# Radial symmetry of solutions
to anisotropic and weighted diffusion equations
with discontinuous nonlinearities

Serena Dipierro  Serena Dipierro: Department of Mathematics and Statistics,
The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA
6009, Australia<EMAIL_ADDRESS>,  Giorgio Poggesi  Giorgio
Poggesi: Department of Mathematics and Statistics, The University of Western
Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
<EMAIL_ADDRESS>and  Enrico Valdinoci  Enrico Valdinoci: Department
of Mathematics and Statistics, The University of Western Australia, 35
Stirling Highway, Crawley, Perth, WA 6009, Australia
<EMAIL_ADDRESS>
###### Abstract.

For $1<p<\infty$, we prove radial symmetry for bounded nonnegative solutions
of

$\begin{cases}-\mathop{\mathrm{div}}\left\\{w(x)\,H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=f(u)\,w(x)\ &\mbox{ in }\
\Sigma\cap\Omega,\\\ u=0\ &\mbox{ on }\ \Gamma_{0},\\\
\langle\nabla_{\xi}H(\nabla u),\nu\rangle=0\ &\mbox{ on }\
\Gamma_{1}\setminus\left\\{0\right\\},\end{cases}$

where $\Omega$ is a Wulff ball, $\Sigma$ is a convex cone with vertex at the
center of $\Omega$, $\Gamma_{0}:=\Sigma\cap\partial\Omega$,
$\Gamma_{1}:=\partial\Sigma\cap\Omega$, $H$ is a norm, $w$ is a given weight
and $f$ is a possibly discontinuous nonnegative nonlinearity.

Given the anisotropic setting that we deal with, the term “radial” is
understood in the Finsler framework, that is, the function $u$ is radial if
there exists a point $x$ such that $u$ is constant on the Wulff shapes
centered at $x$.

When $\Sigma=\mathbb{R}^{N}$, J. Serra obtained the symmetry result in the
isotropic unweighted setting (i.e., when $H(\xi)\equiv|\xi|$ and $w\equiv 1$).
In this case we provide the extension of his result to the anisotropic
setting. This provides a generalization to the anisotropic setting of a
celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new
even for $p=2$ whenever $N>2$.

When $\Sigma\subsetneq\mathbb{R}^{N}$ the results presented are new even in
the isotropic and unweighted setting (i.e., when $H$ is the Euclidean norm and
$w\equiv 1$) whenever $2\neq p\neq N$. Even for the previously known case of
unweighted isotropic setting with $p=2$ and $\Sigma\subsetneq\mathbb{R}^{N}$,
the present paper provides an approach to the problem by exploiting integral
(in)equalities which is new for $N>2$: this complements the corresponding
symmetry result obtained via the moving planes method by Berestycki-Pacella.

The results obtained in the isotropic and weighted setting (i.e., with
$w\not\equiv 1$) are new for any $p$.

###### Key words and phrases:

Symmetry, Convex cones, Weighted anisotropic isoperimetric inequalities

###### 2020 Mathematics Subject Classification:

35B06

##  1\. Introduction

Since the classical works of Aleksandrov [A], Serrin [S] and
Gidas–Ni–Nirenberg [GNN], an intensively studied topic in the theory of
partial differential equations and the calculus of variations focuses on the
radial symmetry of solutions under suitable assumptions on the equation under
consideration, on the boundary conditions and/or on the domain.

In this paper we consider the Finsler framework of a nonlinear anisotropic
equation in divergence form. The setting taken into account is quite general,
since it includes nonlinear operators of $p$-Laplace type, possibly with
weights. The domain considered is obtained from the intersection of a Wulff
ball and a cone $\Sigma\subseteq\mathbb{R}^{N}$ with vertex at its center. The
results provided are of radial symmetry type, where, given the possible
anisotropy of the ambient space, the term “radial” is understood in the sense
that a function $u$ is radial if there exists a point $x$ such that $u$ is
constant on the Wulff shapes centered at $x$ (see Section 2 for the detailed
mathematical setting).

These results will be obtained under suitable homogeneity or concavity
assumptions on the weights $w$ and suitable arithmetic relations between the
dimension of the ambient space, the homogeneous exponent of the weight and the
homogeneity of the nonlinear operator. A pivotal step in our analysis consists
in proving that the level sets of the solution are isoperimetric (and
additionally the anisotropic norm of the gradient of the solution is constant
along these level sets). This cornerstone result will be stated in detail in
Theorem 1.1 below and then combined with a series of isoperimetric
inequalities to establish that the level sets are Wulff shapes: this, together
with the additional information on the constancy of the anisotropic norm of
the gradient of the solution, establishes the radial symmetry of the solution.
Depending on technical conditions on the (an)isotropy of the ambient space and
on the weights, these radial symmetry results will be detailed in Theorems
1.4, 1.5 and 1.6.

The arguments that we use are inspired by those of J. Serra in [Se], which in
turn were inspired by the classical paper of P.-L. Lions [L] in which the
symmetry was obtained in the Euclidean case with $p=N=2$,
$\Sigma=\mathbb{R}^{N}$ and $w\equiv 1$. We remark that the main results in
[L], focusing on elliptic semilinear equations in the plane, can be seen as a
counterpart of those in [GNN], in the sense that the results in [L] weaken the
smoothness assumptions on the source term and on the solution with respect to
the setting in [GNN], at the expense of restricting to positive nonlinearities
and to dimension two only. Interestingly, the method in [L], being based on
integral (in)equalities, is conceptually different from the moving plane
technique used in [GNN] and, in a sense, it is more related, apart from
several important structural differences, to the approach to radial symmetry
inhaugurated by Weinberger in [W]: see also, e.g., [PSc, Re, GL, FGK, FK,
BNST, MP, MP2, MP3, GS, EP, Po, CS, WX, BC, GX, DPV, PT, PT2, CG, CiR, CFR]
for related problems and ramifications.

The method of [L] has been extended by Kesavan–Pacella to the $N$-dimensional
Euclidean setting in presence of the $p$-Laplacian operator in the case $p=N$,
see [KP]. In turn, the techniques of [KP] have triggered the research in the
anisotropic case (with $\Sigma=\mathbb{R}^{N}$ and $p=N$), which was carried
out by Belloni–Ferone–Kawohl in [BFK]. The extension of the method to the case
$p\neq N$ is due to J. Serra ([Se]), who obtained it in the isotropic setting
(with $\Sigma=\mathbb{R}^{N}$).

In [BP], by using the moving planes method, Berestycki–Pacella obtained radial
symmetry in spherical convex cones (in the isotropic unweighted setting), when
$p=2$ and $f$ is Lipschitz. To the authors’ knowledge, symmetry results in
spherical convex cones with $\Sigma\subsetneq\mathbb{R}^{N}$ were considered
only in the unweighted isotropic setting in [BP] (for $p=2$) and [KP] (for
$p=N$).

See also [DFM] and the references therein for further works on symmetry
problems for nonlinear equations.

We point out that, while our method is general enough to work in the
anisotropic weighted setting, the results offered here are new also in some
more classical cases (such as those stated in Theorems 1.4, 1.5 and 1.6).
Moreover, thanks to the available characterizations of the isoperimetric sets,
the conclusions obtained in these cases are stronger and take a less technical
form. As an introductory example, one of the results that we obtain here (see
Theorem 1.5) goes as follows.

Let $\Omega$ be a Wulff ball in $\mathbb{R}^{N}$, $N\geqslant 2$, and let
$1<p<\infty$. Assume that $f\in
L^{\infty}_{\mathrm{loc}}(\left[0,\infty\right))$ is nonnegative. Let $u\in
C^{1}(\overline{\Omega})$ be a weak solution of

$\begin{cases}-\mathop{\mathrm{div}}\left\\{H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=f(u)\ &\mbox{ in }\ \Omega,\\\
u\geqslant 0&\mbox{ in }\ \Omega,\\\ u=0\ &\mbox{ on }\
\partial\Omega,\end{cases}$

Assume that either

$p\geqslant N,$

or

$p<N\quad\text{and, for some nonincreasing function $\phi\geqslant 0$, we have
}\phi\leqslant f\leqslant\frac{Np}{N-p}\phi.$

Then, $u$ is a radially symmetric and radially nonincreasing function.
Moreover,

$u\text{ is radially strictly decreasing on
}\left\\{0<u<\max_{\overline{\Omega}}u\right\\},$

and

$\left\\{0<u<\max_{\overline{\Omega}}u\right\\}\quad\text{is a Wulff annulus
or a punctured Wulff ball.}$

Let us now go into the technical framework of this paper, to present the
results obtained in their full generality.

Let $\Sigma$ be an open cone in $\mathbb{R}^{N}$ with vertex at the origin,
i.e.,

$\Sigma:=\left\\{tx\,:\,x\in\omega,\,t\in(0,\infty)\right\\},$

for some domain $\omega\subseteq\SS^{N-1}$. We stress that the possibility
$\omega=\SS^{N-1}$ (which implies $\Sigma=\mathbb{R}^{N}$) is allowed
throughout the paper. When $\omega\subsetneq\SS^{N-1}$, we assume $\Sigma$ to
be convex and denote by $\nu$ its outward unit normal (which is defined almost
everywhere on $\partial\Sigma$).

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain, and define

$\Gamma_{0}:=\Sigma\cap\partial\Omega\quad{\mbox{ and
}}\quad\Gamma_{1}:=\partial\Sigma\cap\Omega.$

Furthermore, we endow $\mathbb{R}^{N}$ with a norm
$H:\mathbb{R}^{N}\to\mathbb{R}$ such that:

(1.1) $H$ is convex; (1.2) $H(\xi)\geqslant 0$ for $\xi\in\mathbb{R}^{N}$ and
$H(\xi)=0$ if and only if $\xi=0$; (1.3) $H(t\xi)=|t|H(\xi)$ for
$\xi\in\mathbb{R}^{N}$ and $t\in\mathbb{R}$.

We also define

(1.4) $B^{H}:=\left\\{\xi\in\mathbb{R}^{N}:H(\xi)<1\right\\}.$

Throughout the paper, we assume $H\in
C^{2}(\mathbb{R}^{N}\setminus\left\\{0\right\\})$ to be uniformly elliptic. We
say that $H\in C^{2}(\mathbb{R}^{N}\setminus\left\\{0\right\\})$ is uniformly
elliptic if the ball $B^{H}$ is uniformly convex, i.e., such that the
principal curvatures of the boundary of $B^{H}$ are bounded away from zero.
This is a standard assumption on $H$ in order to obtain some regularity of the
solutions, using or adapting the elliptic regularity theory (see, e.g., also
[BC, CFV]).

In this setting, we consider solutions of

(1.5) $\begin{cases}-\mathop{\mathrm{div}}\left\\{w(x)\,H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=f(u)\,w(x)\ &\mbox{ in }\
\Sigma\cap\Omega,\\\ u\geqslant 0&\mbox{ in }\ \Sigma\cap\Omega,\\\ u=0\
&\mbox{ on }\ \Gamma_{0},\\\ \langle\nabla_{\xi}H(\nabla u),\nu\rangle=0\
&\mbox{ on }\ \Gamma_{1}\setminus\left\\{0\right\\}.\end{cases}$

In the case $\Sigma=\mathbb{R}^{N}$, we have $\Gamma_{1}=\varnothing$ and the
last condition in (1.5) is trivially satisfied. In this case, (1.5) simply
becomes

(1.6) $\begin{cases}-\mathop{\mathrm{div}}\left\\{w(x)\,H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=f(u)\,w(x)\ &\mbox{ in }\
\Omega,\\\ u\geqslant 0&\mbox{ in }\ \Omega,\\\ u=0\ &\mbox{ on }\
\partial\Omega.\end{cases}$

In our main results, the assumptions on the weight $w$ are the
following111Using the recent results in [I], assumption (1.8) may be weakened
by requiring that, when $\lambda>0$, either $w$ or $w^{1/\lambda}$ is concave,
or, equivalently, that either $\lambda\in(0,1)$ and $w$ is concave or
$\lambda\in[1,+\infty)$ and $w^{1/\lambda}$ is concave. This generalization is
accomplished by using, when $w$ is concave, the isoperimetric inequality in
Corollary 0.14 (see also Remark 0.15) of [I] instead of the one by [CRS]
recalled here in Theorem 4.1. Strictly speaking, the setting in [I] is
isotropic, but the arguments introduced there have the potential to be
generalized to obtain analogous results in the anisotropic setting as well.
Counterexamples to the isoperimetric inequality are also given in Corollary
0.12 and Remark 0.13 of [I]. We also point out that the constant in [I] (which
is in terms of the volume ratio) improves in several cases the previously
obtained ones.: there exists $\lambda\geqslant 0$ such that

(1.7) $\displaystyle w:\overline{\Sigma}\to\left[0,+\infty\right)\text{ is a
continuous nonnegative function positively homogeneous of
degree~{}$\lambda$,}$ (1.8) $\displaystyle w^{1/\lambda}\text{ is concave in
}\Sigma\text{ (in the case $\lambda>0$)},$ (1.9) $\displaystyle w\text{ is
locally Lipschitz in }\Sigma.$

We notice that when $\lambda>0$, the concavity assumption on $w^{1/\lambda}$
automatically gives that $w$ is positive in $\Sigma$.

To state our main theorem in this setting, we recall the following notation.
We identify the dual space of $\mathbb{R}^{N}$ with $\mathbb{R}^{N}$ itself
via the scalar product $\langle\cdot,\cdot\rangle$. Accordingly, given the
norm $H$ satisfying (1.1), (1.2) and (1.3), the space $\mathbb{R}^{N}$ turns
out to be endowed with the dual norm $H_{0}$, that is the polar function
defined by

(1.10) $H_{0}(x)=\sup_{\xi\neq 0}\frac{\langle
x,\xi\rangle}{H(\xi)}\quad\text{for}\quad x\in\mathbb{R}^{N}.$

Notice that $H$ results to be the support function of the unitary Wulff ball

(1.11) $B^{H_{0}}:=\left\\{x\in\mathbb{R}^{N}:H_{0}(x)<1\right\\}$

of $H$ centered at the origin (see [CrM] and [Sc, Section 1.7]) and, in turn,
$H_{0}$ is the support function of $B^{H}$, defined in (1.4). To ease
notation, the unitary Wulff ball (of $H$) $B^{H_{0}}$ centered at the origin
will be denoted simply by $B$. Similarly, for $r>0$, $B_{r}$ will denote the
Wulff ball (of $H$) of radius $r$ centered at the origin, i.e.,

$B_{r}:=B_{r}^{H_{0}}:=\left\\{x\in\mathbb{R}^{N}:H_{0}(x)<r\right\\}.$

Furthermore, for a set of finite perimeter $E$, we define the weighted
anisotropic perimeter of $E$ in $\Sigma$ as follows

(1.12)
$P_{w,H}(E;\Sigma):=\int_{\Sigma\cap\partial^{*}E}H(\nu)\,w\,d{\mathcal{H}}^{N-1},$

being $\partial^{*}E$ the reduced boundary of $E$ and $\nu$ its outer (measure
theoretical) unit normal vector. Also, for a measurable set $E\subset\Sigma$,
we denote by $w(E)$ the weighted volume of $E$, namely

(1.13) $w(E):=\int_{E}w\,d{\mathcal{H}}^{N}.$

In the unweighted case (i.e., when $w\equiv 1$), $w(E)$ agrees with
${\mathcal{H}}^{N}(E)$.

With this notation, our main theorem is the following:

######  Theorem 1.1.

Let $\Omega$ be a Wulff ball in $\mathbb{R}^{N}$ centered at $0$, $N\geqslant
2$, and let $1<p<\infty$. Assume that $f\in
L^{\infty}_{\mathrm{loc}}(\left[0,\infty\right))$ is nonnegative and the
weight $w$ satisfies (1.7), (1.8), and (1.9).

Let $u\in
C^{1}\left((\Sigma\cap\Omega)\cup\Gamma_{0}\cup(\Gamma_{1}\setminus\left\\{0\right\\})\right)\cap
W^{1,\infty}(\Sigma\cap\Omega)$ be a solution of (1.5) in the weak sense. Set

(1.14) $D:=N+\lambda,$

where $\lambda$ is that appearing in (1.7). Assume that either

(a) $p\geqslant D,$

or

(b) $p<D\quad\text{and, for some nonincreasing function $\phi\geqslant 0$, we
have }\phi\leqslant f\leqslant\frac{Dp}{D-p}\phi.$

Let also

(1.15) $M:=\sup_{\Sigma\cap\Omega}u,$

Then, for a.e. $t\in(0,M)$ the following two conditions are verified:

(i) $\left\\{u>t\right\\}$ satisfies

(1.16)
$\frac{P_{w,H}\big{(}\left\\{u>t\right\\};\Sigma\big{)}}{w\big{(}\Sigma\cap\left\\{u>t\right\\}\big{)}^{\frac{D-1}{D}}}=\frac{P_{w,H}(B;\Sigma)}{w(\Sigma\cap
B)^{\frac{D-1}{D}}},$

(ii) $H(\nabla u)$ is constant on $\left\\{u=t\right\\}$.

We remark that, in the setting of Theorem 1.1, since $u\in
W^{1,\infty}(\Sigma\cap\Omega)$ and $\Sigma\cap\Omega$ is convex, the Sobolev
embedding theorem (see also [H, Theorem 4.1]) ensures that $u\in
C^{0,1}(\overline{\Sigma\cap\Omega})\subset
C^{0}(\overline{\Sigma\cap\Omega})$, and hence an equivalent definition in
(1.15) is to set

$M:=\max_{\overline{\Sigma\cap\Omega}}u.$

We also point out that formula (1.16) says that the sets
$\left\\{u>t\right\\}$ satisfy the equality in the weighted anisotropic
isoperimetric inequality in cones established in [CRS]. This will be recalled
in details in Theorem 4.1.

######  Remark 1.2 (On the regularity of the solutions).

Concerning the regularity assumptions on $u$ in Theorem 1.1, we point out that
the main hypothesis is that $u$ belongs to $W^{1,\infty}(\Sigma\cap\Omega)$,
while the additional smoothness assumptions may be dropped. Moreover, when
$\Sigma=\mathbb{R}^{N}$, also the assumption $u\in
W^{1,\infty}(\Sigma\cap\Omega)$ could be dropped and replaced just by the
assumption that $u$ is bounded.

Indeed, in the case where $\Sigma=\mathbb{R}^{N}$, whenever $\Omega$ is a
$C^{1,\gamma}$ domain, with $0<\gamma\leqslant 1$, then [Li, Theorem 1]
ensures that bounded solutions to (1.6) are automatically
$C^{1,\gamma}(\overline{\Omega})$, for some $0<\gamma\leqslant 1$. Notice
that, our regularity assumption on $H$ ensures that $\Omega$ in Theorem 1.1,
which is a Wulff ball, is of class $C^{2}$. Thus, in the case when
$\Sigma=\mathbb{R}^{N}$, all the regularity assumptions on $u$ in Theorem 1.1
are satisfied by any bounded solution.

When $\Sigma\subsetneq\mathbb{R}^{N}$, we have to consider the mixed boundary
value problem (1.5). For such a problem, regularity up to the (whole) boundary
is a delicate issue, and it strongly depends on how $\Sigma$ and $\Omega$
intersect. Nevertheless, [Li, Theorem 1] still ensures the
$C^{1,\gamma}$-regularity (of bounded solutions) up to $\Gamma_{0}$; the
$C^{1,\gamma}$-regularity up to $\Gamma_{1}\setminus\left\\{0\right\\}$ could
be obtained by [Li, Theorem 2], if we further assume that
$\Gamma_{1}\setminus\left\\{0\right\\}$ is of class $C^{1,\gamma}$, with
$0<\gamma\leqslant 1$, and $w$ satisfy the additional requirement

(1.17) $w\text{ is positive and H\"{o}lder continuous on
}\overline{\Sigma\cap\Omega}\setminus\left\\{0\right\\}.$

We stress that our results are presented without these additional assumptions:
that is, we do not require (1.17) and we only ask $\Sigma$ to be convex (with
no need of the additional assumption that
$\Gamma_{1}\setminus\left\\{0\right\\}$ is of class $C^{1,\gamma}$).

Essentially, the main regularity assumption that we impose on $u$ in Theorem
1.1 (when $\Sigma\subsetneq\mathbb{R}^{N}$) is that $u$ belongs to
$W^{1,\infty}(\Sigma\cap\Omega)$. This is needed in order to ensure the
validity of certain integral identities, such as the Pohozaev-type identity
(3.17), by using an approximation argument. We notice that, when $H$ is the
Euclidean norm and $w\equiv 1$, [CM, Theorem 1.2] guarantees the
$W^{1,\infty}$ regularity up to $\Gamma_{1}$ (including the vertex). As
pointed out in [PT], the assumption $u\in W^{1,\infty}(\Sigma\cap\Omega)$ can
be seen as a gluing condition between the cone and $\Gamma_{0}$. Related to
this, we mention that [PT, Proposition 6.1] guarantees
$C^{2}(\overline{\Omega}\setminus\left\\{0\right\\})$-regularity for weak
solutions of

$\begin{cases}-\Delta u=1\ &\mbox{ in }\ \Sigma\cap\Omega,\\\ u=0\ &\mbox{ on
}\ \Gamma_{0},\\\ \langle\nabla u,\nu\rangle=0\ &\mbox{ on }\
\Gamma_{1}\setminus\left\\{0\right\\},\end{cases}$

in general domains $\Omega$, whenever $\Gamma_{0}$ and $\Gamma_{1}$ intersect
orthogonally.

######  Remark 1.3 (On the weights).

The assumptions (1.7), (1.8) and (1.9) on $w$ guarantee the validity of the
weighted anisotropic isoperimetric inequality (4.1) in convex cones, obtained
in [CRS]. The homogeneity assumption on $w$ contained in (1.7) is also used to
obtain the Pohozaev-type identity in Lemma 3.4.

Various examples of weights satisfying (1.7), (1.8) and (1.9) are provided in
[CRS]. We point out that all the weights provided in those examples also
satisfy assumption (1.17), up to substituting $\Sigma$ with a smaller cone
$\Sigma^{\prime}\subset\Sigma$ (see also item (iii) at page 2983 in [CRS]).

As pointed out in [CRS], the equality holds in the weighted anisotropic
isoperimetric inequality in (4.1) whenever $\Sigma\cap E=\Sigma\cap B_{r}$,
where $r$ is any positive number. That is, Wulff balls (centered at the
origin) intersected with $\Sigma$ are always minimizers of the isoperimetric
inequality (4.1). However, the uniqueness222Whenever a line of direction
$a\in\mathbb{R}^{N}$ is contained in $\Sigma$, uniqueness will be up to
translations in direction $a$ (see Section 4). of those minimizers (i.e., the
characterization of the equality sign in (4.1)) in general is still not
available in the literature (see Section 4). Whenever the uniqueness of those
minimizers is available, it is not difficult to obtain the radial symmetry of
$u$, as a corollary of Theorem 1.1.

The uniqueness of the minimizers can be obtained in the unweighted (i.e., when
$w\equiv 1$) anisotropic setting by adapting the ideas in [FI] to the
anisotropic setting (see Theorem 4.2). Notice that in the unweighted case we
have $D=N$ (in Theorem 1.1), and (1.5) reads as follows:

(1.18) $\begin{cases}-\mathop{\mathrm{div}}\left\\{H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=f(u)\ &\mbox{ in }\
\Sigma\cap\Omega,\\\ u\geqslant 0&\mbox{ in }\ \Sigma\cap\Omega,\\\ u=0\
&\mbox{ on }\ \Gamma_{0},\\\ \langle\nabla_{\xi}H(\nabla u),\nu\rangle=0\
&\mbox{ on }\ \Gamma_{1}\setminus\left\\{0\right\\}.\end{cases}$

In this setting, the uniqueness of the minimizers of the isoperimetric
inequality and Theorem 1.1 lead to

######  Theorem 1.4 (Symmetry in convex cones in the anisotropic unweighted
setting).

Let $\Omega$ be a Wulff ball in $\mathbb{R}^{N}$ centered at $0$, $N\geqslant
2$, and let $1<p<\infty$. Assume that $f\in
L^{\infty}_{\mathrm{loc}}(\left[0,\infty\right))$ is nonnegative. Let $u\in
C^{1}\left((\Sigma\cap\Omega)\cup\Gamma_{0}\cup\Gamma_{1}\setminus\left\\{0\right\\}\right)\cap
W^{1,\infty}(\Sigma\cap\Omega)$ be a solution of (1.18) in the weak sense.
Assume that either

(a′) $p\geqslant N,$

or

(b′) $p<N\quad\text{and, for some nonincreasing function $\phi\geqslant 0$, we
have }\phi\leqslant f\leqslant\frac{Np}{N-p}\phi.$

Then, $u$ is a radially symmetric and radially nonincreasing function.
Moreover, by setting $M$ as in (1.15),

$u\text{ is radially strictly decreasing on }\left\\{0<u<M\right\\},$

and

$\left\\{0<u<M\right\\}\quad\text{is a Wulff annulus or a punctured Wulff ball
centered at }0\text{ intersected with }\Sigma.$

We point out that conditions (a′) and (b′) in Theorem 1.4 correspond,
respectively, to conditions (a) and (b) of Theorem 1.1 when $D=N$.

Also, as usual, when $\Sigma=\mathbb{R}^{N}$, we have $\Gamma_{1}=\varnothing$
and the last condition in (1.18) is trivially satisfied. In this case, (1.18)
simply becomes:

(1.19) $\begin{cases}-\mathop{\mathrm{div}}\left\\{H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=f(u)\ &\mbox{ in }\ \Omega,\\\
u\geqslant 0&\mbox{ in }\ \Omega,\\\ u=0\ &\mbox{ on }\
\partial\Omega,\end{cases}$

and Theorem 1.4 reads as follows.

######  Theorem 1.5 (Symmetry in balls in the anisotropic unweighted setting).

Let $\Omega$ be a Wulff ball in $\mathbb{R}^{N}$, $N\geqslant 2$, and let
$1<p<\infty$. Assume that $f\in
L^{\infty}_{\mathrm{loc}}(\left[0,\infty\right))$ is nonnegative. Let $u\in
C^{1}(\overline{\Omega})$ be a solution of (1.19) in the weak sense. Assume
that either (a′) or (b′) of Theorem 1.4 holds true.

Then, $u$ is a radially symmetric and radially nonincreasing function.
Moreover,

$u\text{ is radially strictly decreasing on
}\left\\{0<u<\max_{\overline{\Omega}}u\right\\},$

and

$\left\\{0<u<\max_{\overline{\Omega}}u\right\\}\quad\text{is a Wulff annulus
or a punctured Wulff ball.}$

For completeness, in light of Remark 1.2, we point out that the regularity
assumption on the solution $u$ taken in Theorem 1.5 can be relaxed by assuming
only that the solution is bounded.

The uniqueness of the minimizers of the isoperimetric inequality is also
available in the weighted isotropic setting (i.e., when $H$ is the Euclidean
norm). In this case, (1.5) reads as follows

(1.20) $\begin{cases}-\mathop{\mathrm{div}}\left\\{w(x)\,|\nabla
u|^{p-2}\,\nabla u\right\\}=f(u)\,w(x)\ &\mbox{ in }\ \Sigma\cap\Omega,\\\
u\geqslant 0&\mbox{ in }\ \Sigma\cap\Omega,\\\ u=0\ &\mbox{ on }\
\Gamma_{0},\\\ \langle\nabla u,\nu\rangle=0\ &\mbox{ on }\
\Gamma_{1}\setminus\left\\{0\right\\}.\end{cases}$

In this case, the uniqueness of the minimizers of the isoperimetric inequality
and Theorem 1.1 lead333The recent examples in [I] for isoperimetric minimizers
that are half-balls instead of full balls may also yield interesting
statements in terms of lack of symmetry of solutions to be compared to Theorem
1.6 here. to

######  Theorem 1.6 (Symmetry in convex cones in the isotropic weighted
setting).

Let $\Omega$ be a (Euclidean) ball in $\mathbb{R}^{N}$ centered at $0$,
$N\geqslant 2$, and let $1<p<\infty$. Assume that $f\in
L^{\infty}_{\mathrm{loc}}(\left[0,\infty\right))$ is nonnegative and the
weight $w$ satisfies (1.7), (1.8), and (1.9). Let $u\in
C^{1}\left((\Sigma\cap\Omega)\cup\Gamma_{0}\cup\Gamma_{1}\setminus\left\\{0\right\\}\right)\cap
W^{1,\infty}(\Sigma\cap\Omega)$ be a solution of (1.20) in the weak sense.

Assume that either (a) or (b) of Theorem 1.1 holds true.

Then, $u$ is a radially symmetric and radially nonincreasing function.
Moreover, by setting $M$ as in (1.15),

$u\text{ is radially strictly decreasing on }\left\\{0<u<M\right\\},$

and

$\left\\{0<u<M\right\\}\quad\text{is a annulus or a punctured ball centered at
}0\text{ intersected with }\Sigma.$

We point out that Theorem 1.5 provides a generalization to the anisotropic
setting of the celebrated result contained in [GNN] and such a generalization
is new even for $p=2$ whenever $N>2$. We recall that [GNN] exploits the moving
planes method, which seems not to be helpful for anisotropic problems.
Instead, the method via integral (in)equalities used here (and based on [L])
turns out to be effective in this setting.

When $\Sigma\subsetneq\mathbb{R}^{N}$, the results presented here are new even
in the isotropic and unweighted setting (i.e., when $H$ is the Euclidean norm
and $w\equiv 1$) whenever $2\neq p\neq N$. In the case $p=2$ and
$\Sigma\subsetneq\mathbb{R}^{N},\,N>2$, the symmetry result was previously
obtained only relying on the moving planes method (and in the unweighted
isotropic setting) in [BP]: even in this special case, the present paper
provides a new approach to the problem via integral (in)equalities (based on
[L]) which complements [BP].

Also, the results obtained here in the isotropic and weighted setting (i.e.,
with $w\not\equiv 1$) are new for any $p$.

Theorems 1.4, 1.5 and 1.6 can be rephrased in terms of the variational
formulation of (1.5), as follows. We restrict this formulation to the case
when $f$ is a continuous nonlinearity to avoid problems with the
differentiability of the functional.

######  Theorem 1.7 (Variational formulation of Theorems 1.4, 1.5 and 1.6).

Let $\Omega$ be a Wulff ball in $\mathbb{R}^{N}$ centered at $0$, $N\geqslant
2$, and let $1<p<\infty$. Assume that $f$ is a continuous nonnegative function
defined on $\left[0,\infty\right)$. Let

$u\in W^{1,\infty}_{\Gamma_{0}}(\Sigma\cap\Omega):=\left\\{v\in
W^{1,\infty}(\Sigma\cap\Omega)\,:\,v=0\text{ on }\Gamma_{0}\right\\}$

be a nonnegative critical point of the functional

$\int_{\Sigma\cap\Omega}\left\\{\frac{H^{p}(\nabla
u)}{p}-F(u)\right\\}w(x)\,d{\mathcal{H}}^{N},\quad\text{where
}\,F(s):=\int_{0}^{s}f(\tau)\,d\tau.$

Assume that $u\in
C^{1}\left((\Sigma\cap\Omega)\cup\Gamma_{1}\setminus\left\\{0\right\\}\right)$
and that, either

$\begin{split}&H\text{ is any uniformly elliptic norm of class
}C^{2}(\mathbb{R}^{N}\setminus\left\\{0\right\\}),\,w\equiv 1,\\\ &\text{and
either \eqref{eq:condizione a} or \eqref{eq:condizione b} of Theorem
\ref{thm:cones} holds true, }\end{split}$

or

$\begin{split}&H\text{ is the euclidean norm, }w\text{ is any weight
satisfying \eqref{eq:assumptions on w for main theorems},
\eqref{eq:assumptions on w for main theorems2} and \eqref{eq:assumptions on w
for main theorems3},}\\\ &\text{and either \eqref{eq:GENERAL condizione a} or
\eqref{eq:GENERAL condizione b} of Theorem \ref{thm:MAIN GENERAL} holds
true.}\end{split}$

Then, $u$ is a radially symmetric and radially nonincreasing function.
Moreover, by setting $M$ as in (1.15), $u$ is radially strictly decreasing on
the set $\left\\{0<u<M\right\\}$, which is a Wulff annulus or a punctured
Wulff ball centered at $0$ intersected with $\Sigma$.

We conclude this introduction with a comment on the proofs of Theorems 1.4,
1.5 and 1.6. As already mentioned, those theorems are obtained by putting
together Theorem 1.1 and the characterization of the equality sign in the
isoperimetric inequality (4.1). When $\Sigma$ is a convex cone containing no
lines, then the characterization of the minimizers of the isoperimetric
inequality forces the superlevel sets to be Wulff balls all centered at $0$
(which is the vertex of $\Sigma$) intersected with $\Sigma$. This immediately
gives the desired symmetry results. When $\Sigma$ contains lines instead, the
characterization of minimizers still informs us that all the superlevel sets
are Wulff balls intersected with $\Sigma$, but a further step is needed in
order to prove that all of those Wulff balls are centered at $0$. This step is
accomplished in Lemma 5.2.

The rest of this paper is organized as follows. Section 2 recalls the basics
of the Finsler framework, placing the known results into a setting convenient
for the applications that we have in mind.

Other useful auxiliary results are collected in Section 3, which contains a
suitable maximum principle in cones, a Stampacchia-type result, some geometric
differential inequalities and a Pohozaev-type identity.

In Section 4 we deal with some isoperimetric inequalities, leveraging on the
existing literature [CGPRS, CR, CRS, FI, FM, FMP] and adapting all the
previous results to the case under consideration in this paper.

The proofs of the main results in Theorems 1.1, 1.4, 1.5 and 1.6 are then
contained in Section 5.

##  2\. Preliminaries and setting

Here we recall some basic facts on the Finsler framework.

Given the norm $H$ satisfying (1.1), (1.2) and (1.3) and the dual norm $H_{0}$
defined in (1.10), we observe that we can reconstruct $H$ in terms of $H_{0}$
as

(2.1) $H(\xi)=\sup_{x\neq 0}\frac{\langle
x,\xi\rangle}{H_{0}(x)}\quad\text{for}\quad\xi\in\mathbb{R}^{N}.$

Consequently, it holds that

(2.2) $|\langle\xi,\eta\rangle|\leqslant H(\xi)H_{0}(\eta)\,,\quad\text{for
any }\,\xi,\,\eta\in\mathbb{R}^{N}.$

The two convex sets $B^{H_{0}}$ and $B^{H}$, defined in (1.11) and (1.4)
respectively, are both centrally symmetric and they are polar of each other.
We denote by $B_{r}^{H_{0}}(x)$ the ball centered at the point $x$ with radius
$r$ in the norm $H_{0}$, i.e.

$B_{r}^{H_{0}}(x):=\left\\{y\in\mathbb{R}^{N}:H_{0}(x-y)<r\right\\}.$

Analogously, we define

$B_{r}^{H}(x):=\left\\{\xi\in\mathbb{R}^{N}:H(x-\xi)<r\right\\}.$

The sets $B_{r}^{H_{0}}(x)$ are named Wulff balls of $H$ centered at $x$ with
radius $r$, and they are homothetic copies of the unitary Wulff ball
$B^{H_{0}}$. We call Wulff shapes of $H$ centered at $x$ with radius $r$ the
boundaries $\partial B_{r}^{H_{0}}(x)$ of $B_{r}^{H_{0}}(x)$, i.e.,

$\partial B_{r}^{H_{0}}(x):=\left\\{y\in\mathbb{R}^{N}:H_{0}(x-y)=r\right\\}.$

In what follows, to ease notation, we will omit the exponent $H_{0}$, that is,
the Wulff ball (of $H$) $B_{r}^{H_{0}}(x)$ centered at $x$ of radius $r$ will
be denoted just by $B_{r}(x)$.

Furthermore, from the homogeneity property of $H$ in (1.3) we have

(2.3) $\langle\nabla_{\xi}H(\xi),\xi\rangle=H(\xi)\,,\quad\text{for any
}\,\xi\in\mathbb{R}^{N},$

where the left-hand side is taken to be $0$ when $\xi=0$.

Given a measurable set $E\subset\mathbb{R}^{N}$, the anisotropic perimeter of
$E$ in $\Sigma$ is defined by

$P_{H}(E;\Sigma):=\sup\left\\{\int_{E}\mathop{\mathrm{div}}\left(\sigma\right)\,dx\,:\,\sigma\in
C_{c}^{1}(\Sigma;\mathbb{R}^{N}),\,H_{0}\left(\sigma\right)\leqslant
1\right\\}.$

When $H$ is the Euclidean norm, we recover the usual Euclidean perimeter of
$E$ in $\Sigma$

$P(E;\Sigma):=\sup\left\\{\int_{E}\mathop{\mathrm{div}}\left(\sigma\right)\,dx\,:\,\sigma\in
C_{c}^{1}(\Sigma;\mathbb{R}^{N}),\,\left|\sigma\right|\leqslant 1\right\\}.$

Notice that

(2.4) $P_{H}(E;\Sigma)$ is finite if and only if $P(E;\Sigma)$ is finite.

Indeed, all norms in $\mathbb{R}^{N}$ are equivalent. Thus, there exist two
positive constants $k_{1}\leqslant k_{2}$ such that $H$ satisfies

$k_{1}|\xi|\leqslant H(\xi)\leqslant k_{2}|\xi|\quad\text{ for any
}\,\xi\in\mathbb{R}^{N}.$

This and the homogeneity of $H$ give that

$\frac{1}{k_{2}}|\xi|\leqslant
H_{0}(\xi)\leqslant\frac{1}{k_{1}}|\xi|\quad\text{ for any
}\,\xi\in\mathbb{R}^{N},$

and hence that

(2.5) $k_{1}P(E;\Sigma)\leqslant P_{H}(E;\Sigma)\leqslant k_{2}P(E;\Sigma),$

which gives (2.4).

If $E$ is a set of finite perimeter, then we can write that

$P(E;\Sigma)={\mathcal{H}}^{N-1}(\Sigma\cap\partial^{*}E)$

and that

(2.6)
$P_{H}(E;\Sigma)=\int_{\Sigma\cap\partial^{*}E}H(\nu)\,d{\mathcal{H}}^{N-1},$

where $\partial^{*}E$ is the reduced (or measure theoretic) boundary of $E$ on
which an outer (measure theoretical) unit normal vector $\nu$ is defined (see
[Gi] and, for the anisotropic setting, [AB]).

Recalling the definition of the weighted anisotropic perimeter in (1.12) and
of the weighted volume of a set in (1.13), we notice that, if $w$ is
positively homogeneous of degree $\lambda\geqslant 0$ and $D$ is the quantity
defined in (1.14), the following444We stress that the right-hand side of (2.7)
means “$D$ multiplied by $w(\Sigma\cap B)$” (recall also the notation in
(1.13)); in particular, no confusion should arise with the derivative of $w$.
relation holds true

(2.7) $P_{w,H}(B;\Sigma)=D\;w(\Sigma\cap B),$

where $B$ is the unitary Wulff ball, see [CRS, Formula (1.14)].

For more details on $P_{w,H}(E;\Sigma)$ and for more general definitions, we
refer the reader to [BBF] and [CRS] (see also [CGPRS]).

##  3\. Additonal ingredients for the proof of the main results

We collect here some other auxiliary results that turn out to be handy for the
proof of the main theorems, such as a maximum principle in cones, a
Stampacchia-type result, some geometric differential inequalities and a
Pohozaev-type identity.

We start by proving the following maximum principle in cones.

######  Lemma 3.1 (Maximum principle in cones).

Let $u$ be a (weak) solution of

$\begin{cases}-\mathop{\mathrm{div}}\left\\{w(x)\,H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=\widetilde{f}(x)\,w(x)\ &\mbox{ in
}\ \Sigma\cap\Omega,\\\ u=0\ &\mbox{ on }\ \Gamma_{0},\\\
\langle\nabla_{\xi}H(\nabla u),\nu\rangle=0\ &\mbox{ on }\
\Gamma_{1}\setminus\left\\{0\right\\},\end{cases}$

where the weight $w$ is continuous in $\overline{\Sigma}$ and positive and
locally Lipschitz in $\Sigma$. If $\widetilde{f}\in
L^{\infty}(\Sigma\cap\Omega)$ is nonnegative, then $u$ is nonnegative. In
particular, if $\widetilde{f}\equiv 0$, then $u\equiv 0$.

###### Proof.

We set $u^{-}:=\max\left\\{0,-u\right\\}$ and we compute

$\begin{split}0\leqslant\int_{\Sigma\cap\Omega}H^{p}(\nabla
u^{-})\,w(x)\,d{\mathcal{H}}^{N}&=\int_{\left\\{u<0\right\\}}H^{p}(\nabla
u)\,w(x)\,d{\mathcal{H}}^{N}\\\ &=\int_{\left\\{u<0\right\\}}H(\nabla
u)^{p-1}\,\langle\nabla_{\xi}H(\nabla u),\nabla
u\rangle\,w(x)\,d{\mathcal{H}}^{N}\\\
&=\int_{\left\\{u<0\right\\}}u\,\widetilde{f}\,w(x)\,d{\mathcal{H}}^{N}\leqslant
0,\end{split}$

where in the second equality we used (2.3) and in the third equality we used
the boundary value problem. Thus, $\nabla u^{-}=0$ a.e. in $\Sigma\cap\Omega$.
Moreover, by Poincaré inequality (which holds true since $u^{-}=0$ on
$\Gamma_{0}$ and ${\mathcal{H}}^{N-1}(\Gamma_{0})>0$) we find that $u^{-}=0$,
and hence $u\geqslant 0$ a.e. in $\Sigma\cap\Omega$. ∎

In what follows, we will exploit the following result regarding the singular
set of solutions to the anisotropic weighted equation considered in the
present paper. In the isotropic unweighted setting, the result is due to Lou
(see [Lo, Theorem 1.1 and Corollary 1.1]). We mention that the results in [Lo]
hold true for more general nonlinearities, namely $f\in L^{q}(\Omega)$ with
$q>p/N$ and $q\geqslant 2$.

######  Lemma 3.2.

Let $E\subset\mathbb{R}^{N}$ be a bounded domain. Let $f\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R})$ and $w\in C(\overline{E})$, with $w$
positive and locally Lipschitz in $E$ . Let $u$ be a bounded weak solution of

(3.1) $-\mathop{\mathrm{div}}\left\\{w(x)\,H(\nabla
u)^{p-1}\,\nabla_{\xi}H(\nabla u)\right\\}=f(u)\,w(x)\ \mbox{ in }\ E,$

that is, $u\in W^{1,p}(E)\cap L^{\infty}(E)$ and

(3.2) $\int_{E}H(\nabla u)^{p-1}\,\langle\nabla_{\xi}H(\nabla
u),\nabla\phi\rangle\,w(x)\,d{\mathcal{H}}^{N}=\int_{E}f\,\phi\,w(x)d{\mathcal{H}}^{N}\quad\text{
for any }\phi\in C_{c}^{\infty}(E).$

Then, $f=0$ a.e. in $\left\\{x\in E\,:\,\nabla u(x)=0\right\\}$.

We remark that we are not making any assumptions on the regularity of $E$ in
Lemma 3.2. As a matter of fact, in what follows we are going to apply this
result with $E:=\Sigma\cap\Omega$.

###### Proof of Lemma 3.2.

We start by proving that

(3.3) $a(\nabla u):=H(\nabla u)^{p-1}\,\nabla_{\xi}H(\nabla u)\in
W^{1,2}_{\mathrm{loc}}(E).$

To prove this, we first observe that

(3.4) if $\widetilde{f}\in L^{\infty}_{\mathrm{loc}}(E)$, then a solution of
$-\mathop{\mathrm{div}}\left(a(\nabla u)\right)=\widetilde{f}$ in $E$
satisfies (3.3).

Indeed, when $H$ is the Euclidean norm, this result can be found in [Lo, Lemma
2.1]. The anisotropic version of this result (for uniformly elliptic norms
$H\in C^{2}(\mathbb{R}^{N}\setminus\left\\{0\right\\})$) will appear in [ACF].

Now, in order to prove (3.3), we notice that (3.1) can be rewritten as

$-\mathop{\mathrm{div}}\left\\{\,H(\nabla u)^{p-1}\,\nabla_{\xi}H(\nabla
u)\right\\}=f(u)+H(\nabla u)^{p-1}\,\left\langle\frac{\nabla
w}{w},\,\nabla_{\xi}H(\nabla u)\right\rangle\ \mbox{ in }\ E.$

Now we set

$\widetilde{f}(x):=f\left(u\left(x\right)\right)+H\left(\nabla
u(x)\right)^{p-1}\,\left\langle\frac{\nabla
w(x)}{w(x)},\,\nabla_{\xi}H\left(\nabla u\left(x\right)\right)\right\rangle$

and we observe that, in light of our assumptions on $f$ and $w$, and the
interior $C^{1,\gamma}$-regularity of $u$ (which holds true by [To]),
$\widetilde{f}\in L^{\infty}_{\mathrm{loc}}(E)$. Here, we used that $u$ is
bounded in order to apply [To]. We also used that $\frac{H(\xi)^{p}}{p}\in
C^{1}(\mathbb{R}^{N})$ (see e.g., [CFV, Lemma 2.2]). Thus, (3.3) follows from
(3.4).

Now, for any $\varphi\in C_{c}^{\infty}(E)$ we set

$\phi(x):=\frac{\left|a(\nabla u(x))\right|}{\varepsilon+\left|a(\nabla
u(x))\right|}\,\varphi(x)$

and we notice that $\phi\in W^{1,2}_{0}(E)$. By using $\phi$ as a test
function in (3.2), we obtain that

(3.5) $\begin{split}\int_{E}\frac{\left|a(\nabla
u)\right|}{\varepsilon+\left|a(\nabla
u)\right|}\,\varphi(x)\,f\,w(x)\,d{\mathcal{H}}^{N}&=\int_{E}\frac{\left|a(\nabla
u)\right|}{\varepsilon+\left|a(\nabla u)\right|}\,\langle a(\nabla
u),\nabla\varphi\rangle\,w(x)\,d{\mathcal{H}}^{N}\\\
&\qquad+\varepsilon\int_{E}\frac{\langle a(\nabla u),\nabla\left(|a(\nabla
u)|\right)\rangle}{\left(\varepsilon+\left|a(\nabla
u)\right|\right)^{2}}\,\varphi(x)\,w(x)\,d{\mathcal{H}}^{N}.\end{split}$

By the Cauchy-Schwartz inequality we have that

$\varepsilon\frac{\langle a(\nabla u),\nabla\left(\left|a(\nabla
u)\right|\right)\rangle}{\left(\varepsilon+\left|a(\nabla
u)\right|\right)^{2}}\leqslant\left|\nabla\left(|a(\nabla u)|\right)\right|,$

and the last term is in $L^{1}(E)$ (actually in $L^{2}(E)$) and does not
depend on $\varepsilon$. Notice also that

(3.6) $\int_{E\setminus\left\\{\nabla u=0\right\\}}\frac{\left|a(\nabla
u)\right|}{\varepsilon+\left|a(\nabla
u)\right|}\,\varphi(x)\,f\,d{\mathcal{H}}^{N}=\int_{E}\frac{\left|a(\nabla
u)\right|}{\varepsilon+\left|a(\nabla
u)\right|}\,\varphi(x)\,f\,d{\mathcal{H}}^{N}.$

Thus, by letting $\varepsilon\to 0^{+}$, (3.5), (3.6) and Lebesgue’s dominated
convergence theorem give that

$\int_{E\setminus\left\\{\nabla
u=0\right\\}}\varphi(x)\,f\,w(x)d{\mathcal{H}}^{N}=\int_{E}\langle a(\nabla
u),\nabla\varphi\rangle\,w(x)\,d{\mathcal{H}}^{N},$

for any $\varphi\in C_{c}^{\infty}(E)$. Hence, this and (3.2) lead to

$\int_{E}\varphi(x)\,f\,w(x)\,d{\mathcal{H}}^{N}=\int_{E\setminus\left\\{\nabla
u=0\right\\}}\varphi(x)\,f\,w(x)\,d{\mathcal{H}}^{N}\quad\text{ for any
}\varphi\in C_{c}^{\infty}(E),$

and the desired result follows. ∎

In what follows, for $t\in\left(-\infty,M\right]$ (where $M$ is that defined
in (1.15)), we consider the sets

$u^{-1}(t):=\left\\{u=t\right\\}:=\left\\{x\in\Sigma\cap\Omega:u(x)=t\right\\}\quad\text{
and }\quad\left\\{u>t\right\\}:=\left\\{x\in\Sigma\cap\Omega:u(x)>t\right\\},$

which are contained in $\Sigma\cap\Omega$ by definition.

Also, we consider the distribution-type functions:

(3.7)
$I(t):=\int_{\left\\{u>t\right\\}}f(u)w(x)\,d{\mathcal{H}}^{N},\qquad\mu(t):=w(\left\\{u>t\right\\}),$

and

(3.8) $K:=I^{\alpha}\mu^{\beta},$

for any $\alpha$, $\beta\in\mathbb{R}$.

In the sequel, we will specify our choice of parameters $\alpha$ and $\beta$
as follows

(3.9) $\alpha:=p^{\prime}=\frac{p}{p-1}\quad{\mbox{ and
}}\quad\beta:=\frac{p-D}{D(p-1)},$

where $D$ is the quantity defined in (1.14).

We notice that, when $w\equiv 1$ (and hence $D=N$), these distribution-type
functions are precisely those used in [Se] in the isotropic unweighted case
(when $\Sigma=\mathbb{R}^{N}$).

######  Lemma 3.3.

Let $\alpha$, $\beta$ be arbitrary real numbers and
$\Omega\subset\mathbb{R}^{N}$ be a bounded smooth domain. Assume that $u\in
C^{0,1}(\overline{\Sigma\cap\Omega})\cap C^{1}(\Sigma\cap\Omega)$ is
nonnegative, $u=0$ on $\Gamma_{0}$, and let $w$ be as in Lemma 3.1. Let $f\in
L^{\infty}_{\mathrm{loc}}(\left[0,\infty\right))$ be nonnegative and let $I$,
$\mu$, $K$ be those defined in (3.7) and (3.8). Then:

  1. (i)

The functions $I$, $\mu$ and $K$ are a.e. differentiable and

(3.10) $-K^{\prime}(t)=\left[\alpha I(t)^{\alpha-1}\mu(t)^{\beta}f(t)+\beta
I(t)^{\alpha}\mu(t)^{\beta-1}\right]\left(-\mu^{\prime}(t)\right)\quad\text{for
a.e. }t.$

  2. (ii)

For a.e. $t\in(0,M)$, it holds that

(3.11) $-\mu^{\prime}(t)\geqslant\int_{\left\\{u=t\right\\}}\frac{w}{|\nabla
u|}\,d{\mathcal{H}}^{N-1}.$

Assume furthermore that $u$ is a weak solution of (1.5). Then:

  1. (iii)

If hypothesis (b) in Theorem 1.1 holds, then $I$, $\mu$ and $K$ are absolutely
continuous functions for $t<M$.

  2. (iv)

Suppose in addition that $\alpha$ and $\beta$ are as in (3.9). Under the
assumptions (a) and (b) in Theorem 1.1, $K(t)$ defined in (3.8) is
nonincreasing for $t\in(0,M)$.

###### Proof.

We take inspiration from the arguments used in [Se, Lemma 4] in the isotropic
unweighted case $\Sigma=\mathbb{R}^{N}$. More details and related questions
can be found, e.g., in [BZ, Tal, Fe].

_(i)_ Since $I$ and $\mu$ are nonincreasing (by definition), they are a.e.
differentiable. Also, they define nonpositive Lebesgue-Stieltjes measures $dI$
and $d\mu$ on $(0,M)$. By definition of Lebesgue integral, we find that

$I(t)=\int_{\left\\{u>t\right\\}}f(u)w(x)\,d{\mathcal{H}}^{N}=-\int_{t}^{M^{+}}f(\tau)\,d\mu(\tau),$

and hence $dI=f\,d\mu$. Thus, $I^{\prime}(t)=f(t)\mu^{\prime}(t)$ for
$d\mu$-a.e. $t$. Now, since $w>0$ in $\Sigma$ and $|\nabla u|$ is bounded in
$\left\\{u\geqslant t\right\\}$, $\mu$ is strictly decreasing. As a
consequence, we have that the Lebesgue measure on $(0,M)$ is absolutely
continuous with respect to $d\mu$. Thus, we get that

$I^{\prime}(t)=f(t)\mu^{\prime}(t)\quad\text{ for a.e. }t,$

and (3.10) easily follows. In all the paper, unless otherwise indicated, a.e.
is always with respect to the Lebesgue measure.

_(ii)_ We define

$\mu_{0}(t):=w\big{(}\left\\{u>t,\,|\nabla u|>0\right\\}\big{)}.$

Let $\varepsilon>0$ and we use coarea formula to compute

(3.12) $\begin{split}w\big{(}\left\\{u>t,\,|\nabla
u|>\varepsilon\right\\}\big{)}&=\int_{\Sigma\cap\Omega}|\nabla
u|\frac{\chi_{\left\\{u>t,\,|\nabla u|>\varepsilon\right\\}}\,w(x)}{|\nabla
u|}\,d{\mathcal{H}}^{N}\\\
&=\int_{t}^{M}\int_{u^{-1}(\tau)}\frac{\chi_{\left\\{u>t,\,|\nabla
u|>\varepsilon\right\\}}\,w(x)}{|\nabla u|}\,d{\mathcal{H}}^{N-1}\,d\tau\\\
&=\int_{t}^{M}\int_{u^{-1}(\tau)\cap\left\\{|\nabla
u|>\varepsilon\right\\}}\frac{w(x)}{|\nabla
u|}\,d{\mathcal{H}}^{N-1}\,d\tau.\end{split}$

By monotone convergence, letting $\varepsilon\to 0$, we find that (3.12) still
holds true for $\varepsilon=0$ (and arbitrary $t$). Thus, $\mu_{0}(t)$ is
absolutely continuous and

(3.13) $-\mu_{0}^{\prime}(t)=\int_{u^{-1}(t)\cap\left\\{|\nabla
u|>0\right\\}}\frac{w(x)}{|\nabla u|}\,d{\mathcal{H}}^{N-1}\,d\tau,\quad\text{
for a.e. }t\in(0,M).$

Again by using coarea formula we also find

$0=\int_{\Sigma\cap\Omega}|\nabla u|\,\chi_{\left\\{u>t,\,|\nabla
u|=0\right\\}}\,d{\mathcal{H}}^{N}=\int_{t}^{M}{\mathcal{H}}^{N-1}\left(u^{-1}(\tau)\cap\left\\{|\nabla
u|=0\right\\}\right)\,d\tau,$

and hence

(3.14) ${\mathcal{H}}^{N-1}\left(u^{-1}(t)\cap\left\\{|\nabla
u|=0\right\\}\right)=0\quad\text{for a.e. }\,t.$

Thus, we can replace (3.13) with

(3.15) $-\mu_{0}^{\prime}(t)=\int_{u^{-1}(t)}\frac{w(x)}{|\nabla
u|}\,d{\mathcal{H}}^{N-1}\,d\tau,\quad\text{ for a.e. }t\in(0,M).$

Also, for a.e. $t\in(0,M)$ (where both $\mu^{\prime}(t)$ and
$\mu_{0}^{\prime}(t)$ exist) we have that

$-\mu^{\prime}(t)=\lim_{s\to t^{+}}\frac{w\left(s\geqslant
u>t\right)}{s-t}\geqslant\lim_{s\to t^{+}}\frac{w\left(s\geqslant
u>t,\,|\nabla u|>0\right)}{s-t}=-\mu_{0}^{\prime}(t).$

Inequality (3.11) now follows from the latter equation and (3.15). We notice
additionally that the equality sign holds for a.e. $t$ if $\left\\{|\nabla
u|=0\right\\}$ has zero ${\mathcal{H}}^{N}$-measure.

_(iii)_ If $p<D$ and $\phi\leqslant f\leqslant\frac{Dp}{D-p}\phi$ for some
nonincreasing function $\phi\geqslant 0$, then a solution of (1.5) satisfies

$-\mathop{\mathrm{div}}\left\\{H(\nabla u)^{p-1}\,\nabla_{\xi}H(\nabla
u)\right\\}=0\quad\text{in }\,\left\\{u\geqslant t_{0}\right\\},$

where $t_{0}\in\left[0,\infty\right]$ satisfies that $\phi(t)>0$ for $t<0$ and
$\phi(t)\equiv 0$ for $t>t_{0}$. Hence, by Lemma 3.1, if $t_{0}<+\infty$, we
will have that $u\equiv t_{0}=M$ in $\left\\{u\geqslant t_{0}\right\\}$.
Therefore, for every $t<M$ it holds that

$-\mathop{\mathrm{div}}\left\\{H(\nabla u)^{p-1}\,\nabla_{\xi}H(\nabla
u)\right\\}=f(u)\geqslant\phi(u)\geqslant\phi(t)>0\quad\text{in
}\,\left\\{0\leqslant u<t\right\\}.$

By using Lemma 3.2 (with $E:=\Sigma\cap\Omega$), we find that $f(u)$ vanishes
a.e. in the set $\left\\{|\nabla u|=0\right\\}\cap\left\\{0\leqslant
u<t\right\\}$. Since $f(u)\geqslant\phi(u)\geqslant\phi(t)>0$ in
$\left\\{0\leqslant u<t\right\\}$, this is only possible if the singular set
$\left\\{|\nabla u|=0\right\\}\cap\left\\{0\leqslant u<t\right\\}$ has zero
measure. Therefore, we have

$\mu(t)=\mu_{0}(t)+w\left(\left\\{u=M\right\\}\right)\quad\text{for every
}\,t<M.$

Since $\mu_{0}(t)$ is absolutely continuous (as noticed in item (ii)), $\mu$
is an absolutely continuous function for $t<M$. Thus, also $I$ and $K$ are
absolutely continuous for $t<M$.

_(iv)_ Under hypothesis (a) of Theorem 1.1, the claim is obvious because
$\alpha$ and $\beta$ in (3.9) are $\geqslant 0$. On the other hand, under
hypothesis (b) we have that item (iii) in the statement of Lemma 3.3 applies
and hence $K$ is absolutely continuous. Moreover, by item (i) we deduce that

(3.16) $-K^{\prime}\geqslant 0\quad\text{a.e.}$

and hence $K$ is nonincreasing also in this case. To prove (3.16), we notice
that in light of item (i), $-K^{\prime}(t)$ has the same sign as $\alpha
f(t)+\beta I(t)/\mu(t)$ , since $I$, $\mu$, $-\mu^{\prime}$ are nonnegative by
definition. Thus, since $\beta<0$, we need
$I(t)/\mu(t)+(\alpha/\beta)f(t)\leqslant 0$ a.e. Observing that $I(t)/\mu(t)$
is the mean of $f(u)$ over the superlevel set ${u>t}$, we easily conclude that
a sufficient condition for $I/\mu+(\alpha/\beta)f\leqslant 0$ is that
$f(s)\leqslant-(\alpha/\beta)f(t)$, whenever $s>t$. And this is satisfied if
$\phi\leqslant f\leqslant-(\alpha/\beta)\phi$ for some nonincreasing
$\phi\geqslant 0$. Replacing $\alpha$, $\beta$ by their values in (3.9) we
obtain the condition (b) in Theorem 1.1 since $-\alpha/\beta=pD/(D-p)$. ∎

######  Lemma 3.4 (Pohozaev-type identity).

Let $\Sigma$ be an open convex cone in $\mathbb{R}^{N}$ and let
$\Omega\subset\mathbb{R}^{N}$ be a bounded smooth domain such that
$\partial\Gamma_{0}=\partial\Gamma_{1}$. Let $u\in
C^{1}\left((\Sigma\cap\Omega)\cup\Gamma_{0}\cup(\Gamma_{1}\setminus\left\\{0\right\\})\right)\cap
W^{1,\infty}(\Sigma\cap\Omega)$ be a solution of (1.5), with $f\in
L^{\infty}_{\mathrm{loc}}([0,\infty))$ nonnegative and $w$ satisfying (1.7)
and (1.9). Let $D$ be as in (1.14).

Then, the following identity holds:

(3.17)
$D\int_{\Sigma\cap\Omega}F(u)\,w(x)\,d{\mathcal{H}}^{N}+\frac{p-D}{p}\int_{\Sigma\cap\Omega}u\,f(u)\,w(x)\,d{\mathcal{H}}^{N}=\frac{1}{p^{\prime}}\int_{\Gamma_{0}}H(\nabla
u)^{p}\,\langle x,\nu\rangle\,w(x)\,d{\mathcal{H}}^{N-1},$

where

$F(s):=\int_{0}^{s}f(\tau)\,d\tau.$

###### Proof.

Due to lack of regularity in our setting, in order to prove (3.17) we argue by
approximation. We approximate $\Sigma\cap\Omega$ by Lipschitz domains
$E_{\delta}$ obtained by chopping off a $\delta$-neighborhood of $0$ (in the
case $0\in\Gamma_{1}$) and a $\delta$-tubular neighborhood of
$\partial\Gamma_{0}$.

_Step 1._ We prove that for any $\eta\in C^{0,1}_{c}(E_{\delta})$ we have that

(3.18)
$-\int_{E_{\delta}}{\mathcal{R}}\,\nabla\eta\,d{\mathcal{H}}^{N}=\int_{E_{\delta}}\eta(x)\,w(x)\left\\{(N+\lambda)\,F(u)+\frac{p-(N+\lambda)}{p}u\,f(u)\right\\}d{\mathcal{H}}^{N},$

where

$\begin{split}{\mathcal{R}}:=w(x)\bigg{\\{}x\Big{(}F(u)-\frac{H\left(\nabla
u\right)^{p}}{p}\Big{)}\bigg{.}&+\langle x,\nabla u\rangle\,H\left(\nabla
u\right)^{p-1}\nabla_{\xi}H(\nabla u)\\\
&\bigg{.}+\left(\frac{(N+\lambda)-p}{p}\right)\,u\,H\left(\nabla
u\right)^{p-1}\nabla_{\xi}H\left(\nabla u\right)\bigg{\\}}.\end{split}$

For any given $\eta\in C^{0,1}_{c}(E_{\delta})$, we consider a smooth (say
$C^{1,\gamma}$) open set $A\subset\subset E_{\delta}$ such that
$\mathrm{supp}(\eta)\subset A$.

Furthermore, by setting

$\psi(t):=\frac{t^{p}}{p},\quad\text{ for }t\geqslant 0,$

we see that $u$ satisfies the equation

$-\mathop{\mathrm{div}}\left(w(x)\,\psi^{\prime}\left(H(\nabla
u)\right)\nabla_{\xi}H(\nabla u)\right)=f(u)\,w(x)\quad\text{ in
}\Sigma\cap\Omega.$

For $t\geqslant 0$ and $\varepsilon\in(0,1)$, we let

$\psi_{\varepsilon}(t):=\psi\left(\sqrt{\varepsilon^{2}+t^{2}}\right)-\psi(\varepsilon).$

We define $\Psi(t):=\psi^{\prime}(t)t$ and
$\Psi_{\varepsilon}(t):=\psi^{\prime}_{\varepsilon}(t)t$. From a standard
argument (see for instance [CFV, Lemma 4.2] or [BC, Lemma 3.2]) we have that

$\psi_{\varepsilon}\to\psi\quad\text{ and
}\quad\Psi_{\varepsilon}\to\Psi\,\text{ uniformly on compact sets of
}[0,+\infty).$

Let $u_{\varepsilon}$ be solution of

$\begin{cases}\mathop{\mathrm{div}}\left(w(x)\,\psi_{\varepsilon}^{\prime}\left(H(\nabla
u_{\varepsilon})\right)\nabla_{\xi}H(\nabla
u_{\varepsilon})\right)=f(u)\,w(x)\quad\text{ in }A\\\ u_{\varepsilon}=u\text{
on }\partial A.\end{cases}$

Standard regularity results give that $u_{\varepsilon}\in
C^{1,\gamma}(\overline{A})\cap W^{2,2}_{\mathrm{loc}}(A)$ and
$u_{\varepsilon}\to u$ in $C^{1}(\overline{A})$, as $\varepsilon\to 0$. For
example, one can obtain $u_{\varepsilon}$ by minimizing the functional

$\int_{A}\left\\{\psi_{\varepsilon}\left(H(\nabla
v)\right)-f(u(x))\,v\right\\}w(x)\,d{\mathcal{H}}^{N}$

among functions $v\in W^{1,p}(A)$ with $v-u\in W^{1,p}_{0}(A)$, and then
exploit arguments similar to those used in [CFV, Proposition 4.3].

Notice that, since $A\subset\subset\Sigma$, we have that $w\in
C^{0,1}(\overline{A})$ and $w>0$ on $\overline{A}$ and hence the results of
regularity (up to the boundary) contained in [Li] can be used on $A$.

Now we set

$\begin{split}{\mathcal{R}}_{\varepsilon}:=w(x)\bigg{\\{}x\Big{(}F(u)-\psi_{\varepsilon}\left(H\left(\nabla
u_{\varepsilon}\right)\right)\Big{)}\bigg{.}&+\langle x,\nabla
u_{\varepsilon}\rangle\,\psi_{\varepsilon}^{\prime}\left(H\left(\nabla
u_{\varepsilon}\right)\right)\nabla_{\xi}H(\nabla u_{\varepsilon})\\\
&\bigg{.}+\left(\frac{(N+\lambda)-p}{p}\right)\,u_{\varepsilon}\,\psi_{\varepsilon}^{\prime}\left(H\left(\nabla
u_{\varepsilon}\right)\right)\nabla_{\xi}H\left(\nabla
u_{\varepsilon}\right)\bigg{\\}}\end{split}$

and we exploit [PS, Proposition 1] and [PS, Remark at page 685] with

$\displaystyle\mathcal{F}:=\mathcal{F}_{\varepsilon}(x,\nabla
u_{\varepsilon}):=\left\\{\psi_{\varepsilon}\left(H(\nabla
u_{\varepsilon})\right)-F(u(x))\right\\}w(x),\qquad h:=x,\qquad
a:=\frac{(N+\lambda)-p}{p}$
$\displaystyle{\mbox{and}}\;\qquad{\mathcal{G}}(x):=-f(u(x))w(x),$

equation (2.3), and the fact that $\langle x,\nabla w\rangle=\lambda\,w(x)$ in
$\Sigma$ (which holds true in light of the $\lambda$-homogeneity of $w$), to
see that

(3.19)
$\begin{split}\mathop{\mathrm{div}}({\mathcal{R}}_{\varepsilon})=&(N+\lambda)\,F(u)\,w(x)+\frac{p-(N+\lambda)}{p}\,u_{\varepsilon}\,f(u)\,w(x)\\\
&\quad+(N+\lambda)\,w(x)\left\\{\frac{\Psi_{\varepsilon}(H(\nabla
u_{\varepsilon}))}{p}-\psi_{\varepsilon}(H(\nabla
u_{\varepsilon}))\right\\}\\\ &\quad+w(x)\,\left\\{F^{\prime}(u)\langle
x,\nabla u\rangle-f(u)\langle x,\nabla
u_{\varepsilon}\rangle\right\\}\end{split}$

in the weak sense in $A$. By recalling that $\eta\in C_{c}^{0,1}(E_{\delta})$,
and that $\mathrm{supp}(\eta)\subset A$, we thus can write that

(3.20)
$\begin{split}-\int_{E_{\delta}}{\mathcal{R}}_{\varepsilon}\,\nabla\eta\,d{\mathcal{H}}^{N}=&\int_{E_{\delta}}\eta(x)\,w(x)\left\\{(N+\lambda)\,F(u)+\frac{p-(N+\lambda)}{p}u_{\delta}^{\varepsilon}\,f(u)\right\\}d{\mathcal{H}}^{N}\\\
&\quad+(N+\lambda)\int_{E_{\delta}}\eta(x)\,w(x)\left\\{\frac{\Psi_{\varepsilon}(H(\nabla
u_{\varepsilon}))}{p}-\psi_{\varepsilon}(H(\nabla
u_{\varepsilon}))\right\\}d{\mathcal{H}}^{N}\\\
&\quad+\int_{E_{\delta}}\eta(x)\,w(x)\,\left\\{F^{\prime}(u)\langle x,\nabla
u\rangle-f(u)\langle x,\nabla
u_{\varepsilon}\rangle\right\\}d{\mathcal{H}}^{N}.\end{split}$

We also note that

$\frac{\Psi(H(\nabla u))}{p}-\psi(H(\nabla u))=0.$

Moreover, since $F^{\prime}(t)=f(t)$ for a.e. $t$, and hence
$F^{\prime}(u(x))=f(u(x))$ for a.e. $x\in\left\\{\nabla u\neq 0\right\\}$, we
thus have that

$\displaystyle\int_{E_{\delta}}\eta(x)\,w(x)\,F^{\prime}(u)\langle x,\nabla
u\rangle\,d{\mathcal{H}}^{N}=\int_{E_{\delta}\cap\left\\{\nabla u\neq
0\right\\}}\eta(x)\,w(x)\,f(u)\langle x,\nabla u\rangle\,d{\mathcal{H}}^{N}$
$\displaystyle\qquad\qquad=\int_{E_{\delta}}\eta(x)\,w(x)\,f(u)\langle
x,\nabla u\rangle\,d{\mathcal{H}}^{N}.$

This implies that

$\int_{E_{\delta}}\eta(x)\,w(x)\,\langle x,\nabla
u\rangle\left\\{F^{\prime}(u)-f(u)\right\\}d{\mathcal{H}}^{N}=0.$

Hence, by taking the limit for $\varepsilon\to 0$ in (3.20), we obtain (3.18).

_Step 2._ For $k\geqslant 1$ we define
$\phi_{k}:\mathbb{R}\to\left[0,1\right]$ as follows

$\phi_{k}(s):=\begin{cases}0\quad&\text{ if }\,s\leqslant\frac{1}{k},\\\
ks-1\quad&\text{ if }\,\frac{1}{k}<s<\frac{2}{k},\\\ 1\quad&\text{ if
}\,s\geqslant\frac{2}{k}.\end{cases}$

Then we define $\eta_{k}\in C_{c}^{0,1}(E_{\delta})$ as

$\eta_{k}(x):=\phi_{k}(\mathrm{dist}(x,\mathbb{R}^{N}\setminus E_{\delta})).$

In this way we have that

$\eta_{k}(x)\to 1\quad\text{ for every }\,x\in E_{\delta}$

and that (see, e.g., [DMS, Formula (21)], [CDLP, Section 7] and [AFP, Chapter
3])

(3.21) $\lim_{k}\int_{E_{\delta}}\langle v,\nabla\eta_{k}\rangle
d{\mathcal{H}}^{N}=-\int_{\partial E_{\delta}}\langle v,\nu\rangle
d{\mathcal{H}}^{N-1},\quad\text{ for any }\,v\in
C(\overline{E}_{\delta};\mathbb{R}^{N}).$

Now we choose $\eta:=\eta_{k}$ in (3.18) and we take the limit as
$k\to\infty$, using (3.21) (with $v={\mathcal{R}}$) and taking into account
that the boundary conditions in (1.5) and the geometry of $\Sigma$ give that

$u=0\quad\text{ and }\quad F(u)=0\quad\text{ on }\Gamma_{0,\delta},$
$\langle\nabla_{\xi}H(\nabla u),\nu\rangle=0\quad\text{ and }\quad\langle
x,\nu\rangle=0\quad\text{ on }\Gamma_{1,\delta},$

where we have set

$\Gamma_{0,\delta}:=\Gamma_{0}\cap\partial
E_{\delta},\quad\Gamma_{1,\delta}:=\Gamma_{1}\cap\partial
E_{\delta}\quad\text{ and }\quad\Gamma_{\delta}:=\partial
E_{\delta}\setminus\left(\Gamma_{0,\delta}\cup\Gamma_{1,\delta}\right).$

Thus, we obtain that

$\begin{split}\int_{\Gamma_{0,\delta}}w(x)&\left\\{\langle x,\nabla
u\rangle\,H^{p-1}\left(\nabla u\right)\langle\nabla_{\xi}H(\nabla
u),\nu\rangle-\langle x,\nu\rangle\frac{H\left(\nabla
u\right)^{p}}{p}\right\\}d{\mathcal{H}}^{N-1}+\int_{\Gamma_{\delta}}\langle{\mathcal{R}},\nu\rangle\,d{\mathcal{H}}^{N-1}\\\
&=\int_{E_{\delta}}w(x)\left\\{(N+\lambda)\,F(u)-\frac{(N+\lambda)-p}{p}u\,f(u)\right\\}d{\mathcal{H}}^{N}.\end{split}$

By using that

$\nabla u=-|\nabla u|\,\nu\quad\text{ on }\,\Gamma_{0,\delta},$

(which holds true in light of the boundary condition on $\Gamma_{0}$ in (1.5))
and (2.3), the previous identity becomes

$\displaystyle\frac{1}{p^{\prime}}\int_{\Gamma_{0,\delta}}w(x)\,H\left(\nabla
u\right)^{p}\,\langle
x,\nu\rangle\,d{\mathcal{H}}^{N-1}+\int_{\Gamma_{\delta}}\langle{\mathcal{R}},\nu\rangle\,d{\mathcal{H}}^{N-1}$
$\displaystyle=$
$\displaystyle\int_{E_{\delta}}w(x)\left\\{(N+\lambda)F(u)-\frac{(N+\lambda)-p}{p}u\,f(u)\right\\}d{\mathcal{H}}^{N},$

where $p^{\prime}=p/(p-1)$. Then we take the limit as $\delta\to 0$. Since
$u\in W^{1,\infty}(\Sigma\cap\Omega)$ and
${\mathcal{H}}^{N-1}(\Gamma_{\delta})$ tends to $0$ as $\delta\to 0$, we have
that the integral over $\Gamma_{\delta}$ tends to $0$. Recalling that
$D=N+\lambda$, (3.17) follows. ∎

##  4\. Isoperimetric inequalities in convex cones

In this section, we revisit the existing literature on isoperimetric
inequalities in convex cones and, providing several technical improvements to
the previous results, we frame the isoperimetric theory into a convenient
setting for our applications.

######  Theorem 4.1 (Weighted anisotropic isoperimetric inequality in cones,
[CRS]).

Let $\Sigma$ be an open convex cone in $\mathbb{R}^{N}$ with vertex at the
origin and let $B$ be the unitary Wulff ball centered at the origin. Let $w$
satisfy (1.7), (1.8), and (1.9). Then, for each measurable set
$E\subset\mathbb{R}^{N}$ with $w(\Sigma\cap E)<\infty$,

(4.1) $\frac{P_{w,H}(E;\Sigma)}{w(\Sigma\cap
E)^{\frac{D-1}{D}}}\geqslant\frac{P_{w,H}(B;\Sigma)}{w(\Sigma\cap
B)^{\frac{D-1}{D}}},$

where $D=N+\lambda$, and $\lambda$ is that in (1.7).

Theorem 4.1 has been obtained in by Cabré–Ros-Oton–Serra in [CRS] in the more
general setting in which $H$ is a gauge. Since we do not need such a
generality here, we have stated it assuming that $H$ is a norm.

As mentioned in the introduction, Wulff balls centered at the origin
intersected with $\Sigma$ are always minimizers of (4.1). Let us discuss about
the uniqueness of those minimizers, or, in other words, about the
characterization of the equality sign in (4.1).

In [CRS] the authors do not provide the characterization of the equality case,
stating that such a characterization will be postponed in a future work. To
the authors’ knowledge such a characterization among general sets in the
unweighted anisotropic setting of Theorem 4.1 is still not available in the
literature.

We recall that the isoperimetric inequality (4.1) in the unweighted isotropic
setting (i.e., when $w\equiv 1$ and $H$ is the Euclidean norm) is due to
Lions–Pacella [LP]. In this setting, they also obtained the characterization
of the equality case when $\Sigma\setminus\left\\{0\right\\}$ is assumed to be
smooth.

Quantitative versions of inequality (4.1) in the unweighted isotropic setting
(i.e., when $w\equiv 1$ and $H$ is the Euclidean norm) including the
characterization of the equality case (in cones that are not necessarily
smooth outside the origin) have been obtained by Figalli-Indrei in [FI]. In
Theorem 4.2 below we show that their proof can be extended to the case of a
general norm $H$.

A quantitative version of inequality (4.1) in the weighted isotropic setting
(i.e., with $w$ as in Theorem 4.1 and $H$ the Euclidean norm) including the
characterization of the equality case (in cones that are not necessarily
smooth outside the origin) has been recently obtained by
Cinti–Glaudo–Pratelli–Ros-Oton–Serra in [CGPRS].

A different proof of Theorem 4.1 has been obtained by Milman-Rotem in [MR].
There, the authors also provide a characterization under the additional
assumption that $\Sigma\cap E$ is convex555[MR] has been published in [Adv.
Math.262, 2014, 867–908]. In the published version, the convexity assumption
in the characterization of minimizers was missing. This has been later fixed
in [MR] (see footnote 1 in [MR]).. Due to the convexity restriction, we cannot
apply their characterization in the present paper. In fact, in order to obtain
Theorms 1.4, 1.5, 1.6 from Theorem 1.1, we need to use such a characterization
on (almost all) the superlevel sets $\left\\{u>t\right\\}$, and we do not know
a priori whether those sets are convex or not.

######  Theorem 4.2 (Characterization of minimizers in the unweighted
anisotropic isoperimetric inequality in convex cones).

Let $\Sigma$ be an open convex cone in $\mathbb{R}^{N}$ with vertex at $0$ and
let $B$ be the unitary Wulff ball centered at $0$. Then,

(4.2) $\frac{P_{H}(E;\Sigma)}{{\mathcal{H}}^{N}(\Sigma\cap
E)^{\frac{N-1}{N}}}\geqslant\frac{P_{H}(B;\Sigma)}{{\mathcal{H}}^{N}(\Sigma\cap
B)^{\frac{N-1}{N}}},$

for every measurable set $E\subset\mathbb{R}^{N}$ with
${\mathcal{H}}^{N}(\Sigma\cap E)<\infty$.

Moreover, up to rotations, we write
$\Sigma=\mathbb{R}^{k}\times\tilde{\Sigma}$, where $0\leqslant k\leqslant N$
and $\tilde{\Sigma}\subset\mathbb{R}^{N-k}$ is an open convex cone containing
no lines, and the equality sign holds in (4.2) if and only if $E$ is a Wulff
ball of some radius $r>0$ centered at
$x_{0}\in\mathbb{R}^{k}\times\left\\{0_{\mathbb{R}^{N-k}}\right\\}$.

As already mentioned, when $H$ is the Euclidean norm, the proof of Theorem 4.2
can be found in [FI]. There, the authors observe that the (isotropic)
isoperimetric inequality in cones (namely formula (5.7) with $H$ being the
Euclidean norm) can be obtained as an immediate corollary of the anisotropic
isoperimetric inequality

(4.3) $\frac{P_{K}(E)}{{\mathcal{H}}^{N}(E)^{\frac{N-1}{N}}}\geqslant
N{\mathcal{H}}^{N}(K)^{\frac{1}{N}}.$

Here, $K$ is an open bounded convex set and, for a set $E$ of finite perimeter
(i.e., $P(E;\mathbb{R}^{N})<\infty$),

$P_{K}(E):=\int_{\partial^{*}E}\|\nu_{E}(x)\|_{K_{0}}\,d{\mathcal{H}}^{N-1},$

where $\partial^{*}E$ is the reduced boundary of $E$, $\nu_{E}$ is the measure
theoretic outer unit normal, and

$\|\nu\|_{K_{0}}:=\sup\left\\{\langle\nu,z\rangle\,:\,z\in K\right\\}.$

Inequality (4.3) and its characterization of the equality case are well known
in the literature. We refer to [Di, Scha, DP, Ta, FM, BM, MS, FI] and
references therein.

As noticed in [FI], the choice $K:=\Sigma\cap B$, being $B$ the unitary
Euclidean ball centered at the origin allows to recover (4.2) when $H$ is the
Euclidean norm. This also allows to obtain the desired characterization. We
will show that their argument still applies when $H$ is any norm in
$\mathbb{R}^{N}$.

###### Proof of Theorem 4.2.

The strategy is to extend to a more general $H$ the arguments used in [FI,
Theorem 2.2]. To this aim, we set

$K:=\Sigma\cap B,\quad\text{ where }B\text{ is the unitary Wulff ball.}$

Let now $E$ be a set contained in $\Sigma$ and with finite perimeter. We
notice that, if $z\in K$, then $H_{0}(z)\leqslant 1$, and hence by (2.2),

$\|\nu_{E}\|_{K_{0}}:=\sup\left\\{\langle\nu_{E},z\rangle\,:\,z\in
K\right\\}\leqslant H(\nu_{E}).$

By definition of $\|\cdot\|_{K_{0}}$ it easily follows that
$\|\nu_{\Sigma}\|_{K_{0}}=0$ for ${\mathcal{H}}^{N-1}$-a.e.
$x\in\partial\Sigma\setminus\left\\{0\right\\}$, and hence also for
${\mathcal{H}}^{N-1}$-a.e. $x\in\partial\Sigma\cap\partial^{*}E$. Thus,

$\begin{split}P_{K}(E)&=\int_{\partial^{*}E}\|\nu_{E}(x)\|_{K_{0}}\,d{\mathcal{H}}^{N-1}\\\
&=\int_{\Sigma\cap\partial^{*}E}\|\nu_{E}(x)\|_{K_{0}}\,d{\mathcal{H}}^{N-1}\\\
&\leqslant\int_{\Sigma\cap\partial^{*}E}H(\nu_{E}(x))\,d{\mathcal{H}}^{N-1}\\\
&=P_{H}(E;\Sigma),\end{split}$

where in the last identity we used (2.6). In light of the inequality

(4.4) $P_{K}(E)\leqslant P_{H}(E;\Sigma)$

and recalling (2.7) (with $w\equiv 1$), it is clear that (4.2) follows from
(4.3).

Now assume that $E$ satisfies the equality in (4.2). In light of (4.4), $E$
will also satisfy the equality in (4.3). By rescaling, if necessary, we may
assume ${\mathcal{H}}^{N}(E)={\mathcal{H}}^{N}(K)$. Thus, we have that

(4.5) $P(K;\Sigma)=N{\mathcal{H}}^{N}(K)=P_{K}(E)=P_{H}(E;\Sigma),$

where the first equality follows from (2.7) (with $w\equiv 1$).

By [FM] (see also [FMP, Theorem A.1]), we obtain that

(4.6) $E=K+a\quad\text{ with }\,a\in\overline{\Sigma}.$

Notice that, for any $v\in\overline{\Sigma}$ it holds that

(4.7) $P_{H}(v+K\,;\Sigma)=P_{H}(K;\Sigma)+P_{H}(S_{v}),$

where

$S_{v}:=\left\\{x\in\partial^{*}\Sigma\cap
B\,:\,\langle\nu_{\Sigma}(x),v\rangle\neq
0\right\\}=\left\\{x\in\partial^{*}\Sigma\cap
B\,:\,\langle\nu_{\Sigma}(x),v\rangle<0\right\\}.$

By putting together (4.5), (4.6) and (4.7), we get that

$P(K;\Sigma)=P_{H}(E;\Sigma)=P_{H}(K;\Sigma)+P_{H}(S_{a}),$

and hence $P_{H}(S_{a})=0$. By recalling (2.5) we find that
${\mathcal{H}}^{N-1}(S_{a})=0$, and hence that
$\langle\nu_{\Sigma}(x),a\rangle=0$ for ${\mathcal{H}}^{N-1}$-a.e.
$x\in\partial^{*}\Sigma$. As in [FI], this gives that the distributional
derivative of $\chi_{\Sigma}$ is zero in the direction defined by $a$ and
hence that

$\chi_{\Sigma}(x)=\chi_{\Sigma}(x+ta)\quad\text{ for all }t\in\mathbb{R}.$

Since $\Sigma=\mathbb{R}^{k}\times\tilde{\Sigma}$, where $0\leqslant
k\leqslant n$ and $\tilde{\Sigma}\subset\mathbb{R}^{N-k}$ contains no lines,
we find that $a\in\mathbb{R}^{k}\times\left\\{0_{\mathbb{R}^{N-k}}\right\\}$.
This concludes the proof. ∎

In the weighted isotropic setting, [CGPRS, Proposition 1.2] provides the
characterization of the equality case.

######  Theorem 4.3 (Characterization of minimizers in the weighted isotropic
isoperimetric inequality in convex cones, [CGPRS]).

Up to rotations, we can write $\Sigma=\mathbb{R}^{k}\times\tilde{\Sigma}$,
where $0\leqslant k\leqslant N$ and $\tilde{\Sigma}\subset\mathbb{R}^{N-k}$ is
an open convex cone containing no lines. Assume that $H$ is the Euclidean
norm. Then, the equality sign holds in (4.1) (with $H$ the Euclidean norm) if
and only if $E$ is a (Euclidean) ball of some radius $r>0$ centered at
$x_{0}\in\mathbb{R}^{k}\times\left\\{0_{\mathbb{R}^{N-k}}\right\\}$.

We point out that [CGPRS, Proposition 1.2] is stated for $0\leqslant k<N$.
However, notice that in the case $k=N$ (i.e., $\Sigma=\mathbb{R}^{N}$), (4.1)
reduces to the unweighted anisotropic isoperimetric inequality in
$\mathbb{R}^{N}$, whose characterization is well known in the literature (and
of course is contained in Theorem 4.2). In fact, the assumptions on the weight
$w$ in Theorem 4.1 force $w$ to be (a positive) constant whenever
$\Sigma=\mathbb{R}^{N}$ (see, e.g., [CR, Remark 3.8 and Lemma 3.9] and [CRS]).

For other results and more information on weighted isoperimetric inequalities,
we refer to Frank Morgan’s blog [Mo], [Ch, CRS, CGPRS], and references
therein.

######  Remark 4.4.

In order to describe the results concerning the characterization of minimizers
we used that, for any open convex cone $\Sigma\subseteq\mathbb{R}^{N}$,
without loss of generality, up to rotations, we can write
$\Sigma=\mathbb{R}^{k}\times\tilde{\Sigma}$, where $0\leqslant k\leqslant N$
and $\tilde{\Sigma}\subset\mathbb{R}^{N-k}$ is an open convex cone containing
no lines.

##  5\. Proof of Theorems 1.1, 1.4, 1.5 and 1.6

In this section, we exploit the results of the previous sections to prove
Theorems 1.1, 1.4, 1.5 and 1.6. To this end, the following Gauss-Green-type
identity and Hölder-isopertimetric-type inequality will be useful.

######  Lemma 5.1.

Suppose that the assumptions of Theorem 1.1 are satisfied. Let $I$ be the
function defined in (3.7). Then, we have the following:

  1. (i)

The Gauss-Green-type identity holds true, namely

(5.1) $I(t)=\int_{\left\\{u=t\right\\}}H(\nabla
u)^{p-1}H(\nu)\,w(x)\,d{\mathcal{H}}^{N-1}\quad\text{for a.e.
}\,t\in\left(0,M\right).$

  2. (ii)

The Hölder-isopertimetric-type inequality holds true, namely

(5.2)
$I(t)^{\frac{1}{p}}\,\left(\int_{\left\\{u=t\right\\}}\frac{w(x)}{|\nabla
u|}\,d{\mathcal{H}}^{N-1}\right)^{\frac{p-1}{p}}\geqslant
c\,\mu(t)^{\frac{D-1}{D}},\quad\text{for a.e. }t\in\left(0,M\right),$

where

(5.3) $c:=\frac{P_{w,H}\left(B;\Sigma\right)}{w(\Sigma\cap
B)^{\frac{D-1}{D}}}$

is the optimal constant in the weighted anisotropic isoperimetric inequality
in cones (4.1), being $B$ the unitary Wulff ball centered at the origin.

Moreover, the equality sign holds in (5.2) if and only if
$\left\\{u>t\right\\}$ is a minimizer of (4.1) and $H(\nabla u)$ is constant
on $\left\\{u=t\right\\}$.

###### Proof.

_(i)_ Since the function $u$ is of bounded variation in $\Sigma\cap\Omega$,
the coarea theorem for BV functions (see e.g. [EG, Theorem 1, Section 5.5])
guarantees that the sets $\left\\{u>t\right\\}$ have finite perimeter for a.e.
$t$. For the measure theoretic boundary $\partial^{*}\left\\{u>t\right\\}$, we
see that

(5.4) $\left\\{u=t\right\\}\cap\left\\{|\nabla
u|>0\right\\}\subset\Sigma\cap\partial^{*}\left(\left\\{u>t\right\\}\right)\subset\left\\{u=t\right\\}.$

By recalling (3.14), we conclude that

${\mathcal{H}}^{N-1}\left(\left\\{u=t\right\\}\cap\left\\{|\nabla
u|>0\right\\}\right)={\mathcal{H}}^{N-1}\left(\Sigma\cap\partial^{*}\left(\left\\{u>t\right\\}\right)\right)\quad\text{for
a.e. }t,$

and the exterior measure theoretical normal vector $\nu$ of
$\partial^{*}\left\\{u>t\right\\}$ satisfies, a.e. in the sense of the measure
${\mathcal{H}}^{N-1}$:

(5.5) $|\nabla u|\,\nu=-\nabla u\quad\quad\quad{\mathcal{H}}^{N-1}-\text{a.e.
on }\Sigma\cap\partial^{*}\left(\left\\{u>t\right\\}\right).$

Accordingly, by using (1.5) (we are using the differential equation together
with the homogeneous boundary condition on $\Gamma_{1}$) and De Giorgi’s
structure theorem (see [Ma, Theorem 15.9] or [Gi]), we have, up to a standard
approximation argument,

$I(t)=\int_{\left\\{u>t\right\\}}f(u)\,w(x)\,d{\mathcal{H}}^{N}=\int_{\Sigma\cap\partial^{*}\left\\{u>t\right\\}}H(\nabla
u)^{p-1}\langle\nabla_{\xi}H(\nabla
u),\nu\rangle\,w(x)\,d{\mathcal{H}}^{N-1},\,\,\text{ for a.e.
}t\in\left(0,M\right)$

and hence, by using (2.3) and (5.5),

(5.6) $I(t)=\int_{\Sigma\cap\partial^{*}\left\\{u>t\right\\}}H(\nabla
u)^{p-1}H\left(\nu\right)\,w(x)\,d{\mathcal{H}}^{N-1},\quad\text{for a.e.
}t\in\left(0,M\right).$

By (3.14) and (5.4), we have that

$\int_{\Sigma\cap\partial^{*}\left\\{u>t\right\\}}H(\nabla
u)^{p-1}H\left(\nu\right)\,w(x)\,d{\mathcal{H}}^{N-1}=\int_{\left\\{u=t\right\\}}H(\nabla
u)^{p-1}H\left(\nu\right)\,w(x)\,d{\mathcal{H}}^{N-1},\,\text{ for a.e.
}t\in\left(0,M\right),$

and the conclusion follows.

_(ii)_ By the isoperimetric inequality (4.1), we have

(5.7) $P_{w,H}\left(\left\\{u>t\right\\};\Sigma\right)\geqslant
c\,\,w\left(\left\\{u>t\right\\}\right)^{\frac{D-1}{D}}=c\,\mu(t)^{\frac{D-1}{D}},\quad\text{for
a.e. }t\in\left(0,M\right),$

where $c$ is the optimal (weighted anisotropic) isoperimetric constant defined
in (5.3).

Now, by the homogeneity of $H$ and (5.5),

$H(\nu)H(\nabla u)^{p-1}=H(\nu)^{p}|\nabla u|^{p-1}.$

By using this and (5.1), we compute that

(5.8)
$\begin{split}I(t)^{\frac{1}{p}}\left(\int_{\left\\{u=t\right\\}}\frac{w(x)}{|\nabla
u|}\,d{\mathcal{H}}^{N-1}\right)^{\frac{p-1}{p}}&=\left(\int_{\left\\{u=t\right\\}}H(\nu)^{p}|\nabla
u|^{p-1}\,w(x)\,d{\mathcal{H}}^{N-1}\right)^{\frac{1}{p}}\left(\int_{\left\\{u=t\right\\}}\frac{w(x)}{|\nabla
u|}\,d{\mathcal{H}}^{N-1}\right)^{\frac{p-1}{p}}\\\
&\geqslant\int_{\left\\{u=t\right\\}}H(\nu)w(x)d{\mathcal{H}}^{N-1}\\\
&=P_{w,H}\left(\left\\{u>t\right\\};\Sigma\right)\\\ &\geqslant
c\,\mu(t)^{\frac{D-1}{D}}.\end{split}$

Here, the first inequality is a consequence of Hölder’s inequality, the second
equality follows from (1.12) (by recalling (3.14) and (5.4)), and the last
inequality comes from (5.7). Thus, (5.2) is proved. The characterization of
the equality sign in (5.2) follows by noting that the equality holds in (5.2)
if and only if the equality sign holds in both the inequalities in (5.8). ∎

We are now ready for the

###### Proof of Theorem 1.1.

We consider the function $K$ defined in (3.8), with $\alpha$ and $\beta$ as in
(3.9). Since $K(t)$ is nonnegative and, by item (iv) of Lemma 3.3,
nonincreasing, we have that

(5.9) $K(0^{-})\geqslant
K(0^{+})-K(M^{-})\geqslant\int_{0}^{M}-K^{\prime}(t)dt.$

Combining this inequality with (3.10) leads to

$K(0^{-})\geqslant\int_{0}^{M}\left[\alpha
I(t)^{\alpha-1}\mu(t)^{\beta}f(t)+\beta
I(t)^{\alpha}\mu(t)^{\beta-1}\right]\left(-\mu^{\prime}(t)\right)\,dt\quad\text{for
a.e. }t.$

Notice that equality here above holds true when $K$ is absolutely continuous.
Being the integrand in the right-hand side (that is $-K^{\prime}(t)$) and
$-\mu^{\prime}(t)$ nonnegative (and hence so is also the factor in the square
bracket), we can use (3.11) to get

$K(0^{-})\geqslant\int_{0}^{M}\left[\alpha
I(t)^{\alpha-1}\mu(t)^{\beta}f(t)+\beta
I(t)^{\alpha}\mu(t)^{\beta-1}\right]\left(\int_{\left\\{u=t\right\\}}\frac{w(x)}{|\nabla
u|}d{\mathcal{H}}^{N-1}\right)\,dt.$

Then we compute

(5.10) $\begin{split}K(0^{-})&\geqslant\int_{0}^{M}\left[\alpha
I(t)^{\alpha-1}\mu(t)^{\beta}f(t)+\beta
I(t)^{\alpha}\mu(t)^{\beta-1}\right]\left(\int_{\left\\{u=t\right\\}}\frac{w(x)}{|\nabla
u|}d{\mathcal{H}}^{N-1}\right)\,dt\\\ &=\int_{0}^{M}\left[\alpha
I(t)^{\alpha-1-\frac{1}{p-1}}\mu(t)^{\beta}f(t)+\beta
I(t)^{\alpha-\frac{1}{p-1}}\mu(t)^{\beta-1}\right]I(t)^{\frac{1}{p-1}}\left(\int_{\left\\{u=t\right\\}}\frac{w(x)}{|\nabla
u|}d{\mathcal{H}}^{N-1}\right)\,dt\\\
&\geqslant\int_{0}^{M}c^{\frac{p}{p-1}}\left[\alpha
I(t)^{\alpha-1-\frac{1}{p-1}}\mu(t)^{\beta}f(t)+\beta
I(t)^{\alpha-\frac{1}{p-1}}\mu(t)^{\beta-1}\right]\mu(t)^{\frac{p(D-1)}{(p-1)D}}\,dt,\end{split}$

where the last inequality follows from (5.2) and the fact that the factor in
the square brackets is nonnegative, being $-K^{\prime}\geqslant 0$.

Since (5.10) has been obtained by applying (5.2) on almost all the level sets,
a necessary condition to get the equality in (5.10) is that, for a.e.
$t\in(0,M)$, $\left\\{u>t\right\\}$ is a minimizer of (4.1) and $H(\nabla u)$
is constant on $\left\\{u=t\right\\}$.

Now we notice that the values of $\alpha$ and $\beta$ in (3.9) are set in
order to satisfy

$\alpha-1-\frac{1}{p-1}=0\quad\text{and}\quad\beta-1+\frac{p(D-1)}{(p-1)D}=0.$

Thus, by setting

$F(s):=\int_{0}^{s}f(\tau)\,d\tau$

and $p^{\prime}=p/(p-1)$, (5.10) leads to

(5.11)
$\begin{split}K(0^{-})&\geqslant\int_{0}^{M}c^{p^{\prime}}\left[p^{\prime}f(t)\,w\left(\left\\{u>t\right\\}\right)+\frac{p-D}{D(p-1)}\int_{\left\\{u>t\right\\}}f(u)\,w(x)\,d{\mathcal{H}}^{N}\right]\,dt\\\
&=c^{p^{\prime}}\int_{0}^{M}\int_{\Sigma\cap\Omega}\chi_{\left\\{u>t\right\\}}w(x)\left(p^{\prime}f(t)+\frac{p-D}{D(p-1)}f(u)\right)\,d{\mathcal{H}}^{N}\,dt\\\
&=c^{p^{\prime}}\left[p^{\prime}\int_{\Sigma\cap\Omega}F(u)\,w(x)\,d{\mathcal{H}}^{N}+\frac{p-D}{D(p-1)}\int_{\Sigma\cap\Omega}u\,f(u)\,w(x)\,d{\mathcal{H}}^{N}\right]\\\
&=c^{p^{\prime}}\frac{p^{\prime}}{D}\left[D\int_{\Sigma\cap\Omega}F(u)\,w(x)\,d{\mathcal{H}}^{N}+\frac{p-D}{p}\int_{\Sigma\cap\Omega}u\,f(u)\,w(x)\,d{\mathcal{H}}^{N}\right].\end{split}$

By noting that

$K(0^{-})=\lim_{t\to
0^{-}}K(t)=w\left(\Sigma\cap\Omega\right)^{\frac{p-D}{D(p-1)}}\left[\int_{\Sigma\cap\Omega}f(u)\,w(x)\,d{\mathcal{H}}^{N}\right]^{p^{\prime}},$

we deduce from (5.11) that

(5.12)
$\frac{D}{p^{\prime}c^{p^{\prime}}}w\left(\Sigma\cap\Omega\right)^{\frac{p-D}{D(p-1)}}\left[\int_{\Sigma\cap\Omega}f(u)\,w(x)\,d{\mathcal{H}}^{N}\right]^{p^{\prime}}\geqslant
D\int_{\Sigma\cap\Omega}F(u)\,w(x)\,d{\mathcal{H}}^{N}+\frac{p-D}{p}\int_{\Sigma\cap\Omega}u\,f(u)\,w(x)\,d{\mathcal{H}}^{N}.$

Now we recall the weighted anisotropic Pohozaev identity (3.17):

$D\int_{\Sigma\cap\Omega}F(u)\,w(x)d{\mathcal{H}}^{N}+\frac{p-D}{p}\int_{\Sigma\cap\Omega}u\,f(u)\,w(x)\,d{\mathcal{H}}^{N}=\frac{1}{p^{\prime}}\int_{\Gamma_{0}}H^{p}(\nabla
u)\,\langle x,\nu\rangle\,w(x)\,d{\mathcal{H}}^{N-1},$

where $\Gamma_{0}=\Sigma\cap\partial\Omega$. By using for the first time that
$\Omega$ is a Wulff ball, (2.1) gives that

$\langle x,\nu\rangle=R\,H(\nu)\quad\text{ for
}x\in\partial\Omega\supseteq\Gamma_{0},$

where $R$ is the radius of the Wulff ball $\Omega$, and hence

(5.13) $\frac{1}{p^{\prime}}\int_{\Gamma_{0}}H^{p}(\nabla u)\,\langle
x,\nu\rangle\,w(x)\,d{\mathcal{H}}^{N-1}=\frac{R}{p^{\prime}}\,\int_{\Gamma_{0}}H^{p}(\nabla
u)\,H(\nu)\,w(x)\,d{\mathcal{H}}^{N-1}.$

By using Hölder’s inequality and (1.12) we obtain

$\begin{split}\int_{\Gamma_{0}}H^{p-1}(\nabla
u)\,H(\nu)\,w(x)d{\mathcal{H}}^{N-1}&=\int_{\Gamma_{0}}\left[H(\nu)\,w(x)\right]^{\frac{1}{p}}\,\left[H^{p-1}(\nabla
u)\,\left(H(\nu)\,w(x)\right)^{\frac{1}{p^{\prime}}}\right]d{\mathcal{H}}^{N-1}\\\
&\leqslant
P_{w,H}(\Omega;\Sigma)^{\frac{1}{p}}\,\left[\int_{\Gamma_{0}}H^{p}(\nabla
u)\,H(\nu)\,w(x)\,d{\mathcal{H}}^{N-1}\right]^{\frac{1}{p^{\prime}}}.\end{split}$

As a result, exploiting (5.1) (which also holds true for $t=0$), we conclude
that

$\int_{\Gamma_{0}}H^{p}(\nabla
u)\,H(\nu)\,w(x)\,d{\mathcal{H}}^{N-1}\geqslant\frac{1}{P_{w,H}(\Omega;\Sigma)^{\frac{1}{p-1}}}\,\left[\int_{\Sigma\cap\Omega}f(u)\,w(x)\,d{\mathcal{H}}^{N}\right]^{p^{\prime}}.$

The last inequality, (5.13) and (3.17) entail that

(5.14)
$D\int_{\Sigma\cap\Omega}F(u)\,w(x)\,d{\mathcal{H}}^{N}+\frac{p-D}{p}\int_{\Sigma\cap\Omega}u\,f(u)\,w(x)\,d{\mathcal{H}}^{N}\geqslant\frac{R}{p^{\prime}\,P_{w,H}(\Omega;\Sigma)^{\frac{1}{p-1}}}\,\left[\int_{\Sigma\cap\Omega}f(u)\,w(x)d{\mathcal{H}}^{N}\right]^{p^{\prime}}.$

By recalling (5.3) and (2.7), the straightforward computation

$\frac{D}{p^{\prime}c^{p^{\prime}}}w\left(\Sigma\cap\Omega\right)^{\frac{p-D}{D(p-1)}}=\frac{R}{p^{\prime}\,P_{w,H}(\Omega;\Sigma)^{\frac{1}{p-1}}}$

shows that (5.12) and (5.14) are opposite inequalities. Thus, they hold with
the equality sign, and hence, for a.e. $t\in\left(0,M\right)$,
$\left\\{u>t\right\\}$ is a is a minimizer of (4.1) and $H(\nabla u)$ is
constant on $\left\\{u=t\right\\}$. ∎

######  Lemma 5.2.

Suppose that the assumptions of Theorem 1.1 are satisfied. Up to rotations, we
write $\Sigma=\mathbb{R}^{k}\times\tilde{\Sigma}$, where $0\leqslant
k\leqslant N$ and $\tilde{\Sigma}\subset\mathbb{R}^{N-k}$ is an open convex
cone containing no lines. Assume that for a.e. $t\in(0,M)$ the following two
conditions are verified:

  * (i)

$\left\\{u>t\right\\}=\Sigma\cap B_{\rho(t)}(x(t))$ where $B_{\rho(t)}(x(t))$
is the Wulff ball of radius $\rho(t)\geqslant 0$ centered at the point
$x(t)\in\mathbb{R}^{k}\times\left\\{0_{\mathbb{R}^{N-k}}\right\\}$, with
$\rho(t)$ and $x(t)$ depending on $t$;

  * (ii)

$H(\nabla u)$ is constant on $\left\\{u=t\right\\}$.

Then, $u$ is a radially symmetric and radially nonincreasing function.
Moreover,

$u\text{ is radially strictly decreasing on }\left\\{0<u<M\right\\},$

and

$\left\\{0<u<M\right\\}\quad\text{is a Wulff annulus or a punctured Wulff ball
centered at }0\text{ intersected with }\Sigma.$

###### Proof.

Let us consider the case $k=N$. We follow the ideas used in [Se, Lemma 6] in
the isotropic setting. By reasoning as in [Se, Lemma 6], we easily find that

$\left\\{u>t\right\\}=B_{\rho(t)}(x(t))\quad\text{ for every }t\in(0,M),$

where $B_{\rho(t)}(x(t))$ denotes the Wulff ball centered at $x(t)$ with
radius $\rho(t)$. Hence, by continuity also (ii) holds for every $t\in(0,M)$,
that is, $H(\nabla u)$ is constant on $\partial B_{\rho(t)}(x(t))$ for every
$t\in(0,M)$. By (5.1), we thus find that

$P_{w,H}(\partial B_{\rho(t)}(x(t)))H\left(\nabla u(\partial
B_{\rho(t)}(x(t)))\right)^{p-1}=\int_{B_{\rho(t)}(x(t))}f(u)\,w(x)\,d{\mathcal{H}}^{N}.$

Notice that, since $f\geqslant 0$, by the maximum principle we cannot have
$f\equiv 0$ on $\left\\{u>t\right\\}$ for $t\in(0,M)$. Hence, $\nabla u$ does
not vanish in $\left\\{0<u<M\right\\}$.

Thus, $u$ is a $C^{1}$ function whose gradient never vanishes in
$\left\\{u<M\right\\}$, and hence
$\widetilde{\mu}(t):={\mathcal{H}}^{N}(\left\\{u>t\right\\})$ is locally
Lipschitz in $(0,M)$. Therefore, also
$\rho(t)=\left(\widetilde{\mu}(t)/{\mathcal{H}}^{N}(B)\right)^{1/N}$ is
locally Lipschitz, being ${\mathcal{H}}^{N}(B)$ the volume of the unitary
Wulff ball.

We observe that $B_{\rho(t)}(x(t))=\left\\{u\geqslant
t\right\\}\supset\left\\{u\geqslant s\right\\}=B_{\rho(s)}(x(s))$ for $t<s$,
and therefore

$x(s)+\rho(s)\frac{x(s)-x(t)}{H_{0}(x(s)-x(t))}\in B_{\rho(s)}(x(s))\subset
B_{\rho(t)}(x(t)).$

As a consequence,

$\begin{split}\rho(t)&\geqslant
H_{0}\left(x(s)+\rho(s)\frac{x(s)-x(t)}{H_{0}(x(s)-x(t))}-x(t)\right)\\\
&=H_{0}\left(\frac{x(s)-x(t)}{H_{0}(x(s)-x(t))}\left(H_{0}(x(s)-x(t))+\rho(s)\right)\right)\\\
&=H_{0}(x(s)-x(t))+\rho(s),\end{split}$

where in the last identity we used the homogeneity of $H_{0}$, and accordingly

(5.15) $H_{0}(x(t)-x(s))\leqslant\rho(t)-\rho(s)\quad\text{ for }t<s,$

that gives that $x(t)$ is also locally Lipschitz.

Now suppose by contradiction that $u$ is not radially symmetric. Then $x(t)$
would not be identically constant in $(0,M)$ and hence we could find some
$t_{0}\in(0,M)$ such that the velocity vector $x^{\prime}(t_{0})$ would exist
and be nonzero. Set

$\overline{\xi}:=\frac{x^{\prime}(t_{0})}{H_{0}\left(x^{\prime}(t_{0})\right)},\qquad
P(t):=x(t)+\rho(t)\overline{\xi}\quad{\mbox{and}}\quad
Q(t):=x(t)-\rho(t)\overline{\xi}.$

By definition, $P(t)$ and $Q(t)$ belong to $\partial\left\\{u>t\right\\}$ for
every $t$, and hence we have

(5.16) $u\left(P(t)\right)\equiv u\left(Q(t)\right)\equiv t\quad\text{ for
every }t,$ (5.17) $\nabla u\left(P(t_{0})\right)=-|\nabla
u\left(P(t_{0})\right)|\,\nu\left(P(t_{0})\right)\quad\text{ and }\quad\nabla
u\left(Q(t_{0})\right)=-|\nabla
u\left(Q(t_{0})\right)|\,\nu\left(Q(t_{0})\right),$

where $\nu$ denotes the exterior unit normal vector field to
$\left\\{u>t_{0}\right\\}=B_{\rho(t_{0})}(x(t_{0}))$. Moreover, since
$\overline{\xi}\in B=\left\\{x\in\mathbb{R}^{N}\,:\,H_{0}(x)=1\right\\}$, by
(2.1) it holds that

(5.18)
$\langle\nu_{B}(\overline{\xi}),\overline{\xi}\rangle=H(\nu_{B}(\overline{\xi})).$

We also observe that

(5.19)
$\nu\left(P(t_{0})\right)=\nu_{B}(\overline{\xi})=-\nu_{B}(-\overline{\xi})=-\nu\left(Q(t_{0})\right).$

To check this, we let $B_{r}=\left\\{H_{0}(x)<r\right\\}$ be the Wulff ball of
radius $r$ centered at the origin, and set $B:=B_{1}$. For any $x\in\partial
B_{r}$ it holds that

$\nu_{B_{r}}(x)=\frac{\nabla_{x}H_{0}(x)}{|\nabla_{x}H_{0}(x)|}\quad\text{and}\quad\nu_{B_{r}}(-x)=\frac{\nabla_{x}H_{0}(-x)}{|\nabla_{x}H_{0}(-x)|}.$

Also, by the homogeneity of $H_{0}$ we have that

$\nabla_{x}H_{0}(tx)=\mathrm{sgn}(t)\nabla_{x}H_{0}(x)\quad\text{for all
}\,t\neq 0,\,x\neq 0$

and consequently

(5.20)
$\nu_{B_{r}}(r\,\xi)=\nu_{B}(\xi)=-\nu_{B}(-\xi)=\nu_{B_{r}}(-r\,\xi)\quad\text{
for any }\xi\in\partial B,\,r\in\mathbb{R}^{+}.$

Additionally, taking into account the translation invariance of the normals,
we obtain

$\nu_{B_{r}(x)}\left(x+r\,\xi\right)=\nu_{B_{r}}\left(r\,\xi\right)\quad\text{
for any }x\in\mathbb{R}^{N},\,r\in\mathbb{R}^{+},\,\xi\in\partial B.$

Combining this information and (5.20) we obtain (5.19), as desired.

Thus, from (5.18) and (5.19) we deduce that

$\langle\nu\left(P(t_{0})\right),\overline{\xi}\rangle=H\left(\nu\left(P(t_{0})\right)\right)\quad\text{
and
}\quad\langle\nu\left(Q(t_{0})\right),\overline{\xi}\rangle=-H\left(\nu\left(Q(t_{0})\right)\right).$

From this and (5.17), it follows that

(5.21) $\langle\nabla
u\left(P(t_{0})\right),\overline{\xi}\rangle=-H\left(\nabla
u\left(P(t_{0})\right)\right)\quad\text{ and }\quad\langle\nabla
u\left(Q(t_{0})\right),\overline{\xi}\rangle=H\left(\nabla
u\left(Q(t_{0})\right)\right).$

By using (5.16) and (5.21), we compute

$1=\frac{d}{dt}u\left(P(t)\right)|_{t=t_{0}}=\langle\nabla
u\left(P(t_{0})\right),\left(H_{0}(x^{\prime}(t_{0}))+\rho^{\prime}(t_{0})\right)\overline{\xi}\rangle=-H\left(\nabla
u\left(P(t_{0})\right)\right)\left(H_{0}(x^{\prime}(t_{0}))+\rho^{\prime}(t_{0})\right)$

and

$1=\frac{d}{dt}u\left(Q(t)\right)|_{t=t_{0}}=\langle\nabla
u\left(Q(t_{0})\right),\left(H_{0}(x^{\prime}(t_{0}))-\rho^{\prime}(t_{0})\right)\overline{\xi}\rangle=H\left(\nabla
u\left(Q(t_{0})\right)\right)\left(H_{0}(x^{\prime}(t_{0}))-\rho^{\prime}(t_{0})\right).$

But by assumption (ii), we have that $H\left(\nabla
u\left(P(t_{0})\right)\right)=H\left(\nabla u\left(Q(t_{0})\right)\right)$.
This leads to $H_{0}(x^{\prime}(t_{0}))=0$, which is a contradiction.

Thus, $u$ is radially symmetric, i.e., $u=u(r)$, with $r=H_{0}(x)$. Since we
showed that $\nabla u$ does not vanish on $\left\\{0<u<M\right\\}$, we thus
have $\partial_{r}u<0$ in this open ring. The set $\left\\{u=M\right\\}$ could
be a point, but also a closed ball (as it happens, e.g., in the example
described in [Se, pag. 1895]). This proves the result when $k=N$.

By direct inspection and recalling that $u$ is $C^{1}$ up to
$\Gamma_{1}\setminus\left\\{0\right\\}$, it is easy to check that the same
proof remains valid when $1\leqslant k\leqslant N-1$. Notice that in this case
we have that
$\overline{\xi}\in\mathbb{R}^{k}\times\left\\{0_{\mathbb{R}^{N-k}}\right\\}$.

The case $k=0$ is trivial. ∎

We have now all the ingredients to complete the

###### Proof of Theorems 1.4, 1.5 and 1.6.

Theorems 1.4 and 1.5 easily follow by putting together Theorems 1.1, 4.2, and
Lemma 5.2.

Theorem 1.6 easily follows by putting together Theorems 1.1, 4.3, and Lemma
5.2. ∎

## Acknowledgements

The authors are members of INdAM/GNAMPA and AustMS and are supported by the
Australian Research Council Discovery Project DP170104880 NEW “Nonlocal
Equations at Work”. The first author is supported by the Australian Research
Council DECRA DE180100957 “PDEs, free boundaries and applications”. The second
and third authors are supported by the Australian Laureate Fellowship
FL190100081 “Minimal surfaces, free boundaries and partial differential
equations”.

The authors thank Xavier Cabré for bringing up to their attention the
reference [Se]. The authors also thank Xavier Ros-Oton and Joaquim Serra for
their useful comments on a preliminary version of this paper, Giulio Ciraolo
for the helpful correspondence about the forthcoming article [ACF], and
Emanuel Indrei for kindly pointing out his recent results in [I].

## References

  * [A] A.D. Aleksandrov, _Uniqueness theorems for surfaces in the large. V_ , Vestnik Leningrad. Univ. 13 1958 no. 19, 5–8.
  * [AB] M. Amar, G. Bellettini, _A notion of total variation depending on a metric with discontinuous coefficients_ , Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 1, 91–133.
  * [AFP] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.
  * [ACF] C.A. Antonini, G. Ciraolo, and A. Farina, _Interior regularity results for anisotropic quasilinear equations_ , forthcoming.
  * [BBF] G. Bellettini, G. Bouchitté, and I. Fragalà, _BV functions with respect to a measure and relaxation of metric integral functionals_ , J. Convex Anal. 6 (1999), 349–366.
  * [BFK] M. Belloni, V. Ferone, and B. Kawohl, _Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators_ , Special issue dedicated to Lawrence E. Payne. Z. Angew. Math. Phys. 54 (2003), no. 5, 771–783.
  * [BP] H. Berestycki, F. Pacella, _Symmetry properties for positive solutions of elliptic equations with mixed boundary conditions_ , J. Funct. Anal. 87 (1989), 177–211.
  * [BC] C. Bianchini, G. Ciraolo, _Wulff shape characterizations in overdetermined anisotropic elliptic problems_ , Commun. Partial Differ. Equ. 43 (2018), 79–820.
  * [BNST] B. Brandolini, C. Nitsch, P. Salani, and C. Trombetti, _Serrin type overdetermined problems: an alternative proof_ , Arch. Rat. Mech. Anal.190(2008), 267–280.
  * [BM] J.E. Brothers, F. Morgan, _The isoperimetric theorem for general integrands_ , Michigan Math. J.41 (1994), no. 3, 419–431.
  * [BZ] J.E. Brothers, W.P. Ziemer, _Minimal rearrangements of Sobolev functions_ , J. Reine Angew. Math. 384 (1988), 153–179.
  * [CRS] X. Cabré, X. Ros-Oton, and J. Serra, _Sharp isoperimetric inequalities via the ABP method_ , J. Eur. Math. Soc. (JEMS) 18 (2016), no. 12, 2971–2998.
  * [CR] A. Ca$\tilde{\mathrm{n}}$ete, C. Rosales, _Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities_ , Calc. Var. Partial Differential Equations 51 (2014), 887–913.
  * [CDLP] M. Carriero, G. Dal Maso, A. Leaci, and E. Pascali: _Relaxation of the non-parametric Plateau problem with an obstacle_ , J. Math. Pures Appl. 67, 359–396 (1988).
  * [Ch] G.R. Chambers, _Proof of the log-convex density conjecture_ , J. Eur. Math. Soc. (JEMS) 21 (2019), no. 8, 2301–-2332.
  * [CM] A. Cianchi, V. Maz’ya, _Global Lipschitz regularity for a class of quasilinear elliptic equations_ , Comm. Part. Diff. Equat. 36 (2011), 100–133.
  * [CS] A. Cianchi, P. Salani, _Overdetermined anisotropic elliptic problems_ , Math. Ann. 345 (2009), 859–881.
  * [CGPRS] E. Cinti, F. Glaudo, A. Pratelli, X. Ros-Oton, and J. Serra, _Sharp quantitative stability for isoperimetric inequalities with homogeneous weights_ , arXiv:2006.13867v1.
  * [CFR] G. Ciraolo, A. Figalli, A. Roncoroni, _Symmetry results for critical anisotropic p-Laplacian equations in convex cones_ , Geom. Funct. Anal., 30 (2020), 770–803.
  * [CG] G. Ciraolo, A. Greco, _An overdetermined problem associated to the Finsler Laplacian_ , Commun. Pure Appl. Anal., 2021, 20 (3) : 1025–1038.
  * [CiR] G. Ciraolo, A. Roncoroni, _Serrin’s type overdetermined problems in convex cones_ , Calc. Var. Partial Differential Equations 59 (2020), no. 1, 21 pp.
  * [CFV] M. Cozzi, A. Farina, and E. Valdinoci, _Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs_ , Adv. Math. 293 (2016), 343–381.
  * [CrM] G. Crasta, A. Malusa, _The distance function from the boundary in a Minkowski space_ , Trans. Amer. Math. Soc. 359 (2007), no. 12, 5725–5759.
  * [DP] B. Dacorogna, C.E. Pfister, _Wulff theorem and best constant in Sobolev inequality_ , J. Math. Pures Appl.(9) 71 (2) (1992), 97–118.
  * [DMS] M. Degiovanni, A. Musesti, and M. Squassina, _On the regularity of solutions in the Pucci-Serrin identity_ , Calc. Var. Partial. Differ. Equations 18 (2003), 317–334.
  * [Di] A. Dinghas, _Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen_ , Z. Kristallogr., Mineral. Petrogr. 105, (1944).
  * [DPV] S. Dipierro, G. Poggesi, and E. Valdinoci, _A Serrin-type problem with partial knowledge of the domain_ , Nonlinear Analysis 208 (2021), 112330.
  * [DFM] J. Dolbeault, P. Felmer, and R. Monneau, _Symmetry and nonuniformly elliptic operators_ , Differential Integral Equations 18 (2005), no. 2, 141–154.
  * [EP] A. Enciso, D. Peralta-Salas, _Symmetry for an overdetermined boundary problem in a punctured domain_ , Nonlinear Anal. 70 (2009), no. 2, 1080–1086.
  * [EG] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Poperties of Functions, Studies in Advanced Mathematics, CRC Press, 1992.
  * [FK] A. Farina, B. Kawohl, _Remarks on an overdetermined boundary value problem_ , Calc. Var. Partial Differential Equations 31(2008), no. 3, 351–357.
  * [Fe] H. Federer, _Curvature measures_ , Trans. Amer. Math. Soc. 93 (1959), 418–491.
  * [FI] A. Figalli, E. Indrei, _A sharp stability result for the relative isoperimetric inequalityinside convex cones_ , J. Geom. Anal. 23 (2013), no. 2, 938–969.
  * [FMP] A. Figalli, F. Maggi, and A. Pratelli, _A mass transportation approach to quantitative isoperimetric inequalities_ , Invent. Math. 182 (2010), no. 1, 167–211.
  * [FM] I. Fonseca, S. Müller, _A uniqueness proof for the Wulff theorem_ , Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), 125–136.
  * [FGK] I. Fragalà, F. Gazzola, and B. Kawohl, _Overdetermined problems with possibly degenerate ellipticity, a geometric approach_ , Math. Z. 254(2006), 117–132.
  * [GL] N. Garofalo, J.L. Lewis, _A symmetry result related to some overdetermined boundary value problems_ , Amer. J. Math. 111 (1989), no. 1, 9–33.
  * [GS] N. Garofalo, E. Sartori, _Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a-priori estimates_ , Adv. Differential Equations 4 (1999), 137–161.
  * [GNN] B. Gidas, W.M. Ni, and L. Nirenberg, _Symmetry and related properties via the maximum principle_ , Comm. Math. Phys. 68 (1979), no. 3, 209–243.
  * [GT] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
  * [Gi] E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, Boston, 1984.
  * [GX] J. Guo, G. Xia, _A partially overdetermined problem in a half ball_ , Calc. Var. Partial Differential Equations 58 (2019), no. 5, Art. 160, 1–15.
  * [H] J. Heinonen, Lectures on Lipschitz analysis. No. 100. University of Jyväskylä, 2005.
  * [I] E. Indrei, _A weighted relative isoperimetric inequality in convex cones_ , Methods Appl. Anal. 28 (2021), no. 1, 1–13.
  * [KP] S. Kesavan, F. Pacella, _Symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities_ , Appl. Anal. 54 (1994), no. 1–2, 27–37.
  * [L] P.-L. Lions, _Two geometrical properties of solutions of semilinear problems_ , Applicable Anal. 12 (1981), no. 4, 267–272.
  * [LP] P.-L. Lions, F. Pacella, _Isoperimetric inequalities for convex cones_ , Proc. Amer. Math. Soc. 109 (1990), no. 2, 477–485.
  * [Li] M. Lieberman, _Boundary regularity for solutions of degenerate elliptic equations_ , Nonlinear Anal. 12 (1988), 1203–1219.
  * [Lo] H. Lou, _On singular sets of local solutions to p-Laplace equations_ , Chin. Ann. Math. 29B(5) (2008), 521–530.
  * [Ma] F. Maggi, Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory, Cambridge University Press, 2012.
  * [MP] R. Magnanini, G. Poggesi, _On the stability for Alexandrov’s Soap Bubble theorem_ , J. Anal. Math., 139(1):179–205, 2019.
  * [MP2] R. Magnanini, G. Poggesi, _Serrin’s problem and Alexandrov’s Soap Bubble Theorem: enhanced stability via integral identities_ , Indiana Univ. Math. J. 69 (2020), no. 4, 1181–1205.
  * [MP3] R. Magnanini, G. Poggesi, _Nearly optimal stability for Serrin’s problem and the Soap Bubble theorem_ , Calc. Var. Partial Differential Equations, 59(1):Paper No. 35, 2020.
  * [MR] E. Milman, L. Rotem, _Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures_ , arXiv:1308.5695v2.
  * [MS] V.D. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. With an appendix by M.Gromov. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986. viii+156 pp.
  * [Mo] F. Morgan, The log-convex density conjecture, Frank Morgan’s blog (April 2010), http://sites.williams.edu/Morgan/2010/04/03/the-log-convex-density-conjecture/
  * [PT] F. Pacella, G. Tralli, _Overdetermined problems and constant mean curvature surfaces in cones_ , Rev. Mat. Iberoam. 36 (2020), 841–867.
  * [PT2] F. Pacella, G. Tralli, _Isoperimetric cones and minimal solutions of partial overdetermined problems_ , Publ. Mat. 65(1): 61–81 (2021).
  * [PSc] L.E. Payne, P.W. Schaefer, _Duality theorems in some overdetermined problems_ , Math. Methods Appl. Sci. 11 (1989), no. 6, 805–819.
  * [Po] G. Poggesi, _Radial symmetry for p-harmonic functions in exterior and punctured domains_ , Appl. Anal. 98 (2019), no. 10, 1785–1798.
  * [PS] P. Pucci, J. Serrin, _A general variational identity_ Indiana Univ. Math. J. 35 (1986), 681–703.
  * [Re] R. C. Reilly, _Applications of the Hessian operator in a Riemannian manifold_ , Indiana Univ. Math. J. 26 (1977), 459–472.
  * [Se] J. Serra, _Radial symmetry of solutions to diffusion equation with discontinuous nonlinearities_ , J. Differential Equations 254 (2013), 1893–1902.
  * [S] J. Serrin, _A symmetry problem in potential theory_ , Arch. Rational Mech. Anal. 43 (1971), 304–318.
  * [Sa] T. Sakai, Riemannian geometry, Translations of Mathematical Monographs, vol. 149, American Mathematical Society, Providence, RI, 1996.
  * [Scha] J. Van Schaftingen, _Anisotropic symmetrization_ , Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 4, 539–565.
  * [Sc] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge (1993).
  * [Tal] G. Talenti, _A weighted version of a rearrangement inequality_. Ann. Univ. Ferrara Sez. VII (N.S.) 43 (1997), 121–133 (1998).
  * [Ta] J. Taylor, _Crystalline variational problems_ , Bull. Amer. Math. Soc. 84 (1978), 568–588.
  * [To] P. Tolksdorf, _Regularity for a more general class of quasilinear elliptic equations_ , J. Diff. Eq. 51 (1984), 126–150.
  * [WX] G. Wang, C. Xia. _A characterization of the Wulff shape by an overdetermined anisotropic PDE_ , Arch. Rational Mech. Anal. 199 (2011) 99–115.
  * [W] H. F. Weinberger, _Remark on the preceding paper of Serrin_ , Arch. Rational Mech. Anal. 43 (1971), 319–320.