# Ricci Inverse Anisotropic Stellar Structures

M. Farasat Shamir<EMAIL_ADDRESS>National University of Computer
and Emerging Sciences,
Lahore Campus, Pakistan.  Mushtaq Ahmad<EMAIL_ADDRESS>National
University of Computer and Emerging Sciences, Islamabad,
Chiniot-Faisalabad Campus, Pakistan.  G. Mustafa<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Zhejiang Normal University,
Jinhua 321004, China  Aisha Rashid<EMAIL_ADDRESS>National
University of Computer and Emerging Sciences,
Lahore Campus, Pakistan.

###### Abstract

This paper offers novel quintessence compact relativistic spherically
symmetrical anisotropic solutions under the recently developed Ricci inverse
gravity Amendola , by employing Krori and Barua gravitational potentials,
$Ar^{2}=\nu(r),~{}\&~{}Br^{2}+C=\mu(r)$ (with A, B, and C being real
constants). For this objective, a specific explicit equation of state,
connecting energy density and radial pressure, i.e., $p_{r}=\omega\rho$, such
that $0<\omega<1$, has been utilized with an anisotripic fluid source. Ricci
inverse field equations are used to find the exclusive expressions of the
energy density, radial and tangential stresses, and the quintessence energy
density, the critical physical attributes reflecting the exceptional conduct
of extremely dense matter configuration. For the observatory source stars
$HerX-1$, $SAXJ1808.4-3658$ & $4U1820-30$, all the important physical
quantities like energy densities, tangential and radial pressures, energy
conditions, gradients, anisotropy, redshift and mass-radius functions, and
stellar compactness have been worked out and analyzed graphically. It has been
concluded that all of the stellar formations under consideration remain free
from any undesirable central singularity and are stable.

Keywords: Anisotropic compact stars; Ricci Inverse Gravity; Quintessence.

##  I Introduction

The self-gravitation mechanism and its effects have a significant impact on
new astrophysics studies and its background. This gravitational collapse
results in the formation of new stellar components known as compact stars.
These compact objects are thought to be exceedingly dense because the are
massive but with smaller volumetric radii. Compact stars have attracted the
interest of researchers in relativistic astronomy due to the relativistic
structures, which are well explained by general relativity and by different
modified theories weber ; delgo ; jimr10 ; jimr11 ; jimr12 ; jimr13 ; jimr14 ;
jimr15 ; jimr16 ; jimr17 ; ad11 ; ad8 ; ad15  as well. Modified and extended
theories have been used by countless researchers to overcome the limitations
and restrictions of general relativity, which occurred because of dark matter
and dark energies kra ; ost ; lid ; brat2 ; wett ; ferr  that effects the
accelerated universe. Each of these approaches is without flaws. Some of the
researcher have used simple cosmological constant wein ; carrl ; brat ; cal ;
armn1 ; armn ; sahn , which has just recently gained dominance in the present
era. Quintessence approach depending upon a scalar field has also been used by
zal ; brax ; barr11 . The emergence of new dimensional constants is an
unappealing feature of most alternative formulations. Another issue is the
addition of emerging fields, either scalar or vector, with no evident
relationship to the curvature tensor, which represents a significant change
from Einstein’s geometrical approach. These difficulties are not necessarily
deal breakers, but they are also not grounds in favor of changing Einstein’s
study of gravity. Moreover, the Starobinsky models staro11 ; barr33 ; staro22
; barr44 ; mull11 ; kosh11 ; mish11 , which include the $R^{2}$ corrected
term, are one of the most promising models. These models are also the
fundamental variants of the $f(R)$ theory feli22 ; nojiri22 . As a result,
having a reliable alternative model may lead to a more revealing picture of
our universe’s early and late time eras appl02 . Furthermore, surprising
findings may occur in alternate gravitational theories due to the availability
of extra $4th$ order corrected terms like $R^{2}$ or
$R_{\alpha\beta}AR^{\alpha\beta}$.

Deser and Woodard deser01  investigated two models that allow for a slow
reaction to cosmic occurrences while avoiding the normal range. They apply the
analysis on FRW space-time by taking the inverse of the D’Alembert operator
along with cosmological constant. Soussa and Woodard sousa  analyzed the
altered Ricci scalar, which allowed them to calculate the gravitational effect
imparted by the metric perturbation trace. They have restricted the matter
dispersion to have the property that its gravitational collapse rate is
similar to the rate of space-time extension. Moreover, Amendola et. al. amen2
have expanded the idea of Deser and Woodard by employing the non-local
function of altered gravity and changing the Einstein-Hilbert equation with a
term $Rf(\square^{-1}R)$, where $f$ is a random function. Furthermore, a non-
local altered gravity nojiri55 ; barv  was obtained by adding the expression
$m^{2}R\Box^{-1}R$, which has caused a self-accelerating expansion by
establishing the equation of state (EoS) of dark energy, $w_{DE}\simeq-1.14$
maggi . Amendola Amendola  has developed a new theory for gravity known as
Ricci inverse (RI) gravity which is dependent on the anti-curvature scalar
$A$. In RI model, the term $A$ is the trace of anti-curvature tensor
$A^{\alpha\beta}$ that holds the property
$A^{\alpha\beta}=R^{-1}_{\alpha\beta}$. Motivated from this recently developed
modified theory of gravity, we aim to investigate anisotropic compact
structures in RI gravity. For this purpose, we investigate the quintessence
structures for compact stars in the RI theory of gravity using $HerX-1$,
$SAXJ1808.4-3658$, and $4U1820-30$ as star sources. Our work is managed as
follows: Section II presents the fundamental concepts, structure, and
equations of the RI gravity. In Section III, the matching constraints for the
RI gravity are developed using Schwarzschild’s geometry. Section IV is devoted
to calculating and analysing physical properties such as energy density,
pressure profiles and quintessence density. The final section contains come
concluding remarks.

##  II Inverse Ricci Gravity

The anticurvature tensor, constructed as inverse of Ricci tensor, enables the
growth of an alternative gravity theory. A theory like this will need to be
examined in order to determine the prevalence of ghosts-like instabilities and
infer the consequences on perturbations like gravitational wave propagation,
perturbation growth, or the Newtonian bound. Hence, a theory like this may
prove a viable option to be assumed as cosmological model after drawing the
general motion equations of the generic function with Lagrangian representing
scalars of the curvature and anticurvature. It has been demonstrated that in
every Lagrangian including a term constituted with some positive or negative
exponent of the anticurvature scalar, a generalized no-go theorem prohibits
cosmological trajectories from joining a decelerated stage to an accelerated
one, ruling out a large group of models. In some especially simple examples of
models with no additional dimensional scales, the no-go theorem is
demonstrated analytically and mathematically. In this novel theory, the
various escape routes from the theorem have been examined, highlighting
several (quite complicated) Lagrangians that do so. A thorough examination of
such concepts in quest of fresh phenomenology and will happen to be quite
fascinating if further investigation is made. A tensor $A^{ab}$ as inverse of
$R_{ab}$ is defined as

$A^{ab}R_{bc}=\delta^{a}_{c},$ (1)

where $A^{ab}$ the anticurvature tensor. The action for Ricci-inverse gravity
is given by Amendola

$S_{Action}=\int\sqrt{-g}d^{4}x(\alpha A+R),$ (2)

where $A$ is a trace of $A^{ab}$, which is defined as

$A^{ab}=R^{-1}_{ab}.$ (3)

The modified field equations are given as Amendola

$R^{ab}-\frac{1}{2}Rg^{ab}-\alpha A^{ab}-\frac{1}{2}\alpha
Ag^{ab}+\frac{\alpha}{2}\left(2g^{\varrho
a}\nabla_{\alpha}\nabla_{\varrho}A^{\alpha}_{\sigma}A^{b\sigma}-\nabla^{2}A^{a}_{\sigma}A^{b\sigma}-g^{ab}\nabla_{\alpha}\nabla_{\varrho}A^{\alpha}_{\sigma}A^{\varrho\sigma}\right)={\mathcal{T}}^{ab}$
(4)

where $A^{\alpha}_{\sigma}A^{b\sigma}=A^{\alpha\tau}g_{\tau\sigma}A^{\sigma
b}=A^{\alpha\tau}A^{b}_{\tau}=A^{\alpha\sigma}A^{b}_{\sigma}=A^{b}_{\sigma}A^{\alpha\sigma}$
with $8\pi G=1$. The stress tensor possessing anisotropic matter content is
read as

${\mathcal{T}}_{ab}=(\varrho+p_{t})u_{a}u_{b}-p_{t}g_{ab}+(p_{r}-p_{t})v_{a}v_{b}.$
(5)

Here, $u_{a}=e^{\frac{b}{2}}\delta_{a}^{0}$ and
$v_{a}=e^{\frac{\lambda}{2}}\delta_{a}^{1}$,
${\mathcal{T}}_{a}^{b}={\mathcal{T}}_{a}^{b}+{\mathcal{D}}_{a}^{b}$,
${\mathcal{D}}_{a}^{b}$ gives the stress tensor for quintessence field
equation exhibiting energy density expressed as $\rho_{q}$ and the parameter
$w_{q}$ represents quintessence field such that ($-1<w_{q}<-\frac{1}{3}$).
Also, $-{\mathcal{D}}_{t}^{t}={\mathcal{D}}_{r}^{r}=-\rho_{q}$, and
$\frac{(3w_{q}+1)\rho_{q}}{2}={\mathcal{D}}_{\theta}^{\theta}={\mathcal{D}}_{\phi}^{\phi}$.
The spherically symmetric metric is given by

$ds^{2}=e^{\mu(r)}dt^{2}-e^{\nu(r)}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2},$ (6)

Now we can choose the Krori and Barua gravitational potentials, which are
defined as

$Ar^{2}=\nu(r)\;\;\;\;\;\;\;\;\;\;\;\;\;\;Br^{2}+C=\mu(r),$ (7)

with $A$, $B$, and $C$ are some arbitrary constants. By using Eqs. (5-7) in
Eq. (4) with quintessence field, we get the following system of modified field
equations

$\displaystyle\rho+\rho_{q}$ $\displaystyle=$ $\displaystyle
e^{-Ar^{2}}\left(\frac{\alpha\left(\chi_{2}+\chi_{3}-\chi_{4}\right)e^{2Ar^{2}}}{B^{2}\left(r^{2}(B-A)+3\right)^{4}}+\frac{1}{2}\alpha\chi_{1}e^{2Ar^{2}}-\frac{2Ar^{2}+e^{Ar^{2}}-1}{r^{2}}\right),$
(8) $\displaystyle p_{r}-\rho_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha\left(\frac{2\chi_{6}}{r\left(B(Br+1)-A\left(Br^{2}+2\right)\right)^{4}}+\chi_{5}\right),$
(9) $\displaystyle p_{t}+\frac{1}{2}(3w_{q}+1)\rho_{q}$ $\displaystyle=$
$\displaystyle-r^{2}\left(-\alpha
e^{Ar^{2}}\left(\frac{\chi_{9}}{\left(r^{2}(A-B)+e^{Ar^{2}}-1\right)^{4}}+\chi_{8}\right)+\alpha\chi_{7}e^{Ar^{2}}+\chi_{10}e^{-Ar^{2}}\right).$
(10)

where $\chi_{i}$, $\\{i=1,...,9\\}$ are given in the Appendix (I). To find the
explicit expressions for the above field equations, we assume a relation
between radial pressure and energy density via $EoS$ parameter, which is
defined as

$p_{r}=\omega\rho,\;\;\;\;\;\;\;\;\;\;\;0<\omega<1.$ (11)

The above relation by Eq. (11) is the particular form of $EoS$, the general
form of this equation was presented by Herrera and Barreto jimr1 . This linear
of equation state, though a simplest choice, has been extensively considered
to discuss the physical properties of astrophysical compact objects jimr2 ;
jimr3 ; jimr4 ; jimr5 ; jimr6 ; jimr7 ; jimr8 ; jimr9 . By using Eq. (11) with
Eqs. (8-10), we get the final expressions for energy density $\rho$,
quintessence energy density $\rho_{q}$, and pressure components
$p_{r},\;p_{t}$ as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{-\frac{1}{2}\alpha\left(2\Psi_{1}\Psi_{2}\right)-e^{-Ar^{2}}\left(\frac{\alpha\Psi_{4}e^{2Ar^{2}}}{B^{2}\left(r^{2}(B-A)+3\right)^{4}}+\frac{1}{2}\alpha\Psi_{3}e^{2Ar^{2}}-\frac{2Ar^{2}+e^{Ar^{2}}-1}{r^{2}}\right)}{\omega+1},$
(12) $\displaystyle\rho_{q}$ $\displaystyle=$
$\displaystyle\frac{e^{-Ar^{2}}\left(-\frac{\alpha\Psi_{4}e^{2Ar^{2}}}{B^{2}\left(r^{2}(B-A)+3\right)^{4}}-\frac{1}{2}\alpha\Psi_{3}e^{2Ar^{2}}+\frac{2Ar^{2}+e^{Ar^{2}}-1}{r^{2}}\right)-\frac{1}{2}\alpha\left(2\Psi_{1}\Psi_{2}\right)}{\omega+1}$
(13) $\displaystyle+$ $\displaystyle
e^{-Ar^{2}}\left(\frac{\alpha\Psi_{4}e^{2Ar^{2}}}{B^{2}\left(r^{2}(B-A)+3\right)^{4}}+\frac{1}{2}\alpha\Psi_{3}e^{2Ar^{2}}-\frac{2Ar^{2}+e^{Ar^{2}}-1}{r^{2}}\right),$
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{\omega\left(-\frac{1}{2}\alpha\left(2\Psi_{1}\Psi_{2}\right)-e^{-Ar^{2}}\left(\frac{\alpha\Psi_{4}e^{2Ar^{2}}}{B^{2}\left(r^{2}(B-A)+3\right)^{4}}+\frac{1}{2}\alpha\Psi_{3}e^{2Ar^{2}}-\frac{2Ar^{2}+e^{Ar^{2}}-1}{r^{2}}\right)\right)}{\omega+1},$
(14) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle-r^{2}\left(e^{-Ar^{2}}\left(\frac{\alpha
e^{2Ar^{2}}}{r^{2}(A-B)+e^{Ar^{2}}-1}+\frac{A-2B}{r^{2}}+B(A-B)\right)-\alpha
e^{Ar^{2}}\right.$ (15) $\displaystyle\times$
$\displaystyle\left.\left(\Psi_{6}\left(\Psi_{8}e^{Ar^{2}}+\Psi_{7}\right)\right)+\alpha\Psi_{5}e^{Ar^{2}}\right)-\frac{1}{2}\rho_{q}(3w_{q}+1),.$

where where $\Psi_{i}$, $\\{i=1,...,10\\}$ are given in the Appendix (I).

##  III Matching Conditions

The interior border metric does not vary irrespective of the star’s
geometrical structure, whether from the outside or the inside. Regardless of
the reference frame, this emergent circumstance mandates that the metric
components must be continuous to the boundary. While evaluating
Schwarzschild’s solution connected with stellar remnants in general
relativity, it is well-thought-out to be the main priority out of all the
accessible different alternatives of matching circumstances. Furthermore,
while engaging with modified gravity theories, it is a good idea to account
for quasi pressure and energy density. Many scholars have done outstanding
work on boundary conditions A ; B . By integrating some unique constraints to
stellar compact structures as well as the thermodynamically related
characteristics, Goswami et al. G  figured out the matching bounds when
exploring the modified gravity. However, the well-known Schwarzschild’s
geometry can be used to solve the various stellar solutions. A few constraints
are imposed at the boundary $r=R$, that is, $p_{r}(r=R)=0$, to get the
expressions for the field equations. To get the quantities of the associated
unknowns, we must first figure out how to align the internal geometry with the
external spacetime. We may employ the Schwarzschild spacetime for this, which
is described as

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (16)

The total mass trapped within the star is denoted by $M$, while the radius of
the compact celestial leftover is denoted by $R$. It is important to mention
here that whatever the geometry of the star is, either derived internally or
externally, the boundary metric must remains the same with the condition that
the components of the metric tensor irrespective of the coordinate system
across the surface of the boundary will remain continuous. No doubt, in theory
of general relativity, the Schwarzschild solution has been really important in
motivating us to choose from the diverse possibilities of the matching
conditions. However, while investigating the compact objects in some
alternative gravity theories, modified Tolman-Oppenheimer-Volkoff equations
with zero pressure and energy density, the solution outside the star can
differ from Schwarzschild’s solution. Thus, it is expected that the solutions
of the modified Tolman-Oppenheimer-Volkoff equations with non-zero energy
density and pressure may accommodate Schwarzschild’s geometry with some
specific choice of modified gravity model R1 ; R2 ; R3 ; R4 ; R5 . Perhaps
this might be the reason or justification that Birkhoff’s theorem may not hold
in alternative theories of gravity. The detailed investigation on the topic in
the context of RI gravity can be an interesting future task.

By examining the metric potentials, the accompanying expressions are obtained
at $r=R$

$\displaystyle
g_{tt}^{+}=g_{tt}^{-},\;\;\;\;\;\;\;g_{rr}^{+}=g_{rr}^{-},~{}~{}~{}~{}\frac{\partial
g_{tt}^{+}}{\partial r}=\frac{\partial g_{tt}^{-}}{\partial r}.$ (17)

The inner and outer signatures respective to the boundary $r=R$ are reflected
as (-) and (+), respectively. Comparison of the inner and outer matrices, the
values of the desired unknown quantities can be determined as follows:

$\displaystyle
A=-\frac{log(1-\frac{2M}{R_{\varepsilon}})}{R_{\varepsilon}^{2}},~{}~{}~{}~{}B=-\frac{Mlog(1-\frac{2M}{R_{\varepsilon}})^{-1}}{R_{\varepsilon}^{2}},~{}~{}~{}C=log(1-\frac{2M}{R_{\varepsilon}})-\frac{M}{R_{\varepsilon}(1-\frac{2M}{R_{\varepsilon}})}.$
(18)

The quantities $M$, $R$, $M/R$, and the calculated values of the involved
parameters like $A$ and $B$, are provided in Tab. 1. Actually, the values of
the parameters $A$ and $B$ come from the Eq. 18. As it’s very clear from the
Eq. 18, there is no effect of the quintessence field $w_{q}$. In fact, Eq. 18
only depends upon the observational values of stars like $HerX-1$,
$SAXJ1808.4-3658$ & $4U1820-30$. Further, it is necessary to mention that
quintessence field $w_{q}$ has an important role in the anisotropic matter,
especially in the tangential pressure component.

Table 1: Approximated values of involved parameters with $(\alpha=1.3\times 10^{-11})$ and $w_{q}=-0.4;)$ . $StrangeQuarkStar$ | $M$ | $R(km)$ | $\frac{M}{R}$ | $A(km^{-2})$ | $B(km^{-2})$
---|---|---|---|---|---
$HerX-1$ | $0.88M_{\odot}$ | $7.7$ | $0.168$ | $0.006906276428$ | $0.004267364618$
$SAXJ1808.4-3658$ | $1.435M_{\odot}$ | $7.07$ | $0.299$ | $0.01823156974$ | $0.01488011569$
$4U1820-30$ | $2.25M_{\odot}$ | $10.0$ | $0.332$ | $0.01090644119$ | $0.009880952381$

Figure 1:  Shows the energy densities for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 2:  Shows the quintessence energy density for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 3:  Shows the radial pressure for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 4:  Shows the tangential pressure for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

##  IV Physical Analysis

We devote this section to investigate a few key characteristics associated
with considered compact stars. Energy density $\rho$, pressure profiles
$p_{r}$, and $p_{t}$, and debates on the quintessence field, as well as their
physical interpretation under RI gravity, are among them. The energy bounds,
anisotropic stress, compactness factor, and the star’s sound speed concerning
the radial and tangential components are also discussed.

###  IV.1 Density and Pressure Profiles

The related evolutions of the energy density, as well as the radial and
tangential pressures, make up the most significant stellar environment
required for the existence of compact structures. The energy density profiles,
and the evolutions of quintessence density, and pressure terms have been
studied for $HerX-1$ (Left), $SAXJ1808.4-3658$ (Middle), and $4U1820-30$
(Right) for diversified values of $\omega=0.01$, $0.03$, $0.05$, $0.07$,
$0.09$. The plots of the related graphs as shown in Figs. (1-4) clearly
demonstrate the related physical analysis. The energy density reaches its
maximum value near the star’s core, indicating the star’s super-dense nature.
Furthermore, the tangential and radial pressure profiles exhibit positive
behavior and reach their maximal values at the surface of star. For our model
under RI gravity, the evolutions of the stars under study also imply the
existence of an anisotropic structure independent of any possible
singularities.

###  IV.2 Energy Bounds

Figure 5:  Shows the energy condition for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 6:  Shows the energy condition for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 7:  Shows the energy condition for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 8:  Shows the energy condition for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 9:  Shows the energy condition for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

The importance of energy bounds among existing physical characteristics in
explaining the presence of anisotropic compact structures is widely recognized
throughout the literature. Furthermore, they serve an important role in
determining the distribution of the matter content of normal as well as exotic
sources inside stellar structures. Because of these energy limitations,
several important conclusions can be reached. Physical attributes for the
energy conditions seem to be fairly advantageous in realistically examining
the matter distribution. The energy constraints have remained fundamentally
vital in addressing cosmological and astrophysical challenges. Below are the
expressions for $NEC$, null energy bounds, $SEC$, the strong energy bounds,
$DEC$, the dominant energy bounds, and $WEC$, the week energy bounds.

$\displaystyle\begin{aligned} NEC:\rho+p_{r}\geq
0,~{}0\leq\rho+p_{t};~{}~{}WEC:\rho\geq 0,~{}\rho+p_{r}\geq,~{}\rho+p_{t}\geq
0,\\\ SEC:0\leq\rho+p_{r},~{}\rho+p_{t}\geq 0,~{}0\leq
2p_{t}+p_{r}+\rho+;~{}~{}DEC:|p_{r}|<\rho,~{}|p_{t}|<\rho.\\\ \end{aligned}$
(19)

In Figs. (5-9), the graphical development of the energy restrictions is
illustrated. The positive profiles of the energy bounds for the stars,
$HerX-1$, $SAXJ1808.4-3658$, and $4U1820-30$ for a variety of choices of $EoS$
parameter show that our solutions are realistically acceptable under RI
gravity.

###  IV.3 Gradients and Anisotropy Analysis

The total derivatives of the physical quantities $\rho$, $p_{r}$, and $p_{t}$
for compact star are described by the expressions $\frac{d\rho}{dr}$,
$\frac{dp_{r}}{dr}$, and $\frac{dp_{t}}{dr}$, respectively. The graphical
representations (10-12) of all these radial derivatives for the compact stars
under discussion suggest a negative evolution, i.e., $\frac{d\rho}{dr}<0$,
$\frac{dp_{r}}{dr}<0$, and $\frac{dp_{t}}{dr}<0$. It is worth noticing here
that the gradients $\frac{d\rho}{dr}$, and $\frac{dp_{r}}{dr}$
$\rightarrow{0}$ at the core of the star $r=0$. This verifies the radial
pressure $p_{r}$ attains the maximum bound as well as the density $\rho$ at
the core of the star. As a result, $\rho$ and $p_{r}$ achieve the maximum
value at $r=0$.

Figure 10:  Shows the derivative of energy density with respect to $r$ for
$HerX-1(Left)$, $SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 11:  Shows the derivative of radial pressure with respect to $r$ for
$HerX-1(Left)$, $SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 12:  Shows the derivative of tangential pressure with respect to $r$
for $HerX-1(Left)$, $SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

The role of anisotropy is critical for the compact sphere modelling in order
to describe the inner geometry of the relativistic stellar configuration under
given conditions, and it is represented by $\Delta$, and is defined as
$\Delta=p_{t}-p_{r}$. It shows the deviation between the tangential and radial
stress quantities for the star configuration’s anisotropic nature. If
$p_{t}>p_{r}$, the anisotropy is deemed non-negative, and it is drawn outwards
and seen as $\Delta>0$. If $p_{r}>p_{t}$, then, the anisotropy remains
negative, resulting in $\Delta>0$, indicating that anisotropy is pulled
inwards. It is evident from Fig. (13) that anisotropy stays positive for our
current investigation, i.e. $\Delta>0$, and, therefore, directed outwards.

Figure 13:  Shows the anisotropy for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

###  IV.4 Stability analysis and Abreau Condition

The sound speeds related with the radial and transversal components with their
corresponding representations of $v^{2}_{r}$ Aand $v^{2}_{t}$ establish the
stability of the compact star solutions obtained in our investigations. For
this purpose, The conditions $0\leq{v^{2}_{t}}=\frac{dp_{t}}{d\rho}\leq 1$ and
$0\leq{v^{2}_{r}}=\frac{dp_{r}}{d\rho}\leq 1$ must be achieved Herrera . The
related plots of sound speeds in Figs. (14) for different values of $EoS$
parameter demonstrate that the radial and transversal sound speeds for the
stellar candidates $HerX-1$, $SAXJ1808.4-3658$, and $4U1820-30$ remain within
the necessary stability restrictions. The limitations of both the radial and
transversal sound velocities are validated for all compact star candidates.
The approximation of the steady and fragile epochs inside the anisotropic
matter configurations can be discussed by modifying the sound speeds
propagations, which have the expression $v^{2}_{t}-v^{2}_{r}$ endorsing the
constraint $0<|v^{2}_{t}-v^{2}_{r}|<1$, and this condition is referred to as
the Abreu condition depicted in Fig.(15). It shows that the for the compact
stars discussed under RI gravity, total stability may be achieved.

Figure 14:  Shows the speeds of sound parameters for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 15:  Shows the Abreu condition for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 16:  Shows the equations of state parameters for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

###  IV.5 Equation of State Parameter

The development of compact star emergence can be predicted using $EoS$
parameter. Furthermore, $EoS$ is a crucial component of any astrophysical
model. The pressure terms $p_{r}$ and $p_{t}$ are used to calculate the $EoS$
for the anisotropic matter distribution. The components of $EoS$ for studying
stellar configuration are $\omega_{r}$ and $\omega_{t}$, which are
mathematically written as

$\omega_{r}=\frac{p_{r}}{\rho_{e}},~{}~{}~{}~{}~{}~{}\omega_{t}=\frac{p_{t}}{\rho_{e}}.$
(20)

The components of the $EoS$ exhibit monotonically decreasing behavior with
increasing radii and is always below 1, which can be witnessed in Fig. 16.
Furthermore,the positive character of both $EoS$ components, $\omega_{r}$ and
$\omega_{t}$, is seen which fulfills the requirement $0\leq\omega_{r}$ and
$\omega_{t}<1$ for the viable stellar solutions.

###  IV.6 Mass-Radius relation, Compactness, and Redshift Analysis

Radii $r$ dependent mass function of a compact star, $m(r)$ is determined by
employing the integral as follows

$m(r)=4\int_{0}^{r}\pi\acute{r^{2}}\rho{d}\acute{r}.$ (21)

The mass-radius graph in Fig. (17) shows that the mass is exactly correlated
to the radius $r$, so that as $r\rightarrow 0$, $m(r)\rightarrow 0$,
indicating that the correct behavior of mass at the star’s core. Also, as
calculated by Bhomer and Harko boht , the mass-radius ratio must remain
$\frac{2M}{r}\leq\frac{8}{9}$ , which in also validated in our study. The
stellar structure’s compactness $\mathcal{K}(r)$ is given by the integral

$\mathcal{K}(r)=\frac{4\pi}{r}\int_{0}^{r}\rho\acute{r^{2}}{d}\acute{r},$ (22)

and plotted in Fig. (18). The redshift $\Re_{S}$ function is given as

$\Re_{S}+1=[1-2\mathcal{K}(r)]^{\frac{-1}{2}}.$ (23)

Fig. (19) shows its graphical depiction and it is concluded that the numerical
approximation of $\Re_{S}$ is still within the required threshold $\Re_{S}\leq
2$, indicating that our worked out solutions in RI gravity are indeed viable.

Figure 17:  Shows the mass function for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 18:  Shows the compactness function for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

Figure 19:  Shows the redshift function for $HerX-1(Left)$,
$SAXJ1808.4-3658(Middle)$, and $4U1820-30(Right)$ with
$\omega=0.01({\color[rgb]{1,0,0}\bigstar})$,
$\omega=0.03({\color[rgb]{.75,0,.25}\bigstar})$,
$\omega=0.05({\color[rgb]{0,0,1}\bigstar})$,
$\omega=0.07({\color[rgb]{1,0,1}\bigstar})$, and
$\omega=0.09({\color[rgb]{0,0,0}\bigstar})$.

##  V Results and discussions

This study is focussed to explore anisotropic compact structures, using a new
alternative theory termed as RI Gravity Amendola . The related field equations
of the RI gravity for compact stars are solved using certain assumptions. We
establish our calculations based on the statistics related to the $HerX-1$,
$SAXJ1808.4-3658$, and $4U1820-30$ as strange star candidates, by choosing
appropriate $EoS$ parametric values. Our current investigation is focused on
the feasibility of the existence of quintessence anisotropic compact stars.
The gravitational breakdown for the emerging universe at diverse epochs has
now been examined by integrating spacetime symmetries with the exclusively
matched Schwarzschild’s geometry. Following are some of the primary findings
from current analysis along with some key points.

  * •

The terms $\rho$, $p_{t}$, $p_{r}$ are critical physical factors for the
creation of stellar formations. The related plots and the statistics connected
to the strange star candidates under our investigation indicate that the
energy density near the star core reaches the maximum value, indicating the
star’s high-density nature. Furthermore, the tangential and radial pressure
terms remain positive and show monotonically decreasing behavior towards the
boundary of the star. These patterns also provide evidence for the presence of
anisotropic matter distribution free from singularities for the RI gravity
under consideration. Additionally, the plots of the quintessence energy
density show a negative trend, indicating that our stellar RI gravity model
supports quintessence.

  * •

The importance of energy restrictions in the literature on compact stellar
structures is fairly clear. All the related plots show that energy constraints
are satisfies proving that our obtained solutions are physically viable under
RI gravity. The gradient profiles $\frac{d\rho}{dr}$, $\frac{dp_{r}}{dr}$ and
$\frac{dp_{t}}{dr}$ of the compact stars are shown in the respective plots. It
is indicated that the first ordere derivative is evolving negatively which
verifies the upper bound of the radial pressure $p_{r}$ with respect to the
density $\rho$. As a result, at $r=0$, $\rho$ and $p_{r}$ attain the maximum
value.

  * •

The associated sound speed graphs are illustrated bythe relevant figures,
which demonstrate that the radial, as well as the transversal sound speeds for
all presented stars, are confined within the required limits of consistency.
The graphical developments demonstrate that the condition
$0<|v^{2}_{st}-v^{2}_{sr}|<1$ is metfor the stable behavior of a compact star.

  * •

The $EoS$ parameters for the stars $HerX-1$, $SAXJ1808.4-3658$, and
$4U1820-30$ have also been plotted for the various values. It is concluded
that the constraints $\omega_{r}$ and $\omega_{t}$ of $EoS$ remain positive in
the stellar interior and are within the stability bounds
$0\leq\omega_{r}$,Aand $\omega_{t}<1$ under RI gravity.

  * •

The anisotropy $\Delta$ with respect to $r$ of the stars under observation
shows a positive increasing behavior as illustrated in their associated
graphs, indicating repelling anisotropic forces accommodated by a dense matter
source.

  * •

Finally, the compactness parameter, mass function, and redshift are also
plotted. It can be observed that $\mu(r)\leq 0.30$ and has an increasing
tendency, whereas $m(r)$ also has an increasing character. Further, $Z_{s}$
has a decreasing attribute and is always $Z_{s}\leq 5$, which is associated
with the stability of the compact structures .

## Appendix (I)

$\displaystyle\chi_{1}$
$\displaystyle=-\frac{2r^{2}}{r^{2}(A-B)+e^{Ar^{2}}-1}+\frac{3}{B\left(r^{2}(B-A)+3\right)}+\frac{1}{Br^{2}(B-A)-2A+B},$
$\displaystyle\chi_{2}$
$\displaystyle=-72A^{3}r^{4}+143A^{2}Br^{4}+114A^{2}r^{2}-88AB^{2}r^{4}-117ABr^{2}+72A+30B^{2}r^{2}-45B,$
$\displaystyle\chi_{3}$
$\displaystyle=10A^{4}r^{6}-29A^{3}Br^{6}+30A^{2}B^{2}r^{6}-13AB^{3}r^{6}+2B^{4}r^{6}+17B^{3}r^{4},$
$\displaystyle\chi_{4}$
$\displaystyle=-2Br^{2}e^{2Ar^{2}}\left(r^{2}(B-A)+3\right)\left(r^{2}\left(2A^{2}-3AB+B^{2}\right)-8A+5B\right),$
$\displaystyle\chi_{5}$
$\displaystyle=\frac{2r^{2}\left(r^{2}\left(2A^{2}-3AB+B^{2}\right)-8A+5B\right)}{B\left(r^{2}(B-A)+3\right)^{3}},$
$\displaystyle\chi_{6}$
$\displaystyle=3A^{4}r^{3}\left(Br^{2}+2\right)^{2}-A^{3}r\left(Br^{2}+2\right)\left(Br^{2}(Br(r+6)+25)-2\right)+A^{2}Br(Br(r(Br(r(B(2r+3)+4)+35)$
$\displaystyle+61)+14)-8)-AB^{2}(r(Br(r(B(r(Br+5)+8)+3)+45)-3)+4)+B^{3}\left(Br\left(Br^{2}+r+11\right)+2\right),$
$\displaystyle\chi_{7}$
$\displaystyle=\frac{\frac{1}{Br^{2}(A-B)+2A-B}-\frac{1}{B\left(r^{2}(B-A)+3\right)}}{2r^{2}}+\frac{1}{r^{2}(A-B)+e^{Ar^{2}}-1},$
$\displaystyle\chi_{8}$
$\displaystyle=\frac{r^{2}\left(2A^{2}-3AB+B^{2}\right)-8A+5B}{B\left(r^{2}(B-A)+3\right)^{3}},$
$\displaystyle\chi_{9}$
$\displaystyle=r^{4}\left(3A^{2}-10AB-B^{2}\right)+e^{Ar^{2}}\left(r^{6}(A-B)\left(3A^{2}-8AB+B^{2}\right)+2Ar^{8}(A-B)^{2}(A+B)-2r^{4}(A-B)\right.$
$\displaystyle\left.(17A-B)+r^{2}(27A-7B)+8\right)-8A^{2}r^{8}(A-B)^{2}+2Ar^{6}(A-B)(9A-B)+2r^{2}(4B-11A)+e^{3Ar^{2}}$
$\displaystyle\times\left(r^{2}(A+B)+2\right)+e^{2Ar^{2}}\left(2Ar^{6}(A-B)(A+B)+2Ar^{4}(A-3B)-2r^{2}(3A+B)-7\right)-3,$
$\displaystyle\chi_{10}$ $\displaystyle=\frac{\alpha
e^{2Ar^{2}}}{r^{2}(A-B)+e^{Ar^{2}}-1}+\frac{A-2B}{r^{2}}+B(A-B).$
$\displaystyle\Psi_{1}$
$\displaystyle=\frac{2r^{2}\left(r^{2}\left(2A^{2}-3AB+B^{2}\right)-8A+5B\right)}{B\left(r^{2}(B-A)+3\right)^{3}}+\frac{1}{r\left(B(Br+1)-A\left(Br^{2}+2\right)\right)^{4}},$
$\displaystyle\Psi_{2}$
$\displaystyle=3A^{4}r^{3}\left(Br^{2}+2\right)^{2}-A^{3}r\left(Br^{2}+2\right)\left(Br^{2}(Br(r+6)+25)-2\right)+A^{2}Br(Br(r(Br(r(B(2r+3)+4)$
$\displaystyle+35)+61)+14)-8)-AB^{2}(r(Br(r(B(r(Br+5)+8)+3)+45)-3)+4)+B^{3}\left(Br\left(Br^{2}+r+11\right)+2\right),$
$\displaystyle\Psi_{3}$
$\displaystyle=-\frac{2r^{2}}{r^{2}(A-B)+e^{Ar^{2}}-1}+\frac{3}{B\left(r^{2}(B-A)+3\right)}+\frac{1}{Br^{2}(B-A)-2A+B},$
$\displaystyle\Psi_{4}$
$\displaystyle=10A^{4}r^{6}-29A^{3}Br^{6}-72A^{3}r^{4}+30A^{2}B^{2}r^{6}-2Br^{2}e^{2Ar^{2}}\left(r^{2}(B-A)+3\right)\left(r^{2}\left(2A^{2}-3AB+B^{2}\right)-8A+5\right.$
$\displaystyle\times\left.B\right)+143A^{2}Br^{4}+114A^{2}r^{2}-13AB^{3}r^{6}-88AB^{2}r^{4}-117ABr^{2}+72A+2B^{4}r^{6}+17B^{3}r^{4}+30B^{2}r^{2}-45B,$
$\displaystyle\Psi_{5}$
$\displaystyle=\frac{\frac{1}{Br^{2}(A-B)+2A-B}-\frac{1}{B\left(r^{2}(B-A)+3\right)}}{2r^{2}}+\frac{1}{r^{2}(A-B)+e^{Ar^{2}}-1},$
$\displaystyle\Psi_{6}$
$\displaystyle=\frac{r^{2}\left(2A^{2}-3AB+B^{2}\right)-8A+5B}{B\left(r^{2}(B-A)+3\right)^{3}}+\frac{1}{\left(r^{2}(A-B)+e^{Ar^{2}}-1\right)^{4}},$
$\displaystyle\Psi_{7}$
$\displaystyle=r^{4}\left(3A^{2}-10AB-B^{2}\right)-8A^{2}r^{8}(A-B)^{2}+2Ar^{6}(A-B)(9A-B)+2r^{2}(4B-11A)+e^{3Ar^{2}}$
$\displaystyle\times\left(r^{2}(A+B)+2\right)+e^{2Ar^{2}}\left(2Ar^{6}(A-B)(A+B)+2Ar^{4}(A-3B)-2r^{2}(3A+B)-7\right)-3,$
$\displaystyle\Psi_{8}$
$\displaystyle=r^{6}(A-B)\left(3A^{2}-8AB+B^{2}\right)+2Ar^{8}(A-B)^{2}(A+B)-2r^{4}(A-B)(17A-B)+r^{2}(27A-7B)+8$

## References

## References

  * (1) L. Amendola, L. Giani, G. Laverda, Ricci-inverse gravity: A novel alternative gravity, its flaws, and how to cure them, Phys. Lett. B 10. 811 (2020), 135923.
  * (2) F. Weber, Strange Quark Matter and Compact Stars, Pro. Part. Nucl. Phys. 54 (2005), 193.
  * (3) G. E. Delgado, J. E. Delgado, Compact stars, Mod. Phys. Lett. A 33 (2018), 1850081.
  * (4) G. Mustafa et al., Realistic stellar anisotropic model satisfying Karmarker condition in $f(R,T)$ gravity, Eur. Phys. J. C 80(2020) 26.
  * (5) G. Mustafa, M. Farasat Shamir, and Xia Tiecheng, Physically viable solutions of anisotropic spheres in $f(R,G)$ gravity satisfying the Karmarkar condition, Phys. Rev. D 101, (2020) 104013.
  * (6) G. Mustafa et al., Realistic solutions of fluid spheres in $f(G,T)$ Gravity under Karmarkar condition, Ann. Phys. 413 (2020) 168059.
  * (7) S. Waheed et al., Physically Acceptable Embedded Class-I Compact Stars in Modified Gravity with Karmarkar Condition, Symmetry 12 (2020) 962.
  * (8) G. Mustafa et al., Anisotropic fluid spheres admitting Karmarkar condition in $f(G,T)$ gravity, Chin. J. Phys. 67 (2020) 576.
  * (9) M. Farasat Shamir et al., Charged anisotropic Bardeen spheres admitting conformal motion, Ann. Phys. 418 (2020) 168184.
  * (10) M. Farasat Shamir et al., Charged anisotropic Finch-Skea-Bardeen spheres, Nucl. Phys. B 967 (2021) 115418.
  * (11) S. Waheed et al., Anisotropic spheres via embedding approach in $R+\beta R^{2}$ gravity with matter coupling, Phys. Dark Uni. 31 (2021) 100747
  * (12) M. F. Shamir, A. Malik, Behavior of Anisotropic Compact Stars in $f(R,\phi)$ Gravity, Commun. Theor. Phys. 71 (2019), 001.
  * (13) A. Malik, A. Ashraf, U. Naqvi, Z. Zhang, Anisotropic spheres via embedding approach in $f(R)$ gravity, Int. J. Geom. Methods Mod. Phys. 19 (2022) 2250073.
  * (14) M. F. Shamir, A. Malik, Bardeen Compact Stars in Modified $f(R)$ Gravity, Chin. J. Phys. 69 (2021), 312-321.
  * (15) L. M. Krauss, M. S. Turner, The cosmological constant is back, Gen. Rel. Grav. 27 (1995), 1137.
  * (16) J. P. Ostriker, P. J. Steinhardt, The observational case for a low-density universe with a non-zero cosmological constant, Nature 377 (1995), 600.
  * (17) A. R. Liddle, D. H. Lyth, P. T. Viana, M. White, Cold dark matter models with a cosmological constant, MNRAS 282 (1996), 281.
  * (18) B. Ratra, P. J. E. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D 37 (1988), 3406.
  * (19) C. Wetterich, Cosmology and the fate of dilatation symmetry, Nucl. Phys. B 302 (1988), 668.
  * (20) P. G.Ferreira, M. Joyce, Structure formation with a self-tuning scalar field, Phys. Rev. Lett. 79 (1997) 4740.
  * (21) S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 (1989) 1.
  * (22) S. M. Carroll, The cosmological constant, Liv. Rev. Rel. 4 (2001) 1.
  * (23) B. Ratra, P. J. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D 37 (1988) 3406.
  * (24) R. R. Caldwell, R. Dave, P. J. Steinhardt, Cosmological imprint of an energy component with general equation of State, Phys. Rev. Lett. 80 (1998), 1582.
  * (25) C. Armendariz-Picon, T. Damour, V. Mukhanov, K-inflation, Phys. Lett. B 458 (1999), 209.
  * (26) C. Armendariz-Picon, V. Mukhanov, P. J. Steinhardt, Essentials of $k$-essence, Phys. Rev. D 63 (2001), 103510.
  * (27) V. Sahni, A. A. Starobinsky, The case for a positive cosmological $\bigwedge$-term, Int. J. Mod. Phys. D 9 (2000), 373.
  * (28) I. Zlatev, L. Wang, P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant, Phys. Rev. Lett. 82 (1999), 896.
  * (29) P. Brax, J. Martin, The Robustness of Quintessence, Phys. Rev. D 61 (2000), 103502.
  * (30) T. Barreiro, E. J. Copeland, and N. J. Nunes, Quintessence arising from exponential potentials, Phys. Rev. D 61 (2000), 127301.
  * (31) A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980), 99.
  * (32) J.D. Barrow, A.C. Ottewill, The stability of general relativistic cosmological theory, J. Phys. A 16 (1983), 2757.
  * (33) A.A. Starobinsky, H.J. Schmidt, On a general vacuum solution of fourth-order gravity, Class. Quant. Grav. 4 (1987), 695.
  * (34) J.D. Barrow, S. Cotsakis, Inflation and the conformal structure of higher-order Gravity Theories, Phys. Lett. B 214 (1988), 515.
  * (35) V. Muller, H.J. Schmidt, A.A. Starobinsky, Power-law inflation as an attractor solution for inhomogeneous cosmological models, Class. Quant. Grav. 7 (1990), 1163.
  * (36) A.S. Koshelev, K. Sravan Kumar, A.A. Starobinsky, $R^{2}$ inflation to probe non-perturbative quantum gravity, JHEP 03 (2018), 071.
  * (37) S.S. Mishra, V. Sahni, A.V. Toporensky, Initial conditions for inflation in an FRW universe, Phys. Rev. D 98 (2018), 083538.
  * (38) A. De Felice, S. Tsujikawa, $f(R)$ theories, Living Rev. Rel. 13 (2010), 3.
  * (39) S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Phys. Rept. 692 (2017), 1.
  * (40) S.A. Appleby, R.A. Battye, A.A. Starobinsky, Curing singularities in cosmological evolution of $f(R)$ gravity, J. Cosmol. Astropart. Phys. 06 (2010), 005.
  * (41) S. Deser, R. P. Woodard, Nonlocal cosmology, Phys. Rev. Lett. 99 (2007), 111301.
  * (42) M. E. Soussa, R. P. Woodard, Letter: The force of gravity from a lagrangian containing inverse powers of the Ricci scalar, Gen. Rel. and Grav. 36 (2004), 855.
  * (43) L. Amendola, Y. Dirian, H. Nersisyan, S. Park, Observational constraints in nonlocal gravity: The deser-woodard case, JCAP 03 (2019), 045.
  * (44) S. I. Nojiri, S. D. Odintsov, Modified non-local- $f(R)$ gravity as the key for the inflation and dark energy Phys. Lett. B 659 (2008), 821.
  * (45) A. O. Barvinsky, Dark energy and dark matter from nonlocal ghost-free gravity theory, Phys. Lett. B 710 (2012), 12.
  * (46) M. Maggiore, M. Michele, Nonlocal gravity and Dark Energy, Phys. Rev. D 90 (2014), 023005.
  * (47) L. Herrera and W. Barreto, General relativistic polytropes for anisotropic matter: The general formalism and applications, Phys. Rev. D 88, 084022 (2013)
  * (48) J. Estevez-Delgado et al.,Strange stars in the presence of quintessence, Eur. Phys. J. Plus 135, 204 (2020).
  * (49) P. Saha and U. Debnath, Study of anisotropic compact stars with quintessence field and modified chaplygin gas in $f(T)$ gravity, Eur. Phys. J. C 79, 919 (2019).
  * (50) G. Abbas and M. R. Shahzad, A New Model of Quintessence Compact Stars in Rastall Theory of Gravity, Eur. Phys. J. A 54, 211 (2018).
  * (51) M. Kalam et al., Anisotropic quintessence stars, Astrophys Space Sci 349, 865,71 (2014)
  * (52) G. Abbas et al., Anisotropic strange quintessence stars in $f(R)$ gravity, Astrophys Space Sci (2015) 358:26
  * (53) A. Ditta et al., Anisotropic stellar structures in the $f(T)$ theory of gravity with quintessence via embedding approach, Chinese Physics C Vol. 45, 4 (2021) 045102
  * (54) G. Mustafa et al., Self-consistent embedded anisotropic quintessence compact stars in Rastall gravity via linear equation of state, Phys. Scr. 96 (2021) 105008
  * (55) G. Mustafa et al., Study on anisotropic star in extended teleparallel gravity with minimal matter coupling, Chinese Journal of Physics 77 (2022) 1742-754
  * (56) G. Abbas, S. Qaisar, M. A Meraj, Anisotropic strange quintessence stars in $f(T)$ gravity, Space Sci. 357 (2015).
  * (57) P. Bhar, Higher dimensional charged gravastar admitting conformal motion, Space Sci. 354 (2014), 457.
  * (58) R. Goswami et al, Collapsing spherical stars in $f(R)$ gravity Phys. Rev. D 90 (2014), 084011.
  * (59) A. Cooney, S. D. Deo, D. Psaltis, Neutron stars in $f(R)$ gravity with perturbative constraints, Phys. Rev. D 82 (2010), 064033.
  * (60) A. Ganguly et al, Neutron stars in the starobinsky model, Phys. Rev. D 89 (2014), 064019.
  * (61) D. Momeni, R. Myrzakulov, Tolman-Oppenheimer-Volkoff equations in modified gauss-bonnet gravity, Methods Mod. Phys. 12 (2015), 1550014.
  * (62) A.V. Astashenok et al, Further stable neutron star models from $f(R)$ gravity, Cosmol.J. Astropart. Phys. 12 (2013), 040.
  * (63) M.F. Shamir and M. Ahmad, Emerging anisotropic compact stars in f(G, T) gravity, Eur. Phys. J. C 77, 674 (2017).
  * (64) L. Herrera, Cracking of self-gravitating compact objects, Phys. Lett. A 165 (1992), 206.
  * (65) C. G. Bohmer, T. Harko, Minimum mass radius ratio for charged gravitational objects, Gen. Rel. Grav. 39 (2007), 757.


# Comparison results for solutions to the anisotropic Laplacian with Robin
boundary conditions

Rossano Sannipoli  Dipartimento di Matematica e Applicazioni “R. Caccioppoli”,
Università degli studi di Napoli Federico II
Via Cintia, Complesso Universitario Monte S. Angelo, 80126 Napoli, Italy.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider PDE’s problems involving the anisotropic Laplacian
operator, with Robin boundary conditions. By means of Talenti techniques,
widely used in the last decades, we prove a comparison result between the
solutions of the above-mentioned problems and the solutions of the symmetrized
ones. As a consequence of these results, a Bossel-Daners type inequality can
be shown in dimension 2.
MSC 2010: 35B51 - 35G20 - 35G30 - 35J60 - 35P15
Keywords: Anisotropic Laplacian - Robin boundary conditions - Talenti
comparison result - Bossel-Daners inequality

##  1\. Introduction

Let $\Omega\subset\mathbb{R}^{n}$ be an open bounded set, with Lipschitz
boundary. Let us consider the following anisotropic problem with Robin
boundary conditions

$\begin{cases}-\mathrm{div}(H(\nabla u)H_{\xi}(\nabla u))=f&\mbox{in}\
\Omega\vspace{0.2cm}\\\ H(\nabla u)H_{\xi}(\nabla u)\cdot\nu+\beta
H(\nu)u=0&\mbox{on}\ \partial\Omega\vspace{0.2cm},\end{cases}$ (1.1)

where $f\geq 0$ (not identically zero) belongs to $L^{2}(\Omega)$, $H$ is a
sufficiently smooth norm in $\mathbb{R}^{n}$, $\nu$ is the outer unit normal
to $\partial\Omega$ and $\beta>0$ is a positive real parameter.
A weak solution to problem (1.1) is a function $u\in H^{1}(\Omega)$ that
satisfies

$\int_{\Omega}H(\nabla u)H_{\xi}(\nabla
u)\cdot\nabla\varphi\,dx+\beta\int_{\partial\Omega}H(\nu)u\varphi\,d\mathcal{H}^{n-1}=\int_{\Omega}f\varphi\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall\varphi\in
H^{1}(\Omega).$ (1.2)

The aim of the paper is to estabilish a comparison result with the solution to
the following symmetrized problem

$\begin{cases}-\mathrm{div}(H(\nabla v)H_{\xi}(\nabla v))=f^{\star}&\mbox{in}\
\Omega^{\star}\vspace{0.2cm}\\\ H(\nabla v)H_{\xi}(\nabla v)\cdot\nu+\beta
H(\nu)v=0&\mbox{on}\ \partial\Omega^{\star}\vspace{0.2cm},\end{cases}$ (1.3)

where $f^{\star}$ is the convex symmetrization of $f$ and $\Omega^{\star}$ is
a set homothetic to the Wulff Shape $\mathcal{W}$ such that
$|\Omega^{\star}|=|\Omega|$ (for the exact definitions, see section 2).

It is well known that, when $H$ is the euclidean norm in $\mathbb{R}^{n}$,
then the convex symmetrization coincides with the Schwarz symmetrization (to
know more about Schwarz symmetrization see [K]). Moreover if the Robin
boundary condition is replaced by the Dirichlet boundary condition, problem
(1.1) becomes

$\begin{cases}-\Delta u=f&\mbox{in}\,\,\,\Omega\\\
u=0&\mbox{on}\,\,\,\partial\Omega,\end{cases}$ (1.4)

with $f\in L^{2}(\Omega)$ (non-negative and not identically zero) and $\Omega$
an open subset of $\mathbb{R}^{n}$. This problem has been widely studied, with
Talenti being the pioneer in this direction. Indeed the author (see [T])
proved, via rearrangements arguments, that the Schwarz symmetrization of the
solution to problem (1.4) is pointwise bounded by the solution to the
following symmetrized problem

$\begin{cases}-\Delta v=f^{\sharp}&\mbox{in}\,\,\,\Omega^{\sharp}\\\
v=0&\mbox{on}\,\,\,\partial\Omega^{\sharp},\end{cases}$ (1.5)

with $f^{\sharp}$ being the Schwarz decreasing rearrangement of $f$ and
$\Omega^{\sharp}$ the ball centered at the origin having the same measure as
$\Omega$. Talenti, with his techniques, gave birth to a series of
generalizations and results that still now take his name. For instance see
[AFLT, ALT, T1, T2] for generalizations to other kind of operators.
When we have Robin boundary conditions with positive parameter, problem (1.4)
becomes

$\begin{cases}-\Delta u=f&\mbox{in}\,\,\,\Omega\\\ \displaystyle\frac{\partial
u}{\partial\nu}+\beta u=0&\mbox{on}\,\,\,\partial\Omega.\end{cases}$ (1.6)

To our knowledge, in literature, there are few comparison results á la Talenti
for this kind of problem. A result of this type has been proved only recently
by [ANT], where they have highlighted the importance of the dependence on the
dimension of the space. The authors, infact, managed to compare the Lorentz
norm (see [L]) of the solution to problem (1.6) with that of the solution to
the symmetrized problem

$\begin{cases}-\Delta v=f^{\sharp}&\mbox{in}\,\,\,\Omega^{\sharp}\\\
\displaystyle\frac{\partial v}{\partial\nu}+\beta
u=0&\mbox{on}\,\,\,\partial\Omega^{\sharp},\end{cases}$ (1.7)

where the exponents of these norms depend on the dimension of the space. In
particular they proved, for $n\geq 2$, that

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

and

$\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\sharp})}\,\,\,\,\,\,\,\text{for
all}\,\,\,0<p\leq\frac{n}{3n-4},$

with $u$ solution to (1.6) and $v$ to (1.7). Moreover, when $f\equiv 1$ in
$\Omega$ and n=2, they showed that

$u^{\sharp}(x)\leq v(x)\;\;\;\;\;x\in\Omega^{\sharp},$ (1.8)

and, for $n\geq 3$, that

$\begin{split}&\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\sharp})}\\\
&\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\sharp})},\end{split}$

for all $0<p\leq\frac{n}{n-2}$.
With a different approach [BG] proved that

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

In [S] one can find the computation of the shape derivative of the
$L^{\infty}$ and $L^{p}$ norms of the solution to the problem (1.6) and the
proof of the stationarity of the ball, when fixing the volume.
The main theorems of this paper show that the proves proposed by [ANT] can be
adapted even when considering the anisotropic Laplacian operator. Let us now
state these results.

######  Theorem 1.1.

Let be $n\geq 2$. If $u$ and $v$ are the solutions to problems (1.1) and (1.3)
respectivley, then

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\star})}\;\;\;\;\;\;\;\text{for
all}\;\;\;\;\;0<p\leq\frac{n}{2n-2}$ (1.9)

and

$\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\star})}\;\;\;\;\;\;\;\text{for
all}\;\;\;\;\;0<p\leq\frac{n}{3n-4}.$ (1.10)

######  Theorem 1.2.

Let $n=2$, $f\equiv 1$ in $\Omega$. If $u$ and $v$ are the solutions to
problems (1.1) and (1.3) respectivley. Then

$u^{\star}(x)\leq v(x)\,\,\,\,\,x\in\Omega^{\star},$ (1.11)

where $u^{\star}$ is the convex symmetrization of $u$.

######  Theorem 1.3.

Let $n\geq 3$ and $f\equiv 1$. If $u$ and $v$ are the solutions to problems
(1.1) and (1.3) respectivley, then

$\|u\|_{L^{p,1}(\Omega)}\leq\|v\|_{L^{p,1}(\Omega^{\star})}$ (1.12)

and

$\|u\|_{L^{2p,2}(\Omega)}\leq\|v\|_{L^{2p,2}(\Omega^{\star})},$ (1.13)

for all $0<p\leq\frac{n}{n-2}$.

The organization of the paper is the following. In Section 2 the reader can
find notations and preliminaries concerning the anisotropic norm, its
properties and the convex symmetrization of a function. Section 3 contains
results about existence, uniqueness and other properties of the solutions to
problem (1.1) and (1.3). In section 4 the main theorems are proved. Section 5
is dedicated to the application of these results to prove, in dimension 2, the
Bossel-Daners inequality for the anisotropic operator. Last section will
include some counterexamples and problems that still are open.

##  2\. Notations and preliminaries

###  2.1. Anisotropy

What follows can be found in [AFLT]. Let
$H:\mathbb{R}^{n}\longrightarrow[0,+\infty]$, $n\geq 2$, be a
$C^{2}(\mathbb{R}^{n}\setminus\\{0\\})$ convex functions that satisfies the
following homogeneity property

$H(t\xi)=|t|H(\xi)\,\,\,\,\,\,\forall\xi\in\mathbb{R}^{n}\,,\,\,\forall
t\in\mathbb{R},$ (2.1)

and such that

$\gamma|\xi|\leq H(\xi)\leq\delta|\xi|,$ (2.2)

for some positive constants $\gamma\leq\delta$. Because of (2.1), we can
assume that the set

$K=\\{\xi\in\mathbb{R}^{n}:H(\xi)\leq 1\\}$

is such that $|K|$ is equal to the measure $\omega_{n}$ of the unit sphere in
$\mathbb{R}^{n}$. We can define the support function of $K$ as

$H^{\circ}(x)=\sup_{\xi\in K}\left<x,\xi\right>,$ (2.3)

where $\left<\cdot,\cdot\right>$ denotes the scalar product in
$\mathbb{R}^{n}$. $H^{\circ}:\mathbb{R}^{n}\longrightarrow[0,+\infty]$ is a
convex, homogeneous function. Moreover $H$ and $H^{\circ}$ are polar to each
other, in the sense that

$H^{\circ}(x)=\sup_{\xi\neq 0}\frac{\left<x,\xi\right>}{H(\xi)}$

and

$H(x)=\sup_{\xi\neq 0}\frac{\left<x,\xi\right>}{H^{\circ}(\xi)}.$

$H^{\circ}$ is the support function of the set

$K^{\circ}=\\{x\in\mathbb{R}^{n}:H^{\circ}(x)\leq 1\\}.$

The set $\mathcal{W}=\\{x\in\mathbb{R}^{n}:H^{\circ}(x)<1\\}$ is the so-called
Wulff shape centered at the origin. We set $k_{n}=|\mathcal{W}|$. More
generally we will denote by $\mathcal{W}_{R}(x_{0})$ the Wulff shape centered
in $x_{0}\in\mathbb{R}^{n}$ with measure $k_{n}R^{n}$ the set
$R\mathcal{W}+x_{0}$, and $\mathcal{W}_{R}(0)=\mathcal{W}_{R}$.
$H$ and $H^{\circ}$ satisfy the following properties:

$H_{\xi}(\xi)\cdot\xi=H(\xi),\;\;\;\;H^{\circ}_{\xi}(\xi)\cdot\xi=H^{\circ}(\xi),$
(2.4)
$H(H^{\circ}_{\xi}(\xi))=H^{\circ}(H_{\xi}(\xi))=1\,\,\,\,\,\forall\xi\in\mathbb{R}^{n}\setminus\\{0\\},$
(2.5)
$H^{\circ}(\xi)H_{\xi}(H^{\circ}_{\xi}(\xi))=H(\xi)H^{\circ}_{\xi}(H_{\xi}(\xi))=\xi\,\,\,\forall\xi\in\mathbb{R}^{n}\setminus\\{0\\}.$
(2.6)

If $\Omega\subset\mathbb{R}^{n}$ is an open bounded set with Lipschitz
boundary and $E$ is an open subset of $\mathbb{R}^{n}$, we can give a
generalized definition of perimeter of $E$ with respect to the anisotropic
norm as follows

$P_{H}(E,\Omega)=\int_{\partial^{*}E\cap\Omega}H(\nu^{E})\,d\mathcal{H}^{n-1},$

where $\partial^{*}E$ is the reduced boundary of $E$ and $\nu^{E}$ is its
outer normal. Clearly, if $E$ is open, bounded and Lipschitz, then the outer
unit normal exists almost everywhere and

$P_{H}(E,\mathbb{R}^{n}):=P_{H}(E)=\int_{\partial
E}H(\nu^{E})\,d\mathcal{H}^{n-1}.$ (2.7)

By (2.2) we have that

$\gamma P(E)\leq P_{H}(E)\leq\delta P(E).$

In [AB, AFLT] it is shown that if $u\in W^{1,1}(\Omega)$, then for a.e. $t>0$

$-\frac{d}{dt}\int_{\\{u>t\\}}H(\nabla
u)\,dx=P_{H}(\\{u>t\\},\Omega)=\int_{\partial^{*}\\{u>t\\}\cap\Omega}\frac{H(\nabla
u)}{|\nabla u|}\,d\mathcal{H}^{n-1}.$ (2.8)

In particular an isoperimetric inequality for the anisotropic perimeter holds
(for instance see [AFLT, Bu, DP, FM])

$P_{H}(E)\geq nk^{\frac{1}{n}}_{n}|E|^{1-\frac{1}{n}}.$ (2.9)

###  2.2. Convex symmetrization

Let $f:\Omega\longrightarrow[0,+\infty]$ be a misurable function. The
decreasing rearrangement $f^{*}$ of $f$ is defined as follows

$f^{*}(s)=\inf\\{t\geq
0:|\\{x\in\Omega:|f(x)|>t\\}|<s\\}\,\,\,\,\,\,s\in[0,|\Omega|],$

which is the generalized inverse function of the distribution function of $f$.
We define the convex symmetrization $f^{\star}$ of $f$ as

$f^{\star}(x)=f^{*}(k_{n}H^{\circ}(x)^{n})\,\,\,\,\,\,\,\,x\in\Omega^{\star}.$

In particular it is well known that the functions $f$, $f^{*}$ and $f^{\star}$
are equimisurable, i.e.

$|\\{f>t\\}|=|\\{f^{*}>t\\}|=|\\{f^{\star}>t\\}|\,\,\,\,\,t\geq 0.$

As a consequence, if $f\in L^{p}(\Omega)$, $p\geq 1$, then $f^{*}\in
L^{p}([0,|\Omega|])$, $f^{\star}\in L^{p}(\Omega^{\star})$ and

$\|f\|_{L^{p}(\Omega)}\|=\|f^{*}\|_{L^{p}([0,|\Omega|])}=\|f^{\star}\|_{L^{p}(\Omega^{\star})}.$

Moreover the Hardy-Littlewood inequality holds (see [K])

$\int_{\Omega}|f(x)g(x)|\,dx\leq\int_{0}^{|\Omega|}f^{*}(s)g^{*}(s)\,ds.$
(2.10)

So, if we consider $g$ as the characteristic function of the set
$\\{x\in\Omega:u(x)>t\\}$, for some misurable function
$u:\Omega\rightarrow\mathbb{R}$ and $t\geq 0$, then we get

$\int_{\\{u>t\\}}f(x)\,dx\leq\int_{0}^{\mu(t)}f^{*}(s)\,ds,$ (2.11)

where, again, $\mu(t)$ is the distribution function of $u$.

##  3\. Existence, uniqueness and properties of the solution

In this section we want to prove that the solution to problem (1.1) exists, it
is unique and non-negative.

###  3.1. Existence

Let us consider the following energy functional

$E[w]=\frac{1}{2}\int_{\Omega}H^{2}(\nabla
w)\,dx+\frac{\beta}{2}\int_{\partial\Omega}H(\nu)w^{2}\,d\mathcal{H}^{n-1}-\int_{\Omega}fw\,dx,\,\,\,\,\,\,w\in
H^{1}(\Omega).$ (3.1)

We notice that the Euler-Lagrange equations of this functional is (1.1), hence
if $E[\cdot]$ has minima, there will exist a solution to the considered
problem.
Let us proceed with the classical Calculus of Variation method to prove the
existence of a minimum.
1) Lower bound. Let us prove that the functional (3.1) is lower bounded. From
(2.2) and the generalized Young’s inequality we have

$\begin{split}E[u]&\geq\frac{\gamma^{2}}{2}\|\nabla
u\|^{2}_{L^{2}(\Omega)}+\frac{\beta\gamma}{2}\|u\|^{2}_{L^{2}(\partial\Omega)}-\frac{\epsilon}{2}\|u\|^{2}_{L^{2}(\Omega)}-\frac{1}{2\epsilon}\|f\|^{2}_{L^{2}(\Omega)}\\\
&\geq C_{1}\bigg{(}\|\nabla
u\|^{2}_{L^{2}(\Omega)}+\|u\|^{2}_{L^{2}(\partial\Omega)}\bigg{)}-\frac{\epsilon}{2}\|u\|^{2}_{L^{2}(\Omega)}-\frac{1}{2\epsilon}\|f\|^{2}_{L^{2}(\Omega)}\\\
&\geq\bigg{(}C_{2}-\frac{\epsilon}{2}\bigg{)}\|u\|^{2}_{L^{2}(\Omega)}-\frac{1}{2\epsilon}\|f\|^{2}_{L^{2}(\Omega)}.\end{split}$

In the last inequality we have used a Poincaré inequality with trace term (see
for instance [BGT]). Here $C_{2}=C_{2}(\beta,\gamma,\Omega)$. If we choose
$\epsilon$ small enough, then

$E[u]\geq-\frac{1}{2\epsilon}\|f\|^{2}_{L^{2}(\Omega)}>-\infty.$

We have proved in this way that the functional is bounded from below. Let be

$m:=\inf_{w\in H^{1}(\Omega)}E[w].$ (3.2)

and let $\\{u_{k}\\}\subset H^{1}(\Omega)$ be a minimizing sequence, i.e.

$\lim_{k\to\infty}E[u_{k}]=m.$

We can suppose that $E[u_{k}]\leq m+1$ for all $k\in\mathbb{N}$.
2) Compactness and lower semicontinuity. Using again the generalized Young’s
inequality and the Poincaré inequality with trace term, we have

$\begin{split}m+1&\geq\frac{\gamma^{2}}{4}\|\nabla
u_{k}\|^{2}_{L^{2}(\Omega)}-\frac{\epsilon}{2}\|u_{k}\|^{2}_{L^{2}(\Omega)}\\\
&+\frac{\gamma^{2}}{4}\|\nabla
u_{k}\|^{2}_{L^{2}(\Omega)}+\frac{\beta\gamma}{2}\|u_{k}\|^{2}_{L^{2}(\partial\Omega)}-\frac{1}{2\epsilon}\|f\|^{2}_{L^{2}(\Omega)}\\\
&\geq\frac{\gamma^{2}}{4}\|\nabla
u_{k}\|^{2}_{L^{2}(\Omega)}-\frac{\epsilon}{2}\|u_{k}\|^{2}_{L^{2}(\Omega)}+C_{3}\|u_{k}\|^{2}_{L^{2}(\Omega)}-\frac{1}{2\epsilon}\|f\|^{2}_{L^{2}(\Omega)},\end{split}$

where $C_{3}=C_{3}(\beta,\gamma,\Omega)$. Choosing $\epsilon$ small enough and
calling $C_{4}=\min(\frac{\gamma^{2}}{4},C_{3}-\frac{\epsilon}{2})$ then

$\|u_{k}\|_{H^{1}(\Omega)}\leq\frac{m+1}{C_{4}}+\frac{1}{2\epsilon
C_{4}}\|f\|^{2}_{L^{2}(\Omega)}<\infty.$

Hence $\\{u_{k}\\}$ is bounded in $H^{1}(\Omega)$, so it exists a subsequence
$\\{u_{k_{j}}\\}\subset\\{u_{k}\\}$ that converges weakly in $H^{1}(\Omega)$
and strongly in $L^{2}(\Omega)$ to a function $u\in H^{1}(\Omega)$. To
simplify the notation let us continue to call the subsequence as
$\\{u_{k}\\}$.
By the strict convexity of the functions $t\longrightarrow t^{2}$ and $H^{2}$,
we have the following inequalities

$u_{k}^{2}\geq u^{2}+2u(u_{k}-u)$ (3.3)

and

$H^{2}(\nabla u_{k})\geq H^{2}(\nabla u)+2H(\nabla u)H_{\xi}(\nabla
u)\cdot(\nabla u_{k}-\nabla u).$ (3.4)

By (3.3) and (3.4), we have

$\begin{split}E[u_{k}]&\geq\frac{1}{2}\int_{\Omega}H^{2}(\nabla
u)\,dx+\frac{\beta}{2}\int_{\partial\Omega}u^{2}\,d\mathcal{H}^{n-1}-\int_{\Omega}fu_{k}\,dx\\\
&+\int_{\Omega}H(\nabla u)H_{\xi}(\nabla u)\cdot(\nabla u_{k}-\nabla
u)\,dx+\int_{\partial\Omega}u(u_{k}-u)\,d\mathcal{H}^{n-1}.\end{split}$

By the weak convergence of $u_{k}$ in $H^{1}(\Omega)$ and being
$H^{1}(\Omega)$ compactly embedded in $L^{2}(\partial\Omega)$, passing to the
limit for $k\to\infty$ we have that

$E[u]\leq m.$

Eventually $E[u]=m$ and $u$ is a minimum.

###  3.2. Uniqueness and non-negativeness

Let us prove now that the minimum of (3.1) is unique. If $u,v\in
H^{1}(\Omega)$, by the strict convexity of the funcion $H^{2}$ we know that if
$t\in[0,1]$, then

$H^{2}(t\nabla u+(1-t)\nabla v)\leq tH^{2}(\nabla u)+(1-t)H^{2}(\nabla v).$
(3.5)

The equality occurs if and only if $t=0$ or $t=1$. Analogously

$[tu+(1-t)v]^{2}\leq tu^{2}+(1-t)v^{2},\,\,\,\,t\in[0,1].$ (3.6)

Let $u\in H^{1}(\Omega)$ be a minimizer of (3.1) and let us suppose that there
exists another minimizer $v\in H^{1}(\Omega)$, such that $u\neq v$. Hence
$E[u]=E[v]=m$. Let us denote by $w=u+v$ and choose $t=\frac{1}{2}$, then by
(3.5) and (3.6)

$E[w]<\frac{E[u]}{2}+\frac{E[v]}{2}=m.$

This fact contraddicts the minimality of $u$ and so the minimum must be
unique.

Let us now show the non-negativeness of the solution. Let $u$ be the unique
minimum of (3.1), namely $E[u]=m$. If we consider $|u|$, by (2.1), we get

$H^{2}(\nabla|u|)=H^{2}\bigg{(}\frac{u}{|u|}\nabla u\bigg{)}=H^{2}(\nabla u).$

Hence

$\begin{split}E[|u|]&=\frac{1}{2}\int_{\Omega}H^{2}(\nabla|u|)\,dx+\frac{\beta}{2}\int_{\partial\Omega}H(\nu)|u|^{2}\,d\mathcal{H}^{n-1}-\int_{\Omega}f|u|\,dx\\\
&=m+\int_{\Omega}f(u-|u|)\,dx=m+2\int_{\\{u\leq 0\\}}fu\,\,\leq m.\end{split}$

By the uniqueness of the minimizer it must be $u=|u|$ in $\Omega$. Eventually
$u\geq 0$ in $\Omega$.

###  3.3. The anisotropic radial case

Let us consider problem (1.3). Because of uniqueness of the solution and the
symmetry of the problem we look for a solution of the type
$v(x)=v(H^{\circ}(x))$. This function solves the following ODE

$\begin{cases}-\frac{1}{r^{n-1}}(r^{n-1}v^{\prime}(r))^{\prime}=f^{*}(k_{n}r^{n})&r\in(0,R)\\\
v^{\prime}(0)=0\\\ v^{\prime}(R)+\beta v(R)=0,\end{cases}$ (3.7)

where $r=H^{\circ}(x)$ and $R$ is such that $\Omega^{\star}=\mathcal{W}_{R}$
and $|\Omega|=|\Omega^{\star}|$.
Integrating the first equation in (3.7), calling $\tilde{t}=k_{n}t^{n}$, we
get

$v^{\prime}(r)=-\frac{1}{r^{n-1}}\int_{0}^{r}t^{n-1}f^{*}(k_{n}t^{n})\,dt+C_{1}=-\frac{1}{nk_{n}r^{n-1}}\int_{0}^{k_{n}r^{n}}f^{*}(\tilde{t})\,d\tilde{t}+C_{1}.$

Since $v^{\prime}(0)=0$ then $C_{1}=0$. By denoting $\tilde{s}=k_{n}s^{n}$,
another integration gives

$\begin{split}v(r)=-\int_{0}^{r}\frac{1}{nk_{n}s^{n-1}}\int_{0}^{k_{n}s^{n}}&f^{*}(\tilde{t})\,d\tilde{t}\,ds+C_{2}=\\\
&-\int_{0}^{k_{n}r^{n}}\frac{1}{n^{2}k_{n}^{\frac{2}{n}}\tilde{s}^{2-\frac{2}{n}}}\int_{0}^{\tilde{s}}f^{*}(\tilde{t})\,d\tilde{t}\,d\tilde{s}+C_{2}.\end{split}$

From $v^{\prime}(R)+\beta v(R)=0$ we compute $C_{2}$, hence

$\begin{split}v(r)=-\int_{0}^{k_{n}r^{n}}&\frac{1}{n^{2}k_{n}^{\frac{2}{n}}\tilde{s}^{2-\frac{2}{n}}}\int_{0}^{\tilde{s}}f^{*}(\tilde{t})\,d\tilde{t}\,d\tilde{s}+\\\
&\int_{0}^{k_{n}R^{n}}\frac{1}{n^{2}k_{n}^{\frac{2}{n}}\tilde{s}^{2-\frac{2}{n}}}\int_{0}^{\tilde{s}}f^{*}(\tilde{t})\,d\tilde{t}\,d\tilde{s}+\frac{1}{\beta
nk_{n}R^{n-1}}\int_{0}^{k_{n}R^{n}}f^{*}(\tilde{t})\,d\tilde{t}.\end{split}$

Therefore

$v(r)=\int_{k_{n}r^{n}}^{k_{n}R^{n}}\frac{1}{n^{2}k_{n}^{\frac{2}{n}}\tilde{s}^{2-\frac{2}{n}}}\int_{0}^{\tilde{s}}f^{*}(\tilde{t})\,d\tilde{t}\,d\tilde{s}+\frac{1}{\beta
nk_{n}R^{n-1}}\int_{0}^{k_{n}R^{n}}f^{*}(\tilde{t})\,d\tilde{t}.$ (3.8)

In this way we have shown that the unique solution to problem (1.3) is
radially symmetric with respect to anisotropic norm and its value on the
boundary is given by

$v(R)=\frac{1}{\beta
nk_{n}R^{n-1}}\int_{0}^{k_{n}R^{n}}f^{*}(\tilde{t})\,d\tilde{t}\geq 0.$

######  Remark 3.1.

We stress that if $f\equiv 1$ in $\Omega$, then by (3.8) the solution to
problem (1.3) can be written explicitly as follows

$v(x)=v(H^{\circ}(x))=\frac{R}{\beta n}+\frac{1}{2n}(R^{2}-H^{\circ}(x)^{2}).$

The solution is a paraboloid with respect to the anisotropic norm. Morover if
$H$ is the euclidean norm in $\mathbb{R}^{n}$, we step back to the classical
torsion problem (or Saint Venant problem) with robin boundary conditions,
whose radial solution is a concave paraboloid.

###  3.4. level sets and distribution functions

If $u$ is a solution to problem (1.1), we will denote by

$U_{t}=\\{x\in\Omega:u(x)>t\\}$

for a non-negative real number $t\geq 0$. It is clear that if $t\leq
u_{\min}$, then $U_{t}=\Omega$ and that if $t>u_{\max}$, then
$U_{t}=\emptyset$. With $u_{\min}$ and $u_{\max}$ we have denoted the minimum
and the maximum of $u$ in $\Omega$. We will denote by

$\partial U^{\text{int}}_{t}=\Omega\cap\partial
U_{t},\,\,\,\,\,\,\,\,\,\,\,\,\partial
U^{\text{ext}}_{t}=\partial\Omega\cap\partial U_{t}$ (3.9)

the internal and external boundary of $U_{t}$ with respect to $\Omega$, and by

$\mu(t)=|U_{t}|$ (3.10)

the distribution function of $u$.
If $v$ is solution to problem (1.3), for $t\geq 0$, we will indicate with

$V_{t}=\\{x\in\Omega^{\star}:v(x)>t\\},\,\,\,\,\,\,\phi(t)=|V_{t}|$

the superlevel sets and the distribution function of $v$, respectively.
Furthermore, for $0\leq t\leq v_{\min}$, $V_{t}=\Omega$, while for
$v_{\min}<t<v_{\max}$, the superlevel sets $V_{t}$ are Wulff shapes homothetic
to $\Omega^{\star}$ and strictly contained in it. Again, $v_{\min}$ and
$v_{\max}$ are the minimum and the maximum of $v$ in $\Omega^{\star}$.

##  4\. Main results

Before proving the main results let us state Gronwall Lemma and prove two
others lemmata that will have a central importance for what will follow.

######  Lemma 4.1.

(Gronwall) Let $\xi(t)$ be a continuously differentiable function satysfing
for some non-negative constant C, the following differential inequality

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

for all $\tau\geq\tau_{0}>0$. Then we have

$\xi(\tau)\leq\tau\frac{\xi(\tau_{0})+C}{\tau_{0}}+C$

and

$\xi^{\prime}(\tau)\leq\frac{\xi(\tau_{0})+C}{\tau_{0}},$

for all $\tau\geq\tau_{0}$.

######  Lemma 4.2.

Let $u$ and $v$ be the solutions to problems (1.1) and (1.3) respectively.
Then for a.e. $t>0$ we have

$n^{2}k_{n}^{\frac{2}{n}}\phi(t)^{\frac{2n-2}{n}}=\left(-\phi^{\prime}(t)+\frac{1}{\beta}\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\right)\int_{0}^{\mu(t)}f^{*}(s)\,ds$
(4.1)

and

$n^{2}k_{n}^{\frac{2}{n}}\mu(t)^{\frac{2n-2}{n}}\leq\left(-\mu^{\prime}(t)+\frac{1}{\beta}\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\right)\int_{0}^{\mu(t)}f^{*}(s)\,ds.$
(4.2)

###### Proof.

Let $t,h>0$ and let us consider the following test function in $H^{1}(\Omega)$

$\varphi_{h}(x)=\begin{cases}0&u\leq t\\\ u-t&t<u\leq t+h\\\ h&u\geq
t+h.\end{cases}$

Substituting this in (1.2) we have

$\begin{split}&\int_{U_{t}\setminus U_{t+h}}H(\nabla u)H_{\xi}(\nabla
u)\cdot\nabla u\,dx+\beta\int_{\partial U^{\text{ext}}_{t}\setminus\partial
U^{\text{ext}}_{t+h}}H(\nu)(u-t)u\,d\mathcal{H}^{n-1}\\\ &+\beta
h\int_{\partial
U^{\text{ext}}_{t+h}}H(\nu)u\,d\mathcal{H}^{n-1}=\int_{U_{t}\setminus
U_{t+h}}f(u-t)dx+h\int_{U_{t+h}}f\,dx.\end{split}$

Applying (2.4) in the first integral, dividing by $h$ and applying Coarea
Formula, we have for a.e. $t>0$

$\begin{split}&\frac{1}{h}\int_{t}^{t+h}\int_{\partial
U^{\text{int}}_{\tau}}\frac{H^{2}(\nabla u)}{|\nabla
u|}\,d\mathcal{H}^{n-1}\,d\tau+\frac{\beta}{h}\int_{\partial
U^{\text{ext}}_{t}\setminus\partial
U^{\text{ext}}_{t+h}}H(\nu)(u-t)u\,d\mathcal{H}^{n-1}\\\ &+\beta\int_{\partial
U^{\text{ext}}_{t+h}}H(\nu)u\,d\mathcal{H}^{n-1}=\frac{1}{h}\int_{U_{t}\setminus
U_{t+h}}f(u-t)dx+\int_{U_{t+h}}f\,dx.\end{split}$

Passing to the limit for $h\rightarrow 0^{+}$ we have

$\int_{\partial U^{\text{int}}_{t}}\frac{H^{2}(\nabla u)}{|\nabla
u|}\,d\mathcal{H}^{n-1}+\beta\int_{\partial
U^{\text{ext}}_{t}}H(\nu)u\,d\mathcal{H}^{n-1}=\int_{U_{t}}f\,dx.$

Let us set

$g(x)=\begin{cases}H(\nabla u)\;\;\;\;\;\;\;\;\partial U^{\text{int}}_{t}\\\
\beta u\;\;\;\;\;\;\;\;\;\;\;\;\;\;\partial U^{\text{ext}}_{t}.\end{cases}$
(4.3)

We want to end the proof using the anisotropic version of the isoperimetric
inequality and to this aim it is necessary to write properly the anisotropic
peremeter of $U_{t}$. Because of the regularity of $\partial\Omega$ we know
that $\partial U^{\text{ext}}_{t}$ is sufficiently regular and a normal vector
can be defined. Being $u\in H^{1}(\Omega)$ and $f\in L^{2}(\Omega)$, then
$\partial U^{\text{int}}_{t}$ can’t have any good regularity property. By
(2.8) we can write for a.e. $t>0$

$P_{H}(U_{t})=\int_{\partial U^{\text{int}}_{t}}\frac{H(\nabla u)}{|\nabla
u|}\,d\mathcal{H}^{n-1}+\int_{\partial
U^{\text{ext}}_{t}}H(\nu)\,d\mathcal{H}^{n-1},$

where $\nu$ is the outer unit normal to $\Omega$. If we set

$h(x)=\begin{cases}\displaystyle\frac{H(\nabla u)}{|\nabla u|}&\partial
U^{\text{int}}_{t}\\\ H(\nu)&\partial U^{\text{ext}}_{t},\end{cases}$ (4.4)

then

$P_{H}(U_{t})=\int_{\partial U_{t}}h(x)\,d\mathcal{H}^{n-1}.$

Furthermore we note that

$\int_{\partial U_{t}}h(x)g(x)\,d\mathcal{H}^{n-1}=\int_{U_{t}}f\,dx.$ (4.5)

Therefore by Schwarz inequality and (4.5), we have for a.e. $t>0$

$\begin{split}P^{2}_{H}(U_{t})&=\left(\int_{\partial
U_{t}}h(x)\,d\mathcal{H}^{n-1}\right)^{2}=\left(\int_{\partial
U_{t}}\sqrt{h(x)g(x)}\sqrt{\frac{h(x)}{g(x)}}\,d\mathcal{H}^{n-1}\right)^{2}\\\
&\leq\int_{\partial U_{t}}h(x)g(x)\,d\mathcal{H}^{n-1}\left(\int_{\partial
U^{\text{int}}_{t}}\frac{1}{|\nabla
u|}\,d\mathcal{H}^{n-1}+\frac{1}{\beta}\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\right)\\\
&\leq\left(-\mu^{\prime}(t)+\frac{1}{\beta}\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\right)\int_{0}^{\mu(t)}f^{*}(s)\,ds.\end{split}$

Hence, by (2.9)

$n^{2}k_{n}^{\frac{2}{n}}\mu(t)^{\frac{2n-2}{n}}\leq\left(-\mu^{\prime}(t)+\frac{1}{\beta}\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\right)\int_{0}^{\mu(t)}f^{*}(s)\,ds.$

If we do the same computations, replacing $v$ with $u$, all the previous
inequalities become equalities and we have (4.1).
In particular, if $f\equiv 1$ in $\Omega$, we have

$n^{2}k_{n}^{\frac{2}{n}}\mu(t)^{\frac{n-2}{n}}\leq-\mu^{\prime}(t)+\frac{1}{\beta}\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1},$ (4.6)

and

$n^{2}k_{n}^{\frac{2}{n}}\phi(t)^{\frac{n-2}{n}}=-\phi^{\prime}(t)+\frac{1}{\beta}\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}.$ (4.7)

∎

######  Remark 4.3.

Let us notice that $u_{\min}\leq v_{\min}$. Indeed, being the level sets of
$v$ homothetic to $\Omega^{\star}$, then, using (1.1), (1.3) and the
isoperimetric inequality

$\begin{split}v_{\min}P_{H}(\Omega^{\star})&=\int_{\partial\Omega^{\star}}H(\nu)v(x)\,d\mathcal{H}^{n-1}=-\frac{1}{\beta}\int_{\partial\Omega^{\star}}H(\nabla
v)H_{\xi}(\nabla v)\cdot\nu\,d\mathcal{H}^{n-1}\\\
&=\frac{|\Omega^{\star}|}{\beta}=\frac{|\Omega|}{\beta}=-\frac{1}{\beta}\int_{\partial\Omega}H(\nabla
u)H_{\xi}(\nabla u)\cdot\nu\,d\mathcal{H}^{n-1}\\\
&=\int_{\partial\Omega}H(\nu)u(x)\,d\mathcal{H}^{n-1}\geq
u_{\min}P_{H}(\Omega)\geq u_{\min}P_{H}(\Omega^{\star}).\end{split}$ (4.8)

As a consequence for all $0<t<v_{\min}$ we have that

$\mu(t)\leq\phi(t)=|\Omega|.$ (4.9)

######  Lemma 4.4.

For all $t\geq v_{\min}$ we have

$\int_{0}^{t}\tau\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\star}}\frac{H(\nu)}{v(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau=\frac{1}{2\beta}\int_{0}^{|\Omega|}f^{*}(s)\,ds$
(4.10)

and

$\int_{0}^{t}\tau\left(\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau\leq\frac{1}{2\beta}\int_{0}^{|\Omega|}f^{*}(s)\,ds$
(4.11)

###### Proof.

By Fubini’s Theorem and using (1.1), we have that

$\begin{split}\int_{0}^{\infty}\tau\left(\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau&=\int_{\partial\Omega}\left(\int_{0}^{u(x)}\frac{H(\nu)}{u(x)}\tau\,d\tau\right)\,d\mathcal{H}^{n-1}\\\
&=\int_{\partial\Omega}\frac{H(\nu)u(x)}{2}\,d\mathcal{H}^{n-1}=\frac{1}{2\beta}\int_{0}^{|\Omega|}f^{*}(s)\,ds.\end{split}$

Analogously,

$\int_{0}^{\infty}\tau\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\star}}\frac{H(\nu)}{v(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau=\frac{1}{2\beta}\int_{0}^{|\Omega|}f^{*}(s)\,ds.$

By monotonicity of the integral we have that for $t\geq 0$

$\int_{0}^{t}\tau\left(\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau\leq\int_{0}^{\infty}\tau\left(\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau$

and if $t\geq v_{\min}$, then $\partial
V_{t}\cap\partial\Omega^{\star}=\emptyset$. Hence

$\int_{0}^{t}\tau\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\star}}\frac{H(\nu)}{v(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau=\int_{0}^{\infty}\tau\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\star}}\frac{H(\nu)}{v(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau,$

and we have (4.11), (4.10).
If $f\equiv 1$ in $\Omega$, then

$\int_{0}^{t}\tau\left(\int_{\partial
U^{\text{ext}}_{t}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau\leq\frac{|\Omega|}{2\beta}$
(4.12)

and

$\int_{0}^{t}\tau\left(\int_{\partial
V_{\tau}\cap\partial\Omega^{\star}}\frac{H(\nu)}{v(x)}\,d\mathcal{H}^{n-1}\right)\,d\tau=\frac{|\Omega|}{2\beta}.$
(4.13)

∎

###### Proof of Theorem 1.1.

Let $0<p\leq\frac{n}{2n-2}$ and let us denote
$K_{n}=n^{2}k_{n}^{\frac{2}{n}}$. Let us multiply (4.2) by $t\mu(t)^{\eta}$,
where $\eta=\frac{1}{p}-\frac{2n-2}{n}\geq 0$, and integrate from $0$ to
$\tau\geq v_{\min}$

$\begin{split}\int_{0}^{\tau}K_{n}t\mu(t)^{\frac{1}{p}}\,dt&\leq\int_{0}^{\tau}-\mu^{\prime}(t)t\mu(t)^{\eta}\bigg{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\bigg{)}\,dt\,\,\,+\\\
&\frac{1}{\beta}\int_{0}^{\tau}t\mu(t)^{\eta}\bigg{(}\int_{\partial
U_{t}^{\text{ext}}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\int_{0}^{\mu(t)}f^{*}(s)\,ds\bigg{)}\,dt\\\
&\leq\int_{0}^{\tau}-t\mu(t)^{\eta}\bigg{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\bigg{)}\,d\mu(t)+\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2},\\\
\end{split}$

where we applied Lemma 4.4 and the fact that $\mu(t)$ is a monotone non
increasing function.
By setting $\displaystyle
F(l)=\int_{0}^{l}w^{\eta}\int_{0}^{w}f^{*}(s)\,ds\,dw$ and integrating by
parts the first and last members in this chain of inequalities we have

$\begin{split}\tau
F(\mu(\tau))+\tau\int_{0}^{\tau}K_{n}\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{\tau}&F(\mu(t))\,dt+\int_{0}^{\tau}\int_{0}^{t}K_{n}\mu(t)^{\frac{1}{p}}\,dr\,dt\\\
&+\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}\end{split}$

By applying Lemma 4.1, with

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

$\displaystyle
C=\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}$
and $\tau_{0}=v_{\min}$, we have that

$\begin{split}F(\mu(\tau))&+\int_{0}^{\tau}K_{n}\mu(t)^{\frac{1}{p}}\,dt\leq\frac{1}{v_{\min}}\bigg{(}\int_{0}^{v_{\min}}F(\mu(t))\,dt\\\
&+\int_{0}^{v_{\min}}\int_{0}^{t}K_{n}\mu(r)^{\frac{1}{p}}\,dr\,dt+\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}\bigg{)}.\end{split}$
(4.14)

Analogously

$\begin{split}F(\phi(\tau))&+\int_{0}^{\tau}K_{n}\phi(t)^{\frac{1}{p}}\,dt=\frac{1}{v_{\min}}\bigg{(}\int_{0}^{v_{\min}}F(\phi(t))\,dt\\\
&+\int_{0}^{v_{\min}}\int_{0}^{t}K_{n}\phi(r)^{\frac{1}{p}}\,dr\,dt+\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}\bigg{)}.\end{split}$
(4.15)

By (4.8) and (4.9), then we can compare directly the righthand sides of (4.14)
and (4.15). So

$F(\mu(\tau))+\int_{0}^{\tau}K_{n}\mu(t)^{\frac{1}{p}}\,dt\leq
F(\phi(\tau))+\int_{0}^{\tau}K_{n}\phi(t)^{\frac{1}{p}}\,dt$

For $\tau\rightarrow+\infty$ we have

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

which is (1.9).
Now we want to prove (1.10). In order to obtain this result let us pass to the
limit as $\tau\rightarrow\infty$ the following inequality:

$\int_{0}^{\tau}K_{n}t\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{\tau}-t\mu(t)^{\eta}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}\,d\mu(t)+\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}.$

After an integration by parts we get

$\int_{0}^{\infty}K_{n}t\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{\infty}F(\mu(t))\,dt+\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}.$

On the other hand

$\int_{0}^{\infty}K_{n}t\phi(t)^{\frac{1}{p}}\,dt=\int_{0}^{\infty}F(\phi(t))\,dt+\frac{|\Omega|^{\eta}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}.$

So we need just to show that

$\int_{0}^{\infty}F(\mu(t))\,dt\leq\int_{0}^{\infty}F(\phi(t))\,dt.$ (4.16)

To this aim we multiply (4.2) by $tF(\mu(t))\mu(t)^{-\frac{2n-2}{2}}$. Since
$F(l)l^{-\frac{2n-2}{2}}$ is a non decreasing function in $l$, when
$0<p\leq\frac{n}{3n-4}$, we can integrate from $0$ to $\tau\geq v_{\min}$, to
obtain

$\begin{split}\int_{0}^{\tau}&K_{n}tF(\mu(t))\,dt\leq\int_{0}^{\tau}-t\mu(t)^{-\frac{2n-2}{n}}F(\mu(t))\bigg{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\bigg{)}\,d\mu(t)\\\
&+\frac{1}{\beta}\int_{0}^{\tau}tF(\mu(t))\mu(t)^{-\frac{2n-2}{n}}\int_{\partial
U_{t}^{\text{ext}}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\bigg{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\bigg{)}\,dt\\\
&\leq\int_{0}^{\tau}-t\mu(t)^{-\frac{2n-2}{n}}F(\mu(t))\bigg{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\bigg{)}\,d\mu(t)\\\
&+\frac{1}{\beta}F(|\Omega|)|\Omega|^{-\frac{2n-2}{n}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}\int_{0}^{\tau}t\int_{\partial
U_{t}^{\text{ext}}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\,dt\\\
&\leq\int_{0}^{\tau}-t\mu(t)^{-\frac{2n-2}{n}}F(\mu(t))\bigg{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\bigg{)}\,d\mu(t)\\\
&+F(|\Omega|)\frac{|\Omega|^{\frac{2n-2}{n}}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2},\end{split}$

where, again, we applied 4.4. Now, if we call $\displaystyle
C=F(|\Omega|)\frac{|\Omega|^{\frac{2n-2}{n}}}{2\beta^{2}}\bigg{(}\int_{0}^{|\Omega|}f^{*}(s)\,ds\bigg{)}^{2}$
and set $\displaystyle
J(l)=\int_{0}^{l}w^{-\frac{2n-2}{n}}F(w)\bigg{(}\int_{0}^{w}f^{*}(s)\,ds\bigg{)}\,dw$,
integrating by parts the first and last member of the previous chain of
inequalities, we have

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

Setting

$\xi(\tau)=\int_{0}^{\tau}\int_{0}^{r}K_{n}F(\mu(z))\,dz\,dr+\int_{0}^{\tau}J(\mu(t))\,dt,$

and applying 4.1 with $\tau_{0}=v_{\min}$ we deduce that

$\begin{split}\int_{0}^{\tau}K_{n}F(\mu(t))\,dt+J(\mu(\tau))\leq\frac{1}{v_{\min}}\bigg{(}\int_{0}^{v_{\min}}&\int_{0}^{r}K_{n}F(\mu(z))\,dz\,dr\\\
&+\int_{0}^{v_{\min}}J(\mu(t))\,dt+C\bigg{)}.\end{split}$

This inequality holds as an equality when we have $\phi$ in place of $\mu$, so
as before

$\int_{0}^{\tau}K_{n}F(\mu(t))\,dt+J(\mu(t))\leq\int_{0}^{\tau}K_{n}F(\phi(t))\,dt+J(\phi(t)).$

For $\tau\rightarrow\infty$ we have (1.10), which concludes the proof. ∎

###### Proof of Theorem 1.2.

Multiplying by $t\geq 0$ inequality (4.6) and integrating from $0$ to
$\tau\geq v_{\min}$, we have that

$2k_{2}\tau^{2}\leq\int_{0}^{\tau}-\mu^{\prime}(t)t\,dt+\frac{|\Omega|}{2\beta^{2}}.$

Here we applied Lemma 4.4. Analogously for (4.7)

$2k_{2}\tau^{2}=\int_{0}^{\tau}-\phi^{\prime}(t)t\,dt+\frac{|\Omega|}{2\beta^{2}}.$

Then

$\int_{0}^{\tau}t(-d\mu(t))\geq\int_{0}^{\tau}t(-d\phi(t)),$

for every $\tau\geq v_{\min}$. Integrating by parts

$\mu(\tau)\leq\phi(\tau)\,\,\,\,\,\,\tau\geq v_{\min}.$ (4.17)

Since $u_{\min}\leq v_{\min}$, inequality (4.17) holds for $t\geq 0$ and the
claim is proved. ∎

###### Proof of Theorem 1.3.

Let $0<p\leq\frac{n}{n-2}$. Let us multiply (4.6) by $t\mu(t)^{\eta}$, where
$\eta=\frac{1}{p}-\frac{n-2}{n}\geq 0$, and integrate from $0$ to $\tau\geq
v_{\min}$

$\begin{split}\int_{0}^{\tau}K_{n}t\mu(t)^{\frac{1}{p}}\,dt&\leq\int_{0}^{\tau}-\mu^{\prime}(t)t\mu(t)^{\eta}\,dt+\frac{1}{\beta}\int_{0}^{\tau}t\mu(t)^{\eta}\int_{\partial
U_{t}^{\text{ext}}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\\\
&\leq\int_{0}^{\tau}-\mu^{\prime}(t)t\mu(t)^{\eta}\,dt+\frac{|\Omega|^{\eta}}{\beta}\int_{0}^{\tau}t\int_{\partial
U_{t}^{\text{ext}}}\frac{H(\nu)}{u(x)}\,d\mathcal{H}^{n-1}\\\
&\leq\int_{0}^{\tau}-\mu^{\prime}(t)t\mu(t)^{\eta}\,dt+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}\leq\int_{0}^{\tau}-t\mu(t)^{\eta}\,d\mu(t)+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}\end{split}$

where, again, $K_{n}=n^{2}k_{n}^{\frac{2}{n}}$, in the third inequality we
applied Lemma 4.4, and in the last the fact that $\mu(t)$ is a monotone non
increasing function.
By setting $\displaystyle G(l)=\int_{0}^{l}w^{\eta}=\frac{l^{\eta+1}}{\eta+1}$
and integrating by parts the first and last members in this chain of
inequalities we have

$\tau
G(\mu(\tau))+\tau\int_{0}^{\tau}K_{n}t\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{\tau}G(\mu(t))\,dt+\int_{0}^{\tau}\int_{0}^{t}K_{n}\mu(t)^{\frac{1}{p}}\,dr\,dt+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}$

By applying Lemma 4.1, with

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

$\displaystyle C=\frac{|\Omega|^{\eta+1}}{2\beta^{2}}$ and
$\tau_{0}=v_{\min}$, we have that

$\begin{split}G(\mu(\tau))&+\int_{0}^{\tau}K_{n}\mu(t)^{\frac{1}{p}}\,dt\leq\frac{1}{v_{\min}}\bigg{(}\int_{0}^{v_{\min}}G(\mu(t))\,dt\\\
&+\int_{0}^{v_{\min}}\int_{0}^{t}K_{n}\mu(r)^{\frac{1}{p}}\,dr\,dt+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}\bigg{)}.\end{split}$
(4.18)

Analogously

$\begin{split}G(\phi(\tau))&+\int_{0}^{\tau}K_{n}\phi(t)^{\frac{1}{p}}\,dt=\frac{1}{v_{\min}}\bigg{(}\int_{0}^{v_{\min}}G(\phi(t))\,dt\\\
&+\int_{0}^{v_{\min}}\int_{0}^{t}K_{n}\phi(r)^{\frac{1}{p}}\,dr\,dt+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}\bigg{)}.\end{split}$
(4.19)

By (4.8) and (4.9), we compare directly the righthand sides of (4.18) and
(4.19). So

$G(\mu(\tau))+\int_{0}^{\tau}K_{n}\mu(t)^{\frac{1}{p}}\,dt\leq
G(\phi(\tau))+\int_{0}^{\tau}K_{n}\phi(t)^{\frac{1}{p}}\,dt$

For $\tau\rightarrow+\infty$ we have

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

which is (1.12). Now we want to prove (1.13). In order to obtain this result
let us pass to the limit as $\tau\rightarrow\infty$ the following inequality:

$\int_{0}^{\tau}K_{n}t\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{\tau}-t\mu(t)^{\eta}\,d\mu(t)+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}.$

After an integration by parts we get

$\int_{0}^{\infty}K_{n}t\mu(t)^{\frac{1}{p}}\,dt\leq\int_{0}^{\infty}G(\mu(t))\,dt+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}.$

On the other hand

$\int_{0}^{\infty}K_{n}t\phi(t)^{\frac{1}{p}}\,dt=\int_{0}^{\infty}G(\phi(t))\,dt+\frac{|\Omega|^{\eta+1}}{2\beta^{2}}.$

So we need just to show that

$\int_{0}^{\infty}G(\mu(t))\,dt\leq\int_{0}^{\infty}G(\phi(t))\,dt.$ (4.20)

To this aim we multiply (4.6) by $tG(\mu(t))\mu(t)^{-\frac{n-2}{n}}$. Since
$G(l)l^{-\frac{n-2}{n}}$ is a non decreasing function in $l$, we can integrate
from $0$ to $\tau\geq v_{\min}$, to obtain

$\begin{split}\int_{0}^{\tau}&K_{n}tG(\mu(t))\,dt\leq\int_{0}^{\tau}-t\mu(t)^{-\frac{n-2}{n}}G(\mu(t))\,d\mu(t)\\\
&+\frac{1}{\beta}\int_{0}^{\tau}tG(\mu(t))\mu(t)^{-\frac{n-2}{n}}\int_{\partial
U_{t}^{\text{ext}}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\,dt\\\
&\leq\int_{0}^{\tau}-t\mu(t)^{-\frac{n-2}{n}}G(\mu(t))\,d\mu(t)+\frac{1}{\beta}G(|\Omega|)|\Omega|^{-\frac{n-2}{n}}\int_{0}^{\tau}t\int_{\partial
U_{t}^{\text{ext}}}\frac{H(\nu)}{u}\,d\mathcal{H}^{n-1}\,dt\\\
&\leq\int_{0}^{\tau}-t\mu(t)^{-\frac{n-2}{n}}G(\mu(t))\,d\mu(t)+G(|\Omega|)\frac{|\Omega|^{\frac{2}{n}}}{2\beta^{2}},\end{split}$

where, again, we applied 4.4. Now, if we call $\displaystyle
C=G(|\Omega|)\frac{|\Omega|^{\frac{2}{n}}}{2\beta^{2}}$ and set $\displaystyle
J(l)=\int_{0}^{l}w^{-\frac{n-2}{n}}G(w)\,dw$, integrating by parts the first
and last member of the previous chain of inequalities, we have

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

Setting

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

and applying 4.1 with $\tau_{0}=v_{\min}$ we deduce that

$\begin{split}\int_{0}^{\tau}K_{n}G(\mu(t))\,dt+J(\mu(\tau))\leq\frac{1}{v_{\min}}\bigg{(}\int_{0}^{v_{\min}}&\int_{0}^{r}K_{n}G(\mu(z))\,dz\,dr\\\
&+\int_{0}^{v_{\min}}J(\mu(t))\,dt,+C\bigg{)}.\end{split}$

This inequality holds as an equality when we have $\phi$ in place of $\mu$, so
as before

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

For $\tau\rightarrow\infty$ we have (4.20), which concludes the proof. ∎

##  5\. Application to PDE’s: Bossel-Daners inequality

Let $\Omega$ be a bounded and smooth open set in $\mathbb{R}^{n}$. Let us
denote by $\nu$ the outer unit normal to $\partial\Omega$ and let $\beta>0$ be
a positive real number. It is well known that for the following Laplacian
eigenvalue problem with Robin boundary conditions

$\begin{cases}-\Delta u=\lambda(\Omega)u&\text{in}\,\,\Omega\\\
\displaystyle\frac{\partial u}{\partial\nu}+\beta
u=0&\text{on}\,\,\partial\Omega,\end{cases}$

a Faber-Krahn type inequality for the first eigenvalue holds. It is famous
under the name of Bossel-Daners inequality and it can be read as follows

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

where $\Omega^{\sharp}$ is the ball centered at the origin with the same
measure as $\Omega$. Equality holds if and only if $\Omega$ is a ball.
Let us consider now the anisotropic case. If $f=\lambda(\Omega)u$, then (1.1)
can be written in this way

$\begin{cases}-\mathrm{div}(H(\nabla u)H_{\xi}(\nabla
u))=\lambda(\Omega)u&\mbox{in}\ \Omega\vspace{0.2cm}\\\ H(\nabla
u)H_{\xi}(\nabla u)\cdot\nu+\beta H(\nu)u=0&\mbox{on}\
\partial\Omega\vspace{0.2cm}.\end{cases}$ (5.1)

The variational characterization for the first eiganvalue is

$\lambda_{1}(\Omega)=\min_{w\in H^{1}(\Omega)\setminus\\{0\\}}J[u],$ (5.2)

where

$\displaystyle J[u]=\frac{\displaystyle\int_{\Omega}H^{2}(\nabla
u)\,dx+\beta\int_{\partial\Omega}u^{2}H(\nu)\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\Omega}u^{2}\,dx}.$
(5.3)

In [DG] the authors proved a Bossel-Daners type inequality for the anisotropic
$p$-Laplacian problem. Indeed, they proved that

$\lambda_{1,\beta}(\Omega)\geq\lambda_{1,\beta}(\Omega^{\star}),$ (5.4)

where $\Omega^{\star}$ is the set homothetic to the Wulff Shape having the
same measure as $\Omega$. In particular the equality case holds if and only if
$\Omega$ is a set homothetic to the Wulff Shape.
In this section we want to give an alternative proof of (5.4) in the planar
case, using the results found in the previous section.

######  Corollary 5.1.

Let $n=2$ and $u$ be the first eigenfunction associated to
$\lambda_{1,\beta}(\Omega)$, solution to problem (5.1). If $z$ is the solution
to the symmetrized problem

$\begin{cases}-\mathrm{div}(H(\nabla z)H_{\xi}(\nabla
z))=\lambda_{1,\beta}(\Omega)u^{\star}&\mbox{in}\
\Omega^{\star}\vspace{0.2cm}\\\ H(\nabla z)H_{\xi}(\nabla z)\cdot\nu+\beta
H(\nu)z=0&\mbox{on}\ \partial\Omega^{\star}\vspace{0.2cm}.\end{cases}$ (5.5)

then we have that

$\lambda_{1,\beta}(\Omega)\geq\lambda_{1,\beta}(\Omega^{\star}).$ (5.6)

Equality occurs if and only if $\Omega$ is homothetic to the Wulff Shape with
same measure as $\Omega$.

###### Proof.

By theorem 1.2, we know that

$\int_{\Omega}u^{2}\,dx=\int_{\Omega^{\star}}(u^{\star})^{2}\,dx\leq\int_{\Omega^{\star}}z^{2}\,dx.$

So, by Cauchy-Schwarz inequality we have

$\int_{\Omega^{\star}}u^{\star}z\,dx\leq\int_{\Omega^{\star}}z^{2}\,dx,$

and this ends the corollary, indeed

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

∎

##  6\. Conclusions and open problems

As in the euclidean case, we have proved that these comparison results depend
on the dimension of the space. In particular if we are in the hypohtesis of
theorem 1.1, when $n=2$, then

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

and

$\|u\|_{L^{2}(\Omega)}\leq\|v\|_{L^{2}(\Omega^{\star})}.$

Therefore a question arises spontaneously. Is it true that

$\|u\|_{L^{p}(\Omega)}\leq\|v\|_{L^{p}(\Omega^{\star})}$ (6.1)

for all values of $p$? In dimension $2$ the answer is negative for large
values of $p$. Next example will show that (6.1) is untrue when $p=\infty$ and
$n=2$.

######  Example 1.

Let $\Omega$ be the union of two disjoint bidimensional Wulff shapes
$\mathcal{W}$ and $\mathcal{W}_{r}$, with radii $1$ and $r$ respectively. If
we choose $\beta=\frac{1}{2}$ and $f$ such that it is constantly $1$ in
$\mathcal{W}$ and constantly zero in $\mathcal{W}_{r}$, then the solutions to
problem (1.1) and (1.3) can be explicitly computed. In particular it is
possible to prove that there exitsts a positive constant $c$ such that

$\|u\|_{L^{\infty}(\Omega)}-\|v\|_{L^{\infty}(\Omega^{\star})}=cr^{2}+o(r^{2}).$

Now someone could ask if (6.1) can be true when $n\geq 3$. Next counterexample
will show its untruthfulness when $n=3$ and $p=2$.

######  Example 2.

If we consider $\Omega$, $\beta$ and $f$ as in the example 1 in the
corresponding three-dimensional case, then, in the hypothesis of theorem 1.1,
the solutions to problem (1.1) and (1.3) can be explicitly computed. It is
possible to prove that there exists a positive constant $d$ such that

$\|u\|_{L^{2}(\Omega)}-\|v\|_{L^{2}(\Omega^{\star})}=dr^{3}+o(r^{3}).$

A problem that is still open is the following

######  Open Problem 1.

In the hypothesis of theorem 1.1, (6.1) is true for $p=1$ and $n\geq 3$?

If we now consider the theorem 1.2, we have proved that when $n=2$ and
$f\equiv 1$ in $\Omega$, then

$u^{\star}(x)\leq v(x)\,\,\,\,x\in\Omega^{\star}.$

In doing so, another question arises:

######  Open Problem 2.

In the hypothesis of theorem 1.2, (6) is true even when $n\geq 3$?

## References

  * [AB] M. Amar, G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. Henri Poincaré, Anal. Nonlinéaire 11 (1994), no.1, 91-133.
  * [AFLT] A. Alvino, V. Ferone, P. L. Lions, G. Trombetti, Convex symmetrization and applicatons, Annales de l’I.H.P. 14 (1997), 275-293-65.
  * [ALT] A. Alvino, P. L. Lions, G. Trombetti, Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Annales de l’I.H.P. 7 (1990), 37-65.
  * [ANT] A. Alvino, C. Nitsch, C. Trombetti, A Talenti comparison result for solutions to elliptic problems with robin boundary conditions, ArXiv (2019): 1-15.
  * [Bu] H. Busemann, The isoperimetric problem for Minkowski area, Amer. J. Math. 71 (1949) 743-762.
  * [BG] D. Bucur, A. Giacomini,  Faber-Krahn inequalities for the Robin-Laplacian: A free discontinuity approach, Arch. Rational Mech. Anal. 218, no. 2, (2015), 757-824.
  * [BGT] D. Bucur, A. Giacomini, P. Trebeschi Best constant in Poincaré inequalities with traces: a free discontinuity approach, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, 1959-1986.
  * [DG] F. Della Pietra, N. Gavitone, Faber-Krahn inequality for anisotropic eigenvalue problems with Robin boundary conditions, Potential Analysis, 41, n. 4 (2014), 1147-1166.
  * [DP] B. Dacorogna, C.E. Pfister, Wulff theorem and best constant in Sobolev inequality, J. Math. Pures Appl. (9) 71 (1992) no. 2, 97-118.
  * [FM] I. Fonseca, S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991) no. 1-2, 125-136.
  * [K] S. Kesavan, Symmetrization & applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd., (2006).
  * [L] G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51, (1950), 37-55. no. 27, Princeton University Press, (1951).
  * [S] R. Sannipoli, Some properties of the torsion function with Robin boundary conditions, ArXiv (2021): 1-18.
  * [T] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3, no. 4, (1976), 697-718.
  * [T1] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura e Appl. 120 (1979), 159-184.
  * [T2] G. Talenti. On the first eigenvalue of the clamped plate, Ann. Mat. Pura e Appl. 129 (1981), 265-280.