# Isoperimetric estimates for solutions to the p-Laplacian with variable Robin
boundary conditions

Vincenzo Amato, Francesco Chiacchio, Andrea Gentile

###### Abstract

In this paper we study the $p$-Poisson equation with Robin boundary
conditions, where the Robin parameter is a function. By means of some weighted
isoperimetric inequalities, we provide various sharp bounds for the solutions
to the problems under consideration. We also derive a Faber-Krahn-type
inequality.

MSC 2020: 35J92, 35P15, 35B45
Keywords: Weighted symmetrization, $p$-Laplacian, variable Robin parameter,
comparison results.

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universita‘ degli
Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via
Cintia - 80126 Napoli, Italy.

e-mail<EMAIL_ADDRESS>
e-mail<EMAIL_ADDRESS>
Mathematical and Physical Sciences for Advanced Materials and Technologies,
Scuola Superiore Meridionale, Largo San Marcellino 10, 80126 Napoli, Italy.

e-mail<EMAIL_ADDRESS>
##  1 Introduction

Symmetrization methods have been revealed to be a very effective and flexible
tool in the study of partial differential equations (in the sequel just PDEs).
The related bibliography is very wide and one of the cornerstone of this
theory is the Talenti’s Theorem (see [Tal76]), which makes use of the Schwarz
symmetrization. The technique he introduced has been refined through the years
and it has been adapted to treat a large class of linear and nonlinear
elliptic and parabolic PDEs (see, e.g., the survey papers [Tal16], [Tro00] and
the monographs [Kaw85], [Kes06] and [Bae19]). However, in most of the
subsequent works, the authors, in order to follow Talenti’s original idea,
consider problems whose solutions have level sets that do not intersect the
boundary of the domain where the problem is defined.

Recently it has been discovered a method to adapt symmetrization techniques to
a class of problems that does not exhibit this feature. More precisely, in
[ANT], the authors were able to obtain a Talenti-type comparison result for
the Laplacian with Robin boundary conditions. For further developments in this
direction see, e.g., [AGM22, CGNT], where the $p$-Laplace and Hermite operator
are studied, respectively, while in [ACNT21] and [ACNT] the authors
investigate linear problems where the Robin parameter is allowed to be a
function.

Here we generalize the above results by considering the following problem

$\begin{cases}-\text{div}\bigl{(}{\left|\nabla u\right|}^{p-2}\nabla
u\bigr{)}=f(x){\left|x\right|}^{\ell}&\text{in }\Omega\\\\[5.16663pt]
\displaystyle{{\left|\nabla u\right|}^{p-2}\,\frac{\partial
u}{\partial\nu}+\beta(x){\left|u\right|}^{p-2}u=0}&\text{on
}\partial\Omega,\end{cases}$ (1.1)

where, here and throughout, $n\geq 2$, $\nu$ denotes the outer unit normal to
$\partial\Omega$ and $\Omega$ is a bounded and Lipschitz domain of
$\operatorname{\mathbb{R}}^{n}$ such that $0\notin\partial\Omega$.

Furthermore we will assume that

$p\geq n,$ ($H_{1}$) $-n<\ell<0,$ ($H_{2}$)
$m=\inf_{\partial\Omega}\beta(x)>0\text{ \ and \
}M=\sup_{\partial\Omega}\beta(x)<+\infty,$ ($H_{3}$)

and, finally,

$0\leq f(x)\in L^{p^{\prime}}(\Omega,{\left|x\right|}^{\ell}\,dx),$ ($H_{4}$)

where $p^{\prime}=\displaystyle\frac{p}{p-1}$.

Hypotheses ($H_{1}$) and ($H_{2}$) ensure the validity of certain
isoperimetric inequalities, where two different weights (which are suitable
powers of the distance from the origin) appear in the perimeter and in the
area element, respectively (see, e.g., [How15, CH15, DHHT12, BBMP99, Csa15,
ABCMP17] and the references therein). Indeed we need to use, in place of the
more common Schwarz symmetrization, some weighted rearrangement, based on
these “double density” isoperimetric inequalities.

A weak solution to problem (1.1) is a function $u\in W^{1,p}(\Omega)$ such
that

$\int_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla
u\nabla\varphi\,dx+\int_{\partial\Omega}\beta(x)\left|u\right|^{p-2}u\varphi
d\mathcal{H}^{n-1}(x)=\int_{\Omega}f\left|x\right|^{\ell}\varphi\,dx$ (1.2)

for any $\varphi\in W^{1,p}(\Omega)$.

Now we need to construct the so-called symmetrized problem. That is a radial
problem, of the same type of (1.1), whose solution will estimate the one to
problem (1.1).

To this aim we introduce the set $\Omega^{\sharp}$ which is the ball centered
at the origin (in the sequel just centered ball) of radius $r^{\sharp}$, with
$r^{\sharp}$ such that

$\lvert\Omega^{\sharp}\rvert_{\ell}=\int_{\Omega^{\sharp}}\left|x\right|^{\ell}dx=\int_{\Omega}\left|x\right|^{\ell}dx=\left|\Omega\right|_{\ell},$

and the function $f^{\sharp}(x)$ defined by the following relation

$\lvert\Set{x\in\Omega:{\left|f(x)\right|}>t}\rvert_{\ell}=\lvert\\{x\in\Omega:f^{\sharp}(x)>t\\}\rvert_{\ell}\text{
\ for a.e. }t\geq 0.$

Our symmetrized problem is the following

$\begin{cases}-\text{div}\bigl{(}{\left|\nabla v\right|}^{p-2}\nabla
v\bigr{)}=f^{\sharp}(x){\left|x\right|}^{\ell}&\text{in
}\Omega^{\sharp}\\\\[5.16663pt] \displaystyle{{\left|\nabla
v\right|}^{p-2}\,\frac{\partial
v}{\partial\nu}+\tilde{\beta}(r^{\sharp})^{\frac{\ell}{p^{\prime}}}{\left|v\right|}^{p-2}v=0}&\text{on
}\partial\Omega^{\sharp}\end{cases}$ (1.3)

where

$\widetilde{\beta}=\inf_{\partial\Omega}\beta(x){\left|x\right|}^{-\frac{\ell}{p^{\prime}}}>0,$
(1.4)

since $0\notin\partial\Omega$ and $H_{3}$ holds. From the (1.4), it
immediately follows that

$\beta(x)\geq\widetilde{\beta}{\left|x\right|}^{\frac{\ell}{p^{\prime}}}.$
(1.5)

Our main results are contained in the following theorems.

######  Theorem 1.1.

Assume that assumptions ($H_{1}$)-($H_{4}$) are in force and let $u$ and $v$
be the solutions to problems (1.1) and (1.3), respectively. Then we have

${\left\|u\right\|}_{L^{1}(\Omega,{\left|x\right|}^{\ell}dx)}=\int_{\Omega}{\left|u(x)\right|}{\left|x\right|}^{\ell}\,dx\leq\int_{\Omega^{\sharp}}{\left|v(x)\right|}{\left|x\right|}^{\ell}\,dx={\left\|v\right\|}_{L^{1}(\Omega^{\sharp},{\left|x\right|}^{\ell}dx)}$
(1.6)

and

${\left\|u\right\|}_{L^{p}(\Omega,{\left|x\right|}^{\ell}dx)}^{p}=\int_{\Omega}{\left|u(x)\right|}^{p}{\left|x\right|}^{\ell}dx\leq\int_{\Omega^{\sharp}}\\!{\left|v(x)\right|}^{p}{\left|x\right|}^{\ell}dx={\left\|v\right\|}_{L^{p}(\Omega^{\sharp},{\left|x\right|}^{\ell}dx)}^{p}.$
(1.7)

If the source term $f(x)$ is constant we get a stronger result, namely a
pointwise comparison between $u^{\sharp}$ and $v$.

######  Theorem 1.2.

Assume that hypotheses ($H_{1}$)-($H_{4}$) are in force and suppose that
$f(x)\equiv 1$ in $\Omega$. If either $p=n=2$ or if $p>2$, $n\geq 2$ and

$\displaystyle{\ell\leq-n+\frac{p-n}{p-2}}$ (1.8)

then

$u^{\sharp}(x)\leq v(x)\qquad x\in\Omega^{\sharp},$ (1.9)

where $u$ and $v$ are the solutions to (1.1) and (1.3) respectively.

######  Remark 1.1.

It is not straightforward to compare our estimates with the ones, see [ANT,
AGM22], obtained by means of the classical Schwarz symmetrization (which
corresponds to choosing $\ell=0$). Nevertheless, in the forthcoming paper
[ACGM], the authors will address the asymptotic as $p\rightarrow+\infty$ of
some inequalities proved in the present note. From such an analysis, it will
be clear that, at least for $p$ large enough and for a special class of
domains, the weighted ($\ell<0$) and the unweighted ($\ell=0$) estimates are
comparable and the first ones are sharper.

We finally note that there is a vast literature on Talenti’s type estimates
obtained via weighted rearrangement. Typically, the weight fulfils an
isoperimetric inequality and it appears in the ellipticity of the differential
operator that one wants to study. In our case, see also [ACNT], the
differential operator is the classical $p$-Laplacian and the weight is somehow
“hidden” in the boundary condition. This feature could represent one of the
novelties of our approach.

######  Remark 1.2.

A straightforward computation shows that the function $v$, appearing in (1.9),
has the following explicit expression

$v(x)=\frac{p-1}{(\ell+p)(n+\ell)^{\frac{1}{p-1}}}\bigl{[}(R^{\sharp})^{\frac{\ell+p}{p-1}}-{\left|x\right|}^{\frac{\ell+p}{p-1}}\bigr{]}+\biggl{(}\frac{(R^{\sharp})^{\frac{\ell}{p}+1}}{\tilde{\beta}(n+\ell)}\biggr{)}^{\frac{1}{p-1}}.$

The first eigenvalue of the $p$-Laplace operator with the boundary conditions
as in (1.1) is the minimum of the following Rayleigh quotient

$\lambda_{1,\beta(x)}(\Omega)=\min_{\begin{subarray}{c}\psi\in
W^{1,p}(\Omega)\\\ \psi\neq
0\end{subarray}}\frac{\displaystyle{\int_{\Omega}{\left|\nabla\psi\right|}^{p}\,dx+\int_{\partial\Omega}\beta(x){\left|\psi\right|}^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega}{\left|\psi\right|}^{p}{\left|x\right|}^{\ell}\,dx}}.$
(1.10)

Finally, Theorem 1.1 allows us to prove the following Faber-Krahn inequality
(see also [ANT, Bos88, BD10, BG10, BG15, BGT19, Csa15, FK15, FNT13] and the
references therein).

######  Theorem 1.3.

Assume hypotheses ($H_{1}$)-($H_{3}$) and let $\tilde{\beta}$ be the constant
defined in (1.4), then we have

$\lambda_{1,\beta(x)}(\Omega)\geq\lambda_{1,\tilde{\beta}}(\Omega^{\sharp}).$

The paper is organized as follows. In the next section, we recall some basic
notion from the theory of rearrangements and the weighted isoperimetric
inequalities needed in the sequel. Moreover, we list some properties of the
solutions to problems (1.1) and (1.3). Section 3 is devoted to the proofs of
Theorem 1.1 and Theorem 1.2. Finally, in Section 4, we prove the Faber-Krahn
inequality.

##  2 Preliminary result

######  Definition 2.1.

Let $\Omega$ be a Lebesgue measurable subset of
$\operatorname{\mathbb{R}}^{n}$ and let $\ell\in(-n,0)$. We define the
$\ell$-weighted measure of $\Omega$ as

${\left|\Omega\right|}_{\ell}:=\int_{\Omega}{\left|x\right|}^{\ell}\,dx,$
(2.1)

and the $k$-weighted perimeter of $\Omega$ as

$P_{k}(\Omega)=\begin{cases}\displaystyle{\int_{\partial\Omega}{\left|x\right|}^{k}\,d\mathcal{H}^{n-1}}&\text{
if }\Omega\text{ is }(n-1)\text{-rectifiable}\\\\[12.91663pt]
+\infty&\text{otherwise}.\end{cases}$ (2.2)

For this family of measures, an isoperimetric inequality holds true if $k$,
$\ell$ and $n$ are suitably related. Here we need the following result (see
either [CH15] Theorem 1.3 or [ABCMP17] Theorem 1.1 case ii).

######  Theorem 2.1.

If assumptions ($H_{1}$) and ($H_{2}$) hold true, then

$P_{\frac{\ell}{p^{\prime}}}(\Omega)\geq
P_{\frac{\ell}{p^{\prime}}}(\Omega^{\sharp}),$ (2.3)

where $\Omega^{\sharp}$ is the centered ball with same $\ell$-measure as
$\Omega$.

######  Remark 2.1.

Since $\lvert\Omega^{\sharp}\rvert_{\ell}$ and $P_{k}(\Omega^{\sharp})$ can be
explicitly computed, is it possible to rewrite (2.3) in the following
equivalent way

$P_{\frac{\ell}{p^{\prime}}}(\Omega)\geq\gamma_{n,\ell,p}\,{\left|\Omega\right|}_{\ell}^{\frac{\ell(p-1)+(n-1)p}{p(n+\ell)}},$
(2.4)

where

$\gamma_{n,\ell,p}=(n\omega_{n})^{\frac{\ell+p}{p(n+\ell)}}(\ell+n)^{\frac{\ell(p-1)+(n-1)p}{p(n+\ell)}}$

and $\omega_{n}$ is the Lebesgue measure of the unit ball of
$\operatorname{\mathbb{R}}^{n}$.

######  Definition 2.2.

Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_weighted distribution function_ of $u$ is the function
$\mu_{\ell}:[0,+\infty[\,\to[0,+\infty[$ defined by

$\mu_{\ell}(t)={\left|\Set{x\in\Omega\,:\,{\left|u(x)\right|}>t}\right|}_{\ell}.$

In the following, we will omit the $\ell$ and just write $\mu$.

######  Definition 2.3.

Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_weighted decreasing rearrangement_ of $u$, denoted by $u^{\ast}$, is the
distribution function of $\mu$.

The _weighted rearrangement_ of $u$ is the function $u^{\sharp}$ whose level
sets are centered balls with the same $\ell$-measure as the level sets of
${\left|u\right|}$. More precisely,

$u^{\sharp}(x)=u^{*}\bigl{(}\,\bigl{\lvert}B_{{\left|x\right|}}\bigr{\rvert}_{\ell}\,\bigr{)}=u^{*}\biggl{(}\frac{n\omega_{n}}{\ell+n}\,{\left|x\right|}^{\ell+n}\biggr{)}.$

with $B_{{\left|x\right|}}$ the centered ball of radius ${\left|x\right|}$.

######  Definition 2.4.

If $p\in[1,+\infty)$ we will denote by
$L^{p}(\Omega,{\left|x\right|}^{\ell}\,dx)$ the space of all measurable
functions such that

${\left\|u\right\|}_{L^{p}(\Omega,{\left|x\right|}^{\ell}\,dx)}:=\biggl{(}\int_{\Omega}{\left|u\right|}^{p}{\left|x\right|}^{\ell}\,dx\biggr{)}^{\frac{1}{p}}<+\infty.$

It is easily checked that $u$, $u^{*}$ e $u^{\sharp}$ are equi-distributed,
hence

$\displaystyle{{\left\|u\right\|}_{L^{p}(\Omega,{\left|x\right|}^{\ell}\,dx)}={\left\|u^{*}\right\|}_{L^{p}(0,{\left|\Omega\right|}_{\ell})}=\lVert{u^{\sharp}}\rVert_{L^{p}(\Omega^{\sharp},{\left|x\right|}^{\ell}\,dx)}}\qquad\forall\,1\leq
p<+\infty.$ (2.5)

We also recall the definition of weighted Lorentz spaces.

######  Definition 2.5.

Let $0<p<+\infty$ and $0<q\leq+\infty$. The weighted Lorentz space
$L^{p,q}(\Omega,{\left|x\right|}^{\ell}\,dx)$ is the space of those functions
$f(x)$ such that the quantity

${\left\|g\right\|}_{L^{p,q}(\Omega,{\left|x\right|}^{\ell}\,dx)}:=\begin{cases}\displaystyle{p^{\frac{1}{q}}\biggl{(}\int_{0}^{\infty}t^{q}\mu(t)^{\frac{q}{p}}\,\frac{dt}{t}\biggr{)}^{\frac{1}{q}}}&0<q<\infty\\\\[8.61108pt]
\displaystyle{\sup_{t>0}\,(t^{p}\mu(t))}&q=\infty\end{cases}$ (2.6)

is finite.

Let us observe that for $p=q$ the Lorentz space coincides with the
$L^{p}(\Omega,{\left|x\right|}^{\ell}\,dx)$ space, as a consequence of the
well-known _Cavalieri’s Principle_

$\int_{\Omega}{\left|g\right|}^{p}{\left|x\right|}^{\ell}\,dx=p\int_{0}^{+\infty}t^{p-1}\mu(t)\,dt.$

By the definition of decreasing rearrangement, the following result holds true
(see for instance [CR71], [Kaw85] and [Kes06]).

######  Proposition 2.2.

Let $u\in L^{1}(\Omega,{\left|x\right|}^{\ell}\,dx)$ be a non negative
function and let $E\subset\Omega$ be a measurable set, then it holds

$\int_{E}u(x){\left|x\right|}^{\ell}\,dx\leq\int_{0}^{{\left|E\right|}_{\ell}}u^{*}(s)\,ds.$
(2.7)

Now we turn our attention on problems (1.1) and (1.3).

######  Proposition 2.3.

If the assumptions ($H_{2}$) and ($H_{3}$) are fulfilled and if $\Omega$ is a
bounded and Lipschitz domain of $\operatorname{\mathbb{R}}^{n}$ then
$W^{1,p}(\Omega)$ is compactly embedded in
$L^{p}(\Omega,{\left|x\right|}^{\ell}\,dx)$.

######  Proposition 2.4.

Problems (1.1) and (1.3) admit a unique solution.

The proofs of the above propositions will be postponed in the Appendix, since
they require rather standard arguments.

Let us observe that $u\geq 0$ in $\Omega$. Indeed, choosing
$\psi=u^{-}=\max\Set{-u,0}$ as test function in (1.2), we have

$0\geq-\int_{\Omega}\lvert\nabla
u^{-}\rvert^{p}\,dx-\int_{\partial\Omega}\beta(x)(u^{-})^{p}\,d\mathcal{H}^{n-1}(x)=\int_{\Omega}(u^{-})f{\left|x\right|}^{\ell}\,dx,$

thus $u^{-}=0$ a.e. in $\Omega$.

Furthermore we have

$u_{m}=\inf_{\Omega}u\leq\min_{\overline{\Omega^{\sharp}}}v=v_{m}.$ (2.8)

Indeed, using the fact that $v(x)=v_{m}$ $\forall x\in\partial\Omega^{\sharp}$
, taking into account of (2.5) and choosing $\psi\equiv 1$ in the weak
formulation of (1.1) and (1.3), respectively, we get

$\displaystyle v_{m}^{p-1}P_{\frac{\ell}{p^{\prime}}}(\Omega^{\sharp})$
$\displaystyle=\int_{\partial\Omega^{\sharp}}{\left|x\right|}^{\frac{\ell}{p^{\prime}}}v(x)^{p-1}\,d\mathcal{H}^{n-1}(x)=\frac{1}{\tilde{\beta}}\int_{\Omega^{\sharp}}f^{\sharp}{\left|x\right|}^{\ell}\,dx$
$\displaystyle=\frac{1}{\tilde{\beta}}\int_{\Omega}f{\left|x\right|}^{\ell}\,dx=\frac{1}{\tilde{\beta}}\int_{\partial\Omega}\beta(x)u(x)^{p-1}\,d\mathcal{H}^{n-1}(x).$

In turn, by (1.5) and using the isoperimetric inequality (2.3), we get

$\displaystyle\frac{1}{\tilde{\beta}}\int_{\partial\Omega}\beta(x)u(x)^{p-1}\,d\mathcal{H}^{n-1}(x)$
$\displaystyle\geq
u_{m}^{p-1}\int_{\partial\Omega}{\left|x\right|}^{\frac{\ell}{p^{\prime}}}\,d\mathcal{H}^{n-1}(x)$
$\displaystyle=u_{m}^{p-1}P_{\frac{\ell}{p^{\prime}}}(\Omega)\geq
u_{m}^{p-1}P_{\frac{\ell}{p^{\prime}}}(\Omega^{\sharp}).$

Gathering the inequalities above, we conclude that

$v_{m}^{p-1}P_{\frac{\ell}{p^{\prime}}}(\Omega^{\sharp})\geq
u_{m}^{p-1}P_{\frac{\ell}{p^{\prime}}}(\Omega^{\sharp})$

and the claim follows.

Furthermore inequality (2.8) implies that

$\mu(t)\leq\phi(t)={\left|\Omega\right|}_{\ell}\quad\forall t\leq v_{m}.$
(2.9)

###  2.1 Some useful Lemmata

Let $u$ be the solution to (1.1). For $t\geq 0$, we define

$U_{t}=\left\\{x\in\Omega:u(x)>t\right\\}\quad\partial U_{t}^{int}=\partial
U_{t}\cap\Omega,\quad\partial U_{t}^{ext}=\partial U_{t}\cap\partial\Omega,$

and

$\mu(t)={\left|U_{t}\right|}_{\ell}\quad
P_{u}(t)=P_{\frac{\ell}{p^{\prime}}}(U_{t}).$

Analogously, if $v$ is the solution to (1.3), we set

$V_{t}=\\{x\in\Omega^{\sharp}:v(x)>t\\},\quad\phi(t)={\left|V_{t}\right|}_{\ell},\quad
P_{v}(t)=P_{\frac{\ell}{p^{\prime}}}(V_{t}).$

As it is easy to check $v(x)\equiv v^{\sharp}(x)$, therefore for $0\leq t\leq
v_{m}$, $V_{t}=\Omega^{\sharp}$, while, for $t>v_{m}$ $V_{t}$ is a centered
ball contained in $\Omega^{\sharp}$.

######  Lemma 2.5 (Gronwall).

Let $\xi(t):[\tau_{0},+\infty[\,\to\operatorname{\mathbb{R}}$ be a continuous
and differentiable function satisfying, for some non negative constant $C$,
the following differential inequality

$\tau\xi^{\prime}(\tau)\leq(p-1)\xi(\tau)+C\quad\forall\tau\geq\tau_{0}>0.$
(2.10)

Then we have

  * (i)

$\displaystyle{\xi(\tau)\leq\left(\xi(\tau_{0})+\frac{C}{p-1}\right)\left(\frac{\tau}{\tau_{0}}\right)^{p-1}-\frac{C}{p-1}\quad\forall\tau\geq\tau_{0}}$;

  * (ii)

$\displaystyle{\xi^{\prime}(\tau)\leq\left(\frac{(p-1)\xi(\tau_{0})+C}{\tau_{0}}\right)\left(\frac{\tau}{\tau_{0}}\right)^{p-2}\quad\forall\tau\geq\tau_{0}}$.

###### Proof.

Dividing both sides of the differential inequality (2.10) by $\tau^{p}$, we
get

$\frac{\xi^{\prime}(\tau)}{\tau^{p-1}}-(p-1)\frac{\xi(\tau)}{\tau^{p}}=\left(\frac{\xi(\tau)}{\tau^{p-1}}\right)^{\prime}\leq\frac{C}{\tau^{p}}.$

Integrating the last inequality on $(\tau_{0},\tau)$ we obtain

$\int_{\tau_{0}}^{\tau}\left(\frac{\xi(\tau)}{\tau^{p-1}}\right)^{\prime}\,d\tau\leq\int_{\tau_{0}}^{\tau}\frac{C}{\tau^{p}}\,d\tau,$

and therefore

$\xi(\tau)\leq\left(\xi(\tau_{0})+\frac{C}{p-1}\right)\left(\frac{\tau}{\tau_{0}}\right)^{p-1}-\frac{C}{p-1},$

which gives _(i)_.

The second statement of the Lemma, claim _(ii)_ , is a direct consequence of
the first one. ∎

######  Lemma 2.6.

Let $u$ and $v$ be the solutions to (1.1) and (1.3) respectively. Then, for
almost every $t>0$, it holds

$\begin{split}&\gamma_{n,\ell,p}\,\mu(t)^{\left(\frac{\ell(p-1)+p(n-1)}{p(\ell+n)}\right)\frac{p}{p-1}}\leq\\\
&\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t)+\frac{1}{\tilde{\beta}^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u}\,d\mathcal{H}^{n-1}(x)\right),\end{split}$
(2.11)

where

$\gamma_{n,\ell,p}=(n\omega_{n})^{\frac{\ell+p}{p(n+\ell)}}(\ell+n)^{\frac{(p-1)\ell+(n-1)p}{p(n+\ell)}}$

and

$\begin{split}&\gamma_{n,\ell,p}\,\phi(t)^{\left(\frac{\ell(p-1)+p(n-1)}{p(\ell+n)}\right)\frac{p}{p-1}}=\\\
&\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\phi^{\prime}(t)+\frac{1}{\tilde{\beta}^{\frac{1}{p-1}}}\int_{\partial
V_{t}^{\text{ext}}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{v}\,d\mathcal{H}^{n-1}(x)\right).\end{split}$
(2.12)

###### Proof.

Let $t>0$ e $h>0$, we use the following test function in (1.2)

$\varphi(x)=\begin{cases}0&\text{ if }u<t\\\ u-t&\text{ if }t<u<t+h\\\
h&\text{ if }u>t+h.\end{cases}$

We get

$\begin{split}\int_{U_{t}\setminus U_{t+h}}{\left|\nabla
u\right|}^{p}\,dx&+h\int_{\partial
U_{t+h}^{ext}}\beta(x)u^{p-1}\,d\mathcal{H}^{n-1}(x)\\\ &+\int_{\partial
U_{t}^{ext}\setminus\partial
U_{t+h}^{ext}}\beta(x)u^{p-1}(u-t)\,d\mathcal{H}^{n-1}(x)\\\
&=\int_{U_{t}\setminus
U_{t+h}}f{\left|x\right|}^{\ell}(u-t)\,dx+h\int_{U_{t+h}}f{\left|x\right|}^{\ell}\,dx.\end{split}$

Dividing by $h$, using coarea formula and letting $h$ go to 0, we have for
a.e. $t>0$

$\int_{\partial
U_{t}}g(x)\,d\mathcal{H}^{n-1}(x)=\int_{U_{t}}f{\left|x\right|}^{\ell}\,dx,$

with

$g(x)=\begin{cases}{\left|\nabla u\right|}^{p-1}&\text{ if }x\in\partial
U_{t}^{int},\\\\[10.00002pt] \beta(x)u^{p-1}&\text{ if }x\in\partial
U_{t}^{ext}.\end{cases}$ (2.13)

Using isoperimetric inequality (2.4) and Hölder inequality, for a.e. $t>0$, we
obtain

$\begin{split}&\gamma_{n,\ell,p}\mu(t)^{\left(\frac{\ell(p-1)+p(n-1)}{p(\ell+n)}\right)}\\\
&\leq P_{u}(t)=\int_{\partial
U_{t}}{\left|x\right|}^{\frac{\ell}{p^{\prime}}}\,d\mathcal{H}^{n-1}(x)\\\
&\leq\left(\int_{\partial
U_{t}}g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
U_{t}}\frac{{\left|x\right|}^{\ell}}{g^{\frac{1}{p-1}}}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}.\end{split}$
(2.14)

Since, for a.e. $t>0$ it holds that

$-\mu^{\prime}(t)=\int_{\partial
U_{t}^{int}}\frac{{\left|x\right|}^{\ell}}{{\left|\nabla
u\right|}}\,d\mathcal{H}^{n-1},$

by (1.5), (2.7), (2.13) and (2.14), we get

$\displaystyle\gamma_{n,\ell,p}\mu(t)^{\left(\frac{\ell(p-1)+p(n-1)}{p(\ell+n)}\right)}$
$\displaystyle\leq\left(\int_{\partial
U_{t}}\\!\\!\\!g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\\!\left(\int_{\partial
U_{t}^{int}}\frac{{\left|x\right|}^{\ell}}{{\left|\nabla
u\right|}}\,d\mathcal{H}^{n-1}(x)+\frac{1}{\tilde{\beta}^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\\!\\!\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u}\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p}}\left(-\mu^{\prime}(t)+\frac{1}{\tilde{\beta}^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u}\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p^{\prime}}}$

for a.e. $t\in[0,\sup_{\Omega}u)$.

Then (2.11) follows. Replacing $u$ with $v$, the solution to (1.3), all the
inequalities are verified as equalities, so we obtain (2.12). ∎

######  Lemma 2.7.

Let $u$ and $v$ be the solutions to (1.1) and (1.3) respectively. Then for all
$\tau\geq v_{m}$ we have

$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
U_{t}^{ext}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt\leq\frac{1}{p\tilde{\beta}}\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds$
(2.15)

and

$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
V_{t}\cap\partial\Omega^{\sharp}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt=\frac{1}{p\tilde{\beta}}\int_{0}^{\lvert\Omega^{\sharp}\rvert_{\ell}}f^{\ast}(s)\,ds.$
(2.16)

###### Proof.

Firstly we show (2.15). Fubini’s Theorem ensures that

$\begin{split}&\int_{0}^{\infty}\\!\\!\tau^{p-1}\left(\int_{\partial
U_{\tau}^{ext}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u(x)}d\mathcal{H}^{n-1}(x)\right)d\tau\\\
&=\int_{\partial\Omega}\left(\int_{0}^{u(x)}\frac{\tau^{p-1}}{u(x)}\,d\tau\right){\left|x\right|}^{\frac{\ell}{p^{\prime}}}\,d\mathcal{H}^{n-1}(x)\\\
&=\frac{1}{p}\int_{\partial\Omega}u(x)^{p-1}{\left|x\right|}^{\frac{\ell}{p^{\prime}}}\,d\mathcal{H}^{n-1}(x).\end{split}$
(2.17)

Choosing $\varphi\equiv 1$ as test function in (1.2) and using (1.5), we
obtain

$\int_{\partial\Omega}\tilde{\beta}{\left|x\right|}^{\frac{\ell}{p^{\prime}}}u^{p-1}\,d\mathcal{H}^{n-1}\leq\int_{\partial\Omega}\beta(x)u^{p-1}\,d\mathcal{H}^{n-1}=\int_{\Omega}f{\left|x\right|}^{\ell}\,dx.$
(2.18)

From (2.17) and (2.18) we immediately get

$\int_{0}^{\infty}\tau^{p-1}\left(\int_{\partial
U_{\tau}^{ext}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau\leq\frac{1}{p\tilde{\beta}}\int_{0}^{\lvert\Omega\rvert_{\ell}}f^{\ast}(s)\,ds,$

and, arguing in the same way, we deduce

$\int_{0}^{\infty}\tau^{p-1}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\frac{1}{p\tilde{\beta}}\int_{0}^{\lvert\Omega\rvert_{\ell}}f^{\ast}(s)\,ds.$

Hence for any $t\geq 0$, we have

$\int_{0}^{t}\tau^{p-1}\left(\int_{\partial
U_{\tau}^{ext}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau\leq\frac{1}{p\tilde{\beta}}\int_{0}^{\lvert\Omega\rvert_{\ell}}f^{\ast}(s)\,ds.$

Since $\partial V_{t}\cap\partial\Omega^{\sharp}$ is empty for any $t\geq
v_{m}$, we get

$\int_{0}^{t}\tau^{p-1}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\frac{1}{p\tilde{\beta}}\int_{0}^{\lvert\Omega\rvert_{\ell}}f^{\ast}(s)\,ds.$

Lemma 2.7 is hence proved. ∎

##  3 Main results

In this Section we prove Theorem 1.1 and Theorem 1.2.

###### Proof of Theorem 1.1.

Let

$\displaystyle{\delta_{1}=1-{{\left(\frac{\ell(p-1)+p(n-1)}{(p-1)(\ell+n)}\right).}}}$

Note that, since $p\geq n$, $\delta_{1}$ is a positive constant.

Multiplying both sides of (2.11) by $t^{p-1}\mu(t)^{\delta_{1}}$ and
integrating over $(0,\tau)$ with $\tau\geq v_{m}$, by Lemma 2.7, we have

$\begin{split}\gamma_{n,\ell,p}\int_{0}^{\tau}t^{p-1}\mu(t)\,dt&\leq\int_{0}^{\tau}\left(-\mu^{\prime}(t)\right)t^{p-1}\mu(t)^{\delta_{1}}\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,dt\\\
&+\frac{{\left|\Omega\right|}_{\ell}^{\delta_{1}}}{p\tilde{\beta}^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$
(3.1)

Setting

$\displaystyle{F(\ell)=\int_{0}^{\ell}\omega^{\delta_{1}}\left(\int_{0}^{\omega}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,d\omega},$

and, integrating by parts both sides of the last inequality, we obtain

$\begin{split}&\tau^{p-1}\left(\gamma_{n,\ell,p}\int_{0}^{\tau}\mu(t)\,dt+F(\mu(\tau))\right)\\\
&\leq(p-1)\int_{0}^{\tau}t^{p-2}\left(\gamma_{n,\ell,p}\int_{0}^{t}\mu(s)\,ds+F(\mu(t))\right)\,dt\\\
&+\frac{{\left|\Omega\right|}_{\ell}^{\delta_{1}}}{p\tilde{\beta}^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$

We observe that all the hypotheses of Gronwall’s Lemma 2.5 are satisfied with

$\xi(\tau)=\xi_{1}(\tau)=\int_{0}^{\tau}t^{p-2}\left(\gamma_{n,\ell,p}\int_{0}^{t}\mu(s)\,ds+F(\mu(t))\right)\,dt$

and

$C=C_{1}=\frac{{\left|\Omega\right|}_{\ell}^{\delta_{1}}}{p\tilde{\beta}^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.$

Applying such a Lemma with $\tau_{0}=v_{m}$, we infer that for any $\tau\geq
v_{m}$ it holds

$\tau^{p-2}\left(\gamma_{n,\ell,p}\int_{0}^{\tau}\mu(s)\,ds+F(\mu(\tau))\right)\leq\left(\frac{(p-1)\xi_{1}(v_{m})+C_{1}}{v_{m}}\right)\left(\frac{\tau}{v_{m}}\right)^{p-2}.$

Replacing $\mu(t)$ with $\phi(t)$ the previous inequality holds as an
equality. Therefore for any $\tau\geq v_{m}$ it holds that

$\gamma_{n,\ell,p}\int_{0}^{\tau}\mu(s)\,ds+F(\mu(\tau))\leq\gamma_{n,\ell,p}\int_{0}^{\tau}\phi(s)\,ds+F(\phi(\tau)).$
(3.2)

On the other hand, since
$\mu(t)\leq\phi(t)={\left|\Omega\right|}_{\ell},\,\forall t\leq v_{m}$, and
$F(\ell)$ is an increasing function, we have that for any $\tau\leq v_{m}$ it
holds that

$\gamma_{n,\ell,p}\int_{0}^{\tau}\mu(s)\,ds+F(\mu(t))\leq\gamma_{n,\ell,p}\int_{0}^{\tau}\phi(s)\,ds+F(\phi(t))$
(3.3)

Combining inequalities (3.2) and (3.3) we conclude that for any $\tau\geq 0$
it holds

$\gamma_{n,\ell,p}\int_{0}^{\tau}\mu(s)\,ds+F(\mu(\tau))\leq\gamma_{n,\ell,p}\int_{0}^{\tau}\phi(s)\,ds+F(\phi(\tau)).$

Since

$\lim_{\tau\rightarrow+\infty}F\left(\mu(\tau)\right)=\lim_{\tau\rightarrow+\infty}F\left(\phi(\tau)\right)=0,$

letting $\tau$ goes to $+\infty$, we conclude that

$\int_{0}^{\infty}\mu(t)\,dt\leq\int_{0}^{\infty}\phi(t)\,dt,$

and hence

${\left\|u\right\|}_{L^{1}(\Omega,{\left|x\right|}^{\ell}\,dx)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp},{\left|x\right|}^{\ell}\,dx)}.$

Now let us show (1.7). Firstly let us observe that it is enough to verify that

$\int_{0}^{\infty}t^{p-1}\mu(t)\,dt\leq\int_{0}^{\infty}t^{p-1}\phi(t)\,dt.$
(3.4)

Integrating by parts the first term on the right-hand side in (3.1), and then
letting $\tau$ go to $+\infty$, we obtain

$\gamma_{n,\ell,p}\int_{0}^{\infty}t^{p-1}\mu(t)\,dt\leq(p-1)\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt+\frac{{\left|\Omega\right|}_{\ell}^{\delta_{1}}}{p\tilde{\beta}^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}},$

and similarly we have

$\gamma_{n,\ell,p}\int_{0}^{\infty}t^{p-1}\phi(t)\,dt=\
(p-1)\int_{0}^{\infty}t^{p-2}F(\phi(t))\,dt+\frac{{\left|\Omega\right|}_{\ell}^{\delta_{1}}}{p\tilde{\beta}^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.$

Thus if we prove

$\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt\leq\int_{0}^{\infty}t^{p-2}F(\phi(t))\,dt$
(3.5)

we get the claim (3.4).

Let

$\theta_{1}=-\frac{\ell(p-1)+p(n-1)}{(p-1)(\ell+n)}.$

Multiplying both sides of (2.11) by
$\displaystyle{t^{p-1}F(\mu(t))\mu(t)^{\theta_{1}}}$ and then integrating over
$(0,\tau)$, taking into account that the function
$\displaystyle{h(\ell)=F(\ell)\ell^{\theta_{1}}}$ is non decreasing, we get

$\begin{split}&\gamma_{n,\ell,p}\int_{0}^{\tau}t^{p-1}F(\mu(t))\,dt\\\
&\leq\int_{0}^{\tau}\bigl{(}-\mu^{\prime}(t)\bigr{)}t^{p-1}\mu(t)^{\theta_{1}}F(\mu(t))\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\\!dt\\\
&+F({\left|\Omega\right|}_{\ell})\frac{{\left|\Omega\right|}_{\ell}^{\theta_{1}}}{p\tilde{\beta}^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$
(3.6)

Let

$C_{2}=F({\left|\Omega\right|}_{\ell})\frac{{\left|\Omega\right|}_{\ell}^{\theta_{1}}}{p\tilde{\beta}^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}_{\ell}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.$

Integrating by parts both sides of (3.6) we derive

$\begin{split}\gamma_{n,\ell,p}\,\tau\int_{0}^{\tau}t^{p-2}F(\mu(t))\,dt+\tau
H_{\mu}(\tau)&\leq\gamma_{n,\ell,p}\int_{0}^{\tau}\int_{0}^{t}r^{p-2}F(\mu(r))\,drdt\\\
&+\int_{0}^{\tau}H_{\mu}(t)\,dt+C_{2},\end{split}$ (3.7)

where

$H_{\mu}(\tau)=-\int_{\tau}^{+\infty}t^{p-2}\mu(t)^{\theta_{1}}F(\mu(t))\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\biggr{)}^{\frac{1}{p-1}}\,d\mu(t).$

Setting

$\begin{multlined}\xi_{2}(\tau)=\gamma_{n,\ell,p}\int_{0}^{\tau}\int_{0}^{t}r^{p-2}F(\mu(r))\,dr+\int_{0}^{t}H_{\mu}(t)\,dt,\end{multlined}\xi_{2}(\tau)=\gamma_{n,\ell,p}\int_{0}^{\tau}\int_{0}^{t}r^{p-2}F(\mu(r))\,dr+\int_{0}^{t}H_{\mu}(t)\,dt,$

then (3.7) reads as

$\tau\xi_{2}^{\prime}(\tau)\leq\xi_{2}(\tau)+C_{2}.$

Lemma 2.5, with $\tau_{0}=v_{m}$, ensures that the following inequality holds
true for any $\tau\geq v_{m}$

$\displaystyle\gamma_{n,\ell,p}\int_{0}^{\tau}t^{p-2}F(\mu(t))\,dt+H_{\mu}(\tau)$
$\displaystyle\leq\frac{\displaystyle{(p-1)\int_{0}^{v_{m}}t^{p-2}F(\mu(t))\,dt+H_{\mu}(v_{m})+C_{2}}}{v_{m}}\left(\frac{\tau}{v_{m}}\right)^{p-2}.$

The previous inequality becomes an equality if we replace $\mu(t)$ with
$\phi(t)$, so, recalling that $\mu(t)\leq\phi(t)={\left|\Omega\right|}_{\ell}$
for $t\leq v_{m}$, we obtain

$\gamma_{n,\ell,p}\int_{0}^{\tau}t^{p-2}F(\mu(t))\,dt+H_{\mu}(\tau)\leq\gamma_{n,\ell,p}\int_{0}^{\tau}t^{p-2}F(\phi(t))\,dt+H_{\phi}(\tau).$

Letting $\tau\to\infty$, we have

$\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt\leq\int_{0}^{\infty}t^{p-2}F(\phi(t))\,dt,$

since, as we will show,

$\lim_{\tau\to 0}H_{\mu}(\tau)=\lim_{\tau\to 0}H_{\phi}(\tau)=0.$ (3.8)

This proves (3.5), and hence (1.7).

To prove (3.8), recalling that $p\geq n$, we observe that

$t^{p-2}\mu(t)=\int_{\Set{u>t}}t^{p-2}{\left|x\right|}^{\ell}\,dx\leq\int_{\Set{u>t}}u^{p-2}{\left|x\right|}^{\ell}\,dx\leq{\left\|u\right\|}_{L^{p}(\Omega,{\left|x\right|}^{\ell}\,dx)}^{p-2}\,\mu(t)^{\frac{2}{p}},$

therefore

$\displaystyle{\left|H_{\mu}(\tau)\right|}$
$\displaystyle=\int_{\tau}^{+\infty}t^{p-2}F(\mu(t))\mu(t)^{\theta_{1}}\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\biggr{)}(-\mu^{\prime}(t))\,dt$
$\displaystyle\leq\biggl{(}\int_{0}^{{\left|\Omega\right|}_{\ell}}\\!f^{*}(s)\,ds\biggr{)}{\left\|u\right\|}_{L^{p}(\Omega,{\left|x\right|}^{\ell}dx)}^{p-2}\int_{\tau}^{+\infty}\\!\\!\\!F(\mu(t))\mu(t)^{\frac{2}{p}+\theta_{1}-1}(-\mu^{\prime}(t))\,dt.$

The claim follows by observing that the right-hand side of the above
inequality goes to $0$ as $\tau\to+\infty$.

∎

######  Remark 3.1.

We emphasize that, with the same argument in [ANT, AGM22], it is possible to
obtain a stronger comparison in Theorem 1.1. It can be shown

${\left\|u\right\|}_{L^{k,1}(\Omega,{\left|x\right|}^{\ell}\,dx)}\,\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp},{\left|x\right|}^{\ell}\,dx)}\qquad\forall\,0<k\leq\frac{(\ell+n)(p-1)}{\ell(p-1)+p(n-1)}$

and

${\left\|u\right\|}_{L^{pk,p}(\Omega,{\left|x\right|}^{\ell}\,dx)}\,\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp},{\left|x\right|}^{\ell}\,dx)}\qquad\forall\,0<k\leq\frac{(\ell+n)(p-1)}{\ell(p-1)+p(n-2)+n},$

where ${\left\|\cdot\right\|}_{L^{p,q}(\Omega,{\left|x\right|}^{\ell}\,dx)}$
is defined in (2.6).

###### Proof of Theorem 1.2.

. Let

$\theta_{2}=\frac{\ell(p-1)+p(n-1)}{p(\ell+n)}$

and

$\delta_{2}=-\left(\theta_{2}-\frac{1}{p}\right)\frac{p}{p-1}.$

We point out that $\delta_{2}\geq 0$ by assumption (1.8).

Obviously in this case we have

$\displaystyle{\int_{0}^{\mu(t)}f^{\ast}(s)\,ds=\mu(t)}.$

Hence (2.11) can be written as

$\gamma_{n,\ell,p}\,\mu(t)^{\left(\theta_{2}-\frac{1}{p}\right)\frac{p}{p-1}}\leq-\mu^{\prime}(t)+\frac{1}{\tilde{\beta}^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{{\left|x\right|}^{\frac{\ell}{p^{\prime}}}}{u}\,d\mathcal{H}^{n-1}(x).$
(3.9)

Multiplying both sides of (3.9) by $t^{p-1}\mu(t)^{\delta_{2}}$ and, then,
integrating from $0$ to $\tau\geq v_{m}$, we obtain

$\begin{split}\gamma_{n,\ell,p}\int_{0}^{\tau}t^{p-1}\,dt&\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta_{2}}(-\mu^{\prime}(t))\,dt\\\
&+\frac{1}{\tilde{\beta}^{\frac{1}{p-1}}}\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta_{2}}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\\\
&\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta_{2}}(-\mu^{\prime}(t))\,dt+\frac{{\left|\Omega\right|}_{\ell}^{1+\delta_{2}}}{p\tilde{\beta}^{\frac{p}{p-1}}}.\end{split}$
(3.10)

Taking into account of Lemma 2.7, if we replace $\mu(t)$ with $\phi(t)$, the
previous inequality holds as equality. Hence, we get

$\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta_{2}}(-\mu^{\prime}(t))\,dt\geq\int_{0}^{\tau}t^{p-1}\phi(t)^{\delta_{2}}(-\phi^{\prime}(t))\,dt.$

In turn an integration by parts yields

$\begin{split}&-\tau^{p-1}\,\frac{\mu(\tau)^{1+\delta_{2}}}{1+\delta_{2}}+(p-1)\int_{0}^{\tau}t^{p-2}\,\frac{\mu(t)^{1+\delta_{2}}}{1+\delta_{2}}\,dt\geq\\\
&-\tau^{p-1}\,\frac{\phi(\tau)^{1+\delta_{2}}}{1+\delta_{2}}+(p-1)\int_{0}^{\tau}t^{p-2}\,\frac{\phi(t)^{1+\delta_{2}}}{1+\delta_{2}}\,dt.\end{split}$

The above inequality allows us to use claim (ii) of the Gronwall’s Lemma, this
time to the function

$\xi_{3}(\tau)=\int_{0}^{\tau}s^{p-2}\left(\frac{\mu(s)^{1+\delta_{2}}-\phi(s)^{1+\delta_{2}}}{1+\delta_{2}}\right)\,ds.$

Hence for any $\tau\geq v_{m}$ it holds

$\tau^{p-2}\left(\frac{\mu^{1+\delta_{2}}(\tau)-\phi^{1+\delta_{2}}(\tau)}{1+\delta_{2}}\right)\leq(p-1)\frac{\tau^{p-2}}{v_{m}^{p-2}}\int_{0}^{v_{m}}\\!\\!\\!s^{p-2}\left(\frac{\mu^{1+\delta_{2}}(s)-\phi^{1+\delta_{2}}(s)}{1+\delta_{2}}\right)ds.$

Inequality (2.8) ensures us that the right-hand side of the inequality above
is non-positive, therefore

$\mu(\tau)\leq\phi(\tau)\quad\forall\tau\geq v_{m}.$

Finally, since by (2.8) we have

$\mu(\tau)\leq\phi(\tau)={\left|\Omega\right|}_{\ell}\quad\forall\tau\leq
v_{m},$

and, therefore, the inequality above holds true in $\left[0,+\infty\right)$.
Claim (1.9) is hence proven.

∎

##  4 Some applications

In this section we give some applications of our previous results. First of
all, as already mentioned in the introduction, it is possible to derive a
Faber-Krahn inequality for the Robin $p$-Laplacian operator with non-constant
boundary parameter.

###### Proof of Theorem 1.3.

Let $u$ be a positive minimizer of (1.10), then

$\begin{cases}-\Delta_{p}u=\lambda_{1,\beta}(\Omega)\,u^{p-1}{\left|x\right|}^{\ell}&\text{
in }\Omega\\\\[3.00003pt] {\left|\nabla
u\right|}^{p-2}\displaystyle{\frac{\partial
u}{\partial\nu}}+\beta(x)u^{p-1}=0&\text{ on }\partial\Omega.\end{cases}$

Now, let $z$ be a solution to the following problem

$\begin{cases}-\Delta_{p}z=\lambda_{1,\beta}(\Omega)\,(u^{\sharp})^{p-1}\,{\left|x\right|}^{\ell}&\text{
in }\Omega^{\sharp}\\\\[3.00003pt] {\left|\nabla
z\right|}^{p-2}\displaystyle{\frac{\partial
z}{\partial\nu}}+\tilde{\beta}(r^{\sharp})^{\frac{\ell}{p^{\prime}}}z^{p-1}=0&\text{
on }\partial\Omega^{\sharp}.\end{cases}$ (4.1)

Therefore, Theorem 1.1 ensures that

$\int_{\Omega}u^{p}{\left|x\right|}^{\ell}\,dx=\int_{\Omega^{\sharp}}(u^{\sharp})^{p}{\left|x\right|}^{\ell}\,dx\leq\int_{\Omega^{\sharp}}z^{p}{\left|x\right|}^{\ell}\,dx,$

while Hölder inequality gives

$\displaystyle\int_{\Omega^{\sharp}}(u^{\sharp})^{p-1}z{\left|x\right|}^{\ell}\,dx$
$\displaystyle\leq\left(\int_{\Omega^{\sharp}}(u^{\sharp})^{p}{\left|x\right|}^{\ell}\,dx\right)^{\frac{p-1}{p}}\left(\int_{\Omega^{\sharp}}z^{p}{\left|x\right|}^{\ell}\,dx\right)^{\frac{1}{p}}$
$\displaystyle\leq\int_{\Omega^{\sharp}}z^{p}{\left|x\right|}^{\ell}\,dx.$

Hence, writing $\lambda_{1,\beta(x)}(\Omega)$ according to (4.1), we get

$\begin{split}\lambda_{1,\beta(x)}(\Omega)&=\frac{\displaystyle{\int_{\Omega^{\sharp}}{\left|\nabla
z\right|}^{p}\,dx+\int_{\partial\Omega^{\sharp}}\tilde{\beta}(r^{\sharp})^{\frac{\ell}{p^{\prime}}}z^{p}\
\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega^{\sharp}}(u^{\sharp})^{p-1}z{\left|x\right|}^{\ell}\,dx}}\\\
&\geq\frac{\displaystyle{\int_{\Omega^{\sharp}}{\left|\nabla
z\right|}^{p}\,dx+\int_{\partial\Omega^{\sharp}}\tilde{\beta}(r^{\sharp})^{\frac{\ell}{p^{\prime}}}z^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega^{\sharp}}z^{p}{\left|x\right|}^{\ell}\,dx}}\geq\lambda_{1,\tilde{\beta}}(\Omega^{\sharp}).\end{split}$

∎

##  5 Appendix

Here we provide the proofs of Propositions 2.3 and 2.4 just for $p=n$, since
the remaining cases can be treated analogously.

###### Proof of Proposition 2.3.

We recall that, as well-known, $W^{1,n}(\Omega)$ is compactly embedded in
$L^{q}(\Omega)$ for every $q$ finite. Then Hölder inequality gives

$\left(\int_{\Omega}{\left|\psi\right|}^{n}{\left|x\right|}^{\ell}\right)^{\frac{1}{n}}\leq\left(\int_{\Omega}{\left|\psi\right|}^{q}\right)^{\frac{1}{q}}\left(\int_{\Omega}{\left|x\right|}^{\frac{\ell}{n}s}\right)^{\frac{1}{s}}\quad\text{
where }\quad\frac{1}{n}=\frac{1}{q}+\frac{1}{s}.$

If $q\geq 0$, then $s\geq n$ and so

$q>\frac{n^{2}}{n-{\left|\ell\right|}}\quad\implies\quad\frac{{\left|\ell\right|}}{n}s<n.$

Hence

${\left\|\psi\right\|}_{L^{n}(\Omega,{\left|x\right|}^{\ell}\,dx)}\leq
D_{1}{\left\|\psi\right\|}_{L^{q}(\Omega)}\leq
C_{1}{\left\|\psi\right\|}_{W^{1,n}(\Omega)},$

and, therefore, the embedding is at least continuous.

Now, if we take a sequence $\\{\psi_{k}\\}_{k}$ bounded in $W^{1,n}(\Omega)$,
by the compact embedding of $W^{1,n}(\Omega)$ in $L^{q}(\Omega)$, up to a
subsequence, $\\{\psi_{k}\\}_{k}$ strongly converges in $L^{q}(\Omega)$. The
continuous embedding of $L^{q}(\Omega)$ in
$L^{n}(\Omega,{\left|x\right|}^{\ell}\,dx)$ guarantees us the strong
convergence in the second space as well. ∎

###### Proof of Proposition 2.4.

The solution to (1.1) is the unique minimum of the functional

$\begin{split}G:\psi\in
W^{1,n}(\Omega)&\rightarrow\frac{1}{n}\int_{\Omega}{\left|\nabla\psi\right|}^{n}\,dx+\frac{1}{n}\int_{\partial\Omega}\beta(x){\left|\psi\right|}^{n}\,d\mathcal{H}^{n-1}(x)\\\
&-\int_{\Omega}f\psi{\left|x\right|}^{\ell}\,dx.\end{split}$ (5.1)

Existence can be achieved by means of direct methods of calculus of variation,
see for instance [Dac08, Giu94, Lin06, AGM22], while one can prove uniqueness
by using the same arguments contained in [BK02, DS87].

Here we just recall the proof of the existence.

First of all we notice that functional $G$ is bounded from below. Indeed by
(1.5) and since $0\not\in\partial\Omega$, we have

$\displaystyle G(\psi)$
$\displaystyle=\frac{1}{n}\int_{\Omega}{\left|\nabla\psi\right|}^{n}\,dx+\frac{1}{n}\int_{\partial\Omega}\beta(x){\left|\psi\right|}^{n}\,d\mathcal{H}^{n-1}(x)-\int_{\Omega}f\psi{\left|x\right|}^{\ell}\,dx$
$\displaystyle\geq\frac{1}{n}\int_{\Omega}{\left|\nabla\psi\right|}^{n}\,dx+C\int_{\partial\Omega}{\left|\psi\right|}^{n}-\int_{\Omega}f\psi{\left|x\right|}^{\ell}\,dx$
$\displaystyle\geq
C{\left\|\psi\right\|}_{W^{1,n}(\Omega)}^{n}-\int_{\Omega}f\psi{\left|x\right|}^{\ell}\,dx,$

where in the last inequality we have used the well-known trace embedding
Theorem. Using Hölder inequality and Proposition 2.3, we have

$G(\psi)\geq
C{\left\|\psi\right\|}_{W^{1,n}(\Omega)}^{n}-{\left\|\psi\right\|}_{W^{1,n}(\Omega)}{\left\|f\right\|}_{L^{n^{\prime}}(\Omega,{\left|x\right|}^{\ell}\,dx)}$

By Young parametric inequality we finally get

$\displaystyle
G(\psi)\geq\left(C-\frac{\varepsilon^{n}}{n}\right){\left\|\psi\right\|}_{W^{1,n}(\Omega)}^{n}-\frac{C}{\varepsilon^{n^{\prime}}n^{\prime}}{\left\|f\right\|}_{L^{n^{\prime}}(\Omega,{\left|x\right|}^{\ell}\,dx)}$
(5.2)

For $\varepsilon$ small enough we have

$G(\psi)\geq-\frac{1}{\varepsilon
n^{\prime}}\,{\left\|f\right\|}_{L^{n^{\prime}}(\Omega,{\left|x\right|}^{\ell}\,dx)}>-\infty.$

Now let $\Set{u_{k}}_{k}$ a minimizing sequence for $G$, we can assume,
without loss of generality, that $G(u_{k})\leq m_{G}+1$ where

$m_{G}=\inf_{u\in W^{1,n}(\Omega)}G(u).$

So, by (5.2) for a fixed $\varepsilon$ small, we have

$m_{G}+1+\frac{C}{\varepsilon^{n^{\prime}}n^{\prime}}{\left\|f\right\|}_{L^{n^{\prime}}(\Omega,{\left|x\right|}^{\ell}\,dx)}\geq\left(C-\frac{\varepsilon^{n}}{n}\right){\left\|u_{k}\right\|}_{W^{1,n}(\Omega)}^{n}.$

So the sequence $\Set{u_{k}}_{k}$ is bounded in $W^{1,n}(\Omega)$ hence, up to
a subsequence, it converges weakly in $W^{1,n}(\Omega)$ and strongly in
$L^{n}(\Omega,{\left|x\right|}^{\ell}\,dx)$ and in $L^{n}(\partial\Omega)$ to
a function $u$.

Thus, we have

$\displaystyle\liminf_{k}\int_{\Omega}{\left|\nabla u_{k}\right|}^{n}\,dx$
$\displaystyle\geq\int_{\Omega}{\left|\nabla u\right|}^{n}\,dx$
$\displaystyle\liminf_{k}\int_{\Omega}\beta(x)u_{k}^{n}\,d\mathcal{H}^{n-1}(x)$
$\displaystyle=\int_{\partial\Omega}\beta(x)u^{n}\,d\mathcal{H}^{n-1}(x)$
$\displaystyle\liminf_{k}\int_{\Omega}u_{k}f{\left|x\right|}^{\ell}\,dx$
$\displaystyle=\int_{\Omega}uf{\left|x\right|}^{\ell}\,dx$

and finally

$m_{G}=\liminf_{k}G(u_{k})\geq G(u)=m_{G}.\qed$

Founding This work has been partially supported by the PRIN project 2017JPCAPN
(Italy) Grant: “Qualitative and quantitative aspects of nonlinear PDEs” and by
GNAMPA of INdAM.

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# A Dataset and BERT-based Models for Targeted Sentiment Analysis on Turkish
Texts

M. Melih Mutlu
Department of Computer Engineering
Boğaziçi University
<EMAIL_ADDRESS>
&Arzucan Özgür
Department of Computer Engineering
Boğaziçi University
<EMAIL_ADDRESS>

###### Abstract

Targeted Sentiment Analysis aims to extract sentiment towards a particular
target from a given text. It is a field that is attracting attention due to
the increasing accessibility of the Internet, which leads people to generate
an enormous amount of data. Sentiment analysis, which in general requires
annotated data for training, is a well-researched area for widely studied
languages such as English. For low-resource languages such as Turkish, there
is a lack of such annotated data. We present an annotated Turkish dataset
suitable for targeted sentiment analysis. We also propose BERT-based models
with different architectures to accomplish the task of targeted sentiment
analysis. The results demonstrate that the proposed models outperform the
traditional sentiment analysis models for the targeted sentiment analysis
task.

##  1 Introduction

Tweet |  Sentence
Sentiment |  Targeted
Sentiment
---|---|---
coca cola daha iyi lezzet olarak
(coca cola’s taste is better) |  positive |  positive
whatsapp çöktü de biraz rahatladım bildirimlerden kurtuldum
(whatsapp is crashed so I’m little relieved, got rid of notifications) |  positive |  negative
Table 1:  Sample tweets from the dataset. Targets are shown in italics.
Sentences are annotated with respect to overall sentence sentiment and
targeted sentiment which represent the sentiment towards the target. English
translations are provided in parenthesis.

The increasing availability of the Internet and the growing number of online
platforms allowed people to easily create online content. Because of the value
of mining the people’s opinions, the sentimental information contained in this
online data makes sentiment analysis (SA) an interesting topic. It is an area
that is attracting the attention not only of academic researchers, but also of
businesses and governments Birjali et al. (2021) and has become a rapidly
growing field, as evidenced by the number of recent SA papers published
Mäntylä et al. (2018).

The problem with traditional sentiment analysis is that it cannot capture the
different attitudes toward multiple aspects in a given text. For example, if
the given text is “Phones from this brand are great, but I don’t really like
their laptops”, the sentiment towards the two targets “phone” and “laptop” are
positive and negative, respectively. Traditional sentiment analysis methods
would not be able to detect this opposing sentiment for “phone” and “laptop”,
but would assign an overall sentiment for the text. Targeted Sentiment
Analysis (TSA) aims to overcome this challenge and extracts sentiment from a
given text with respect to a specific target. One of the challenges of TSA is
the lack of available datasets. Both TSA and SA require labeled datasets.
Collecting data from various sources and labeling them, which is mostly done
manually, is an expensive process. Although the number of datasets suitable
for SA has recently increased due to new studies in the SA area, not all SA
datasets are usable for TSA Pei et al. (2019). TSA requires more refined
datasets. The labels should reflect the sentiment toward targets rather than
the overall sentiment of the sentences.

English is the most studied language for sentiment analysis Dashtipour et al.
(2016). SA models that perform satisfactorily for English do not seem to
always work with similar performance for Turkish Kaya et al. (2012). In this
work, we create a manually annotated dataset from Twitter specifically labeled
for both traditional and targeted sentiment analysis in Turkish. Then, we
experiment with different model architectures for the Turkish TSA task.
Experimental results demonstrate that our techniques outperform traditional
sentiment analysis models.

###  1.1 Problem Definition

Let $E$ denotes all entities in a given document $D$ such that:

$\displaystyle D$ $\displaystyle=\\{w_{1},\dots,w_{k}\\}\text{ each }w\text{
is a word; }k\in\mathbb{Z^{+}}$ $\displaystyle E$
$\displaystyle=\\{e_{1},\dots,e_{l}\\}\text{ each }e\text{ is an entity;
}l\in\mathbb{Z^{+}}$ $\displaystyle T$
$\displaystyle=\\{t_{1},\dots,t_{m}\\}\text{ }t_{i}\text{ is a target;
}t_{i}\in E\text{; }m,i\in\mathbb{Z^{+}}$

The objective of targeted sentiment analysis is to find all sentiment
$(s_{i},t_{i})$ pairs in document $D$ where $t_{i}$ is a target from $T$ and
$s_{i}$ is the sentiment toward $t_{i}$.

##  2 Related Work

One of the challenges of targeted sentiment analysis is identifying contexts
associated with target words in the sentiment classification. Early methods
for understanding the relationship between the target and the rest of the
sentence rely on hand-crafted feature extractions and rule-based techniques
Ding et al. (2008); Jiang et al. (2011). Recurrent neural networks (RNN) have
been implemented for sentiment analysis in the recent years. It achieved
improved results compared to earlier methods Dong et al. (2014); Nguyen and
Shirai (2015); Baktha and Tripathy (2017). Two RNNs are used to obtain the
context from both left and right and combine the context knowledge in Tang et
al. (2016). Attention mechanisms are recently added into RNN-based methods to
model the connection between each word and the target Wang et al. (2016); Ma
et al. (2017); Zhang et al. (2020).

Vaswani et al. (2017) introduced the transformer architecture consisting of
encoder and decoder blocks based on self-attention layers. Bidirectional
Encoder Representations from Transformers (BERT) has been introduced and shown
to achieve the state-of-the-art in various NLP tasks in Devlin et al. (2019).
BERT has recently become a widely used approach for sentiment analysis in many
languages Sun et al. (2019); Li et al. (2019). Köksal and Özgür (2021) provide
a Twitter dataset in Turkish for sentiment analysis called BounTi. It consists
of Twitter data which are about predefined universities and manually annotated
by considering sentimental polarities towards these universities. They propose
a BERT model fine-tuned using the BounTi dataset to identify sentiment in
Turkish tweets.

##  3 Dataset

(a) [CLS] output

(b) Max-pooling outputs of target tokens

(c) The output of the first target marker

Figure 1: An overview of architectures to get and handle outputs from BERT

Twitter is a commonly used source of sentiment classification dataset in the
literature Jiang et al. (2011); Severyn and Moschitti (2015); Kruspe et al.
(2020). In this study, we also create a Twitter dataset with 3952 tweets whose
timestamps span a six-month period between January 2020 and June 2020. The
tweets are collected via the official Twitter API by separately searching our
6 targets selected from famous companies and brands.

This dataset is manually annotated with three labels, positive, negative, and
neutral. Two factors are considered in the annotation process, namely sentence
sentiment and targeted sentiment. Each tweet has the following two labels. The
sentence sentiment label expresses the overall sentiment of the sentence,
regardless of the target word, as in traditional sentiment analysis
techniques. On the other hand, the targeted sentiment label reflects the
sentiment for the target in that sentence. The collected tweets are annotated
separately by two annotators (one of the authors and a volunteer annotator)
who are native Turkish speakers. Cohen’s $\kappa$ Cohen (1960) is used to
demonstrate inter-annotator agreement and is calculated as 0.855. In case of
conflict between annotators, they re-evaluated the conflicting tweets. After
re-evaluation, tweets on which the annotators agree are retained and
conflicting tweets are removed from the dataset.

Table 1 shows example sentences from the dataset. The first tweet is a
positive comment about the target and the sentence is also positive overall.
The second tweet indicates a negative opinion about the target, since it has
stated as crashed, although the sentence expresses a positive situation
overall. Both sentence and targeted sentiment are the same for most of the
tweets as in the first example. Only in 21% of the tweets, targeted sentiment
differs from the overall sentence sentiment. This means that the rest of the
dataset is similar to a standard sentiment analysis dataset. The number of
negative tweets in the dataset is significantly higher than the number of
positive and neutral tweets for each target. The strikingly high number of
negative tweets may be caused by the tendency of customers to write a review
when they have had a bad experience. The total percentages of positive,
negative and neutral classes are 19%, 58% and 23%, respectively. The dataset
is randomly divided into train, test, and validation sets by 65%, 20% and 15%,
respectively. The distribution of labels for each subset is kept similar to
the distribution of labels for the entire dataset.

The dataset contains ungrammatical text, slang, acronyms, as well as special
Twitter characters. During pre-processing URLs and mentions (@) are deleted.
Hashtag signs (#) are removed, but hashtags are kept for two reasons: hashtags
have been shown to express sentiment Alfina et al. (2017); Celebi and Özgür
(2018) and some tweets contain the targets as hashtags.

##  4 Methodology

Baldini Soares et al. (2019) has introduced a novel method with transformer
structure in the field of relation extraction. The key idea behind this work
is to tag the entities with additional tokens before feeding the BERT model
with the input. Different combinations of input and output types are
evaluated. The best results are obtained when entity markers are added to the
input and when the output of the starting entity markers are taken as the
output from BERT. Motivated by the results of Baldini Soares et al.’s work,
this paper evaluates several BERT architectures with different input and
output techniques for the targeted sentiment analysis task.

Two input representation techniques are investigated. In the standard input
representation, the inputs are simply entered into the model without
modification. In the second input representation approach, the targets are
highlighted by adding additional special target tokens [TAR] at the beginnings
and ends of targets, as shown in Table 2. These target tokens are expected to
collect information about the target, just as the [CLS] token collects
knowledge about the entire sentence. The three approaches for outputs explored
in this study are shown in Figure 1. The [CLS] output approach uses only the
output of the first token from the last hidden state of BERT, as proposed for
classification in the original paper Devlin et al. (2019). In the second
approach, the outputs of the tokens originating from the target, including the
outputs of the [TAR] tokens, are max-pooled. The first target marker approach
considers only the output of the first [TAR] token in the input instead of the
output of the standard [CLS]. All output approaches utilize a softmax layer at
the end for classification.

Tweets with [TAR] tokens
---
[TAR]whatsapp[TAR] çöktü de biraz rahatladım bildirimlerden kurtuldum
([TAR]whatsapp[TAR] is crashed so I’m little relieved, got rid of
notifications)
[TAR]coca cola[TAR] daha iyi lezzet olarak
([TAR]coca cola[TAR]’s taste is better)
Table 2:  Example tweets with target marker representation

###  4.1 Model Descriptions

First, two baseline models are defined in order to show the drawbacks of the
traditional SA models. One baseline is the BERT-based BounTi model Köksal and
Özgür (2021). The second baseline is also a BERT-based traditional SA model,
but fine-tuned with our new dataset using sentence sentiment. Both have
similar architectures and use the [CLS] output for sentiment classification.

Four other variants of BERT-based models are proposed for targeted sentiment
analysis. T-BERT is a model with a similar architecture to our baseline
models. It makes no changes to the input and takes its output from the [CLS]
token. The main difference is that targeted sentiment labels are used in the
training phase. Therefore, the model is trained to learn targeted sentiment,
whereas the baseline models are not aware of the target. T-BERTmarked employs
only the target marker representation on top of T-BERT and adds [TAR] tokens
into the input. [TAR] token is introduced to BERT’s tokenizer and the
vocabulary is resized. Hence, the tokenizer accepts [TAR] as one of its
special tokens such as [SEP]. T-BERTmarked-MP is another model with target
marker representation, additionally it max-pools all outputs of target tokens.
T-BERTmarked-TS also utilizes target markers. However, it takes its output
only from the first target token [TAR] unlike T-BERTmarked-MP.

In the training phase of all models, BERTurk Schweter (2020) is chosen as the
base BERT model. Class weights are set inversely proportional to the class
distribution to reduce the effects of an unbalanced data set. The batch size
is chosen as 24. Hyperparameters like weight decay, learning rate, and warm-up
steps are selected as 0.1, $1e-5$, and $300$ respectively. As optimizer, AdamW
is used.

##  5 Results

All proposed BERT variants and baselines are evaluated for targeted sentiment
analysis over our introduced dataset. Macro averaged F1-Score is used as the
evaluation metric in these experiments. The results are presented in Table 3.
All targeted BERT variants outperform both baseline models for TSA.
T-BERTmarked-MP achieves the best results with 67% F1-score, while T-BERT is
relatively the worst performing targeted model with 61% F1-score.
T-BERTmarked-TS and T-BERTmarked obtain performance quite close to each other,
the difference between those models is insignificant. They both have
approximately 65% F1-scores.

Model |  F1-Score
---|---
Baseline Model |  0.591
BounTi Model |  0.498
T-BERT |  0.610
T-BERTmarked |  0.659
T-BERTmarked-TS |  0.653
T-BERTmarked-MP |  0.669
Table 3:  Performance of all models for TSA with test dataset against targeted sentiment labels  Model |  F1-Score
---|---
Baseline Model |  0.256
BounTi Model |  0.233
T-BERT |  0.401
T-BERTmarked |  0.428
T-BERTmarked-TS |  0.459
T-BERTmarked-MP |  0.444
Table 4:  Performance of all models for TSA with data whose targeted and
sentence sentiment are different.

Only 21% of the dataset has different sentence and targeted sentiment. These
portion of data can demonstrate the distinction between targeted and sentence
sentiment classification better. If both labels are the same, then traditional
SA models may seem to accurately predict targeted sentiment. However, such
sentences do not show how accurate the predictions from neither TSA nor SA
models are. For this reason, a subset of our dataset such that all sentences
have different targeted and sentence sentiment is used for another round of
experiments. Table 4 shows the results for the TSA task with this subset.
Baseline models’ F1-score decreases dramatically to 25%, and it’s 23% for
BounTi model. Targeted BERT model with the lowest score (40% F1-score)
outperforms both models. T-BERTmarked-TS achieves better targeted sentiment
predictions with 46% F1-score. T-BERTmarked-TS improves the baseline
performance by 79% on F1-score.

##  6 Discussion

Our results suggest that target oriented models can significantly improve the
performance for targeted sentiment analysis. BERT architectures that perform
successfully in the relation extraction field are shown to be successful for
the targeted sentiment analysis task. Target markers make BERT models
understand target related context better compared to the [CLS] token. All
three models with target markers outperform the baselines and T-BERT. Hence,
adding target markers is an effective approach for improving TSA performance.

T-BERTmarked-TS and T-BERTmarked-MP are shown to perform slightly better than
the other target oriented models. The common aspect of these models, apart
from the target tokens, is that they both focus on the outputs of the target-
related tokens rather than the [CLS] tokens. Therefore, it can be concluded
that target outputs improves the performance for the TSA task.

We only considered one target in each sentence and annotated according to that
target. Other targets in the sentence, if any, are ignored. Multiple targets
with conflicting targeted sentiment in the same sentence can be a problem to
consider. There are cases where a sentence has more than one target, and each
target has a different targeted sentiment. For example, in a comparison, the
sentiment toward one target may actually depend on the sentiment of another
target in the same sentence. In this work, the scope is limited to only one
target in each sentence. Target markers are also used only for this one target
in the sentence and other possible targets are ignored. The lack of proper
treatment of such cases in this work may affect the performance of all models.

Sentence and targeted sentiment are identical for 79% of the dataset. Thus, if
a traditional SA model, which is designed to predict the overall sentence
sentiment, is used for the TSA task, its success for this task would be
overestimated. The results demonstrate that targeted sentiment analysis models
perform significantly better than traditional sentiment analysis models on the
TSA task. However, the performance of the TSA models increases when they are
tested on the entire test dataset, rather than on a subset containing only
tweets with different sentence and targeted sentiment labels. This highlights
that they may still be biased in favor of sentence sentiment to some extent.

##  7 Ethical Considerations and Limitations

The dataset contains public tweets in Turkish that are provided by the
official Twitter API for research. Only tweet ID’s and labels of the tweets
are shared publicly to follow Twitter’s terms and conditions. The annotators
have no affiliation with any of the companies that are used as targets in the
dataset, so there is no potential bias due to conflict of interest.

The models developed in this work are not yet satisfactory to use their
results without human monitoring. It is recommended to manually check the
predictions of these models before using them.

##  8 Conclusion and Future Work

We presented a manually annotated Turkish Twitter dataset specifically created
for targeted sentiment analysis and is also suitable for the traditional
sentiment analysis task. This allowed us to develop and evaluate novel models
for targeted sentiment analysis in a low-resource language such as Turkish.

We adapted and investigated BERT-based models with different architectures for
targeted sentiment analysis. Experiments show significant improvement on
baseline performance.

As future work, we plan to expand our dataset so that it contains more
sentences with different sentence and targeted sentiment. Moreover, novel
methods for sentences with multiple targets will be investigated.

## Acknowledgements

We would like to thank Abdullatif Köksal for helpful discussions and Merve
Yılmaz Mutlu for annotations. GEBIP Award of the Turkish Academy of Sciences
(to A.Ö.) is gratefully acknowledged.

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