c,r)$ and
$V_{r}\subset\Omega\setminus B(c,r)$ such that its big part is mapped inside
the topological images of $B(c,r)$. Therefore, $f_{m}$ creates bubbles inside.
We can define $F_{j}$, $F$, $E_{j}$ and $E$ in a similar way and conclude.

∎

Recall that the symmetric difference of two sets $S,T\subset\mathbb{R}^{n}$ is
defined as

$S\triangle T:=(S\setminus T)\cup(T\setminus S).$

######  Lemma 3.1.

Let $n\geq 3$, $\Omega,\Omega^{\prime}\subset\mathbb{R}^{n}$ be bounded
domains and let $\varphi$ satisfy (1.1) and (1.2). Let $f_{m}\in
W^{1,n-1}(\Omega,\Omega^{\prime})$, $m=0,1,2\dots$, be a sequence of
homeomorphisms of $\overline{\Omega}$ onto $\overline{\Omega^{\prime}}$ with
$J_{f_{m}}>0$ a.e. such that

$\sup_{m}\int_{\Omega}\Bigl{(}|Df_{m}|^{n-1}+\varphi(J_{f_{m}})\Bigr{)}\,dx<\infty.$

Assume further that $f_{m}=f_{0}$ on $\partial\Omega$ for all
$m\in\mathbb{N}$. Let $f$ be a weak limit of $f_{m}$ in
$W^{1,n-1}(\Omega,\mathbb{R}^{n})$ and $B\subseteq\Omega$ be a ball such that
$f$ satisfies $\mathrm{(INV)}$ in $B$,

  * •

$f_{m}\to f$ weakly in $W^{1,n-1}(\partial B,\mathbb{R}^{n})$,

  * •

$f_{m}\to f$ $\mathcal{H}^{n-1}$-a.e. on $\partial B$,

  * •

$\int_{\partial
B}\left(|D_{\tau}f|^{n-1}+|D_{\tau}f_{m}|^{n-1}\right)\,d\mathcal{H}^{n-1}<C.$

Then it holds that

(3.23)
$|f_{k}(B)\triangle\operatorname{im}_{T}(f,B)|\overset{k\to\infty}{\to}0.$

###### Proof.

Let us assume by contradiction that we have a subsequence $f_{m_{k}}$ such
that

$|f_{m_{k}}(B)\triangle\operatorname{im}_{T}(f,B)|>4\lambda>0.$

We show that then $f$ does not satisfy $\mathrm{(INV)}$.

For simplicity, we pass to that subsequence and keep the notation $f_{m}$. We
consequentially choose a further subsequence such that $f_{m}(x)\to f(x)$ a.e.
on $B$ and, again passing to subsequences if neccessary, given $\varepsilon>0$
we find $g$ as in the proof of Theorem 1.1 such that

(3.24)
$|\operatorname{im}_{T}(f,B)\triangle\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}|\leq|\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq\operatorname{Deg}(f,B,y)\\}|\leq\varepsilon.$

Let us split into two cases: either $|\operatorname{im}_{T}(f,B)\setminus
f_{m}(B)|>2\lambda$ or
$|f_{m}(B)\setminus\operatorname{im}_{T}(f,B)|>2\lambda$ for infinitely many
$m$ and thus we can assume that this is true for all $m$.

In the first case we find

$U_{m}\subseteq\Omega\setminus B\text{ such that
}f_{m}(U_{m})\subseteq\operatorname{im}_{T}(f,B)\text{ and
}|f_{m}(U_{m})|>2\lambda.$

We can assume that $\varepsilon<\lambda$ and hence we can find
$U_{m}^{\prime}\subseteq U_{m}$ such that

$f_{m}(U_{m}^{\prime})\cap\bigl{(}\operatorname{im}_{T}(f,B)\triangle\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}\bigr{)}=\varnothing$

and $|f_{m}(U_{m}^{\prime})|>\lambda$. We know from (2.5) that
$|U_{m}^{\prime}|>\Psi^{-1}(\lambda)$. Therefore,

$|U|>\Psi^{-1}(\lambda)/2,\text{ where }U=\limsup
U_{m}^{\prime}=\bigcap_{k=1}^{\infty}\bigcup_{m=k}^{\infty}U_{m}^{\prime}.$

Using Theorem 2.16 (as $|h(S)|=0$ there) and (3.24) we have

$\bigl{|}\partial\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\})\bigr{|}=0\text{ and
}\bigl{|}\operatorname{im}_{T}(f,B)\triangle\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}\bigr{|}<\varepsilon$

and according to Lemmata 2.3 and 2.2 their preimages under $f$ are arbitrarily
small (depending on $\varepsilon$). Hence we can set $\varepsilon$ to be small
enough such that $|U^{\prime}|>\Psi^{-1}(\lambda)/2$, where

$U^{\prime}=U\setminus\Bigl{[}f^{-1}\Bigl{(}\partial\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}\cup\bigl{(}\operatorname{im}_{T}(f,B)\triangle\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}\bigr{)}\Bigr{)}\Bigr{]}.$

For every $x\in U^{\prime}$ we have a subsequence $f_{m_{k}}$ such that $x\in
U_{m_{k}}^{\prime}$ and thus

$f_{m_{k}}(x)\in\operatorname{im}_{T}(f,B)\cap\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}.$

Since $f_{m}\to f$ pointwise a.e., we have for a.e. $x\in U^{\prime}$ that

$f(x)\in\overline{\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq 0\\}}.$

However, from the definition of $U^{\prime}$ we know that actually

$f(x)\in\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq 0\\},$

and since

$x\notin
f^{-1}\bigl{(}\operatorname{im}_{T}(f,B)\triangle\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}\bigr{)},$

we have that $f(x)\in\operatorname{im}_{T}(f,B)$ for every $x\in U^{\prime}$.
That contradicts $\mathrm{(INV)}$.

We deal with the second case analogously. We find

$U_{m}\subseteq B\text{ such that
}f_{m}(U_{m})\cap\operatorname{im}_{T}(f,B)=\varnothing\text{ and
}|f_{m}(U_{m})|>2\lambda.$

We define the following sets in the same way and arrive to

$f_{m_{k}}(x)\in\bigl{(}\Omega^{\prime}\setminus\operatorname{im}_{T}(f,B)\bigr{)}\cap\bigl{(}\Omega^{\prime}\setminus\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\}\bigr{)}.$

Again, $x$ is not in the $f$-preimage of

$\partial\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq 0\\}\text{ nor
of
}\operatorname{im}_{T}(f,B)\triangle\\{y\in\Omega^{\prime}:\operatorname{deg}(g,B,y)\neq
0\\},$

and therefore $f(x)\notin\operatorname{im}_{T}(f,B)$ for a set of a positive
measure, which contradicts $\mathrm{(INV)}$. ∎

##  4\. Proof of Theorem 1.1: $\mathrm{(N)}$ condition, lower semicontinuity
and injectivity a.e.

###  4.1. Lusin $\mathrm{(N)}$ condition

######  Lemma 4.1.

Let $f_{m}\in W^{1,n-1}(\Omega,\mathbb{R}^{n})$ be a sequence of
homeomorphisms with $J_{f_{m}}>0$ a.e. such that $f_{m}$ satisfies the Lusin
$\mathrm{(N)}$ condition and the sequence of Jacobians $J_{f_{m}}$ is
equiintegrable. Assume that $f\in W^{1,n-1}(\Omega,\mathbb{R}^{n})\cap
L^{\infty}(\Omega,\mathbb{R}^{n})$ is a weak limit of $f_{m}$ in
$W^{1,n-1}(\Omega,\mathbb{R}^{n})$ such that for every $a\in\Omega$ there is
$r_{a}>0$ such that for $\mathcal{H}^{1}$-a.e. $r\in(0,r_{a})$ it satisfies
(3.23) and $\mathrm{(INV)}$ in $B(a,r)$. Then,

  1. (i)

the distributional Jacobian $\operatorname{Det}Df\geq 0$ is a Radon measure;

  2. (ii)

$\operatorname{Det}Df$ is absolutely continuous w.r.t. Lebesgue measure: for
any set $E\subset\Omega$ with $|E|=0$, it holds that
$\operatorname{Det}Df(E)=0$;

  3. (iii)

$f$ satisfies the Lusin $\mathrm{(N)}$ condition.

###### Proof.

The first item $(i)$ is stated in Lemma 2.15 $(i)$.

Let $E\subset\Omega$ with $|E|=0$ be given. Fix $\delta>0$ and let $c(n)$ be a
constant from Besicovitch covering theorem. Since $E$ is a set of measure
zero, there exists an open set $U\subset\Omega$ such that $E\subset A$ and
$|U|<\frac{\delta}{c(n)}$. Consider a covering of $E$ by balls
$B(a,\tilde{r}_{a})$ for all $a\in E$ such that $B(a,\tilde{r}_{a})\subset U$,
$f$ satisfies $\mathrm{(INV)}$ in $B(a,\tilde{r}_{a})$ and (3.23) holds on
this ball. By the Bezicovitch Theorem (e.g. [26, Theorem A.2]), we can find at
most countable collection of balls $B_{k}:=B(a_{k},\tilde{r}_{k})$ such that

$E\subset\bigcup_{k}B_{k}\subset U\text{ and
}\bigcup_{k}B_{k}=\bigcup_{j=1}^{c(n)}\bigcup_{B_{i}\in A_{j}}B_{i},$

where subcollections $A_{j}$ consists of disjoined balls $B_{i}$ and a
constant $c(n)$ depends only on the dimension $n$.

Note that since (3.23) is valid for one ball of the covering, it stays true
for a finite union of such balls $\bigcup_{k=1}^{M}B_{k}$: for any
$\varepsilon>0$ and $m$ big enough it holds that

(4.1)
$\Bigl{|}f_{m}\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}\triangle\Bigl{(}\bigcup_{k=1}^{M}\operatorname{im}_{T}\left(f,B_{k}\right)\Bigr{)}\Bigr{|}\leq\frac{\varepsilon}{3c(n)}.$

Then, by Lemma 2.15 $(iii)$ we have

$\displaystyle\operatorname{Det}Df\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}$
$\displaystyle=\operatorname{Det}Df\Bigl{(}\bigcup_{j=1}^{c(n)}\bigcup_{B_{i}\in
A_{j},i\leq
M}B_{i}\Bigr{)}\leq\sum_{j=1}^{c(n)}\operatorname{Det}Df\Bigl{(}\bigcup_{B_{i}\in
A_{j},i\leq M}B_{i}\Bigr{)}$
$\displaystyle\leq\sum_{j=1}^{c(n)}\Bigl{|}\bigcup_{B_{i}\in A_{j},i\leq
M}\operatorname{im}_{T}(f,B_{i})\Bigr{|}\leq
c(n)\Bigl{|}\bigcup_{k=1}^{M}\operatorname{im}_{T}(f,B_{k})\Bigr{|}.$

To prove $(ii)$, we fix $\varepsilon>0$ and $\delta>0$ such that

(4.2) $\Psi(t)<\frac{\varepsilon}{3\,c(n)}\quad\text{ for any }t<\delta,$

where $\Psi$ is given by Lemma 2.1 (note that in the proof of this part of
lemma we have used only equiintegrability of $J_{f_{m}}$ and Lusin
$\mathrm{(N)}$ condition for $f_{m}$). Since $\bigcup_{k}B_{k}\subset U$ and
$|U|<\delta$ we have using (2.5)

$\sum_{k\in\mathbb{N}}\left|B_{k}\right|<\delta,\text{ and therefore
}\Bigl{|}f_{m}\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}\Bigr{|}<\dfrac{\varepsilon}{3\,c(n)}\text{
for any }m\in\mathbb{N}.$

Relying on (4.1)–(4.2), for $M$ big enough there exists $m$ such that

$\displaystyle\operatorname{Det}Df(E)$
$\displaystyle\leq\operatorname{Det}Df\Bigl{(}\bigcup_{k\in\mathbb{N}}B_{k}\Bigr{)}\leq\operatorname{Det}Df\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}+\frac{\varepsilon}{3}\leq
c(n)\Bigl{|}\bigcup_{k=1}^{M}\operatorname{im}_{T}(f,B_{k})\Bigr{|}+\frac{\varepsilon}{3}$
$\displaystyle\leq
c(n)\Bigl{|}f_{m}\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}\Bigr{|}+c(n)\Bigl{|}f_{m}\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}\triangle\Bigl{(}\bigcup_{k=1}^{M}\operatorname{im}_{T}\left(f,B_{k}\right)\Bigr{)}\Bigr{|}+\frac{\varepsilon}{3}\leq\varepsilon.$

For $(iii)$ it is enough to notice that
$|f(B)|\leq|\operatorname{im}_{T}f(B)|$ by Lemma 2.14. Hence

$\displaystyle|f(E)|$
$\displaystyle\leq\Bigl{|}\bigcup_{k=1}^{M}\operatorname{im}_{T}(f,B_{k})\Bigr{|}+\frac{\varepsilon}{3}$
$\displaystyle\leq\Bigl{|}f_{m}\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}\Bigr{|}+\Bigl{|}f_{m}\Bigl{(}\bigcup_{k=1}^{M}B_{k}\Bigr{)}\triangle\Bigl{(}\bigcup_{k=1}^{M}\operatorname{im}_{T}\left(f,B_{k}\right)\Bigr{)}\Bigr{|}+\frac{\varepsilon}{3}\leq\varepsilon.$

Since $\varepsilon>0$ has been chosen arbitrary, we conclude
$\operatorname{Det}Df(E)=|f(E)|=0$.

∎

###  4.2. Lower semicontinuity

The main obstacle to obtain the lower semicontinuity of $\mathcal{F}$ is to
ensure weak convergence of Jacobians. One usually has to assume higher
regularity of $f\in W^{1,n}$ (see e.g. [9, 15, 31]) but this is not available
for us.

######  Lemma 4.2.

Let $f_{m}\in W^{1,n-1}(\Omega,\mathbb{R}^{n})$ be a sequence of
homeomorphisms with $J_{f_{m}}>0$ a.e., such that $f_{m}$ satisfies the Lusin
$\mathrm{(N)}$ condition and the sequence of Jacobians $J_{f_{m}}$ is
equiintegrable. Assume that $f\in W^{1,n-1}(\Omega,\mathbb{R}^{n})\cap
L^{\infty}(\Omega,\mathbb{R}^{n})$ is a weak limit of $f_{m}$ in
$W^{1,n-1}(\Omega,\mathbb{R}^{n})$ such that for every $a\in\Omega$ there is
$r_{a}>0$ such that for $\mathcal{H}^{1}$-a.e. $r\in(0,r_{a})$ it satisfies
(3.23) and $\mathrm{(INV)}$ in $B(a,r)$. Then there exists a subsequence
$\\{J_{f_{k_{m}}}\\}_{m\in\mathbb{N}}$ of $\\{J_{f_{k}}\\}_{k\in\mathbb{N}}$
such that $J_{f_{k_{m}}}\rightharpoonup J_{f}$ weakly in $L^{1}(\Omega)$.

###### Proof.

By the Dunford–Pettis Theorem (e.g. [30, Theorem B.103]) there exist a (non-
relabeled) subsequence $J_{f_{m}}$ and a function $j\in L^{1}(\Omega)$ s.t.
$J_{f_{m}}\to j$ weakly in $L^{1}(\Omega)$, and hence for any measurable set
$E\subset\Omega$

$\int_{E}J_{f_{m}}(x)\,dx\to\int_{E}j(x)\,dx.$

Let $B$ be such that $f$ satisfies the $\mathrm{(INV)}$ condition and (3.23)
in $B$, then by the area formula (2.1) and (3.23) we have

$\int_{B}J_{f_{m}}(x)\,dx=|f_{m}(B)|\to|\operatorname{im}_{T}(f,B)|.$

On the other hand, by Lemma 2.15 $(ii)$–$(iii)$ and Lemma 4.1 $(ii)$, we know
that

$\operatorname{Det}f(B)=\int_{B}J_{f}(x)\,dx=|\operatorname{im}_{T}f(B)|.$

Therefore, for all such balls, it holds that

$\int_{B}j(x)\,dx=|\operatorname{im}_{T}(f,B)|=\int_{B}J_{f}(x)\,dx,$

which in turn implies $j(x)=J_{f}(x)$ for a.e. $x\in\Omega$. ∎

######  Lemma 4.3.

Under conditions of Theorem 1.1, the functional $\mathcal{F}$ is lower
semicontinuous with respect to weak convergence, i.e., (1.5) holds.

###### Proof.

Weak convergence $f_{k}\to f$ in $W^{1,n-1}$ implies weak convergence (up to
subsequence) in $L^{1}$ of all minors $\operatorname{det}_{l}(Df_{m})$ of
order $l\leq n-2$ (see e.g. [35, Lemma 5.10]):

$\|\operatorname{det}_{l}(Df_{m})\|_{L^{1}(\Omega)}\leq
C\|Df_{m}\|_{L^{n-1}}^{l}\leq CM^{\frac{l}{n-1}}.$

Weak convergence of
$\operatorname{det}_{n-1}(Df_{k})=\operatorname{cof}Df_{k}$ in $L^{1}$ follows
by the standard argument, provided uniform integrability (1.3) (see [2,
Theorem 6.2] or [10, Theorem 7.5-1]). The equiintegrability of $J_{f_{m}}$
follows from (1.2) by the de la Valée Poussin Theorem (e.g. [30, Theorem
B.104]). Moreover, conditions of Lemma 3.1 are fulfilled (see Step 2 of the
proof in Section 3). Therefore, Lemmata 3.1 and 4.2 ensure $J_{f_{m}}\to
J_{f}$ weakly in $L^{1}$, which now allows us to use De Giorgi Theorem [14,
Theorem 3.23] to conclude the lower semicontinuity (1.5) of $\mathcal{F}$. ∎

###  4.3. Injectivity almost everywhere

One of the main reasons to consider the $\mathrm{(INV)}$ condition is that it
implies injectivity a.e. [12, Lemma 3.7], [33, Lemma 3.4], as discussed in
Introduction. In the setting of this paper, we can say even more.

######  Lemma 4.4.

Let conditions of Theorem 1.1 be fulfilled and let $h$ be a weak limit of
$f_{m}^{-1}$ in $W^{1,1}(\Omega^{\prime},\mathbb{R}^{n})$. Then $h(f(x))=x$
for a.e. $x\in\Omega$ and under additional assumption
$|\partial\Omega^{\prime}|=0$ we have $f(h(y))=y$ for a.e.
$y\in\Omega^{\prime}$.

###### Proof.

From Theorem 2.5 we obtain that there is a subsequence of $f_{m}^{-1}$ which
converges weakly in $W^{1,1}$ to some $h$ and we work with this subsequence
here.

Recall that we know that $J_{f}(x)>0$ for a.e. $x\in\Omega$. Since $f$
satisfies $\mathrm{(N)}$ we can use (2.2) (for $A=f^{-1}(E)$) to obtain that

(4.3) $|f^{-1}(E)|=0\text{ for every }E\subset\mathbb{R}^{n}\text{ with
}|E|=0.$

Therefore, for a.e. $x\in\Omega$ we know that $f(x)$ is a Lebesgue point of
$h$.

Moreover, we claim that for a.e. $x\in\Omega$ and every $\eta>0$ there exists
a ball $B=B(a,r)$ such that $f$ satisfies $\mathrm{(INV)}$ and (3.23) in $B$
and

(4.4) $x\in B,\>r<\eta,\text{ and }f(x)\in\operatorname{im}_{T}(f,B)\text{ is
a point of density $1$ of }\operatorname{im}_{T}(f,B).$

Indeed, choose a countable set of balls

$\displaystyle\mathcal{B}:=\bigl{\\{}B(c,r_{i}):$ $\displaystyle
c\in\mathbb{Q}^{n}\cap\Omega,\ r_{i}\in[2^{-i-1},2^{-i})\text{ for all
}i\in\mathbb{N}$ $\displaystyle\text{ and }f\text{ satisfies
}\mathrm{(INV)}\text{ and }\eqref{eq:fk-ft}\text{ in }B(c,r_{i})\bigr{\\}}.$

For every $B_{j}\in\mathcal{B}$ we know that a.e. point of
$\operatorname{im}_{T}(f,B_{j})$ is a point of density $1$ and with the help
of (4.3) we can find a null set $\Sigma_{j}$ such that

$f(x)\in\operatorname{im}_{T}(f,B_{j})\text{ for each }x\in
B_{j}\setminus\Sigma_{j}\text{ and }f(x)\text{ is a point of density of
}\operatorname{im}_{T}(f,B_{j}).$

Then $\Sigma:=\bigcup_{j}\Sigma_{j}$ is a null set and for every
$x\in\Omega\setminus\Sigma$ we have (4.4) for some ball $B_{j}$.

Let us first prove that $h(f(x))=x$ a.e. We pick $x$ such that $f(x)$ is a
Lebesgue point of $h$ and $f(x)$ is a point of density of
$\operatorname{im}_{T}(f,B)$ for some ball $B=B(a,r)$ satisfying (4.4). Now we
can find a ball $\tilde{B}$ around $f(x)$ so that

$\frac{1}{|\tilde{B}|}\int_{\tilde{B}}|h(z)-h(f(x))|\,dz<r/2\qquad\text{
and}\qquad\left|\bigl{\\{}y\in\tilde{B}:\
y\in\operatorname{im}_{T}(f,B)\bigr{\\}}\right|>0.9|\tilde{B}|.$

Using convergence $f_{m}^{-1}$ to $h$ in $L^{1}_{{\rm
loc}}(\Omega^{\prime},\mathbb{R}^{n})$ and (3.23), we fix $m$ big enough so
that

$\int_{\tilde{B}}|f_{m}^{-1}(z)-h(z)|\,dz<r|\tilde{B}|/2\qquad\text{and}\qquad|f_{m}(B)\triangle\operatorname{im}_{T}(f,B)|<0.1|\tilde{B}|.$

Combining the estimates, we obtain

$\frac{1}{|\tilde{B}|}\int_{\tilde{B}}|f_{m}^{-1}(z)-h(f(x))|\,dz<r\qquad\text{and}\qquad\left|\bigl{\\{}y\in\tilde{B}:\
y\in f_{m}(B)\bigr{\\}}\right|>0.8|\tilde{B}|.$

We claim that this implies

(4.5) $h(f(x))\in B(a,4r),$

since otherwise we get a contradiction from

$\displaystyle\frac{1}{|\tilde{B}|}\int_{\tilde{B}}|f_{m}^{-1}(z)-h(f(x))|\,dz$
$\displaystyle\geq\frac{1}{|\tilde{B}|}\int_{\\{y\in\tilde{B}:\ y\in
f_{m}(B)\\}}\bigl{|}h(f(x))-a-(f_{m}^{-1}(z)-a)\bigr{|}\,dz$
$\displaystyle\geq 0.8(4r-r)>r.$

Since $r>0$ is chosen arbitrary, we conclude from (4.5) that $h(f(x))=x$ for
a.e. $x\in\Omega$.

It is not difficult to see that $f(\Omega)\subset\overline{\Omega^{\prime}}$.
From Lemma 4.2 and change of variables (2.2) we know that

$|\Omega^{\prime}|=\lim_{k\to\infty}|f_{k}(\Omega)|=\lim_{k\to\infty}\int_{\Omega}J_{f_{k}}(x)\,dx=\int_{\Omega}J_{f}(x)\,dx=\int_{\mathbb{R}^{n}}N(f,\Omega,y)\,dy.$

From the a.e.-injectivity [12, Lemma 3.7] of $f$ together with the
$\mathrm{(N)}$ condition for $f$ we now obtain

$|\Omega^{\prime}|=\int_{\mathbb{R}^{n}}N(f,\Omega,y)\,dy=\int_{f(\Omega)}1\,dy=|f(\Omega)|.$

Since $f(\Omega)\subset\overline{\Omega^{\prime}}$ and
$|\partial\Omega^{\prime}|=0$ we obtain that a.e. point $y\in\Omega^{\prime}$
lies in $f(\Omega)$ and $N(f,\Omega,y)=1$ there.

The other equality $f(h(y))=y$ for a.e. $y\in\Omega^{\prime}$ now follows
easily. We know $h(f(x))=x$ holds for a.e. $x\in\Omega$ and that $f$ satisfies
the $\mathrm{(N)}$ condition. Hence for a.e. $y\in f(\Omega)$ we can pick
$x\in\Omega$ such that $f(x)=y$ and $h(f(x))=x$. Now

$f(h(y))=f(h(f(x)))=f(x)=y.$

Note that in this proof we do not need a Sobolev regularity but only
$f_{m}^{-1}\to h$ in $L^{1}_{{\rm loc}}$. ∎

###### Proof of Theorem 1.1.

Theorem 1.1 now follows from result in Section 3, Lemma 4.1, Lemma 4.4 and
Lemma 4.3. ∎

###  4.4. Counterexamples to lower semicontinuity

The following example shows that one has to ask the condition $\mathrm{(N)}$
for $f_{m}$ to conclude lower semicontinuity of a quasiconvex functional, even
if $\varphi$ and $A$ satisfy (1.1) and (1.3).

######  Lemma 4.5 (Counterexample for lsc).

Let $p<n$, then there exist $\varphi$ and $A$ that satisfy (1.1) and (1.3) and
homeomorphisms $f_{m}$, $f\colon[0,1]^{n}\to[0,1]^{n}$ such that $J_{f_{m}}$,
$J_{f}>0$ a.e., (1.4) is fulfilled, $f_{m}=id$ on $\partial([0,1]^{n})$ and
$f_{m}$ converge to $f$ weakly in $W^{1,p}([0,1]^{n},\mathbb{R}^{n})$.
However, $f_{m}$ does not satisfy the Lusin $\mathrm{(N)}$ condition for all
$m\in\mathbb{N}$ and

$\int_{(0,1)^{n}}J_{f}(x)\,dx>\liminf_{m\to\infty}\int_{(0,1)^{n}}J_{f_{m}}(x)\,dx.$

###### Proof.

Take any $A(t)\leq Ct^{\beta}$, where $C>0$ and $\beta>1$ are some constants,
and take $\varphi$ which behaves like an identity around $1$ and satisfies
(1.1). Consider a Ponomarev-type map $g\colon[0,1]^{n}\to[0,1]^{n}$, $g\in
W^{1,p}([0,1]^{n},\mathbb{R}^{n})$ which maps a Cantor-set $\mathcal{C}_{A}$
of measure zero to a Cantor-set $\mathcal{C}_{B}=g(\mathcal{C}_{A})$ of
positive measure, and which is identical on $\partial([0,1]^{n})$. Such a map
can be found by the standard construction, see [26, Chapter 4.3] with

$a_{k}=\frac{1}{k^{\alpha}}\text{ and }b_{k}=1+\frac{1}{k^{\alpha n}}\text{
where }0<\alpha<\min\left\\{\frac{n}{p},\frac{n}{(n-1)\beta}\right\\}.$

Referring the reader to [26, Chapter 4.3] for details, we just notice that on
the $k$-th level we have

$\displaystyle|Dg|\approx\max\left\\{\frac{b_{k}}{a_{k}},\frac{b_{k-1}-b_{k}}{a_{k-1}-a_{k}}\right\\}\approx
k^{\alpha},\qquad
J_{g}\approx\left(\frac{b_{k}}{a_{k}}\right)^{n-1}\cdot\frac{b_{k-1}-b_{k}}{a_{k-1}-a_{k}}\approx
1,$ $\displaystyle|\operatorname{cof}Dg|\leq C|Dg|^{n-1}\approx
k^{\alpha(n-1)}\quad\text{ and }\quad\left|\\{k\text{-th
level}\\}\right|\approx 2^{-kn}\frac{1}{k^{n+1}}.$

It follows that the map $g$ has finite energy since

(4.6) $\displaystyle\int_{(0,1)^{n}}|Dg(x)|^{p}\,dx\leq
C\sum_{k=1}^{\infty}2^{kn}2^{-kn}\frac{1}{k^{n+1}}k^{\alpha p}<\infty,$
$\displaystyle\int_{(0,1)^{n}}A(|\operatorname{cof}Dg(x)|)\,dx\leq
C\int_{(0,1)^{n}}|\operatorname{cof}Dg(x)|^{\beta}\,dx\leq
C\sum_{k=1}^{\infty}2^{kn}2^{-kn}\frac{1}{k^{n+1}}k^{\alpha\beta(n-1)}<\infty,$
$\displaystyle\int_{(0,1)^{n}}\varphi(J_{g}(x))\,dx<\infty,\qquad\text{and}\qquad\int_{(0,1)^{n}}K_{g}^{\frac{1}{n-1}}(x)\,dx\leq\int_{(0,1)^{n}}|Dg(x)|^{\frac{n}{n-1}}\,dx<\infty.$

We set $f_{1}=g$ and we divide the cube $[0,1]^{n}$ into $m^{n}$ equal cubes
both in the domain and in the target. Fix one of those $m^{n}$ small cubes
$Q_{z}:=\\{x\in[0,1]^{n}:\|x-z\|_{\infty}<\frac{1}{2m}\\}$ with a center point
$z$ and define $f_{m}|_{Q_{z}}\colon Q_{z}\to Q_{z}$ as a scaled and
translated copy of $g$

$f_{m}(x):=\frac{1}{m}g\bigl{(}m(x-z)\bigr{)}+z.$

It is easy to see by change of variables that

$\int_{(0,1)^{n}}|Df_{m}|^{p}\,dx=\int_{(0,1)^{n}}|Dg|^{p}\,dx$

and analogously for other integrals in (4.6). It follows that
$\sup_{m}\mathcal{F}(f_{m})=\mathcal{F}(g)<\infty$ and hence there is a
subsequence which converges weakly in $W^{1,n-1}$. Since $f_{m}\to
f:=\operatorname{id}$ pointwise, identity is a weak limit of $f_{m}$.

By construction, $g$ maps the Cantor-set $\mathcal{C}_{A}$ to the Cantor-set
$\mathcal{C}_{B}$, where $\mathcal{C}_{A}$ is a set where $g$ fails the Lusin
$\mathrm{(N)}$ condition. Hence, for every $m\in\mathbb{N}$ and for each
$Q_{z}$ it holds that

$\int_{Q_{z}}J_{f_{m}}(x)\,dx=\frac{1-|\mathcal{C}_{B}|}{m^{n}}.$

Therefore,

$\int_{(0,1)^{n}}J_{f_{m}}(x)\,dx=1-|\mathcal{C}_{B}|<1=\int_{(0,1)^{n}}J_{f}(x)\,dx,$

so the lower semicontinuity fails at least for this quasiconvex functional. ∎

Figure 4. $f_{3}$ and its action on
$Q_{\left(\frac{1}{6},\frac{1}{2}\right)}$.

###  4.5. $\mathrm{(N)}$ condition and lower semicontinuity in the context of
[17]

In our previous result [17] we have shown that limit of homeomorphisms in
$W^{1,n-1}$ satisfy the $\mathrm{(INV)}$ condition under different
assumptions. Instead of $\int A(|\operatorname{cof}Df_{m}|)\leq C$ we assumed
the integrability of the distortion function. Here we show that even in that
case we can obtain that the corresponding functional is lower semicontinuous
and that $f$ satisfies $\mathrm{(N)}$ under the additional assumptions that
all $f_{m}$ satisfy $\mathrm{(N)}$ and that (1.2) holds (these assumptions
were not in [17]). As in [17] we assume that there is $A>0$ with

(4.7) $A^{-1}\varphi(t)\leq\varphi(2t)\leq A\varphi(t),\qquad t\in(0,\infty).$

######  Theorem 4.6.

Let $n\geq 3$, $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be Lipschitz
domains and let $\varphi$ satisfy (1.1) and (4.7). Let $f_{m}\in
W^{1,n-1}(\Omega,\Omega^{\prime})$, $m=0,1,2\dots$, be a sequence of
homeomorphisms of $\overline{\Omega}$ onto $\overline{\Omega^{\prime}}$ with
$J_{f_{m}}>0$ a.e. such that

$\sup_{m}\int_{\Omega}\Bigl{(}|Df_{m}(x)|^{n-1}+\varphi(J_{f_{m}}(x))+\Bigl{(}\frac{|Df_{m}(x)|^{n}}{J_{f_{m}(x)}}\Bigr{)}^{\frac{1}{n-1}}\Bigr{)}\,dx<\infty.$

Assume further that $f_{m}=f_{0}$ on $\partial\Omega$ for all
$m\in\mathbb{N}$. Let $f$ be a weak limit of $f_{m}$ in
$W^{1,n-1}(\Omega,\mathbb{R}^{n})$ , then $f$ satisfies the $\mathrm{(INV)}$
condition.

Moreover, under the additional assumptions (1.2) and that all $f_{m}$ satisfy
the Lusin $\mathrm{(N)}$ condition we obtain that our $f$ satisfies the Lusin
$\mathrm{(N)}$ condition and we have lower semicontinuity of energy

(4.8)
$\mathcal{G}(f):=\int_{\Omega}\Bigl{(}|Df(x)|^{n-1}+\varphi(J_{f}(x))+\Bigl{(}\frac{|Df(x)|^{n}}{J_{f}(x)}\Bigr{)}^{\frac{1}{n-1}}\Bigr{)}\leq\liminf_{m\to\infty}\mathcal{G}(f_{m}).$

Further

$\text{ for a.e.\ }x\in\Omega\text{ we have }h(f(x))=x\text{ and for a.e.\
}y\in\Omega^{\prime}\text{ we have }f(h(y))=y,$

where $h$ is a weak-$*$ limit of (some subsequence of) $f_{m}^{-1}$ in
$BV(\Omega^{\prime},\mathbb{R}^{n})$.

###### Proof.

The fact that the limit $f$ satisfies the $\mathrm{(INV)}$ condition follows
from [17, Theorem 3.1 a)] and Lemma 2.3.

As in Step 2 of Section 3, we know that for every center and almost every
radius the corresponding ball satisfies the conditions of Lemma 3.1. Thus,

(4.9)
$|f_{m}(B)\triangle\operatorname{im}_{T}(f,B)|\overset{k\to\infty}{\to}0.$

Further, the $\mathrm{(N)}$ condition of $f$ and a lower semicontinuity of
$\mathcal{G}$ follow from Lemmata 4.1–4.2, provided with (4.9). To prove the
lower semicontinuity of $\mathcal{G}$, we just note that the function

$g(x,y)=x^{n-1}+x^{\frac{n}{n-1}}y^{-\frac{1}{n-1}}+\varphi(y)$

is convex, and as in Lemma 4.3 use the De Giorgi Theorem [14, Theorem 3.23]
again. Let us note that our functional depends only on $|Df|$ and
$\operatorname{det}Df$, but not on $\operatorname{cof}Df$, so we do not need
to care about convergence of $(n-1)\times(n-1)$ subdeterminants here.

Further, from [18, Corollary 4.2] we conclude that there exists a subsequence
of $f_{m}^{-1}$ which converges weakly-$*$ in $BV$ to some $h$, in particular
$f_{m}^{-1}\to h$ in $L^{1}_{{\rm loc}}$. Injectivity almost everywhere of
both $f$ and $h$ is then obtained by following the lines of proof of Lemma
4.4. ∎

##  5\. Application to Calculus of Variations

Let $n\geq 3$ and $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be bounded
domains, e.g. representing the reference and deformed configurations in
nonlinear elasticity. Define the energy functional

(5.1) $\mathcal{E}(f):=\int_{\Omega}W(Df(x))\,dx,$

where $W\colon\mathbb{R}^{n\times n}\to\mathbb{R}$ is a polyconvex function,
i.e., $W$ can be expressed as a convex function of the minors of its argument,
satisfying

(5.2)
$W(F)\geq\begin{cases}C\bigl{(}|F|^{n-1}+\varphi(\operatorname{det}F)+A(|\operatorname{cof}F|)-1\bigr{)},&\text{if
}\operatorname{det}F>0,\\\ \infty,&\text{if }\operatorname{det}F\leq
0,\end{cases}$

for some $C>0$ and for some positive functions $A$ and $\varphi$. Consider a
homeomorphism $f_{0}$ from $\overline{\Omega}$ onto
$\overline{\Omega^{\prime}}$ such that $\mathcal{E}(f_{0})<\infty$, and the
following sets of admissible functions:

$\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n}):=\Bigl{\\{}f:\overline{\Omega}\to\mathbb{R}^{n}:f\text{
is a homeomorphism of }\overline{\Omega}\text{ onto
}\overline{\Omega^{\prime}}\text{ satisfying }\\\ \text{ the Lusin
}\mathrm{(N)}\text{ condition, }f=f_{0}\text{ on }\partial\Omega,\text{ and
}\mathcal{E}(f)\leq\mathcal{E}(f_{0})\Bigr{\\}}$

and

$\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n}):=\Bigl{\\{}f:\Omega\to\mathbb{R}^{n}:\text{
there are }f_{m}\in\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n})\text{ with }\\\
f_{m}\rightharpoonup f\text{ weakly in
}W^{1,n-1}(\Omega,\mathbb{R}^{n})\Bigr{\\}}.$

Note that $\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$ is
weakly (sequentially) closed and hence it is a suitable set of mappings for
variational approach:

######  Proposition 5.1.

Let $g_{m}\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$ and
assume that $g_{m}\rightharpoonup g$ weakly in
$W^{1,n-1}(\Omega,\mathbb{R}^{n})$. Then
$g\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$. In
particular, there exists a sequence
$f_{m}\in\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n})$ such that
$f_{m}\rightharpoonup g$ weakly in $W^{1,n-1}(\Omega,\mathbb{R}^{n})$,
$g=f_{0}$ on $\partial\Omega$, and
$\sup_{m}\mathcal{E}(f_{m})\leq\mathcal{E}(f_{0})<\infty$.

###### Proof.

Since $W^{1,n-1}$ is reflexive and separable we can find
$\\{L_{i}\\}_{i\in\mathbb{N}}\subset(W^{1,n-1})^{*}$ which is dense. We can
assume (passing to a subsequence) that

$|L_{i}(g_{m}-g)|<\frac{1}{k}\text{ for every }i\in\\{1,\ldots,m\\}.$

For every $g_{m}$ we can find a sequence in
$\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n})$ which converges weakly and thus
we can fix $f_{m}\in\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n})$ such that

$|L_{i}(g_{m}-f_{m})|<\frac{1}{m}\text{ for every }i\in\\{1,\ldots,m\\}.$

It follows that for every $i\in\mathbb{N}$ we have

$\lim_{m\to\infty}L_{i}(f_{m})=L_{i}(g).$

Since $f_{m}(x)\in\Omega^{\prime}\subset B(0,R)$ for all $x\in\Omega$,
$\|Df_{m}\|_{L^{n-1}}\leq\mathcal{E}(f_{m})\leq\mathcal{E}(f_{0})$ result in
$\|f_{m}\|_{W^{1,n-1}}\leq M$ for all $m$ and some constant $M>0$, so we
easily obtain that

$\lim_{m\to\infty}L(f_{m})=L(g)\text{ for every }L\in(W^{1,n-1})^{*}.$

Note further that the set $f_{0}+W_{0}^{1,n-1}(\Omega,\mathbb{R}^{n})$ is
closed and convex and thus weakly closed, therefore, $g=f_{0}$ on
$\partial\Omega$. ∎

######  Proposition 5.2.

Let $g$, $g_{m}\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$
and assume that $g_{m}\rightharpoonup g$ weakly in
$W^{1,n-1}(\Omega,\mathbb{R}^{n})$. Then (up to subsequence)
$J_{g_{m}}\rightharpoonup J_{g}$ weakly in $L^{1}(\Omega)$.

###### Proof.

Let us first prove that for every ball $B$, such that $f$ satisfies the
$\mathrm{(INV)}$ condition and (3.23) in $B$,

(5.3)
$\lim_{m\to\infty}|\operatorname{im}_{T}(g_{m},B)|=|\operatorname{im}_{T}(g,B)|.$

Fix such a ball $B$ for all $m\in\mathbb{N}$ and $\varepsilon>0$. In view of
Lemma 3.1, for any $m\in\mathbb{N}$ there exists a sequence
$f_{m,k}\in\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n})$ such that

$|f_{m,k}(B)\triangle\operatorname{im}_{T}(g_{m},B)|\leq\frac{\varepsilon}{2}.$

Using the diagonal procedure as in Proposition 5.1, we find a sequence
$f_{m,k(m)}\in\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n})$ with
$f_{m,k(m)}\rightharpoonup g$ weakly in $W^{1,n-1}$. Again by Lemma 3.1 it
holds that

$|f_{m,k(m)}(B)\triangle\operatorname{im}_{T}(g,B)|\leq\frac{\varepsilon}{2}$

for $m$ big enough. Combining these two inequalities, we obtain (5.3).

Now the proof follows proof of Lemma 4.2, since for every $a\in\Omega$ there
is $r_{a}>0$ such that for $\mathcal{H}^{1}$-a.e. $r\in(0,r_{a})$ the mapping
$g$ satisfies (3.23) and $\mathrm{(INV)}$ in $B(a,r)$. ∎

######  Theorem 5.3.

Let $n\geq 3$ and $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be bounded
domains, let also $W\colon\mathbb{R}^{n\times n}\to\mathbb{R}$ be a polyconvex
function satisfying (5.2) for some functions $A$ and $\varphi$ satisfying
(1.1)–(1.3) and a constant $C>0$. Assume further that $f_{0}$ is a
homeomorphism from $\overline{\Omega}$ onto $\overline{\Omega^{\prime}}$ such
that $\mathcal{E}(f_{0})<\infty$, where $\mathcal{E}$ is the energy defined by
(5.1). Then there exists
$f\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$ such that

$\mathcal{E}(f)=\inf\bigl{\\{}\mathcal{E}(h):\
h\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})\bigr{\\}}.$

Moreover, $f$ satisfies the $\mathrm{(INV)}$ condition and the Lusin
$\mathrm{(N)}$ condition.

###### Proof.

Let $f_{m}$ be a minimizing sequence for $\mathcal{E}$, then $f_{m}$ form a
bounded sequence in $W^{1,n-1}$, and hence using Proposition 5.1 there is
$f\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$ such that (up
to a subsequence) $f_{m}\rightharpoonup f$ weakly in $W^{1,n-1}$. Provided
with Proposition 5.2, we obtain that $\mathcal{E}$ is lower semicontinuous in
$\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$ following the
proof of Lemma 4.3.

Proposition 5.1 and Theorem 1.1 imply thus that $f$ satisfies the
$\mathrm{(INV)}$ and the $\mathrm{(N)}$ conditions and also that

$\mathcal{E}(f)\leq\liminf_{m\to\infty}\mathcal{E}(f_{m})=\lim_{m\to\infty}\mathcal{E}(f_{m})=\inf\bigl{\\{}\mathcal{E}(h):\
h\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})\bigr{\\}}\leq\mathcal{E}(f).$

∎

######  Remark 5.4.

Let us note that it is not clear if the two following infima

$\inf\bigl{\\{}\mathcal{E}(h):\
h\in\mathcal{H}_{f_{0}}(\Omega,\mathbb{R}^{n})\bigr{\\}}\quad\text{ and
}\quad\inf\bigl{\\{}\mathcal{E}(h):\
h\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})\bigr{\\}},$

are equal or not since the space
${\mathcal{H}}_{f_{0}}(\Omega,\mathbb{R}^{n})$ is not compact.

Analogously we can use the results of [17] and Section 4.5 to obtain the
following theorem.

######  Theorem 5.5.

Let $n\geq 3$ and $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ be
Lipschitz domains, let also $W\colon\mathbb{R}^{n\times n}\to\mathbb{R}$ be a
polyconvex function satisfying

$W(F)\geq\begin{cases}C\Bigl{(}|F|^{n-1}+\varphi(\operatorname{det}F)+\Bigl{(}\frac{|F|^{n}}{\operatorname{det}F}\Bigr{)}^{\frac{1}{n-1}}-1\Bigr{)},&\text{if
}\operatorname{det}F>0,\\\ \infty,&\text{if }\operatorname{det}F\leq
0,\end{cases}$

for some function $\varphi$ satisfying (1.1)–(1.2) and (4.7), and a constant
$C>0$. We assume that $W$ may be represented as a convex function of
subdeterminants of order strictly less than $n-1$ and of
$\operatorname{det}F$, i.e., it is not a function of subdeterminants of order
$n-1$. Assume further that $f_{0}$ is a homeomorphism from $\overline{\Omega}$
onto $\overline{\Omega^{\prime}}$ such that $\mathcal{E}(f_{0})<\infty$, where
$\mathcal{E}$ is the energy defined by (5.1). Then there exists
$f\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})$ such that

$\mathcal{E}(f)=\inf\bigl{\\{}\mathcal{E}(h):\
h\in\overline{\mathcal{H}}^{w}_{f_{0}}(\Omega,\mathbb{R}^{n})\bigr{\\}}.$

Moreover, $f$ satisfies the $\mathrm{(INV)}$ condition and the Lusin
$\mathrm{(N)}$ condition.

###### Proof.

The proof is analogous to the proof of Theorem 5.3. The only difference is
that in the proof of lower semicontinuity we do not have (1.3) and therefore
we cannot prove weak convergence of $\operatorname{cof}Df_{m}$ as in the proof
of Lemma 4.3. However, we do not need this as our $W$ “does not depend” on
$(n-1)\times(n-1)$ subdeterminants. ∎

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