# A Talenti comparison result for a class of Neumann
boundary value problems

A. Celentano, C. Nitsch, C. Trombetti

###### Abstract

In this paper, we establish a comparison principle in terms of Lorentz norms
and point-wise inequalities between a positive solution $u$ to the Poisson
equation with non-homogeneous Neumann boundary conditions and a specific
positive solution $v$ to the Schwartz symmetrized problem, which is related to
$u$ through an additional boundary condition.

MSC 2020: 35B09; 35B51; 35D30; 35J05.

Keywords: Talenti comparison, Poisson equation, Neumann conditions.

E-mail address, A. Celentano<EMAIL_ADDRESS>
E-mail address, C. Nitsch<EMAIL_ADDRESS>
E-mail address, C. Trombetti<EMAIL_ADDRESS>
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.
Angelo, 80126 Napoli, Italy.

##  1 Introduction

In the paper [20], Talenti proved a key result in the context of
symmetrization techniques, which are tools used to deduce information about a
problem, through the study of a simplified one. Given an open and bounded set
$\Omega\subset\mathbb{R}^{n}$ and a function $f$ such that $f\in
L^{\frac{2n}{n+2}}(\Omega)$ if $n>2$, $f\in L^{p}(\Omega)$ for some $p>1$ if
$n=1$, a comparison principle is established between the solutions of the two
following problems

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega$},\\\ \\\
u=0&\mbox{on
$\partial\Omega$},\end{array}\right.\qquad\left\\{\begin{array}[]{ll}-\Delta
v=f^{\sharp}&\mbox{in $\Omega^{\sharp}$},\\\ \\\ v=0&\mbox{on
$\partial\Omega^{\sharp}$},\end{array}\right.$ (1)

where $\Omega^{\sharp}$ denotes the ball, centered at the origin, with the
same Lebesgue measure $\mathcal{L}^{n}$ as $\Omega$ and $f^{\sharp}$ is the
decreasing Schwarz rearrangement of $f$ (see Section 2). Specifically, Talenti
showed that $u^{\sharp}\leq v$ holds $\mathcal{L}^{n}$ almost everywhere in
$\Omega^{\sharp}$.

Afterwards, many generalizations in different directions appeared in the
literature. Under Dirichlet boundary conditions, nonlinear elliptic operators
are considered in [21], parabolic operators in [3], anisotropic elliptic
operators in [2] and higher order operators in [22, 6]. Different boundary
conditions were taken into account in [4], where the authors established a
comparison result between the Lorentz norms of the solutions of the elliptic
problems, obtained by replacing the Dirichlet conditions in (1) with a Robin
one. In the planar case, if $f\equiv 1$ in $\Omega$, they are able to recover
a point-wise inequality as the one proved by Talenti. The proof is based on
the isoperimetric inequality, rearrangement properties and a careful analysis
of superlevel sets of the solutions, which can touch the boundary in the case
of Robin conditions. Further extensions are possible. In [19], the anisotropic
case is treated; in [5], the authors deal with the $p$-Laplace operator; in
[8], with the Hermite operator and in [1], with mixed Robin and Dirichlet
boundary conditions.

In the end, we mention some results concerning the Neumann boundary
conditions. In [18], the authors proved a point-wise comparison between the
Schwarz rearrangements of positive and negative parts of a solution of an
elliptic problem with homogeneous Neumann condition and the solutions of two
symmetrized problems, both defined on a ball having half measure of $\Omega$,
with Dirichlet boundary conditions, obtained by taking the Schwarz
rearrangements of the positive and negative parts of $f$. Comparisons with
solutions to problems that are not obtained by symmetrizing the original PDE
are considered in [10, 11]. In [15], the author established a comparison
between the $L^{p}$ norm of a solution of Poisson equations with homogeneous
Neumann boundary conditions and a solution of the problem with cap-symmetrized
data when $\Omega$ is a ball or a spherical shell. In [16], similar results
are proved also for pure Robin condition and mixed Robin and Neumann boundary
conditions. Other kinds of rearrangements have been investigated, for
instance, in [14, 7]. Eventually, we mention the work [9], where comparison
principles for Neumann problems are established on Riemannian manifolds.

Using the methods presented in [4], we provide in this paper a comparison
result in terms of Lorentz norms (see Definition 2.2) and point-wise
inequalities between a positive solution $u$ to the Poisson equation with non-
homogeneous Neumann boundary conditions and a particular positive solution $v$
to the Schwartz symmetrized problem, which is linked to $u$ through an
appropriate boundary condition.

For $n\geq 2$, let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded and
connected set with Lipschitz boundary. Given $f\in L^{r}(\Omega)$ with
$r=\max\\{2,2^{-1}n\\}$, we consider the following problem

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega$},\\\ \\\
\displaystyle\frac{\partial u}{\partial\nu}=c&\mbox{on
$\partial\Omega$,}\end{array}\right.$ (2)

where $\nu$ denotes the outer unit normal to $\partial\Omega$ and $c$ is a
real constant. A function $u\in W^{1,2}(\Omega)$ is a weak solution to (2) if

$\int_{\Omega}\nabla
u\cdot\nabla\phi\,dx-c\int_{\partial\Omega}\phi\,d\mathcal{H}^{n-1}=\int_{\Omega}f\phi\,dx,\quad\forall\phi\in
H^{1}(\Omega).$ (3)

It is known [23, Proposition 7.7] that (2) admits weak solutions, all defined
modulo constants, if and only if the following compatibility condition is
satisfied

$c=-\frac{1}{\mathcal{H}^{n-1}(\partial\Omega)}\int_{\Omega}f\,dx.$ (4)

A Lipschitz domain satisfies the exterior cone property [13, Theorem 1.2.2.2],
then standard results [12, Theorems 8.22, 8.27] imply that a solution $u$ to
(2) is bounded on $\Omega$ and locally Holder continuous. Hence, it exists
$k_{0}\in\mathbb{R}$ such that for $k>k_{0}$, $u_{k}=u+k$ is a positive weak
solution to (2). We take into account the Schwartz symmetrized problem

$\left\\{\begin{array}[]{ll}-\Delta v=f^{\sharp}&\mbox{in
$\Omega^{\sharp}$},\\\ \\\ \displaystyle\frac{\partial
v}{\partial\nu}=c^{\star}&\mbox{on
$\partial\Omega^{\sharp}$},\end{array}\right.$ (5)

By applying rearrangement properties (see Section 2), we deduce from the
compatibility condition in problems (2) and (5) that

$c^{\star}=-\frac{1}{\mathcal{H}^{n-1}(\partial\Omega^{\sharp})}\int_{\Omega^{\sharp}}f^{\sharp}\,dx\leq-\frac{1}{\mathcal{H}^{n-1}(\partial\Omega^{\sharp})}\int_{\Omega}f\,dx=\frac{\mathcal{H}^{n-1}(\partial\Omega)}{\mathcal{H}^{n-1}(\partial\Omega^{\sharp})}c.$
(6)

From now on, we assume

$\displaystyle\int_{\Omega}f\,dx>0,$ (7)

which implies $c,c^{*}<0$. Moreover, the isoperimetric inequality ensures that
$c^{*}\leq c$. A solution to problem (5) is radial and non-increasing along
the radius. A comparison principle can be established between a positive weak
solution $u$ to (2) and the unique positive weak solution $v\in
W^{1,2}(\Omega^{\sharp})$ to problem (5) which satisfies one of the following
equalities

$\frac{c^{\star}}{c}\int_{\partial\Omega}u\,d\mathcal{H}^{n-1}=\int_{\partial\Omega^{\sharp}}v\,d\mathcal{H}^{n-1},$
(8)
$\frac{c^{\star}}{c}\int_{\partial\Omega}u^{2}\,d\mathcal{H}^{n-1}=\int_{\partial\Omega^{\sharp}}v^{2}\,d\mathcal{H}^{n-1}.$
(9)

For sign changing solutions, some counterexamples are possible (see Example 4
in Section 4).
We state our main results.

######  Theorem 1.1.

Let $u$ be a positive solution to problem (2). If $v$ is the positive solution
to problem (5) satisfying the equality (8), then

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{for all
}0<p\leq\frac{n}{2n-2};$ (10)

If $v$ is the positive solution to problem (5) satisfying the equality (9),
then

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{for all
}0<p\leq\frac{n}{2n-2},$ (11)
$\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\sharp})}\quad\mbox{for
all }0<p\leq\frac{n}{3n-4}.$ (12)

######  Theorem 1.2.

Let assume that $f\equiv 1$ in $\Omega$. Let $u$ be a positive solution to
problem (2). If $v$ is the positive solution to problem (5) satisfying the
equality (8), it holds

$u^{\sharp}(x)\leq v(x)\quad\forall\,x\in\>\mbox{in
$\Omega^{\sharp}$},\quad\>\mbox{for $n=2$}.$

When $n\geq 3$, we have

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{for all
}\>0<p\leq\frac{n}{n-2}.$

If $v$ is the positive solution to problem (5) satisfying the equality (9), it
holds

$u^{\sharp}(x)\leq v(x)\quad\forall\,x\in\>\mbox{in
$\Omega^{\sharp}$},\quad\>\mbox{for $n=2$}$

When $n\geq 3$, we have

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{and}\quad\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\sharp})}\quad\mbox{for
all }\>0<p\leq\frac{n}{n-2}.$

In the hypothesis of Theorem 1.1, when the dimension is $n=2$, we recover the
$L^{1}$ comparison under both conditions (8) and (9), but the $L^{2}$
comparison only under condition (9). A stronger result occurs in the case
$f\equiv 1$ in $\Omega$. From Theorem 1.2, we get
$\|u\|_{L^{1}(\Omega)}\leq\|v\|_{L^{1}(\Omega^{\sharp})}$ and
$\|u\|_{L^{2}(\Omega)}\leq\|v\|_{L^{2}(\Omega^{\sharp})}$ in any dimension.
The former holds true under both conditions (8) and (9), whereas the latter
only under condition (9).

This paper is organized as follows. In Section 2, we introduce definitions and
some properties of rearrangements. In Section 3, we prove Theorems 1.1 and
1.2. In Section 4, we discuss some generalizations and the optimality of our
results, providing several examples.

##  2 Notations and Preliminaries

We review the definitions of decreasing and Schwartz rearrangements, using
[17] as reference.

######  Definition 2.1.

Let $h:x\in\Omega\rightarrow[0,+\infty)$ be a measurable function, then the
decreasing rearrangement $h^{*}$ of $h$ is defined as

$h^{*}(s)=\inf\\{t\in\mathbb{R}:|\\{x\in\Omega:h(x)>t\\}|<s\\},\quad
s\in[0,|\Omega|].$

Let $\Omega^{\sharp}$ be the ball centered at the origin and having the same
measure as $\Omega$. The Schwartz rearrangement of $h$ is defined as follows

$h^{\sharp}(x)=h^{*}(\omega_{n}|x|^{n}),\quad x\in\Omega^{\sharp}.$

where $\omega_{n}$ is the measure of the unit ball in $\mathbb{R}^{n}$.

######  Remark 1.

Given a measurable function $h:\Omega\rightarrow\mathbb{R}$, we define the
decreasing rearrangement $h^{\star}$ of $h$ to be the decreasing rearrangement
of $|h|$, that is $h^{\star}:=(|h|)^{\star}$. In turn, we define the Schwartz
rearrangement $h^{\sharp}$ of $h$ to be the Schwartz rearrangement of $|h|$.

The functions $|h|$, $h^{*}$ and $h^{\sharp}$ are equi-distributed, that is

$|\\{x\in\Omega:|h(x)|>t\\}|=|\\{s\in(0,|\Omega|:h^{*}(s)>t\\}|=|\\{x\in\Omega^{\sharp}:h^{\sharp}(x)>t\\}|,\quad
t\in\mathbb{R}.$

As a consequence, if $h\in L^{p}(\Omega)$, $1\leq p\leq\infty$, then $h^{*}\in
L^{p}(0,|\Omega|)$, $h^{\sharp}\in L^{p}(\Omega^{\sharp})$ and

$||h||_{L^{p}(\Omega)}=||h^{*}||_{L^{p}(0,|\Omega|)}=||h^{\sharp}||_{L^{p}(\Omega^{\sharp})}.$

We shall make use of the following fundamental inequality involving
rearrangements, which is the Hardy-Littlewood inequality. Given $h\in
L^{p}(\Omega)$ and $g\in L^{q}(\Omega)$ such that $\frac{1}{p}+\frac{1}{q}=1$,
then

$\int_{\Omega}h(x)g(x)dx\leq\int_{0}^{|\Omega|}h^{*}(s)g^{*}(s)ds.$ (13)

The Hardy-Littlewood inequality is useful in evaluating the integral of a
function $h\in L^{p}(\Omega)$, $1\leq p\leq+\infty$ on the level sets of a
measurable function $u$, which is a common task when applying rearrangement
theory to the study of partial differential equations. By choosing
$g=\chi_{\\{u>t\\}}$ in (13), we get

$\int_{\\{u>t\\}}h(x)dx\leq\int_{0}^{\mu(t)}h^{\ast}(s)ds.$ (14)

If we take $h=u\geq 0$ in (14), it follows

$\int_{\\{u>t\\}}u(x)dx=\int_{0}^{\mu(t)}u^{\ast}(s)ds.$ (15)

For $U\subset\Omega$, we define

$\partial_{i}U:=\partial U\cap\Omega,\quad\partial_{e}U:=\partial
U\cap\partial\Omega.$

Let $u$ and $v$ be solutions to problems (2) and (5) respectively. For
$t\in\mathbb{R}$, we denote by

$U_{t}=\\{x\in\Omega:|u(x)|>t\\},\quad\mu(t)=|U_{t}|.$
$V_{t}=\\{x\in\Omega^{\sharp}:|v(x)|>t\\},\quad\phi(t)=|V_{t}|.$

Finally, we recall the definition of Lorentz space.

######  Definition 2.2.

For $0<p<\infty$ and $0<q\leq\infty$, the Lorentz space $L^{p,q}(\Omega)$
consists of all measurable functions $g$ in $\Omega$ such that it is finite
the quantity

$\|g\|_{L^{p,q}(\Omega)}=\left\\{\begin{array}[]{ll}\displaystyle
p^{\frac{1}{q}}\left(\int_{0}^{\infty}t^{q}\left|\left\\{x\in\Omega:|g(x)|>t\right\\}\right|^{\frac{q}{p}}\frac{dt}{t}\right)^{\frac{1}{q}}&0<q<\infty,\\\
\\\
\displaystyle\sup_{t>0}\left(t^{p}\left|\left\\{x\in\Omega:|g(x)|>t\right\\}\right|\right)&q=\infty.\end{array}\right.$

When $p=q$, the Lorentz space $L^{p,q}(\Omega)$ coincides with the
$L^{p}(\Omega)$ space and $\|g\|_{L^{p,p}(\Omega)}=\|g\|_{L^{p}(\Omega)}$.

##  3 Proof of the main Theorems

Several lemmas are necessary to provide proofs of Theorems (1.1) and (1.2).

######  Lemma 3.1 (Gronwall).

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

$\tau\xi^{\prime}(\tau)\leq\xi(\tau)+C\qquad\mbox{for all
$\tau\geq\tau_{0}>0$.}$

Then we have

  * (i)

$\xi(\tau)\leq\tau\frac{\xi(\tau_{0})+C}{\tau_{0}}-C\qquad\mbox{for all
$\tau\geq\tau_{0}$},$

  * (ii)

$\xi^{\prime}(\tau)\leq\frac{\xi(\tau_{0})+C}{\tau_{0}}\qquad\mbox{for all
$\tau\geq\tau_{0}$}.$

The following analysis of the superlevel sets of positive solutions to
problems (2) and (5), which may touch the boundary of $\Omega$ and
$\Omega^{\sharp}$ respectively, is critical for the comparison results.

######  Lemma 3.2.

Let $u$ and $v$ be positive solutions to problems (2) and (5) respectively.
For a.e. $t>0$, we have

$\gamma_{n}\phi(t)^{\frac{2n-2}{n}}=\left(-\phi^{\prime}(t)-\frac{1}{c}\int_{\partial_{e}V_{t}}d\mathcal{H}^{n-1}\right)\int_{0}^{\phi(t)}f^{*}(s)ds$
(16)

and

$\gamma_{n}\mu(t)^{\frac{2n-2}{n}}\leq\left(-\mu^{\prime}(t)-\frac{1}{c}\int_{\partial_{e}U_{t}}d\mathcal{H}^{n-1}\right)\int_{0}^{\mu(t)}f^{*}(s)\,ds,$
(17)

where $\gamma_{n}=n^{2}\omega_{n}^{-\frac{2}{n}}$.

###### Proof.

Given $t>0$ and $h>0$, we choose the test function in (3)

$\varphi_{h}^{i}(x)=\left\\{\begin{array}[]{ll}0&\mbox{if $0<u<t$},\\\ \\\
h&\mbox{if $u>t+h$},\\\ \\\ u-t&\mbox{if $t<u<t+h$}.\end{array}\right.$

Then,

$\begin{array}[]{ll}\displaystyle\int_{U_{t}\setminus U_{t+h}}|\nabla
u|^{2}\,dx-
ch\displaystyle\int_{\partial_{e}U_{t+h}}&d\mathcal{H}^{n-1}-c\displaystyle\int_{\partial_{e}U_{t}\setminus\partial_{e}U_{t+h}}(u-t)\,d\mathcal{H}^{n-1}=\\\
\\\ &\displaystyle\int_{U_{t}\setminus
U_{t+h}}f(u-t)\,dx+h\displaystyle\int_{U_{t+h}}f\,dx,\end{array}$

dividing by $h$ and letting $h$ go to $0$, an application of the coarea
formula shows that for a.e. $t>0$

$\int_{\partial U_{t}}g(x)\,d\mathcal{H}^{n-1}=\int_{\partial_{i}U_{t}}|\nabla
u|\,d\mathcal{H}^{n-1}-c\int_{\partial_{e}U_{t}}d\mathcal{H}^{n-1}=\int_{U_{t}}f\,dx,$

where

$g(x)=\left\\{\begin{array}[]{ll}|\nabla u|&\mbox{if
$x\in\partial_{i}U_{t}$,}\\\ \\\ -c&\mbox{if $x\in\partial_{e}U_{t}$.}\\\
\end{array}\right.$

Using the isoperimetric inequality, we get that for a.e. $t>0$

$\displaystyle\gamma_{n}\mu(t)^{\frac{2n-2}{n}}\leq\left(\mathcal{H}^{n-1}(U_{t})\right)^{2}=\left(\int_{\partial
U_{t}}\,d\mathcal{H}^{n-1}\right)^{2}$ $\displaystyle\leq\left(\int_{\partial
U_{t}}|g|\,d\mathcal{H}^{n-1}\right)\left(\int_{\partial
U_{t}}\frac{1}{|g|}\,d\mathcal{H}^{n-1}\right)$
$\displaystyle\leq\int_{U_{t}}f\,dx\left(-\mu^{\prime}(t)-\frac{1}{c}\int_{\partial_{e}U_{t}}d\mathcal{H}^{n-1}\right)\leq\int_{0}^{\mu(t)}f^{*}(s)\,ds\left(-\mu^{\prime}(t)-\frac{1}{c}\int_{\partial_{e}U_{t}}d\mathcal{H}^{n-1}\right)$

For the function $v$, all the previous inequalities hold as equalities. ∎

######  Remark 2.

From now on, we denote by

$v_{m}=\inf_{\Omega^{\sharp}}v,\>u_{m}=\inf_{\Omega}u.$

Assuming hypothesis (7), whether any condition between (8) and (9) is in
force, it holds

$u_{m}\leq v_{m},$ (18)

Let us assume hypothesis (8), then

$\frac{1}{c^{*}}v_{m}\mathcal{H}^{n-1}(\partial\Omega^{\sharp})=\frac{1}{c^{*}}\int_{\partial\Omega^{\sharp}}v(x)\,d\mathcal{H}^{n-1}=\frac{1}{c}\int_{\partial\Omega}u(x)\,d\mathcal{H}^{n-1}\leq\frac{1}{c}u_{m}\mathcal{H}^{n-1}(\partial\Omega)\leq
0.$

From the compatibility conditions in problems (2) and (5), we get

$\frac{c^{\star}}{c}=\frac{\mathcal{H}^{n-1}(\partial\Omega)}{\mathcal{H}^{n-1}(\partial\Omega^{\sharp})}\left(\int_{\Omega}f\,dx\right)^{-1}\int_{\Omega}|f|\,dx$

and (18) follows from the isoperimetric inequality. Assuming hypothesis (9), a
similar argument can be made. We also observe that

$\mu(t)\leq\phi(t)=|\Omega|\qquad\mbox{for all $0\leq t\leq v_{m}$.}$ (19)

As an important consequence of the reasoning above, inequality (19) holds
strictly for some $0\leq t\leq v_{m}$, unless $\Omega$ is a ball and $f$ is a
non negative function.

We can estimate the boundary integral on the right-hand side of (16) and (17),
using two fundamental lemmas.

######  Lemma 3.3.

Let $u$ and $v$ be positive solutions to problems (2) and (5) respectively.
For all $t\geq v_{m}$ we have

$\int_{0}^{t}\int_{\partial_{e}V_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau=\int_{\partial\Omega^{\sharp}}v(x)\,d\mathcal{H}^{n-1},$
(20)

while

$\int_{0}^{t}\int_{\partial_{e}U_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau\leq\int_{\partial\Omega}u(x)\,d\mathcal{H}^{n-1}.$
(21)

###### Proof.

By Fubini’s theorem, we have

$\displaystyle\int_{0}^{+\infty}\int_{\partial_{e}U_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau=\int_{\partial\Omega}\left(\int_{0}^{u(x)}\,d\tau\,\right)d\mathcal{H}^{n-1}=\int_{\partial\Omega}u(x)\,d\mathcal{H}^{n-1}.$

Analogously,

$\displaystyle\int_{0}^{+\infty}\int_{\partial_{e}V_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau=\int_{\partial\Omega^{\sharp}}\left(\int_{0}^{v_{m}}\,d\tau\,\right)d\mathcal{H}^{n-1}=\int_{\partial\Omega^{\sharp}}v(x)\,d\mathcal{H}^{n-1}.$

A trivial inequality for $t\geq 0$ is

$\int_{0}^{t}\int_{\partial_{e}U_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau\leq\int_{0}^{+\infty}\int_{\partial_{e}U_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau,$

while we observe that for $t\geq v_{m}=\min_{\partial\Omega^{\sharp}}v$ then
$\partial V_{t}\cap\partial\Omega^{\sharp}=\emptyset$ and

$\int_{0}^{t}\int_{\partial_{e}V_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau=\int_{0}^{+\infty}\int_{\partial_{e}V_{\tau}}\,d\mathcal{H}^{n-1}\,d\tau.$

∎

######  Lemma 3.4.

Let $u$ and $v$ be positive solutions to problems (2) and (5) respectively.
For all $t\geq v_{m}$ we have

$2\int_{0}^{t}\tau\left(\displaystyle\int_{\partial_{e}V_{\tau}}\>d\mathcal{H}^{n-1}\right)\,d\tau=\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1},$
(22)

while

$2\int_{0}^{t}\tau\left(\displaystyle\int_{\partial_{e}U_{\tau}}\>d\mathcal{H}^{n-1}\right)\,d\tau\leq\int_{\partial\Omega}u^{2}\,d\mathcal{H}^{n-1}.$
(23)

###### Proof.

By Fubini’s theorem, we have

$\begin{array}[]{ll}2\displaystyle\int_{0}^{\infty}\tau\left(\displaystyle\int_{\partial_{e}U_{\tau}}\>d\mathcal{H}^{n-1}\right)\,d\tau&=2\displaystyle\int_{\partial\Omega}\left(\int_{0}^{u(x)}\tau\>d\tau\,\right)d\mathcal{H}^{n-1}\\\
&=\displaystyle\int_{\partial\Omega}u^{2}(x)\>d\mathcal{H}^{n-1}.\end{array}$

Analogously,

$2\displaystyle\int_{0}^{\infty}\tau\displaystyle\int_{\partial_{e}V_{\tau}}\>d\mathcal{H}^{n-1}\,d\tau=\displaystyle\int_{\partial\Omega}v^{2}(x)\>d\mathcal{H}^{n-1}.$

Reasoning as above, we obtain the thesis. ∎

For simplicity, we set

$\displaystyle\gamma_{1}$
$\displaystyle=\int_{\partial\Omega}u\,d\mathcal{H}^{n-1},\qquad\gamma_{1}^{\star}=\int_{\partial\Omega^{\sharp}}v\,d\mathcal{H}^{n-1},$
$\displaystyle\gamma_{2}$
$\displaystyle=\int_{\partial\Omega}u^{2}\,d\mathcal{H}^{n-1},\qquad\gamma_{2}^{\star}=\int_{\partial\Omega^{\sharp}}v^{2}\,d\mathcal{H}^{n-1}.$

###### Proof of Theorem 1.1.

We start by proving inequality (10), assuming condition (8). Let
$0<p\leq\frac{n}{2n-2}$. Multiplying (17) by
$\mu(t)^{\frac{1}{p}-\frac{2n-2}{n}}$, integrating from $0$ to some $\tau\geq
v_{m}$ and applying Lemma 3.3, we deduce that

$\displaystyle\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{\tau}-\mu^{\prime}(t)\mu(t)^{\delta}\left(\int_{0}^{\mu(t)}f^{*}(s)\,ds\right)\,dt-\frac{\gamma_{1}|\Omega|^{\delta}}{c}\int_{0}^{|\Omega|}f^{*}(s)\,ds,$

where we have set $\displaystyle\delta=\frac{1}{p}-\frac{2n-2}{n}$. Using the
monotonicity of $\mu$ and applying a change of variables [17][Theorem 6.14,
Proposition 15.2], we get

$\displaystyle\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{p}}\,dt\leq\int_{\mu(\tau)}^{|\Omega|}w^{\delta}\left(\int_{0}^{w}f^{*}(s)ds\right)\,dw-\frac{\gamma_{1}|\Omega|^{\delta}}{c}\int_{0}^{|\Omega|}f^{*}(s)\,ds.$

Taking $\tau\to+\infty$, it follows

$\displaystyle\int_{0}^{+\infty}\gamma_{n}\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{|\Omega|}w^{\delta}\left(\int_{0}^{w}f^{*}(s)ds\right)\,dw-\frac{\gamma_{1}|\Omega|^{\delta}}{c}\int_{0}^{|\Omega|}f^{*}(s)\,ds.$

Arguing in the same way with the equality (16), we have

$\displaystyle\int_{0}^{+\infty}\gamma_{n}\phi(t)^{\frac{1}{p}}\,dt=\int_{0}^{|\Omega|}w^{\delta}\left(\int_{0}^{w}f^{*}(s)ds\right)\,dw-\frac{\gamma_{1}^{\star}|\Omega|^{\delta}}{c^{\star}}\int_{0}^{|\Omega|}f^{*}(s)\,ds.$

Hence

$\displaystyle\int_{0}^{+\infty}\gamma_{n}\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{+\infty}\gamma_{n}\phi(t)^{\frac{1}{p}}\,dt,$

which means

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}.$

We will now assume condition (9) and verify inequalities (11) and (12).
Let $0<p\leq\frac{n}{2n-2}$. Multiplying (17) by
$t\mu(t)^{\frac{1}{p}-\frac{2n-2}{n}}$, integrating from $0$ to some $\tau\geq
v_{m}$ and applying Lemma 3.4, we get

$\begin{array}[]{rl}\displaystyle\int_{0}^{\tau}\gamma_{n}t\mu(t)^{\frac{1}{p}}\,dt\leq&\displaystyle\int_{0}^{\tau}-\mu^{\prime}(t)t\mu(t)^{\delta}\left(\int_{0}^{\mu(t)}f^{*}(s)ds\right)\,dt-\displaystyle\frac{\gamma_{2}|\Omega|^{\delta}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds,\end{array}$

where $\displaystyle\delta=\frac{1}{p}-\frac{2n-2}{n}$. Since $\mu(t)$ is a
monotone non increasing function, we have

$\begin{array}[]{rl}\displaystyle\int_{0}^{\tau}\gamma_{n}t\mu(t)^{\frac{1}{p}}dt\leq&\displaystyle\int_{0}^{\tau}-t\mu(t)^{\delta}\left(\int_{0}^{\mu(t)}f^{*}(s)ds\right)\,d\mu(t)-\displaystyle\frac{\gamma_{2}|\Omega|^{\delta}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds.\end{array}$
(24)

Therefore, it is possible to integrate by parts both sides of the last
inequality. Introducing the function
$F(\ell)=\displaystyle\int_{0}^{\ell}w^{\delta}\left(\int_{0}^{w}f^{*}(s)ds\right)\,dw$,
we obtain

$\begin{array}[]{rl}\tau
F(\mu(\tau))+\tau\displaystyle\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{p}}dt\leq&\displaystyle\int_{0}^{\tau}F(\mu(t))dt+\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}\mu(t)^{\frac{1}{p}}dr\,dt\displaystyle-\displaystyle\frac{\gamma_{2}|\Omega|^{\delta}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds.\end{array}$

Adopting the same notation of Lemma 3.1, we set

$\xi(\tau)=\int_{0}^{\tau}F(\mu(t))dt+\int_{0}^{\tau}\left(\int_{0}^{t}\gamma_{n}\mu(r)^{\frac{1}{p}}dr\right)\,dt,$

$C=-\displaystyle\frac{\gamma_{2}|\Omega|^{\delta}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds$
and $\tau_{0}=v_{m}$. An application of Lemma 3.1 yields

$\begin{array}[]{ll}F(\mu(\tau))&+\displaystyle\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{p}}dt\leq\displaystyle\frac{1}{v_{m}}\Bigg{(}\int_{0}^{v_{m}}F(\mu(t))dt\\\
\\\
&+\displaystyle\int_{0}^{v_{m}}\int_{0}^{t}\gamma_{n}\mu(r)^{\frac{1}{p}}dr\,dt-\displaystyle\frac{\gamma_{2}|\Omega|^{\delta}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds\,\Bigg{)}.\end{array}$
(25)

Arguing in the same way with the equality (16), we have

$\begin{array}[]{ll}F(\phi(\tau))&+\displaystyle\int_{0}^{\tau}\gamma_{n}\phi(t)^{\frac{1}{p}}dt=\displaystyle\frac{1}{v_{m}}\Bigg{(}\int_{0}^{v_{m}}F(\phi(t))dt\\\
\\\
&+\displaystyle\int_{0}^{v_{m}}\displaystyle\int_{0}^{t}\gamma_{n}\phi(r)^{\frac{1}{p}}dr\,dt-\displaystyle\frac{\gamma^{*}_{2}|\Omega|^{\delta}}{2c^{*}}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds\,\Bigg{)}=\\\
&F(|\Omega|)+\displaystyle\frac{\gamma_{n}|\Omega|^{\frac{1}{p}}v_{m}}{2}-\displaystyle\frac{\gamma^{*}_{2}|\Omega|^{\delta}}{2v_{m}c^{*}}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds.\end{array}$
(26)

Recalling inequality (19) and using the monotonicity property of function $F$,
a comparison between the right-hand sides in (25) and in (26) can be
established, yielding

$F(\phi(\tau))+\int_{0}^{\tau}\gamma_{n}\phi(t)^{\frac{1}{p}}\geq
F(\mu(\tau))+\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{p}}.$

Passing to the limit as $\tau\to\infty$, we get

$\int_{0}^{\infty}\mu(t)^{\frac{1}{p}}dt\leq\displaystyle\int_{0}^{\infty}\phi(t)^{\frac{1}{p}}dt,$

and hence

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}.$

Finally, we prove inequality (12). It can be rewritten as follow

$\int_{0}^{\infty}t\mu(t)^{\frac{1}{p}}dt\leq\int_{0}^{\infty}t\phi(t)^{\frac{1}{p}}dt,\quad\forall\,p\in\left(0,\frac{n}{3n-4}\right].$

In (24), we take the limit as $\tau\to\infty$, and then we integrate by parts
the first term on the right-hand side to get

$\int_{0}^{\infty}\gamma_{n}t\mu(t)^{\frac{1}{p}}dt\leq\int_{0}^{\infty}F(\mu(t))dt-\displaystyle\frac{\gamma_{2}|\Omega|^{\delta}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds.$

On the other hand

$\int_{0}^{\infty}\gamma_{n}t\phi(t)^{\frac{1}{p}}dt=\int_{0}^{\infty}F(\phi(t))dt-\displaystyle\frac{\gamma^{*}_{2}|\Omega|^{\delta}}{2c^{*}}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds.$

Hence, it is enough to show that

$\int_{0}^{\infty}F(\mu(t))dt\leq\int_{0}^{\infty}F(\phi(t))dt.$ (27)

In order to prove inequality (27), we multiply (17) by
$tF(\mu(t))\mu(t)^{-\frac{2n-2}{n}}.$ Since the function
$F(\ell)\ell^{-\frac{2n-2}{n}}$ is non decreasing in $\ell$ $\mbox{for
}0<p\leq\frac{n}{3n-4}$, an integration from $0$ to any $\tau\geq v_{m}$
yields

$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\gamma_{n}tF(\mu(t))dt\leq&\displaystyle\int_{0}^{\tau}-t\mu^{-\frac{2n-2}{n}}F(\mu(t))\left(\displaystyle\int_{0}^{\mu(t)}f^{*}(s)ds\right)\,d\mu(t)\\\
&-F(|\Omega|)\displaystyle\frac{\gamma_{2}|\Omega|^{-\frac{2n-2}{n}}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds.\end{array}$
(28)

We now integrate by parts both sides of (28). After setting
$C=-F(|\Omega|)\displaystyle\frac{\gamma_{2}|\Omega|^{-\frac{2n-2}{n}}}{2c}\displaystyle\int_{0}^{|\Omega|}f^{*}(s)ds$
and
$H(\ell)=\displaystyle\int_{0}^{\ell}w^{-\frac{2n-2}{n}}F(w)\int_{0}^{w}f^{*}(s)ds\,dw,$
we get

$\tau\int_{0}^{\tau}\gamma_{n}F(\mu(t))dt+\tau
H(\mu(\tau))\leq\int_{0}^{\tau}\int_{0}^{r}\gamma_{n}F(\mu(z))dz\,dr+\int_{0}^{\tau}H(\mu(t))dt+C.$

Lemma 3.1 can be applied with
$\xi(\tau)=\displaystyle\int_{0}^{\tau}\int_{0}^{r}\gamma_{n}F(\mu(z))dz\,dr+\int_{0}^{\tau}H(\mu(t))dt$
and $\tau_{0}=v_{m}$, obtaining

$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\gamma_{n}F(\mu(t))dt+H(\mu(\tau))\\\
\\\
\leq&\displaystyle\frac{1}{v_{m}}\left(\displaystyle\int_{0}^{v_{m}}\displaystyle\int_{0}^{r}\gamma_{n}F(\mu(z))dz\,dr+\int_{0}^{v_{m}}H(\mu(t))dt+C\right).\end{array}$

The above inequality holds as an equality whenever $\mu$ is replaced by
$\phi$. Since (19) is in force, it holds

$\int_{0}^{\tau}\gamma_{n}F(\mu(t))dt+H(\mu(\tau))\leq\int_{0}^{\tau}\gamma_{n}F(\phi(t))dt+H(\phi(\tau))$

and for $\tau\to\infty$ we get inequality (27) which concludes the proof. ∎

###### Proof of Theorem 1.2.

First, we prove the point-wise comparison in dimension $n=2$, assuming
condition (8). After multiplying by $t$ inequality (17), we integrate from $0$
to $\tau\geq v_{m}$ both sides. Observing that now

$\int_{0}^{\mu(t)}f^{*}(s)ds=\mu(t),$ (29)

we get

$4\pi\tau\leq\int_{0}^{\tau}-\mu^{\prime}(t)\,dt-\displaystyle\frac{\gamma_{1}}{c}\quad\mbox{for
}\tau\geq v_{m}.$

The equality holds whenever $\mu$ is replaced by $\phi$

$4\pi\tau=\int_{0}^{\tau}-\phi^{\prime}(t)\,dt-\displaystyle\frac{\gamma_{1}^{\star}}{c^{\star}}\quad\mbox{for
}\tau\geq v_{m}.$

Therefore,

$\int_{0}^{\tau}\,(-d\mu(t))\geq\int_{0}^{\tau}\,(-d\phi(t))\quad\mbox{for
}\tau\geq v_{m},$

which means

$\mu(\tau)\leq\phi(\tau)\quad\mbox{for }\tau\geq v_{m}.$ (30)

We get the thesis recalling inequality (19). Now we consider $n\geq 3$.
Inequality (17) becomes

$\gamma_{n}\mu(t)^{\frac{n-2}{n}}\leq\left(-\mu^{\prime}(t)-\displaystyle\frac{1}{c}\int_{\partial_{e}U_{t}}\,d\mathcal{H}^{n-1}\right).$

Let $q\leq\frac{n}{n-2}$. Multiplying (17) by
$\mu(t)^{\frac{1}{q}-\frac{n-2}{n}}$, integrating from $0$ to some $\tau\geq
v_{m}$ and applying Lemma 3.3, we deduce that

$\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{q}}\,dt\leq\int_{0}^{\tau}-\mu^{\prime}(t)\mu(t)^{\eta}dt-\displaystyle\frac{\gamma_{1}|\Omega|^{\eta}}{c},$

where $\displaystyle\eta=\frac{1}{q}-\frac{n-2}{n}$. Using the monotonicity of
$\mu$ and applying a change of variables, we obtain

$\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{q}}dt\leq\int_{\mu(\tau)}^{|\Omega|}w^{n}\,dw-\displaystyle\frac{\gamma_{1}|\Omega|^{\eta}}{c}.$

Taking $\tau\to+\infty$

$\int_{0}^{+\infty}\gamma_{n}\mu(t)^{\frac{1}{q}}\,dt\leq\int_{0}^{|\Omega|}w^{n}\,dw-\displaystyle\frac{\gamma_{1}|\Omega|^{\eta}}{c}.$

Arguing in the same way with (16), we get

$\int_{0}^{+\infty}\gamma_{n}\phi(t)^{\frac{1}{q}}\,dt=\int_{0}^{|\Omega|}w^{n}\,dw-\displaystyle\frac{\gamma_{1}^{\star}|\Omega|^{\eta}}{c^{\star}}$

and the thesis follows. We will now assume condition (9) and prove the point-
wise comparison in dimension $n=2$. We multiply by $t$ the inequality (17) and
we integrate from $0$ to $\tau\geq v_{m}$. Since inequality (29) holds, we
have

$2\pi\tau^{2}\leq\displaystyle\int_{0}^{\tau}-\mu^{\prime}(t)t\,dt-\displaystyle\frac{\gamma_{2}}{2c}\quad\mbox{for
}\tau\geq v_{m}.$

At the same time, equality holds true whenever $\mu$ is replaced by $\phi$

$2\pi\tau^{2}=\displaystyle\int_{0}^{\tau}-\phi^{\prime}(t)t\,dt-\displaystyle\frac{\gamma_{2}^{*}}{2c^{*}}\quad\mbox{for
}\tau\geq v_{m}.$

Then,

$\displaystyle\int_{0}^{\tau}t\,(-d\mu(t))\geq\displaystyle\int_{0}^{\tau}t\,(-d\phi(t)),\quad\mbox{for
}\tau\geq v_{m}.$ (31)

An integration by parts gives

$\mu(\tau)\leq\phi(\tau),\quad\mbox{for }\tau\geq v_{m}.$ (32)

Since (18) is in force, inequality (32) follows for $t\geq 0$ and the claim is
proved. We consider $n\geq 3$. Inequality (17) reads as follows

$\gamma_{n}\mu(t)^{\frac{n-2}{n}}\leq\left(-\mu^{\prime}(t)-\displaystyle\frac{1}{c}\displaystyle\int_{\partial_{e}U_{t}}\>d\mathcal{H}^{n-1}\right).$

Let $q\leq\frac{n}{n-2}$. Multiplying (17) by
$t\mu(t)^{\frac{1}{q}-\frac{n-2}{n}}$, integrating from $0$ to some $\tau\geq
v_{m}$ and applying Lemma 3.4, we deduce that

$\int_{0}^{\tau}\gamma_{n}t\mu(t)^{\frac{1}{q}}\,dt\leq\int_{0}^{\tau}-\mu^{\prime}(t)t\mu(t)^{\eta}dt-\displaystyle\frac{\gamma_{2}|\Omega|^{\eta}}{2c}.$

Here we have set $\displaystyle\eta=\frac{1}{q}-\frac{n-2}{n}$. Since $\mu(t)$
is a monotone non increasing function, we can write

$\int_{0}^{\tau}\gamma_{n}t\mu(t)^{\frac{1}{q}}dt\leq\int_{0}^{\tau}-t\mu(t)^{\eta}\,d\mu(t)-\displaystyle\frac{\gamma_{2}|\Omega|^{\eta}}{2c}.$
(33)

Integrating by parts the last inequality and introducing the function
$G(\ell)=\displaystyle\int_{0}^{\ell}w^{\eta}=\frac{\ell^{\eta+1}}{\eta+1}$,
we obtain

$\tau
G(\mu(\tau))+\tau\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{q}}dt\leq\int_{0}^{\tau}G(\mu(t))dt+\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}\mu(t)^{\frac{1}{q}}dr\,dt-\displaystyle\frac{\gamma_{2}|\Omega|^{\eta}}{2c}.$

After setting

$\xi(\tau)=\int_{0}^{\tau}G(\mu(t))dt+\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}\mu(t)^{\frac{1}{q}}dr\,dt,$

$C=-\displaystyle\frac{\gamma_{2}|\Omega|^{\eta}}{2c}$ and $\tau_{0}=v_{m}$,
Lemma 3.1 can be applied, yielding

$\begin{array}[]{ll}\displaystyle
G(\mu(\tau))&+\displaystyle\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{q}}dt\leq\frac{1}{v_{m}}\Bigg{(}\int_{0}^{v_{m}}G(\mu(t))dt\\\
\\\
&\displaystyle+\int_{0}^{v_{m}}\int_{0}^{t}\gamma_{n}\mu(r)^{\frac{1}{q}}dr\,dt-\displaystyle\frac{\gamma_{2}|\Omega|^{\eta}}{2c}\Bigg{)}.\end{array}$
(34)

Arguing in the same way with equality (16), we can conclude that

$\begin{array}[]{ll}\displaystyle
G(\phi(\tau))&+\displaystyle\int_{0}^{\tau}\gamma_{n}\phi(t)^{\frac{1}{q}}dt=\frac{1}{v_{m}}\Bigg{(}\int_{0}^{v_{m}}G(\phi(t))dt\\\
\\\
&\displaystyle+\int_{0}^{v_{m}}\int_{0}^{t}\gamma_{n}\phi(r)^{\frac{1}{q}}dr\,dt-\displaystyle\frac{\gamma_{2}^{*}|\Omega|^{\eta}}{2c^{*}}\Bigg{)}.\end{array}$
(35)

Recalling inequality (19) and using the monotonicity property of function $G$,
a comparison between the right-hand sides in (34) and in (35) can be
established, yielding

$G(\mu(\tau))+\int_{0}^{\tau}\gamma_{n}\mu(t)^{\frac{1}{q}}\leq
G(\phi(\tau))+\int_{0}^{\tau}\gamma_{n}\phi(t)^{\frac{1}{q}}.$

Passing to the limit as $t\to\infty$ we get

$\int_{0}^{\infty}\mu(t)^{\frac{1}{q}}dt\leq\int_{0}^{\infty}\phi(t)^{\frac{1}{q}}dt$

and hence

$\|u\|_{L^{q,1}(\Omega)}\leq\|v\|_{L^{q,1}(\Omega^{\sharp})}.$

In order to conclude the proof, we have to show that

$\int_{0}^{\infty}t\mu(t)^{\frac{1}{q}}dt\leq\int_{0}^{\infty}t\phi(t)^{\frac{1}{q}}dt.$

We consider the limit as $\tau\to\infty$ in (33), and then we integrate by
parts the first term on the right-hand side to obtain

$\int_{0}^{\infty}\gamma_{n}t\mu(t)^{\frac{1}{q}}\leq\int_{0}^{\infty}G(\mu(t))dt-\displaystyle\frac{\gamma_{2}|\Omega|^{\eta}}{2c}.$

At the same time

$\int_{0}^{\infty}\gamma_{n}t\phi(t)^{\frac{1}{q}}=\int_{0}^{\infty}G(\phi(t))dt-\displaystyle\frac{\gamma_{2}^{*}|\Omega|^{\eta}}{2c^{*}}.$

Therefore, it is enough to prove that

$\int_{0}^{\infty}G(\mu(t))dt\leq\int_{0}^{\infty}G(\phi(t))dt.$ (36)

To show inequality (36), we multiply the inequality (17) by
$tG(\mu(t))\mu(t)^{-\frac{n-2}{n}}.$ Since the function
$G(\ell)\ell^{-\frac{n-2}{n}}=\ell^{\eta+2/n}$ is non decreasing in $\ell$ for
$0<q\leq\frac{n}{n-2}$, an integration from $0$ to any $\tau\geq v_{m}$ yields

$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\gamma_{n}tG(\mu(t))dt\leq&\displaystyle\int_{0}^{\tau}-t\mu^{-\frac{n-2}{n}}G(\mu(t))\,d\mu(t)\\\
\\\
&\displaystyle-G(|\Omega|)\displaystyle\frac{\gamma_{2}|\Omega|^{-\frac{2-n}{n}}}{2c}.\end{array}$
(37)

We now integrate by parts both sides of (37). After setting
$C=-G(|\Omega|)\displaystyle\frac{\gamma_{2}|\Omega|^{-\frac{2-n}{n}}}{2c}$
and $J(\ell)=\displaystyle\int_{0}^{\ell}w^{-\frac{n-2}{n}}G(w)\,dw$, we have

$\tau\int_{0}^{\tau}\gamma_{n}G(\mu(t))dt+\tau
J(\mu(\tau))\leq\int_{0}^{\tau}\int_{0}^{r}\gamma_{n}G(\mu(z))dz\,dr+\int_{0}^{\tau}J(\mu(t))dt+C.$

As above, we can apply Lemma 3.1 with

$\displaystyle\xi(\tau)=\int_{0}^{\tau}\int_{0}^{r}\gamma_{n}G(\mu(z))dz\,dr+\int_{0}^{\tau}J(\mu(t))dtdt,$

and $\tau_{0}=v_{m}$, deducing that

$\int_{0}^{\tau}\gamma_{n}G(\mu(t))dt+J(\mu(\tau))\leq\frac{1}{v_{m}}\left(\int_{0}^{v_{m}}\int_{0}^{r}\gamma_{n}G(\mu(z))dz\,dr+\int_{0}^{v_{m}}J(\mu(t))dt+C\right).$

The previous inequality holds as an equality whenever $\mu$ is replaced by
$\phi$. Recalling inequality (19), it holds

$\int_{0}^{\tau}\gamma_{n}G(\mu(t))dt+J(\mu(\tau))\leq\int_{0}^{\tau}\gamma_{n}G(\phi(t))dt+J(\phi(\tau))$

and for $\tau\to\infty$, we get inequality (36) which concludes the proof. ∎

##  4 Further remarks

The proofs of Theorems 1.1, 1.2 can be generalized to the case of an open and
bounded Lipschitz set $\Omega$ which has a finite number $k\geq 1$ of
connected components $\Omega_{j}$. Since $\Omega$ is a Lipschitz set, the
boundaries of its connected components must be disjoint. We generalize the
boundary condition in problem (2) with a piecewise constant one, by imposing
$\mathcal{H}^{n-1}$-a.e on $\partial\Omega$, the equality between the normal
derivative of a solution $u$ and
$g(x)=\sum_{j=1}^{k}c_{j}\chi_{\partial\Omega_{j}}(x)$. We assume that the
integral of the function $f$ on $\Omega_{j}$ is positive for every $j$. If the
compatibility condition (4) is satisfied for every connected component
$\Omega_{j}$, then the results of Theorems 1.2, 1.1 can be recovered. We
establish a comparison between a positive solution $u$ to the following
problem

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega$},\\\ \\\
\displaystyle\frac{\partial u}{\partial\nu}=g&\mbox{on
$\partial\Omega$}.\end{array}\right.$ (38)

and the positive solution $v$ to the Schwartz symmetrized problem, having the
same form of problem (5), which satisfies one of the following equalities

$\sum_{j=1}^{k}\frac{c^{*}}{c_{j}}\int_{\partial\Omega_{j}}u\,d\mathcal{H}^{n-1}=\int_{\partial\Omega^{\sharp}}v\,d\mathcal{H}^{n-1},$
(39)
$\sum_{j=1}^{k}\frac{c^{*}}{c_{j}}\int_{\partial\Omega_{j}}u^{2}\,d\mathcal{H}^{n-1}=\int_{\partial\Omega^{\sharp}}v^{2}\,d\mathcal{H}^{n-1}.$
(40)

######  Corollary 4.1.

Let $u$ be a positive solution to problem (38). If $v$ is the positive
solution to problem (5) satisfying the equality (39), then

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{for all
}0<p\leq\frac{n}{2n-2};$ (41)

If $v$ is the positive solution to problem (5) satisfying the equality (40),
then

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{for all
}0<p\leq\frac{n}{2n-2},$ (42)
$\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\sharp})}\quad\mbox{for
all }0<p\leq\frac{n}{3n-4}.$ (43)

######  Corollary 4.2.

Let assume that $f\equiv 1$ in $\Omega$. Let $u$ be a positive solution to
problem (38). If $v$ is the positive solution to problem (5) satisfying the
equality (39), it holds

$u^{\sharp}(x)\leq v(x)\quad\forall\,x\in\>\mbox{in
$\Omega^{\sharp}$},\quad\>\mbox{for $n=2$}.$

When $n\geq 3$, we have

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{for all
}\>0<p\leq\frac{n}{n-2}.$

If $v$ is the positive solution to problem (5) satisfying the equality (40),
it holds

$u^{\sharp}(x)\leq v(x)\quad\forall\,x\in\>\mbox{in
$\Omega^{\sharp}$},\quad\>\mbox{for $n=2$}$

When $n\geq 3$, we have

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\quad\mbox{and}\quad\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\sharp})}\quad\mbox{for
all }\>0<p\leq\frac{n}{n-2}.$

In the end, we provide five examples. The first shows that under condition
(39), we can’t recover the comparison (43) in $L^{2}$ norm, even in dimension
2.

######  Example 1.

Let $\Omega\subset\mathbb{R}^{2}$ be the union of two disjoint balls $B_{1}$
and $B_{2}$ of radius 1 and $1>\epsilon>0$. Let $u,v\geq 0$ be the unique
solution to the problems

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega$},\\\ \\\
\displaystyle\frac{\partial u}{\partial\nu}=-\frac{1}{2}\chi_{\partial
B_{1}}-\frac{\epsilon}{2}\chi_{\partial B_{2}}&\mbox{on $\partial\Omega$},\\\
\\\ u=\chi_{\partial B_{1}}&\mbox{on
$\partial\Omega$},\end{array}\right.\qquad\left\\{\begin{array}[]{ll}-\Delta
v=f^{\sharp}&\mbox{in $\Omega^{\sharp}$},\\\ \\\ \displaystyle\frac{\partial
v}{\partial\nu}=-\frac{\sqrt{2}}{4}(1+\epsilon)&\mbox{on
$\partial\Omega^{\sharp}$},\\\ \\\ v=\frac{1+\epsilon}{2}&\mbox{on
$\partial\Omega^{\sharp}$},\end{array}\right.$ (44)

where $f=\chi_{B_{1}}+\epsilon\chi_{B_{2}}$. Then

$\|u\|_{L^{2}(\Omega)}^{2}-\|v\|_{L^{2}(\Omega^{\sharp})}^{2}=\frac{\pi}{16}(4+\ln
2)-\frac{1}{32}\pi\epsilon(47+\log(16))+o(\epsilon)$

###### Proof.

The solutions $u$ and $v$ are positive for the maximum principle. A
straightforward calculation shows that condition (39) is satisfied but

$\sum_{j=1}^{2}\frac{c^{*}}{c_{j}}\int_{\partial\Omega_{j}}u^{2}\,d\mathcal{H}^{n-1}>\int_{\partial\Omega^{\sharp}}v^{2}\,d\mathcal{H}^{n-1}.$

Moreover

$u_{|B_{1}}=\frac{5-r^{2}}{4},\qquad u_{|B_{2}}=\frac{\epsilon}{4}(1-r^{2}).$

Since $\Omega^{\sharp}$ is a ball of radius $\sqrt{2}$, $f^{\sharp}(r)=1$ for
$0\leq r<1$ and $f^{\sharp}(r)=\epsilon$ for $1<r<\sqrt{2}$, we get

$v=\left\\{\begin{array}[]{ll}c_{1}-\frac{r^{2}}{4}&\mbox{for $0\leq r\leq
1$},\\\ \\\ c_{2}+c_{3}\ln{r}-\frac{\epsilon}{4}r^{2}&\mbox{for
$1<r\leq\sqrt{2}$}.\end{array}\right.$

The solution $v$ is $C^{1}$ and by imposing the continuity of $v^{\prime}$,
$v$ and the second boundary condition in (44), we obtain

$c_{3}=\frac{1}{2}(\epsilon-1),\quad
c_{2}=\frac{1}{2}+\epsilon+\frac{1}{2}(1-\epsilon)\ln{\sqrt{2}},\quad
c_{1}=\frac{3}{4}(1+\epsilon)+\frac{1}{2}(1-\epsilon)\ln{\sqrt{2}}.$

By means of a first order expansion, we get

$\displaystyle\|u\|_{L^{2}(\Omega)}^{2}=\frac{61}{48}\pi+o(\epsilon),\quad\|v\|_{L^{2}(\Omega^{\sharp})}^{2}=\frac{\pi}{48}(49-3\log(2))+\frac{1}{32}\pi\epsilon(47+\log(16))+o(\epsilon).$

In conclusion

$\displaystyle\|u\|_{L^{2}(\Omega)}^{2}-\|v\|_{L^{2}(\Omega^{\sharp})}^{2}=\frac{\pi}{16}(4+\ln
2)-\frac{1}{32}\pi\epsilon(47+\log(16))+o(\epsilon).$

∎

Next, we set the previous example in dimension 3. It shows that the $L^{1}$
comparison (41) becomes false in dimension greater than 2, if the function $f$
is not identically 1 in $\Omega$.

######  Example 2.

Let $\Omega\subset\mathbb{R}^{3}$ be the union of two disjoint balls $B_{1}$
and $B_{2}$ of radius 1 and $\epsilon>0$. Let $u,v\geq 0$ be the unique
solutions to the problems

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega$},\\\ \\\
\displaystyle\frac{\partial u}{\partial\nu}=-\frac{1}{3}\chi_{\partial
B_{1}}-\frac{\epsilon}{3}\chi_{\partial B_{2}}&\mbox{on $\partial\Omega$},\\\
\\\ u=\chi_{\partial B_{1}}&\mbox{on
$\partial\Omega$},\end{array}\right.\qquad\left\\{\begin{array}[]{ll}-\Delta
v=f^{\sharp}&\mbox{in $\Omega^{\sharp}$},\\\ \\\ \displaystyle\frac{\partial
v}{\partial\nu}=-\frac{1+\epsilon}{3\sqrt[3]{4}}&\mbox{on
$\partial\Omega^{\sharp}$},\\\ \\\ v=\frac{1+\epsilon}{2\sqrt[3]{2}}&\mbox{on
$\partial\Omega^{\sharp}$},\end{array}\right.$ (45)

where $f=\chi_{B_{1}}+\epsilon\chi_{B_{2}}$. Then

$\|u\|_{L^{1}(\Omega)}-\|v\|_{L^{1}(\Omega^{\sharp})}=\frac{14\pi}{9}-\frac{8\sqrt[3]{4}}{9}\pi-\frac{2\pi}{45}\epsilon-\frac{28\sqrt[3]{4}}{45}\pi\epsilon.$

###### Proof.

The solutions $u$ and $v$ are positive for the maximum principle. A
straightforward calculation shows that condition (39) is satisfied. Moreover

$u_{|B_{1}}=\frac{7-r^{2}}{6},\qquad u_{|B_{2}}=\frac{\epsilon}{6}(1-r^{2}).$

Since $\Omega^{\sharp}$ is a ball of radius $\sqrt[3]{2}$, $f^{\sharp}(r)=1$
for $0\leq r<1$ and $f^{\sharp}(r)=\epsilon$ for $1<r<\sqrt[3]{2}$, we get

$v=\left\\{\begin{array}[]{ll}c_{1}-\frac{r^{2}}{6}&\mbox{for $0\leq r\leq
1$},\\\ \\\ c_{2}+\frac{c_{3}}{r}-\frac{\epsilon}{6}r^{2}&\mbox{for
$1<r\leq\sqrt[3]{2}$}.\end{array}\right.$

The solution $v$ is $C^{1}$ and by imposing the continuity of $v^{\prime}$,
$v$ and the second boundary condition in (44), we obtain

$c_{3}=\frac{1-\epsilon}{3},\quad
c_{2}=\frac{(1+7\epsilon)}{6\sqrt[3]{2}},\quad
c_{1}=\frac{1}{2}+\frac{1}{6\sqrt[3]{2}}-\frac{\epsilon}{2}+\frac{7\epsilon}{6\sqrt[3]{2}}.$

Evaluating the $L^{1}$-norms, we get

$\displaystyle\|u\|_{L^{1}(\Omega)}=\frac{64}{45}\pi+\frac{4\pi}{45}\epsilon,\quad\|v\|_{L^{1}(\Omega^{\sharp})}=-\frac{4}{30}\pi+\frac{8\sqrt[3]{4}}{9}\pi+4\pi\left(\frac{1}{30}+\frac{7\sqrt[3]{4}}{45}\right)\epsilon.$

In conclusion

$\displaystyle\|u\|_{L^{1}(\Omega)}-\|v\|_{L^{1}(\Omega^{\sharp})}=\frac{14\pi}{9}-\frac{8\sqrt[3]{4}}{9}\pi-\frac{2\pi}{45}\epsilon-\frac{28\sqrt[3]{4}}{45}\pi\epsilon.$

∎

With the following example, we show that even under condition (40) and
dimension 2, an $L^{p}$ comparison for $p>2$ is not satisfied if $p$ is large
enough.

######  Example 3.

Let $\Omega\subset\mathbb{R}^{2}$ be the union of two disjoint balls $B_{1}$
and $B_{2}$ of radius 1 and $\epsilon>0$. Let $u,v\geq 0$ be the unique
solutions to the problems

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega$},\\\ \\\
\displaystyle\frac{\partial u}{\partial\nu}=-\frac{1}{2}\chi_{\partial
B_{1}}-\frac{\epsilon}{2}\chi_{\partial B_{2}}&\mbox{on $\partial\Omega$},\\\
\\\ u=\chi_{\partial B_{1}}&\mbox{on
$\partial\Omega$},\end{array}\right.\qquad\left\\{\begin{array}[]{ll}-\Delta
v=f^{\sharp}&\mbox{in $\Omega^{\sharp}$},\\\ \\\ \displaystyle\frac{\partial
v}{\partial\nu}=-\frac{\sqrt{2}}{4}(1+\epsilon)&\mbox{on
$\partial\Omega^{\sharp}$},\\\ \\\ v=\sqrt{\frac{1+\epsilon}{2}}&\mbox{on
$\partial\Omega^{\sharp}$},\end{array}\right.$ (46)

where $f=\chi_{B_{1}}+\epsilon\chi_{B_{2}}$. Then

$\|u\|_{L^{6}(\Omega)}^{6}-\|v\|_{L^{6}(\Omega^{\sharp})}^{6}\approx
2.494-11.134\epsilon+o(\epsilon)$

###### Proof.

The solutions $u$ and $v$ are positive for the maximum principle. A
straightforward calculation shows that condition (40) is satisfied. We have

$u_{|B_{1}}=\frac{5-r^{2}}{4},\qquad u_{|B_{2}}=\frac{\epsilon}{4}(1-r^{2}),$

$\Omega^{\sharp}$ is a ball of radius $\sqrt{2}$, $f^{\sharp}(r)=1$ for $0\leq
r<1$ and $f^{\sharp}(r)=\epsilon$ for $1<r<\sqrt{2}$ and

$v=\left\\{\begin{array}[]{ll}c_{1}-\frac{r^{2}}{4}&\mbox{for $0\leq r\leq
1$},\\\ \\\ c_{2}+c_{3}\ln{r}-\frac{\epsilon}{4}r^{2}&\mbox{for
$1<r\leq\sqrt{2}$}.\end{array}\right.$

The solution $v$ is $C^{1}$ and by imposing the continuity of $v^{\prime}$,
$v$ and the second boundary condition in (46), we obtain

$c_{3}=\frac{1}{2}(\epsilon-1),\quad
c_{2}=\sqrt{\frac{1+\epsilon}{2}}+\frac{\epsilon}{2}+\frac{1}{2}(1-\epsilon)\ln{\sqrt{2}},\quad
c_{1}=\sqrt{\frac{1+\epsilon}{2}}+\frac{1}{4}(1+\epsilon)+\frac{1}{2}(1-\epsilon)\ln{\sqrt{2}}.$

By means of a first order expansion, we get

$\displaystyle\|u\|_{L^{6}(\Omega)}^{6}\approx
6.765+o(\epsilon),\quad\|v\|_{L^{6}(\Omega^{\sharp})}^{6}\approx
4.271+11.134\epsilon+o(\epsilon).$

In conclusion

$\displaystyle\|u\|_{L^{6}(\Omega)}^{6}-\|v\|_{L^{6}(\Omega^{\sharp})}^{6}\approx
2.494-11.134\epsilon+o(\epsilon).$

∎

Finally, a slightly modified version of an example from [18] is taken into
consideration. This allows us to prove that, when $c<0$, the comparison result
(10) no longer holds if $u$ and $v$ fulfill condition (8) but are not both
positive.

######  Example 4.

For $a,\epsilon\in\mathbb{R}^{+}$, let $u_{a,\epsilon},v_{a,\epsilon}$ be the
unique solutions to the problems

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega_{a,\epsilon}$},\\\
\\\ \displaystyle\frac{\partial u}{\partial\nu}=-\epsilon&\mbox{on
$\partial\Omega_{a,\epsilon}$},\\\ \\\
\displaystyle\int_{\partial\Omega_{a,\epsilon}}u=0,\\\
\end{array}\right.\qquad\left\\{\begin{array}[]{ll}-\Delta
v=f^{\sharp}&\mbox{in $\Omega_{a,\epsilon}^{\sharp}$},\\\ \\\
\displaystyle\frac{\partial v}{\partial\nu}=c^{*}&\mbox{on
$\partial\Omega_{a,\epsilon}^{\sharp}$},\\\ \\\
\displaystyle\int_{\partial\Omega_{a,\epsilon}^{\sharp}}v=0,\end{array}\right.$

where $f(x,y)=-(\operatorname{sgn}(y)+\operatorname{sgn}(x))$ and
$\Omega_{a,\epsilon}=\left(-2\epsilon-\frac{1}{2a},\frac{1}{2a}\right)\times\left(-2\epsilon-\frac{a}{2},\frac{a}{2}\right)$.
Then, it exists $a_{0}>0$ such that
$\|u_{a,\epsilon}\|_{L^{1}(\Omega_{a,\epsilon})}>\|v_{a,\epsilon}\|_{L^{1}(\Omega_{a,\epsilon}^{\sharp})}$
for $a>a_{0}$ and $\epsilon<a_{0}^{-1}$.

###### Proof.

The solution $u_{a,\epsilon}$ can be written as
$u_{a,\epsilon}=z_{a,\epsilon}+k(a,\epsilon)$, where
$z_{a,\epsilon}=\frac{1}{2}y^{2}\operatorname{sgn}(y)-\left(\epsilon+\frac{a}{2}\right)y+\frac{1}{2}x^{2}\operatorname{sgn}(x)-\left(\epsilon+\frac{1}{2a}\right)x$.
Since

$\int_{\partial\Omega_{a,\epsilon}}z_{a,\epsilon}\,d\mathcal{H}^{1}\geq 0,$

then $k(a,\epsilon)\leq 0$. For $x,y\in\Omega$ such that $x,y\geq 0$, we have

$\displaystyle\frac{1}{2}y^{2}\operatorname{sgn}(y)-\left(\epsilon+\frac{a}{2}\right)y$
$\displaystyle\leq 0,$
$\displaystyle\frac{1}{2}x^{2}\operatorname{sgn}(x)-\left(\epsilon+\frac{1}{2a}\right)x$
$\displaystyle\leq 0.$

Therefore

$\|u_{a,\epsilon}\|_{L^{1}(\Omega)}\geq\frac{1}{2a}\int_{0}^{a/2}\left[\left(\epsilon+\frac{a}{2}\right)y-\frac{1}{2}y^{2}\operatorname{sgn}(y)\right]dy=\frac{a^{2}}{48}+\frac{a\epsilon}{16}>\frac{a^{2}}{48}.$

We consider the solution $w$ to the problem

$\left\\{\begin{array}[]{ll}-\Delta w=2&\mbox{in $U$},\\\ \\\ w=0&\mbox{on
$\partial U$},\\\ \end{array}\right.$

where U is the ball centered at the origin with radius 4. It exists $a_{1}>0$
such that $|\Omega_{a,\epsilon}^{\sharp}|<4$ for $a>a_{1}$ and
$\epsilon<a_{1}^{-1}$, then $\Omega_{a,\epsilon}^{\sharp}\subset U$ and
applying the maximum principle, we get $w\geq v_{a,\epsilon}$ in
$\Omega_{a,\epsilon}^{\sharp}$, which implies
$\|w\|_{L^{1}(U)}\geq\|v_{a,\epsilon}\|_{L^{1}(\Omega_{a,\epsilon}^{\sharp})}$.
∎

As we can see, the conditions (8) and (9) become meaningless when

$\int_{\Omega}f\,dx=0.$

On the other hand, the comparisons proved before are not generally verified if
we take into account the solutions whose traces’ mean value equals 0. To this
aim, we consider the example given in [18].

######  Example 5.

For $a\in\mathbb{R}^{+}$, let $u_{a},v_{a}$ be the unique solutions to the
problems

$\left\\{\begin{array}[]{ll}-\Delta u=f&\mbox{in $\Omega_{a}$},\\\ \\\
\displaystyle\frac{\partial u}{\partial\nu}=0&\mbox{on
$\partial\Omega_{a}$},\\\ \\\ \displaystyle\int_{\partial\Omega_{a}}u=0,\\\
\end{array}\right.\qquad\left\\{\begin{array}[]{ll}-\Delta
v=f^{\sharp}&\mbox{in $\Omega_{a}^{\sharp}$},\\\ \\\
\displaystyle\frac{\partial v}{\partial\nu}=-\frac{1}{4\sqrt{\pi}}&\mbox{on
$\partial\Omega_{a}^{\sharp}$},\\\ \\\
\displaystyle\int_{\partial\Omega_{a}^{\sharp}}v=0,\end{array}\right.$

where $f(x,y)=-\operatorname{sgn}(y)$ and
$\Omega_{a}=\left(-\frac{1}{2a},\frac{1}{2a}\right)\times\left(-\frac{a}{2},\frac{a}{2}\right)$.
Then, it exists $a_{0}>0$ such that
$\|u_{a}\|_{L^{1}(\Omega)}>\|v_{a}\|_{L^{1}(\Omega^{\sharp})}$ for $a>a_{0}$.

###### Proof.

The solution $u_{a}$ is the function
$u_{a}=\frac{1}{2}y^{2}\operatorname{sgn}(y)-\frac{1}{2}ay$. Evaluating its
$L^{1}$ norm, we get

$\|u_{a}\|_{L^{1}(\Omega)}=\frac{2}{a}\int_{-\frac{a}{2}}^{0}\left(-\frac{1}{2}y^{2}-\frac{1}{2}ay\right)dy=\frac{a^{2}}{12}.$

We observe that the symmetrized problem do not depend on the parameter $a$,
because $|\Omega_{a}^{|}=1$ and $f(x,y)=\operatorname{sgn}(y)$ is equal to
$\pm 1$ on sets of measure $\frac{1}{2}$ for every $a>0$. Therefore, $v_{a}=v$
and $\|v\|_{L^{1}(\Omega^{\sharp})}$ is a positive constant. ∎

## Acknowledgements

All the authors are members of Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta
Matematica (INdAM). The author Carlo Nitsch was partially supported by the
INdAM-GNAMPA project 2023 "Symmetry and asymmetry in PDEs", cod. CUP
E53C22001930001. The author Antonio Celentano was partially supported by the
INdAM-GNAMPA project 2023 "Modelli matematici di EDP per fluidi e strutture e
proprieta’ geometriche delle soluzioni di EDP", cod. CUP E53C22001930001, and
by the INdAM-GNAMPA project 2024 "Problemi frazionari: proprietà quantitative
ottimali, simmetria, regolarità", cod. CUP E53C23001670001.

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# TOPIC IDENTIFICATION FOR SPONTANEOUS SPEECH: ENRICHING AUDIO FEATURES WITH
EMBEDDED LINGUISTIC INFORMATION
††thanks: This work was supported by the Foundation for Aalto University
Science and Technology, by awarding the Huawei doctoral student grant. We are
grateful for the Academy of Finland projects funding number 337073 “FIN-CLARIN
as a Collaborative Platform for Speech Processing” and 345790 ”Understanding
speech and scene with ears and eyes”. The computational resources were
provided by Aalto ScienceIT.

Dejan Porjazovski  Aalto University
Espoo, Finland
<EMAIL_ADDRESS>Tamás Grósz  Aalto University
Espoo, Finland
<EMAIL_ADDRESS>Mikko Kurimo  Aalto University
Espoo, Finland
<EMAIL_ADDRESS>
###### Abstract

Traditional topic identification solutions from audio rely on an automatic
speech recognition system (ASR) to produce transcripts used as input to a
text-based model. These approaches work well in high-resource scenarios, where
there are sufficient data to train both components of the pipeline. However,
in low-resource situations, the ASR system, even if available, produces low-
quality transcripts, leading to a bad text-based classifier. Moreover,
spontaneous speech containing hesitations can further degrade the performance
of the ASR model. In this paper, we investigate alternatives to the standard
text-only solutions by comparing audio-only and hybrid techniques of jointly
utilising text and audio features. The models evaluated on spontaneous Finnish
speech demonstrate that purely audio-based solutions are a viable option when
ASR components are not available, while the hybrid multi-modal solutions
achieve the best results.

###### Index Terms:

topic identification, multi-modal, spontaneous speech, linguistically-enhanced
embeddings

##  I Introduction

With the rapid increase of data generation and collection, there is a need to
organise it in a meaningful way for efficient retrieval and analysis. One way
to organise the data is by automatically categorising documents based on their
topic. Categorising documents by topic is an active research area and
automatic topic identification (topic ID) is a positive step forward in that
direction. Topic ID is also beneficial in enhancing the human-computer
interaction devices by improving their efficiency [1]. Having such topic ID
systems for high-resource languages can be taken for granted, whereas for
languages with scarce resources, those systems might not be available. This
lack of resources has shifted the focus of research towards alternative topic
ID solutions.

To date, topic ID using text input is a well-explored area, unlike audio,
where only a few solutions exist. The text-based approaches typically rely on
distributed representations of words or bag-of-words for extracting meaningful
information [2, 3]. However, such techniques are not directly applicable to
audio.

The existing research on topic ID from audio typically explores a multi-model
pipeline where an automatic speech recognition (ASR) system generates
transcripts, and a separate classifier identifies the topics from the
transcripts. One such pipeline system was explored in [4], where the authors
extracted features from word-based ASR lattices and used them together with a
novel minimum classification error technique. The authors in [5], on the other
hand, focused on exploring different techniques for selecting which
recognition lattices are most useful for topic ID. In [6], the authors
explored a different direction by utilising the gender information. They
employed two gender-dependent acoustic models and a bi-gram language model to
generate transcripts for the various text-based topic ID systems. Another
study [7] utilised multiple text features. In their experiments, the words and
graphemes, generated by HMM and CTC ASR systems, were used as two different
inputs to a CNN model, showcasing the effectiveness of the multi-stream input.
Although these pipeline systems for topic ID work well, they nevertheless rely
on a separate ASR system to generate the transcripts.

Obtaining a satisfying performance with an ASR system, however, is challenging
due to the difficulties of collecting enough data in low-resource or domain-
specific scenarios. Thus, some research performs topic ID directly from audio,
without relying on an ASR system. In [8], the authors applied an unsupervised
approach of acoustic unit discovery to generate features for training an SVM
system. A similar approach for acoustic unit discovery was explored in [9], in
comparison to a dynamic time-warping solution of unsupervised term discovery.
Other research [10] explored an end-to-end approach for topic ID from audio.
The acoustic unit discovery techniques differ from the proposed solutions in
this study since we do not discover acoustic features but instead rely on
audio embeddings.

While previous research [e.g., 4-10] performs topic ID on either text or
audio, it is possible to combine those features. For example, the authors in
[11] combined audio and text features, showcasing the benefits of both
modalities. Another approach for combining audio and textual modalities was
explored in [12]. In their study, the authors presented a solution to learn
audio-linguistic embeddings, where the acoustic information is modelled by
reconstructing the spectrogram and the linguistic information by learning to
transcribe the audio. Generally, by combining both audio and text features, a
performance improvement is expected at a cost of increased complexity and
computational time.

Figure 1: Overview of the topic ID systems.

Inspired by the audio-linguistic embeddings [12], we experimented with
learning linguistically-enhanced audio embeddings by reconstructing the filter
banks and aligning the audio and text embedding vector spaces. This approach
is different from the audio-linguistic embeddings proposed in [12] since
instead of learning to transcribe the audio, we minimise the distance between
the audio and text vector spaces. In addition to the audio and linguistic
losses, we further include the topic ID loss as part of the learning
framework.

This work aims to investigate audio-only and hybrid techniques of jointly
using audio and text-based inputs for performing topic ID from spontaneous
Finnish speech and compare them to the standard text-based solutions. The
text-based approaches include semi-supervised, manual transcript, and pipeline
systems. The first system uses the manually and automatically (by ASR)
transcribed portions of the data. The second system utilises only the manually
transcribed data. The third one is a commonly-used pipeline technique for
doing topic ID on the ASR-generated transcripts. As audio-only solutions, we
experimented with three models. The first one uses the audio features, whereas
the second one is an embedding transfer approach that learns audio embeddings,
enhanced with linguistic information. The last audio-only model is a self-
supervised Wav2vec2 [13] solution. To jointly utilise both the audio and text
information, we experimented with two models. The first one simply
concatenates the audio and text embeddings, while the second one implements
multi-task training by simultaneously learning the ASR and topic ID tasks. To
the best of our knowledge, this multi-task approach has not been explored for
the topic ID task. The proposed audio-only and hybrid techniques are evaluated
and compared against the standard text-only approaches on a colloquial Finnish
dataset, reflecting a real-world scenario. An overview of the explored topic
ID systems is given in Figure 1.

Contributions. In this paper, we present the effectiveness of the proposed
linguistically-enhanced audio embeddings, outperforming the other audio-only
solution. Furthermore, we present a multi-task approach that jointly learns
the ASR and topic ID tasks as an alternative to the pipeline system.

##  II Dataset

For conducting the experiments, we used the newly collected Lahjoita puhetta
(LP) [14] conversational Finnish corpus. It is a diverse corpus with over
20.000 different speakers covering different age groups and speaking
colloquial Finnish. The colloquial and spontaneous nature of the corpus allows
us to evaluate the models in a real-world scenario. The length of the audio
recordings varies significantly, from a few seconds to several minutes. In our
experiments, we used only the samples which are up to 50 seconds long and
discarded the rest. This choice reduces the computational cost that long
utterances impose. The duration of the transcribed part of the dataset used in
this study for the train, dev, and test splits is 408, 3, and 3 hours,
respectively, whereas the untranscribed part is 405 hours (used exclusively
for training). The 8 topics used in the experiments are: ”Animal friends”,
”Sports moments”, ”My surroundings”, ”Summer”, ”The cursed Covid”, ”Media
skills”, ”Rated R”, and ”Nature”. In the original dataset, the ”Media skills”
topic is divided into three categories, which we merged into one. The topic
distribution is given in Figure 2.

Figure 2: Topic distribution in the LP dataset.

##  III Topic identification approaches

All the explored approaches consist of one or multiple of the following
blocks: audio encoder, text encoder, and attention-based decoder. The CRDNN
audio encoder uses filter banks with 40 filters as input and produces audio
embeddings. The text encoder extracts word embeddings using the uncased
FinBERT model [15]. The embeddings are averaged to get an utterance-level
representation. The GRU decoder is used in cases where the model additionally
performs the ASR task. As a decoding strategy for the ASR, we used beam search
with a size of 10. During training, we optimised the negative log-likelihood.
In the experiments where additionally the ASR decoder is trained, the CTC
objective [16] is used as an auxiliary loss for the first 20 epochs with a
weighting factor of 0.5. We used the SpeechBrain toolkit to conduct all the
experiments and the recipes are publicly available111https://github.com/aalto-
speech/Topic-identification-for-spontaneous-Finnish-speech.

We compare our proposed audio-only and hybrid solutions to the baselines
presented in [14]. The audio-only baseline model (referred to as audio BL in
the table) is a 5-layer TDNN, followed by a statistics pooling and two linear
layers. The audio+text baseline (referred to as audio+text BL) additionally
utilises the text by extracting FinBERT embeddings and using them as input to
a BLSTM network. The predictions are made by concatenating the audio and text
embeddings.

###  III-A Text-based approaches

Three different text-based models are used to compare the performance against
the audio-only and hybrid solutions. The first one is trained on part of the
data that have manually annotated transcripts. We will refer to this model as
text man_trn. The second model combines manually and automatically generated
transcripts. The automatically generated transcripts for the untranscribed
portion of the data are obtained using a separate ASR system. We will call
this approach text semi-supervised. The ASR system is trained with the
standard end-to-end sequence-to-sequence technique [17], where the model
outputs SentencePiece unigram subword units [18]. The WER and CER on the test
set are 34.63% and 12.55%, respectively. The reason for the high WER and CER
might be the agglutinative nature of the Finnish language, which has long
words. Furthermore, the colloquial Finnish does not have a fixed written form,
making it hard to transcribe even by humans [14]. Lastly, we trained a text
pipeline system where all the transcripts are generated by the ASR model.

###  III-B Audio-based approaches

The audio-based solutions rely on the audio encoder to extract meaningful
acoustic embeddings. The audio-only system embeds the utterances using the
CRDNN encoder and uses those embeddings for classification (see Figure 1a).
Since we do not rely on the transcripts, the whole audio data is used,
combining the transcribed and untranscribed parts.

As mentioned earlier, extracting meaningful information from audio is more
difficult than from text. To bridge the gap between audio and text embeddings,
we make use of the audio and text data by learning linguistically-enhanced
audio embeddings. We do that by pre-training the embedding transfer model
using three objectives (see Figure 1c). The linguistic information is modelled
by minimising the L1 loss between the audio and word embeddings. Due to the
different embedding dimensions, the audio embeddings are upscaled using a
feed-forward layer to match the dimension of the text embeddings. To model the
acoustic information, we minimise the L1 loss between the audio embeddings and
the original filter bank features. The audio embeddings are downscaled using a
feed-forward layer to match the dimension of the filter banks. This acoustic
loss is used to get a system similar to an audio autoencoder. We opted for
that to ensure the model preserves the acoustic information rather than solely
adapting to the text embeddings. When computing the L1 loss between the
different embeddings, we are using utterance-level representations, obtained
by averaging. Lastly, the negative log-likelihood loss is used to learn the
topic ID task from the audio embeddings. During pre-training, the three
objective functions are interpolated with equal contribution. The negative
log-likelihood loss is defined as:

$Loss_{nll}=-\sum^{N}_{i=1}log(p(y_{i}|a_{i};\theta))$ (1)

where, $N$ is the number of samples, $y_{i}$ is the ground truth label for the
sample $i$, $a_{i}$ is the audio embedding for the sample $i$, and $\theta$
are the model parameters. The acoustic and linguistic L1 losses are defined
as:

$Loss_{L1}=\sum_{i=1}^{N}|a_{i}-k_{i}|$ (2)

where $k$ is either the filter bank features or the text embedding for the
sample $i$. After the model is pre-trained, a new model is initialised using
the pre-trained weights and trained on the whole audio data in the same way as
the audio-only system (see Figure 1d). We put the embedding transfer approach
under the audio-based systems because during inference it uses only the audio
features.

The last audio-only solution is the pre-trained multilingual Wav2vec2
222https://huggingface.co/facebook/Wav2vec2-xls-r-300m model. We fine-tuned
this model for 10 epochs on a 100-hour subset of the Lahjoita puhetta corpus.
The samples were chosen so that they would reflect the class distribution of
the whole dataset. We did not use the whole dataset during fine-tuning due to
the computational time. Additionally, at least in terms of WER, there are no
big differences observed when fine-tuning on a whole dataset, in comparison to
a 10-hour subset [19].

###  III-C Hybrid audio and text-based approaches

The hybrid systems exploit the information from both the audio and text
modalities. The first solution employs a multi-task approach where the ASR and
topic ID tasks are learned jointly using the transcribed portion of the data
(see Figure 1e). This way, the ASR part is optimised for the topic ID task and
vice-versa. In the process, hard parameter sharing is employed [20], where the
same audio encoder output is used in both tasks. For the topic ID task, the
embeddings produced from the audio and text encoders are concatenated and
passed to the classification layer. To learn the ASR task, only the audio
embeddings are used. During training, both loss functions are interpolated
with equal contribution. The epoch resulting in the lowest WER of 35.11% is
chosen as a starting point for the fine-tuning step. In the fine-tuning
process, in addition to the transcribed portion, the untranscribed part is
used by first generating transcripts using the pre-trained model. The model is
fine-tuned only on the topic ID task by concatenating the audio and text
modalities, using the whole training data (see Figure 1f). In the tables, we
refer to this model as multi-task.

In the last experiment, only the transcribed portion of the data is exploited.
The topic ID task is learned by concatenating the audio and text embeddings
and using the output for classification. This audio+text approach does not
incorporate any pre-training and is used to show whether utilising the audio
data along with the text is better than just the text.

##  IV Results and discussion

To account for the imbalanced class distribution in the Lahjoita puhetta
dataset, the systems are evaluated using the micro $F_{1}$ and unweighted
average recall (UAR) metrics. The results obtained by the text-only, audio-
only, and hybrid models are presented in Table I.

TABLE I: Micro $F_{1}$ and UAR scores on the dev and test sets for the text-based (rows 1-3), audio-based (rows 4-7), and hybrid (rows 8-10) models. Model | Params | F1 dev | F1 test | UAR dev | UAR test
---|---|---|---|---|---
Text pipeline | 7.7K | 85.99 | 80.38 | 80.30 | 75.15
Text man_trn | 7.7K | 87.22 | 83.97 | 81.40 | 78.86
Text semi-sup | 7.7K | 87.96 | 84.68 | 83.70 | 80.40
Audio BL | 4.2M | 68.28 | 65.65 | 58.34 | 58.34
Emb transfer | 21.2M | 75.67 | 76.31 | 69.32 | 72.56
Audio-only | 20.8M | 76.90 | 75.83 | 69.23 | 68.50
Wav2vec2 | 317M | 78.62 | 71.77 | 70.37 | 67.29
Audio+text BL | 15.8M | 89.95 | 81.34 | 85.80 | 78.70
Audio+text | 20.8M | 86.48 | 85.64 | 81.69 | 79.80
Multi-task | 25M | 88.69 | 86.12 | 83.33 | 80.40

In the first three rows of Table I, we can see the $F_{1}$ and UAR results for
the text-based approaches. The text pipeline, which is the most commonly used
approach in the literature, achieves an $F_{1}$ score of 80.38% on the test
set. Even though this model uses the ASR-generated transcripts from the whole
data, it still falls behind the text man_trn system, trained only on the
manually transcribed portion, highlighting the importance of accurate
transcriptions. The benefits of the additional non-transcribed data are shown
by the text semi-supervised model, achieving the best performance of 84.68%
$F_{1}$ score on the test set. A similar trend can be observed by looking at
the UAR metric, where the text semi-supervised model got 80.40%, which is a
5.25% absolute improvement over the pipeline system. The gap between UAR and
$F_{1}$ showcases that the models somewhat overfitted to the over-represented
topics, but the overfit is not severe.

From the results produced by the audio-based approaches (rows 4-7), we can
notice that the embedding transfer model slightly outperforms the audio-only
model in terms of $F_{1}$ score on the test set, while in terms of the UAR
metric, the difference is more substantial. On the dev set, the F1 score of
the audio-only model is better, but its UAR value is close to the one yielded
by the embedding transfer. Additionally, both models outperformed the baseline
system, showcasing the benefits of the larger CRDNN model. Even though the
Wav2vec2 model is pre-trained on 436K hours of data and has over 300 million
parameters, it still falls behind the proposed audio-only solutions when
evaluated on the test set. On the dev set, on the other hand, the Wav2vec2
model achieves the best results, possibly due to overfitting.

Next, we analyse the results achieved by combining the audio and text-based
features, shown in rows 8-10. The audio+text approach, which uses only the
transcribed part of the data, achieves an $F_{1}$ score of 85.64% on the test
set. This is a 1.67% absolute improvement over the text man_trn model that
utilises only the manually-transcribed text features. Similarly, an
improvement can be noticed by looking at the UAR metric. These results tell us
that the model additionally benefits from the audio-based features. An
improvement over the audio+text approach is realised by the multi-task system,
which uses the manually and automatically transcribed portions of the data. In
terms of WER, the multi-task system does not suffer too much, performing only
0.48% worse than the stand-alone ASR model. The baseline audio+text BL model
achieved the best $F_{1}$ and UAR results on the dev set, but the worst on the
test set, indicating a severe overfit.

Generally, by looking at the results presented in Table I, we can conclude
that the audio-only approaches fall behind the text-only ones. When combining
audio and text-based features, on the other hand, we can see an improvement
over the commonly-used pipeline approach. The best results for the $F_{1}$
metric are realised by the multi-task system, whereas for the UAR, both multi-
task and text semi-supervised achieve equal scores. Another point worth noting
is that the text-based approaches tend to perform better on the dev than on
the test set when evaluated using the $F_{1}$ metric, indicating that they
might be overfitting. Meanwhile, the models that additionally use the audio
features tend to perform similarly on the dev and test sets, suggesting that
the audio modality serves as a regularizer for the models.

Figure 3: Embeddings produced for the test set by the audio-only, text-only, and embedding transfer systems. TABLE II: Model agreement in terms of micro $F_{1}$ on the test set. |  Audio-only |  Emb trnf |  Text semi-sup
---|---|---|---
Multi-task | 73.20 | 76.55 | 85.40
Audio-only | / | 72.24 | 72.72
Emb trnf | / | / | 75.59

By using the input modalities in different ways, we expect that the systems
would recognise the same examples with varying difficulties. To investigate
how much the models agree with each other, we treated one model’s predictions
as true labels and compared them against the predictions of the other models.
The model agreements are given in Table II. These results demonstrate that the
embedding transfer solution has a higher agreement with the multi-task and
text semi-supervised approaches, in comparison to the audio-only, showcasing
the effectiveness of modelling the linguistic information. This is further
confirmed by plotting the embeddings for audio-only, text-only, and embedding
transfer systems, as shown in Figure 3. The embeddings are mapped in a 2D
space using the t-SNE algorithm [21]. From the scatter plot, we can observe
that the embedding transfer samples (green) are mostly following the
distribution of the audio samples (orange), while still gaining some
linguistic insights, showing that the embedding transfer approach is capable
of capturing audio and linguistic information.

##  V Conclusions

In this paper, we experimented with text-only, audio-only, and hybrid
approaches for topic ID on spontaneous Finnish speech. The experiments showed
that the solutions using only audio features fall slightly behind the standard
pipeline approach, but are still a viable option when the ASR system is not
available. The popular Wav2vec2 model falls behind the other audio-only
solutions, indicating that for the colloquial Finnish language, custom
solutions trained from scratch are a better option. The embedding transfer
system, proposed in this study, was able to effectively model the audio-
linguistic information, outperforming the other audio-only solutions. By
combining audio and text-based modalities, we observed a significant
improvement over the pipeline system and a slight improvement over the other
text-based solutions. The best results are realised using the multi-task
approach of jointly learning the ASR and topic ID tasks. In the future, we
plan to apply the experiments to the English language. Additionally, we plan
to replace the filter banks with features extracted from a pre-trained self-
supervised model.

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