# Anisotropic $p$-capacity and anisotropic Minkowski inequality

Chao Xia, Jiabin Yin  School of Mathematical Sciences, Xiamen University,
Xiamen, 361005, China<EMAIL_ADDRESS>School of Mathematical Sciences,
Xiamen University, Xiamen, 361005, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a sharp anisotropic $L^{p}$ Minkowski inequality
involving the total $L^{p}$ anisotropic mean curvature and the anisotropic
$p$-capacity, for any bounded domains with smooth boundary in
$\mathbb{R}^{n}$. As consequences, we obtain an anisotropic Willmore
inequality, a sharp anisotropic Minkowski inequality for outward
$F$-minimising sets and a sharp volumetric anisotropic Minkowski inequality.
For the proof, we utilize a nonlinear potential theoretic approach which has
been recently developed in [3].

###### Key words and phrases:

Minkowski inequality; anisotropy mean curvature; anisotropic $p$-Laplacian;
nonlinear potential theory; $p$-capacity.

2020 Mathematics Subject Classification. 53C21, 31C15, 52A39, 49Q10.

CX was supported by NSFC (Grant No. 11871406).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Minkowski norm, Wulff shape, anisotropic area
    2. 2.2 Anisotropic $p$-Laplacian
    3. 2.3 Anisotropic curvature for level sets
  3. 3 Anisotropic $p$-capacitary potential
  4. 4 Essential monotonicity
  5. 5 Effective monotonicity and $L^{p}$ anisotropic Minkowski inequality
    1. 5.1 First effective inequality
    2. 5.2 Second effective inequality
    3. 5.3 $L^{p}$ anisotropic Minkowski inequality
  6. 6 Anisotropic Minkowski Inequality for outward $F$-minimising sets

##  1\. Introduction

The Minkowski inequality in the theory of convex geometry says that if
$\Omega\subset\mathbb{R}^{n}$ is a convex domain with smooth boundary, then

$\displaystyle\frac{1}{\omega_{n-1}}\int_{\partial\Omega}\frac{H}{n-1}d\sigma\geq\left(\frac{|\partial\Omega|}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}},$
(1.1)

with equality holding if and only if $\Omega$ is a round ball. Here $H$ is the
mean curvature of $\partial\Omega$ with respect to the outward unit normal and
$\omega_{n-1}=|\mathbb{S}^{n-1}|,$ the area of unit $(n-1)$-sphere.

In recent decades, many mathematicians are interested in the question whether
(1.1) holds for non-convex domains. By using smooth solutions to inverse mean
curvature flow, Guan-Li [20] proved that (1.1) holds for star-shaped domains
with mean convex boundary. Huisken-Ilmanen [23] developed a level set weak
formulation for inverse mean curvature flow, which was adopted by Huisken [22]
to prove (1.1) for outward minimising sets (see also [19]). Quite recently, a
powerful and elegant approach based on nonlinear potential theory has been
developed by Agostiniani-Fogagnolo-Mazzieri in [3] (see also related papers
[2, 6, 18]), which enables them to establish a sharp $L^{p}$ version of
Minkowski inequality involving the $p$-capacity of $\Omega$ ($1<p<n$) for
general bounded domain with smooth boundary

$\frac{1}{\omega_{n-1}}\int\limits_{\partial\Omega}\Big{|}\frac{H}{n-1}\Big{|}^{p}d\sigma\geq\left[\big{(}\frac{p-1}{n-p}\big{)}^{p-1}\frac{1}{\omega_{n-1}}{\rm
Cap}_{p}(\Omega)\right]^{\frac{n-p-1}{n-p}}.$ (1.2)

Here ${\rm Cap}_{p}(\Omega)$ is called by the $p$-capacity of $\Omega$, given
by

${\rm Cap}_{p}(\Omega)=\inf\Big{\\{}\int_{\mathbb{R}^{n}}|\nabla u|^{p}dx:u\in
C_{c}^{\infty}(\mathbb{R}^{n}),u\geq 1\hbox{ on }\Omega\Big{\\}}.$

We remark that various geometric inequalities involving the mean curvature
integral, the capacity and the surface area have been considered by Xiao [40].
In particular, (1.2) has been proved in [40, Theorem 3.1] in the case of star-
shaped, mean convex domains. As $p\to 1$ in (1.2), Agostiniani-Fogagnolo-
Mazzieri have showed that the limit version of (1.2) recovers (1.1) for
outward minimising sets by Huisken [22]. Moreover, they are able to obtain a
volumetric version of Minkowski inequality

$\frac{1}{\omega_{n-1}}\int\limits_{\partial\Omega}\Big{|}\frac{H}{n-1}\Big{|}d\sigma\geq\left(\frac{|\Omega|}{|\mathbb{B}^{n}|}\right)^{\frac{n-2}{n}}.$
(1.3)

for any bounded domains. Previously, (1.3) has been shown by Trudinger [33]
and Chang-Wang [14] (see also [27]) for bounded domains with $2$-convex
boundary. The idea of Agostiniani-Fogagonolo-Mazzieri in [3] is based on a
conformal change of the Euclidean metric and a monotone integral over level
sets of $p$-capacitary potentials under the conformal metric. Actually, it
turns out that the results can be obtained by working directly on the
Euclidean metric rather than the conformal metric. It is worth highlighting
that their approach can be used to obtain sharp geometric inequalities for
hypersurfaces in manifolds with nonnegative Ricci curvature [2, 6, 18] as well
as for static manifolds [7, 11, 12], see also [13, 39].

In this paper, we are concerned with an anisotropic version of Minkowski
inequality. Let $F\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be a
Minkowski norm on $\mathbb{R}^{n}$. The unit Wulff ball $\mathcal{W}_{F}$ with
respect to $F$ (centered at the origin) is given by

$\mathcal{W}_{F}=\\{F^{o}(x)<1\\},$

where $F^{o}$ is the dual norm of $F$. For a bounded domain $\Omega$ with
smooth boundary, the anisotropic area of $\partial\Omega$ is defined by

$|\partial\Omega|_{F}=\int_{\partial\Omega}F(\nu)d\sigma.$

The well-known Wulff theorem (see e.g Theorem 20.8 in [25]) says that Wulff
balls are the only minimizers for the anisotropic isoperimetric problem.
Equivalently, the Wulff inequality holds true:

$|\partial\Omega|_{F}\geq
n|\mathcal{W}_{F}|^{\frac{1}{n}}|\Omega|^{1-\frac{1}{n}}.$ (1.4)

Equality in (1.4) holds if and only if $\Omega$ is a Wulff ball.

The anisotropic mean curvature $H_{F}$ of $\partial\Omega$ arises from the
first variational formula for $|\partial\Omega|_{F}$. The anisotropic
Minkowski inequality says that if $\Omega\subset\mathbb{R}^{n}$ is a convex
domain with smooth boundary, then

$\displaystyle\frac{1}{\kappa_{n-1}}\int_{\partial\Omega}\frac{H_{F}}{n-1}F(\nu)d\sigma\geq\left(\frac{|\partial\Omega|_{F}}{\kappa_{n-1}}\right)^{\frac{n-2}{n-1}},$
(1.5)

with equality holding if and only if $\Omega$ is a Wulff ball. Here
$\kappa_{n-1}=n|\mathcal{W}_{F}|$. When $F$ is the Euclidean norm
$F(\xi)=|\xi|$, then (1.4) and (1.5) reduce to the classical isoperimetric
inequality and the Minkowski inequality (1.1) respectively. Inequality (1.5)
is in fact equivalent to the following Minkowski inequality for mixed volumes

$\displaystyle V_{(2)}(\Omega,\mathcal{W}_{F})\geq
V_{1}(\Omega,\mathcal{W}_{F})^{\frac{n-2}{n-1}}|\mathcal{W}_{F}|^{\frac{1}{n-1}}$
(1.6)

where $V_{i}(\Omega,\mathcal{W}_{F})$ is the mixed volume for two convex
bodies $\bar{\Omega}$ and $\bar{\mathcal{W}_{F}}$, which is given by

$\left|\Omega+t\mathcal{W}_{F}\right|=\sum_{k=0}^{n}\binom{n}{k}t^{k}V_{(k)}(\Omega,\mathcal{W}_{F}).$

Inequality (1.6) is a special one for a class of Alexandrov-Fenchel
inequalities for mixed volumes in the theory of convex bodies, see e.g.
Schneider’s celebrated book [30] Section 7.3, or the Introduction in [38].

Regarding (1.5), the first named author [38] showed that (1.5) holds for star-
shaped domains with $F$-mean convex ($H_{F}\geq 0$) boundary, by using the
smooth solution to the inverse anisotropic mean curvature flow, in the same
spirit of Guan-Li [20]. Della Pietra, Gavitone and the first named author [15]
has considered the level set formulation of the inverse anisotropic mean
curvature flow, in the spirit of Huisken-Ilmanen [23], and proved the
existence of the weak solutions by using $L^{p}$-type approximation. However,
such approximation seems not sufficient to prove (1.5) for outward
$F$-minimising sets.

In this paper, we will use the method developed in [3, 6, 18] to study (1.5)
for non-convex domains. For $1<p<n$, the anisotropic $p$-capacity of $\Omega$
is defined by

${\rm Cap}_{F,p}(\Omega)=\inf\Big{\\{}\int_{\mathbb{R}^{n}}F^{p}(\nabla
u)dx:u\in C_{c}^{\infty}(\mathbb{R}^{n}),u\geq 1\hbox{ on }\Omega\Big{\\}}.$

We remark that there are early consideration for basic properties of the
anisotropic $p$-capacity, see e.g. Maz’ya’s book [26], Section 2.2. Also, the
Brunn-Minkowski theory has been extended to the anisotropic $p$-capacity [4,
5].

In the following we generalize (1.2) to the anisotropic case.

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{n}(n\geq 3)$ be a bounded domain with smooth
boundary. Then, for every $1<p<n$, the following inequality holds

$\frac{1}{\kappa_{n-1}}\int\limits_{\partial\Omega}\Big{|}\frac{H_{F}}{n-1}\Big{|}^{p}F(\nu)d\sigma\geq\left[\big{(}\frac{p-1}{n-p}\big{)}^{p-1}\frac{1}{\kappa_{n-1}}{\rm
Cap}_{F,p}(\Omega)\right]^{\frac{n-p-1}{n-p}}.$ (1.7)

Moreover, equality holds in (1.7) if and only if $\Omega$ is a Wulff ball.

As a direct consequence (with $p=n-1$), we get the anisotropic Willmore
inequality.

######  Theorem 1.2.

Let $\Omega\subset\mathbb{R}^{n}(n\geq 3)$ be a bounded domain with smooth
boundary. Then

$\int\limits_{\partial\Omega}\Big{|}\frac{H_{F}}{n-1}\Big{|}^{n-1}F(\nu)d\sigma\geq\kappa_{n-1}.$
(1.8)

Moreover, equality holds in (1.8) if and only if $\Omega$ is a Wulff ball.

Theorem 1.1 are a special case of Theorem 5.4. Consider the anisotropic
$p$-capacitary potential $u$, which satisfies

$\left\\{\begin{aligned} \Delta_{F,p}u=&0\ \ {\rm in}\ \
\mathbb{R}^{n}\setminus\bar{\Omega}\\\ u=&1\ \ {\rm on}\ \ \partial\Omega\\\
u(x)\rightarrow&0\ \ {\rm as}\ \ |x|\rightarrow\infty,\end{aligned}\right.$

where $\Delta_{F,p}$ is the anisotropic $p$-Laplace operator (see Section 2).
It is not hard to see that

${\rm
Cap}_{F,p}(\Omega)=\int_{\mathbb{R}^{n}\setminus\bar{\Omega}}F^{p}(\nabla
u)dx=\int_{\partial{\Omega}}F^{p-1}(\nabla u)F(\nu)d\sigma.$

Let $1<p<n$ and $q\geq 1+\frac{1}{p^{*}}$, where
$p^{*}:=\frac{(n-1)(p-1)}{n-p}$. Let $\Phi_{p,q}:[1,\infty)\to\mathbb{R}$ be
defined by

$\Phi_{p,q}(\tau):=\tau^{(q-1)p^{*}}\int\limits_{\\{u=1/\tau\\}}F^{q(p-1)}(\nabla
u)F(\nu)d\sigma.$

The asymptotic behavior of $\Phi_{p,q}$ corresponds to ${\rm
Cap}_{F,p}(\Omega)$ and the key ingredient to prove Theorem 5.4 is that
$\Phi_{p,q}$ is an (essentially) monotone non-increasing function. See
Sections 4 and 5.

Next by letting $p\to 1$ in (1.7), we get the following extended anisotropic
Minkowski Inequality.

######  Theorem 1.3.

If $\Omega\subset\mathbb{R}^{n}(n\geq 3)$ is a bounded domain with smooth
boundary, then

$\frac{1}{\kappa_{n-1}}\int\limits_{\partial\Omega}\Big{|}\frac{H_{F}}{n-1}\Big{|}F(\nu)d\sigma\geq\left(\frac{1}{\kappa_{n-1}}\inf_{\begin{subarray}{c}\Omega\subset
U\\\ \partial U\hbox{smooth}\end{subarray}}|\partial
U|_{F}\right)^{\frac{n-2}{n-1}}.$ (1.9)

Moreover, equality in (1.9) is achieved by Wulff balls.

As consequences, we are able to deduce a sharp anisotropic Minkowski
inequality for outward $F$-minimising sets (see Definition 6.2) and a sharp
volumetric anisotropic Minkowski inequality.

######  Corollary 1.1.

If $\Omega\subset\mathbb{R}^{n}(n\geq 3)$ is a bounded outward $F$-minimising
set with smooth boundary, then

$\frac{1}{\kappa_{n-1}}\int\limits_{\partial\Omega}\frac{H_{F}}{n-1}F(\nu)d\sigma\geq\left(\frac{|\partial\Omega|_{F}}{\kappa_{n-1}}\right)^{\frac{n-2}{n-1}}.$
(1.10)

Moreover, equality in (1.10) is achieved by Wulff balls. Conversely, if the
equality holds in (1.10) for some bounded strictly outward $F$-minimising open
set with smooth and strictly $F$-mean-convex boundary, then $\Omega$ is a
Wulff ball.

######  Corollary 1.2.

Let $\Omega\subset\mathbb{R}^{n}(n\geq 3)$ be a bounded set with smooth
boundary. Then

$\frac{1}{\kappa_{n-1}}\int\limits_{\partial\Omega}\Big{|}\frac{H_{F}}{n-1}\Big{|}F(\nu)d\sigma\geq\left(\frac{n|\Omega|}{\kappa_{n-1}}\right)^{\frac{n-2}{n}}.$
(1.11)

Moreover, equality holds in (1.11) if and only if $\Omega$ is a Wulff ball.

To end the introduction, we make some remark on the strictly outward
$F$-minimising hull. Let $\Omega$ be a bounded domain with smooth boundary. It
is well-known that the infimum

$\inf\limits_{\begin{subarray}{c}\Omega\subset U\\\ \partial
U\hbox{smooth}\end{subarray}}|\partial U|_{F}$

may not be achieved in general. In the setting of set of finite perimeter, the
infimum

$\displaystyle\inf_{\begin{subarray}{c}\Omega\subset E\\\ \ E\hbox{ set of
perimeter}\end{subarray}}P_{F}(E)$ (1.12)

can be achieved, by the direct method in Calculus of Variations. Here

$P_{F}(E)=\int_{\partial^{*}E}F(\nu_{E})d\mathcal{H}^{n-1}$

is the anisotropic perimeter of $E$, where $\partial^{*}E$ is the reduced
boundary of $E$ and $\nu_{E}$ is the outward measure theoretical unit normal
to $E$, see e.g. [25, Chapter 20]. There is a so-called strictly outward
$F$-minimising hull corresponding to $\Omega$ which attains the infimum, see
[3, 17, 23]. Similarly as in [17, Definition 2.12], the strictly outward
$F$-minimising hull of $\Omega$ denoted by $\Omega^{*}$, can be reformulated
as the measure theoretic interior Int$(E)$ of any set $E$ that solves the
minimisation problem

$|E|=\inf\limits_{G\in{\rm SOFMBE(\Omega)}}|G|$

where

${\rm SOFMBE}(\Omega)=\\{G\,|\,\Omega\subseteq G\ \hbox{ and }\ G\hbox{ is
bounded and strictly outward}\ F-{\rm minimising}\\}$

is the Strictly Outward $F$-Minimizing Bounded Envelopes of $\Omega$. It turns
out that $\Omega^{*}$ is the unique maximal volume solution to the least
anisotropic perimeter problem with obstacle $\Omega$, up to zero-measure
modification, see [17, Theorem 2.16] (In our case, the assumption on the
existence of an exhaustion of $\mathbb{R}^{n}$ by a sequence of strictly
outward $F$-minimising sets is automatically satisfied. In fact, such
exhaustion is given by the Wulff balls). In particular, $\Omega^{*}$ attains
the infimum in (1.12), namely,

$P_{F}(\Omega^{*})=\inf_{\begin{subarray}{c}\Omega\subset E\\\ \ E\hbox{ set
of perimeter}\end{subarray}}P_{F}(E).$

It is direct to see that

$\inf_{\begin{subarray}{c}\Omega\subset U\\\ \partial
U\hbox{smooth}\end{subarray}}|\partial
U|_{F}\geq\inf_{\begin{subarray}{c}\Omega\subset E\\\ \ E\hbox{ set of
perimeter}\end{subarray}}P_{F}(E)=P_{F}(\Omega^{*}).$

Thus it follows from (1.9) that it holds

$\frac{1}{\kappa_{n-1}}\int\limits_{\partial\Omega}\Big{|}\frac{H_{F}}{n-1}\Big{|}F(\nu)d\sigma\geq\left(\frac{P_{F}(\Omega^{*})}{\kappa_{n-1}}\right)^{\frac{n-2}{n-1}}.$
(1.13)

######  Remark 1.1.

In [3], the isotropic version of (1.13) has been shown. In their argument, a
regularity result in [31] and an exterior smooth approximation in [29] for the
strictly outward minimising hull $\Omega^{*}$ have to play a role. In this
paper, by showing a stronger inequality (1.9), we can avoid of using the
regularity and an exterior smooth approximation for the anisotropic case,
which seems unavailable in the reference.

The rest of the paper is organized as follows. In Section 2 we review the
basic concepts in anisotropic setting, including the anisotropic mean
curvature for level sets and the anisotropic $p$-laplace operator. In
particular, we prove a Kato-type identity for anisotropic $p$-harmonic
functions. In Section 3 we make a systematic study of anisotropic
$p$-capacitary potential, focusing on its asymptotic behavior. In Sections 4
and 5, we establish the essential and effective monotonicity properties for
the crucial function $\Phi_{p,q}(\tau)$, and in turn, prove the $L^{p}$
anisotropic Minkowski inequality, Theorem 1.1. In Section 6, we study the
outward $F$-minimising sets, the limit of ${\rm Cap}_{F,p}$ as $p\to 1^{+}$
and finally prove Theorem 1.3 and Corollaries 1.1 and 1.2.

##  2\. Preliminaries

###  2.1. Minkowski norm, Wulff shape, anisotropic area

Let $F\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be a Minkowski norm on
$\mathbb{R}^{n}$, in the sense that

  * (i)

$F$ is a norm in $\mathbb{R}^{n}$, i.e., $F$ is a convex, even,
$1$-homogeneous function satisfying $F(x)>0$ when $x\neq 0$ and $F(0)=0$;

  * (ii)

$F$ satisfies a uniformly elliptic condition: $\nabla^{2}(\frac{1}{2}F^{2})$
is positive definite in $\mathbb{R}^{n}\setminus\\{0\\}$.

The dual norm $F^{o}\colon\mathbb{R}^{n}\rightarrow[0,+\infty[$ of $F$ is
defined as

$F^{o}(x)=\sup_{\xi\neq 0}\frac{\langle\xi,x\rangle}{F(\xi)}.$

$F^{o}$ is also a Minkowski norm on $\mathbb{R}^{n}$.

Furthermore,

$F(\xi)=\sup_{x\neq 0}\frac{\langle\xi,x\rangle}{F^{o}(x)}.$

We remark that, throughout this paper, we use conventionally $\xi$ as the
variable for $F$ and $x$ as the variable for $F^{o}$.

Denote

$\mathcal{W}_{F}=\\{x\in\mathbb{R}^{n}\colon F^{o}(x)<1\\}.$

For the simplicity of notations, we will denote by
$\mathcal{W}_{F}=\mathcal{W}$. We call $\mathcal{W}$ the unit Wulff ball
centered at the origin, and $\partial\mathcal{W}$ the Wulff shape.

More generally, we denote

$\mathcal{W}_{r}(x_{0})=r\mathcal{W}+x_{0},$

and call it the Wulff ball of radius $r$ centered at $x_{0}$. We simply denote
$\mathcal{W}_{r}=\mathcal{W}_{r}(0)$.

The following properties of $F$ and $F^{o}$ hold true and will be frequently
used in this paper (see e.g. [15, 37]).

######  Proposition 2.1.

Let $F:\mathbb{R}^{n}\rightarrow[0,\infty)$ be a Minkowski norm. Then for any
$x,\xi\in\mathbb{R}^{n}\setminus\\{0\\}$, the following holds:

  1. (1)

$\langle F_{\xi}(\xi),\xi\rangle=F(\xi),\quad\langle
F^{o}_{x}(x),x\rangle=F^{o}(x).$

  2. (2)

$\sum_{j}F_{\xi_{i}\xi_{j}}(\xi)\xi_{j}=0$ for any $i=1,\ldots,n$.

  3. (3)

$F(F_{x}^{o}(x))=F^{o}(F_{\xi}(\xi))=1.$

  4. (4)

$F^{o}(x)F_{\xi}(F_{x}^{o}(x))=x,\quad
F(\xi)F_{x}^{o}\left(F_{\xi}(\xi)\right)=\xi.$

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with smooth boundary
$\partial\Omega$ and $\nu$ be its unit outward normal of $\partial\Omega$. The
anisotropic area $|\partial\Omega|_{F}$ of $\Omega$ is defined by

$|\partial\Omega|_{F}=\int_{\partial\Omega}F(\nu)d\sigma.$ (2.1)

Note that when $\Omega=\mathcal{W}$, the unit Wulff ball, one can check by the
divergence theorem that

$|\partial\mathcal{W}|_{F}=\int_{\partial\mathcal{W}}\frac{1}{|\nabla
F^{o}|}d\sigma=\int_{\mathcal{W}}{\rm div}(x)dx=n|\mathcal{W}|.$ (2.2)

For notation simplicity, we denote

$\kappa_{n-1}=|\partial\mathcal{W}|_{F}=n|\mathcal{W}|.$

###  2.2. Anisotropic $p$-Laplacian

Let $u$ be twice continuous differentiable at $x\in\mathbb{R}^{n}$. We denote
by $F_{i},F_{ij},\ldots$ the partial derivatives of $F$ and by
$u_{i},u_{ij},\ldots$ the partial derivatives of $u$,

$F_{i}=\frac{\partial F}{\partial\xi_{i}},\
F_{ij}=\frac{\partial^{2}F}{\partial\xi_{i}\partial\xi_{j}},\
u_{i}=\frac{\partial u}{\partial x_{i}},\ u_{ij}=\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}.$

For $x$ such that $\nabla u(x)\neq 0$, denote

$\displaystyle a_{ij}(\nabla u)(x):=$
$\displaystyle\frac{\partial^{2}}{\partial\xi_{i}\partial\xi_{j}}(\frac{1}{2}F^{2})(\nabla
u(x))=(F_{i}F_{j}+FF_{ij})(\nabla u(x)),$ (2.3) $\displaystyle a_{ij,p}(\nabla
u)(x):=$
$\displaystyle\frac{\partial^{2}}{\partial\xi_{i}\partial\xi_{j}}(\frac{1}{p}F^{p})(\nabla
u(x))=F^{p-2}(a_{ij}+(p-2)F_{i}F_{j})(\nabla u(x)).$

The anisotropic Laplacian and $p$-Laplacian of $u$ at regular points ($|\nabla
u|\neq 0$) is given by

$\displaystyle\Delta_{F}u:=$ $\displaystyle a_{ij}(\nabla u)u_{ij},$ (2.4)
$\displaystyle\Delta_{F,p}u:=$ $\displaystyle a_{ij,p}(\nabla
u)u_{ij}=F^{p-2}(\Delta_{F}u+(p-2)F_{i}F_{j}u_{ij}),$

See e.g. [10, 34]. For notation simplicity, we also introduce

$V(\xi)=\frac{1}{p}F^{p}(\xi).$

and

$\displaystyle W_{ij}=\partial_{x_{j}}\left(\partial_{\xi_{i}}V(\nabla
u)\right)=a_{ik,p}(\nabla u)u_{kj}.$ (2.5)

One sees easily that

${\rm tr}(W)=\Delta_{F,p}u.$

Let $1\leq k\leq n$, be an integer. For a $n$-vector
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$, the $k$-th elementary symmetric
function on $\lambda$ is defined by

$\sigma_{k}(\lambda)=\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq
n}\lambda_{i_{1}}\cdots\lambda_{i_{k}}.$

Given a real matrix $B=(b_{ij})\in\mathbb{R}^{n\times n}$, we denote by
$S_{k}(B)$ the sum of all the principal minors of $B$ of order $k$. In
particular, $S_{1}(B)={\rm tr}(B)$, the trace of $B$, and $S_{n}(B)={\rm
det}(B)$, the determinant of $B$. When the eigenvalues $\lambda_{B}$ of $B$
are all real, it is clear that

$S_{k}(B)=\sigma_{k}(\lambda_{B}).$

For more properties on $S_{k}(B)$, we refer to [16, 28].

For our purpose, we consider $k=2$. By setting

$S^{ij}_{2}(B)={\rm tr}B\,\delta_{ij}-b_{ji},$

then

$S_{2}(B)=\frac{1}{2}S^{ij}_{2}(B)b_{ij}=\frac{1}{2}\big{(}({\rm
tr}B)^{2}-{\rm tr}B^{2}\big{)}.$ (2.6)

Of particular interest for our purpose is $S_{2}(W)$, where $W$ is given by
(2.5).

Since ${\rm tr}W=\Delta_{F,p}u$, it holds

$S^{ij}_{2}(W)=\Delta_{F,p}u\,\delta_{ij}-a_{jk,p}(\nabla u)u_{ki}.$ (2.7)

Moreover, in this setting $S^{ij}_{2}(W)$ is divergence free, see e.g.[8, 10,
16]

$\partial_{x_{j}}S^{ij}_{2}(W)=0.$ (2.8)

It follows from (2.6) and (2.8) that if $\Delta_{F,p}u=0$, then

$\partial_{x_{j}}(S^{ij}_{2}(W)V_{\xi_{i}}(\nabla
u))=2S_{2}(W)=-a_{ik,p}a_{jl,p}u_{kj}u_{il}.$ (2.9)

###  2.3. Anisotropic curvature for level sets

We recall the concept of anisotropic curvature for a hypersurface in
$\mathbb{R}^{n}$. See e.g. [34, 38].

Let $M$ be a smooth embedded hypersurface in $\mathbb{R}^{n}$ and $\nu$ be one
unit normal of $M$. The corresponding anisotropic normal of $M$ is defined by

$\nu_{F}=\nabla F(\nu).$

The anisotropic principal curvatures
$\kappa_{F}=(\kappa_{1}^{F},\ldots,\kappa_{n-1}^{F})\in\mathbb{R}^{n-1}$ are
defined as the eigenvalues of the map

$d\nu_{F}\colon T_{x}M\to T_{\nu_{F}(x)}\partial\mathcal{W}.$

The mean curvature (with respect to $\nu$) is defined to be

$H_{F}=\sigma_{1}(\kappa_{F})=\sum_{i}\kappa_{i}^{F}.$

The second mean curvature (with respect to $\nu$) is defined to be

$\sigma_{2}(\kappa_{F})=\sum_{i<j}\kappa_{i}^{F}\kappa_{j}^{F}.$

A well-known variational characterization for $H_{F}$ is that it arises from
the first variation of the anisotropic area functional $|\partial\Omega|_{F}$,
(see e.g. Proposition 2.1 in [38]), namely,

$\frac{d}{dt}\int_{\partial\Omega_{t}}F(\nu)d\sigma=\int_{\partial\Omega_{t}}H_{F}\langle
Y,\nu\rangle d\sigma,$

where $Y$ is the variation vector filed for $\partial\Omega_{t}$.

In this paper we are interested in the case when $M$ is given by a regular
level set of a smooth function $u$, that is $M=\\{u=t\\}$ for some regular
value $t$. For our purpose, we choose the unit normal $\nu=-\frac{\nabla
u}{|\nabla u|}$ and

$\nu_{F}=-F_{\xi}(\nabla u),\ \ H_{F}=-{\rm div}(F_{\xi}(\nabla u)).$

In this case, we have that

$H_{F}=-{\rm div}(F_{\xi}(\nabla u))=-F_{ij}u_{ij},$ (2.10)

Here $\rm div$ is the Euclidean divergence. See e.g. [16].

From Theorem 2.5 in [16], we also have that

$\sigma_{2}(\kappa_{F})=S_{2}(F_{ik}u_{kj}).$

It follows that

$\displaystyle\frac{n-2}{n-1}S_{1}^{2}(F_{ik}u_{kj})-2S_{2}(F_{ik}u_{kj})=$
$\displaystyle\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F}).$
(2.11)

We remark that the right hand side of (2.11) is exactly the square of
traceless part of anisotropic second fundamental form, which is nonnegative.

Next we give two propositions to the connection between an anisotropic
$p$-harmonic function $u$ and the anisotropic curvatures of regular level sets
of $u$. The first one is a formula for the anisotropic mean curvature of
regular level set of $u$.

######  Proposition 2.2.

Let $u$ satisfy $\Delta_{F,p}u=0$. Then the anisotropic mean curvature of
regular level set of $u$ is given by

$H_{F}=(p-1)F^{-1}F_{i}F_{j}u_{ij}.$ (2.12)

###### Proof.

Note that $\Delta_{F,p}u=0$ implies $FF_{ij}u_{ij}+(p-1)F_{i}F_{j}u_{ij}=0$.
The assertion follows from (2.10). ∎

The second is a Kato-type identity for anisotropic $p$-harmonic functions,
which generalizes Proposition 4.4 in [18].

######  Proposition 2.3.

Let $u$ satisfy $\Delta_{F,p}u=0$. Let $\kappa_{F}$ denote the anisotropic
principal curvature of regular level set of $u$. Then, at any point where
$F(\nabla u)\not=0$, the following identity holds true:

$\displaystyle a_{ij}a_{kl}u_{ik}u_{jl}=$ $\displaystyle F^{2}(\nabla
u)\left[\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})\right]$
(2.13) $\displaystyle+(1+\frac{(p-1)^{2}}{n-1})|\nabla(F(\nabla
u))|^{2}_{a_{F}}$ $\displaystyle+(1-\frac{(p-1)^{2}}{n-1})|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}},$

where

$|\nabla(F(\nabla u))|^{2}_{a_{F}}:=a_{ij}(F_{k}u_{ki})(F_{l}u_{lj})\geq 0,$
(2.14)

and

$|\nabla^{T}(F(\nabla u))|^{2}_{a_{F}}:=FF_{ij}(F_{k}u_{ki})(F_{l}u_{lj})\geq
0.$ (2.15)

######  Remark 2.1.

$|\cdot|_{a_{F}}$ can be viewed as the norm with respect to the positive
definite matrix $(a_{ij})$. Also by the definition, we have

$\displaystyle|\nabla(F(\nabla u))|^{2}_{a_{F}}=|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}+(F_{i}F_{j}u_{ij})^{2}.$ (2.16)

###### Proof.

The proof is inspired by [37]. Note that $a_{ij}=FF_{ij}+F_{i}F_{j}$, we can
separate $\Delta_{F}u=a_{ij}u_{ij}=A+B$ where

$A=F_{i}F_{j}u_{ij}\ \ {\rm and}\ \ B=FF_{ij}u_{ij}.$

Since

$a_{ij}a_{kl}u_{ik}u_{jl}=A^{2}+2FF_{kl}F_{i}F_{j}u_{ik}u_{jl}+F^{2}F_{ij}F_{kl}u_{ik}u_{jl},$

and

$\frac{(A+B)^{2}}{n}+\frac{n}{n-1}\Big{(}\frac{B}{n}-\frac{n-1}{n}A\Big{)}^{2}=A^{2}+\frac{1}{n-1}B^{2},$

we have

$\displaystyle a_{ij}a_{kl}u_{ik}u_{jl}=$
$\displaystyle\frac{(A+B)^{2}}{n}+\frac{n}{n-1}\Big{(}\frac{B}{n}-\frac{n-1}{n}A\Big{)}^{2}-\frac{1}{n-1}B^{2}$
(2.17)
$\displaystyle+2FF_{kl}F_{i}F_{j}u_{ik}u_{jl}+F^{2}F_{ij}F_{kl}u_{ik}u_{jl}$
$\displaystyle=$
$\displaystyle\frac{(a_{ij}u_{ij})^{2}}{n}+\frac{n}{n-1}\Big{(}\frac{FF_{ij}u_{ij}}{n}-\frac{n-1}{n}F_{i}F_{j}u_{ij}\Big{)}^{2}$
$\displaystyle-\frac{1}{n-1}(FF_{ij}u_{ij})^{2}+2FF_{kl}F_{i}F_{j}u_{ik}u_{jl}+F^{2}F_{ij}F_{kl}u_{ik}u_{jl}.$

On the on hand, it follows $\Delta_{F,p}u=0$ that

$a_{ij}u_{ij}=-(p-2)F_{i}F_{j}u_{ij}.$ (2.18)

On the other hand, we have the following computations

$2FF_{kl}F_{i}F_{j}u_{ik}u_{jl}=2a_{kl}F_{i}F_{j}u_{ik}u_{jl}-2F_{i}F_{j}F_{k}F_{l}u_{ik}u_{jl}.$
(2.19)

Substituting (2.18) and (2.19) into (2.17) yields to

$\displaystyle a_{ij}a_{kl}u_{ik}u_{jl}=$
$\displaystyle\Big{(}\frac{(p-1)^{2}}{n-1}-1\Big{)}(F_{i}F_{j}u_{ij})^{2}+2a_{kl}F_{i}F_{j}u_{ik}u_{jl}$
(2.20)
$\displaystyle+F^{2}F_{ij}F_{kl}u_{ik}u_{jl}-\frac{1}{n-1}(FF_{ij}u_{ij})^{2}$
$\displaystyle=$
$\displaystyle\Big{(}\frac{(p-1)^{2}}{n-1}-1\Big{)}(F_{i}F_{j}u_{ij})^{2}+(1-\frac{(p-1)^{2}}{n-1})|\nabla(F(\nabla
u))|^{2}_{a_{F}}$
$\displaystyle+F^{2}F_{ij}F_{kl}u_{ik}u_{jl}-\frac{1}{n-1}(FF_{ij}u_{ij})^{2}$
$\displaystyle+\left(1+\frac{(p-1)^{2}}{n-1}\right)|\nabla(F(\nabla
u))|^{2}_{a_{F}}$ $\displaystyle=$
$\displaystyle\left(1-\frac{(p-1)^{2}}{n-1}\right)|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}+\left(1+\frac{(p-1)^{2}}{n-1}\right)|\nabla(F(\nabla
u))|^{2}_{a_{F}}$
$\displaystyle+F^{2}F_{ij}F_{kl}u_{ik}u_{jl}-\frac{1}{n-1}(FF_{ij}u_{ij})^{2},$

where the last equality we have used (2.16). Finally using (2.11), we obtain

$\displaystyle F_{ij}F_{kl}u_{ik}u_{jl}-\frac{1}{n-1}(F_{ij}u_{ij})^{2}$
$\displaystyle=$
$\displaystyle\frac{n-2}{n-1}S_{1}^{2}(F_{ik}u_{kj})-2S_{2}(F_{ik}u_{kj})$
$\displaystyle=$
$\displaystyle\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F}).$

This completed the proof of Proposition 2.3. ∎

We also have some facts as following

######  Proposition 2.4.

Let $u$ satisfy $\Delta_{F,p}u=0$. Then, at any point where $F(\nabla
u)\not=0$, the following identity holds true:

$\displaystyle F^{2-p}a_{jk,p}F_{i}u_{ik}F_{l}u_{lj}=$
$\displaystyle|\nabla^{T}F(\nabla
u)|^{2}_{a_{F}}+\frac{1}{p-1}H_{F}^{2}F^{2}(\nabla u),$ (2.21) $\displaystyle
a_{jk,p}a_{il,p}u_{ik}u_{lj}F^{4-2p}=$
$\displaystyle\frac{n}{n-1}H_{F}^{2}F^{2}+2(p-1)|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}$ (2.22) $\displaystyle+F^{2}(\nabla
u)\left[\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})\right].$

###### Proof.

From Proposition 2.3, we have that

$\displaystyle a_{ij}a_{kl}u_{ik}u_{jl}=$ $\displaystyle F^{2}(\nabla
u)\left[\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})\right]$
(2.23)
$\displaystyle+\Big{(}1+\frac{(p-1)^{2}}{n-1}\Big{)}\frac{1}{(p-1)^{2}}F^{2}H_{F}^{2}+2|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}.$

By using (2.3) and (2.23), directing computation gives that

$\displaystyle a_{jk,p}a_{il,p}u_{ik}u_{lj}F^{4-2p}=$
$\displaystyle(a_{jk}+(p-2)F_{j}F_{k})(a_{il}+(p-2)F_{i}F_{l})u_{ik}u_{lj}$
$\displaystyle=$ $\displaystyle
a_{jk}a_{il}u_{ik}u_{lj}+2(p-2)|\nabla(F(\nabla
u))|^{2}_{a_{F}}+\frac{(p-2)^{2}}{(p-1)^{2}}H_{F}^{2}F^{2}$ $\displaystyle=$
$\displaystyle\frac{n}{n-1}H_{F}^{2}F^{2}+2(p-1)|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}$ $\displaystyle+F^{2}(\nabla
u)\left[\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})\right].$

This completes the assertion. ∎

##  3\. Anisotropic $p$-capacitary potential

In this section, we will introduce the anisotropic $p$-capacity and its
related properties.

######  Definiton 3.1 (anisotropic $p$-capacity).

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with smooth boundary.
For $p\in[1,n)$, the anisotropic $p$-capacity of $\Omega$ is defined as

${\rm Cap}_{F,p}(\Omega)={\rm inf}\Big{\\{}\int_{\mathbb{R}^{n}}F^{p}(\nabla
u)dx\,\Big{|}\,u\in C_{c}^{\infty}(\mathbb{R}^{n}),\,u\geq 1\ \ {\rm on}\ \
\Omega\Big{\\}},$ (3.1)

It is easy to check that (see e.g. [26])

${\rm Cap}_{F,p}(\Omega)={\rm inf}\Big{\\{}\int_{\mathbb{R}^{n}}F^{p}(\nabla
u)\,dx\,\Big{|}\,u\geq\chi_{\Omega},\,u\in{C}^{\infty}_{c}(\mathbb{R}^{n})\Big{\\}}.$
(3.2)

Next we consider the case $p\in(1,n)$. The variational structure of the above
definition leads naturally to the formulation of the problem

$\left\\{\begin{aligned} \Delta_{F,p}u=&0\ \ {\rm in}\ \
\mathbb{R}^{n}\setminus\bar{\Omega}\\\ u=&1\ \ {\rm on}\ \ \partial\Omega\\\
u(x)\rightarrow&0\ \ {\rm as}\ \ |x|\rightarrow\infty,\end{aligned}\right.$
(3.3)

we say a function $u\in W^{1,p}_{loc}(U)$ is a weak solution of
$\Delta_{F,p}u=0$ in an open set $U$ if

$\int_{U}\langle F^{p-1}(\nabla u)F_{\xi}(\nabla u),\nabla\psi\rangle dx=0.$

for any $\psi\in C^{\infty}_{c}(U)$. A weak solution satisfying (3.3) is
called the anisotropic $p$-capacitary potential associated with $\Omega$.

We collect the existence and regularity results for the anisotropic
$p$-capacity potentials in the following theorem, see e.g. [8, 9].

######  Theorem 3.1 (Existence and regularity of anisotropic $p$-capacity
potentials).

Let $\Omega$ be a bounded open set with smooth boundary, and let $1<p<n$. Then

  1. (1)

There exists a unique weak solution $u\in
C^{1,\alpha}_{loc}(\mathbb{R}^{n}\setminus\bar{\Omega})\cap
C(\mathbb{R}^{n}\setminus\Omega)$ to (3.3);

  2. (2)

$u$ is smooth away from ${\rm Crit}(u):=\\{\nabla u=0\\}$;

  3. (3)

The solution u fullfills

${\rm
Cap}_{F,p}(\Omega)=\int_{\mathbb{R}^{n}\setminus\bar{\Omega}}F^{p}(\nabla
u)dx.$ (3.4)

######  Remark 3.1.

Note that since $\partial\Omega$ is assumed to be smooth, by the Hopf Lemma
for anisotropic $p$-harmonic functions, we have that $\nabla u\not=0$ in a
neighborhood of $\partial\Omega$. In particular, $u$ is smooth in
$\partial\Omega$. Coupled with this fact, the asymptotic expansions below
imply that ${\rm
Crit}(u)=\big{\\{}x\in\mathbb{R}^{n}\setminus\bar{\Omega}\,|\,Du(x)=0\big{\\}}$
is a compact subset of $\mathbb{R}^{n}\setminus\bar{\Omega}$ (generically
depending on $p$), and in turn that $u$ is analytic outside this set.

For $1<p<n$, let

$\Gamma_{F,p}(x)=\frac{p-1}{n-p}\left(\frac{1}{\kappa_{n-1}}\right)^{\frac{1}{p-1}}F^{o}(x)^{\frac{p-n}{p-1}}.$

One can check that

$\Delta_{F,p}\Gamma_{F,p}(x)=\delta_{0}\hbox{ in }\mathbb{R}^{n},$

where $\delta_{0}$ is the Dirac Delta function about the origin. We call
$\Gamma_{F,p}$ the fundamental solution to $\Delta_{F,p}u=0$ in
$\mathbb{R}^{n}$. See [36].

######  Proposition 3.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with smooth boundary,
and $1<p<n$. Let $u$ is a weak solution of $\Delta_{F,p}u=0$ in
$\mathbb{R}^{n}\setminus\Omega$. Then $u$ satisfies

  1. (1)

$\lim_{|x|\rightarrow+\infty}\frac{u(x)}{\Gamma_{F,p}(x)}={{\rm
Cap}_{F,p}}(\Omega)^{\frac{1}{p-1}},$

  2. (2)

$\nabla u(x)=\,{{\rm
Cap}_{F,p}}(\Omega)^{\frac{1}{p-1}}\nabla\Gamma_{F,p}(x)+o(|x|^{-\frac{n-1}{p-1}}),\;{\rm
as}\;|x|\to+\infty.$

###### Proof.

If $u$ is solution of (3.3), it is a standard argument by using comparison
theorem to show that there exists two positive constants $C_{1},C_{2}$ such
that

$C_{1}\Gamma_{F,p}\leq u\leq C_{2}\Gamma_{F,p}.$

Following the argument of [24], Theorem 1.1 and Remark 1.5, (see [36], Theorem
4.1 and Remark 4.1 for anisotropic case), we conclude that there exists
$\gamma\in\mathbb{R}$ such that

$\displaystyle\lim_{|x|\rightarrow+\infty}\frac{u(x)}{\Gamma_{F,p}(x)}=\gamma,$
(3.5)
$\displaystyle\lim_{|x|\rightarrow+\infty}(F^{o}(x))^{\frac{n-1}{p-1}}\left(\nabla
u-\gamma\nabla\Gamma_{F,p}\right)=0.$ (3.6)

It remains to show

$\gamma={\rm Cap}_{F,p}(\Omega)^{\frac{1}{p-1}}.$

Through the following integration by parts that holds true by the anisotropic
$p$-harmonicity of $u$:

$\int\limits_{\partial\Omega}F^{p-1}(\nabla
u)F(\nu)d\sigma=-\lim_{R\rightarrow\infty}\int\limits_{\partial\mathcal{W}_{R}}F^{p-1}(\nabla
u)\langle F_{\xi}(\nabla u),\nu_{\partial\mathcal{W}_{R}}\rangle d\sigma,$
(3.7)

where $\nu=-\frac{\nabla u}{|\nabla u|}$ on $\partial\Omega$ and
$\nu_{\partial\mathcal{W}_{R}}$ is the outward unit normal to
$\mathcal{W}_{R}$, we shall see in Proposition 3.2 below that the left-hand
side of (3.7) is exactly ${\rm Cap}_{F,p}(\Omega)$. We next compute the limit
in the right-hand side of (3.7). Since

$\nabla u=\gamma\nabla\Gamma_{F,p}+o(|x|^{-\frac{n-1}{p-1}}),$

we have

$F(\nabla u)=\gamma
F(\nabla\Gamma_{F,p})+o(|x|^{-\frac{n-1}{p-1}})=\gamma\left(\frac{1}{\kappa_{n-1}}\right)^{\frac{1}{p-1}}F^{o}(x)^{-\frac{n-1}{p-1}}+o(|x|^{-\frac{n-1}{p-1}})$

On $\partial\mathcal{W}_{R}$,

$\nu_{\partial\mathcal{W}_{R}}=\frac{\nabla F^{o}}{|\nabla
F^{o}|}=-\kappa_{n-1}^{\frac{1}{p-1}}F^{o}(x)^{\frac{n-1}{p-1}}\frac{\nabla\Gamma_{F,p}}{|\nabla
F^{o}|}=-\kappa_{n-1}^{\frac{1}{p-1}}F^{o}(x)^{\frac{n-1}{p-1}}\frac{\gamma^{-1}\nabla
u}{|\nabla F^{o}|}+o(1).$

It follows that on $\partial\mathcal{W}_{R}$

$\displaystyle\langle F_{\xi}(\nabla u),\nu_{\partial\mathcal{W}_{R}}\rangle$
$\displaystyle=$
$\displaystyle-\kappa_{n-1}^{\frac{1}{p-1}}F^{o}(x)^{\frac{n-1}{p-1}}\frac{\gamma^{-1}F(\nabla
u)}{|\nabla F^{o}|}+o(1)$ $\displaystyle=$ $\displaystyle-\frac{1}{|\nabla
F^{o}|}+o(1).$

Hence

$\displaystyle F^{p-1}(\nabla u)\langle F_{\xi}(\nabla
u),\nu_{\partial\mathcal{W}_{R}}\rangle=-\gamma^{p-1}\frac{1}{\kappa_{n-1}}\frac{(F^{o}(x))^{1-n}}{|\nabla
F^{o}|}+o(|x|^{1-n}).$

Combining the fact that

$\int\limits_{\partial\mathcal{W}_{R}}\frac{(F^{o}(x))^{1-n}}{|\nabla
F^{o}|}d\sigma=\kappa_{n-1},$

we deduce that

$\displaystyle\lim_{R\rightarrow\infty}\int\limits_{\partial\mathcal{W}_{R}}F^{p-1}(\nabla
u)\langle F_{\xi}(\nabla u),\nu_{\partial\mathcal{W}_{R}}\rangle
d\sigma=-\gamma^{p-1}.$

Thus $\gamma={\rm Cap}_{F,p}(\Omega)^{\frac{1}{p-1}}$, and $(1),(2)$ of
Proposition 3.1 follows. ∎

We need the following expression for ${\rm Cap}_{F,p}(\Omega)$ in terms of an
integral on $\partial\Omega$.

######  Proposition 3.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with smooth boundary,
and let $1<p<n$. Then, the solution $u$ to (3.3) satisfies

${\rm Cap}_{F,p}(\Omega)=\int_{\partial{\Omega}}F^{p-1}(\nabla
u)F(\nu)d\sigma.$ (3.8)

where $\nu=-\frac{\nabla u}{|\nabla u|}$ on $\partial\Omega$.

###### Proof.

For $\varepsilon>0$, let $U_{\varepsilon}$ be the $\varepsilon$-tubular
neighborhood of Crit$(u)$, namely

$U_{\varepsilon}=\Big{\\{}x\in\mathbb{R}^{n}\setminus\bar{\Omega}\,\big{|}\,{\rm
dist}(x,{\rm Crit(u)})<\varepsilon\Big{\\}},$

where dist$(x,{\rm Crit}(u))$ is the distance of $x$ from ${\rm Crit}(u)$. By
the compactness of Crit$(u)$ in $\mathbb{R}^{n}\setminus\bar{\Omega}$, we have
that $U_{\varepsilon}\subset\\{u\geq t\\}$, for $\varepsilon>0$ and $t>0$
small enough. Since $F(\nabla u)=0$ on Crit$(u)$, we have from identity (3.4)
and by Monotone Convergence Theorem that

${\rm Cap}_{F,p}(\Omega)=\lim_{\varepsilon\rightarrow 0^{+}}\lim_{t\rightarrow
0^{+}}\int\limits_{\\{u\geq t\\}\setminus U_{\varepsilon}}F^{p}(\nabla u)dx.$

Then, for $\varepsilon$ and $t$ small enough, by intergration by parts,

$\displaystyle\int\limits_{\\{u\geq t\\}\setminus U_{\varepsilon}}F^{p}(\nabla
u)dx=$ $\displaystyle\int\limits_{\\{u\geq t\\}\setminus U_{\varepsilon}}{\rm
div}(uF^{p-1}(\nabla u)F_{\xi}(\nabla u))dx$ $\displaystyle=$
$\displaystyle-t\int\limits_{\\{u=t\\}}\frac{F^{p}(\nabla u)}{|\nabla
u|}d\sigma+\int\limits_{\partial\Omega}F^{p-1}(\nabla u)F(\nu)d\sigma$
$\displaystyle+\int\limits_{\partial U_{\varepsilon}}uF^{p-1}(\nabla u)\langle
F_{\xi}(\nabla u),\nu_{\partial U_{\varepsilon}}\rangle d\sigma.$

By the computation in the proof of Proposition 3.1, we see
$\int\limits_{\\{u=t\\}}\frac{F^{p}(\nabla u)}{|\nabla u|}d\sigma$ is bounded
when $t\to 0+$, which implies

$-t\int\limits_{\\{u=t\\}}\frac{F^{p}(\nabla u)}{|\nabla u|}d\sigma\to 0\hbox{
as }t\to 0.$

On the other hand, we have

$\int\limits_{\partial U_{\varepsilon}}uF^{p-1}(\nabla u)\langle
F_{\xi}(\nabla u),\nu_{\partial U_{\varepsilon}}\rangle d\sigma\to 0\hbox{ as
}\varepsilon\to 0.$

since $F(\nabla u)$ tends to zero as $x$ approaches Crit$(u)$. The assertion
follows. ∎

##  4\. Essential monotonicity

In this section, we assume $u$ is anisotropic $p$-harmonic, that is,
$\Delta_{F,p}u=0$. We will establish some Lemmas for later using.

Let

$p^{*}:=\frac{(n-1)(p-1)}{n-p}.$

and

$\Lambda=\Big{\\{}(p,q)\in\mathbb{R}^{2}\Big{|}\,1<p<n,\ {\rm and}\ q\geq
1+\frac{1}{p^{*}}\Big{\\}}.$

For $(p,q)\in\Lambda$, we consider the function
$\Phi_{p,q}:[1,+\infty)\rightarrow\mathbb{R}^{n}$ given by

$\Phi_{p,q}(\tau):=\tau^{(q-1)p^{*}}\int\limits_{\\{u=1/\tau\\}}F^{q(p-1)}(\nabla
u)F(\nu)d\sigma.$ (4.1)

By the asymptotic behavior given in Lemma 3.1, it is direct to check the
limits of $\Phi_{p,q}(\tau)$.

######  Lemma 4.1.

$\lim_{\tau\rightarrow+\infty}\Phi_{p,q}(\tau)=\big{(}\frac{n-p}{p-1}\big{)}^{(q-1)p^{*}}\big{(}\kappa_{n-1}\big{)}^{\frac{(q-1)(p-1)}{n-p}}({\rm
Cap}_{F,p}(\Omega))^{1-\frac{(q-1)(p-1)}{n-p}}.$ (4.2)

###### Proof.

From the asymptotic behavior of $u$ in Proposition 3.1, we know that For
$\varepsilon>0$, there exists $\tau_{0}$ such that for $\tau>\tau_{0}$ and
$u(x)=\frac{1}{\tau}$,

$\displaystyle(1-\varepsilon)\gamma\Gamma_{F,p}\leq
u(x)\leq(1+\varepsilon)\gamma\Gamma_{F,p},$
$\displaystyle(1-\varepsilon)\gamma\left(\frac{1}{\kappa_{n-1}}\right)^{\frac{1}{p-1}}F^{o}(x)^{-\frac{n-1}{p-1}}\leq
F(\nabla
u(x))\leq(1+\varepsilon)\gamma\left(\frac{1}{\kappa_{n-1}}\right)^{\frac{1}{p-1}}F^{o}(x)^{-\frac{n-1}{p-1}}.$

From this one deduces that on such $\\{u(x)=\frac{1}{\tau}\\}$

$\displaystyle(1-\tilde{\varepsilon})\gamma^{-\frac{(q-1)(p-1)^{2}}{n-p}}\big{(}\frac{n-p}{p-1}\big{)}^{(q-1)p^{*}}\big{(}\kappa_{n-1}\big{)}^{\frac{(q-1)(p-1)}{n-p}}\tau^{-(q-1)p^{*}}$
$\displaystyle\leq F^{(q-1)(p-1)}(\nabla u(x))$
$\displaystyle\leq(1+\tilde{\varepsilon})\gamma^{-\frac{(q-1)(p-1)^{2}}{n-p}}\big{(}\frac{n-p}{p-1}\big{)}^{(q-1)p^{*}}\big{(}\kappa_{n-1}\big{)}^{\frac{(q-1)(p-1)}{n-p}}\tau^{-(q-1)p^{*}}.$

where $\tilde{\varepsilon}$ is some function on $\varepsilon$ with the
property $\lim_{\varepsilon\to 0}\tilde{\varepsilon}=0.$ Arguing as in
Proposition 3.2, we see that

$\displaystyle\int\limits_{\\{u=1/\tau\\}}F^{p-1}F(\nu)d\sigma={\rm
Cap}_{F,p}(\Omega).$

Since

$\tau^{(q-1)p^{*}}\int\limits_{\\{u=1/\tau\\}}F^{q(p-1)}F(\nu)d\sigma=\tau^{(q-1)p^{*}}\int\limits_{\\{u=1/\tau\\}}F^{(q-1)(p-1)}(F^{p-1}F(\nu)d\sigma$

It follows that

$\displaystyle(1-\tilde{\varepsilon})\big{(}\frac{n-p}{p-1}\big{)}^{(q-1)p^{*}}\big{(}\kappa_{n-1}\big{)}^{\frac{(q-1)(p-1)}{n-p}}({\rm
Cap}_{F,p}(\Omega))^{1-\frac{(q-1)(p-1)}{n-p}}$ $\displaystyle\leq$
$\displaystyle\tau^{(q-1)p^{*}}\int\limits_{\\{u=1/\tau\\}}F^{q(p-1)}F(\nu)d\sigma$
$\displaystyle\leq$
$\displaystyle(1+\tilde{\varepsilon})\big{(}\frac{n-p}{p-1}\big{)}^{(q-1)p^{*}}\big{(}\kappa_{n-1}\big{)}^{\frac{(q-1)(p-1)}{n-p}}({\rm
Cap}_{F,p}(\Omega))^{1-\frac{(q-1)(p-1)}{n-p}}.$

The assertion follows by letting $\tau\to\infty$ and then $\varepsilon\to 0$.
∎

For $(p,q)\in\Lambda$. Let us consider the vector field $X$ with component

$\displaystyle X^{j}=$
$\displaystyle-(q-1)u^{-{(q-1)p^{*}}+2}F^{q(p-1)-1}\left(a_{jk,p}F^{1-p}F_{i}u_{ik}-{p^{*}}\frac{FF_{j}}{u}\right)$
(4.3)

where

$\displaystyle a_{jk,p}=F^{p-2}(a_{jk}+(p-2)F_{j}F_{k}).$

Next, we have the following Lemma.

######  Lemma 4.2.

$\Phi_{p,q}$ is differentiable at the regular values of $u$ and

$\displaystyle\frac{d}{d\tau}\Phi_{p,q}(\tau)=\int\limits_{\\{u=1/\tau\\}}\langle
X,\frac{\nabla u}{|\nabla u|}\rangle d\sigma=\int\limits_{\\{u<1/\tau\\}}{\rm
div}Xdx.$ (4.4)

###### Proof.

By co-area formula and divergence theorem we have

$\displaystyle\frac{d}{d\tau}\Phi_{p,q}(\tau)=$
$\displaystyle\frac{d}{d\tau}\Big{\\{}\int\limits_{\\{u=1/\tau\\}}u^{-{(q-1)p^{*}}}F^{q(p-1)}F(\nu)d\sigma\Big{\\}}$
(4.5) $\displaystyle=$
$\displaystyle\frac{d}{d\tau}\Big{\\{}\int\limits_{\\{u<1/\tau\\}}\partial_{x_{i}}\left(u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}F^{p-1}F_{i}\right)dx\Big{\\}}$
$\displaystyle=$
$\displaystyle\frac{d}{d\tau}\Big{\\{}\int\limits_{\\{u<1/\tau\\}}(q-1)u^{-{(q-1)p^{*}}}F^{q(p-1)}\left(-{p^{*}}\frac{F}{u}+(p-1)\frac{F_{i}F_{k}u_{ik}}{F}\right)dx\Big{\\}}$
$\displaystyle=$
$\displaystyle-(q-1)\tau^{{(q-1)p^{*}}-2}\int\limits_{\\{u=1/\tau\\}}F^{q(p-1)-1}\left((p-1)\frac{F_{i}F_{k}u_{ik}}{F}-{p^{*}}\frac{F}{u}\right)F(\nu)d\sigma.$

By using the definition of $X$ and the fact

$a_{jk,p}u_{j}=(p-1)F^{p-2}FF_{k},$

we deduce

$\displaystyle\langle X,\,\nabla u\rangle=$
$\displaystyle-(q-1)u^{-{(q-1)p^{*}}+2}F^{q(p-1)-1}\left((p-1)F_{k}F_{i}u_{ik}-{p^{*}}\frac{F^{2}}{u}\right).$

Then, we obtain the first equality. The second equality follows from the
divergence theorem,. ∎

Finally, we compute the divergence of $X$.

######  Lemma 4.3 (Divergence of $X$).

For any $(p,q)\in\Lambda$, let $u$ be a solution of (3.3) and $X$ be the
vector field defined in (4.3). Then, the following identity holds at any point
$x\in\mathbb{R}^{n}\setminus\Omega$ such that $F(\nabla u)\not=0$.

$\displaystyle{\rm div}X=$
$\displaystyle-(q-1)u^{-{(q-1)p^{*}}+2}F^{q(p-1)-1}(\nabla
u)\Big{\\{}\left[\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})\right]$
(4.6) $\displaystyle+\Big{(}q(p-1)-1\Big{)}\frac{|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}}{F^{2}(\nabla u)}$
$\displaystyle+\Big{(}q-1-\frac{1}{p^{*}}\Big{)}\left|H_{F}-{p^{*}}\frac{F(\nabla
u)}{u}\right|^{2}\Big{\\}}\leq 0.$

###### Proof.

Using (2.7) and $\Delta_{F,p}u=0$, we can write $X$ in the following way

$\displaystyle X^{j}=(q-1)\Big{\\{}$ $\displaystyle
u^{-{(q-1)p^{*}}+2}F^{(q-2)(p-1)-1}S^{ij}_{2}V_{\xi_{i}}(\nabla u)$ (4.7)
$\displaystyle+p^{*}u^{-{(q-1)p^{*}}+1}F^{(q-1)(p-1)}F^{p-1}F_{j}\Big{\\}}.$

Separating

${\rm div}X=(q-1)(I+II),$ (4.8)

where

$\displaystyle I=$
$\displaystyle\partial_{x_{j}}\Big{(}u^{-{(q-1)p^{*}}+2}F^{(q-2)(p-1)-1}S^{ij}_{2}V_{\xi_{i}}(\nabla
u)\Big{)},$ $\displaystyle II=$
$\displaystyle{p^{*}}\partial_{x_{j}}\Big{(}u^{{-(q-1)p^{*}}+1}F^{(q-1)(p-1)}F^{p-1}F_{j}\Big{)},$

By using Proposition 2.2, (2.9) and (2.12), we have the following computations

$\displaystyle I=$
$\displaystyle-(2-{(q-1)p^{*}})u^{-{(q-1)p^{*}}+1}F^{q(p-1)}H_{F}-u^{-{(q-1)p^{*}}+2}F^{q(p-1)-3}\times$
(4.9)
$\displaystyle\times\Big{[}\big{(}(q-2)(p-1)-1\big{)}F^{2-p}a_{jk,p}F_{i}F_{l}u_{ki}u_{lj}+F^{4-2p}a_{jk,p}a_{il,p}u_{ik}u_{lj}\Big{]},$

and

$\displaystyle
II=\Big{(}1-{(q-1)p^{*}}\Big{)}{p^{*}}u^{-(q-1){p^{*}}}F^{q(p-1)+1}+{{(q-1)p^{*}}}u^{1-{(q-1)p^{*}}}F^{q(p-1)}{H_{F}}.$
(4.10)

Next, by using (2.21) and (2.22), we have

$\displaystyle\Big{[}(q-2)(p-1)-1\Big{]}F^{2-p}a_{jk,p}F_{i}F_{l}u_{ki}u_{lj}+F^{4-2p}a_{jk,p}a_{il,p}u_{ik}u_{lj}$
(4.11) $\displaystyle=$
$\displaystyle\Big{[}(q-2)(p-1)-1\Big{]}\Big{(}|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}+\frac{1}{p-1}H_{F}^{2}F^{2}\Big{)}+\frac{n}{n-1}H_{F}^{2}F^{2}$
$\displaystyle+2(p-1)|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}+F^{2}\left[\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})\right]$
$\displaystyle=$
$\displaystyle\Big{[}(q-1)-\frac{1}{p^{*}}\Big{]}F^{2}H^{2}_{F}+\Big{(}q(p-1)-1\Big{)}|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}$
$\displaystyle+F^{2}\left[\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})\right].$

Substituting (4.9)-(4.11) into (4.8) yields (4.6). ∎

##  5\. Effective monotonicity and $L^{p}$ anisotropic Minkowski inequality

The aim of this section is to given a complete proof of Theorem 1.1. For this
purpose, we will establish the following two effective monotonicity
inequalities:

$\Phi^{\prime}_{p,q}(1)\leq 0\ \ {\rm and}\ \
\Phi_{p,q}(+\infty)=\lim\limits_{\tau\to\infty}\Phi_{p,q}(\tau)\leq\Phi_{p,q}(1).$

###  5.1. First effective inequality

Combining Lemma 4.2 and 4.3, we see that
$\Phi^{\prime}_{p,q}(s)\leq\Phi^{\prime}_{p,q}(S)$ for any $s<S$ if ${\rm
Crit}(u)=\emptyset$. However, in general ${\rm Crit}(u)\not=\emptyset$.
Nevertheless, we are able to provide an effective version of the considered
monotonicity, showing that is actually in force, provided $S$ is large enough
and $s$ is closed to $1$. This is enough to get the desired effective
inequality $\Phi^{\prime}_{p,q}(1)\leq 0$.

######  Theorem 5.1 (Effective Monotonicity Formula–I).

For $(p,q)\in\Lambda$, let $u$ be the solution to (3.3) and let
$1<\bar{s}<\bar{S}<+\infty$ be such that ${\rm
Crit}(u)\subset\\{\bar{S}^{-1}<u<\bar{s}^{-1}\\}$. Then, for every $1\leq
s\leq\bar{s}\leq\bar{S}\leq S$, the inequality

$\Phi^{\prime}_{p,q}(s)\leq\Phi^{\prime}_{p,q}(S),$ (5.1)

holds true, where $\Phi_{p,q}$ is defined in (4.1). Moreover, one has that
$\Phi^{\prime}_{p,q}(1)\leq 0$.

######  Remark 5.1.

The existence of $\bar{s}$ and $\bar{S}$ follows from Remark 3.1 and
Proposition 3.1.

###### Proof.

Fix $s$ and $S$ such that $1\leq s\leq\bar{s}\leq\bar{S}\leq S$. For given
$\varepsilon>0$, let $\chi:[0,+\infty)\rightarrow\mathbb{R}$ be a smooth
nonnegative cut-off-function, such that

$\left\\{\begin{aligned} \chi(t)=&0\ \ {\rm in}\ \
t<\frac{1}{2}{\varepsilon},\\\ {\chi}^{\prime}(t)\geq&0\ \ {\rm in}\ \
\frac{1}{2}{\varepsilon}\leq t\leq\frac{3}{2}\varepsilon,\\\ \chi(t)=&1\ \
{\rm in}\ \ t>\frac{3}{2}\varepsilon.\end{aligned}\right.$ (5.2)

Define a smooth vector field

$\tilde{X}=\chi\big{(}u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla u)\big{)}X.$

Since $F(\nabla u)=0$ on Crit$\,(u)$, we have $X=0\hbox{ in }{\rm Crit}(u)$.
By choosing $\varepsilon$ small enough, we can make sure that

$\tilde{X}=X\hbox{ on }\\{u=s^{-1}\\}\hbox{ and }\\{u=S^{-1}\\},$

since ${\rm
Crit}(u)\subset\\{\bar{S}^{-1}<u<\bar{s}^{-1}\\}\subset\\{S^{-1}<u<s^{-1}\\}$.
Denote

$\displaystyle\Theta:=$ $\displaystyle\langle
X,\nabla(u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla u))\rangle$ $\displaystyle=$
$\displaystyle(q-1)(p-1)u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)-1}\Big{\langle}X,\nabla\big{(}F(\nabla
u)\big{)}-\frac{n-1}{n-p}\frac{\nabla u}{u}F\Big{\rangle}.$

Then we have the following computations

$\displaystyle\int\limits_{\\{u=1/s\\}}\langle X,\frac{\nabla u}{|\nabla
u|}\rangle d\sigma-\int\limits_{\\{u=1/S\\}}\langle X,\frac{\nabla u}{|\nabla
u|}\rangle d\sigma=\int\limits_{\\{1/S<u<1/s\\}}{\rm div}{\tilde{X}}dx$ (5.3)
$\displaystyle=$ $\displaystyle\int\limits_{\\{1/S<u<1/s\\}\setminus
U_{\varepsilon/2}}\chi{\rm div}Xdx+\int\limits_{U_{3\varepsilon/2}\setminus
U_{\varepsilon/2}}{\chi}^{\prime}\Theta dx.$

where in the last identity we have used the tubular neighborhood of
Crit$\,(u)$ defined for every $\delta>0$ as
$U_{\delta}=\\{u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla u)\leq\delta\\}$.

From Lemma 4.2, we can see that the first term on the right hand side of (5.3)
is non-positive, we next prove that $\Theta\leq 0$. By using (4.3), (2.12) and
(2.15), we have that

$\displaystyle\Theta=$
$\displaystyle-(q-1)^{2}(p-1)u^{-2{(q-1)p^{*}}+2}F^{(2q-1)(p-1)-2}\times$
(5.4)
$\displaystyle\times\Big{\\{}\frac{a_{jk,p}F_{i}u_{ik}}{F}F^{2-p}-{p^{*}}\frac{FF_{j}}{u}\Big{\\}}\Big{(}F_{l}u_{lj}-\frac{(n-1)}{n-p}\frac{F}{u}u_{j}\Big{)}$
$\displaystyle=$
$\displaystyle-(q-1)^{2}(p-1)u^{-2{(q-1)p^{*}}+2}F^{(2q-1)(p-1)-1}\times$
$\displaystyle\times\Big{\\{}\frac{a_{jk,p}F_{i}u_{ik}F_{l}u_{lj}}{F^{p}(\nabla
u)}+\frac{(n-1)^{2}(p-1)}{(n-p)^{2}}\frac{F^{2}}{u^{2}}-2{p^{*}}\frac{F_{k}F_{i}u_{ik}}{u}\Big{\\}}$
$\displaystyle=$
$\displaystyle-(q-1)^{2}u^{-2{(q-1)p^{*}}+2}F^{(2q-1)(p-1)-1}\times$
$\displaystyle\times\Big{\\{}\Big{|}H_{F}-{p^{*}}\frac{F}{u}\Big{|}^{2}+(p-1)\frac{|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}}{F^{2}}\Big{\\}}\leq 0.$

This completes the proof of (5.1).

It follows from (5.1) that, for every $S\geq\bar{S}$,

$\Phi^{\prime}_{p,q}(1)\leq\Phi^{\prime}_{p,q}(S).$

Integrating both sides of the above inequality on an interval of the form
$(\bar{S},S)$, we obtain

$\Phi^{\prime}_{p,q}(1)(S-\bar{S})+\Phi_{p,q}(\bar{S})\leq\Phi_{p,q}(S)$

If by contradiction, $\Phi^{\prime}_{p,q}(1)>0$, then, letting
$S\rightarrow+\infty$ above inequality, we would deduce
$\lim_{S\rightarrow+\infty}\Phi_{p,q}(S)\rightarrow+\infty$ against the
boundedness of $\Phi_{p,q}$ by Lemma 4.1. ∎

###  5.2. Second effective inequality

$\Phi_{p,q}(\infty)\leq\Phi_{p,q}(1)$.

For a given $(p,q)\in\Lambda$ and a given $0<\lambda<1$, we consider the
vector field

$Y_{\lambda}=(u^{-1}-\lambda)X-F^{q(p-1)}u^{-(q-1)p^{*}}\nabla_{\xi}F(\nabla
u),$ (5.5)

It is convenient to observe that at a regular value of $u$ it holds

$(\tau-\lambda)\Phi^{\prime}_{p,q}(\tau)-\Phi_{p,q}(\tau)=\int\limits_{\\{u=1/\tau\\}}\Big{\langle}Y_{\lambda},\frac{\nabla
u}{|\nabla u|}\Big{\rangle}$ (5.6)

Next we compute the divergence of $Y_{\lambda}$.

######  Lemma 5.1 (Divergence of $Y_{\lambda}$).

For any $(p,q)\in\Lambda$ and any $0<\lambda<1$, let $u$ be the solution of
(3.3) and $Y_{\lambda}$ be the vector field defined in (5.5). Then, the
following identity holds at any point
$x\in\mathbb{R}^{n}\setminus\bar{\Omega}$ such that $F(\nabla u)\not=0$

${\rm div}{Y_{\lambda}}=(u^{-1}-\lambda){\rm div}X\leq 0,$

where ${\rm div}X$ is non-positive defined in (4.6)

###### Proof.

By the very definition of $Y_{\lambda}$, we obtain that

$\displaystyle{\rm div}Y_{\lambda}=(u^{-1}-\lambda){\rm div}X-u^{-2}\langle
X,\,\nabla u\rangle-{\rm div}\big{(}F^{q(p-1)}u^{-(q-1)p^{*}}F_{\xi}(\nabla
u)\big{)}$

By using the definition of $X$, we compute

$\displaystyle u^{-2}\langle X,\,\nabla u\rangle=$
$\displaystyle-(q-1)u^{-{(q-1)p^{*}}}F^{q(p-1)-1}\left((p-1)F_{k}F_{i}u_{ik}-{p^{*}}\frac{F^{2}}{u}\right).$

By using $\Delta_{F,p}u={\rm div}(F^{p-1}F_{\xi}(\nabla u))=0$, we get

$\displaystyle{\rm div}\big{(}F^{q(p-1)}u^{-(q-1)p^{*}}F_{\xi}(\nabla
u)\big{)}=(q-1)u^{-{(q-1)p^{*}}}F^{q(p-1)-1}\left((p-1)F_{k}F_{i}u_{ik}-{p^{*}}\frac{F^{2}}{u}\right).$

The assertion follows. ∎

######  Theorem 5.2 (Effective Monotonicity Formula–II).

For any $1<p<n$, let $u$ be the solutions of (3.3) and let
$1<\bar{s}<\bar{S}<+\infty$ be such that ${\rm
Crit}(u)\subset\\{\bar{S}^{-1}<u<\bar{s}^{-1}\\}$. Then, for every $1\leq
s\leq\bar{s}\leq\bar{S}\leq S$, the inequality

$(s-\lambda)\Phi^{\prime}_{p,q}(s)-\Phi_{p,q}(s)\leq(S-\lambda)\Phi^{\prime}_{p,q}(S)-\Phi_{p,q}(S),$
(5.7)

holds true. Moreover, one has that $\Phi_{p,q}(+\infty)\leq\Phi_{p,q}(1)$.

###### Proof.

Fix $s$ and $S$ such that $1\leq s\leq\bar{s}\leq\bar{S}\leq S$. Let
$\chi:[1,+\infty)\rightarrow\mathbb{R}$ be the same smooth nonnegative cut-off
function as in the proof of Theorem 5.1, so that the properties (5.2) holds.
To simplify the notation, let us also set

$\eta_{\lambda}(u)=(u^{-1}-\lambda)^{-1}.$

Then, let us consider the smooth vector field

$\tilde{Y}_{\lambda}=\chi\Big{(}\eta_{\lambda}(u)u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla
u)\Big{)}Y_{\lambda},$

where $Y_{\lambda}$ has been defined in (5.4). Again, choosing $\varepsilon$
small enough, we can suppose $\tilde{Y}_{\lambda}=Y_{\lambda}$ on
$\\{u=1/s\\}$ and $\\{u=1/S\\}$, with $s$ and $S$ as in the statement.

Denote

$\Psi:=\Big{\langle}\nabla\Big{(}\eta_{\lambda}(u)u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla
u)\Big{)},Y_{\lambda}\Big{\rangle}.$

By applying the divergence theorem to the smooth vector field
$\tilde{Y}_{\lambda}$ on the region $\\{1/S<u<1/s\\}$ gives

$\displaystyle\int\limits_{\\{u=\frac{1}{s}\\}}\langle
Y_{\lambda},\,\frac{\nabla u}{|\nabla u|}\rangle
d\sigma-\int\limits_{\\{u=\frac{1}{S}\\}}\langle Y_{\lambda},\,\frac{\nabla
u}{|\nabla u|}\rangle
d\sigma=\int\limits_{\\{\frac{1}{S}<u<\frac{1}{s}\\}}{\rm
div}\tilde{Y}_{\lambda}dx$
$\displaystyle=\int\limits_{\\{\frac{1}{S}<u<\frac{1}{s}\\}\setminus
U_{\varepsilon/2}}\chi\Big{(}\eta_{\lambda}(u)u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla
u)\Big{)}{\rm div}Y_{\lambda}dx+\int\limits_{U_{3\varepsilon/2}\setminus
U_{\varepsilon/2}}\chi^{\prime}\Psi dx$

where this time the tubular neighborhoods of ${\rm Crit}(u)$ are defined, for
every $\delta>0,$ as
$U_{\delta}=\\{\eta_{\lambda}(u)u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla
u)\leq\delta\\}$. Since, as observed in Lemma 5.1, the divergence of
$Y_{\lambda}$ is non-positive on $\\{1/S\leq u\leq 1/s\\}\setminus
U_{\varepsilon/2}$, where clearly $F(\nabla u)\not=0$. Next we prove $\Psi$ is
non-positive on $U_{3\varepsilon/2}\setminus U_{\varepsilon/2}$.

Noticing that

$\displaystyle\nabla\Big{(}\eta_{\lambda}(u)u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)}(\nabla
u)\Big{)}$ (5.8) $\displaystyle=$ $\displaystyle
u^{-2-{(q-1)p^{*}}}\eta_{\lambda}^{2}(u)F^{(q-1)(p-1)}\nabla
u+(q-1)(p-1)\eta_{\lambda}\times$ $\displaystyle\times
u^{-{(q-1)p^{*}}}F^{(q-1)(p-1)-1}\Big{(}\nabla(F(\nabla
u))-\frac{n-1}{n-p}\frac{F}{u}\nabla u\Big{)}$

By using Proposition 2.2, (5.4) and (5.8), a direct computation gives

$\displaystyle\Psi=$
$\displaystyle\Theta-2(q-1)\eta_{\lambda}(u)u^{-2{(q-1)p^{*}}}F^{(2q-1)(p-1)-1}\Big{(}(p-1)F_{i}F_{k}u_{ik}-p^{*}\frac{F^{2}}{u}\Big{)}$
$\displaystyle-\eta^{2}_{\lambda}(u)u^{-2{(q-1)p^{*}}}F^{(2q-1)(p-1)-1}\frac{F^{2}}{u^{2}}$
$\displaystyle=$
$\displaystyle-u^{-2{(q-1)p^{*}}}F^{(2q-1)(p-1)-1}\Big{\\{}\Big{[}(q-1)u\Big{(}H_{F}-{p^{*}}\frac{F}{u}\Big{)}+\eta_{\lambda}(u)\frac{F}{u}\Big{]}^{2}+$
$\displaystyle+(q-1)^{2}(p-1)\frac{u^{2}}{F^{2}}|\nabla^{T}(F(\nabla
u))|^{2}_{a_{F}}\Big{\\}}\leq 0,$

This completes the proof of (5.7).

It remains to show that
$\lim_{\tau\rightarrow+\infty}\Phi_{p,q}(\tau)\leq\Phi_{p,q}(1)$. Applying the
inequality (5.7) with $0<\lambda<1$, $s=1$ and $S\geq\bar{S}$, we get

$\Phi_{p,q}(S)-\Phi_{p,q}(1)\leq(S-\lambda)\Phi_{p,q}^{\prime}(S)-(1-\lambda)\Phi_{p,q}^{\prime}(1),$

Observe now that (5.1) holds also for $\bar{S}<s<S$, because that ${\rm
Crit}(u)\cap[\bar{S},S]=\emptyset$. Then, the same argument in Theorem 5.1 to
deduce that $\Phi_{p,q}^{\prime}(1)\leq 0$ gives also
$\Phi_{p,q}^{\prime}(s)\leq 0$ for any $s>\bar{S}$. In particular,
$\Phi_{p,q}^{\prime}$ is a definitely bounded monotone function, and this
implies $\lim{\rm inf}_{S\rightarrow+\infty}\Phi_{p,q}^{\prime}(S)\leq 0$.
Hence, passing to the inferior limit as $S\rightarrow+\infty$ in the above
inequality yields

$\lim_{S\rightarrow+\infty}\Phi_{p,q}(S)-\Phi_{p,q}(1)\leq-(1-\lambda)\Phi_{p,q}^{\prime}(1).$

Letting $\lambda\rightarrow 1^{-}$ on the right hand side leads to the second
effective inequality
$\lim_{S\rightarrow+\infty}\Phi_{p,q}(S)\leq\Phi_{p,q}(1)$. ∎

###  5.3. $L^{p}$ anisotropic Minkowski inequality

We are ready to prove the following geometric inequalities between anisotropic
$p$-capacity and total anisotropic mean curvatures.

######  Theorem 5.3.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary.
Let $(p,q)\in\Lambda$. Then

$\left(\int_{\partial\Omega}\big{|}\frac{H_{F}}{n-1}\big{|}^{q(p-1)}F(\nu)d\sigma\right)^{\frac{1}{q}}\geq\big{(}\frac{p-1}{n-p}\big{)}^{p-1}\frac{{\rm
Cap}_{F,p}(\Omega)}{|\partial\Omega|_{F}^{1-\frac{1}{q}}}.$ (5.9)

Moreover, equality holds in (5.9) if only if $\Omega$ is a Wulff ball.

###### Proof.

Recall the formula (4.5) and also the identity (2.10). Thus the first
effective inequality $\Phi_{p,q}^{\prime}(1)\leq 0$ implies that

$0\leq\int\limits_{\partial\Omega}F^{q(p-1)-1}(\nabla
u)\big{(}H_{F}-{p^{*}}F(\nabla u)\big{)}F(\nu)d\sigma,$

and thus, by the Hölder inequality, one gets

$\int\limits_{\partial\Omega}F^{q(p-1)}(\nabla
u)F(\nu)d\sigma\leq\Big{(}\frac{n-p}{p-1}\Big{)}^{q(p-1)}\int\limits_{\partial\Omega}\Big{|}\frac{H_{F}}{n-1}\Big{|}^{q(p-1)}F(\nu)d\sigma.$
(5.10)

On the other hand, by Hölder inequality, we obtain

$\int\limits_{\partial\Omega}F^{p-1}F(\nu)d\sigma\leq\Big{(}\int\limits_{\partial\Omega}F^{q(p-1)}F(\nu)d\sigma\Big{)}^{\frac{1}{q}}\Big{(}\int_{\partial\Omega}F(\nu)d\sigma\Big{)}^{\frac{q-1}{q}}.$

This combining with (3.8), (5.10) yields (5.9).

Assume now that equality holds in (1.7), Then, equality holds in (5.9), and
consequently $\Phi^{\prime}_{p,q}(1)=0$. it follows that ${\rm div}X=0$ along
$\partial\Omega$. From (4.6), we see that

$\frac{n-2}{n-1}\sigma_{1}^{2}(\kappa_{F})-2\sigma_{2}(\kappa_{F})=0.$

Then $\partial\Omega$ is anisotropic umbilical, which yields that
$\partial\Omega$ is of Wullf shape. ∎

######  Theorem 5.4.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary.
Let $(p,q)\in\Lambda$. Then

$\frac{1}{\kappa_{n-1}}\int\limits_{\partial\Omega}\Big{|}\frac{H_{F}}{n-1}\Big{|}^{q(p-1)}F(\nu)d\sigma\geq\Big{[}\big{(}\frac{p-1}{n-p}\big{)}^{p-1}\frac{1}{\kappa_{n-1}}{\rm
Cap}_{p,F}(\Omega)\Big{]}^{1-\frac{(q-1)(p-1)}{n-p}}.$ (5.11)

Equality holds in (5.11) if only if $\Omega$ is a Wullf ball.

###### Proof.

The second effective inequality $\Phi_{p,q}(+\infty)\leq\Phi_{p,q}(1)$,
combining with the asymptotic behavior (4.2), implies that

$\displaystyle(\frac{n-p}{p-1})^{(q-1)p^{*}}(\kappa_{n-1})^{\frac{(q-1)(p-1)}{n-p}}({\rm
Cap}_{F,p}(\Omega))^{1-\frac{(q-1)(p-1)}{n-p}}$ $\displaystyle=$
$\displaystyle\lim_{\tau\rightarrow+\infty}\Phi_{p,q}(\tau)\leq\Phi_{p,q}(1)=\int\limits_{\partial\Omega}F^{q(p-1)}(\nabla
u)F(\nu)d\sigma.$

Then, by using (5.10), we obtain (5.11). The rigidity follows from the proof
as in Theorem 5.3. ∎

Proof of Theorems 1.1 and 1.2.

Let $q=p/(p-1)$ in (5.11), we obtain the anisotropic $L^{p}$ Minkowski
inequality (1.7) in Theorem 1.1. Let $q=\frac{n-1}{p-1}$ in (5.11), we obtain
the anisotropic Willmore inequality (1.8) in Theorem 1.2.

##  6\. Anisotropic Minkowski Inequality for outward $F$-minimising sets

######  Definiton 6.1.

Let $E\subset\mathbb{R}^{n}$ be a bounded set of finite perimeter (s.f.p. for
short). The anisotropic perimeter of $E$ is defined by

$P_{F}(E)=\int_{\partial^{*}E}F(\nu_{E})d\mathcal{H}^{n-1}$

where $\partial^{*}E$ is the reduced boundary of $E$ and $\nu_{E}$ is the
outward measure theoretical unit normal to $E$.

######  Remark 6.1.

When $E$ has smooth boundary, then

$P_{F}(E)=|\partial E|_{F}=\int_{\partial E}F(\nu)d\sigma.$

######  Definiton 6.2 (Outward $F$-minimising and strictly outward
$F$-minimising sets).

Let $E\subset\mathbb{R}^{n}$ be a bounded set of finite perimeter. We say that
$E$ is outward $F$-minimising if

$\displaystyle P_{F}(E)\leq P_{F}(G)\hbox{ for }G\subset\mathbb{R}^{n}\hbox{
bounded s.f.p. with }E\subset G.$ (6.1)

We say that $E$ is strictly outward $F$-minimising if it is outward
$F$-minimising and equality in (6.1) holds only if $|G\setminus E|=0$.

######  Proposition 6.1.

Let $\Omega$ be an outward $F$-minimising set with smooth boundary. Then

$\displaystyle\inf_{\begin{subarray}{c}\Omega\subset U\\\ \partial
U\hbox{smooth}\end{subarray}}|\partial U|_{F}=|\partial\Omega|_{F}.$ (6.2)

###### Proof.

It follows directly by the definition. ∎

######  Proposition 6.2.

Let $\Omega$ be an outward $F$-minimising set with smooth boundary. Then
$H_{F}\geq 0$.

###### Proof.

It follows from (6.2) and the first variational formula of $|\partial U|_{F}$.
∎

######  Proposition 6.3.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with smooth boundary.
Then

$\lim\limits_{p\rightarrow 1^{+}}{\rm Cap}_{F,p}(\Omega)={\rm
Cap}_{F,1}(\Omega)=\inf_{\begin{subarray}{c}\Omega\subset U\\\ \partial
U\hbox{smooth}\end{subarray}}|\partial U|_{F}.$ (6.3)

###### Proof.

The second equality can be found in [26], Lemma 2.2.5. See also [2]. For the
convenience of the reader, we provide a proof here.

Step 1.

${\rm Cap}_{F,1}(\Omega)\geq\inf_{\begin{subarray}{c}\Omega\subset U\\\
\partial U\hbox{smooth}\end{subarray}}|\partial U|_{F}.$ (6.4)

By co-area formula, for any $u\in{C}^{\infty}_{c}(\mathbb{R}^{n})$ with
$u\geq\chi_{\Omega}$ we have that

$\int\limits_{\mathbb{R}^{n}}F(\nabla
u)dx\geq\int^{1}_{0}|\partial\\{u>t\\}|_{F}dt\geq\inf_{\begin{subarray}{c}\Omega\subset
U\\\ \partial U\hbox{smooth}\end{subarray}}|\partial U|_{F}.$

In particular, taking the infimum over any such $u$, we get (6.4).

Step 2.

${\rm Cap}_{F,1}(\Omega)\leq\liminf_{p\rightarrow 1^{+}}{\rm
Cap}_{F,p}(\Omega).$ (6.5)

For every $u\in{C}^{\infty}_{c}(\mathbb{R}^{n})$ with $u\geq\chi_{\Omega}$ and
any $q>0$, by the definition of ${\rm Cap}_{F,1}(\Omega)$ and the Hölder
inequality, we have

$\displaystyle{\rm Cap}_{F,1}(\Omega)$
$\displaystyle\leq\int_{\mathbb{R}^{n}}F(\nabla u^{q})dx$ $\displaystyle\leq
q\Big{(}\int_{\mathbb{R}^{n}}u^{\frac{(q-1)p}{p-1}}dx\Big{)}^{(p-1)/p}\Big{(}\int_{\mathbb{R}^{n}}F^{p}(\nabla
u)dx\Big{)}^{1/p}.$ (6.6)

Choose $q=\frac{p(n-1)}{n-p}>1$, then $\frac{(q-1)p}{p-1}=\frac{np}{n-p}$
which is the Sobolev critical exponent of $p$. The anisotropic Sobolev
inequality with best constant [1] says that

$\displaystyle\left(\int_{\mathbb{R}^{n}}u^{\frac{np}{n-p}}dx\right)^{\frac{n-p}{np}}\leq\left(\frac{\omega_{n-1}}{\kappa_{n-1}}\right)^{\frac{1}{n}}T_{n,p}\left(\int_{\mathbb{R}^{n}}F^{p}(\nabla
u)dx\right)^{\frac{1}{p}}.$ (6.7)

where $T_{n,p}$ is the best constant in the classical Sobolev inequality, due
to Talenti [32].

It follows from (6.6) and (6.7) that

$\displaystyle{\rm
Cap}_{F,1}(\Omega)\leq\frac{p(n-1)}{n-p}\left(\frac{\omega_{n-1}}{\kappa_{n-1}}\right)^{\frac{p-1}{n-p}}T_{n,p}^{\frac{n(p-1)}{n-p}}\Big{(}\int_{\mathbb{R}^{n}}F^{p}(\nabla
u)dx\Big{)}^{\frac{n-1}{n-p}},$ (6.8)

Taking the infimum over any $u\in{C}^{\infty}_{c}(\mathbb{R}^{n})$ with
$u\geq\chi_{\Omega}$ in (6.8), we obtain

${\rm
Cap}_{F,1}(\Omega)\leq\frac{p(n-1)}{n-p}\left(\frac{\omega_{n-1}}{\kappa_{n-1}}\right)^{\frac{p-1}{n-p}}T_{n,p}^{\frac{n(p-1)}{n-p}}{\rm
Cap}_{F,p}(\Omega)^{\frac{n-1}{n-p}}.$ (6.9)

Passing to the limit in (6.9) as $p\to 1^{+}$, since $\lim_{p\to
1^{+}}T_{n,p}={n}^{\frac{1}{n}-1}\omega_{n-1}^{-\frac{1}{n}}$, we conclude
(6.5).

Step 3.

$\limsup_{p\rightarrow 1^{+}}{\rm
Cap}_{F,p}(\Omega)\leq\inf_{\begin{subarray}{c}\Omega\subset U\\\ \partial
U\hbox{smooth}\end{subarray}}|\partial U|_{F}.$ (6.10)

Let $U\subset\mathbb{R}^{n}$ be any open and bounded set with smooth boundary
such that $\Omega\subset U$. Let $d_{U}$ be the distance function
$d_{U}(x)={\rm dist}(x,U)$. It is well-known that $d_{U}$ is smooth in a
neighborhood of $U$. Introduce a smooth cut-off function $\chi_{\varepsilon}$
satisfying

$\left\\{\begin{aligned} \chi_{\varepsilon}&=1\ \ {\rm in}\ t<\varepsilon,\\\
-\frac{1}{\varepsilon}<&\chi^{\prime}_{\varepsilon}(t)<0\ \ {\rm in}\
\varepsilon\leq t\leq 2\varepsilon,\\\ \chi_{\varepsilon}&=0\ \ {\rm in}\
t>2\varepsilon,\end{aligned}\right.$

and set $\eta_{\varepsilon}=\chi_{\varepsilon}(d_{U})$. Choosing $\varepsilon$
small enough so that $\eta_{\varepsilon}$ is smooth. Hence

${\rm
Cap}_{F,p}(\Omega)\leq\int\limits_{\mathbb{R}^{n}}F^{p}(\nabla\eta_{\varepsilon})dx$

for any $p\geq 1$. Letting $p\rightarrow 1^{+}$, by using the co-area formula
and the mean value theorem, we get

$\displaystyle\limsup_{p\rightarrow 1^{+}}{\rm Cap}_{F,p}(\Omega)$
$\displaystyle\leq$
$\displaystyle\int\limits_{\mathbb{R}^{n}}F(\nabla\eta_{\varepsilon})dx=\int^{2\varepsilon}_{\varepsilon}|\chi^{\prime}_{\varepsilon}(t)|\int_{\\{d_{U}=t\\}}F(\nu)d\sigma
dt$ $\displaystyle\leq$
$\displaystyle\int_{\\{d_{U}=r_{\varepsilon}\\}}F(\nu)d\sigma,$

for some $r_{\varepsilon}\in(\varepsilon,2\varepsilon)$. Let
$r_{\varepsilon}\rightarrow 0^{+}$, we conclude that

$\limsup_{p\rightarrow 1^{+}}{\rm Cap}_{F,p}(\Omega)\leq|\partial U|_{F}$

for any bounded set $U$ with smooth boundary such that $\Omega\subset U$.
(6.10) follows.

Step 4. The assertion (6.3) follows from (6.4), (6.5) and (6.10). ∎

Proof of Theorem 1.3. The inequality (1.9) follows from Theorem 1.1 and
Proposition 6.3 by letting $p\to 1^{+}$.

When $\Omega$ is a Wulff ball, then in view of (6.13), we see that

$\displaystyle\inf_{\begin{subarray}{c}\Omega\subset U\\\ \partial
U\hbox{smooth}\end{subarray}}|\partial U|_{F}=|\partial\Omega|_{F}.$

Hence the equality in (1.9) holds. ∎

Proof of Corollary 1.1. The first assertion follows directly from Theorem 1.3,
Propositions 6.1 and 6.2.

Next we consider the equality case in (1.10) for some strictly outward
$F$-minimising set $\Omega$ with smooth and strictly $F$-mean-convex boundary.
Let $\\{\partial\Omega_{t}\\}_{t\in[0,T)}$ be the evolution of
$\partial\Omega$ under smooth inverse anisotropic mean curvature flow, for
some $T>0$. By [15], we know that the weak inverse anisotropic mean curvature
flow starting at $\partial\Omega$ coincides with the smooth flow
$\\{\Omega_{t}\\}_{t\in[0,T^{*})}$ for some $0<T^{*}\leq T$. In view of
Proposition 3.4 of [15] (see also Lemma 2.4 of [23]), $\Omega_{t}$ is strictly
outward $F$-minimising and strictly $F$-mean-convex for every $t\in[0,T^{*})$
. Then (1.10) holds for every $\partial\Omega_{t}$ with $t\in[0,T^{*})$. For
$t\in[0,T^{*})$, we can define monotonic quantity as following

$\Psi(t)=|\partial\Omega_{t}|_{F}^{-\frac{n-2}{n-1}}\int_{\partial\Omega_{t}}H_{F}F(\nu)d\sigma.$

by using Proposition 2.1 of [38],

$\Psi^{\prime}(0)=-|\partial\Omega|_{F}^{-\frac{n-2}{n-1}}\int\limits_{\partial\Omega}\frac{1}{H_{F}}\left|h_{F}-\frac{H_{F}}{n-1}g_{F}\right|^{2}d\sigma\leq
0.$ (6.11)

Here $h_{F}-\frac{H_{F}}{n-1}g_{F}$ is the traceless anisotropic second
fundamental form. If the strict inequality $\Psi^{\prime}(0)<0$ holds, it
follows that

$\Psi(t)<\Psi(0)=(n-1)\kappa_{n-1}^{1/(n-1)}\hbox{ for some }t\in(0,T^{*}),$

which contradicts (1.10) for some outward anisotropic minimising $\Omega_{t}$
with strictly anisotropic mean-convex boundary. Therefor $\Psi^{\prime}(0)=0$.
It follows from (6.11) that $\partial\Omega$ is totally anisotropic umbilical
and in turn, $\partial\Omega$ is of a Wulff shape. ∎

Proof of Corollary 1.2. For any $U\subset\mathbb{R}^{n}$ bounded with smooth
boundary and $\Omega\subset U$, by the Wulff inequality (1.4), we have

$\displaystyle\frac{|\partial
U|_{F}}{\kappa_{n-1}}\geq\left(\frac{n|U|}{\kappa_{n-1}}\right)^{\frac{n-1}{n}}\geq\left(\frac{n|\Omega|}{\kappa_{n-1}}\right)^{\frac{n-1}{n}}.$
(6.12)

Thus

$\displaystyle\frac{1}{\kappa_{n-1}}\inf_{\begin{subarray}{c}\Omega\subset
U\\\ \partial U\hbox{smooth}\end{subarray}}|\partial
U|_{F}\geq\left(\frac{n|\Omega|}{\kappa_{n-1}}\right)^{\frac{n-1}{n}}.$ (6.13)

The inequality follows from Theorem 1.3 and (6.13). The equality
classification for (1.11) follows from that of the Wulff inequality. ∎

Acknowledgements. This work was supported by NSFC (Grant No. 11871406). We
would like to thank Dr. M. Fogagnolo for attracting our attention to their
recent preprint [17] and explaining the new reformation of strictly outward
minimizing hull to us. We would also like to thank Professors V. Agostiniani,
L. Mazzieri and Deping Ye for their interest.

## References

  * [1] Alvino A, Ferone V, Trombetti G, Lions P L. Convex symmetrization and applications. Ann Inst H Poincaré Anal Non Linéaire, (1997), 14: 275-293
  * [2] Agostiniani V, Fogagnolo M, Mazzieri L. Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. Invent Math, 2020, 222: 1033-1101
  * [3] Agostiniani V, Fogagnolo M, Mazzieri L. Minkowski inequaltiy via nonlinear potentail theory. ArXiv:1906.00322v4, 2019
  * [4] Akman M, Gong J, Hineman J, Lewis J. Vogel A, The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity. ArXiv:1709.00447v2, 2017
  * [5] Akman M, Lewis J, Saari O, Vogel A. The Brunn-Minkowski inequality and a Minkowski problem for $\mathcal{A}$-harmonic Green’s function. ArXiv:1810.03752, 2018
  * [6] Agostiniani V, Mazzieri L. Monotonicity formulas in potential theory. Calc Var Partial Differential Equations, 2020, 59: 1, 32pp.
  * [7] Agostiniani V, Mazzieri L. On the geometry of the level sets of bounded static potentials. Comm Math Phys, 2017, 355: 261-301
  * [8] Bianchini C, Ciraolo G. Wulff shape characterizations in overdetermined anisotropic elliptic problems. Comm Partial Differential Equations, 2018, 43: 790-820
  * [9] Bianchini C, Ciraolo G, Salani P. An overdetermined problem for the anisotropic capacity. Calc Var Partial Differential Equations, 2016, 55: 4, 24pp.
  * [10] Cianchi A, Salani P. Overdetermined anisotropic elliptic problems. Math Ann, 2009, 345: 859-881.
  * [11] Borghini S, Mazzieri L. On the mass of static metrics with positive cosmological constant: I. Classical Quantum Gravity, 2018, 35: 12, 125001, 43pp.
  * [12] Borghini S, Mazzieri L. On the mass of static metrics with positive cosmological constant: II. Comm Math Phys, 2020, 377: 2079-2158
  * [13] Bray H, Miao P Z, On the capacity of surfaces in manifolds with nonnegative scalar curvature. Invent Math, 2008, 172: 459-475
  * [14] Chang S-Y A, Wang Y. Inequalities for quermassintegrals on k-convex domains. Advance in Mathematics, 2013, 248: 335-377
  * [15] Della Pietra F, Gavitone N, Xia C. Motion of level sets by inverse anisotropic mean curvature. ArXiv:1804.06639 to appear in Comm Anal Geom, 2018
  * [16] Della Pietra F, Gavitone N, Xia C. Symmetrization with respect to mixed volumes. ArXiv:1902.04965v2, 2019
  * [17] Fogagnolo M, Mazzieri L. Minimising hulls $p$-capacisty and isoperimetric inequality on complete Riemannian manifolds. ArXiv:2012.09490v1, 2020
  * [18] Fogagnolo M, Mazzieri L, Pinamonti A. Geometric aspects of $p$-capacitary potentials. Ann Inst H Poincaré Anal Non Linéaire, 2019, 36: 1151-1179
  * [19] Freire A, Schwartz F. Mass-capacity inequalities for conformally flat manifolds with boundary. Comm Partial Differential Equations, 2014, 39: 98-119
  * [20] Guan P F, Li J F. The quermassintegral inequalities for $k$-convex starshaped domains. Adv Math, 2009, 221: 1725-1732
  * [21] Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
  * [22] Huisken G. An isoperimetric concept for the mass in general relativity. Video, available at https://video.ias.edu/node/234.
  * [23] Huisken G, Ilmanen T. The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality. J Differential Geom, 2001, 59: 353-437
  * [24] Kichenassamy S, Véron L. Singular solutions of the $p$-Laplace equation. Math Ann, 1986, 275: 599-615
  * [25] Maggi F. Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2012.
  * [26] Maz’ya V. Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. xxviii+866 pp.
  * [27] Qiu G. A family of higher-order isoperimetric inequalities. Comm Cont Math, 17 (2015) no.3, 1450015.
  * [28] Reilly R C. On the Hessian of a function and the curvatures of its graph. Michigan Math J, 1973, 20: 373-383
  * [29] Schmidt T. Strict interior approximation of sets of finite perimeter and functions of bounded variation. Proc Amer Math Soc, 2015, 143: 2069-2084
  * [30] Schneider R. Convex bodies: the Brunn–Minkowski theory. Number 151. Cambridge University Press, 2013.
  * [31] Sternberg P, Ziemer W P, Williams G. $C^{1,1}$-regularity of constrained area minimizing hypersurfaces. Journal of Differential Equations, 1991, 94: 83-94
  * [32] Talenti G. Best constant in Sobolev inequality. Ann Mat Pura Appl, 1976, 110: 353-372
  * [33] Trudinger N S. Isoperimetric inequalities for quermassintegrals. Annales de l’IHP Analyse Non Lineaire, 1994, 11: 411-425
  * [34] Wang G F, Xia C. A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch Ration Mech Anal, 2011, 199: 99-115
  * [35] Wang G F, Xia C. A Brunn-Minkowski inequality for a Finsler-Laplacian. Analysis (Munich), 2011, 31: 103-115
  * [36] Wang G F, Xia C. Blow-up analysis of a Finsler-Liouville equation in two dimensions. J Differential Equations, 2012, 252: 1668-1700.
  * [37] Wang G F, Xia C. An optimal anisotropic Poincaré inequality for convex domains. Pacific J Math, 2012, 258: 305-325
  * [38] Xia C. Inverse anisotropic mean curvature flow and a Minkowski type inequality. Adv Math, 2017, 315: 102-129
  * [39] Xiao J. The $p$-harmonic capacity of an asymptotically flat $3$-manifold with non-negative scalar curvature. Ann Henri Poincaré, 2016, 17: 2265-2283
  * [40] Xiao J. $p$-capacity vs surface area. Adv Math, 2017, 308: 1318-1336.