# Statistics of a granular cluster ensemble at a liquid-solid-like phase
transition

Enrique Navarro<EMAIL_ADDRESS>Claudio Falcón
<EMAIL_ADDRESS>Departamento de Física, Facultad de Ciencias Físicas y
Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile

###### Abstract

We report on the construction of a granular network of particles to study the
formation, evolution and statistical properties of clusters of particles
developing at the vicinity of a liquid-solid-like phase transition within a
vertically vibrated quasi two-dimensional granular system. Using the data of
particle positions and local order from Castillo et al [Phys. Rev. Lett. 109,
095701 (2012)], we extract granular clusters taken as communities of the
granular network via modularity optimization. Each one of these communities is
a patch of particles with a very well defined local orientational order
embedded within an array of other patches forming a complex cluster network.
The distribution of cluster sizes and life-spans for the cluster network
depend on the distance to the liquid-solid-like phase transition of the quasi
two-dimensional granular system. Specifically, the cluster size distribution
displays a scale-invariant behavior for at least a decade in cluster sizes,
while cluster lifespans grow monotonically with each cluster size. We believe
this systematic community analysis for clustering in granular systems can
serve to study and understand the spatio-temporal evolution of mesoscale
structures in systems displaying out-of-equilibrium phase transitions.

##  I Introduction

Dry granular matter, i.e. a large collection of macroscopic particles
interacting via dissipative collisions, can be driven into different phases
(such as solid, liquid and gas-like ones) which depend on the energy
injection-dissipation balance occurring within the system [1, 2, 3, 4]. These
out-of-equilibrium equilibrium phases display interesting transitions, which
have been studied using very well-known theoretical tools relying on symmetry,
dimensionality and conservation arguments [5, 6, 7]. These arguments allow a
generic and universal way to characterize, in particular, the granular
system’s macroscopic evolution and properties usually linked to the large
scale, slow modes that dominate the dynamics of the system. In this regard,
the locality of interactions between grains are smeared out on the large scale
dynamics. Thus, the local granular information (such as local force
fluctuations and/or particle agglomerations) is lost within this modeling.
Nevertheless, this local information is of paramount importance for the
mechanical stability of granular matter when force chains are present [2, 8,
9, 10], as well as for the description of defects in vibrated granular matter
[11, 12, 13, 14], specially in the case of structured granular systems (such
as the case of nonisometric grains [15, 16, 17, 18, 19, 20]).

Recently, network science methodologies and techniques have been implemented
in the study of granular systems [21, 22, 23, 24, 25, 26, 27, 28, 29] in order
to understand the effect and importance of the local information (encoded into
a granular network of forces, positions or bond orientational order) on the
overall dynamics of the system. Quasi-two dimensional granular systems have
been the main subject of study in this approach due to the direct
accessibility to the information of each particle (such as position, velocity
and force). It must be noticed that this feature has has already been
exploited extensively to track out-of-equilibrium two-dimensional phase
transitions [30, 31, 32, 33, 34, 35, 36]. From the local information,
different types of networks can be constructed, defined solely by the
definition of nodes and their connections. In this paradigm, encoding the
complex relations of the granular system’s particles in a rather simple
networks can be extremely useful in the study of the granular system’s static
and/or dynamical properties, particularly the way it creates mesoscopic
structures such as force chains [37, 38] or clusters [39, 40].

In this work, we study the formation, evolution and statistics of clusters of
particles at the vicinity of a liquid-solid-like phase transition vía
community detection by optimizing a quality function called modularity [41].
Experimental data of particle orientational and spatial order taken from Ref.
[33] is used to construct a network which encodes within its links the local
orientational order of the granular system, enabling the use of the tools of
network science within the framework of non-equilibrium phase transitions. To
wit, we relate clusters of ordered particles of the non-equilibrium granular
system to hard partitions of its respective network, which are computed using
a null model for granular matter, validated via an entropic argument.

##  II Bond-orientational network

Figure 1: Image of steel particles (1 mm in diameter) used to construct our
clusters for $\Gamma=4.95$. a) $|Q_{n}^{4}|$ of each each particle. Color bar
shows available values of $|Q_{n}^{4}|$. b) Links between particles used to
construct the order network (in red) following Eq. (2).

A set of bond-orientational networks were constructed from data sets used in
Ref. [33], where $\mathcal{N}=11504$ stainless steel spherical particles of
diameter $d=1$ mm are confined in a box of lateral dimensions $100d\times
100d$ and a vertical dimension of $1.94d$. The box is vibrated vertically with
in a sinusoidal acceleration $a(t)=\Gamma g\cos{(\omega t)}$, where $g$ is
gravity, $1<\Gamma<6$ and $\omega=200\pi$ rad/s. 3000 images of the in-plane
motion of the particle ensemble were acquired at 10 fps and used to detect
them with subpixel accuracy using a particle tracking algorithm. From each
image, the position of each particle in the horizontal plane $\vec{r}_{n}$ and
the fourfold bond-orientational order parameter per particle

$Q^{4}_{n}=\frac{1}{\mathcal{M}_{n}}\sum_{m\in\mathcal{M}_{n}}e^{i4\alpha_{nm}}$
(1)

were calculated. Here $\mathcal{M}_{n}$ is set of nearest neighbors of
particle $n$ computed via a Voronoi partition and $\alpha_{nm}$ is the angle
between the neighbor $m$ of particle $n$ and a given axis. From this point on,
we will set $Q_{n}^{4}\equiv Q_{n}$, as we will only consider four-fold
orientational order.

For a given $\Gamma$, a (simple) network represented by an adjacency matrix

$A_{nm}^{(p)}=\begin{cases}1&\mbox{if
}\,|Q_{n}|,|Q_{m}|\geq\langle|Q|\rangle\,\,\mbox{and }\,m\in\mathcal{M}_{n}\\\
0&\mbox{otherwise }\\\ \end{cases},$ (2)

is computed for each image using $\langle|Q|\rangle$ as a threshold for the
solid-liquid transition [33]. The results presented here display no change
when $\langle|Q|\rangle$ is varied within $\pm 10\%$. Non-directionality for
the network connections is enforced by the symmetric nature of $A_{nm}^{(p)}$.
Tadpoles are directly removed from the adjacency matrix as no self-links are
available ($A_{nn}\equiv 0$ for all $n$) [42].

The binary adjacency matrix computed above encodes the connections (edges)
between particles (nodes) $n$ and $m$ with a large local orientational order.
In Fig 1a) a snapshot of particles at a given $\Gamma$ is presented displaying
the local value of $|Q_{4}|$ as a color map over each particle, showing that
they arrange themselves in cluster of similar $|Q_{4}|$ values. Using Eq.(2),
the connections given by the adjacency matrix are shown with red lines between
particles in Fig. 1b) We relate the particle clusters displayed in Fig 1a) via
the array of connections displayed in Fig 1b) to the network’s communities
[42], which are indivisible subgroups within the network (what is called a
hard partition). In a similar fashion one can encode the strength of the
connections between nodes vía a weighted adjacency matrix $W_{nm}^{(p)}$
between nodes $n$ and $m$, which in our case can be readily defined as
$W_{nm}^{(p)}=|Q_{n}||Q_{m}|A_{nm}^{(p)}$.

In our work, the clusters are found by maximizing a certain quality function,
the network’s modularity $\mathcal{Q}$,

$\mathcal{Q}=\sum_{n,m\in\mathcal{N}}(W_{nm}^{(p)}-\gamma
P_{nm})\delta_{C_{n},C_{m}}$ (3)

where node $n$ resides in community $C_{n}$ and node $m$ resides in community
$C_{m}$, $\gamma$ is called the resolution parameter, $P_{nm}$ is a matrix
term stemming from a null model [43] for the edge distribution within the
network, and $\delta_{x,y}$ is Kronecker’s delta function. Modularity has been
proposed as a direct way to measure and quantify the community structure
within large networks [41], where communities (or modules) are nodes that have
a larger ammount of non-zero connections among themselves than with the rest
of the network’s nodes [44]. One can understand $\gamma$ as the ratio between
the spatial densities of two communities in an optimal hard partition
following the null model selection. When $\gamma<1$ the hard partition tends
to favor the selection of larger communities rather than smaller ones, and
vice versa [43]. We have set $\gamma$=1 as we try to find the cluster dynamics
at the solid-liquid-like transition where the system tends to agglomerate into
clusters with a large span in scales (ideally, in a scale invariant way).

Figure 2: Examples of communities found from optimizing $\mathcal{Q}$ using
(a) the Newman-Girvan, (b) the geographic, and (c) our modified null model for
$\gamma=1$ and $\Gamma=4.95$. Patches of particles with the same color
represent communities. Particles with no connections (edges) as well as
singleton clusters are not shown as they do not contribute to the adjacency
matrix.

We have tested 3 different null models to track the structure of particle
clusters: the Newman-Girvan null model [45], the geographic null model [28]
and a modification of the former one, adapted to the observed data. The
Newman-Girvan null model, which has been extensively used in community
detection [43] is based on edges which are placed at random on each node. The
randomness of this edge configuration is quantified by the degree $k_{n}$ of
each of the network’s nodes (i.e. the number of edges connecting it) [42]. To
wit, $P_{nm}=k_{n}k_{m}/\kappa$ where
$\kappa=2\sum_{n}k_{n}=2\mathcal{N}\langle k\rangle$ is the total number of
edges of the network and $\langle k\rangle$ is the average number of edges per
node on the network. For granular matter, the hypothesis of random connections
for each and every node does not correspond to the reality of the local
granular network connectivity. To correct this hypothesis, Daniels and co-
workers [28] proposed a geographic null model, where nodes represent particles
and edges represent, for instance, forces between them, which are encoded into
$P_{nm}=\langle f\rangle A_{nm}^{(p)}$ with $\langle f\rangle$ the mean inter-
particle force. This null model displays communities which follow the local
force chains of the granular material.

Figure 3: A given cluster of particles (in the bottom plane) is represented by
a cluster node (in the upper plane) for $\Gamma=4.83$. The red lines show the
correspondance between the two networks. The black lines (in the upper plane)
show the edges of the cluster network which represent the amount of particles
at the edges between particle clusters (in the bottom plane).

These two null models display very different community structures when we
maximize $\mathcal{Q}$ for a given $\Gamma$ and $\gamma=1$, as shown in Fig.
2. Here only connected particles are depicted, which contribute to non-zero
terms to the adjacency matrix. Using the Newman-Girvan null model,(cf. Fig.
2a)) the cluster array found from the $\mathcal{Q}$ optimization displays
clusters with sizes that are exponentially distributed with several inter-
cluster connections, much more than the ones found using the geographic null
model (cf. Fig. 2b)). A particular characteristic of the cluster array arising
from the use of the geographic model is the appearance of single particle
clusters within clusters of 4 to 10 particles, which develop as a consequence
of the nature of a model constructed to follow force chains in compacted
granular matter [28]. Following this feature, we propose a modified model to
take into account the quality of the ordering between neighboring particles.
We propose the null model $P_{nm}=\beta\langle|Q|\rangle^{2}A_{nm}^{(p)}/2$
following the geographic model approach, where $\beta$ is the fraction of
ordered particles within the system (see the Appendix for an entropic
justification of this null model).

Figure 4: a) Distribution of cluster sizes $\rho(n_{c})$ vs $n_{c}$ for
$0.02<\epsilon<0.2$. Inset: Best fit power-law exponent $\alpha$ as a function
of $\epsilon$ for $N_{c}\in\\{50,500\\}$. Horizontal dashed line shows the
limiting value of $\alpha$ as a function of $\epsilon$. Vertical dashed line
shows the larges value of $\epsilon$ from which $\alpha saturates$. Color bar
shows values of $\epsilon$. b) Cluster cut-off $n_{c}^{\star}$ as a function
of $\epsilon$. c) the Largest cluster distance $\bar{n}_{c}$ as a function of
$\epsilon$.

To find algorithmically these communities we optimize the community partition
by maximizing $\mathcal{Q}$ using a local greedy maximization algorithm [46,
47] in such a way that the total edge weight within the communities is as
large as possible with respect to a chosen null model [43]. The optimally
computed community partition is then recomputed a number of times (chosen
heuristically [48]) in order to assure a certain convergence as the
maximization process is NP-hard [49]. In our maximization process we have
repeated the calculation at least 20 times per images by permuting the nodes,
finding the same clusters.

##  III Cluster network

After the above optimization process, we find a set of communities (clusters)
$C_{\alpha}$ of particles which are connected by intra-communal edges
($A_{nm}^{(p)}$ for $n,m$ $\in C_{\alpha}$). These clusters are also connected
between them via inter-communal edges ($A_{nm}^{(p)}$ for $n$ $\in
C_{\alpha}$, $m\in C_{\beta}$ and $\alpha\neq\beta$) thus forming a new
coarse-grained network. We construct a new adjacency matrix for the (much
smaller) cluster network

$A_{\alpha\beta}^{(C)}=\begin{cases}1&\mbox{if }\,A_{nm}^{(p)}=1\,\,\mbox{for
any }\,n\in C_{\alpha},m\in C_{\beta},\\\ 0&\mbox{otherwise. }\\\ \end{cases}$
(4)

which is used to describe the cluster evolution as a function of $\Gamma$ near
the solid-liquid transition occurring at $\Gamma_{c}=5.09\pm 0.07$ [33]. This
means that the inter-communal edge between two clusters is nonzero if there is
at least one particle shared at the edges of these clusters. A representation
of this clusterisation process is shown in Fig. 3 using a small region of an
image at $\Gamma=4.95$. The detected clusters of particles, represented by
large nodes with sizes scaling with the number of particles per cluster,
display inter-cluster connections (black lines) which represents that there
are particles at the edges between clusters.As reported by Castillo [33], the
granular system displays an increasing global orientational order as
$\langle|Q|\rangle$ grows linearly with $\Gamma<\Gamma_{c}$. As this occurs,
particles arrange themselves into clusters with a larger and larger number of
particles that adjoin other clusters via shared particles at their edges.
Using the normalized acceleration
$0.02<\epsilon=(\Gamma_{c}-\Gamma)/\Gamma_{c}<0.2$, we compute the
distribution of sizes $n_{c}$, and lifespan $\tau_{c}$ of the clusters from
these network sets as a function of $\epsilon$.

###  III.1 Size and lifespan properties of the cluster network

The distribution of cluster sizes $\rho(n_{c})$ using our modified null model
is shown in Fig. 4 for different values of $\epsilon$. It displays a power-law
behavior for $\rho(n_{c})$ as a function of $n_{c}$, $\rho(n_{c})\sim
n_{c}^{\alpha}$ for at least a decade in cluster sizes between 20 to 200
particles per cluster. The best fit values for $\alpha$ within the above range
are depicted in Fig. 4 a) (inset), showing a limiting behavior for low
$\epsilon$. This power law behavior saturates displaying a cut-off for cluster
sizes larger than a given value $n_{c}^{\star}(\epsilon)$ which is larger and
larger as we approach the liquid-solid-like transition. Using a simple
exponential correction $\rho(n_{c})\sim
n_{c}^{\alpha}\times\text{exp}(-n_{c}/n_{c}^{\star})$, we compute
$n_{c}^{\star}=n_{c}^{\star}(\epsilon)$ which is displayed in Fig. 4 b).
Similarly, one can compute the largest cluster distance found for each value
of $\epsilon$, $\bar{n}{{}_{c}}=\langle n_{c}^{2}\rangle$ which is shown in
Fig. 4 c). As in the case of $n_{c}^{\star}$, $\bar{n}_{c}$ decreases with
$\epsilon$. One can probe a power-law behavior for both $n_{c}^{\star}$ and
$\bar{n}_{c}$ in $\epsilon$ in the range where $\alpha$ converges towards
-2.0, as it is depicted in Fig. 4 a) (inset). Within this range,
$n_{c}^{\star}\sim\epsilon^{\zeta^{\star}}$ with $\zeta^{\star}=0.23+0.05$ and
$\bar{n}_{c}\sim\epsilon^{\bar{\zeta}}$ with $\bar{\zeta}=0.21+0.05$. One
might adscribe the above power-law behavior for cluster size distribution to
percolation-related problems [50, 51, 52, 53] as it displays the predicted
exponent for our cluster distribution. Recently, a couple of different second-
order phase transitions (one of activity and one of orientational order) have
been found to appear in a system of two-dimensional sheared granular discs at
the same critical value of the control parameter [35], which might be the case
of the data from Ref. [33]. We have checked that for our cluster network, this
is not the case, as neither $n_{c}^{\star}$ nor $\bar{n}_{c}$ display the
critical behavior expected in percolation-related problems as a function of
$\epsilon$ within the experimental range of the present data.

Figure 5: Temporal cluster evolution procedure for two consecutive images. (a)
Cluster network detection for both images. (b) Area projection of one cluster
network onto the other. (c) Link creation between clusters. (d) Link trimming
by weight. (e) Backward link trimming. (f) Forward link trimming.

Using these cluster network sets, we track the temporal cluster evolution for
a given $\epsilon$ to compute the average lifespan of clusters of size
$n_{c}$. The way we track the cluster temporal evolution is depicted in Fig.
5. The process starts by tracking the cluster networks within two consecutive
images $j$ and $j+1$ (Fig. 5a)). The area of each cluster from image $j$ is
projected onto image $j+1$ (Fig. 5b)) and a link is created between a cluster
$C_{n}^{j}$ of image $j$ and a cluster $C_{m}^{j+1}$ of image $j+1$ if nodes
from $C_{m}^{j+1}$ correspond to the projected area of $C_{n}^{j}$ (Fig. 5c))
.This link has a directed weight $p^{(f)}_{nm}$ equal to the ratio of the
number of nodes of $C_{n}^{j}$ and $C_{m}^{j+1}$. In the same fashion there is
a directed weight $p^{(b)}_{mn}$ equal to the ratio of the number of nodes of
$C_{m}^{j+1}$ and $C_{n}^{j}$. We keep only the links with
$(p^{(b)}_{nm},p^{(f)}_{nm})>0.5$, which means that both clusters share at
least half of the nodes (Fig. 5d)). As $C_{m}^{j+1}$ can be linked with more
than one cluster $C_{n}^{j}$, we restrict the linkage between clusters by
keeping the one with the largest $p^{(b)}_{mn}$ (Fig. 5e)). The same procedure
is done then forwards in time with $p^{(f)}_{nm}$, to link only one cluster
$C_{n}^{j}$ to only cluster $C_{m}^{j+1}$ (Fig. 5f)). Following this link path
for each cluster from image $j=1$ to $j=\mathcal{N}$ we can track its lifespan
as the number of links it holds until the cluster is no longer traceable.

From the above procedure, the average lifespan $\tau_{c}=\tau(n_{c})$ as a
function of $n_{c}$ is shown in Fig. 6 using the experimental sample frequency
of 10 Hz. Clusters with sizes in $(n_{c}-\Delta n_{c},n_{c}+\Delta n_{c})$ are
binned together using $\Delta n_{c}=$20 which is the average size of the small
clusters found around the edges of the images (see Fig. 1). For a fixed
$\epsilon$, $\tau_{c}$ grows monotonically with $n_{c}\in\\{50,500\\}$ as a
power law with exponent $\eta=0.24\pm 0.05$ as a best fit slope. Clusters with
larger $n_{c}$ are sustained with a larger $\tau_{c}$, displaying a large
dispersion as $n_{c}$ is of the order of 1000 particles for all values of
$\epsilon$ as shown in Fig. 6a). Lifespan fluctuations $\sigma(\tau_{c})$ are
depicted in Fig. 6b) as a function of $n_{c}$, which grow (on average)
monotonically with $\tau_{c}$. A power law can be fitted as
$\sigma(\tau_{c})\sim\tau_{c}^{\mu}$, with $\mu=0.55\pm 0.05$ as the best fit
slope. This shows that as clusters grow in size and thus $\tau_{c}$ increases,
$\sigma(\tau_{c})/\tau_{c}\sim\tau_{c}^{-1/2}$ decreases, which means that
$\tau_{c}$ can be used a proper time scale to describe the slow dynamics of
granular cluster evolution. It is important to notice that this findings are
also observed when $(p_{nm}^{(f)},p_{nm}^{(b)})$ are reduced from 0.5 to 0.1,
albeit larger larger fluctuations in $\tau_{c}$ are found.

Figure 6: a) Average lifespan $\tau_{c}$ as a function of $N_{c}$ as a
function $\epsilon$. b) Standard deviation of the average lifespan
$\sigma(\tau_{c})$ vs the average lifespan $\tau_{c}$.

##  IV Conclusions

The results presented above display a community detection scheme via a
$\mathcal{Q}$ optimization which allows the computation of a cluster ensemble
of ordered particles in a quasi-two dimensional vibrated granular system close
to a solid-to-liquid-like phase transition. These clusters are constructed as
hard partitions using an entropic null model in $\mathcal{Q}$ which takes into
account both the bond-orientational and the spatial order of the granular
system. From the computed cluster ensemble as we increase the normalized
acceleration $\epsilon=(\Gamma_{c}-\Gamma)/\Gamma_{c}$, we track the cluster
size distribution $\rho(n_{c})$ and the cluster mean lifespan $\tau_{c}$. The
cluster size distribution displays a power-law dependence on $n_{c}$ with an
exponent close to -2.0 which independent of $\epsilon$ in the range
$n_{c}\in\\{50,500\\}$, and an exponential cut-off with a slope equal to
$1/n^{\star}_{c}(\epsilon)$ which increases with $\epsilon$. The mean lifespan
$\tau_{c}$ of clusters with size $n_{c}$ increases with a power-law as a
function $n_{c}$ with an exponent close to $1/4$ for $n_{c}\in\\{50,500\\}$.
For larger values of $n_{c}$, large fluctuations of $\tau_{c}$ are observed.

We believe that this community analysis vía $\mathcal{Q}$ optimization for
cluster detection in granular systems can be of use to study and understand
the spatio-temporal evolution of mesoscale structures in systems, specially
ones displaying out-of-equilibrium phase transitions. Furthermore, the
application of an entropic null model in the $\mathcal{Q}$ optimization scheme
enables a systematic computation of mesoscale structures in out-of-equilibrium
granular systems and their dynamics without the necessity of defining ad-hoc
parameter values [28]. It is our hope that this scheme will be used on other
quasi-two dimensional granular systems to study, compare and contrast the
dynamics of their mesoscale structures as phase transitions develop in such
systems.

##  V Acknowledgements

The authors wish to thank G. Castillo, N. Mujica and R. Soto for kindly
allowing access to the database used in Ref. [33]. This work was partially
supported by FONDECYT Regular Grants 1190005 and 1210656.

##  Appendix A A Null model for ordered phases in granular matter

The null model presented in the present work to find communities was found by
an entropic maximization scheme. We start by considering a multiplex network
[54, 55, 56], which means that two nodes within one network can belong also to
a different one at the same time. In our case we will consider a multiplex
formed by $N$ labeled nodes from $n=1,...,N$ and $M$ layers represented by a
$\vec{G}=(G^{(1)},...,G^{(M)})$, where $G^{(\alpha)}$ indicates the set of all
possible networks at layer $\alpha=1,...,M$ of the multiplex. Nodes can be
connected by specifying an adjacency matrix $A_{n,m}^{\alpha}$ (as defined in
the text) which can be weighted, depending on the type of network under study.

With this in mind, we task ourselves to find the ensemble properties of the
multiplex by specifying the probability $P(\vec{G})$ for every possible
multiplex. For a given set of multiplex probabilities $P(\vec{G})$ we can
compute the entropy of the multiplex ensemble

$\mathcal{S}=-\sum_{\vec{G}}P(\vec{G})\log{(P(\vec{G}))},$ (5)

which is proportional to logarithm of the number of possible multiplexes of
the ensemble. As in statistical mechanics, we can find an ensemble of
multiplexes that maximize $\mathcal{S}$, subjected to a given set of
restrictions (a Gibbs probability ensemble). If we assume that the layers of
the multiplex are uncorrelated, the probability of the multiplex can be
written as

$P(\vec{G})=\prod_{\alpha=1}^{M}P(G^{\alpha})$ (6)

which simplifies greatly our task.

We now specify our problem within this framework. First we will set $M=2$ and
$\alpha=o,s$. One layer, called the orientational layer ($o$), connects pairs
of nodes $(n,m)$ with intralayer weights $w^{(o)}_{n,m}=|Q_{n}||Q_{m}|$ as
long as $|Q_{n}|,|Q_{m}|>\langle|Q|\rangle$. A second layer, called the
spatial layer ($s$), connects pairs of nodes $(k,l)$ with intralayer weights
$w^{(s)}_{k,l}=f(d_{k,l})$ where $d_{k,l}=|\vec{r}_{n}-\vec{r}_{m}|$ is the
spatial distance (measured in number of diameters $d$) between nodes and
$f(x)$ is a scalar function that goes to zero with increasing $x$. Note that
in the orientational layer any pair of nodes $(n,m)$ that satisfy the
thresholding scheme for $|Q_{n}|$ and $|Q_{m}|$ is connected. In this
approximation, local orientational order does not couple directly with spatial
order [33], we will assume for simplicity that $\langle
w^{(o)}_{n,m}w^{(s)}_{n,m}\rangle=\langle w^{(o)}_{n,m}\rangle\langle
w^{(s)}_{n,m}\rangle$, i.e., we neglect the overlap between layers [56]. This
means that the probability of finding a link between nodes $(n,m)$ in both
layers of the multiplex is simply the multiplication of the intralayer link
probabilities for these nodes, following Eq.(6).

In a canonical multiplex ensemble for our data, the set of multiplexes
$\vec{G}=(G^{(o)},G^{(s)})$ satisfy constraints on average, informed by the
dynamics of the granular layer both in the orientational layer and in the
spatial one. In this case, for the orientational layer we set the global
constrain

$\displaystyle L^{(o)}$ $\displaystyle=$
$\displaystyle\sum_{\vec{G}}P(\vec{G})\sum_{n<m}^{N}w_{n,m}^{(o)}$
$\displaystyle=$
$\displaystyle\beta\frac{N(N-1)}{2}\langle|Q^{\star}|\rangle^{2}$

which can be understood as setting on average a fraction $\beta$ of nodes
(particles) within the system with a fourfold bond-orientational order
parameter per particle $\langle|Q^{\star}|\rangle$. For the spatial layer the
constraint

$0=\sum_{\vec{G}}P(\vec{G})\sum_{n<m}^{N}\Theta(w_{n,m}^{(s)})(\Theta(d_{nm}-d))$
(8)

sets in average the spatial interaction of particles only the nearest
neighbors for each particle. Here $\Theta(x)$ is the Heaviside function [57].
Thus, maximizing $\mathcal{S}$ from Eq.(5) constraint to Eqs.(A)-(8) gives

$P(\vec{G})=\frac{e^{\sum_{n<m}^{N}-\Lambda
w_{n,m}^{(o)}-\Delta\Theta(w_{n,m}^{(s)})(\Theta(d_{nm}-d))}}{\mathcal{Z}}$
(9)

where $\mathcal{Z}$ is the partition function of our system and
$(\Lambda,\Delta)$ are Lagrange multiplier enforcing the orientational
($\Lambda$) and spatial ($\Delta$) constraints of the system, respectively. As
the layers are assumed to be uncorrelated, we calculate
$\mathcal{Z}=\mathcal{Z}^{(o)}\times\mathcal{Z}^{(s)}$ with

$\displaystyle\mathcal{Z}^{(o)}$ $\displaystyle=$
$\displaystyle\sum_{G^{(o)}}e^{-\Lambda\sum_{n<m}^{N}w_{n,m}^{(o)}}$ (10)
$\displaystyle=$
$\displaystyle\prod_{n<m}\sum_{\\{w_{n,m}^{(o)}\\}}e^{-\Lambda w_{n,m}^{(o)}}$
$\displaystyle=$ $\displaystyle\prod_{n<m}(1-e^{-\Lambda})^{-1}$

and

$\displaystyle\mathcal{Z}^{(s)}$ $\displaystyle=$
$\displaystyle\sum_{G^{(s)}}e^{-\Delta\sum_{n<m}^{N}\Theta(w_{n,m}^{(s)})(\Theta(d_{nm}-d))}$
(11) $\displaystyle=$
$\displaystyle\prod_{n<m}\sum_{\\{\Theta(w_{n,m}^{(s)})\\}=0,1}e^{-\Delta\Theta(w_{n,m}^{(s)})(\Theta(d_{nm}-d))}$
$\displaystyle=$ $\displaystyle\prod_{n<m}(1+e^{-\Delta(\Theta(d_{nm}-d))}).$

It is now straightforward to compute the link probability (i.e., the null
model) for our mulitplex between nodes $(n,m)$ from the product partition
function $P_{nm}$. The link probability between two nodes $P_{nm}^{(o)}$ for
the orientational layer can be computed from the weight probability
$\pi_{nm}(w)$ of the link for a given weight $w^{(s)}_{nm}=w$

$\displaystyle\pi_{nm}(w)$ $\displaystyle=$
$\displaystyle\sum_{\vec{G}}P(\vec{G})\delta(w^{(o)}_{nm}=w)$ (12)
$\displaystyle=$ $\displaystyle e^{-\Lambda w}\times(1-e^{-\Lambda}),$

as $P_{nm}^{(o)}=1-\pi_{nm}(w=0)=e^{-\Lambda}$. Using Eq.(A) and Eq.(10), we
link $\Lambda$ and the restriction via

$\displaystyle L^{(o)}$ $\displaystyle=$
$\displaystyle-\frac{\partial\log{(\mathcal{Z}^{(o)})}}{\partial\Lambda}=\sum_{n<m}\frac{e^{-\Lambda}}{1-e^{-\Lambda}}$
(13) $\displaystyle=$
$\displaystyle\frac{N(N-1)}{2}\frac{e^{-\Lambda}}{1-e^{-\Lambda}}$
$\displaystyle=$
$\displaystyle\beta\frac{N(N-1)}{2}\langle|Q^{\star}|\rangle^{2},$

which sets
$e^{-\Lambda}=\beta\langle|Q^{\star}|\rangle^{2}/(1+\beta\langle|Q^{\star}|\rangle^{2})=P_{nm}^{(o)}$.
The link probability between two nodes $p_{nm}^{(s)}$ in the spatial layer can
be computed similarly imposing that the weigh of the link t is larger than
zero, i.e.,

$P_{nm}^{(s)}=\sum_{\vec{G}}P(\vec{G})\delta(w^{(s)}_{nm}>0)=\frac{e^{-\Delta(\Theta(d_{nm}-d))}}{1+e^{-\Delta(\Theta(d_{nm}-d))}}.$
(14)

Similarly as before, we link $\Delta$ with the restriction vía

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{\partial\log{(\mathcal{Z}^{(s)})}}{\partial\Delta}$ (15)
$\displaystyle=$
$\displaystyle\sum_{n<m}\frac{e^{-\Delta(\Theta(d_{nm}-d))}}{1+e^{-\Delta(\Theta(d_{nm}-d))}}(\Theta(d_{nm}-d)),$

which is fulfilled only if $\Delta\to\infty$ as it needs that the sum over all
non-neighbors yields zero. In that case,
$P_{nm}^{(s)}=\frac{1}{2}(1-\Theta(d_{nm}-d))$ for every neighboring pair
$(n.m)$ in the layer and $P_{nm}^{(s)}=0$ otherwise. Following Eq.(6), the
link probability for our multiplex (and thus our null model for the granular
network) is

$\displaystyle P_{nm}^{(p)}$ $\displaystyle=$ $\displaystyle
P_{nm}^{(s)}\times P_{nm}^{(o)}$ (16) $\displaystyle=$
$\displaystyle\frac{\beta\langle|Q^{\star}|\rangle^{2}}{1+\beta\langle|Q^{\star}|\rangle^{2}}\frac{(1-\Theta(d_{nm}-d))}{2},$

which can be approximated to
$\beta\langle|Q^{\star}|\rangle^{2}(1-\Theta(d_{nm}-d))/2$ for $\beta\ll 1$.
In this expression the term $(1-\Theta(d_{nm}-d))$ can be understood as the
spatial adjacency matrix $A_{nm}^{(s)}$ for particles that are in contact
(nearest neighbors). An important consequence of this approach is the entropic
justification for the use of a geographic model proposed in Ref. [28] to find
community structure in a granular system subjected to simple local and global
restrictions.

Figure 7: Cluster network connection statistics. a) Distribution of the number
of connections for the entire network $\rho(\kappa)$ versus $\kappa$ as a
function of $\epsilon$. b) Distribution of connections per node $\rho(k)$
versus $k$ as a function of $\epsilon$. c) Mean value of the number of
connections per node $\langle k(n_{c})\rangle$ versus $n_{c}$ as a function of
epsilon. Same colorbar as in previous figures.

Furthermore, using these techniques, one can compute directly the entropy of
the granular network via the application Eq.(5) to the problem at hand. As the
partition function for the problem in the limit of negligible overlap is
$\mathcal{Z}=\mathcal{Z}^{(o)}\times\mathcal{Z}^{(s)}$,
$\mathcal{S}=\mathcal{S}(\epsilon)=\mathcal{S}^{(o)}+\mathcal{S}^{(e)}=-\Lambda
L^{(o)}+\log{(\mathcal{Z}^{(o)})}+\log{(\mathcal{Z}^{(e)})}$, which can be
used to compute the entropy for the ensemble of granular networks as a
function of $\epsilon$, which is depicted in Fig. 8. As $\epsilon$ is the
normalized acceleration which injects energy into the granular system, one
could use the calculation of the granular network entropy
$\mathcal{S}(\epsilon)$ as a proxy to the an out-of-equilibrium specific heat
$\frac{d\mathcal{S}}{d\epsilon}\simeq C_{V}(\epsilon)$ which shows no
divergence as $\epsilon$ approaches zero close to the solid-liquid type
transition. This is consistent with the $\alpha$=0 exponent (cf. Fig. 8b)) in
the language of dynamical phase transitions [6] within the $C-$model used to
describe this granular system [33].

Figure 8: a) Semilogarithmic horizontal plot of $\mathcal{S}(\epsilon)$ vs
$\epsilon$ for the granular network. b) Semilogarithmic horizontal plot of
$\frac{d\mathcal{S}(\epsilon)}{d\epsilon}$ vs $\epsilon$ for the granular
network.

##  Appendix B Cluster network properties

In this Appendix, we will focus on the statistical properties of the cluster
network constructed above. We characterize these networks as a function of the
normalized acceleration $\epsilon=(\Gamma_{c}-\Gamma)/\Gamma_{c}$. We will
first deal with the statistics of the edges within the network. In Fig. 7a) we
depict the distribution of the number of connections $\kappa$ as a function of
$\epsilon$. The mean number of connections $\langle\kappa\rangle$ decreases
linearly with $\epsilon$, which is expected as smaller clusters merge into
larger ones, and thus less connections are formed within the network. The
distribution can be reasonable fitted by a chi-squared distribution where the
number of degrees of freedom $\chi$ increases with decreasing $\epsilon$. For
the case of connections per node $k$ (cf. Fig. 7b)), the distribution can be
reasonable fitted with an exponential function for $k>4$, with a slope which
decreases with $\epsilon$, allowing larger values for $k$ as we approach the
solid-liquid transition. We also computed the amount of connections as a
function of $n_{c}$, $k=k(n_{c})$ grows as a power law with $n_{c}$ as
$n_{c}^{\zeta_{k}}$ with $\zeta_{k}=0.55\pm 0.05$ and is independent of
$\epsilon$. This scaling, shown in Fig. 7c), is to be expected as the
connections between clusters depends on the number of particles at the edges
of the clusters, and thus $k\sim n_{c}^{1/2}$.

## References

  * [1] I. Aranson, L. Tsimring, Granular Patterns, (Oxford University Press; 2nd edition, 2014).
  * [2] H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod. Phys. 68, 1259–1273 (1996).
  * [3] P. -G. de Gennes, Rev. Mod. Phys. 71,374–382 (1999).
  * [4] I. Aranson, and L. S. Tsimring, Rev. Mod. Phys. 78, 641–692 (2006).
  * [5] M. Henkel, H. Hinrichsen, and S. Lübeck, Non-Equilibrium Phase Transitions: Volume 1: Absorbing Phase Transitions (Theoretical and Mathematical Physics) (Springer Dordrecht, First Edition, 2009).
  * [6] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
  * [7] J. D. Brock ,Bond-Orientational Order in Condensed Matter Systems, pp. 1-29 (edkted by K. J. Strandburg, Springer-Verlag, New York, 1992).
  * [8] R.M. Nedderman, Statics and Kinematics of Granular Materials (Cambridge University, Cambridge, UK, 1992).
  * [9] C. Goldenberg and I. Goldhirsch, Phys. Rev. Lett. 89, 084302 (2002).
  * [10] J. H. Snoeijer, T. J. H. Vlugt, M. van Hecke, and W. van Saarloos, Phys. Rev. Lett. 92, 054302 (2004).
  * [11] S. Douady, S. Fauve, and C. Laroche, Europhys. Lett. 8, 621–627 (1989).
  * [12] F. Melo, P. Umbanhowar, and H. L. Swinney, Phys. Rev. Lett. 75, 3838–3841 (1995).
  * [13] K, Watanabe, and H. Tanaka, Phys. Rev. Lett. 100, 158002 (2008).
  * [14] T. Schindler and S. C. Kapferm, Phys. Rev. E 99, 022902 (2019).
  * [15] J. Galanis, R. Nossal, W. Losert, and D. Harries, Phys. Rev. Lett. 105, 168001 (2010).
  * [16] M. González-Pinto, F. Borondo, Y. Martínez-Ratón, and E. Velasco, Soft Matter 13,2571-2582 (2017).
  * [17] M. González-Pinto, F. Borondo, Y. Martínez-Ratón, and E. Velasco, Soft Matter, 13, 2571-2582, (2017).
  * [18] M.González-Pinto et al, New J. Phys. 21, 033002 (2019).
  * [19] E. Basurto, P. Gurin, S. Varga, and G. Odriozola, Phys. Rev. Res. 2, 013356 (2020).
  * [20] A. Díaz-De Armas, M. Maza-Cuello, Y. Martínez-Ratón, and E. Velasco, Phys. Rev. Res. 2, 033436 (2020).
  * [21] P. R. et al, Granul. Matter 1, 203–211 (1999).
  * [22] J. S. Andrade, H. J. Herrmann, R. F. S. Andrade and L. R. da Silva, Phys. Rev. Lett. 94 18702 (2005).
  * [23] A. Smart, P. Umbanhowar and J. Ottino, EPL 79, 24002 (2007).
  * [24] D. M. Walker, and A. Tordesillas, Int, J, Solids Struct. 47 624–639 (2010).
  * [25] S. Ardanza-Trevijano, I. Zuriguel, R. Arévalo, and D. Maza, Phys. Rev. E 89, 052212 (2014).
  * [26] D.M. Walker, A. Tordesillas, M. Small, R. P. Behringer, and C. K. Tse, Chaos, 24, 013132 (2014).
  * [27] J. A. Dijksman, et al, Phys. Rev. E 97, 042903 (2018).
  * [28] L. Papadopoulos, M. A. Porter, K. E. Daniels, and D. S. Bassett, J. Compl. Net. 6, 485–565 (2018).
  * [29] S. Nauer, L.Böttcher, and M. A. Porter, J. Comp. Net. 8 1–27 (2019).
  * [30] J. S. Olafsen, and J. S. Urbach, Phys. Rev. Lett 81,4369–4372 (1998).
  * [31] A. Prevost, P. Melby, D. A. Egolf, and J. S. Urbach, Phys. Rev. E 70, 050301 (2004).
  * [32] P. M. Reis,R. A. Ingale, and M. D. Shattuck, Phys. Rev. Lett. 96, 258001 (2006).
  * [33] G. Castillo, N. Mujica, and R. Soto, Phys. Rev. Lett. 109, 095701 (2012).
  * [34] X. Sun, Y. Li, Y. Ma. and Z. Zhang, Sci. Rep. 6, 24056 (2016).
  * [35] A. Ghosh et al, Phys. Rev. Lett. 129, 188002 (2022).
  * [36] L. Galliano, M. E. Cates, and L. Berthier, Phys. Rev. Lett. 131, 047101 (2023).
  * [37] D. S. Bassett et al, Soft Matter 11, 2731-2744 (2015).
  * [38] L. Papadopoulos, J. Puckett, K. E. Daniels, and D. S. Bassett, Phys. Rev. E 94, 032908 (2016); Y. Huang and K. E. Daniels, Granul. Matter 18 85 (2016).
  * [39] B. Kou et al, Phys. Rev. Lett. 121, 018002 (2018).
  * [40] Yi. Sun et al, Granul. Matter 22, 70 (2020).
  * [41] M.E.J. Newman, Proc. Nat. Acad. Sci. 103(23), 8577-82 (2006).
  * [42] M. E. J. Newman, Networks: An Introduction (Oxford University Press, 1st Ed., 2010).
  * [43] M. Sarzynska, E. A. Leicht, G. Chowell and M. A. Porter, J. Comp.x Net. 4, 363–406 (2016).
  * [44] M. Barthélemy, Phys. Rep. 499 1–101 (2011).
  * [45] M. E. J. Newman and M. Girvan, Phys. Rev. E. 69, 026113 (2004).
  * [46] A. Clauset, M. E. J. Newman, and C. Moore, Phys. Rev. E 70(6) (2004).
  * [47] V. D. Blondel et al, J. Stat. Mech. P10008 (2008).
  * [48] L. G. S. Jeub, O. Sporns and S. Fortunato, Sci Rep. 8, 3259 (2018).
  * [49] U. Brandes et al, IEEE Trans. Knowl. Data Eng. 20, 172-188 (2008).
  * [50] D. Stauffer, Phys. Rep. 54, 1-74 (1979).
  * [51] M. B. Isichenko, Rev. Mod. Phys. 64 961-1043 (1992).
  * [52] A. A. Saberi, Phys. Rep. 578, 1-32 (2015)
  * [53] M. Li, et al, Phys. Rep. 907, 1-68 (2021).
  * [54] G. Bianconi, Multilayer Networks: Structure and Function (Oxford University Press, 2023).
  * [55] G. Bianconi, Phys. Rev. E 87, 062806 (2013).
  * [56] G. Menichetti, D. Remondini, and G. Bianconi, Phys. Rev. E 90, 062817 (2014).
  * [57] G. B. Arfken, H. J. Weber and F. E. Harris, Mathematical Methods for Physicists: A Comprehensive Guide (Academic Press; 7th edition, 2012).


# Comparison results for solutions to $p$-Laplace equations with Robin
boundary conditions

Vincenzo Amato, Andrea Gentile, Alba Lia Masiello

###### Abstract

In the last decades comparison results of Talenti type for Elliptic Problems
with Dirichlet boundary conditions have been widely investigated. In this
paper, we generalize the results obtained in [3] to the case of $p$-Laplace
operator with Robin boundary conditions.

The point-wise comparison, obtained in [3] only in the planar case, holds true
in any dimension if $p$ is sufficiently small.

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>
000 2010 Mathematics Subject Classification: 35J92, 35P15. 000 Key words and
phrases: $p$-Lapacian, Robin boundary conditions.

##  1 Introduction

Let $\beta$ be a positive parameter and let $\Omega$ be a bounded open set of
$\operatorname{\mathbb{R}}^{n}$, $n\geq 2$, with Lipschitz boundary.

Let $f\in L^{p^{\prime}}(\Omega)$ be a non-negative function. We consider the
following problem

$\begin{cases}-\text{div}({\left|\nabla u\right|}^{p-2}\nabla u)=f&\text{ in
}\Omega\\\ {\left|\nabla u\right|}^{p-2}\displaystyle{\frac{\partial
u}{\partial\nu}}+\beta{\left|u\right|}^{p-2}u=0&\text{ on
}\partial\Omega.\end{cases}$ (1)

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

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

We want to establish a comparison principle with the solution to the following
symmetrized problem

$\begin{cases}-\text{div}({\left|\nabla v\right|}^{p-2}\nabla
v)=f^{\sharp}&\text{ in }\Omega^{\sharp}\\\ {\left|\nabla
v\right|}^{p-2}\displaystyle{\frac{\partial
v}{\partial\nu}}+\beta{\left|v\right|}^{p-2}v=0&\text{ on
}\partial\Omega^{\sharp},\end{cases}$ (3)

where $\Omega^{\sharp}$ is the ball centered in the origin with the same
measure of $\Omega$ and $f^{\sharp}$ is the Schwarz rearrangement of $f$ (see
next section for its definition).

This kind of problems has been widely investigated in the last decades. The
first step is contained in [10], where Talenti proved a pointwise comparison
result between $u^{\sharp}$ and $v$ in the case of Dirichlet Laplacian. After
this, several papers generalized the result of Talenti: for instance, the one
by Talenti himself [11], in which the operator is a generic non-linear
operator in divergence form, or the one by Alvino, Lions and Trombetti [2] in
which the authors deal with both elliptic and parabolic cases: in both papers,
Dirichlet boundary conditions are considered.

Different kind of boundary conditions are considered by Alvino, Nitsch and
Trombetti in [3], where they establish a comparison between a suitable norm of
$u$ and $v$, respectively solution to

$\begin{cases}-\Delta u=f&\text{in}\,\Omega\\\ \displaystyle{\frac{\partial
u}{\partial\nu}}+\beta
u=0&\text{on}\,\partial\Omega.\end{cases}\quad\quad\begin{cases}-\Delta
v=f^{\sharp}&\text{in}\,\Omega^{\sharp}\\\ \displaystyle{\frac{\partial
u}{\partial\nu}}+\beta u=0&\text{on}\,\partial\Omega^{\sharp}.\end{cases}$

They found out that if $f$ is a non-negative function in $L^{2}(\Omega)$, then

$\displaystyle{\left\|u\right\|}_{L^{k,1}(\Omega)}\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}\quad$
$\displaystyle\forall 0<k\leq\frac{n}{2n-2}$
$\displaystyle{\left\|u\right\|}_{L^{2k,2}(\Omega)}\leq{\left\|v\right\|}_{L^{2k,2}(\Omega^{\sharp})}\quad$
$\displaystyle\forall 0<k\leq\frac{n}{3n-4}$

where ${\left\|\cdot\right\|}_{L^{k,q}}$ is the so called _Lorentz norm_ ,
whose definition can be found in next section. Moreover, the authors in [3]
were able to establish a comparison á la Talenti,

$u^{\sharp}(x)\leq v(x),\quad\forall x\in\Omega^{\sharp}$

in the case $f\equiv 1$ and $n=2$. This will be the starting point of our
work: it will be clear that our results coincide with the one in [3] in the
case $p=2$.

For completeness sake, we cite that this wasn’t the first result in this
sense, indeed in [6] the authors study a comparison result for the
$p$-torsion, that is the case $f\equiv 1$, with a completely different
argument, obtaining

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

Another work that is worth to be mentioned is [1], where the authors obtained
similar results to [3] in the case of mixed Dirichlet and Robin boundary
conditions.

This paper is organized as follows. In the next section, we give some basic
notions about rearrangements of functions and Lorentz spaces. Moreover, we
list some properties of the solutions to problems (1) and (3). In section 3,
we prove the main results about comparison of the two solutions in terms of
the Lorentz norm.

In particular, we prove

######  Theorem 1.1.

Let $u$ and $v$ be the solutions to problem (1) and (3) respectively. Then we
have

${\left\|u\right\|}_{L^{k,1}(\Omega)}\,\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}\,\;\forall\,0<k\leq\frac{n(p-1)}{(n-1)p},$
(4)
${\left\|u\right\|}_{L^{pk,p}(\Omega)}\,\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})}\,\;\forall\,0<k\leq\frac{n(p-1)}{(n-2)p+n}.$
(5)

We observe that from Theorem 1.1, we have that, if $p\geq n$

${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}.$

######  Theorem 1.2.

Assume that $f\equiv 1$ and let $u$ and $v$ be the solutions to (1) and (3)
respectively.

  1. $(i)$

If $\displaystyle{1\leq p\leq\frac{n}{n-1}}$ then

$u^{\sharp}(x)\leq v(x)\qquad x\in\Omega^{\sharp},$ (6)

  2. $(ii)$

if $\displaystyle{p>\frac{n}{n-1}}$ and
$\displaystyle{0<k\leq\frac{n(p-1)}{n(p-1)-p}}$ , then

$\displaystyle{\left\|u\right\|}_{L^{k,1}(\Omega)}$
$\displaystyle\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}$ (7)
$\displaystyle{\left\|u\right\|}_{L^{pk,p}(\Omega)}$
$\displaystyle\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})}.$

Then we explicitly observe that in the case $f\equiv 1$ from Theorem 1.2 we
have that

${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}\qquad\text{
for }p>1.$

While we have the point-wise comparison only for
$\displaystyle{p\leq\frac{n}{n-1}}$.

In section 4, using tools from section 3, we give a new proof of the Faber-
Krahn inequality with Robin boundary conditions in the case $p\geq n$. This
topic was already studied in the papers by Bucur, Giacomini, Daners and
Trebeschi, [4], [5] , [6] and [7] where the authors proved the Faber-Krahn
inequality for the eigenvalues of the Laplacian, or of the $p$-Laplacian, with
Robin boundary conditions, for every $p>1$. Actually, the results in [4] are
more general, since they hold for every $p>1$, but they are obtained with
completely different tools than the ones contained in our paper.

Finally, in section 5, we provide some examples and open problems and we
discuss the optimality of our results.

##  2 Notions and preliminaries

######  Definition 2.1.

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

$\mu(t)={\left|\Set{x\in\Omega\,:\,{\left|u(x)\right|}>t}\right|}$

Here, and in the whole paper, ${\left|A\right|}$ stands for the
$n$-dimensional Lebesgue measure of the set $A$.

######  Definition 2.2.

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

The _Schwarz rearrangement_ of $u$ is the function $u^{\sharp}$ whose level
sets are balls with the same measure as the level sets of $u$. The functions
$u^{\sharp}$ and $u^{*}$ are linked by the relation

$u^{\sharp}(x)=u^{*}(\omega_{n}{\left|x\right|}^{n})$

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

$\displaystyle{{\left\|u\right\|}_{L^{p}(\Omega)}={\left\|u^{*}\right\|}_{L^{p}(0,{\left|\Omega\right|})}=\lVert{u^{\sharp}}\rVert_{L^{p}(\Omega^{\sharp})}}.$

An important propriety of the decreasing rearrangement is the Hardy-
Littlewood inequaliy, that is

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

So, by choosing $h=\chi_{\left\\{{\left|u\right|}>t\right\\}}$, one has

$\int_{{\left|u\right|}>t}{\left|h(x)\right|}\,dx\leq\int_{0}^{\mu(t)}h^{*}(s)\,ds.$

######  Definition 2.3.

Let $0<p<+\infty$ and $0<q\leq+\infty$. The Lorentz space $L^{p,q}(\Omega)$ is
the space of those functions such that the quantity:

${\left\|g\right\|}_{L^{p,q}}=\begin{cases}\displaystyle{p^{\frac{1}{q}}\left(\int_{0}^{\infty}t^{q}\mu(t)^{\frac{q}{p}}\,\frac{dt}{t}\right)^{\frac{1}{q}}}&0<q<\infty\\\
\displaystyle{\sup_{t>0}\,(t^{p}\mu(t))}&q=\infty\end{cases}$

is finite.

Let us observe that for $p=q$ the Lorentz space coincides with the $L^{p}$
space, as a consequence of the well known _Cavalieri’s Principle_

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

See [12] for more details on Lorentz space.

Let us consider the functional

$\mathfrak{F}(w)=\frac{1}{p}\int_{\Omega}{\left|\nabla
w\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|w\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\int_{\Omega}fw\,dx$

defined on $W^{1,p}(\Omega)$. This functional is well defined and its Euler-
Lagrange equation is exactly (1). If we show that the functional admits a
minimum, our problem will always have a solution.

  1. $1)$

Let us show that the functional is bounded from below, indeed using the
parametric Young inequality, we have

$\displaystyle\mathfrak{F}(u)$
$\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|u\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
$\displaystyle\geq\frac{1}{p}\left(\int_{\Omega}{\left|\nabla
u\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|u\right|}^{p}\,d\mathcal{H}^{n-1}(x)\right)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
$\displaystyle\geq\frac{\lambda_{1,\beta}(\Omega)-\varepsilon^{p}}{p}\int_{\Omega}{\left|u\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$

In the last inequality we used the Sobolev inequality with trace term

$\int_{\Omega}{\left|\nabla
u\right|}^{p}+\beta\int_{\partial\Omega}{\left|u\right|}^{p}\geq\lambda_{1,\beta}(\Omega)\int_{\Omega}{\left|u\right|}^{p}.$

In general, the quantity $\lambda_{1,\beta}(\Omega)$ denotes the first
eigenvalue of the $p$-Laplacian with Robin boundary conditions, whose
definition is given in (28), which can be also seen as a trace constant of the
set $\Omega$.

If $\varepsilon$ is small enough, then the quantity

$\frac{\lambda_{1,\beta}(\Omega)-\varepsilon^{p}}{p}$

is non negative, and then

$\mathfrak{F}(u)\geq-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}$

so

$m=\inf_{W^{1,p}}\mathfrak{F}(u)>-\infty.$

  2. $2)$

Compactness and lower semicontinuity.

Let $\left\\{u_{i}\right\\}$ be a minimizing sequence. We can assume that
$\mathfrak{F}(u_{i})\leq m+1$, $\forall i$. Using again the Young inequality,
we have

$\displaystyle m+1$ $\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\int_{\Omega}fu_{i}\,dx$
$\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\frac{\beta}{p}\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u_{i}\right|}^{p}\,dx-\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$

Then

$\displaystyle
m+1+\frac{1}{p^{\prime}\varepsilon^{p^{\prime}}}\int_{\Omega}{\left|f\right|}^{p^{\prime}}\,dx$
$\displaystyle\geq\frac{1}{2p}\left(\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)\right)-\frac{\varepsilon^{p}}{p}\int_{\Omega}{\left|u_{i}\right|}^{p}\,dx$
$\displaystyle+\frac{1}{2p}\left(\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|u_{i}\right|}^{p}\,d\mathcal{H}^{n-1}(x)\right)$
$\displaystyle\geq\frac{1}{2p}\int_{\Omega}{\left|\nabla
u_{i}\right|}^{p}\,dx+\left(\frac{\lambda_{1,\beta}(\Omega)-2\varepsilon^{p}}{2p}\right)\int_{\Omega}{\left|u_{i}\right|}^{p}\,dx.$

Then, the minimizing sequence $\left\\{u_{i}\right\\}$ is bounded in
$W^{1,p}(\Omega)$, so there exists a subsequence $\Set{u_{i_{k}}}$ weakly
converging in $W^{1,p}(\Omega)$ and strongly in $L^{p}(\Omega)$ to a function
$u$. Let us show that $u$ is the minimum.

The function $t^{p}$ is strictly convex for $p>1$, so

$\displaystyle{\left|u_{i_{k}}\right|}^{p}\geq{\left|u\right|}^{p}+p{\left|u\right|}^{p-2}u(u_{i_{k}}-u)$
(8) $\displaystyle{\left|\nabla u_{i_{k}}\right|}^{p}\geq{\left|\nabla
u\right|}^{p}+p{\left|\nabla u\right|}^{p-2}\nabla u(\nabla u_{i_{k}}-\nabla
u)$ (9)

Putting (8) e (9) in $\mathfrak{F}(u_{i_{k}})$, we obtain

$\displaystyle\int_{\Omega}fu_{i_{k}}\,dx+\mathfrak{F}(u_{i_{k}})$
$\displaystyle\geq\frac{1}{p}\int_{\Omega}{\left|\nabla
u\right|}^{p}\,dx+\int_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla u(\nabla
u_{i_{k}}-\nabla u)$
$\displaystyle+\frac{\beta}{p}\int_{\partial\Omega}{\left|u\right|}^{p}\,d\mathcal{H}^{n-1}(x)+\beta\int_{\partial\Omega}{\left|u\right|}^{p-2}u(u_{i_{k}}-u)\,d\mathcal{H}^{n-1}$

Passing to the limit for $k\to\infty$, by the weak convergence of
$\left\\{u_{i_{k}}\right\\}$ the integral over $\Omega$ on the right-hand side
goes to 0. The integral over $\partial\Omega$ goes to 0 as well. Indeed, the
space $W^{1,p}(\Omega)$ is compactly embedded in $L^{p}(\partial\Omega)$ (for
more details, see [9] 2.5), and $u_{i_{k}}-u\to 0$ in $L^{p}(\partial\Omega)$.
So, we obtain

$m\geq\mathfrak{F}(u).$

This ensures us that $u$ is the minimum of the functional.

The uniqueness of the minimum follows from the fact that $\mathfrak{F}(u)$ is
the sum of a strictly convex part and a linear part.

We observe that the solutions $u$ and $v$ to (1) and (3) respectively are both
$p$-superharmonic and then, by the strong maximum principle in [13], it
follows that they achieve their minima on the boundary. Denoting by $u_{m}$
and $v_{m}$ the minimum of $u$ and $v$ respectively, thanks to the
positiveness of $\beta$ and the Robin boundary conditions, we have that
$u_{m}\geq 0$ and $v_{m}\geq 0$. Hence $u$ and $v$ are strictly positive in
the interior of $\Omega$. Moreover we can observe that

$u_{m}=\min_{\Omega}u\leq\min_{\Omega^{\sharp}}v=v_{m},$ (10)

in fact, if we consider

$\begin{split}v_{m}^{p-1}\text{Per}(\Omega^{\sharp})&=\int_{\partial\Omega^{\sharp}}v(x)^{p-1}\,d\mathcal{H}^{n-1}(x)=\frac{1}{\beta}\int_{\Omega^{\sharp}}f^{\sharp}\,dx=\frac{1}{\beta}\int_{\Omega}f\,dx\\\
&=\int_{\partial\Omega}u(x)^{p-1}\,d\mathcal{H}^{n-1}(x)\\\ &\geq
u_{m}^{p-1}\text{Per}(\Omega)\geq{\left|u\right|}_{m}^{p-1}\text{Per}(\Omega^{\sharp}).\end{split}$

A consequence of (10) that will be used in what follows is that

$\mu(t)\leq\phi(t)={\left|\Omega\right|}\quad\forall t\leq v_{m}.$ (11)

###  2.1 Useful lemmas

Let $u$ be the solution to (1). For $t\geq 0$, we denote by

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

and by

$\mu(t)={\left|U_{t}\right|}\quad P_{u}(t)=Per(U_{t})$

where ${\left|\cdot\right|}$ is the Lebesgue measure on
$\operatorname{\mathbb{R}}^{n}$ and $Per(\cdot)$ is the perimeter.
If $v$ is the solution to (3), using the same notations, we set

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

Because of the invariance of the $p$-Laplacian and of the Schwarz
rearrangement of $f$ by rotation, there exists a radial solution to (3) and,
by uniqueness of solutions, this solution is $v$.

Since $v$ is radial, positive and decreasing along the radius then, for $0\leq
t\leq v_{m}$, $V_{t}=\Omega^{\sharp}$, while, for
$v_{m}<t<\max_{\Omega^{\sharp}}v$, $V_{t}$ is a ball, concentric to
$\Omega^{\sharp}$ and strictly contained in it.

######  Lemma 2.1 (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

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

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 by $\tau^{p}$, we obtain

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

Now, we integrate from $\tau_{0}$ to $\tau$ and we obtain

$\begin{split}\int_{\tau_{0}}^{\tau}\left(\frac{\xi(\tau)}{\tau^{p-1}}\right)^{\prime}\,d\tau&\leq\int_{\tau_{0}}^{\tau}\frac{C}{\tau^{p}}\,d\tau\\\
\implies\xi(\tau)&\leq\left(\xi(\tau_{0})+\frac{C}{p-1}\right)\left(\frac{\tau}{\tau_{0}}\right)^{p-1}-\frac{C}{p-1},\end{split}$

which gives _(i)_.

In order to obtain _(ii)_ , we just take into account _(i)_ in the
differential inequality. ∎

######  Lemma 2.2.

Let $u$ and $v$ be solutions to (1) and (3) respectively. Then for almost
every $t>0$ we have

$\gamma_{n}\mu(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}\leq\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right),$ (12)

where
$\displaystyle{\gamma_{n}=\left(n\omega_{n}^{1/n}\right)^{\frac{p}{p-1}}}$.

And

$\gamma_{n}\phi(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}=\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\phi^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
V_{t}^{\text{ext}}}\frac{1}{v}\,d\mathcal{H}^{n-1}(x)\right).$ (13)

###### Proof.

Let $t>0$ e $h>0$, we choose the test function

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

Then,

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

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

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

where

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

So, using the isoperimetric inequality, for a. e. $t>0$ we have

$\begin{split}n\omega_{n}^{1/n}\mu(t)^{\left(1-\frac{1}{n}\right)}&\leq
P_{u}(t)=\int_{\partial U_{t}}\,d\mathcal{H}^{n-1}(x)\leq\left(\int_{\partial
U_{t}}g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
U_{t}}\frac{1}{g^{\frac{1}{p-1}}}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}\\\
&=\left(\int_{\partial
U_{t}}g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
U_{t}^{int}}\frac{1}{{\left|\nabla
u\right|}}\,d\mathcal{H}^{n-1}(x)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}\\\
&\leq\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}\quad
t\in[0,\max_{\Omega}u),\end{split}$

Then (12) follows. We notice that if $v$ is the solution to (3), than all the
inequalities are verified as equalities, so we have (13). ∎

######  Lemma 2.3.

For all $\tau\geq v_{m}$ we have

$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
U_{t}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt\leq\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds.$
(14)

Moreover,

$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
V_{t}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt=\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds,$
(15)

###### Proof.

If we integrate the quantity

$t^{p-1}\left(\int_{\partial
U_{t}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right),$

from 0 to $+\infty$, by Fubini theorem, we obtain

$\begin{split}\int_{0}^{\infty}\tau^{p-1}\left(\int_{\partial
U_{\tau}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau&=\int_{\partial\Omega}\left(\int_{0}^{u(x)}\frac{\tau^{p-1}}{u(x)}\,d\tau\right)\,d\mathcal{H}^{n-1}(x)\\\
&=\frac{1}{p}\int_{\partial\Omega}u(x)^{p-1}\,d\mathcal{H}^{n-1}(x)\\\
&=\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds,\end{split}$

where the last equality follows from the fact that $u$ solves (1).

Analogously

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

Since $u$ is positive, we obtain, $\forall t\geq 0$,

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

on the other hand, since $\partial V_{t}\cap\partial\Omega^{\sharp}$ is empty
for $t\geq v_{m}$, we have

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

and the proof of lemma 2.3 is complete. ∎

######  Remark 2.1.

It can be observed that, since $\partial V_{t}\cap\partial\Omega^{\sharp}$ is
empty for $t\geq v_{m}$ and $\phi(t)={\left|\Omega\right|}$ for $t\leq v_{m}$,
for all $\delta>0$ and for all $t$, we have

$\begin{split}&\int_{0}^{t}\tau^{p-1}\phi(\tau)^{\delta}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\\\
&\int_{0}^{v_{m}}\tau^{p-1}\phi(\tau)^{\delta}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\\\
&\int_{0}^{+\infty}\tau^{p-1}\phi(\tau)^{\delta}\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,d\tau=\frac{{\left|\Omega\right|}^{\delta}}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds.\end{split}$

##  3 Main results

Now we prove Theorem 1.1 and Theorem 1.2 .

###### Proof of Theorem 1.1.

Let $\displaystyle{0<k\leq\frac{n(p-1)}{p(n-1)}}$, so
$\displaystyle{\delta=\frac{1}{k}-\frac{(n-1)p}{n(p-1)}}$ is positive.

Multiplying (12) by $t^{p-1}\mu(t)^{\delta}$ and integrating from $0$ to
$\tau\geq v_{m}$, by the previous Lemma, we obtain

$\begin{split}\int_{0}^{\tau}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt&\leq\int_{0}^{\tau}\left(-\mu^{\prime}(t)\right)t^{p-1}\mu(t)^{\delta}\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,dt\\\
&+\frac{{\left|\Omega\right|}^{\delta}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$
(16)

Setting
$\displaystyle{F(l)=\int_{0}^{l}\omega^{\delta}\left(\int_{0}^{\omega}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,d\omega}$,
we can integrate by parts both sides of the last inequality, getting

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

Setting
$\displaystyle{\xi(\tau)=\int_{0}^{\tau}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(t))\right)\,dt}$
and
$\displaystyle{C=\frac{{\left|\Omega\right|}^{\delta}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}}$,
we are in the hypothesis of Lemma 2.1 (Gronwall), namely

$\tau\xi^{\prime}(\tau)\leq(p-1)\xi(\tau)+C,$

so, choosing $\tau_{0}=v_{m}$, we have

$\begin{multlined}\tau^{p-2}\left(\int_{0}^{\tau}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(\tau))\right)\leq\left(\frac{(p-1)\xi(v_{m})+C}{v_{m}}\right)\left(\frac{\tau}{v_{m}}\right)^{p-2},\end{multlined}\tau^{p-2}\left(\int_{0}^{\tau}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(\tau))\right)\leq\left(\frac{(p-1)\xi(v_{m})+C}{v_{m}}\right)\left(\frac{\tau}{v_{m}}\right)^{p-2},$

where

$\xi(v_{m})=\int_{0}^{v_{m}}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(t))\right)\,dt.$

The previous inequality becomes an equality if we replace $\mu(t)$ with
$\phi(t)$. Since $\mu(t)\leq\phi(t)={\left|\Omega\right|},\quad\forall t\leq
v_{m}$, and $F(l)$ is monotone, we obtain

$\int_{0}^{v_{m}}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(t))\right)\,dt\leq\int_{0}^{v_{m}}t^{p-2}\left(\int_{0}^{t}\gamma_{n}\phi(s)^{\frac{1}{k}}\,ds+F(\phi(t))\right)\,dt,$

hence

$\int_{0}^{\tau}\gamma_{n}\mu(s)^{\frac{1}{k}}\,ds+F(\mu(\tau))\leq\int_{0}^{\tau}\gamma_{n}\phi(s)^{\frac{1}{k}}\,ds+F(\phi(\tau)).$

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

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

and hence

${\left\|u\right\|}_{L^{k,1}(\Omega)}\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}\quad\forall\,0\,<k\leq\frac{n(p-1)}{p(n-1)}.$

To prove the inequality (5), it is enough to show that

$\int_{0}^{\infty}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{\infty}t^{p-1}\phi(t)^{\frac{1}{k}}\,dt.$
(17)

Let us consider equation (16), let us integrate by parts the first term on the
right-hand side from 0 to $\tau$ and then let us pass to the limit as
$\tau\to\infty$, we have

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

Therefore, if we show that

$\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt\leq\int_{0}^{\infty}t^{p-2}F(\phi(t))\,dt$
(18)

we obtain the (17). To this aim, we multiply (12) by
$\displaystyle{t^{p-1}F(\mu(t))\mu(t)^{-\frac{(n-1)p}{n(p-1)}}}$ and
integrate. First, we observe that, by the choice
$\displaystyle{k\leq\frac{n(p-1)}{(n-2)p+n}}$, it follows that the function
$\displaystyle{h(l)=F(l)l^{-\frac{(n-1)p}{n(p-1)}}}$ is non decreasing. Hence,
we obtain

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

If we integrate by parts both sides of the last expression and set

$\displaystyle{C=F({\left|\Omega\right|})\frac{{\left|\Omega\right|}^{-\frac{p(n-1)}{n(p-1)}}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}},$

we obtain

$\tau\int_{0}^{\tau}\gamma_{n}t^{p-2}F(\mu(t))\,dt+\tau
H_{\mu}(\tau)\leq\int_{0}^{\tau}\int_{0}^{t}r^{p-2}F(\mu(r))\,drdt+\int_{0}^{\tau}H_{\mu}(t)\,dt+C$
(19)

where

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

Setting

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

then (19) becomes

$\tau\xi^{\prime}(\tau)\leq\xi(\tau)+C.$

So lemma 2.1, with $\tau_{0}=v_{m}$, gives

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

Of course, the inequality holds as an equality if we replace $\mu(t)$ with
$\phi(t)$, so we get, keeping in mind that
$\mu(t)\leq\phi(t)={\left|\Omega\right|}$ for $t\leq v_{m}$,

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

Letting $\tau\to\infty$, one has

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

as $H_{\mu}(\tau),H_{\phi}(\tau)\to 0$. This proves (18), and hence (5).

The fact that both $H_{\mu}$ and $H_{\phi}$ go to 0 as $\tau$ goes to infinity
can be easily deduced distinguishing the cases.

  * •

If $p\geq 2$

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

  * •

If $p<2$

$\displaystyle{\left|H_{\mu}(\tau)\right|}$
$\displaystyle=\int_{\tau}^{+\infty}t^{p-2}F(\mu(t))\mu(t)^{-\frac{p(n-1)}{n(p-1)}}\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,s\biggr{)}(-\mu^{\prime}(t))\,dt$
$\displaystyle\leq\tau^{p-2}\int_{\tau}^{+\infty}F(\mu(t))\mu(t)^{-\frac{p(n-1)}{n(p-1)}}\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,s\biggr{)}(-\mu^{\prime}(t))\,dt\xrightarrow{\tau\to+\infty}0.$

and analogously for $H_{\phi}$, which concludes the proof. ∎

###### Proof of Theorem 1.2.

.

  1. $(i)$

Firstly, we observe that
$\displaystyle{\int_{0}^{\mu(t)}f^{\ast}(s)\,ds=\mu(t)}$, so (12) becomes

$\gamma_{n}\mu(t)^{\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\leq-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x).$ (20)

Let us multiply both sides by $t^{p-1}\mu(t)^{\delta}$, where
$\delta=-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}$. We point out
that $\delta\geq 0$ for $p\leq\frac{n}{n-1}$. Hence, integrating from $0$ to
$\tau\geq v_{m}$, we have

$\begin{split}\int_{0}^{\tau}\gamma_{n}t^{p-1}&\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta}(-\mu^{\prime}(t))\,dt+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\\\
&\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\delta}(-\mu^{\prime}(t))\,dt+\frac{{\left|\Omega\right|}^{\delta+1}}{p\beta^{\frac{p}{p-1}}}\end{split}$
(21)

Taking into account remark 2.1, if we replace $\mu(t)$ with $\phi(t)$ the
previous inequality holds as equality.

Hence, we get

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

Then an integration by parts gives

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

Finally, using Gronwall’s Lemma with the function
$\displaystyle{\xi(\tau)=\int_{0}^{\tau}s^{p-2}\left(\frac{\mu(s)^{\delta+1}-\phi(s)^{\delta+1}}{\delta+1}\right)\,ds}$
we obtain

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

The quantity on the right-hand side is non-positive, thanks to (10), so

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

and, remembering that,

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

we get the point-wise inequality of the functions.

  2. $(ii)$

Now we want to show that

${\left\|u\right\|}_{L^{k,1}(\Omega)}\leq{\left\|v\right\|}_{L^{k,1}(\Omega^{\sharp})}$

so it is enough to show

$\int_{0}^{+\infty}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{+\infty}\phi(t)^{\frac{1}{k}}\,dt$
(22)

We multiply (20) by
$t^{p-1}\mu(t)^{\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}$
and integrate from $0$ to $\tau\geq v_{m}$. Then using Lemma 2.3 and Remark
2.1, we obtain

$\int_{0}^{\tau}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{\tau}t^{p-1}\mu(t)^{\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}(-\mu^{\prime}(t))\,dt+\frac{{\left|\Omega\right|}^{\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}+1}}{p\beta^{\frac{p}{p-1}}}$
(23)

and equality holds if we replace $\mu$ with $\phi$. In order to be shorter, we
set

$\eta=\frac{1}{k}-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1},\quad
C=\frac{{\left|\Omega\right|}^{\eta+1}}{p\beta^{\frac{p}{p-1}}}.$

We point out that (23) follows by (21) if $\eta\geq 0$, namely

$0<k\leq\frac{n(p-1)}{n(p-1)-p}$

With these notations and keeping in mind that $\mu$ is a non increasing
function, we have from (23) taht

$\int_{0}^{\tau}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq\int_{0}^{\tau}-t^{p-1}\mu(t)^{\eta}\,d\mu(t)+C$
(24)

Let us set
$\displaystyle{G(\ell)=\int_{0}^{\ell}w^{\eta}\,dw=\frac{\ell^{\eta+1}}{\eta+1}}$,
let us integrate by parts both sides of (24) in order to obtain

$\begin{split}&\gamma_{n}\tau^{p-1}\int_{0}^{\tau}\mu(t)^{\frac{1}{k}}\,dt+\tau^{p-1}G(\mu(\tau))\\\
&\leq(p-1)\biggl{[}\int_{0}^{\tau}\gamma_{n}t^{p-2}\int_{0}^{t}\mu(r)^{\frac{1}{k}}\,drdt+\int_{0}^{\tau}t^{p-2}G(\mu(t))\,dt\biggr{]}+C\end{split}$
(25)

Setting

$\xi(\tau)=\int_{0}^{\tau}\left(\gamma_{n}t^{p-2}\int_{0}^{t}\mu(r)^{\frac{1}{k}}\,dr\right)\,dt+\int_{0}^{\tau}t^{p-2}G(\mu(t))\,dt$

(25) reads as follows

$\tau\xi^{\prime}(\tau)\leq(p-1)\xi(\tau)+C$

Hence, using Gronwall’s Lemma 2.1 with $\tau_{0}=v_{m}$, we get

$\gamma_{n}\tau^{p-2}\int_{0}^{\tau}\mu(t)^{\frac{1}{k}}\,dt+\tau^{p-2}G(\mu(\tau))\leq\biggl{(}\frac{(p-1)\xi(v_{m})+C}{v_{m}}\biggr{)}\biggl{(}\frac{\tau}{v_{m}}\biggr{)}^{p-2}$

where

$\displaystyle{\xi(v_{m})=\int_{0}^{v_{m}}\gamma_{n}t^{p-2}\int_{0}^{t}\mu(r)^{\frac{1}{k}}\,dr\,dt+\int_{0}^{v_{m}}t^{p-2}G(\mu(t))\,dt}$

Again, if we replace $\mu$ with $\phi$, the previous inequality holds as an
equality and $\xi(v_{m})$ is less or equal than the same quantity obtained by
replacing $\mu$ with $\phi$, as (10) holds. Keeping in mind (11), we have

$\tau^{p-2}\left(\gamma_{n}\int_{0}^{\tau}\mu(t)^{\frac{1}{k}}\,dt+G(\mu(\tau))\right)\leq\tau^{p-2}\left(\gamma_{n}\int_{0}^{\tau}\phi(t)^{\frac{1}{k}}\,dt+G(\phi(\tau))\right)$

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

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

namely (22).

To conclude the proof, we have to show that

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

that is to say

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

We consider (24), pass to the limit as $\tau\to+\infty$ and integrate by parts
the first term on the right-hand side

$\int_{0}^{+\infty}\gamma_{n}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt\leq(p-1)\int_{0}^{+\infty}t^{p-2}G(\mu(t))\,dt+C.$

Hence it is enough to show that

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

To this aim, we multiply (20) by
$t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}$
and integrate from $0$ to $\tau\geq v_{m}$

$\displaystyle\int_{0}^{\tau}\gamma_{n}t^{p-1}G(\mu(t))\,dt$
$\displaystyle\leq\int_{0}^{\tau}t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\,d\mu(t)$
$\displaystyle+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{0}^{\tau}t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\biggl{(}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}\biggr{)}\,dt$

Since $\displaystyle{k\leq\frac{n(p-1)}{n(p-1)-p}}$, using Lemma 2.3 and the
fact that the function
$G(\ell)\ell^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}$ is non
decreasing, we obtain

$\int_{0}^{\tau}\gamma_{n}t^{p-1}G(\mu(t))\,dt\leq\int_{0}^{\tau}t^{p-1}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\,d\mu(t)+C$
(26)

with

$C=\frac{1}{p\beta^{\frac{p}{p-1}}}G({\left|\Omega\right|}){\left|\Omega\right|}^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}+1}$

If we replace $\mu$ with $\phi$ the previous inequality holds as an equality,
thanks to (2.1). Now, let us integrate by parts both sides of (26), obtaining

$\tau\int_{0}^{\tau}\gamma_{n}t^{p-2}G(\mu(t))\,dt+\tau
H(\tau)\leq\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}t^{p-2}G(\mu(r))\,drdt+\int_{0}^{\tau}H_{\mu}(t)\,dt+C$
(27)

where

$H_{\mu}(\tau)=-\int_{\tau}^{+\infty}t^{p-2}G(\mu(t))\mu(t)^{-\left(1-\frac{1}{n}-\frac{1}{p}\right)\frac{p}{p-1}}\,d\mu(t)$

Setting

$\xi(\tau)=\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}t^{p-2}G(\mu(r))\,drdt+\int_{0}^{\tau}H_{\mu}(t)\,dt$

the (27) reads as follows

$\tau\xi^{\prime}(\tau)\leq\xi(\tau)+C$

Again, using Gronwall’s Lemma 2.1, we get

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

with

$\xi(v_{m})=\int_{0}^{v_{m}}\int_{0}^{t}\gamma_{n}t^{p-2}G(\mu(r))\,drdt+\int_{0}^{v_{m}}H_{\mu}(t)\,dt.$

Keeping in mind that for $\phi$ the previous inequalities hold as equality and
the fact that $G$ is not decreasing, $\xi(v_{m})$ is less or equal to the same
quantity obtained by replacing $\mu$ with $\phi$. Hence, we obtain

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

and passing to the limit as $\tau\to+\infty$, we finally get

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

indeed, as in the proof of Theorem 1.1, $H_{\mu}(\tau)$ and $H_{\phi}(\tau)$
go to 0 as $\tau\to\infty$. That concludes the proof. ∎

######  Corollary 3.1.

Let $u$ and $v$ be the solutions to (1) and (3) respectively. Then, if $p\geq
n$, we have

${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}.$

Moreover in the case $f\equiv 1$, Theorem 1.2 gives

${\left\|u\right\|}_{L^{1}(\Omega)}\leq{\left\|v\right\|}_{L^{1}(\Omega^{\sharp})}\quad\text{and}\quad{\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})}\qquad\forall
p>1$

and the point-wise comparison for $\displaystyle{p\leq\frac{n}{n-1}}$.

###### Proof.

If $p\geq n$ the upper bounds of $k$, in both cases (4) e (5), are greater
than $1$ and so we can choose $k=1$. The assertion follows from the fact that

${\left\|\cdot\right\|}_{L^{p,p}(\Omega)}={\left\|\cdot\right\|}_{L^{p}(\Omega)}.$

Analogously if $f\equiv 1$. ∎

##  4 Faber–Krahn inequality

We recall that the first eigenvalue of $p$-Laplace operator with Robin
boundary conditions is obtained as the minimum of the Rayleigh quotients,
i.e.,

$\lambda_{1,\beta}(\Omega)=\min_{\begin{subarray}{c}\omega\in
W^{1,p}(\Omega)\\\ \omega\neq
0\end{subarray}}\frac{\displaystyle{\int_{\Omega}{\left|\nabla\omega\right|}^{p}\,dx+\beta\int_{\partial\Omega}{\left|\omega\right|}^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega}{\left|\omega\right|}^{p}\,dx}}.$
(28)

We can observe that if $u$ achieves the minimum of Rayleigh quotients, so does
${\left|u\right|}$. From this we have that $u$ in non-negative. Furthermore,
we have if $u_{\lambda_{1}}\geq 0$, as a consequence of Harnack inequality,
$u_{\lambda_{1}}>0$.

Another important thing is that the eigenvalue is simple. Indeed, as shown in
[8], if $\Omega$ is smooth enough and $u$ and $v$ are to eigenfunctions
referred to the first eigenvalue, we can choose as test function
$\displaystyle{\varphi_{1}=\frac{u^{p}-v^{p}}{u^{p-1}}}$ in

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

and $\displaystyle{\varphi_{2}=\frac{v^{p}-u^{p}}{v^{p-1}}}$ in

$\int_{\Omega}{\left|\nabla v\right|}^{p-2}\nabla
v\nabla\varphi_{2}\,dx+\beta\int_{\partial\Omega}v^{p-1}\varphi_{2}\,d\mathcal{H}^{n-1}(x)=\int_{\Omega}\lambda_{1}v^{p-1}\varphi_{2}\,dx.$

Summing the two equations, we have

$\begin{split}0&=\int_{\Omega}\left\\{1+(p-1)\left(\frac{v}{u}\right)^{p}\right\\}{\left|\nabla
u\right|}^{p}+\left\\{1+(p-1)\left(\frac{u}{v}\right)^{p}\right\\}{\left|\nabla
v\right|}^{p}\\\ &-\int_{\Omega}p\left(\frac{v}{u}\right)^{p-1}{\left|\nabla
u\right|}^{p-2}\nabla u\nabla v+p\left(\frac{u}{v}\right)^{p-1}{\left|\nabla
v\right|}^{p-2}\nabla v\nabla u\\\
&=\int_{\Omega}(u^{p}-v^{p})\left({\left|\nabla\log{u}\right|}^{p}-{\left|\nabla\log{v}\right|}^{p}\right)\\\
&-\int_{\Omega}pv^{p}{\left|\nabla\log{u}\right|}^{p-2}{\left|\nabla\log{u}\right|}\left(\nabla\log{v}-\nabla\log{u}\right)\\\
&-\int_{\Omega}pu^{p}{\left|\nabla\log{v}\right|}^{p-2}{\left|\nabla\log{v}\right|}\left(\nabla\log{u}-\nabla\log{v}\right)\end{split}$

Now, using the well known inequalities, which hold true for each $w_{1}$ and
$w_{2}\in\operatorname{\mathbb{R}}^{n}$ ,

$\begin{gathered}{\left|w_{2}\right|}^{p}\geq{\left|w_{1}\right|}^{p}+p{\left|w_{1}\right|}^{p-2}w_{1}\cdot(w_{2}-w_{1})+\frac{{\left|w_{2}-w_{1}\right|}^{p}}{2^{p-1}-1}\quad\text{if}\quad
p\geq 2\\\
{\left|w_{2}\right|}^{p}\geq{\left|w_{1}\right|}^{p}+p{\left|w_{1}\right|}^{p-2}w_{1}\cdot(w_{2}-w_{1})+C(p)\frac{{\left|w_{2}-w_{1}\right|}^{2}}{({\left|w_{1}\right|}+{\left|w_{2}\right|})^{2-p}}\quad\text{if}\quad
1<p<2,\end{gathered}$ (29)

if we consider the case $p\geq 2$, we choose $w_{2}=\nabla\log u$ and
$w_{1}=\nabla\log v$, we obtain

$\frac{1}{2^{p-1}-1}\int_{\Omega}\left(\frac{1}{v^{p}}+\frac{1}{u^{p}}\right){\left|v\nabla
u-u\nabla v\right|}^{p}=0.$

Hence, we obtain that $v\nabla u=u\nabla v$ a.e. in $\Omega$, and so there
exists a constant $K$ for which $u=Kv$. This means that $\lambda_{1}$ is
simple.

For the proof of (29), we refer to [8].

The following corollary of Theorem 1.1 holds true, that is Faber-Krahn
inequality.

######  Corollary 4.1.

Let $u$ and $v$ be the solutions to (1) and (3), respectively. Then , if
$p\geq n$, we have

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

###### Proof.

Let $u$ an eigenfunction referred to the first eigenvalue of (1), then it
solves

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

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

$\begin{cases}-\Delta_{p}z=\lambda_{1,\beta}(\Omega)\,\lvert
u^{\sharp}\rvert^{p-2}u^{\sharp}&\text{ in }\Omega^{\sharp}\\\ {\left|\nabla
z\right|}^{p-2}\displaystyle{\frac{\partial
z}{\partial\nu}}+\beta{\left|z\right|}^{p-2}z=0&\text{ on
}\partial\Omega^{\sharp}.\end{cases}$

In that case, corollary 3.1 gives

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

and hence, by Hölder inequality

$\int_{\Omega^{\sharp}}(u^{\sharp})^{p-2}u^{\sharp}z\,dx\leq\left(\int_{\Omega^{\sharp}}{\left|u^{\sharp}\right|}^{p}\,dx\right)^{\frac{p-1}{p}}\left(\int_{\Omega^{\sharp}}z^{p}\,dx\right)^{\frac{1}{p}}\leq\int_{\Omega^{\sharp}}z^{p}\,dx.$

Therefore, observing that we can write the eigenvalue
$\lambda_{1,\beta}(\Omega)$ in the following way, we obtain

$\begin{split}\lambda_{1,\beta}(\Omega)&=\frac{\displaystyle{\int_{\Omega^{\sharp}}{\left|\nabla
z\right|}^{p}\,dx+\beta\int_{\partial\Omega^{\sharp}}z^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega^{\sharp}}(u^{\sharp})^{p-2}u^{\sharp}z\,dx}}\\\
&\geq\frac{\displaystyle{\int_{\Omega^{\sharp}}{\left|\nabla
z\right|}^{p}\,dx+\beta\int_{\partial\Omega^{\sharp}}z^{p}\,d\mathcal{H}^{n-1}(x)}}{\displaystyle{\int_{\Omega^{\sharp}}z^{p}\,dx}}\geq\lambda_{1,\beta}(\Omega^{\sharp}).\end{split}$

∎

##  5 Conclusions

We have been able to extend the results obtained for the Laplacian to the
$p$-Laplacian. Many problems remain open, such as

Open Problem In the assumptions of Theorem 1.2, does the point-wise comparison
hold also for $p>\frac{n}{n-1}$?

We have already observed in the corollary 3.1 that if $p\geq n$ we have an
estimate on the $L^{p}$ norms of $u$ and $v$. Can we generalize this estimate
also for $q\neq p$? We know for sure that for $q=\infty$ this can’t be done,
as it can be seen in the following example.

######  Example 5.1.

Let $\Omega\subseteq\operatorname{\mathbb{R}}^{n}$ be the union of two
disjoint balls, $B_{1}$ and $B_{r}$ with radii 1 and $r$ respectively. We
choose $\displaystyle{\beta<\left(\frac{n-1}{p-1}\right)^{p-1}}$ with $p\neq
n$, and we fix $f=1$ on $B_{1}$ and $f=0$ on $B_{r}$. Both $u$ and $v$ can be
explicitly computed. We have
${\left\|u\right\|}_{\infty}-{\left\|v\right\|}_{\infty}=Cr^{n}+o(r^{n})$,
where $C$ is a positive constant.

_Proof._ We want an explicit expression of $u$ and $v$ respectively. Starting
from $u$, it is a solution to

$\begin{cases}-\text{div}({\left|\nabla u\right|}^{p-2}\nabla u)=f&\text{ in
}\Omega\\\ {\left|\nabla u\right|}^{p-2}\displaystyle{\frac{\partial
u}{\partial\nu}}+\beta{\left|u\right|}^{p-2}u=0&\text{ on
}\partial\Omega.\end{cases}$

with $f\rvert_{B_{1}}=1$ and $f\rvert_{B_{r}}=0$.

It’s clear that $u\rvert_{B_{r}}=0$ and $u(x)=u({\left|x\right|})$ it’s radial
on $B_{1}$.

So the equation (1) becomes

$s^{n-1}\Delta_{p}u(s)=\frac{d}{ds}\left(s^{n-1}{\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)\right)$

and then

$\displaystyle\frac{d}{ds}\left(s^{n-1}{\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)\right)$
$\displaystyle=s^{n-1}\Delta_{p}u(s)=-s^{n-1}$ $\displaystyle
s^{n-1}{\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)$
$\displaystyle=-\frac{s^{n}}{n}+c.$

We set $c=0$, in order to have a $C^{1}$-solution.

${\left|u^{\prime}(s)\right|}^{p-2}u^{\prime}(s)=-\frac{s}{n}\implies
u^{\prime}(s)=-\frac{s^{\frac{1}{p-1}}}{n^{\frac{1}{p-1}}}\qquad\alpha=\frac{1}{p-1}.$

If we integrate, we obtain

$u(s)=-\frac{p-1}{n^{\alpha}p}s^{\frac{p}{p-1}}+A.$

The Robin boundary conditions become

${\left|u^{\prime}(1)\right|}^{p-2}u^{\prime}(1)+\beta u(1)^{p-1}=0\quad(u\geq
0),$

now we can compute the value of $A$

$-\frac{1}{n}+\beta\left(-\frac{p-1}{n^{\alpha}p}+A\right)^{p-1}=0\implies
A=\frac{1}{(n\beta)^{\alpha}}+\frac{p-1}{n^{\alpha}p}.$

So

$u(s)=\frac{p-1}{n^{\alpha}p}\left(1-s^{\frac{p}{p-1}}\right)+\frac{1}{(n\beta)^{\alpha}}.$

As $u$ is decreasing, we have

${\left\|u\right\|}_{\infty}=u(0)=\frac{p-1}{n^{\alpha}p}+\frac{1}{(n\beta)^{\alpha}}.$

Now, let us compute $v(s)$. We will do this firstly for $s\in(0,1)$, then for
$s\in(1,\overline{r})$ where $\overline{r}=(1+r^{n})^{\frac{1}{n}}$ is
determined by the condition ${\left|\Omega\right|}=|\Omega^{\sharp}|$.

Let $s<1$

$\frac{d}{ds}\left(s^{n-1}{\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)\right)=-s^{n-1}$
${\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)=-\frac{s}{n}\implies
v^{\prime}(s)=-\frac{s^{\frac{1}{p-1}}}{n^{\alpha}}$
$v(s)=-\frac{p-1}{n^{\alpha}p}s^{\frac{p}{p-1}}+B.$

Now we can’t determine $B$ as before, as $v$ is not identically 0 in the
anulus $B_{\overline{r}}\backslash B_{1}$.

Let $s>1$ and $p\neq n$

$\frac{d}{ds}\left(s^{n-1}{\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)\right)=0$
${\left|v^{\prime}(s)\right|}^{p-2}v^{\prime}(s)=\frac{C}{s^{n-1}}$

by imposing the continuity of the derivative for $s=1$, we obtain that
$C=-1/n$

$\displaystyle v^{\prime}(s)$
$\displaystyle=-\frac{s^{-\frac{n-1}{p-1}}}{n^{\alpha}},$ $\displaystyle v(s)$
$\displaystyle=-\frac{p-1}{n^{\alpha}(p-n)}s^{\frac{p-n}{p-1}}+D,$

and by Robin conditions

$\displaystyle{\left|v^{\prime}(\overline{r})\right|}^{p-2}v^{\prime}(\overline{r})+\beta
v(\overline{r})^{p-1}=0,$
$\displaystyle-\frac{\overline{r}^{-n-1}}{n}+\beta\left(-\frac{p-1}{n^{\alpha}(p-n)}\overline{r}^{\frac{p-n}{p-1}}+D\right)^{(p-1)}=0,$
$\displaystyle
D=\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}+\frac{p-1}{n^{\alpha}(p-n)}\overline{r}^{\frac{p-n}{p-1}}.$

By imposing the continuity of $v$ for $s=1$, we have

$B=\frac{p-1}{n^{\alpha}p}+\frac{\overline{r}^{-\frac{n-1}{p-1}}}{(n\beta)^{\alpha}}+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)$

that is to say

$v(s)=\begin{cases}u(s)+\frac{1}{(n\beta)^{\alpha}}\left(\overline{r}^{-\frac{n-1}{p-1}}-1\right)+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)&\text{
if }s<1\\\
\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-s^{\frac{p-n}{p-1}}\right)&\text{
if }1<s<\overline{r}\end{cases}$

For convenience’s sake, we set
$\displaystyle{h=\frac{1}{(n\beta)^{\alpha}}\left(\overline{r}^{-\frac{n-1}{p-1}}-1\right)+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)}$.

So we have

${\left\|v\right\|}_{L^{\infty}(\Omega^{\sharp})}={\left\|v\right\|}_{L^{\infty}(B_{1})}={\left\|u\right\|}_{L^{\infty}(\Omega)}+h=u(0)+h$

By using Taylor expansion of the function $(1+r^{n})^{\delta}$ we get

$h=\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)r^{n}+o(r^{n}),$

so, if we choose $\beta<\left(\frac{n-1}{p-1}\right)^{p-1}$, we get

${\left\|v\right\|}_{L^{\infty}(\Omega^{\sharp})}={\left\|u\right\|}_{L^{\infty}(\Omega)}-Cr^{n}+o(r^{n})\text{
where }C>0.$

Next example 5.2 is a counterexample to the corollary 3.1 in the case $n>p$.

######  Example 5.2.

Let $\Omega\subseteq\operatorname{\mathbb{R}}^{n}$, $p<n$ be the union of two
disjoint balls $B_{1}$ and $B_{r}$ with radii 1 and $r$ respectively. We
choose $\displaystyle{\beta\leq\left(\frac{n-p}{p(p-1)}\right)^{p-1}}$ and we
fix $f=1$ on $B_{1}$ and $f=0$ on $B_{r}$. Both $u$ and $v$ can be explicitly
computed. We have
${\left\|u\right\|}_{p}^{p}-{\left\|v\right\|}_{p}^{p}=Cr^{n}+o(r^{n})$, where
$C$ is a positive constant.

_Proof._ Let us consider the Taylor expansion of $(1+y)^{p}$, we get

${\left\|v\right\|}^{p}_{L^{p}(B_{1})}=\int_{B_{1}}(u+h)^{p}={\left\|u\right\|}^{p}_{L^{p}(B_{1})}+p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}h+o(r^{n})$

Moreover

${\left\|v\right\|}^{p}_{L^{p}(B_{\overline{r}}\backslash
B_{1})}=\frac{\omega_{n}}{(n\beta)^{\alpha p}}r^{n}+o(r^{n})$

as if $1<s<\overline{r}$

$\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}\leq
v(s)\leq\frac{1}{(n\beta)^{\alpha}}\overline{r}^{-\frac{n-1}{p-1}}+\frac{p-1}{n^{\alpha}(p-n)}\left(\overline{r}^{\frac{p-n}{p-1}}-1\right)$

thus

$v(s)=\frac{1}{(n\beta)^{\alpha}}+O(r^{n})$

and by integration we obtain the value of the norm in
$L^{p}(B_{\overline{r}}\backslash B_{1})$.

So

${\left\|v\right\|}^{p}_{L^{p}(\Omega^{\sharp})}={\left\|v\right\|}^{p}_{L^{p}(B_{1})}+{\left\|v\right\|}^{p}_{L^{p}(B_{\overline{r}}\backslash
B_{1})}={\left\|u\right\|}^{p}_{L^{p}(\Omega)}+p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}h+\frac{\omega_{n}}{(n\beta)^{\alpha
p}}r^{n}+o(r^{n})$

and recalling that

$h=\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)r^{n}+o(r^{n})$

we get

${\left\|v\right\|}^{p}_{L^{p}(\Omega^{\sharp})}={\left\|u\right\|}^{p}_{L^{p}(\Omega)}+\left[p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)+\frac{\omega_{n}}{(n\beta)^{\alpha
p}}\right]r^{n}+o(r^{n}).$

We have to understand whether

$p{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}\left(-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}\right)+\frac{\omega_{n}}{(n\beta)^{\alpha
p}}<0.$ (30)

If we choose $\displaystyle{\beta<\left(\frac{n-1}{p-1}\right)^{p-1}}$ we have
$\displaystyle{-\frac{1}{(n\beta)^{\alpha}}\frac{n-1}{n(p-1)}+\frac{1}{n^{\alpha+1}}<0}$.
In order to have (30), we need

$\displaystyle{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}$
$\displaystyle>\frac{\omega_{n}}{(n\beta)^{\alpha
p}}\left[\frac{n(p-1)n^{\alpha}\beta^{\alpha}}{p(n-1)-p\beta^{\alpha}(p-1)}\right]$
$\displaystyle{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}$
$\displaystyle>\frac{\omega_{n}}{(n\beta)^{\alpha(p-1)}}\left[\frac{n(p-1)}{p(n-1)-p\beta^{\alpha}(p-1)}\right].$

If we show that

$\left[\frac{n(p-1)}{p(n-1)-p\beta^{\alpha}(p-1)}\right]\leq 1$ (31)

then

$u(s)>\frac{1}{(n\beta)^{\alpha}}\implies{\left\|u\right\|}^{p-1}_{L^{p-1}(B_{1})}>\frac{\omega_{n}}{(n\beta)^{\alpha(p-1)}}.$

We just have to verify (31)

$\begin{split}\left[\frac{n(p-1)}{p(n-1)-p\beta^{\alpha}(p-1)}\right]\leq
1&\iff n(p-1)\leq p(n-1)-p\beta^{\alpha}(p-1)\\\ &\iff
p-n\leq-p(p-1)\beta^{\alpha}<0\quad\text{ (if and only if $p<n!$)}\\\
&\iff\beta\leq\left(\frac{n-p}{p(p-1)}\right)^{p-1}\end{split}$

## References

  * [1] Alvino A., Chiacchio F., Nitsch C., and Trombetti C. Sharp estimates for solutions to elliptic problems with mixed boundary conditions. _J. Math. Pures Appl._ , 152:251—261 (2021).
  * [2] A. Alvino, P.L. Lions, and G. Trombetti. Comparison results for elliptic and parabolic equations via Schwarz symmetrization. _Ann. Inst. H. Poincaré Anal. Non Linéaire_ , 7(2):37–65 (1990).
  * [3] A. Alvino, C. Nitsch, and C. Trombetti. A Talenti comparison result for solutions to elliptic problems with robin boundary conditions. _to appear on Comm. Pure Appl. Math._
  * [4] Dorin Bucur and Daniel Daners. An alternative approach to the Faber-Krahn inequality for Robin problems. _Calc. Var. Partial Differential Equations_ , 37(1-2):75–86 (2010).
  * [5] Dorin Bucur and Alessandro Giacomini. A variational approach to the isoperimetric inequality for the Robin eigenvalue problem. _Arch. Ration. Mech. Anal._ , 198(3):927–961 (2010).
  * [6] Dorin Bucur and Alessandro Giacomini. Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity approach. _Arch. Ration. Mech. Anal._ , 218(2):757–824 (2015).
  * [7] Dorin Bucur, Alessandro Giacomini, and Paola Trebeschi. Best constant in Poincaré inequalities with traces: a free discontinuity approach. _Ann. Inst. H. Poincaré Anal. Non Linéaire_ , 36(7):1959–1986 (2019).
  * [8] Peter Lindqvist. Addendum: “On the equation ${\rm div}(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=0$” [Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164; MR1007505 (90h:35088)]. _Proc. Amer. Math. Soc._ , 116(2):583–584 (1992).
  * [9] Jindřich Nečas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. _Springer, Heidelberg_ (2012). Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader.
  * [10] Giorgio Talenti. Elliptic equations and rearrangements. _Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)_ , 3(4):697–718 (1976).
  * [11] Giorgio Talenti. Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. _Ann. Mat. Pura Appl. (4)_ , 120:160–184 (1979).
  * [12] Giorgio Talenti. Inequalities in rearrangement invariant function spaces. In _Nonlinear anal- ysis, function spaces and applications, Vol. 5 (Prague, 1994)_ , pages 177–230. Prometheus, Prague (1994).
  * [13] J. L. Vázquez. A strong maximum principle for some quasilinear elliptic equations. _Appl. Math. Optim._ , 12(3):191–202 (1984).