# Local Sobolev-Poincaré imbedding domains

Tian Liang and Zheng Zhu  Tian Liang
School of Mathematics and Statistics
Huizhou University
Guangdong 516007
P.R. China<EMAIL_ADDRESS>Zheng Zhu
School of Mathematical Science
Beihang University
Changping District Shahe Higher Education Park South Third Street No. 9
Beijing 102206
P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this article, we study local Sobolev-Poincaré imbedding domains. The main
result reads as below.

  1. (1)

for $1\leq p\leq n$, a bounded uniform domain is also a local Sobolev-Poincaré
imbedding domain of order $p$; conversely a local Sobolev-Poincaré imbedding
domain of order $p$ is locally linearly connected $(LLC)$. A uniform domain is
$(LLC)$. Conversely, with some very weak connecting assumption, a $(LLC)$
domain is uniform.

  2. (2)

for $n<p<\infty$, a bounded domain is a local Sobolev-Poincaré imbedding
domain of order $p$ if and only if it is an $\alpha$-cigar domain for
$\alpha=(p-n)/(p-1)$. Hence, a domain is a local Sobolev-Poincaré imbedding
domain of oder $p$ if and only if it is a (global) Sobolev-Poincaré imbedding
domain.

###### Key words and phrases:

Local sobolev-Poincaré inequalities, Uniform domains, Cigar domains, LLC

The authors would like to thank Prof. Yuan Zhou for lots of valuable
discussions which helps this paper essentially. The authors also thank Prof.
Tapio Rajala and Prof. Qingshan Zhou for pointing out some typos in the first
version of manuscript and making the manuscript more readable. The first
author is supported by the NSFC grant (No.12201238) and GuangDong Basic and
Applied Basic Research Foundation (Grant No. 2022A1515111056). The second
author is supported by the NSFC grant (No. 12301111).

##  1\. Introduction

In this paper, we want to give geometrical characterizations to domains which
support local Sobolev-Poincaré imbeddings. If we do not explain deliberately,
we always assume domains are bounded. First, we review some existing results
about domains that support (global) Sobolev-Poincaré imbeddings. If a bounded
domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ has some sufficiently nice
geometrical property, for example the boundary of ${\Omega}$ is smooth, the
classical Sobolev-Poincaré imbeddings for this bounded domain depend on
weather the exponent $p$ is less than, equal to, or larger than the dimension
$n$. In the case $1\leq p<n$, for every function $u$ in the Sobolev class
$W^{1,p}({\Omega})$, we have the following Sobolev-Poincaré inequality

(1.1)
${}\inf\limits_{c\in{\mathbb{R}}}\left(\int_{\Omega}\left|u(x)-c\right|^{\frac{np}{n-p}}dx\right)^{\frac{n-p}{np}}\leq
C\left(\int_{\Omega}|\nabla u(x)|^{p}dx\right)^{\frac{1}{p}},$

with a constant $C$ independent of $u$. By making use of the triangle
inequality, the inequality is equivalent to the following inequality

(1.2)
$\left(\int_{\Omega}|u(x)-u_{\Omega}|^{\frac{np}{n-p}}dx\right)^{\frac{n-p}{np}}\leq
C\left(\int_{\Omega}|\nabla u(x)|^{p}dx\right)^{\frac{1}{p}}$

with maybe a different constant $C$ which is also independent of $u$. This
inequality was first proved by Sobolev in [33, 34] for $1<p<n$; for $p=1$, it
was proved by Gagliardo [7] and Nirenberg [31] (also see [28] by Maz’ya).

For $p=n$, for sufficiently nice domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$
and every $u\in W^{1,n}({\Omega})$, one gets the following Trudinger
inequality [36],

(1.3) $\|u-u_{\Omega}\|_{\phi(L)({\Omega})}\leq C\left(\int_{\Omega}|\nabla
u(x)|^{n}dx\right)^{\frac{1}{n}}$

with a constant $C$ independent of $u$. Here

$\phi(t)=\exp\left(t^{\frac{n}{n-1}}\right)-1$

and $\|\cdot\|_{\phi(L)({\Omega})}$ is the corresponding Orlicz norm on
${\Omega}$ defined by

$\|f\|_{\phi(L)({\Omega})}:=\inf\left\\{s>0:\int_{\Omega}\phi\left(\frac{|f(x)|}{s}\right)dx\leq
1\right\\}.$

It is well-known that the inequality (1.3) is equivalent to the following
inequality

(1.4) $\inf\limits_{c\in{\mathbb{R}}}\|u-c\|_{\phi(L)({\Omega})}\leq
C\left(\int_{\Omega}|\nabla u(x)|^{n}dx\right)^{\frac{1}{n}}$

with maybe a different constant $C$ independent of $u$, see [29].

For $n<p<\infty$, for certain domains ${\Omega}$ (including all balls) and
$u\in W^{1,p}({\Omega})$, one gets the following Hölder imbedding inequality
(Morry’s inequality)

(1.5) $|u(x_{1})-u(x_{2})|\leq
C|x_{1}-x_{2}|^{1-\frac{n}{p}}\left(\int_{\Omega}|\nabla(x)|^{p}dx\right)^{\frac{1}{p}}$

for every $x_{1},x_{2}\in{\Omega}$ and a constant $C$ independent of $u,x_{1}$
and $x_{2}$. To be more precise, the inequality (1.5) holds for a
representative of $u$. To simplify the notation, we still use $u$ for this
representative.

As we know, Sobolev-Poincaré imbedding inequalities (1.2), (1.3) and (1.5)
hold on sufficiently nice domains (like balls). We say a domain is a Sobolev-
Poincaré imbedding domain of order $p$, if the corresponding Sobolev-Poincaré
inequality holds on the domain. Obviously, $W^{1,p}$-extension domains for
$1\leq p<\infty$ are Sobolev-Poincaré imbedding domains of order $p$. A domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called a $W^{1,p}$-extension domain,
if every function in $W^{1,p}({\Omega})$ can be extended to be a function in
$W^{1,p}({{{\mathbb{R}}}^{n}})$ whose norm is controlled from above uniformly
by the norm of the original function. Then one natural question that arises is
if can we find geometric characterization of Sobolev-Poincaré imbedding
domains or not.

For $1\leq p<n$, Bojarski proved that John domains are Sobolev-Poincaré
imbedding domains of order $p$, see [2]. Also see [13] for this result.
Roughly, a domain is called a John domain, if every point in the domain can be
connected to a center point by a twisted cone. Please see the precise
definition below. Conversely, under some very weak connecting assumption that
is the slice condition, Buckley and Koskela [4] proved that a Sobolev-Poincaré
imbedding domain of order $p$ for some $1\leq p<n$ must be a John domain. By
the result in [10] proved by Gehring and Martio, if a domain is
quasiconformally equivalent to the unit ball, it satisfies the slice condition
(see definition below). The Riemann Mapping Theorem implies that every proper
simply connected domain in ${\mathbb{R}}^{2}$ satisfies the slice condition.
Indeed, every finitely connected planar domain still satisfies the slice
condition. For $p=n$, Smith and Stegenga proved that $0$-carrot domains are
Sobolev-Poincaré imbedding domains of order $n$. Conversely, Buckley and
Koskela proved that if a domain satisfying the slice condition is a Sobolev-
Poincaré imbedding domain of order $n$, then it must be a $0$-carrot domain.
See the precise definition of $\alpha$-carrot domain for $0\leq\alpha<1$
below. By the definition, it is easy to verify that John domains are
$0$-carrot domains. For $n<p<\infty$, under the assumption that the domain
satisfies the slice condition, Buckley and Koskela [4] proved that a domain is
a Sobolev-Poincaré imbedding domain of order $p$ if and only if it is an
$\alpha$-cigar domain for $\alpha=(p-n)/(p-1)$. See the precise definition of
cigar domains below. In the paper [4], Buckley and Koskela always said weak
carrot domains and weak cigar domains. To simplify the notation, we just say
carrot domains and cigar domains here. In [40], Zhou generalized this result
to the fractional Hajłasz-Sobolev spaces.

As we explained above, the geometric characterization for Sobolev-Poincaré
imbedding domains is already clear. However, one can also consider the local
version of this result. To be more precise, what is the geometric
characterization of local Sobolev-Poincaré imbedding domains? A domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called a local Sobolev-Poincaré
imbedding domain of order $p$, if it satisfies a local version of one of the
corresponding inequalities (1.1), (1.4) and (1.5). To be more precise, a
domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called a local Sobolev-
Poincaré imbedding domain of order $p$ for $1\leq p<n$, if for every
$u\in{W}^{1,p}(\Omega)$ and every $x\in\Omega$ and
$0<r<{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})$, the inequality

(1.6)
$\displaystyle\inf\limits_{c\in\mathbb{R}}\left(\int_{B(x,r)\cap{\Omega}}|u(y)-c|^{\frac{np}{n-p}}dy\right)^{\frac{n-p}{np}}\leq
C_{1}\left(\int_{B(x,C_{2}r)\cap{\Omega}}|\nabla
u(y)|^{p}dy\right)^{\frac{1}{p}}$

holds with constants $C_{1},C_{2}\geq 1$ independent of $u$ and $B(x,r)$; for
$p=n$, if for every $u\in{W}^{1,n}(\Omega)$ and every $x\in\Omega$ and
$0<r<{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})$, the inequality

(1.7)
$\displaystyle\inf\limits_{c\in\mathbb{R}}\int_{B(x,r)\cap\Omega}\exp\left(\frac{A|u(y)-c|}{\|\nabla
u\|_{L^{n}(B(x,C_{2}r)\cap\Omega)}}\right)^{\frac{n}{n-1}}\,dy\leq
C_{1}|B(x,r)|$

holds with constants $C_{1},C_{2}\geq 1$ and $A>0$ independent of $u$ and
$B(x,r)$; for $n<p\leq\infty$, if for every $u\in{W}^{1,p}(\Omega)$ and every
$x\in\Omega$ and $0<r<{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})$, the
inequality

(1.8) $\displaystyle|u(x_{1})-u(x_{2})|\leq
C_{1}|x_{1}-x_{2}|^{1-\frac{n}{p}}\left(\int_{B(x,C_{2}r)\cap{\Omega}}|\nabla
u(y)|^{p}dy\right)^{\frac{1}{p}}.$

holds for almost every $x_{1},x_{2}\in B(x,r)\cap{\Omega}$ and constants
$C_{1},C_{2}\geq 1$ independent of $u$ and $B(x,r)$.

For bounded domains, by taking the radius $r$ to be large enough, one can see
that a local Sobolev-Poincaré imbedding domain of order $p$ is also a (global)
Sobolev-Poincaré imbedding domain of order $p$, for every $1\leq p<\infty$.
But the converse is not correct for $1\leq p\leq n$. The planar slit disk is
the unit disk with one radial segment removed. The slit disk is a typical John
domain with slice condition but which is not have a local Sobolev-Poincaré
imbedding domain of order $p$ for any $1\leq p<\infty$. The following theorem
states that uniform domains should be a correct characterization to local
Sobolev-Poincaré imbedding domains of order $p$ for $1\leq p\leq n$. Uniform
domains were first introduced by Martio and Sarvas [27]. Roughly, a domain is
called a uniform domain, if arbitrary two points inside the domain can be
connected by a twisted double cone whose size is comparable to the distance
between these two points. Please see the precise definition below. In [19],
Jones proved that uniform domains are $W^{k,p}$-extension domains for
arbitrary $k\in\mathbb{N}$ and $1\leq p\leq\infty$. Inspired by the results
due to Bojarski [2], Smith and Stegenga [32], Buckley and Koskela [4, 5], a
local Sobolev-Poincaré imbedding domain of order $p$, for $1\leq p\leq n$,
locally should be not far from a John domain (for $1\leq p<n$) or a $0$-carrot
domain (for $p=n$). From this point of view, one can expect it should be a
uniform domain, see Lemma 2.2 below and [37, Lemma 1.9]. Here we will prove
uniform domains are local Sobolev-Poincaré imbedding domains of order $p$ for
every $1\leq p\leq n$ and a local Sobolev-Poincaré imbedding domain of order
$p$ for $1\leq p\leq n$ is locally linearly connected $(LLC)$. $(LLC)$ is
slightly weaker than the uniform condition. See the definition of $(LLC)$ in
next section. A uniform domain must be $(LLC)$; conversely, with some extra
weak connecting assumption, a $(LLC)$ domain is also a uniform domain, see
[3].

######  Theorem 1.1.

If ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is a uniform domain, then it is also
a local Sobolev-Poincaré imbedding domain of order $p$ for every $1\leq p\leq
n$. Conversely, a local Sobolev-Poincaré imbedding domain of order $p$ for
$1\leq p\leq n$ is locally linearly connected ($LLC$).

By the result in [5, Theorem 3.2], a domain quasiconformally equivalent to a
uniform domain satisfies the slice condition. By the Rieman Mapping Theorem,
every proper simply connected domain in $\mathbb{R}^{2}$ is conformally
equivalent to the unit disk. For example, see [1]. Obviously, the unit disk is
a uniform domain. Hence, the second part of Theorem 1.1 holds for every proper
simply connected domain in ${\mathbb{R}}^{2}$.

By the definitions, one can easily check that the class of uniform domains is
a proper subclass of the class of John domains or the class of carrot domains.
For example, the slit disk (the unit disk minus a radial segment) in the plane
is both a John domain and a carrot domain. However, it cannot be a uniform
domain. Hence, for $1\leq p\leq n$, the class of local Sobolev-Poincaré
imbedding domains of order $p$ does not equal to the class of (global)
Sobolev-Poincaré imbedding domains of order $p$. However, the next theorem
says they are the same for $n<p<\infty$.

######  Theorem 1.2.

Let ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ be a bounded domain which satisfies
the slice condition with respect to every point $x\in{\Omega}$. Then
${\Omega}$ is a local Sobolev-Poincaré imbedding domain of order $p$ for
$n<p<\infty$ if and only if it is an $\alpha$-cigar domain with
$\alpha=(p-n)/(p-1)$.

The paper is arranged as following. In section $2$, we give definitions,
recall some known results and prove some lemmas which will be used later. In
section 3, we prove Theorem 1.1 and Theorem 1.2.

##  2\. Preliminary

Let us start this section by giving some definitions. First, let us recall the
definition of John domains.

######  Definition 2.1.

A domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called a $c_{0}$-John domain
for some constant $0<c_{0}\leq 1$, if it is bounded and there exists a
distinguished center point $x_{0}\in{\Omega}$ such that for every
$x\in{\Omega}$, there exists a path $\gamma:[0,l]\to{\Omega}$ parametrized by
arclength such that $\gamma(0)=x$, $\gamma(l)=x_{0}$ such that for every
$t\in[0,l]$, we have

${\mathop{\mathrm{missing}}{\,dist\,}}(\gamma(t),\partial{\Omega})\geq
c_{0}t.$

Roughly, every point inside a John domain can be connected to a center by a
twisted cone. Next we give the definition of uniform domain.

######  Definition 2.2.

A domain $\Omega\subset{{{\mathbb{R}}}^{n}}$ is called $\epsilon_{0}$-uniform
for $0<\epsilon_{0}\leq 1$, if for all $x,y\in\Omega$, there exists a
rectifiable curve $\gamma\subset\Omega$ joining $x,y$ with

$l(\gamma)\leq\frac{1}{\epsilon_{0}}|x-y|,$

and for every $z\in\gamma$, we have

$d(z,\partial\Omega)\geq\epsilon_{0}\frac{|x-z||y-z|}{|x-y|}.$

Roughly, every pair of points inside the uniform domain can be connected by a
twisted double cone. By the definition, it is easy to see that every uniform
domain is also a John domain. The planar slit disk implies the converse is not
correct. Indeed, to prove a local Sobolev-Poincaré imbedding domain of order
$p$ for $1\leq p\leq n$ is also a uniform domain, we will prove that it
satisfies the following locally linear connection (for short $LLC$).

######  Definition 2.3.

We say a domain $\Omega\subset{{{\mathbb{R}}}^{n}}$ satisfies locally linear
connection (for short $LLC$) if there exists a constant $b\in(0,1]$ such that
for any $z\in{{{\mathbb{R}}}^{n}}$ and $r>0$, we have

  1. $LLC(1)$: Points in $\Omega\cap B(z,r)$ can be jointed by a locally rectifiable curve in $\Omega\cap B(z,\frac{r}{b})$;

  2. $LLC(2)$: Points in $\Omega\setminus B(z,r)$ can be jointed by a locally rectifiable curve in $\Omega\setminus B(z,br)$.

A homeomorphism
$h:{\Omega}\subset{{{\mathbb{R}}}^{n}}\to{\Omega}^{\prime}\subset{{{\mathbb{R}}}^{n}}$
in the class $W^{1,n}_{\rm loc}({\Omega},{\Omega}^{\prime})$ is called a
quasiconformal mapping, if there exists a constant $K>1$ such that for almost
every $z\in{\Omega}$, we have

$|Dh(z)|^{n}\leq K|J_{h}(z)|,$

where $|Dh(z)|$ means the operator norm of the differential matrix $Dh(z)$ and
$|J_{h}(z)|$ is the absolutely value of the Jacobian determinate. A domain
${\Omega}$ is called quasiconformally equivalent to another domain
${\Omega}^{\prime}$, if there exists a quasiconformal mapping from ${\Omega}$
onto ${\Omega}^{\prime}$. According to results [10] by Gehring and Martio, a
domain quasiconformally equivalent to a uniform domain is $LLC$; conversely,
if a $LLC$ domain has also the following slice property, it is also a uniform
domain. Now, let us give the definition to slice property.

######  Definition 2.4.

Let ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ be a domain with a distinguished
point $x_{0}$ and $C>1$ be a constant. We say that ${\Omega}$ has the
$C$-slice property with respect to $x_{0}$, if for every $x\in{\Omega}$, there
is a path $\gamma:[0,1]\to{\Omega}$ with $\gamma(0)=x_{0}$ and $\gamma(1)=x$,
and there is a pairwise disjoint collection of open subsets
$\\{U_{i}\subset{\Omega}:i=0,\cdots,j\\}$ such that:

  1. (1)

$x_{0}\in U_{0}$, $x\in U_{j}$, and $x_{0}$ and $x$ are in different
components of ${\Omega}\setminus\overline{U_{i}}$ for all $0<i<j$;

  2. (2)

If $\tilde{\gamma}\subset\subset{\Omega}$ is a curve containing both $x$ and
$x_{0}$, then for all $0<i<j$, we have

${\mathop{\mathrm{missing}}{\,diam\,}}(U_{i})\leq Cl(\tilde{\gamma}\cap
U_{i});$

  3. (3)

For every $t\in[0,1]$, we have

$B\left(\gamma(t),\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(\gamma(t),\partial{\Omega})}{C}\right)\subset\bigcup_{i=0}^{j}\overline{U_{i}};$

  4. (4)

For every $0\leq i\leq j$, we have

${\mathop{\mathrm{missing}}{\,diam\,}}(U_{i})\leq
C{\mathop{\mathrm{missing}}{\,dist\,}}(\gamma(t),\partial{\Omega})$

for all $\gamma(t)\in\gamma\cap U_{i}.$ Also there exists $x_{i}\in\gamma\cap
U_{i}$ as $x_{0}$ is previously defined and $x_{j}=x$ such that

$B\left(x_{i},\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x_{i},\partial{\Omega})}{C}\right)\subset
U_{i}.$

We say that ${\Omega}$ has a slice property if it has the $C$-slice property
for some $C>1$. The following lemma was proved by Buckley and Herron in [3].

######  Lemma 2.1.

A domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is a uniform domain if and only
if it is quasiconvex, $LLC$ and satisfies the slice condition with respect to
every $x\in{\Omega}$.

Let us give the definition of cigar domains and carrot domains. We follow the
definition given by Buckley and Koskela in [5] where they called them weak
cigar domains and weak carrot domains. To simplify the notation, we just say
cigar domains and carrot domains. First, let us give the definition to cigar
domains. Let us give the definition of cigar domains.

######  Definition 2.5.

Let $0<\alpha<1$ and $0<\beta\leq\alpha$. A domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called an $(\alpha,\beta)$-cigar
domain, if for every pair of points $x,y\in{\Omega}$, there exists a path
$\gamma:[0,1]\to{\Omega}$ such that $\gamma(0)=x$, $\gamma(1)=y$ and the
inequality

$\int_{\gamma}{\mathop{\mathrm{missing}}{\,dist\,}}(z,\partial{\Omega})^{\alpha-1}|dz|\leq
C|x-y|^{\beta}$

holds with a constant $C$ independent of $x,y$. A domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called a $(0,0)$-cigar domain, if for
every pair of points $x,y\in{\Omega}$, there exists a path
$\gamma:[0,1]\to{\Omega}$ such that $\gamma(0)=x$, $\gamma(1)=y$ and the
inequality

$\int_{\gamma}\frac{1}{{\mathop{\mathrm{missing}}{\,dist\,}}(z,\partial{\Omega})}|dz|\leq
C\log\left(1+\frac{|x-y|}{\min\\{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\partial{\Omega}),{\mathop{\mathrm{missing}}{\,dist\,}}(y,\partial{\Omega})\\}}\right)$

holds with a constant $C$ independent of $x,y$.

To simplify the notation, we use $\alpha$-cigar domain for
$(\alpha,\alpha)$-cigar domain. The class of $(0,0)$-cigar domains coincides
with the class of uniform domains, see [8, 18]. It was shown in [9] that when
$\alpha>0$, the class of $(\alpha,\beta)$-cigar domains is strictly larger
than the class of uniform domains. In [26], Lappalainen proved that every
$\alpha$-Hölder continuous function defined on an $(\alpha,\beta)$-cigar
domain can be extended to be a global $\beta$-Hölder continuous function. Now,
let us give the definition of carrot domains.

######  Definition 2.6.

Let $0\leq\alpha<1$. A domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called
an $\alpha$-carrot domain, if there exists a distinguished center
$x_{0}\in{\Omega}$ such that for every $x\in{\Omega}$, there exists a path
$\gamma:[0,1]\to{\Omega}$ such that $\gamma(0)=x$, $\gamma(1)=1$ and the
inequality

(2.1)
$\int_{\gamma}{\mathop{\mathrm{missing}}{\,dist\,}}(z,\partial{\Omega})^{\alpha-1}|dz|\leq\begin{cases}C,&\
\ {\rm if}\ 0<\alpha<1,\\\
C\left(1+\log\left(\frac{C_{1}}{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\partial{\Omega})}\right)\right),&\
\ {\rm if}\ \alpha=0,\end{cases}$

holds for constants $C,C_{1}$ independent of $x$.

By the definition, for arbitrary $0\leq\alpha<1$, an $\alpha$-carrot domain
must be bounded. It is easy to verify that John domains are $\alpha$-carrot
domains for all $0\leq\alpha<1$. The converse is not correct. It is not hard
to construct a carrot domain which is not a John domain, see [9, 5]. The
condition in the definition of carrot domains is widely known as
quasihyperbolic boundary condition, see [21] and references therein.

The definitions of uniform domains and John domains show these two classes of
domains are closely relative. The next lemma shows that a uniform domain is
not far from John domain locally. The proof of the case
${\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})=\infty$ was already given in
[37, Lemma 1.9]. It seems their argument is not easy to extend to the bounded
case. For the completeness and the convenience of readers, we give a proof
below for the case ${\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})<\infty$.

######  Lemma 2.2.

Suppose that $\Omega$ is an $\epsilon_{0}$-uniform domain with
${\mathop{\mathrm{missing}}{\,diam\,}}\Omega<\infty$. Given any
$x_{0}\in\Omega$ and $r>0$, there is a $c_{0}$-John domain
$\Omega_{r}\subset\Omega$ and a constant $C_{2}>1$ such that

$B(x_{0},r)\cap\Omega\subset\Omega_{r}\subset B(x_{0},C_{2}r)\cap\Omega.$

Moreover, the constant $c_{0}$ only depends on the constant $\epsilon_{0}$.

###### Proof.

Assume there exists a point $y_{0}\in\Omega$ such that
$|x_{0}-y_{0}|\geq\frac{4r}{\epsilon_{0}}$. Otherwise, let
$C_{2}=\frac{4}{\epsilon_{0}}$, we have

$B(x_{0},C_{2}r)\cap\Omega=\Omega.$

The result in [9, Lemma 2.18] says that a bounded uniform domain is always a
John domain. Hence, we can just take $\Omega_{r}=\Omega$ to obtain

$B(x_{0},r)\cap\Omega\subset\Omega_{r}\subset B(x_{0},C_{2}r)\cap\Omega.$

In the other case, fix $y_{0}\in\Omega$ with
$|x_{0}-y_{0}|=\frac{4r}{\epsilon_{0}}$. There exists a path
$\gamma_{0}:[0,1]\to{\Omega}$ with $\gamma_{0}(0)=x_{0}$,
$\gamma_{0}(1)=y_{0}$ and $\gamma_{0}$ satisfies the conditions in Definition
2.2. Set

$\displaystyle
G_{0}:=\left\\{z\in\Omega\,:\exists\,v\in\gamma_{0}\,such\,\,that\,|z-v|<\frac{\epsilon_{0}|x_{0}-v||y_{0}-v|}{|x_{0}-y_{0}|}\right\\}.$

Then $G_{0}\subset\Omega$ contains a point $z_{0}\in\gamma_{0}$ with
$|x_{0}-z_{0}|=\frac{2r}{\epsilon_{0}}$ and $B(z_{0},r)\subset G_{0}$. Indeed,
the triangle inequality gives

$\displaystyle\epsilon_{0}\frac{|x_{0}-z_{0}||y_{0}-z_{0}|}{|x_{0}-y_{0}|}\geq\epsilon_{0}\frac{|x_{0}-z_{0}|(|x_{0}-y_{0}|-|x_{0}-z_{0}|)}{|x_{0}-y_{0}|}\geq
r.$

Fix $x\in(B(x_{0},r)\cap\Omega)\backslash B(z_{0},r)$. There exists a path
$\gamma_{x}:[0,1]\to{\Omega}$ with $\gamma_{x}(0)=z_{0}$ and $\gamma_{x}(1)=x$
and the path $\gamma_{x}$ satisfies the conditions in Definition 2.2. Similar
to $G_{0}$, define a domain $G_{x}\subset\Omega$ by setting

$\displaystyle
G_{x}:=\left\\{z\in\Omega\,:\,\exists\,v\in\gamma_{x}\,such\,that\,|v-z|<\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}\right\\}.$

Next we prove that

$\displaystyle\Omega_{r}:=\bigcup_{x\in(B(x_{0},r)\cap\Omega)\backslash
B(z_{0},r)}G_{x}\cup B(z_{0},r)$

is a John domain. For every $x\in(B(x_{0},r)\cap{\Omega})\setminus
B(z_{0},r)$, we have

$r\leq|x-z_{0}|\leq|x-x_{0}|+|x_{0}-z_{0}|\leq
r\left(1+\frac{2}{\epsilon_{0}}\right).$

Hence, for every $z\in G_{x}$, there exists $v\in\gamma_{x}$ such that

$\displaystyle|x_{0}-z|\leq|x_{0}-x|+|x-v|+|v-z|.$

Obviously, $|x_{0}-x|<r$. Denote $\gamma_{x}^{[x,v]}$ to be the subcurve of
$\gamma_{x}$ from $x$ to $v$ and $\gamma_{x}^{[v,z_{0}]}$ to be the subcurve
of $\gamma_{x}$ from $v$ to $z_{0}$. Since $\gamma_{x}$ satisfies the
conditions in Definition 2.2, we have

$|x-v|\leq l\left(\gamma_{x}^{[x,v]}\right)\leq
l(\gamma_{x})\leq\frac{1}{\epsilon_{0}}|x-z_{0}|.$

Since

$l\left(\gamma_{x}^{[x,v]}\right)+l\left(\gamma_{x}^{[v,z_{0}]}\right)=l(\gamma_{x}),$

by the definition of the set $G_{x}$, we have

$|v-z|<\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}\leq\frac{\epsilon_{0}l\left(\gamma_{x}^{[x,v]}\right)l\left(\gamma_{x}^{z_{0},v}\right)}{|x-z_{0}|}\leq\frac{\epsilon_{0}\left(l(\gamma_{x})\right)^{2}}{4|x-z_{0}|}\leq\frac{\epsilon_{0}}{4}|x-z_{0}|.$

Combine these inequalities together, we obtain

$|x_{0}-z|\leq
r+\frac{1}{\epsilon_{0}}|x-z_{0}|+\frac{\epsilon_{0}}{4}|x-z_{0}|\leq
r\left(1+\frac{(\epsilon_{0}+4)^{2}}{4\epsilon_{0}^{2}}\right).$

Hence,

$\Omega_{r}\subset
B\left(x_{0},r\left(1+\frac{(\epsilon_{0}+4)^{2}}{4\epsilon_{0}^{2}}\right)\right)\cap\Omega.$

Now, let us provethat ${\Omega}_{r}$ is a John domain with the distinguished
center $z_{0}$. If $z\in B(z_{0},r)$, it is not difficult to check that the
segment $[z,z_{0}]$ between $z$ and $z_{0}$ satisfies the condition in
Definition 2.1 with the constant $c=1$. Let us consider the case that $z\in
G_{x}\backslash B(z_{0},r)$ for some $x\in(B(x_{0},r)\cap{\Omega})\setminus
B(z_{0},r)$. By the definition of $G_{x}$, there exists a point
$v\in\gamma_{x}$ such that

$z\in B\left(v,\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}\right)\subset
G_{x}.$

So $[z,v]\subset G_{x}$ and

$[z,v]\cup\gamma^{[v,z_{0}]}_{x}\subset{\Omega}_{r}$

is a rectifiable curve from $z$ to $z_{0}$. There exists a unique $t\in(0,1)$
such that

$|\gamma_{x}(t)-v|=\frac{\epsilon_{0}|x-v||z_{0}-v|}{2|x-z_{0}|}$

and for every $\tilde{t}\in(0,t)$, we have

$|\gamma_{x}(\tilde{t})-v|>\frac{\epsilon_{0}|x-v||z_{0}-v|}{2|x-z_{0}|}.$

Denote $w:=\gamma_{x}(t)$. Then

$\tilde{\gamma}:=[z,v]\cup[v,w]\cup\gamma_{x}^{[w,z_{0}]}\subset{\Omega}_{r}$

is also a rectifiable curve from $z$ to $z_{0}$.

Now, let us check the curve $\tilde{\gamma}$ satisfies the conditions in
Definition 2.1 with a constant $c=\frac{6(\epsilon_{0}+2)}{\epsilon_{0}^{3}}$
depending only on $\epsilon_{0}$. For every $s\in[z,v]\subset\tilde{\gamma}$,
since $G_{x}\cup B(z_{0},r)\subset{\Omega}_{r}$ and

$z\in B\left(v,\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}\right)\subset
G_{x},$

we have

$\displaystyle d(s,\partial{\Omega}_{r})$ $\displaystyle\geq
d(s,\partial(G_{x}\cup B(z_{0},r)))\geq d(s,\partial G_{x})$
$\displaystyle\geq\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}-|s-v|$
$\displaystyle\geq\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}-\left(|z-s|-|v-z|\right)$
$\displaystyle\geq|z-s|=l(\tilde{\gamma}^{[z,s]})\geq\frac{\epsilon_{0}^{3}l(\tilde{\gamma}^{[z,s]})}{6(\epsilon_{0}+2)}.$

For every $s\in[v,w]$ we know

$\displaystyle d(s,\partial{\Omega}_{r})$ $\displaystyle\geq
d(s,\partial(G_{x}\cup B(z_{0},r)))\geq d\left(s,\partial
B\left(v,\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}\right)\right)$
$\displaystyle\geq\frac{\epsilon_{0}|x-v||z_{0}-v|}{|x-z_{0}|}-|v-s|\geq\frac{\epsilon_{0}|x-v||z_{0}-v|}{2|x-z_{0}|}$
$\displaystyle\geq\frac{1}{3}\left(|z-v|+|v-s|\right)>\frac{\epsilon_{0}^{3}l(\tilde{\gamma}^{[z,s]})}{6(\epsilon_{0}+2)}.$

For every $s\in\tilde{\gamma}^{[w,z_{0}]}=\gamma^{[w,z_{0}]}_{x}$, we have

$|s-v|\geq\frac{\epsilon_{0}|x-v||z_{0}-v|}{2|x-z_{0}|},$

we have

$|v-z|\leq 2|s-v|\leq 2l(\tilde{\gamma}^{[v,s]}).$

We have

$l(\tilde{\gamma}^{[z,s]})\leq
l([z,v])+l(\gamma_{x}^{[x,s]})-l\gamma_{x}^{[x,v]}\leq
3l(\gamma_{x}^{[x,s]}).$

If $|s-z_{0}|\geq\frac{r}{2}$, then we have

$\displaystyle d(s,\partial{\Omega}_{r})$ $\displaystyle\geq
d(s,\partial(G_{x}\cup B(z_{0},r)))\geq d(s,\partial G_{x})$
$\displaystyle\geq\frac{\epsilon_{0}|x-s||z_{0}-s|}{|x-z_{0}|}\geq\frac{\epsilon_{0}^{2}|x-s|}{2(\epsilon_{0}+2)}\geq\frac{\epsilon_{0}^{3}l(\gamma_{x}^{[x,s]})}{6(\epsilon_{0}+2)}\geq\frac{\epsilon_{0}^{3}l(\tilde{\gamma}^{[z,s]})}{6(\epsilon_{0}+2)}.$

If $|s-z_{0}|\leq\frac{r}{2}$, since $B(z_{0},r)\subset\Omega$, we have

$\displaystyle d(s,\partial(G_{x}\cup B(z_{0},r)))\geq d(s,\partial
B(z_{0},r))\geq r-|s-z_{0}|\geq\frac{r}{2}.$

Combine it with

$|x-z_{0}|\leq r\left(1+\frac{2}{\epsilon_{0}}\right),$

we have

$\displaystyle d(s,\partial{\Omega}_{r})$ $\displaystyle\geq
d(s,\partial(G_{x}\cup B(z_{0},r)))\geq\frac{r}{2}$
$\displaystyle\geq\frac{\epsilon_{0}|x-z_{0}|}{2(\epsilon_{0}+2)}\geq\frac{\epsilon_{0}^{2}l(\gamma_{x})}{2(\epsilon_{0}+2)}\geq\frac{\epsilon_{0}^{3}l(\tilde{\gamma}^{[z,s]})}{6(\epsilon_{0}+2)}.$

Combine all the estimates above, we can take
$c_{0}={\epsilon_{0}^{3}}/{6(\epsilon_{0}+2)}$ and obtain the desired result.
∎

Let ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ be a domain. We define the intrinsic
metric $d_{\Omega}$ on ${\Omega}$ by setting

$d_{\Omega}(x_{1},x_{2}):=\left\\{l(\gamma):\gamma\subset{\Omega}\ {\rm is\ a\
locally\ rectifiable\ curve\ joining}\ x_{1}\ {\rm and}\ x_{2}\right\\}.$

For arbitrary $x\in{\Omega}$ and $r>0$, we define the intrinsic ball
$B_{\Omega}(x,r)$ by setting

$B_{\Omega}(x,r):=\left\\{y\in{\Omega}:d_{\Omega}(x,y)<r\right\\}$

and define the intrinsic boundary $\partial_{\Omega}B_{\Omega}(x,r)$ of the
intrinsic ball $B_{\Omega}(x,r)$ by setting

$\partial_{\Omega}B_{\Omega}(x,r):=\left\\{y\in{\Omega}:d_{\Omega}(x,y)=r\right\\}.$

Obviously, $B_{\Omega}(x,r)$ is a subset of $B(x,r)\cap{\Omega}$. A domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is called Ahlfors $n$-regular, if there
exists a constant $0<c<1$ and some $r_{o}>0$ such that for every
$x\in\overline{\Omega}$ and every $0<r<r_{o}$, we have

$|B(x,r)\cap{\Omega}|\geq c|B(x,r)|.$

Similarly, we say that ${\Omega}$ is intrinsic Ahlfors $n$-regular, if there
exists a constant $0<c<1$ and some $r_{o}>0$ such that for every
$x\in\overline{\Omega}$ and $0<r<r_{o}$, we have

$|B_{\Omega}(x,r)|\geq c|B(x,r)|.$

It is easy to see that intrinsic Ahlfors $n$-regularity can imply Ahlfors
$n$-regularity, since $B_{\Omega}(x,r)$ is a subset of $B(x,r)\cap{\Omega}$.

By the result in [15] due to Hajłasz, Koskela and Tuominen, we can conclude
that a local Sobolev-Poincaré imbedding domain of order $p$ for $1\leq
p<\infty$ is Ahlfors $n$-regular. This observation may not come from their
result in the paper but comes from the proof in the paper directly.
Nevertheless, we will prove a stronger result below. Of course, the idea
behind the argument comes from the paper [15] by Hajłasz, Koskela and
Tuominen.

######  Lemma 2.3.

Let ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ be a local Sobolev-Poincaré
imbedding domain of order $p$ for $1\leq p<\infty$. Then ${\Omega}$ is
intrinsic Ahlfors $n$-regular.

###### Proof.

Fix $x\in{\Omega}$ and $0<r<r_{o}$ with

$r_{o}=\min\left\\{1,\left(\frac{|{\Omega}|}{\omega_{n}2}\right)^{\frac{1}{n}}\right\\},$

where $\omega_{n}$ means the volume of $n$-dimensional unit ball. Let
$b_{0}=1$ and $b_{j}\in(0,1)$ for $j\in N$ such that

(2.2)
$|B_{\Omega}(x,b_{j}r)|=2^{-1}|B_{\Omega}(x,b_{j-1}r)|=2^{-j}|B_{\Omega}(x,r)|.$

For each $j\geq 0$, define a function $u_{x,b_{j+1},b_{j}}$ on $\Omega$ by
setting

$\displaystyle u_{x,b_{j+1},b_{j}}(y):=\left\\{\begin{array}[]{ll}1,&y\in
B_{\Omega}(x,b_{j+1}r),\\\
\frac{b_{j}r-d_{\Omega}(y,x)}{b_{j}r-b_{j+1}r},&y\in
B_{\Omega}(x,b_{j}r)\setminus B_{\Omega}(x,b_{j+1}r),\\\
0,&elsewhere.\end{array}\right.$

Some simple computation gives

(2.3) $|\nabla
u_{x,b_{j+1},b_{j}}(y)|\leq\begin{cases}\frac{1}{b_{j}r-b_{j+1}r},\ &\ y\in
B_{\Omega}(x,b_{j}r)\setminus B_{\Omega}(x,b_{j+1}r),\\\ 0,\ &\ {\rm
elsewhere}.\end{cases}$

Some simple computation gives

$\int_{\Omega}|u_{x,b_{j+1},b_{j}}(y)|^{p}dy\leq|B_{{\Omega}}(x,b_{j}r)|.$

For every $C_{2}\geq 1$, we have

(2.4) $\displaystyle\left(\int_{\Omega}|\nabla
u_{x,b_{j+1},b_{j}}(y)|^{p}\,dy\right)^{\frac{1}{p}}=\left(\int_{B(x,C_{2}r)\cap\Omega}|\nabla
u_{x,b_{j+1},b_{j}}(y)|^{p}\,dy\right)^{\frac{1}{p}}\leq\frac{|B_{\Omega}(x,b_{j}r)|^{\frac{1}{p}}}{b_{j}r-b_{j+1}r}.$

Hence, $u_{x,b_{j+1},b_{j}}\in W^{1,p}({\Omega})$. According to the
relationship between $p$ and $n$, we divide the argument below into three
cases.

Case ($1\leq p<n$):  Let $j\geq 0$. For each $c\in\mathbb{R}$, by the
definition of $u_{x,b_{j+1},b_{j}}$, we have

$u_{x,b_{j+1},b_{j}}-c\geq\frac{1}{2}$

either on $B_{\Omega}(x,b_{j+1}r)$ or on $\Omega\setminus
B_{\Omega}(x,b_{j}r)$. Since $0<r<r_{o}$, obviously,

$|{\Omega}\setminus B_{\Omega}(x,b_{j}r)|\geq|B_{\Omega}(x,b_{j+1}r)|.$

Hence, we have

(2.5)
$\inf\limits_{c\in\mathbb{R}}\left(\int_{B(x,r)\cap\Omega_{0}}|u_{x,b_{j+1},b_{j}}(y)-c|^{\frac{np}{n-p}}\,dy\right)^{\frac{n-p}{np}}\geq\frac{1}{2}|B_{\Omega}(x,b_{j+1}r)|^{\frac{n-p}{np}}.$

By combining (1.6), (2.2), (2.4) and (2.5), we obtain

(2.6)
$\displaystyle\left|B_{\Omega}(x,b_{j}r)\right|^{\frac{n-p}{np}}\leq\frac{C|B_{\Omega}(x,b_{j}r)|^{\frac{1}{p}}}{b_{j}r-b_{j+1}r},$

that is,

$\displaystyle b_{j}r-b_{j+1}r\leq C|B_{\Omega}(x,b_{j}r)|^{\frac{1}{n}}\leq
C2^{-j/n}|B_{\Omega}(x,r)|^{\frac{1}{n}}.$

Since $b_{j}\rightarrow 0$ as $j\rightarrow\infty$, we have

$\displaystyle r=\sum\limits_{j\geq 0}(b_{j}r-b_{j+1}r)\leq
C\sum\limits_{j\geq 0}2^{-j/n}|B_{\Omega}(x,r)|^{\frac{1}{n}}\leq
C|B_{\Omega}(x,r)|^{\frac{1}{n}}.$

It implies ${\Omega}$ is intrinsic Ahlfors $n$-regular.

Case ($p=n$):  For each $c\in{\mathbb{R}}$, we have

$|u_{x,b_{j+1},b_{j}}-c|\geq\frac{1}{2}$

either on $B_{\Omega}(x,b_{j+1}r)$ or on ${\Omega}\setminus
B_{\Omega}(x,b_{j}r)$. Since

$|{\Omega}\setminus B_{\Omega}(x,b_{j}r)|\geq|B_{\Omega}(x,b_{j+1}r)|,$

combine (2.4) and (1.7), we obtain

$|B_{\Omega}(x,b_{j}r)|\exp\left(\frac{A(b_{j}r-b_{j+1}r)}{2|B_{\Omega}(x,b_{j}r)|^{\frac{1}{n}}}\right)^{\frac{n}{n-1}}\leq
C_{1}|B(x,r)|.$

It implies that

$b_{j}-b_{j+1}\leqslant\frac{2|B_{\Omega}(x,b_{j}r)|^{\frac{1}{n}}}{Ar}\log^{\frac{n-1}{n}}\left(\frac{C_{1}|B(x,r)|}{|B_{\Omega}(x,b_{j}r)|}\right).$

This leads to

$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle\sum\limits_{j\geqslant
0}(b_{j}-b_{j+1})$ $\displaystyle\leq$ $\displaystyle C\sum\limits_{j\geqslant
0}\frac{|B_{\Omega}(x,b_{j}r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}\log^{\frac{n-1}{n}}\left(\frac{|B(x,r)|}{|B_{\Omega}(x,b_{j}r)|}\right)$
$\displaystyle\leq$ $\displaystyle C\sum\limits_{j\geqslant
0}2^{-j/n}\frac{|B_{\Omega}(x,r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}\log^{\frac{n-1}{n}}\left(\frac{2^{nj}|B(x,r)|}{|B_{\Omega}(x,r)|}\right)$
$\displaystyle\leq$ $\displaystyle C\sum\limits_{j\geq
0}2^{-j/n}\frac{|B_{\Omega}(x,r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}\left(\log(2^{jn})+\log\left(\frac{|B(x,r)|}{|B_{\Omega}(x,r)|}\right)\right)^{\frac{n-1}{n}}$
$\displaystyle\leq$ $\displaystyle C\sum\limits_{j\geq
0}2^{-j/n}\log^{\frac{n-1}{n}}(2^{jn})\frac{|B_{\Omega}(x,r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}$
$\displaystyle+C\sum\limits_{j\geq
0}2^{-j/n}\frac{|B_{\Omega}(x,r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}\log^{\frac{n-1}{n}}\left(\frac{|B(x,r)|^{\frac{1}{n}}}{|B_{{\Omega}}(x,r)|^{\frac{1}{n}}}\right)$
$\displaystyle\leq$ $\displaystyle
C\frac{|B_{\Omega}(x,r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}+C\frac{|B_{\Omega}(x,r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}\log^{\frac{n-1}{n}}\left(\frac{|B(x,r)|^{\frac{1}{n}}}{|B_{\Omega}(x,r)|^{\frac{1}{n}}}\right)=:A_{1}+A_{2}.$

Hence, at least one of $A_{1}$ and $A_{2}$ must be not less than
$\frac{1}{2}$. If $A_{1}\geq\frac{1}{2}$, then we have

$|B_{\Omega}(x,r)|\geq cr^{n}$

for some small constant $0<c<1$. It implies that ${\Omega}$ is intrinsic
Ahlfros $n$-regular. If $A_{2}\geq\frac{1}{2}$, then we have

$\frac{|B_{\Omega}(x,r)|^{\frac{1}{n}}}{|B(x,r)|^{\frac{1}{n}}}\log^{\frac{n-1}{n}}\left(\frac{|B(x,r)|^{\frac{1}{n}}}{|B_{\Omega}(x,r)|^{\frac{1}{n}}}\right)\geq\tilde{c}$

for a sufficiently small $0<\tilde{c}<1$. Since

$|B_{\Omega}(x,r)|\leq|B(x,r)|,$

it implies that there exists a small enough constant $c>0$ such that

$|B_{\Omega}(x,r)|\geq cr^{n}.$

Hence, ${\Omega}$ is intrinsic Ahlfors $n$-regular.

Case ($n<p<\infty$):  For every $j\geq 0$, fix a point
$x_{j}\in\partial_{\Omega}B_{\Omega}(x,b_{j}r)$. By the definition of
intrinsic metric, for arbitrary $\epsilon>0$, there exists a curve
$\gamma_{\epsilon}\subset{\Omega}$ joining $x$ and $x_{j}$ such that

$l(\gamma_{\epsilon})<b_{j}r+\epsilon.$

Choose one point $\tilde{x}_{j}\in\gamma_{\epsilon}$ such that

$l\left(\gamma_{\epsilon}^{[\tilde{x}_{j},x_{j}]}\right)=b_{j}r-b_{j+1}r+\epsilon.$

Then the triangle inequality gives

$\tilde{x}_{j}\in B_{\Omega}(x,b_{j+1}r)\ {\rm and}\ |x_{j}-\tilde{x}_{j}|\leq
b_{j}r-b_{j+1}r+\epsilon.$

Apply inequalities (1.8) and (2.4) to the function $u_{x,b_{j+1},b_{j}}$, we
obtain

$1\leq
C_{1}\left(b_{j}r-b_{j+1}r+\epsilon\right)^{1-\frac{n}{p}}\frac{|B_{\Omega}(x,b_{j}r)|^{\frac{1}{p}}}{b_{j}r-b_{j+1}r}.$

Since $\epsilon>0$ is arbitrary, we have

$b_{j}r-b_{j+1}r\leq C|B_{\Omega}(x,b_{j}r)|^{\frac{1}{n}}.$

Sum up $j$, we obtain

$\displaystyle b_{0}r=\sum\limits_{j\geq 0}(b_{j}r-b_{j+1}r)\leq
C\sum\limits_{j\geq 0}|B_{\Omega}(x,b_{j}r)|^{\frac{1}{n}}\leq
C\sum\limits_{j\geq 0}2^{-j/n}|B_{\Omega}(x,r)|^{\frac{1}{n}}.$

It implies that there exists a small enough constant $0<c<1$ such that

$|B_{\Omega}(x,r)|\geq cr^{n}.$

So ${\Omega}$ is intrinsic Ahlfors $n$-regular. ∎

##  3\. Proof of Theorem 1.1

###  3.1. From uniform condition to local Sobolev-Poincaré imbedding

In this section, we prove the sufficient part of Theorem 1.1. We will show a
uniform domain is also a local Sobolev-Poincaré imbedding domain of order $p$
for all $1\leq p\leq n$.

######  Theorem 3.1.

Let ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ be a uniform domain. Then it is also
a local Sobolev-Poincaré imbedding domain of order $p$ for every $1\leq p\leq
n$.

###### Proof.

Let $\Omega$ be a $\epsilon_{0}$-uniform domain for $0<\epsilon_{0}\leq 1$.
Fix $x_{0}\in{\Omega}$ and $0<r<\infty$. By Lemma 2.2 for the case
${\mathop{\mathrm{missing}}{\,diam\,}}\Omega<\infty$ and [37, Lemma 1.9] for
the case ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega=\infty$, there always
exists a $c_{0}$-John domain $\Omega_{r}\subset\Omega$ with $c_{0}$ only
depends on $\epsilon_{0}$ and

$B(x_{0},r)\cap\Omega\subset\Omega_{r}\subset B(x_{0},C_{2}r)\cap\Omega.$

Depending on if $p$ equals to $n$ or not, we divide the argument below into
two cases.

Case ($1\leq p<n$): Let $u\in{W}^{1,p}(\Omega)$ be arbitrary. Since
$B(x_{0},r)\cap\Omega\subset\Omega_{r}$, we have

$\displaystyle\inf\limits_{c\in\mathbb{R}}\left(\int_{B(x_{0},r)\cap\Omega}|u(x)-c|^{\frac{np}{n-p}}\,dx\right)^{\frac{n-p}{np}}$
$\displaystyle\leq\inf\limits_{c\in\mathbb{R}}\left(\int_{\Omega_{r}}|u(x)-c|^{\frac{np}{n-p}}\,dx\right)^{\frac{n-p}{np}}.$

Since ${\Omega}_{r}$ is a $c_{0}$-John domain with some constant $c_{0}$ only
depends on $\epsilon_{0}$, by the result in [2] by Bojarski, there exists
$C\geq 1$ independent of $u,x_{0}$ and $r$ such that

$\displaystyle\inf\limits_{c\in\mathbb{R}}\left(\int_{\Omega_{r}}|u(x)-c|^{\frac{np}{n-p}}\,dx\right)^{\frac{n-p}{np}}\leq
C\left(\int_{\Omega_{r}}|\nabla u(x)|^{p}\,dx\right)^{\frac{1}{p}}\leq
C\left(\int_{B(x_{0},C_{2}r)\cap\Omega}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}.$

Hence we have

$\displaystyle\inf\limits_{c\in\mathbb{R}}\left(\int_{B(x_{0},r)\cap\Omega}|u(x)-c|^{\frac{np}{n-p}}\,dx\right)^{\frac{n-p}{np}}\leq
C\left(\int_{B(x_{0},C_{2}r)\cap\Omega}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}.$

It implies ${\Omega}$ is a local Sobolev-Poincaré imbedding domain of order
$p$ for every $1\leq p<n$.

Case ($p=n$): Let $u\in{W}^{1,n}(\Omega)$ be arbitrary. Since

$B(x_{0},r)\cap\Omega\subset\Omega_{r}\subset B(x_{0},C_{2}r)\cap{\Omega},$

we have

$\displaystyle\inf\limits_{c\in\mathbb{R}}\int_{B(x_{0},r)\cap\Omega}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(B(x,C_{2}r)\cap\Omega)}}\right)^{\frac{n}{n-1}}\,dx$
$\displaystyle\leq\inf\limits_{c\in\mathbb{R}}\int_{\Omega_{r}}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(\Omega_{r})}}\right)^{\frac{n}{n-1}}\,dx.$

By [9, Lemma 3.11], we know $\Omega_{r}$ is also a $0$-carrot domain. By the
result dues to Smith and Stegenga [32], we have

$\displaystyle\inf\limits_{c\in\mathbb{R}}\int_{\Omega_{r}}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(\Omega_{r})}}\right)^{\frac{n}{n-1}}\,dx\leq C|\Omega_{r}|$

for a constant $C$ independent of $x_{0},r,u$. Hence, we have

$\displaystyle\inf\limits_{c\in\mathbb{R}}\int_{B(x_{0},r)\cap\Omega}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(B(x_{0},C_{2}r)\cap\Omega)}}\right)^{\frac{n}{n-1}}\,dx$
$\displaystyle\leq C|\Omega_{r}|\leq C|B(x_{0},C_{2}r)\cap\Omega|$
$\displaystyle\leq C|B(x_{0},C_{2}r)|\leq C|B(x_{0},r)|.$

It implies the uniform domain ${\Omega}$ is also a local Sobolev-Poincaré
imbedding domain of order $n$. ∎

###  3.2. From local Sobolev-Poincaré imbedding to $(LLC)$

In this section, we prove the necessary part of Theorem 1.1. We will show
under the assumption that the domain satisfies the slice condition with
respect to every point inside the domain, for $1\leq p\leq n$, a local
Sobolev-Poincaré imbedding domain of order $p$ is also a uniform domain. As we
know, for a domain which satisfies the slice condition with respect to every
point inside the domain, it is a uniform domain if and only if it is
quasiconvex and $LLC$. Hence, instead of showing the local Sobolev-Poincaré
imbedding domain of order $p$ with $1\leq p\leq n$ is uniform, we will prove
it is quasiconvex and $LLC$. By the definition, a domain is quasiconvex
implies this domain is $LLC(1)$. According to $p$ equals to $n$ or not, we
divide the argument into two separated cases.

First, let us consider the case that $1\leq p<n$.

######  Theorem 3.2.

Let $u\in W^{1,p}(\Omega)$ be arbitrary for $1\leq p<n$. If there exist
constants $C_{1},C_{2}>1$ such that the inequality

(3.1)
$\displaystyle\inf\limits_{c\in\mathbb{R}}\left(\int_{B(x_{0},r)\cap\Omega}|u(x)-c|^{\frac{np}{n-p}}\,dx\right)^{\frac{n-p}{np}}\leq
C_{1}\left(\int_{B(x_{0},C_{2}r)\cap\Omega}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}$

holds for every $x_{0}\in\Omega$ and $0<r<\infty$, then $\Omega$ is
quasiconvex and $LLC$.

###### Proof.

To prove $LLC(1)$: Let $x_{1}$, $x_{2}$ be a pair of points in $\Omega$ and
$x_{1},x_{2}\in B(z,r)\cap{\Omega}$ for some $z\in{{{\mathbb{R}}}^{n}}$ and
$r>0$. We need to show the existence of a rectifiable curve $\gamma\subset
B(z,r/b)\cap{\Omega}$ jointing $x_{1}$ and $x_{2}$ for some constant $0<b<1$
independent of $x_{1}$ and $x_{2}$. If

$|x_{1}-x_{2}|\leq\max\left\\{d(x_{1},\partial\Omega),d(x_{2},\partial\Omega)\right\\},$

the line segment $[x_{1},x_{2}]$ gives the desired curve. So we always assume

$|x_{1}-x_{2}|\geq\max\left\\{d(x_{1},\partial\Omega),d(x_{2},\partial\Omega)\right\\}$

below.

By Lemma 2.3, ${\Omega}$ is intrinsic Ahlfors $n$-regular. There exists
$r_{o}>0$ such that for every $x\in\overline{\Omega}$ and every $0<r<r_{o}$,
we have

$|B_{\Omega}(x,r)|\geq cr^{n}$

for a sufficiently small $0<c<1$ independent of $x$ and $r$. First, we
consider the case that

$|x_{1}-x_{2}|\geq\frac{r_{o}}{2C_{2}}$

where the constant $C_{2}$ comes from the inequality (1.6). Since
$x_{1},x_{2}\in B(z,r)\cap{\Omega}$, we have $r>\frac{r_{o}}{4C_{2}}$.
Obviously, $x_{1}$ and $x_{2}$ can always be connected by a locally
rectifiable curve in $B(z,r+{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega}))$.
Hence, we can choose a sufficient small $b>0$ such that

$B(z,r+{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega}))\subset B(z,r/b)\subset
B(z,r+2{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})).$

Some simple computation implies we can choose

$b=\frac{r_{o}}{5C_{2}{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})+r_{o}}.$

From now on, we always assume

$|x_{1}-x_{2}|<\frac{r_{o}}{2C_{2}}$

below. For $i=1,2$, Let $\Omega_{i}$ be the connected component of $\Omega\cap
B\left(x_{i},\frac{|x_{1}-x_{2}|}{8}\right)$ which contains $x_{i}$. Assume
$\Omega_{1}\cap\Omega_{2}=\emptyset$. Otherwise, we can choose
$b=\frac{1}{8}$. We claim that there exists a positive constant $N\geq
100C_{2}$ with an upper bound independent of $x_{1},x_{2}$ such that
$\Omega_{2}\subset\Omega\cap B(x_{1},N|x_{1}-x_{2}|)$. Assume not, define
$\tilde{\Omega}_{1}$ to be the connected component of ${\Omega}\cap
B(x_{1},N|x_{1}-x_{2}|)$ which contains ${\Omega}_{1}$ and define a function
$u$ on $\Omega$ by setting

$\displaystyle u(x):=\left\\{\begin{array}[]{ll}1,&x\in
B\left(x_{1},\frac{|x_{1}-x_{2}|}{8}\right)\cap\tilde{\Omega}_{1}\\\
\frac{1}{\log
8N}\log\frac{N|x_{1}-x_{2}|}{|x-x_{1}|},&x\in\left(B\left(x_{1},N|x_{1}-x_{2}|\right)\backslash
B\left(x_{1},\frac{|x_{1}-x_{2}|}{8}\right)\right)\cap\tilde{\Omega}_{1}\\\
0,&elsewhere.\end{array}\right.$

Some simple computations gives

$\displaystyle|\nabla u(x)|=\left\\{\begin{array}[]{ll}\frac{1}{\log
8N}\frac{1}{|x-x_{1}|},&x\in\left(B\left(x_{1},N|x_{1}-x_{2}|\right)\backslash
B\left(x_{1},\frac{|x_{1}-x_{2}|}{8}\right)\right)\cap\tilde{\Omega}_{1}\\\
0,&elsewhere.\end{array}\right.$

Hence, we have

(3.2) $\displaystyle\left(\int_{B(x_{1},2C_{2}|x_{1}-x_{2}|)\cap\Omega}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle\leq\left(\int_{\left(B(x_{1},2C_{2}|x_{1}-x_{2}|)\backslash
B\left(x_{1},\frac{|x_{1}-x_{2}|}{8}\right)\right)\cap\tilde{\Omega}_{1}}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle\leq\frac{C}{|x_{2}-x_{1}|\log
8N}\left|\left(B(x_{1},2C_{2}|x_{1}-x_{2}|)\backslash
B\left(x_{1},\frac{|x_{1}-x_{2}|}{8}\right)\right)\cap\tilde{\Omega}_{1}\right|^{\frac{1}{p}}$
$\displaystyle\leq C\frac{|x_{1}-x_{2}|^{\frac{n-p}{p}}}{\log 8N}.$

Observe that for each $c\in\mathbb{R}$, we have $|u-c|>\frac{1}{2}$ either on
${\Omega}_{1}$ or on ${\Omega}_{2}$. By Lemma 2.3, ${\Omega}$ is intrinsic
Ahlfors $n$-regular. Since

$|x_{1}-x_{2}|<\frac{r_{o}}{2C_{2}},$

we have

$\min\\{|{\Omega}_{1}|,|{\Omega}_{2}|\\}\geq c|x_{1}-x_{2}|^{n}$

for some small enough constant $0<c<1$. Since

${\Omega}_{1},{\Omega}_{2}\subset B(x_{1},2|x_{1}-x_{2}|)\cap{\Omega},$

we have

(3.3)
$\displaystyle\inf\limits_{c\in\mathbb{R}}\left(\int_{B(x_{1},2|x_{1}-x_{2}|)\cap\Omega}|u(x)-c|^{\frac{np}{n-p}}\,dx\right)^{\frac{n-p}{np}}\geq
c|x_{1}-x_{2}|^{\frac{n-p}{n}}$

for some small enough constant $c>0$.

By combining inequalities (1.6), (3.2) and (3.3), we obtain that there exists
a sufficiently small constant $c>0$ such that

$\frac{1}{\log 8N}>c.$

It gives an upper bounded $N_{o}$ of $N$. Hence, there exists a locally
rectifiable curve

$\gamma_{0}\subset\Omega\cap B(x_{1},(N_{0}+1)|x_{1}-x_{2}|)$

joining $\Omega_{1}$ and $\Omega_{2}$. Fix
$\tilde{x_{i}}\in\Omega_{i}\cap\gamma_{0}$, for $i=1,2$. Then there exists a
locally rectifiable curve $\gamma_{i}\subset{\Omega}_{i}$ which joins $x_{i}$
and $\tilde{x_{i}}$, for $i=1,2$. Set

$\gamma:=\gamma_{0}\cup\gamma_{1}\cup\gamma_{2},$

then the locally rectifiable curve $\gamma$ joins $x_{1}$ and $x_{2}$ and

$\gamma\subset\Omega\cap B(x_{1},(N_{0}+2)|x_{1}-x_{2}|)\subset
B(z,2(N_{0}+2)r).$

Hence, we can choose $b=\frac{1}{2(N_{0}+2)}$. Conclude two cases above, we
can choose

$b=\min\left\\{\frac{r_{o}}{5C_{2}{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})+r_{o}},\frac{1}{2(N_{0}+2)}\right\\}$

and we proved ${\Omega}$ satisfies $LLC(1)$.

To prove $LLC(2)$: Without loss of generality, we can only consider the case
that $x_{1},x_{2}\in\Omega\backslash B(x_{0},r)$ for some $x_{0}\in\Omega$ and
$0<r<{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})$. Furthermore, we can
assume that $x_{1},x_{2}\in\partial B(x_{0},r)\cap\Omega$. Assume that $x_{1}$
and $x_{2}$ are not in the same component of $\Omega\backslash{B(x_{0},br)}$
for $b\in(0,1)$, it suffices to find a uniform lower bound for $b$. We may
assume $b<\frac{1}{16}$. Denote by $\Omega_{i}$ the connected component of
$\Omega\backslash{B(x_{0},br)}$ which contains $x_{i}$ for $i=1,2$, then
$\Omega_{1}\cap\Omega_{2}=\emptyset$. Without loss of generality, we can only
consider the case that $0<r\leq r_{o}$ where $r_{o}$ comes from the fact that
${\Omega}$ is intrinsic Ahlfors $n$-regular. If we proved that for every
$0<r\leq r_{o}$, there exists $0<a<1$ such that $x_{1}$ and $x_{2}$ can be
connected by a locally rectifiable curve $\gamma\subset{\Omega}\setminus
B(x_{0},ar)$. Then for every
$r_{o}<r<{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})$, $x_{1},x_{2}$ can
be connected by a locally rectifiable curve

$\gamma\subset{\Omega}\setminus
B\left(x_{0},\frac{ar_{o}}{{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})}r\right).$

Fix $0<r\leq r_{o}$. Define $F_{i}$ to be the connected component of
$\Omega\cap B(x_{i},\frac{r}{2})$ which contains $x_{i}$ for $i=1,2$.
Obviously, we have $F_{1},F_{2}\subset B(x_{0},2r)$, $F_{1}\cap
F_{2}=\emptyset$ and $F_{i}\cap B(x_{0},\frac{r}{2})=\emptyset$ for $i=1,2$.
Since ${\Omega}$ is intrinsic Ahlfors $n$-regular, there exists a sufficient
small constant $0<c<1$ such that we have

$|F_{1}|\geq cr^{n}\ {\rm and}\ |F_{2}|\geq cr^{n}.$

Define a function $u$ on ${\Omega}$ by setting

$\displaystyle u(x)=\left\\{\begin{array}[]{ll}1,&x\in\Omega_{2}\backslash
B(x_{0},\frac{r}{2})\\\
\frac{1}{\log\frac{1}{2b}}log\frac{|x-x_{0}|}{br},&x\in\Omega_{2}\cap\left(B(x_{0},\frac{r}{2})\backslash
B(x_{0},br)\right)\\\ 0,&elsewhere.\end{array}\right.$

Obviously, this function is well defined. Noting that

$\displaystyle|\nabla
u(x)|=\frac{1}{\log\frac{1}{2b}}\frac{\chi_{\Omega_{2}\cap\left(B(x_{0},\frac{r}{2})\backslash
B(x_{0},br)\right)}(x)}{|x-x_{0}|},$

we have

$\displaystyle\left(\int_{B(x_{0},c_{2}r)\cap\Omega}|\nabla
u(x)|^{p}dx\right)^{\frac{1}{p}}$
$\displaystyle\leq\frac{1}{\log\frac{1}{2b}}\left(\int_{\Omega_{2}\cap\left(B(x_{0},\frac{r}{2})\backslash
B(x_{0},br)\right)}|x-x_{0}|^{-p}dx\right)^{\frac{1}{p}}$
$\displaystyle\leq\frac{1}{\log\frac{1}{2b}}r^{\frac{n-p}{p}}\left[\left(\frac{1}{2}\right)^{n-p}-b^{n-p}\right]^{\frac{1}{p}}$
$\displaystyle\leq\frac{Cr^{\frac{n-p}{p}}}{\log\frac{1}{2b}}.$

For each $c\in{\mathbb{R}}$, $|u-c|\geq\frac{1}{2}$ holds either on
$\Omega_{2}\cap\left(B(x_{0},r)\setminus
B\left(x_{0},\frac{r}{2}\right)\right)$ or on
$\Omega_{1}\cap\left(B(x_{0},r)\setminus
B\left(x_{0},\frac{r}{2}\right)\right)$. Since ${\Omega}$ is intrinsic Ahlfors
$n$-regular, we have both

$\left|{\Omega}_{i}\cap\left(B(x_{0},r)\setminus
B\left(x_{0},\frac{r}{2}\right)\right)\right|\geq cr^{n}$

for $i=1,2$ and a sufficient small constant $0<c<1$. Hence, we have

(3.5)
$\displaystyle\inf\limits_{c\in\mathbb{R}}\left(\int_{B(x_{0},r)\cap\Omega}|u(x)-c|^{\frac{np}{n-p}}\,dx\right)^{\frac{n-p}{np}}\geq
cr^{\frac{n-p}{n}},$

for a sufficient small $0<c<1$. By combining inequalities (1.6), (3.2) and
(3.5), we obtain

$\frac{1}{\log\frac{1}{2b}}>c$

for some sufficient small $0<c<1$. It gives a lower bound to $b$ and proves
${\Omega}$ is $LLC(2)$. ∎

Next, we consider the case that $p=n$.

######  Theorem 3.3.

If for every $u\in W^{1,n}(\Omega)$, there exist constants $C_{1},C_{2}>1$ the
following inequality

(3.6)
$\displaystyle\inf\limits_{c\in\mathbb{R}}\int_{B(x_{0},r)\cap\Omega}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(B(x_{0},C_{2}r)\cap\Omega)}}\right)^{\frac{n}{n-1}}\,dx\leq
C_{1}|B(x_{0},r)|$

holds with every $x_{0}\in\Omega$ and
$0<r<{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$, then $\Omega$ is $LLC$.

###### Proof.

To prove $LLC(1)$: All the argument is almost as same as the argument to show
${\Omega}$ is $LLC(1)$ in the proof of Theorem 3.2. The difference is that we
need to replace the inequality (1.6) by the inequality (1.7) and to estimate
the $n$-Dirichlet energy of the test function $u$. Some simple computation
gives

$\displaystyle\left(\int_{B(x_{1},2C_{2}|x_{1}-x_{2}|)\cap\Omega}|\nabla
u(x)|^{n}\,dx\right)^{\frac{1}{n}}$
$\displaystyle\leq\left(\int_{\left(B(x_{1},2|x_{1}-x_{2}|)\backslash
B\left(x_{1},\frac{|x_{1}-x_{2}|}{8}\right)\right)\cap\tilde{\Omega}_{1}}|\nabla
u(x)|^{n}\,dx\right)^{\frac{1}{n}}$ $\displaystyle\leq\frac{C}{\log 8N},$

and

$\displaystyle\int_{B(x_{1},2|x_{1}-x_{2}|)\cap\Omega}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(B(x_{0},Cr)\cap\tilde{\Omega}_{1})}}\right)^{\frac{n}{n-1}}\,dx$
$\displaystyle\geq\int_{B(x_{1},2|x_{1}-x_{2}|)\cap\tilde{\Omega}_{1}}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(B(x_{0},Cr)\cap\tilde{\Omega_{1}})}}\right)^{\frac{n}{n-1}}\,dx$
$\displaystyle\geq\exp(C\log
8N)^{\frac{n}{n-1}}|B(x_{1},2|x_{1}-x_{2}|)\cap\tilde{\Omega}_{1}|$
$\displaystyle\geq c\exp(C\log 8N)^{\frac{n}{n-1}}|x_{1}-x_{2}|^{n},$

for some sufficient small $0<c<1$. Then the inequality (3.6) gives

$\displaystyle 1>c\exp(C\log 8N)^{\frac{n}{n-1}}$

It implies a uniform upper bound to $N$ and ${\Omega}$ is $LLC(1)$.

Case $LLC$(2) : Similar to the proof of $LLC(1)$, all argument is almost as
same as the argument to show ${\Omega}$ is $LLC(1)$ in the proof of Theorem
3.2. The difference is that we need to replace the inequality (1.6) by the
inequality (1.7) and to estimate the $n$-Dirichlet energy of the test function
$u$. Some simple computation gives

$\displaystyle\left(\int_{B(x_{0},C_{2}r)\cap\Omega}|\nabla
u(x)|^{n}dx\right)^{\frac{1}{n}}$
$\displaystyle\leq\frac{C}{\log\frac{1}{2b}}\left(\int_{\Omega_{2}\cap\left(B(x_{0},\frac{r}{2})\backslash
B(x_{0},br)\right)}|x-x_{0}|^{n}dx\right)^{\frac{1}{n}}$ $\displaystyle\leq
C\left(\log\frac{1}{2b}\right)^{\frac{1-n}{n}}$

and

$\displaystyle\int_{B(x_{0},r)\cap\Omega}\exp\left(\frac{A|u(x)-c|}{\|\nabla
u\|_{L^{n}(B(x_{0},C_{2}r)\cap\tilde{\Omega}_{1})}}\right)^{\frac{n}{n-1}}\,dx\geq
c\exp\left(C\log\frac{1}{2b}\right)r^{n}$

for some sufficient small $0<c<1$. Then the inequality (3.6) gives

$1>c\exp\left(C\log\frac{1}{2b}\right).$

It implies a uniform lower bound to $b$ and ${\Omega}$ is $LLC(2)$. ∎

##  4\. Proof of Theorem 1.2

In this section, we prove Theorem 1.2.

###### Proof of Theorem 1.2.

First ,we show a local Sobolev-Poincaré imbedding domain of order $p$ for
$n<p<\infty$ with slice condition is also an $\alpha$-cigar domain for
$\alpha=(p-n)/(p-1)$. For arbitrary $x_{1},x_{2}\in{\Omega}$, we can find
$x\in{\Omega}$ and $0,r<{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})$ with
$x_{1},x_{2}\in B(x,r)\cap{\Omega}$. Then, we have

$\displaystyle|u(x_{1})-u(x_{2})|$ $\displaystyle\leq$ $\displaystyle
C_{1}|x_{1}-x_{2}|^{1-\frac{n}{p}}\left(\int_{B(x,C_{2}r)\cap{\Omega}}|\nabla
u(y)|^{p}dy\right)^{\frac{1}{p}}$ $\displaystyle\leq$ $\displaystyle
C_{1}|x_{1}-x_{2}|^{1-\frac{n}{p}}\left(\int_{{\Omega}}|\nabla
u(y)|^{p}dy\right)^{\frac{1}{p}}.$

So ${\Omega}$ is also a global Sobolev-Poincaré imbedding domain of order $p$.
By the result in [5], ${\Omega}$ is an $\alpha$-cigar domain for
$\alpha=(p-n)/(p-1)$.

Next, we show an $\alpha$-cigar domain ${\Omega}$ with $\alpha=(p-n)/(p-1)$ is
a local Sobolev-Poincaré imbedding domain of order $p$ for $n<p<\infty$. By
[5, Lemma 2.2], for every pair of points $x_{1},x_{2}\in{\Omega}$, there
exists curve $\gamma\subset{\Omega}$ joining $x_{1}$ and $x_{2}$ with

$l(\gamma)\leq C|x_{1}-x_{2}|$

for some uniform constant $C>1$, and

$\int_{\gamma}{\mathop{\mathrm{missing}}{\,dist\,}}(z,\partial{\Omega})^{\alpha-1}|dz|\leq
C|x_{1}-x_{2}|^{\alpha}.$

It implies that there exists a large enough $C>1$ such that for every
$x\in{\Omega}$ and $0<r<{\mathop{\mathrm{missing}}{\,diam\,}}({\Omega})$,
$B(x,Cr)\cap{\Omega}$ contains a connected component, which contains
$B(x,r)\cap{\Omega}$, is an $\alpha$-cigar domain with $\alpha=(p-n)/(p-1)$.
Hence, for every $x_{1},x_{2}\in B(x,r)\cap{\Omega}$, we have

$|u(x_{1})-u(x_{2})|\leq
C_{1}|x_{1}-x_{2}|^{1-\frac{n}{p}}\left(\int_{B(x,Cr)\cap{\Omega}}|\nabla
u(y)|^{p}dy\right)^{\frac{1}{p}}.$

It implies ${\Omega}$ is a local Sobolev-Poincará imbedding domain of order
$p$. ∎

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