# On a variational problem of nematic liquid crystal droplets

Qinfeng Li  School of Mathematics, Hunan University, Changsha, Hunan, China
410012<EMAIL_ADDRESS>and  Changyou Wang  Department of Mathematics,
Purdue University, West Lafayette, IN 47907, USA<EMAIL_ADDRESS>
###### Abstract.

Let $\mu>0$ be a fixed constant, and we prove that minimizers to the following
energy functional

$\displaystyle E_{f}(u,\Omega):=\int_{\Omega}|\nabla u|^{2}+\mu P(\Omega)$

exist among pairs $(\Omega,u)$ such that $\Omega$ is an $M$-uniform domain
with finite perimeter and fixed volume, and $u\in
H^{1}(\Omega,\mathbb{S}^{2})$ with $u=\nu_{\Omega}$, the measure-theoretical
outer unit normal, almost everywhere on the reduced boundary of $\Omega$. The
uniqueness of optimal configurations in various settings is also obtained. In
addition, we consider a general energy functional given by

$\displaystyle E_{f}(u,\Omega):=\int_{\Omega}|\nabla
u(x)|^{2}\,dx+\int_{\partial^{*}\Omega}f\big{(}u(x)\cdot\nu_{\Omega}(x)\big{)}\,d\mathcal{H}^{2}(x),$

where $\partial^{*}\Omega$ is the reduced boundary of $\Omega$ and $f$ is a
convex positive function on $\mathbb{R}$. We prove that minimizers of $E_{f}$
also exist among $M$-uniform outer-minimizing domains $\Omega$ with fixed
volume and $u\in H^{1}(\Omega,\mathbb{S}^{2})$.

###### Key words and phrases:

Liquid crystal droplets, M-uniform domains, Outer minimal sets

###### 2010 Mathematics Subject Classification:

Primary 35J50; Secondary 58E20, 58E30

The first author is partially supported by the National Science Fund for Youth
Scholars (No. 1210010723) and the Fundamental Research Funds for the Central
Universities, Hunan Provincial Key Laboratory of intelligent information
processing and Applied Mathematics.

The second author is partially supported by NSF grants 1764417 and 2101224.

## Contents

1\. Introduction

2\. Prerequisite: sets of finite perimeter and traces of functions

3\. Compactness of $M$-uniform domains

4\. Existence of equilibrium liquid crystal droplets in Problem A-C

5\. On the uniqueness of Problem C

References

##  1\. Introduction

In this paper we study the existence of liquid crystal droplets
$(\Omega_{0},u_{0})$, consisting of a domain $\Omega_{0}\subset\mathbb{R}^{3}$
representing the shape of a liquid crystal drop and a unit vector field
$u_{0}\in H^{1}(\Omega,\mathbb{S}^{2})$ representing the average orientation
field of liquid crystal molecules within the liquid crystal drop $\Omega$,
that minimizes the total energy functional, including both the elastic energy
in the bulk and the interfacial energy defined by

$\displaystyle E_{f}(u,\Omega):=\int_{\Omega}|\nabla
u(x)|^{2}\,dx+\int_{\partial^{*}\Omega}f\big{(}u(x)\cdot\nu_{\Omega}(x)\big{)}\,d\mathcal{H}^{2}(x),$
(1.1)

among all pairs $(\Omega,u)$, where $\Omega$ is a domain of finite perimeter
with a fixed volume that is compactly contained in the ball
$B_{R_{0}}\subset\mathbb{R}^{3}$ with center $0$ and radius $R_{0}$ for some
fixed constant $R_{0}>0$, and $u\in H^{1}(\Omega,\mathbb{S}^{2})$, which is
defined by

$H^{1}(\Omega,\mathbb{S}^{2})\equiv\Big{\\{}v\in
H^{1}(\Omega,\mathbb{R}^{3}):\ |v(x)|=1\ \mbox{a.e.}\ x\in\Omega\Big{\\}}.$

The functional $E_{f}(u,\Omega)$ should be understood in the sense that the
surface integral is taken over the reduced boundary $\partial^{*}\Omega$ of
$\Omega$, $u\ \lfloor_{\partial^{*}\Omega}$ is the trace of $u$ on
$\partial^{*}\Omega$, $\nu_{\Omega}$ is the measure theoretical outer unit
normal of $\partial^{*}\Omega$, and $f$ is usually assumed to have a
nonnegative lower bound (with a typical choice of
$f(t)=\mu(1+wt^{2}),t\in[-1,1],$ for some constants $\mu>0$ and $-1<w<1$).

We will study the following minimization problem of (1.1).

Problem A. Find a pair $(\Omega,u)$ that minimizes $E_{f}(u,\Omega)$ over all
pairs $(\Omega,u)$ where $\Omega$ is a domain of finite perimeter in a fixed
ball $B_{R_{0}}\subset\mathbb{R}^{3}$, with a fixed volume $V_{0}>0$, and
$u\in H^{1}(\Omega,\mathbb{S}^{2})$, when $f:[-1,1]\to\mathbb{R}$ is a
nonnegative, continuous convex function.

We are also interested in the case when there is a constant contact angle
condition between the liquid crystal orientation field $u$ and the reduced
boundary of liquid crystal drop $\partial^{*}\Omega$, i.e.,
$u\cdot\nu_{\Omega}\equiv c$ on $\partial^{*}\Omega$, for some constant
$c\in[-1,1]$. In this case, the energy functional $E_{f}(u,\Omega)$ in (1.1)
reduces to

$\displaystyle{E}(u,\Omega):=\int_{\Omega}|\nabla
u(x)|^{2}\,dx+\mu\mathcal{H}^{2}(\partial^{*}\Omega)$ (1.2)

for some constant $\mu\geq 0$. Problem A can be reformulated as follows.

Problem B. Find a pair $(\Omega,u)$ that minimizes ${E}(u,\Omega)$ over all
pairs $(\Omega,u)$ where $\Omega$ is a domain of finite perimeter in a fixed
ball $B_{R_{0}}\subset\mathbb{R}^{3}$, with a fixed volume $V_{0}>0$, and
$u\in H^{1}(\Omega,\mathbb{S}^{2})$ satisfies $u\cdot\nu_{\Omega}\equiv c$ on
$\partial^{*}\Omega$ for some $c\in[-1,1]$.

We would like to mention that the contact angle condition in Problem B is
referred as

  * (i)

the planar anchoring condition when the constant $c=0$, and

  * (ii)

the homeotropic anchoring condition when the constant $c=1$.

We would like to point out that recently Geng and Lin in a very interesting
paper [23] studied Problem B under the planar anchoring condition (i) in
dimension two, and proved the existence of a minimizer $(\Omega,u)$ such that
the optimal shape $\partial\Omega$ of the droplet is a chord-arc curve with
two cusps, which can be parametrized in $H^{\frac{3}{2}}$ and has its unit
normal vector field $\nu_{\Omega}$ belongs to VMO.

Because the homeotropic anchoring condition is an important physical
condition, we are also interested in the following problem.

Problem C. Find a solution to Problem B when the contact angle condition
corresponds to $c=1$.

Motivation. The main difficulty of the minimization problems A, B, and C lies
in showing the sequential lower semicontinuity of $E_{f}(u,\Omega)$ (or
${E}(u,\Omega)$) when both domains $\Omega$ and vector fields $u\in
H^{1}(\Omega,\mathbb{S}^{2})$ vary. It is even a difficult question to ask
whether the configuration space is closed under weak convergence of liquid
crystal pairs $(\Omega,u)$.

In [34], under the assumption that all admissible domains $\Omega\subset
B_{R_{0}}$ are convex domains, Lin and Poon have proved that there exists a
minimizing pair $(\Omega_{0},u_{0})$ of Problem A. Moreover, $u_{0}$ enjoys a
partial regularity property similar to that of minimizing harmonic maps by
Schoen and Uhlenbeck [40, 41]. It was further proven by [34] that, up to
translations, $(\Omega_{0},u_{0})=(B_{R},\frac{x}{|x|})$ is a unique minimizer
of Problem C among convex domains with $|B_{R}|=V_{0}$.

We would like to point out that the convexity assumption of admissible domains
$\Omega$ plays a crucial role in [34], since a minimizing sequence
$(\Omega_{i},u_{i})$ of convex domains $\Omega_{i}\subset B_{R_{0}}$ with
$|\Omega_{i}|=V_{0}$ has a subsequence $\Omega_{i_{k}}\rightarrow\Omega$ in
$L^{1}$, for some bounded convex domain $\Omega\subset B_{R_{0}}$ with
$|B_{R_{0}}|=V_{0}$, such that
$\mathcal{H}^{2}(\partial\Omega_{i_{k}})\rightarrow\mathcal{H}^{2}(\partial\Omega)$
and $\nu_{\Omega_{i_{k}}}\rightarrow\nu_{\Omega}$ almost everywhere with
respect to a spherical coordinate system111For example, one can parametrize
$\partial\Omega_{i_{k}}$ and $\partial\Omega$ over the unit sphere
$\mathbb{S}^{2}$.. Moreover, there exists $u\in H^{1}(\Omega,\mathbb{S}^{2})$
such that $\nabla u_{i_{k}}\chi_{\Omega_{i_{k}}}\rightarrow\nabla
u\chi_{\Omega}$ weakly in $L^{2}(\mathbb{R}^{3})$. The uniqueness of minimizer
of Problem C among convex domains relies on the following important
inequalities:

$\displaystyle\int_{\Omega}|\nabla
u(x)|^{2}\,dx\geq\int_{\partial\Omega}H(x)\,d\mathcal{H}^{2}(x),\ \forall u\in
H^{1}(\Omega,\mathbb{S}^{2})\ {\rm{with}}\ u=\nu_{\Omega}\ {\rm{a.e.\ on}}\
\partial\Omega,$ (1.3)

and

$\displaystyle\int_{\partial\Omega}H(x)\,d\mathcal{H}^{2}(x)\geq\sqrt{4\pi\mathcal{H}^{2}(\partial\Omega})\
\mbox{for convex $\Omega$},\ {\rm{equality\ holds\ iff}}\ \Omega=B_{R},$ (1.4)

where $H$ denotes the mean curvature of $\partial\Omega$. In [34], (1.3) is
derived for any $\Omega\in W^{2,1}$, while (1.4) is proven by the Brunn-
Minkowski inequality for convex domains.

In this paper, we would like to relax the convexity assumption from [34] and
investigate Problems A, B, and C over a larger class of domains possibly
containing non-convex domains with less regular boundaries. The class of
domains contains Sobolev extension domains with some uniform parameters, as
well outer minimal domains.

The main theorems of this paper arose from the Ph.D. thesis of the first
author [32]. The interested reader can refer to [32] for more related results.

Outline of this paper:
In section 2, we will review certain classes of domains in $\mathbb{R}^{n}$,
including $M$-uniform domains, which are Sobolev extension domains with
constants depending on $M$ and $n$; the outer minimal domains, which are a
generalization of convex domains.

In section 3, we will show in Theorem 3.5 that, up to a set of measure zero,
the $L^{1}$-limit of $M$-uniform domains is $M$-uniform. A few other results
on the relation between $L^{1}$-convergence and Hausdorff convergence are also
derived.

In section 4, we will establish the weak lower semicontinuity of bulk elastic
energy of $(\Omega,u)$ for two classes of domains: a) the admissible sets of
$M$-uniform domains, and b) the admissible sets of outer minimal $M$-uniform
domains. It is more subtle to prove the lower semicontinuity of surface energy
for Problem A. We will only consider outer minimal sets and our proof is
inspired by Reshetnyak’s lower semicontinuity theorem (see [36, Theorem
20.11]) and the perimeter convergence Lemma 4.1. Thus combining the
compactness of $M$-uniform domain results and the lower-semicontinuity
results, the existence Theorem 4.2 on problems A, B and C is proved among
these classes of admissible sets.

In section 5, we will apply results by [10], [16], [26] and [27] to show
$(B_{R},\frac{x}{|x|})$ is the unique minimizer of Problem C over strictly
star-shaped mean convex $C^{1,1}$-domains, $C^{1,1}$-outer minimal sets, and
$C^{1,1}$-revolutionary domains, see Theorem 5.5 and Remark 5.6.

##  2\. Prerequisite: sets of finite perimeter and traces of functions

We first stipulate some notations. Let $V_{0}>0$ be the fixed volume in the
Problems A, B, C. Since the admissible domains in the problems have this fixed
volume, we will use the convention that any minimizing sequences have their
diameters larger than a universal constant $c_{0}=c_{0}(V_{0})>0$ because of
the isodiametric inequality (see [13, Theorem 2.2.1]).

We will denote by $B_{r}(x):=\\{y\in\mathbb{R}^{n}:|y-x|<r\\}$ and
$B_{r}:=B_{r}(0)$. Throughout this paper all sets under consideration are
contained in a large ball $B_{R_{0}}$, where $R_{0}>0$ is fixed. For any set
$A\subset\mathbb{R}^{n}$, denote by $A_{\epsilon}$ the interior
$\epsilon$-neighborhood $\\{x\in A:B_{\epsilon}(x)\subset A\\}$, and
$A^{\epsilon}$ the exterior $\epsilon$-neighborhood $\bigcup_{x\in
A}B_{\epsilon}(x)$. Denote by ${\rm{int}}(A)$ the topological interior part of
$A$, $A^{c}=\mathbb{R}^{n}\setminus A$, and ${\rm{diam}}(A)$ the diameter of
$A$. For $0\leq d\leq n$, $\mathcal{H}^{d}$ denotes the $d$-dimensional
Hausdorff measure in $\mathbb{R}^{n}$. Let $d^{H}(\cdot,\cdot)$ denote the
Hausdorff distance in $\mathbb{R}^{n}$. $P(A;D)$ denotes the distributional
perimeter of $A$ in $D\subset\mathbb{R}^{n}$. For a set $A$ of finite
perimeter, let $\nu_{A}$ denote the measure theoretical outer unit normal of
the reduced boundary $\partial^{*}A$, and $\mu_{A}$ denotes the Gauss-Green
measure of $A$, that is,
$\mu_{A}=\nu_{A}\cdot\mathcal{H}^{n-1}\lfloor_{\partial^{*}A}$. Denote by
$\omega_{n}$ the volume of unit ball in $\mathbb{R}^{n}$ and $|A|$ the
Lebesgue measure of $A$.. For any open set $\Omega\subset\mathbb{R}^{n}$ and
$u\in BV(\Omega)$, denote by $Du$ the distributional derivative of $u$, that
is a vector-valued Radon measure, and $\|Du\|(\Omega)$ the total variation of
$u$ on $\Omega$.

In this paper, $``\lesssim_{c}"$ denotes an inequality up to constant
multiplier $c>0$. For any measurable set $E$ and $0\leq\alpha\leq 1$, we
define

$E^{\alpha}=\Big{\\{}x\in\mathbb{R}^{n}:\lim_{r\rightarrow 0}\frac{|E\cap
B_{r}(x)|}{|B_{r}(x)|}=\alpha\Big{\\}},$

and refer $E^{1}$ and $E^{0}$ as the measure theoretical interior and exterior
part of $E$ respectively. Denote by
$\partial_{*}E:=\mathbb{R}^{n}\setminus(E^{0}\cup E^{1})$ the measure
theoretical boundary of $E$, which is also called the essential boundary. In
this paper, we will need the following theorem, due to Federer (see [13,
Chapter 5]).

######  Theorem 2.1.

For any measurable set $E$, if $\mathcal{H}^{n-1}(\partial_{*}E)<\infty$, then
$E$ is a set of finie perimeter. Furthermore, if $E$ is a set of finite
perimeter, then $\mathbb{R}^{n}=E^{0}\cup E^{1}\cup\partial_{*}E$,
$\partial^{*}E\subset E^{(1/2)}\subset\partial_{*}E$, and
$\partial^{*}E=\partial_{*}E\,({\rm{mod}}\ \mathcal{H}^{n-1})$.

Next, we recall the definition of $M$-uniform domains.

######  Definition 2.2.

For $M\geq 1$, a domain $\Omega\subset\mathbb{R}^{n}$ is called an $M$-uniform
domain, if for any two points $x,y\in\Omega$, there is a rectifiable curve
$\gamma:[0,1]\rightarrow\Omega$ such that $\gamma(0)=x,\ \gamma(1)=y$, and

$\displaystyle\mathcal{H}^{1}(\gamma([0,1]))\leq M|x-y|,$ (2.1) $\displaystyle
d(\gamma(t),\partial\Omega)\geq\frac{1}{M}\min\big{\\{}|\gamma(t)-x|,|\gamma(t)-y|\big{\\}},\,\forall
t\in[0,1].$ (2.2)

######  Remark 2.3.

P. Jones [29] introduced the notion of $(\epsilon,\delta)$-domain. One can
check that any $(\epsilon,\infty)$-domain is an $M$-domain, with
$M=\frac{2}{\epsilon}$. On the other hand, any $M$-uniform domain is a
$(\frac{1}{M^{2}},\infty)$-domain222Since (2.1) and (2.2) imply
$d(\gamma(t),\partial\Omega)\geq\frac{1}{M}\frac{|\gamma(t)-x||\gamma(t)-y|}{\mathcal{H}^{1}(\gamma([0,1]))}\geq\frac{1}{M^{2}}\frac{|\gamma(t)-x||\gamma(t)-y|}{|x-y|},\
\forall t\in[0,1].$ . It was also proven by [29] that any $(\epsilon,\delta)$
domain is a Sobolev extension domain, and the converse is true when $n=2$. We
refer to [20] and [29] for more details on $M$-uniform domains.

Since we will study minimization problems involving traces of bounded $H^{1}$
vector fields in this paper, we will need the following Gauss-Green formula.

######  Theorem 2.4.

Let $\Omega$ be a bounded uniform domain of finite perimeter in
$\mathbb{R}^{n}$ and $u\in H^{1}(\Omega)\cap L^{\infty}(\Omega)$. Then for any
$\phi\in C_{0}^{1}(\mathbb{R}^{n},\mathbb{R}^{n})$, we have

$\displaystyle\int_{\Omega}u{\rm{div}}\phi+\int_{\Omega}\phi
Du=\int_{\partial^{*}\Omega}(\phi\cdot\nu_{\Omega})u^{*}d\mathcal{H}^{n-1},$
(2.3)

where $\nu_{\Omega}$ is the measure-theoretic unit outer normal to
$\partial^{*}\Omega$, and $u^{*}$ is given by the formula

$\displaystyle\lim_{r\rightarrow
0}\displaystyle\frac{\int_{B_{r}(x)\cap\Omega}|u-u^{*}(x)|}{r^{n}}=0,\
\mbox{$\mathcal{H}^{n-1}$-a.e. $x\in\partial^{*}\Omega$}.$ (2.4)

###### Proof.

According to [29], we may let $\hat{u}\in H^{1}_{0}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$ be an extension of $u$ such that $\hat{u}=u$ in
$\Omega$ and

$\displaystyle\|\hat{u}\|_{H^{1}(\mathbb{R}^{n})}\leq
C(n,\Omega)\|u\|_{H^{1}(\Omega)}.$

Hence $\hat{u}\in BV(\mathbb{R}^{n})$, and thus according to [2, Theorem
3.77], the interior trace of $\hat{u}$, denoted by $\hat{u}^{*}$ here, is
well-defined for $\mathcal{H}^{n-1}$-a.e. on $\partial^{*}\Omega$, and equals
to $u^{*}$, given by (2.4), for $\mathcal{H}^{n-1}$-a.e. on
$\partial^{*}\Omega$. Let $\tilde{u}=\hat{u}\chi_{\Omega}$. Since $\hat{u}$ is
bounded, $u^{*}\in L^{1}(\partial^{*}\Omega)$ and thus by [2, Theorem 3.84],
$\tilde{u}=\hat{u}\chi_{\Omega}\in BV(\mathbb{R}^{n})$, with

$\displaystyle
D\tilde{u}=D\hat{u}\lfloor_{\Omega^{1}}-u^{*}\nu_{\Omega}\mathcal{H}^{n-1}\lfloor_{\partial^{*}\Omega}.$

Hence for any $\phi\in C_{0}^{1}(\mathbb{R}^{n},\mathbb{R}^{n})$, we have

$\displaystyle\int_{\mathbb{R}^{n}}\phi D\tilde{u}=\int_{\Omega^{1}}\phi
D\hat{u}-\int_{\partial^{*}\Omega}(\phi\cdot\nu_{\Omega})u^{*}\,d\mathcal{H}^{n-1}.$
(2.5)

Since

$\displaystyle\int_{\mathbb{R}^{n}}\phi
D\tilde{u}=-\int_{\mathbb{R}^{n}}\tilde{u}\div\phi=-\int_{\Omega}u\div\phi,$

from (2.5) we have

$\displaystyle\int_{\Omega}u\div\phi+\int_{\Omega^{1}}\phi
D\hat{u}=\int_{\partial^{*}\Omega}(\phi\cdot\nu_{\Omega})u^{*}d\mathcal{H}^{n-1}.$
(2.6)

Since $\Omega$ is equivalent to $\Omega^{1}$ up to a set of Lebesgue measure
zero and $\hat{u}\in H^{1}(\mathbb{R}^{n})$, we have

$\displaystyle
D\hat{u}\lfloor_{\Omega^{1}}=D\hat{u}\lfloor_{\Omega}=Du\lfloor_{\Omega}$
(2.7)

Hence (2.6) and (2.7) imply (2.3). ∎

For the purpose later in this paper, we also introduce the following
definition.

######  Definition 2.5.

For any $c>0$, we denote by $\mathcal{D}_{c}$ the class of bounded sets in
$\mathbb{R}^{n}$ such that for any set $E\in\mathcal{D}_{c}$,

$\displaystyle|B_{r}(x)\cap E|>cr^{n}$ (2.8)

holds for any $x\in\partial E$ and $0<r<{\rm{diam}}(E)$.

Recall that two sets $E,F\subset\mathbb{R}^{n}$ are said to be
$\mathcal{H}^{n}$-equivalent, denoted by $E\approx F$, if $E\Delta
F=(E\setminus F)\cup(F\setminus E)$ has zero Lebesgue measure. Note that by
the Lebesgue density theorem, if $E\in\mathcal{D}_{c}$, then $|\partial E\cap
E^{c}|=0$. Hence $\partial E\subset E$ (mod $\mathcal{H}^{n}$) and
$\overline{E}\approx E$. In particular, we have

######  Remark 2.6.

Any $E\in\mathcal{D}_{c}$ is equivalent to its closure $\overline{E}$.

We also have

######  Remark 2.7.

For $c>0$, if $E\in\mathcal{D}_{c}$ is a set of finite perimeter, then there
is $c^{\prime}>0$ depending only on $c$ and $n$ such that for any
$x\in\overline{E}$ and $0<r<{\rm{diam}}(E)$, $|B_{r}(x)\cap E|\geq
c^{\prime}r^{n}$.

###### Proof.

For $x\in\overline{E}$ and $0<r<{\rm{diam}}(E)$, there are two cases:
(a) If $r\geq 2d(x,\partial E)$, then there is $z\in\partial E$ such that
$B_{\frac{r}{2}}(z)\subset B_{r}(x)$. Hence

$|B_{r}(x)\cap E|\geq|B_{\frac{r}{2}}(z)\cap E|\geq
c(\frac{r}{2})^{n}=\frac{c}{2^{n}}r^{n}.$

(b) If $r\leq 2d(x,\partial E)$, then $B_{\frac{r}{2}}(x)\subset E$ and hence

$|B_{r}(x)\cap E|\geq|B_{\frac{r}{2}}(x)|=\frac{\omega_{n}}{2^{n}}r^{n}.$

Hence the conclusion holds with
$c^{\prime}=\min\\{\frac{c}{2^{n}},\frac{\omega_{n}}{2^{n}}\\}$. ∎

The following proposition shows that any $M$-uniform domain belongs to
$\mathcal{D}_{c}$ for some $c>0$.

######  Proposition 2.8.

For any $M\geq 1$ and $c_{0}>0$, if $\Omega\subset\mathbb{R}^{n}$ is an
$M$-uniform domain, with ${\rm{diam}}(\Omega)\geq c_{0}>0$, then
$\Omega\in\mathcal{D}_{c}$ for some $c>0$ depending only on $M$, $n$ and
$c_{0}$.

###### Proof.

For any $x\in\partial\Omega$ and $0<r<{\rm{diam}}(\Omega)$, we claim that
there is a constant $c_{1}=c_{1}(M)>0$ such that $B_{r}(x)\cap\Omega$ contains
a ball of radius $c_{1}r$. Indeed, since $0<r<{\rm{diam}}(\Omega)$, there is
$y\in\Omega\setminus B_{\frac{r}{2}}(x)$. Let $\gamma$ be the curve joining
$x$ and $y$ given by the definition of $M$-uniform domain. Choose
$z\in\partial B_{\frac{r}{3}}(x)\cap\gamma$. Then we have that $z\in\Omega$
and

$d(z,\partial\Omega)\geq\frac{1}{M}\min\big{\\{}|z-x|,|z-y|\big{\\}}\geq\frac{1}{M}\min\big{\\{}\frac{r}{3},\frac{r}{2}-\frac{r}{3}\big{\\}}=\frac{r}{6M}.$

Hence $B_{c_{1}r}(z)\subset\Omega$, with $c_{1}=\frac{1}{6M}$. From this
claim, we see that for any $x\in\partial\Omega$ and any
$r<{\rm{diam}}(\Omega)$,

$|B_{r}(x)\cap\Omega|\geq|B_{c_{1}r}(z)|\geq\omega_{n}c_{1}^{n}r^{n}.$

This completes the proof. ∎

The following remark will be used in the proof of compactness of $M$-uniform
domains.

######  Remark 2.9.

For $M>0$ and $c_{0}>0$, if $\Omega\subset\mathbb{R}^{n}$ is an $M$-uniform
domain, with $|\Omega|\geq c_{0}$, then there is $r_{0}>0$ depending only on
$M,n,c_{0}$ such that $\Omega$ contains a ball of radius $r_{0}$.

###### Proof.

It follows directly from the isodiametric inequality and Proposition 2.8. ∎

Similar to $\mathcal{D}_{c}$, we also define the class $\mathcal{D}^{c}$ as
follows.

######  Definition 2.10.

For $c>0$, the set class $\mathcal{D}^{c}$ consists of all bounded set
$E\subset\mathbb{R}^{n}$ such that

$\displaystyle|B_{r}(x)\cap E^{c}|>cr^{n}$ (2.9)

holds for any $x\in\partial E$ and $0<r<{\rm{diam}}(E)$.

The following proposition from [36, Proposition 12.19] yields that we can
always find an $\mathcal{H}^{n}$-equivalent set $\widetilde{E}$ of any set $E$
of finite perimeter with slightly better topological boundary.

######  Proposition 2.11.

For any Borel set $E\subset\mathbb{R}^{n}$, there exists an
$\mathcal{H}^{n}$-equivalent set $\widetilde{E}$ of $E$ such that for any
$x\in\partial\widetilde{E}$ and any $r>0$,

$\displaystyle 0<|\widetilde{E}\cap B_{r}(x)|<\omega_{n}r^{n}.$ (2.10)

In particular,
${\rm{spt}}\mu_{E}={\rm{spt}}\mu_{\widetilde{E}}=\partial\widetilde{E}$.

In order to illustrate the construction of such an equivalent set, which is
needed in later sections, we will sketch the proof.

###### Proof.

First, we define two disjoint open sets

$A_{1}:=\big{\\{}x\in\mathbb{R}^{n}\ |\ \mbox{there exists $r>0$ such that
$|E\cap B_{r}(x)|=0$}\big{\\}},$

and

$A_{2}:=\big{\\{}x\in\mathbb{R}^{n}\ |\ \mbox{there exists $r>0$ such that
$|E\cap B_{r}(x)|=\omega_{n}r^{n}$}\big{\\}}.$

Then by simple covering arguments we have that $|E\cap A_{1}|=0$ and
$|A_{2}\setminus E|=0$. Set $\widetilde{E}=(A_{2}\cup E)\setminus A_{1}$. Then

$|\widetilde{E}\Delta E|\leq|A_{2}\setminus E|+|E\cap A_{1}|=0.$

Moreover, since $A_{2}\subset{\rm{int}}(\widetilde{E})$ and
$\overline{\widetilde{E}}\subset\mathbb{R}^{n}\setminus A_{1}$, we have that
$\partial\widetilde{E}\subset\mathbb{R}^{n}\setminus(A_{1}\cup A_{2})$ and
hence (2.10) holds. ∎

We now recall the notion of outer minimal sets, which can be viewed as a
subsolution of area minimizing sets. It is a generalization of convex sets,
see for example [22, Definition 15.6] and related results therein.

######  Definition 2.12.

A set $E\subset\mathbb{R}^{n}$ of finite perimeter is an outer minimal set, if
$P(E)\leq P(F)$ holds for any set $F\supset E$.

We would like to point out that an outer-minimal set is also called as a
pseudo-convex set by [33]. Thus by [33, Corollary 7.16] we have

######  Remark 2.13.

If $E\subset\mathbb{R}^{n}$ is an outer-minimizing and
${\rm{spt}}\mu_{E}=\partial E$, then $E\in\mathcal{D}^{c}$, for some $c>0$
depending only on $n$ and $E$. Consequently, $E={\rm{int}}(E)\
({\rm{mod}}\mathcal{H}^{n})$.

######  Remark 2.14.

Since the boundary of an outer minimal set (domain) can have positive
$\mathcal{H}^{n}$ measure (see [4]), an outer minimal domain may not be an
$M$-uniform domain for any $M\geq 1$.

Combining Proposition 2.8 and Remark 2.13, we have

######  Remark 2.15.

Let $\Omega$ be an $M$-uniform outer minimal domain with
$\rm{spt}\mu_{\Omega}=\partial\Omega$, then
$\Omega\in\mathcal{D}_{c}\cap\mathcal{D}^{c}$ for some $c>0$, and hence
$\partial_{*}\Omega=\partial\Omega$.

We would like to state the following proposition, which is a consequence of
[24, Corollary 1.10], since for any $E\in\mathcal{D}_{c}$,
$\mathcal{H}^{n-1}(\partial E\cap E^{0})=0$.

######  Proposition 2.16.

Let $c>0$ and $E\in\mathcal{D}_{c}$. Then there exists bounded smooth sets
$E_{i}$ such that $E_{i}\Supset E$, $E_{i}\rightarrow E$ in $L^{1}$ and
$P(E_{i})\rightarrow P(E_{i})$.

##  3\. Compactness of $M$-uniform domains

In this section, we will establish in Theorem 3.5 the $L^{1}$-compactness
property of $M$-uniform domains. We begin with

######  Lemma 3.1.

For $c>0$, suppose that $\\{D_{i}\\}\subset\mathcal{D}_{c}$ satisfies
$D_{i}\rightarrow D$ in $L^{1}(\mathbb{R}^{n})$ as $i\rightarrow\infty$. Then
after modifying over a set of Lebesgue measure zero, $D\in\mathcal{D}_{c}$.
Moreover, for any $\epsilon>0$, there is $N=N(\epsilon)>0$ such that for any
$i>N$, the following properties hold:
(i) $D\subset D_{i}^{\epsilon}$.
(ii) $(D_{i})_{\epsilon}\subset D$.
(iii) $D_{i}\subset D^{\epsilon}$.
In particular, $d^{H}(D_{i},D)\rightarrow 0$ as $i\rightarrow\infty$.

###### Proof.

We first identify $D$ with its $\mathcal{H}^{n}$-equivalent set in the sense
of Proposition 2.11. We argue by contradiction.

If (i) were false, then there would exist $\epsilon_{0}>0$, $x_{0}\in D$ and a
sequence $k\rightarrow\infty$ such that $B_{\epsilon_{0}}(x_{0})\cap
D_{k}=\emptyset$. Hence by the hypothesis and Proposition 2.11, we obtain that

$0=|B_{\epsilon}(x_{0})\cap D_{k}|\rightarrow|B_{\epsilon}(x_{0})\cap D|>0,$

this is impossible.

If (ii) were false, then there would exist $\epsilon_{0}>0$ and a sequence of
points $x_{i}\in(D_{i})_{\epsilon_{0}}\setminus D$. Assume that
$x_{i}\rightarrow x_{0}$. Then $x_{0}\in\partial D\cup D^{c}$. Hence by the
proof of Proposition 2.11, we have that
$\omega_{n}\epsilon_{0}^{n}>|B_{\epsilon_{0}}(x_{0})\cap D|$. On the other
hand, since $B_{\epsilon_{0}}(x_{i})\subset D_{i}$, we have that

$\displaystyle\big{|}B_{\epsilon_{0}}(x_{0})\cap D\big{|}$ $\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}\big{|}B_{\epsilon_{0}}(x_{i})\cap
D\big{|}\geq\liminf_{i\rightarrow\infty}\big{(}|B_{\epsilon_{0}}(x_{i})\cap
D_{i}|-|D_{i}\Delta D|\big{)}$ $\displaystyle=$
$\displaystyle\omega_{n}\epsilon_{0}^{n}-\limsup_{i\rightarrow\infty}|D_{i}\Delta
D|=\omega_{n}\epsilon_{0}^{n}.$

We get a desired contradiction.

If (iii) were false, then there would exist $\epsilon_{0}>0$ and a subsequence
of $x_{i}\in D_{i}\setminus D^{\epsilon_{0}}$. Without loss of generality,
assume $x_{i}\rightarrow x_{0}$ and thus $x_{0}\in\mathbb{R}^{n}\setminus
D^{\epsilon_{0}}$. By Remark 2.7, there is a $c^{\prime}>0$ depending only on
$c$ and $n$ such that

$c^{\prime}\epsilon_{0}^{n}\leq\big{|}B_{\epsilon_{0}}(x_{i})\cap
D_{i}\big{|}.$

On the other hand, it follows from $|B_{\epsilon_{0}}(x_{0})\cap D|=0$ that

$\displaystyle\liminf_{i\rightarrow\infty}\big{|}B_{\epsilon_{0}}(x_{i})\cap
D_{i}\big{|}$ $\displaystyle\leq$
$\displaystyle\limsup_{i\rightarrow\infty}\big{(}|B_{\epsilon_{0}}(x_{i})\cap
D|+|D\Delta D_{i}|\big{)}$ $\displaystyle\leq$
$\displaystyle|B_{\epsilon_{0}}(x_{0})\cap
D|+\limsup_{i\rightarrow\infty}|D_{i}\Delta D|=0.$

This yields a desired contradiction.

It remains to show $D\in\mathcal{D}_{c}$. Indeed, by Proposition 2.11,
$x\in\partial D$ implies that $x\in{\rm{spt}}\mu_{D}$. Note $D_{i}\rightarrow
D$ in $L^{1}(\mathbb{R}^{n})$ implies that
$\mu_{D_{i}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu_{D}$ as
convergence of Radon measures. Hence there exists
$x_{i}\in{\rm{spt}}\mu_{D_{i}}\subset\partial D_{i}$ such that
$x_{i}\rightarrow x$ so that for any $r>0$, it holds that

$\big{|}B_{r}(x)\cap D\big{|}=\lim_{i}\big{|}B_{r}(x_{i})\cap
D\big{|}\geq\liminf_{i}\big{|}B_{r}(x_{i})\cap
D_{i}\big{|}-\limsup_{i}\big{|}D_{i}\Delta D\big{|}\geq cr^{n}.$

This implies $D\in\mathcal{D}_{c}$. ∎

The following remark follows directly from (i) and (iii).

######  Remark 3.2.

If $D_{i}$ and $D$ satisfy the same assumptions as in Lemma 3.1, and if
${\rm{int}}(D)\neq\emptyset$, then ${\rm{int}}(D)$ is connected.

Similar to Lemma 3.1, for a set in the class $\mathcal{D}^{c}$ we have

######  Lemma 3.3.

For $c>0$, if $\\{D_{i}\\}\subset\mathcal{D}^{c}$ and $D_{i}\rightarrow D$ in
$L^{1}(\mathbb{R}^{n})$, then after modifying a set of zero
$\mathcal{H}^{n}$-measure, $D\in\mathcal{D}^{c}$. Moreover, for any
$\epsilon>0$, there is $N=N(\epsilon)>0$ such that if $i>N$, the following
properties holds:
(i) $D\subset D_{i}^{\epsilon}$.
(ii) $(D_{i})_{\epsilon}\subset D$.
(iii) $D_{\epsilon}\subset D_{i}$.

The following corollary follows directly from Lemma 3.3.

######  Corollary 3.4.

For any $c>0$ and a sequence $\\{D_{i}\\}\subset\mathcal{D}^{c}$ with
uniformly bounded perimeters, there is an open set $D\in\mathcal{D}^{c}$ such
that $D_{i}\rightarrow D$ in $L^{1}(\mathbb{R}^{n})$. Moreover, $D$ and
$D_{i}$ satisfy the properties (i),(ii) and (iii) of Lemma 3.3.

Now we are ready to prove the main theorem of this section.

######  Theorem 3.5.

For $M>0$, $R_{0}>0$, and $c_{0}>0$, if $\\{\Omega_{i}\\}$ is a sequence of
$M$-uniform domains in $B_{R_{0}}$ such that $|\Omega_{i}|\geq c_{0}>0$ and
$\Omega_{i}\rightarrow D$ in $L^{1}(\mathbb{R}^{n})$, then there is an
$M$-uniform domain $\Omega$ such that $\Omega_{i}\rightarrow\Omega$ in
$L^{1}(\mathbb{R}^{n})$.

###### Proof.

As in Proposition 2.11, we assume ${\rm{spt}}\mu_{D}=\partial D$. We first
prove that ${\rm{int}}(D)\neq\emptyset$. Indeed, notice that by Remark 2.9,
there exists a $r_{0}>0$ depending only on $c_{0},n$ and $M$ such that each
$\Omega_{i}$ contains a ball of radius $r_{0}$. Therefore, for each
$\Omega_{i}$, if $\epsilon<\frac{r_{0}}{2}$, then by definition
$(\Omega_{i})_{\epsilon}$ contains a ball of radius $\frac{r_{0}}{2}$. By
Lemma 3.1 (ii), $D$ also contains a ball of radius $\frac{r_{0}}{2}$ and hence
${\rm{int}}(D)\neq\emptyset$.

Set $\Omega={\rm{int}}(D)$. It suffices to show that $\Omega$ is an
$M$-uniform domain, since the $L^{1}$ convergence of $\Omega_{i}$ to $\Omega$
follows directly from Remark 2.6, Proposition 2.8, and the fact $\Omega\subset
D\subset\overline{\Omega}$.

Fix any $x,y\in\Omega$, then given any $N>>M$, say $N>2M$, we may choose
$0<\epsilon<\frac{1}{N}$ so small that
$k\epsilon<d(x,\partial\Omega)\leq(k+1)\epsilon$, $k>>N$ (say
$k>(1+1/M)(N+1)$), and $|x-y|>2(N+1)\epsilon$. From Lemma 3.1 (i) and (iii),
and since $int(\Omega)\neq\emptyset$, we know that
$d^{H}(\Omega_{i},\Omega)\rightarrow 0$, hence we we may choose
$x_{i},y_{i}\in\Omega_{i}\cap\Omega$, with
$|x_{i}-x|<\epsilon,|y_{i}-y|<\epsilon$ for $i$ large. By Lemma 3.1 (ii), we
may also choose $i$ large such that

$\displaystyle(\Omega_{i})_{\epsilon}\subset\Omega.$ (3.1)

Also we choose $\gamma_{i}\subset\Omega_{i}$ to be the rectifiable curve
connecting $x_{i}$ and $y_{i}$ in $\Omega_{i}$ as in the definition of
$M$-uniform domain. For any $p\in\gamma_{i}$, if $p\in
B_{N\epsilon}(x_{i})\cup B_{N\epsilon}(y_{i})$, then clearly $p\in
B_{(N+1)\epsilon}(x)\cup B_{(N+1)\epsilon}(y)\subset\Omega$. Moreover, this
implies

$d(p,\partial\Omega)\geq
k\epsilon-(N+1)\epsilon>\frac{1}{M}(N+1)\epsilon\geq\frac{1}{M}\min\\{|p-x|,|p-y|\\}.$
(3.2)

Clearly (3.2) also holds for any $p$ on the line segment between $x_{i}$ and
$x$, and between $y_{i}$ and $y$.

If $p\notin B_{N\epsilon}(x_{i})\cup B_{N\epsilon}(y_{i})$, then
$d(p,\partial\Omega_{i})\geq\frac{1}{M}\min\\{|p-x_{i}|,|p-y_{i}|\\}>\frac{1}{M}N\epsilon$,
thus
$p\in(\Omega_{i})_{N\epsilon/M}\subset(\Omega_{i})_{\epsilon}\subset\Omega\cap\Omega_{i}$.
Moreover, let $r=d(p,\partial((\Omega_{i})_{\epsilon}))$, then by (3.1),
$B_{r}(p)\subset\Omega$, so $d(p,\partial\Omega)\geq
r=d\left(p,\partial((\Omega_{i})_{\epsilon})\right)\geq
d(p,\partial\Omega_{i})-\epsilon$. Therefore,

$\frac{d(p,\partial\Omega)}{\min\\{|p-x_{i}|,|p-y_{i}|\\}}\geq\frac{d(p,\partial\Omega_{i})-\epsilon}{\min\\{|p-x_{i}|,|p-y_{i}|\\}}\geq\frac{1}{M}-\frac{\epsilon}{N\epsilon}\geq\frac{1}{M}-\frac{1}{N}.\\\
$ (3.3)

Hence by the choice of $\epsilon$ and $N$ we have that

$d(p,\partial\Omega)\geq(\frac{1}{M}-\frac{1}{N})(\min\\{|p-x|,|p-y|\\}-\epsilon)\geq(\frac{1}{M}-\frac{1}{N})(\min\\{|p-x|,|p-y|\\})-\frac{1}{MN}.$
(3.4)

Therefore, we may let $\gamma^{N}$ be the curve with three parts. The first
part connects $x$ and $x_{i}$ with line segment, the second part connects
$x_{i}$ and $y_{i}$ with $\gamma_{i}$ as above and the third part connects
$y_{i}$ and $y$ with line segment. It is clear that $\gamma^{N}\subset\Omega$
and $\gamma^{N}$ connects $x$ and $y$, then from (3.2) and (3.4) and the
choice of $\epsilon$, we obtain
(i) $\mathcal{H}^{1}(\gamma^{N})\leq M|x-y|+2\frac{M+1}{N}$, and
(ii)
$d(p,\partial\Omega)\geq(\frac{1}{M}-\frac{1}{N})\min\\{|p-x|,|p-y|\\}-\frac{1}{MN}\quad\forall
p\in\gamma^{N}$.
Then by compactness of $(\overline{\Omega},d^{H})$, and since $\gamma^{N}$ is
connected, there is a compact connected set $E\subset\overline{\Omega}$ such
that $d^{H}(\gamma^{N},E)\rightarrow 0$ as $N\rightarrow\infty$. Then by [14,
Theorem 3.18],

$\mathcal{H}^{1}(E)\leq\liminf_{N\rightarrow\infty}\mathcal{H}^{1}(\gamma^{N})\leq
M|x-y|.$

Then by [14][Lemma 3.12], $E$ is path connected, thus we can choose a curve
$\gamma\subset E$ joining $x$ and $y$. For any $p\in\gamma$, we can choose
sequence $p_{N}\in\gamma^{N},p_{N}\rightarrow p$. Since

$d(p_{N},\partial\Omega)\geq(\frac{1}{M}-\frac{1}{N})\min\\{|p_{N}-x|,|p_{N}-y|\\}-\frac{1}{2MN},$

we have, after sending $N\rightarrow\infty$,

$d(p,\partial\Omega)\geq\frac{1}{M}\min\\{|p-x|,|p-y|\\},$

which also clearly implies $\gamma\subset int\,\Omega$. Then $\gamma$
satisfies both properties in the definition of $M$-uniform domain, thus
$\Omega$ is $M$-uniform. By Remark 3.2 and Proposition 2.8, $\Omega$ is a
domain. This completes the proof. ∎

######  Remark 3.6.

The full generality of compactness of $M$-uniform domains is obtained in [11,
Theorem 1.2], where it is shown that any sequence of $M$-uniform domains with
fixed volume must have uniformly bounded fractional perimeters, and thus have
an $L^{1}$ limit up to a subsequence, and the limit is also $M$-uniform.

##  4\. Existence of equilibrium liquid crystal droplets in Problem A-C

In this section we will study the existence of minimizers to Problems A-C,
which can be extended in $n$-dimensions. We begin with the following Lemma,
which plays a crucial role in Problems A-C over outer minimal sets.

######  Lemma 4.1.

For $c>0$, let $\\{E_{i}\\}_{i=1}^{\infty}\in\mathcal{D}_{c}$ be a sequence of
outward-minimizing sets such that $E_{i}\rightarrow E$ in $L^{1}$ as
$i\to\infty$. Then $E\in\mathcal{D}_{c}$ is also an outward-minimizing set.
Moreover, $P(E_{i})\rightarrow P(E)$ and
$\mathcal{H}^{n-1}(\partial_{*}E_{i})\rightarrow\mathcal{H}^{n-1}(\partial_{*}E)$
as $i\to\infty$.

###### Proof.

Let $F\supset E$. Then by [2, Proposition 3.38(d)] and the outward-minimality
of $E_{i}$ we have

$P(E_{i}\cap F)\leq P(F)+P(E_{i})-P(E_{i}\cup F)\leq P(F).$

This implies

$P(E)=P(E\cap F)\leq\liminf_{i}P(E_{i}\cap F)\leq F(F).$

Hence $E$ is outward-minimizing. By Lemma 3.1 and Remark 2.13,
$E\in\mathcal{D}_{c}\cap\mathcal{D}^{c}$. It follows from Proposition 2.16
that for any $\epsilon>0$, there exists a smooth open set $O_{\epsilon}\Supset
E$ such that

$P(O_{\epsilon})\leq P(E)+\epsilon.$

Applying Lemma 3.1 (iii), we have that there exists a sufficiently large
$i_{0}\geq 1$ such that

$E_{i}\subset O_{\epsilon},\forall i\geq i_{0}.$

This, combined with the outward minimality of $E_{i}$, implies

$P(E_{i})\leq P(O_{\epsilon})\leq P(E)+\epsilon,\ \forall i\geq i_{0}.$

Thus

$\limsup_{i}P(E_{i})\leq P(E).$

On the other hand, by lower semicontinuity we have

$P(E)\leq\liminf_{i}P(E_{i}).$

Therefore $P(E_{i})\rightarrow P(E)$ as $i\to\infty$.

Since $E_{i},E\in\mathcal{D}_{c}\cap\mathcal{D}^{c}\ $, the last statement
follows from Theorem 2.1.

∎

Now we are ready to state the main theorem of this section.

######  Theorem 4.2.

The following statements hold:

  * i)

For $M\geq 1$, the infimum of Problem C in the class of $M$-uniform domains of
finite perimeter is attained.

  * ii)

For $M>1$, the infimum of Problems A, B, C can be attained in the class of
$M$-uniform outer minimal domains.

###### Proof.

We first prove i). For a minimizing sequence $(\Omega_{i},u_{i})$, where
$\Omega_{i}$ are $M$-uniform domains with finite perimeter and $u_{i}\in
H^{1}(\Omega_{i},\mathbb{S}^{2})$. Let $\hat{u}_{i}\in
H^{1}(B_{R_{0}},\mathbb{R}^{3})$ be an extension of $u_{i}$ such that

$\|\hat{u}_{i}\|_{H^{1}(B_{R_{0}})}\leq C(n,M)\|u_{i}\|_{H^{1}(\Omega_{i})}.$

Hence there is a $\hat{u}\in H^{1}(B_{R_{0}},\mathbb{R}^{3})$ such that

$\hat{u}_{i}\rightharpoonup\hat{u}\ {\rm{in}}\ H^{1}(B_{R_{0}}).$

By Theorem 3.5, there is an $M$-uniform domain $\Omega\subset B_{R_{0}}$ such
that $\Omega_{i}\rightarrow\Omega$ in $L^{1}$. Since
$\nabla\hat{u}_{i}\rightharpoonup\nabla\hat{u}$ in $L^{2}(B_{R_{0}})$ and
$\chi_{\Omega_{i}}\rightarrow\chi_{\Omega}$ in $L^{1}(B_{R_{0}})$, by the
lower semicontinuity we have that

$\displaystyle\int_{\Omega}|\nabla\hat{u}|^{2}\leq\liminf_{i\rightarrow\infty}\int_{\Omega_{i}}|\nabla\hat{u}_{i}|^{2}=\liminf_{i\rightarrow\infty}\int_{\Omega_{i}}|\nabla{u}_{i}|^{2}.$
(4.1)

Denote $u=\hat{u}\big{|}_{\Omega}$. Then it is not hard to see $|u|=1$ for
a.e. $x\in\Omega$ so that $u\in H^{1}(\Omega,\mathbb{S}^{2})$. In order to
show $(\Omega,u)$ is a minimizer of Problem (C) among $M$-uniform domains of
finite perimeter, we have to verify that $u^{*}=\nu_{\Omega}$ for
$\mathcal{H}^{n-1}$-a.e. on $\partial^{*}\Omega$. In fact, it follows from
$\chi_{\Omega_{i}}\rightarrow\chi_{\Omega}$ in $L^{2}(B_{R_{0}})$ and
$div(\hat{u}_{i})\rightharpoonup div(\hat{u})$ in $L^{2}(B_{R_{0}})$ and
Theorem 2.4 that

$\displaystyle
P(\Omega_{i})=\int_{\Omega_{i}}div({u}_{i})=\int_{B_{R_{0}}}\chi_{\Omega_{i}}div(\hat{u}_{i})$
$\displaystyle\rightarrow$
$\displaystyle\int_{B_{R_{0}}}\chi_{\Omega}div(\hat{u})=\int_{\Omega}div(u)$
$\displaystyle=$
$\displaystyle\int_{\partial^{*}\Omega}u^{*}\cdot\nu_{\Omega}\,d\mathcal{H}^{n-1}\leq
P(\Omega).$

This, combined with the lower semicontinuity property of perimeter, implies
that $u^{*}=\nu_{\Omega}$ for $\mathcal{H}^{n-1}$-a.e. on
$\partial^{*}\Omega$. Hence the proof of i) is complete.

Next, we prove ii). For Problem A in part ii), let $(\Omega_{h},u_{h})$ be a
minimizing sequence among $M$-uniform, outer minimal domains and $H^{1}$-unit
vector fields on $\Omega_{h}$. Since $\Omega_{h}$ are outward-minimizing sets
in $B_{R_{0}}$, $P(\Omega_{h})$ are uniformly bounded. By Lemma 4.1 and
Theorem 3.5, we may assume that there exists an $M$-uniform, outer minimal
domain $\Omega$ such that up to a subsequence, $\Omega_{h}\rightarrow\Omega$
in $L^{1}$ and $P(\Omega_{h})\rightarrow P(\Omega)$. As in the proof of i)
above, we may extend $u_{h}$ in $B_{R_{0}}$, still denoted as $u_{h}$, so that
$u_{h}\rightharpoonup u$ in $H^{1}(B_{R_{0}},\mathbb{R}^{3})$ for some $u\in
H^{1}(B_{R_{0}},\mathbb{R}^{3})$. Thus we have

$\int_{\Omega}|\nabla u|^{2}\leq\liminf_{h}\int_{\Omega_{h}}|\nabla
u_{h}|^{2},$

and $u(x)\in\mathbb{S}^{2}$ for a.e. $x\in\Omega$.

Since $f$ is convex, we can write

$f(x)=\sup_{i}(a_{i}x+b_{i}).$

In the following, we do not distinguish $u$ with $u^{*}$ on
$\partial^{*}\Omega$, and we do not distinguish
$\partial^{*}\Omega_{h},\partial^{*}\Omega$ with
$\partial\Omega_{h},\partial\Omega$ due to Remark 2.15. Define

$\tau_{h}(A):=\mathcal{H}^{n-1}(\partial^{*}\Omega_{h}\cap A),\
\tau(A):=\mathcal{H}^{n-1}(\partial^{*}\Omega\cap A),\ {\rm{and}}\
\mu_{h}(A):=\int_{A}f(u_{h}\cdot\nu_{h})d\tau_{h},$

for any measurable $A\subset\mathbb{R}^{n}$, where $\nu_{h}$ is the measure
theoretical outer unit normal of $\Omega_{h}$. Then Lemma 4.1 implies that

$\displaystyle\tau_{h}(A)\rightarrow\tau(A)\quad\mbox{as
$h\rightarrow\infty$}.$ (4.2)

Since $f$ is bounded and nonnegative, $\mu_{h}$ are nonnegative Radon measures
so that we may assume there is a nonnegative Radon measure $\mu$ such that
after passing to a subsequence, $\mu_{h}\rightharpoonup\mu$ as $h\to\infty$ as
weak convergence of Radon measures. Decompose $\mu$ as
$\mu=(D_{\tau}\mu)\tau+\mu^{s},\mu^{s}\perp\tau$, and $\mu^{s}\geq 0$. Then

$\displaystyle\liminf_{h\rightarrow\infty}\mu_{h}(A)\geq\mu(A)\geq\int_{A}D_{\tau}\mu
d\tau.$ (4.3)

It follows from Theorem 2.1 that $x\in\partial^{*}\Omega$ holds for
$\tau$-a.e. $x\in B_{R_{0}}$. Now any such $x\in\partial^{*}\Omega$, we claim
that there exists $r_{j}\to 0$ such that for $B_{j}=B_{r_{j}}(x)$, it holds
that

  * (a)

$\mathcal{H}^{n-1}(\partial B_{j}\cap\partial\Omega)=0$ and
$\mathcal{H}^{n-1}(\partial B_{j}\cap\partial\Omega_{h})=0,\,\forall h\geq 1$.

  * (b)

$\displaystyle\int_{\partial
B_{j}\cap\Omega_{h}}u_{h}\cdot\nu_{B_{j}}\,d\mathcal{H}^{n-1}\rightarrow\int_{\partial
B_{j}\cap\Omega}u\cdot\nu_{B_{j}}\,d\mathcal{H}^{n-1}$ as
$h\rightarrow\infty$.

  * (c)

$\mu(\partial B_{j})=0$.

  * (d)

$D_{\tau}\mu(x)=\displaystyle\lim_{j}\frac{\mu(B_{j})}{\tau(B_{j})}$ and
$\displaystyle\lim_{j\rightarrow\infty}\frac{\int_{B_{j}}u\cdot\nu_{B_{j}}d\tau}{\tau(B_{j})}=u(x)\cdot\nu(x)$.

Indeed, (a) and (c) are true because $\tau,\tau_{h},$ and $\mu$ are
nonnegative Radon measures. (d) follows from the Lebesgue differentiation
Theorem. To see (b), let $\tilde{u}_{h}=u_{h}\chi_{\Omega_{h}}$ and
$\tilde{u}=u\chi_{\Omega}$. Since $\tilde{u}_{h}\rightarrow\tilde{u}$ in
$L^{1}$, we have

$\displaystyle\int_{B_{1}(x)}|\tilde{u_{h}}-\tilde{u}|=\int_{0}^{1}\int_{\partial
B_{r}(x)}|\tilde{u}_{h}-\tilde{u}|d\mathcal{H}^{n-1}dr\rightarrow
0\quad\mbox{as $h\rightarrow\infty$}.$

Therefore by Fatou’s Lemma,

$\int_{0}^{1}\liminf_{h\rightarrow\infty}\int_{\partial
B_{r}(x)}|\tilde{u}_{h}-\tilde{u}|d\mathcal{H}^{n-1}\,dr=0,$

hence for almost every $r\in(0,1)$ and for a subsequence of
$h\rightarrow\infty$,

$\displaystyle\big{|}\int_{\partial
B_{r}(x)\cap\Omega_{h}}u_{h}\cdot\nu_{B_{r}(x)}\,d\mathcal{H}^{n-1}-\int_{\partial
B_{r}(x)\cap\Omega}u\cdot\nu_{B_{r}(x)}\,d\mathcal{H}^{n-1}\big{|}$
$\displaystyle\leq\int_{\partial
B_{r}(x)}|\tilde{u}_{h}-\tilde{u}|d\mathcal{H}^{n-1}\rightarrow 0.$

This finishes the proof of (b). Now we return to the proof of v). By (c),

$\displaystyle\mu(B_{j})=\lim_{h\rightarrow\infty}\mu_{h}(B_{j})=\lim_{h\rightarrow\infty}\int_{\partial\Omega_{h}\cap
B_{j}}f(u_{h}\cdot\nu_{h})\,d\mathcal{H}^{n-1}.$

Also as $h\rightarrow\infty$, up to a subsequence we have

$\displaystyle\int_{\partial\Omega_{h}\cap
B_{j}}u_{h}\cdot\nu_{h}\,d\mathcal{H}^{n-1}$
$\displaystyle=\int_{\partial(\Omega_{h}\cap
B_{j})}u_{h}\cdot\nu_{\Omega_{h}\cap B_{j}}\,d\mathcal{H}^{n-1}-\int_{\partial
B_{j}\cap\Omega_{h}}u_{h}\cdot\nu_{B_{j}}\,d\mathcal{H}^{n-1},$
$\displaystyle=\int_{\Omega_{h}\cap B_{j}}divu_{h}-\int_{\partial
B_{j}\cap\Omega_{h}}u_{h}\cdot\nu_{B_{j}}\,d\mathcal{H}^{n-1},$
$\displaystyle\rightarrow\int_{\Omega\cap B_{j}}divu-\int_{\partial
B_{j}\cap\Omega}u\cdot\nu_{B_{j}}\,d\mathcal{H}^{n-1},$
$\displaystyle=\int_{\partial(\Omega\cap B_{j})}u\cdot\nu_{\Omega\cap
B_{j}}\,d\mathcal{H}^{n-1}-\int_{\partial
B_{j}\cap\Omega}u\cdot\nu_{B_{j}}\,d\mathcal{H}^{n-1}$
$\displaystyle=\int_{\partial\Omega\cap
B_{j}}u\cdot\nu_{\Omega}\,d\mathcal{H}^{n-1}.$

Therefore, for $\tau$-a.e. $x\in B_{R_{0}}$, it follows

$\displaystyle D_{\tau}\mu(x)$ $\displaystyle=$
$\displaystyle\lim_{j}\frac{\mu(B_{j})}{\tau(B_{j})}$ $\displaystyle=$
$\displaystyle\lim_{j}\lim_{h}\frac{\int_{\partial\Omega_{h}\cap
B_{j}}f(u_{h}\cdot\nu_{h})\,d\mathcal{H}^{n-1}}{\mathcal{H}^{n-1}(\partial\Omega\cap
B_{j})}$ $\displaystyle\geq$
$\displaystyle\lim_{j}\lim_{h}\frac{\int_{\partial\Omega_{h}\cap
B_{j}}(a_{i}u_{h}\cdot\nu_{h}+b_{i})\,d\mathcal{H}^{n-1}}{\mathcal{H}^{n-1}(\partial\Omega\cap
B_{j})}$ $\displaystyle=$ $\displaystyle\lim_{j}\frac{\int_{\partial\Omega\cap
B_{j}}(a_{i}u\cdot\nu_{\Omega}+b_{i})\,d\mathcal{H}^{n-1}}{\mathcal{H}^{n-1}(\partial\Omega\cap
B_{j})},\quad\mbox{also by}\,\eqref{0}$ $\displaystyle=$ $\displaystyle
a_{i}u(x)\cdot\nu_{\Omega}(x)+b_{i}.$

Hence $D_{\tau}\mu\geq f(u\cdot\nu_{\Omega})$ for $\tau$\- a.e. $x\in
B_{R_{0}}$, and

$\displaystyle\liminf_{h}\int_{\partial\Omega_{h}}f(u_{h}\cdot\nu_{h})\,d\mathcal{H}^{n-1}$
$\displaystyle=$
$\displaystyle\liminf_{h}\mu_{h}(B_{R_{0}})\geq\int_{B_{R}}D_{\tau}\mu d\tau$
(4.5) $\displaystyle\geq$
$\displaystyle\int_{B_{R_{0}}}f(u\cdot\nu)d\tau=\int_{\partial\Omega}f(u\cdot\nu)\,d\mathcal{H}^{n-1}.$

Therefore, $(\Omega,u)$ is a minimizer.

To complete the proof of statements in ii), it remains to show if
$(\Omega_{i},u_{i})$ are a minimizing sequence in Problem (B) and converges
weakly to $(\Omega,u)$, then $u\cdot\nu=c$ for $\mathcal{H}^{n-1}$-a.e. on
$\partial^{*}\Omega$. This can be seen from

$\liminf_{i\rightarrow\infty}\int_{\partial^{*}\Omega_{i}}f(u_{i}\cdot\nu_{i})d\mathcal{H}^{n-1}\geq\int_{\partial^{*}\Omega}f(u\cdot\nu)d\mathcal{H}^{n-1}.$

In fact, by choosing $f(t)=\mu(t-c)^{2}$ we have that

$\displaystyle\int_{\partial^{*}\Omega}(u\cdot\nu-c)^{2}\,d\mathcal{H}^{n-1}\leq\liminf_{i\rightarrow\infty}\int_{\partial^{*}\Omega_{i}}(u_{i}\cdot\nu_{i}-c)^{2}\,d\mathcal{H}^{n-1}=0.$
(4.6)

Hence $u\cdot\nu\equiv c$ for $\mathcal{H}^{n-1}$-a.e. on
$\partial^{*}\Omega$. This completes the proof. ∎

##  5\. On the uniqueness of Problem C

In this section, we will show the uniqueness of Problem C in the class of
$C^{1,1}$-star-shaped, mean convex domains in $\mathbb{R}^{3}$. We will assume
the domains has volume $V_{0}=|B_{1}|$, where $B_{1}\subset\mathbb{R}^{3}$ is
the unit ball centered at $0$. We begin with

######  Lemma 5.1.

For any bounded $C^{1,1}$-domain $\Omega\subset\mathbb{R}^{3}$,

$\inf\big{\\{}\int_{\Omega}|\nabla u|^{2}\ \big{|}\ u\in
H^{1}(\Omega,\mathbb{S}^{2}),u=\nu_{\Omega}\,\mbox{on
$\partial\Omega$}\big{\\}}\geq\int_{\partial\Omega}H_{\partial\Omega}\,d\mathcal{H}^{2},$
(5.1)

where $H_{\partial\Omega}$ is the mean curvature of $\partial\Omega$.

###### Proof.

Let $u\in H^{1}(\Omega,\mathbb{S}^{2})$, with $u=\nu_{\Omega}$ on
$\partial\Omega$, be such that

$\int_{\Omega}|\nabla u|^{2}=\inf\big{\\{}\int_{\Omega}|\nabla u|^{2}\
\big{|}\ u\in H^{1}(\Omega,\mathbb{S}^{2}),u=\nu_{\Omega}\,\mbox{on
$\partial\Omega$}\big{\\}}.$

Then by [40, 41], $u\in
C^{\infty}(\Omega\setminus\\{a_{i}\\}_{i=1}^{N},\mathbb{S}^{2})$ for a finite
set $\displaystyle\cup_{i=1}^{N}\\{a_{i}\\}\Subset\Omega$. Observe that

$(div(u))^{2}-tr(\nabla u)^{2}=div(div(u)u-(\nabla u)u)\ \ \ {\rm{in}}\ \
\Omega\setminus\cup_{i=1}^{N}\\{a_{i}\\}.$

By [35, Proposition 2.2.1], we have that

$|\nabla u|^{2}\geq(divu)^{2}-tr(\nabla u)^{2}\ \ \ {\rm{in}}\ \
\Omega\setminus\cup_{i=1}^{N}\\{a_{i}\\}.$

By [3, Theorem 1.9], near each $a_{i}$, $u(x)\sim
R(\frac{x-a_{i}}{|x-a_{i}|})$ for some rotation $R\in O(3)$. In particular,
one has that for $r>0$ sufficiently small,

$\Big{|}\int_{\partial B_{r}(a_{i})}(div(u)u-(\nabla
u)u)\cdot\nu_{B_{r}(a_{i})}\,d\mathcal{H}^{2}\Big{|}=O(r).$

Hence

$\displaystyle\int_{\Omega}|\nabla u|^{2}$
$\displaystyle\geq\int_{\Omega\setminus\cup_{i=1}^{N}B_{r}(a_{i})}(div(u))^{2}-tr(\nabla
u)^{2}$
$\displaystyle=\int_{\Omega\setminus\cup_{i=1}^{N}B_{r}(a_{i})}div((divu)u-(\nabla
u)u)$ $\displaystyle=\int_{\partial\Omega}(div(u)u-(\nabla
u)u)\cdot\nu_{\Omega}\,d\mathcal{H}^{2}$ $\displaystyle\
-\sum_{i=1}^{n}\int_{\partial B_{r}(a_{i})}(div(u)u-(\nabla
u)u)\cdot\nu_{B_{r}(a_{i})}\,d\mathcal{H}^{2}$
$\displaystyle\geq\int_{\partial\Omega}\big{(}div(u)-((\nabla
u)\nu_{\Omega})\cdot\nu_{\Omega}\big{)}\,d\mathcal{H}^{2}-CNr$
$\displaystyle=\int_{\partial\Omega}\big{(}div_{\partial\Omega}\nu_{\Omega}\big{)}\,d\mathcal{H}^{2}-CNr=\int_{\partial\Omega}H_{\partial\Omega}\,d\mathcal{H}^{2}-CNr.$

This implies (5.1) after sending $r\rightarrow 0$. ∎

The inequality (5.1) leads us to study the minimization of the total mean
curvatures. It is well-known that

$\int_{\partial\Omega}H_{\partial\Omega}\,d\mathcal{H}^{2}\geq 4\sqrt{\pi
P(\Omega)}$ (5.2)

is true if $\Omega$ is convex, and the equality holds if and only if $\Omega$
is a ball. Very recently, Dalphin-Henrot-Masnou-Takahashi [10] proved that if
$\Omega$ is a revolutionary solid and $H\geq 0$, then (5.2) is true, and the
equality holds if and only if $\Omega$ is a ball. Without the mean convexity,
(5.2) is false, see [10]. In the next lemma we present a proof that (5.2) is
true if $\Omega$ is a $C^{1,1}$ star-shaped and mean convex domain. The key
ingredient of the proof is based on the result by Gerhardt [21]. We remark
that a more general version of (5.2) has been proven by Guan-Li [23]. Here we
will sketch the proof, since it is elementary in $\mathbb{R}^{3}$.

######  Lemma 5.2.

The inequality (5.2) holds, if $\Omega$ is $C^{1,1}$-strictly star-shaped and
mean convex.

###### Proof.

By the remark below, we may assume $\Omega\in C^{\infty}$. By a standard
argument, we can perturb $\Omega$ so that $H>0$ everywhere. Indeed, represent
$\partial\Omega$ as an embedding $F^{0}:\mathbb{S}^{2}\to\mathbb{R}^{3}$ and
consider the mean curvature flow
$\\{F_{t}:\mathbb{S}^{2}\to\mathbb{R}^{3}:t\in[0,T)\\}$, which is a family of
embeddings so that

$\frac{\partial F}{\partial t}=H\nu_{t}\ \ 0<t<T;\ \ F_{0}=F^{0},$

where $\nu_{t}$ is the inward unit normal of the embedding $F_{t}$. It is
well-known that the solution exists for a short time $T>0$. If $t>0$ is small,
then $F_{t}(\mathbb{S}^{2})$ remains to be star-shaped. The evolution of the
mean curvature $H$ of $F_{t}(\mathbb{S}^{2})$ is given by

$\frac{\partial H}{\partial t}=\Delta H+|A|^{2}H,$

where $A$ is the second fundamental form of $F_{t}(\mathbb{S}^{2})$. Then the
strong maximum principle implies that $H>0$ everywhere on
$F_{t}(\mathbb{S}^{2})$ for $t>0$. It is clear that after a small perturbation
in $C^{1}$-norm, $\Omega$ is still strictly star-shaped.

Hence it suffices to prove (5.2) by assuming $H>0$ everywhere on
$\partial\Omega$. We argue it by contradiction. Suppose there were a strictly
star-shaped domain $\Omega$ with $H>0$ everywhere on $\partial\Omega$ such
that

$\frac{\int_{\partial\Omega}H\,d\mathcal{H}^{2}}{4\sqrt{\pi P(\Omega)}}<1.$

Representing $\partial\Omega$ as an embedding
$G_{0}:\mathbb{S}^{2}\to\mathbb{R}^{3}$. Now consider the inverse mean
curvature flow $\\{G_{t}:\mathbb{S}^{2}\to\mathbb{R}^{3}:t\in[0,\infty)\\}$,
which is a family of embeddings that solves

$\frac{\partial G}{\partial t}=\frac{1}{H}\nu_{t},$

where $\nu_{t}$ is the inward unit normal of the embedding $G_{t}$. It has
been shown by Gerhardt [21] that $S_{t}:=G_{t}(\partial\Omega)$ converges to
the unit sphere $\mathbb{S}^{2}$, up to rescalings by $e^{-t/2}$, as
$t\to\infty$. Set

$y(t)=\frac{\int_{S_{t}}H\,d\mathcal{H}^{2}}{4\sqrt{\pi Area(S_{t})}},\ t>0.$

Observe that $y(t)$ is scaling-invariant. Therefore, $y(0)<1$ and
$y(t)\rightarrow 1$ as $t\to\infty$. On the other hand, using the evolution
equations under the inverse mean curvature flow we have that

$\frac{d}{dt}H=-\Delta H-\frac{|A|^{2}}{H},$

and

$\frac{d}{dt}\sqrt{g}=\sqrt{g},$

where $\Delta$ is the surface Laplacian and $g$ is the metric on surface
$S_{t}$ induced by Euclidean metric in $\mathbb{R}^{3}$. Direct calculations
imply

$\displaystyle\frac{d}{dt}\left(\frac{\int_{S_{t}}H\,d\mathcal{H}^{2}}{4\sqrt{\pi
P(\Omega)}}\right)$ $\displaystyle=$
$\displaystyle\Big{(}\int_{S_{t}}\big{(}H-\frac{|A|^{2}}{H}\big{)}\,d\mathcal{H}^{2}\Big{)}\frac{1}{4\sqrt{\pi
Area(S_{t})}}-\frac{\int_{S_{t}}H\,d\mathcal{H}^{2}}{8\sqrt{\pi Area(S_{t})}}$
$\displaystyle=$ $\displaystyle\frac{1}{4\sqrt{\pi
Area(S_{t})}}\left(\int_{S_{t}}\frac{2K}{H}\,d\mathcal{H}^{2}-\frac{1}{2}\int_{S_{t}}H\,d\mathcal{H}^{2}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{4\sqrt{\pi
Area(S_{t})}}\int_{S_{t}}\frac{4K-H^{2}}{2H}\,d\mathcal{H}^{2}\leq 0,$

since $H^{2}\geq 4K$, here $K$ is the Gauss curvature of $S_{t}$. Therefore,
$y(t)\leq y(0)<1$ for all $t>0$. We get a desired contradiction. ∎

######  Remark 5.3.

(5.2) is actually true for any $C^{1}$-strictly star-shaped surface with
bounded nonnegative generalized mean curvature, in particular for a
$C^{1,1}$-mean convex surface. Indeed, by [27, Lemma 2.6], we can find a
family of smooth strictly star-shaped mean convex hypersurfaces converging to
the surface uniformly in $C^{1,\alpha}\cap W^{2,p}$ for $0<\alpha<1$ and
$1<p<\infty$ so that the total mean curvature of the smooth surfaces converges
to the total mean curvature of the original surface. We refer the reader to
[27] for the detail.

By Lemma 5.2 and the isoperimetric inequality $P(\Omega)\geq
4\pi(\frac{3}{4\pi}|\Omega|)^{2/3}$, we immediately have

######  Corollary 5.4.

It holds that

$\displaystyle\inf\\{\int_{\Omega}|\nabla u|^{2}:\mbox{$\Omega$ is
$C^{1,1}$-star-shaped, mean convex},|\Omega|=|B_{1}|,u\in
H^{1}(\Omega,\mathbb{S}^{2}),$ $\displaystyle\qquad\qquad
u=\nu_{\Omega}\,\mbox{on $\partial\Omega$}\\}\geq 8\pi,$

and the equality holds if and only if $\Omega=B_{1}$, up to translation and
rotation.

As a consequence, we have

######  Theorem 5.5.

The Problem (C) over $C^{1,1}$-star-shaped and mean convex domains is uniquely
achieved at $\Omega=B_{1}$ and $u(x)=\frac{x}{|x|}$.

###### Proof.

By direct calculations,

$\int_{B_{1}}|\nabla(\frac{x}{|x|})|^{2}=\int_{B_{1}}\frac{2}{|x|^{2}}=8\pi.$

Hence by the first statement in Corollary 5.4, (4.2) is attained at
$(B_{1},\frac{x}{|x|})$. The uniqueness follows from the last statement of
Corollary 5.4 and [8, Theorem 7.1]. ∎

######  Remark 5.6.

Huisken first proves that (5.2) holds if $\Omega$ is $C^{1,1}$-outer minimal
(not necessarily connected), though it seems that he didn’t publish it. See
also Freire-Schwartz [16, Theorem 5]. Hence the same result as in Theorem 5.5
holds in the class of $C^{1,1}$-outer minimal open sets. By [10], the same
result as in Theorem 5.5 holds in the class of smooth domains of revolution.

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# An Interactive Web-Based System for Creating Single Panel Cartoons with
Visually Valid Compositions

Ergun Akleman
Visual Computing & Computational Media,
Texas A&M University, College Station, TX, 77831
<EMAIL_ADDRESS>
Akhilesh Vijaykumar
Visualization Department,
Texas A&M University, College Station, TX, 77831
<EMAIL_ADDRESS>
Richard Furuta
Computer Science and Engineering Department,
Texas A&M University, College Station, TX, 77831
<EMAIL_ADDRESS>
Derya Akleman
Statistics Department,
Texas A&M University, College Station, TX, 77831
<EMAIL_ADDRESS>
Joint with Computer Science and Engineering Department.

###### Abstract

The creation of cartoon-based stories (comics) requires a lot of creativity
and hard work for naive users Akleman et al. (2015). We observe that single-
panel cartoons are the building blocks of any comic story. To develop a strong
comic story, it is critical to obtain visually valid single panels Liu et al.
(2012). In this work, we have developed a methodology to guarantee the
placement of characters to obtain a valid cartoon frame based on the methods
used by professional cartoonists. Using this methodology, we have developed a
web-based system to create single-panel cartoons from a given set of character
images. We have made this system available in GitHub as open-source so that
this basic single-panel cartoon can be used as infrastructure to develop more
complex structures. Our web-based system for single-panel cartoons can be
viewed at http://storytelling.viz.tamu.edu.

_Keywords_ Cartoon Panels, Caricature

|  |  |
---|---|---|---
(A) Internal Left | (B) External Left 1 | (C) External Left 2 | (D) Apex
|  |  |
(F) External Right 1 | (G) External Right 2 | (H) External Right 3 | (I) Internal Right
Figure 1: An example of character placement in a single panel with two
characters using our single parameter solution. This one-parameter character
placement is based on a continuous interpolation of conceptual 2D versions of
3D camera views: from internal view to external view and from external view to
apex.

Figure 2: Our motivation for visual validity came from William Hogart’s
Engraving called “Satire on False Perspective. Hogart demonstrated that
anybody who does not have expert illustration knowledge is liable to create
non-valid compositions.

##  1 Introduction and Motivation

Comics and cartoons as a distinct form of art emerged during $17^{th}$ and
$18^{th}$ centuries by great artists such as William Hogarth, James Gillray,
Thomas Rowlandson, and George Cruikshank Smolderen (2014); Rowson (2015); Horn
(1976). Despite their long history, comics and cartoons are one of the least
understood art forms that require not only a significant amount of
illustration and drawing experience but also a good understanding of visual
storytelling. One of the key elements of cartoons and comics is the creation
of valid single cartoon or comic panels that can provide strong visual
storytelling experience Carrier and Oliker (2001); Eisner (2008); McCloud
(1993); McLoud (2000); McCloud (2006); Akleman and Celik (2020). There exist
cartoon interfaces such as ToonyTool Schoonhoven (2021) that provide complete
freedom to place images in a panel. Users translate, scale, and rotate
multiple images to produce a single panel. However, with such complete
freedom, it is hard for naive users to compose visually valid panels that can
be visually compelling. There is a need for a simple approach to guarantee
that such naive users can obtain visually valid and compelling panels by
specifying only a few parameters.

The concept of visual validity was introduced by English painter and
cartoonist William Hogarth (1697-1764) in his 1754 engraving called “Satire on
False Perspective” Cornew (2001); Hogarth (1754). In the subtitle, Hogarth
wrote that “Whoever makes a Design without the Knowledge of Perspective will
be liable to such Absurdities as are shown in this [engraving]”. There are
more than 20 Absurdities in this image and it is even hard to find all
absurdities. In this paper, we call these absurdities visually non-valid
compositions. Our goal is to guarantee the creation of the only visually valid
compositions.

In this work, our major contribution is the identification of a single
parameter family of visually valid single-panel cartoon compositions based on
professional cartoon expertise. Using this family, we have identified an
approach to obtain appealing single cartoon panels using only one parameter
and developed a web-based prototype. Our approach is based on five essential
3D camera views. These five views are widely used in movies to create dynamic
dialogues. We observe that these particular camera views can be adopted for 2D
character placement. Based on this observation, we have identified 2D
conceptual versions of these five views for any given two images that
represent characters/objects. Our one-parameter solution is a continuous
interpolation between these five essential views. Our solution can be obtained
simply by translation and scaling of these two images. The operation can
recursively be extended to any number of images by creating a binary tree.

|  |  |
---|---|---|---
(A) Line of Interest. | (B) Internal Left. | (C) External Left. |
|  |  |
(E) Apex. | (F) External Right. | (G) Internal Right. |
Figure 3: In 3D, any two objects/characters define a a line of interest as
shown on A. If the camera stays only one side of the line of interest, the
objects/characters consistently stay at the left and right of the panel. In
this case, the yellow object/character will always stay at the left and the
blue object/character will always stay at the right. There are only five
qualitatively distinct camera positions as shown from (B) to (G). We have
observed that we can create a continuous path between 2D conceptual views that
corresponds to these five camera views.

##  2 On the Importance of 3D Camera Views For Storytelling

Five qualitatively different 3D views, internal left view, external left view,
apex view, external right view, and internal right view, are widely used in
movies and comics to create dynamic dialogues Arijon (1991); He et al. (1996)
(see Figure 3). We observed that these five views are key elements to
analyzing any movie and photographic scene in terms of the storytelling point
of view and producing visually valid compositions in cartoons. Assume that the
scene has two essential elements, objects or characters, that are given by
their positions $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$; or a motion that is
given a position and direction vector, $\mathbf{p}_{1}$ and $\vec{v}_{2}$.
These two can be considered exactly the same way by assuming
$\vec{v}_{2}=\mathbf{p}_{2}-\mathbf{p}_{1}$. Now, let the camera position be
given by $\mathbf{p}_{0}$.

|
---|---
Figure 4: Two examples of visually non-valid compositions.

If $|\mathbf{p}_{1}-\mathbf{p}_{0}|=|\mathbf{p}_{2}-\mathbf{p}_{0}|$, we call
this an apex view. In this case, the two objects/characters have exactly the
same importance. On the other hand, If
$|\mathbf{p}_{1}-\mathbf{p}_{0}|>|\mathbf{p}_{2}-\mathbf{p}_{0}|$, the camera
is closer to the object/character 1 and object/character 1 is emphasized.
However, there are two possible cases based on the orientation of the camera:
(1) External view, in which object 2 is still visible but smaller; and (2)
Internal view, in which object 2 is outside of the frame and is invisible in
the frame. In other words, although the external view emphasizes one of the
object(s)/character(s) by giving it more screen space, the internal view has
even stronger emphasis since that particular object/character is the center of
attention by being completely alone in the scene.

Based on these five views, we can control the emphasis given any object or
character; even a single frame. Assume that we have a human and a key. We can
make either the human more important or the key more important in a single
frame. When we have multiple frames, different views create an additional
dynamic and the order of images creates a story that can easily be interpreted
by the viewer. Note that the analysis can be applied to multiple objects and
characters by grouping them using a binary tree. For 2D scene generation, we
must also apply the same rules as if there exists a conceptual camera that can
help us decipher the meaning of the image.

##  3 2D Conceptual Versions of 3D Camera Views

Let $I_{i,j}$ denote $j^{th}$ image of $i^{th}$ object/character given with
transparent background. We also assume that we only know the width and height
of each of these images, denoted by $w_{i,j}$ and $h_{i,j}$ respectively. Each
image is expected to be tightly packed in the sense that there is no space
left at the top, bottom, left, and right. We also expect that objects or
characters can only be cut from the bottom. Our goal is to place them in a
reasonably successful way with this limited amount of rules to produce
visually valid compositions.

|
---|---
Figure 5: Two examples of visually valid compositions that can be used as Apex
views.

Our key observation comes from a fundamental perspective rule. If there is a
ground plane and all objects/characters stand on this plane, the bottom of the
objects that are closer to the camera will be lower in the image. Therefore,
the placement in Figure 4 is not acceptable since the characters that appear
closer to the camera stay higher in the image. On the other hand, the bottom
positions can be the same regardless of their closeness to the camera. We can
always obtain this acceptable solution by putting the camera exactly in the
bottom plane as shown in Figure 5. This method also works with objects cut
from the bottom as shown in Figure 8. Note that the placements shown in
Figures 5 and 8 are actually external views since one of the
objects/characters are closer to the camera than the other. Moreover, the
sizes of objects/characters also matter to classify a view as Apex as shown in
Figure 8. On the other hand, since we do not have any information about
physical objects/characters that images represent, it is not possible to
identify exact Apex views. Therefore, our solution is to classify all cases in
which the bottoms of the two images are exactly in the same position as the
Apex view.

|
---|---

Figure 6: Two examples that demonstrate objects cut from the bottom can be
used as visually valid compositions if they are placed at the bottom. These
frames are considered Apex, although they are really external.

|
---|---

Figure 7: Two examples that demonstrate the actual objects’ sizes do not
matter to obtain visually valid compositions if we place the bottoms of both
images approximately at the same level. These views are actually external.
However, we consider them Apex.

|
---|---

Figure 8: Actual Apex views corresponding to characters in Figure 8. We do not
consider this Apex since the bottoms of images are not at the same level. Note
that the images at the left are cropped. Their bottom positions are below.

Our method is not really an interpolation of two end views, but it is an
extrapolation of an Apex view into both external directions. We start with an
acceptable Apex position. We then extrapolate this Apex view towards both the
external right and external left. This extrapolation can simply be done by
continuous translation and scaling. In the next section, we present a special
Apex view without a loss of generality.

|
---|---
(A) Apex View. | (B) External right.
Figure 9: The specific Apex view that we use in the presentation of the
methodology and an Exterior view that is obtained by scaling, translating and
cropping the two images.

##  4 Methodology

We present our extrapolation process starting with the Apex view into two
images that have the same height and that are placed side by side without a
gap as shown in Figure 9(A). We can always make any two images the same
height, say $H$ by resizing them appropriately. Let these two images be
denoted by $I_{1}$ and $I_{2}$; and their initial width and height be denoted
by $(w_{1},h_{1})$ and $(w_{2},h_{2})$ respectively. Now, let $s_{1}$ and
$s_{2}$ denote uniform scaling parameters with the property $s_{1}=H/h_{1}$
and $s_{2}=H/h_{2}$, we can form a new composite image with width and height
and $W=s_{1}w_{1}+s_{2}w_{2}$ and $H=s_{1}h_{1}=s_{2}h_{2}$. This gives us a
well-defined Apex view panel shown at the left in Figure 9(A).

Now, to produce an external view, we scale down one of the images, say
$I_{1}$. Let $t_{1}$ denote the new scaling parameter, then the new width of
image 1, $w^{\prime}_{1}$ becomes $w^{\prime}_{1}=t_{1}s_{1}w_{1}$. This
leaves a space between image 1 and image 2. We can scale up image 2 just to
fill the space. However, the external view requires that a part of the image
goes out of the panel. If we scale to fill the gap as shown in Figure 9(B), we
obtain an exterior view. The gap created by scaling down image 1 is
$s_{1}w_{1}-w^{\prime}_{1}=(1-t_{1})s_{1}w_{1}$. As a result, the width of
image 2 must be at least $s_{2}w_{2}+(1-t_{1})s_{1}w_{1}$. In other words, the
scaling $t_{2}$ must be at least the following:

$t_{2}\geq\frac{s_{2}w_{2}+(1-t_{1})s_{1}w_{1}}{s_{2}w_{2}}=1+\frac{(1-t_{1})s_{1}w_{1}}{s_{2}w_{2}}$

In this case, to make image 2 cropped on the right side, we use the following
equation for scaling image 2:

$t_{2}=1+\frac{a(1-t_{1})s_{1}w_{1}}{s_{2}w_{2}}$

where $a$ is a number larger than 1. In our implementation, we use $a=2$ to
make the images cropped exactly the same amount as the new space created as
shown in Figure 9(B).

We never crop the top of the image, although cropping the top is common in
movies and comics. To keep the top of the image at exactly the same level, we
simply scale image 2 using the upper left corner as the origin of scaling. To
remove the gap between the two images, we further translate image 2 in the $x$
direction, in the following amount:

$x_{2}=-(1-t_{1})x_{1}w_{1}$

An important property of this operation is that it is independent of what we
choose for $a$ and $t_{2}$. So, a single parameter $t_{1}$ between $0$ and $1$
is sufficient to produce extrapolation from the Apex view to the External
Right view. Internal Right is obtained in the limit when $t_{1}=0$.

For extrapolation from Apex to External Left view, we change $t_{2}$ from $1$
to $0$. We compute the scaling and translation of image 1 in a reciprocal way.
We compute $t_{1}$ using $t_{2}$ and $a$ as

$t_{1}=1+\frac{a(1-t_{2})s_{2}w_{2}}{s_{1}w_{1}}$

Using $t_{1}$, we scale image 1 upper right corner of image 1 as the origin of
the scaling. Then, to remove the gap between the two images, we translate the
image 1 in the $x$ direction by the following amount:

$x_{1}=-(1-t_{2})x_{2}w_{2}$

In this case, a single parameter $t_{2}$ between $0$ and $1$ is sufficient to
produce an extrapolation from the Apex view to the External Left view.
Internal Left is obtained in the limit when $t_{2}=0$.

This idea can be extended to other Apex views. Note that it is also possible
to create a wide variety of width/height ratios for Apex views by choosing
$s_{0}=H/h_{0}$, $s_{1}<H/h_{1}$ and given a $w_{0}$ amount of gap between two
images. Then $W=w_{0}+s_{1}w_{1}+s_{2}w_{2}$. The previous equations will
still hold for these cases with some minor adjustments. This flexibility is
particularly useful to produce a comic page that consists of multiple panels.
This is why it is easy to design comic pages.

Multi-object/character external views can simply be obtained by scaling and
translating only the characters in the extreme right and extreme left in the
original Apex views.

##  5 Implementation

We have implemented this single parameter interface as a single-panel cartoon
creation tool on the Web using JavaScript and webGL. This tool was developed
to provide only a minimal amount of control to users. We have only one slider
to obtain all views from Apex to External and Internal. The advantage of
minimal control is that the user need not worry about too many settings in
order to obtain the desired aesthetic.

We have provided multiple expressions for objects/characters such that the
users can choose different emotions that can be appropriate for the text. We
have also included sentiment analysis to choose appropriate expressions for a
given text. However, sentiment analysis currently works only offline so has
not yet been included in the Web-based system. We allow horizontal flips to
make two characters look at each other or different directions. Such flips can
also change the overall impression of dynamics between two objects/characters.
We have also added word balloons that point to the images.

Using this tool as an underlying infrastructure, we have provided four
different selections for creating science, mathematics, animal, and political
cartoons. Each uses a different database of images. We allow to use of all
images together. Our web-based system for single-panel cartoons can be viewed
at http://storytelling.viz.tamu.edu.

Having single panels is useful for creating comic pages. Figures 10, 11 and 12
show three comic pages constructed from single-panel cartoons generated by our
prototype. A comic artist randomly created the panels in the prototype system
and then put the individual panels together in order in Photoshop resizing as
needed to make a row’s pair of panels the same height and the rows the same
width but without changing the width and height ratios of the generated
panels. This exercise suggests that such pages can eventually be created
automatically by packing rectangles appropriately. Although the general
problem of 2D packing problems is hard Lodi et al. (2002), this particular
case has extra flexibilities and constraints that simplify the problem. It is
even possible to easily change the width and height ratios of panels by
changing the distances between the characters and the number of characters.
Our next goal is to provide full pages automatically.

Figure 10: A potential comic page featuring Frances Allen, Edsger W. Dijkstra,
Dennis Ritchie, Ivan Sutherland, and Alan Turing. We have composited the
$8.5\times 11$ page using single cartoon panels. One can try to fill out
balloons to see if it can turn into a meaningful story.

Our experience is that once a page is provided almost anybody can come up with
an interesting story just by filling out balloons. We conjecture that the
inherent knowledge about the characters plays an important role in the
creation of these stories. The reader is invited to try to fill in the
balloons in Figures 10, 11 and 12 to test this conjecture. In the prototype,
we also provided links for each character to their corresponding Wikipedia
page to enable viewers to obtain more information. Another feature of this
system is that information on each artwork can be represented in a simple
database. This information, in particular, can include the name of the
creator. In the current system, all images are created by only one
professional cartoonist. We think that is important to motivate other artists
to contribute with their work since they will be recognized.

##  6 Discussion and Future Work

In this work, we have demonstrated that one continuous parameter can be used
to obtain visually valid placement of characters in single cartoon panels.
This parameter can be considered a continuous interpolation from five
qualitatively different 3D cameras for two characters: Internal Left, External
Left, Apex, External Right, and Internal Right views He et al. (1996). These
five views are widely used in movies and comics to create dynamic dialogues
Arijon (1991). We have also demonstrated that we can obtain a significant
range of width and height ratios for composited panels. These results suggest
that the actual design space is significantly rich and it is possible to
easily obtain visually valid cartoon frames.

It is important to note that we do not create all visually valid compositions.
We only create some compositions that are guaranteed to be visually valid. It
is possible to provide more degrees of freedom. For instance, we can allow
overlapping the characters with an additional slider. Since we know which
character is front, we can order them in space. However, can cause some non-
valid compositions. It is hard to guarantee to obtain only valid cases.

This type of limitation also explains the difficulty faced by naive users.
Potential placements of objects/characters in a given cartoon frame define a
very large design space. Most of these placements are useless. If you randomly
search for a solution without the experience, it is impossible to find a valid
one. Therefore, cartoon expertise is critical to obtain visually valid
solutions. We have developed this solution with significant input from a
professional cartoonist.

Figures 10 11 and 12 are good examples of the importance of cartoon and comic
experience. It did not take much time for a cartoonist to create those pages
from a set of 6-7 cartoon panels. There was only a need to decide which panels
would come next to each other. Then, it was possible to quickly put together
an $8.5\times 11$ comic page. Here the main constraint is the size of the
page. The panels are chosen randomly just to fit this space. Despite that, it
is clear that some sort of story emerges.

This particular solution for single-panel cartoon creation is developed to
serve as an infrastructure for the automatic creation of comic pages that
consist of multiple panels. Our results demonstrate that it is possible to
randomly create single panels that can fit nicely into any given comic page
format from $8.5\times 11$ to A4. We can also use a Markov model to pick up
expressions automatically Liu et al. (2012). This work also suggests that it
is possible to create panels with changing camera views that can provide a
dynamic look. Such dynamism is critical for effective storytelling. It can
also be possible to create these panels automatically from a given text.

We can also choose expressions automatically using sentiment analysis. We
initially added sentiment analysis to obtain character expressions
automatically based on text. However, we could not include it web-based
interactive tool as our initial attempts made the interface a little too
complicated. On the other hand, we can later include sentiment analysis.

In this system, all characters and their expressions are created only by one
professional cartoonist. Our current implementation also allows the inclusion
of other artists by providing acknowledgment per image. We think that we have
to rely on artists since It is still hard to automatically create very
expressive characters. On the other hand, we can still use automatically
created characters even if they are not that expressive. This will be helpful
to turn texts into comic stories automatically. Such a system can also be used
to create storyboards from screenplays for movies. Such storyboards can be
used to identify weaknesses in stories.

One of the main problems is the creation of expressive balloons. Comic book
artists spend a significant amount of time placing balloons. The placement and
orientation of balloons play an important role in easy-to-read panels. Our
current balloon placement solution is visually valid, but it is very
conservative and not visually interesting.

Figure 11: Another potential comic page featuring an octopus, a sea lion, and
a shark. We again have composited the $8.5\times 11$ page using single cartoon
panels. Note that we added an empty rectangle for the “title” page, which also
simplifies the rectangle packing problem.

Figure 12: Another potential comic page featuring a dodecahedron, a sphere,
and a Moebius strip. We again have composited the $8.5\times 11$ page using
single cartoon panels. Note that we added an empty rectangle for the “title”
page, which also simplifies the rectangle packing problem.

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