# Quarkonium Masses in a hot QCD Medium Using Conformable Fractional of the
Nikiforov-Uvarov Method

M. Abu-Shady  Department of Applied Mathematics, Faculty of Science, Menoufia
University, Shbien El-Kom, Egypt

###### Abstract

By using conformable fractional of the Nikiforov-Uvarov (CF-NU) method, the
radial Schrödinger equation is analytically solved. The energy eigenvalues and
corresponding functions are obtained, in which the dependent temperature
potential is employed. The effect of fraction-order parameter is studied on
heavy-quarkonium masses such as charmonium and bottomonium in a hot QCD medium
in the 3D and the higher dimensional space. A comparison is studied with
recent works. We conclude that the fractional-order plays an important role in
a hot QCD medium in the 3D and higher-dimensional space.

Keywords: Fractional derivative, Schrödinger equation, Nikiforov-Uvarov
method, finite temperature, quarkonium spectra

††preprint:

LABEL:FirstPage1 LABEL:LastPage#1102

##  I Introduction

The color screening of the static chromo-electric fields is one of the most
remarkable features of the quark-gluon plasma formation. The Debye screening
in quantum chromodynamic theory plasma has been studied as a probe of
deconfinement in a hot medium which shows a reduction in the interaction
between heavy quarks and antiquarks due to color screening leading to a
suppression in $J/\Psi\ $yields [1, 2]. Thereby the quarkonium in the hot
medium is a good tool to examine the confined/deconfined state of matter. The
dissociation of heavy quarkonium states in a hot QCD medium is studied by
investigating the medium modifications to a heavy-quark potential [3].

The development of the Schrödinger equation (SE) plays a major role at finite
temperature. Matsui and Satz $\left[4\right]$ have investigated the formation
of a hot quark-gluon plasma by calculating the $J/\Psi$ radius of charmonium.
In Ref. $\left[5\right]$, the SE is solved by using the Funke-Hecke theorem to
describe the electron and proton media. Wong $\left[6\right]$ has studied the
binding energies and wave functions of heavy quarkonia in quark-gluon plasma
by using a temperature dependent potential inferred from lattice gauge
calculations. Thus, the study shows that the model with the new Q-Q̄ potential
gives dissociation temperatures that agree with the spectral function
analyses. The SE in a screened Coulomb heavy-quark potential is solved to
study the temperature dependence of the heavy-quarkonium interaction based on
the Bhanot–Peskin leading order perturbative QCD analysis in which the 1S
charmonium thermal width is determined and compared to recent lattice QCD
results $\left[7\right]$. Wong $\left[8\right]$ has investigated the Q-Q̄
potential by using the thermodynamic quantities to give spontaneous
dissociation temperatures for quarkonium and has also found the quark drip
lines which separate the region of bound color-singlet QQ̄ states from the
unbound region. In Ref. $\left[9\right],$ the authors numerically solved the
SE at finite temperature by employing temperature-dependent effective
potential given by a linear combination of color singlet and internal
energies. By using thermodynamic laws, the SE equation is derived at finite
temperature $\left[10\right]$.

Some authors $\left[11-17\right]$ focus to extend the SE to the higher-
dimensional space which gives more detail about the systems under study.
Moreover, the energy eigenvalues and wave functions are obtained in the
higher-dimensional space and their applications on quarkonium properties are
studied in the vacuum, hot and dense media.

Recently, the fractional calculus has attracted attention in the different
fields of physics which the nonlinear, complex effects are included such as in
Refs. $\left[18-21\right]$. In high energy physics, the description of heavy-
quarkonium energy spectra and complex phenomena of the standard model as in
Ref. $\left[22\right]$, in which, the author used the conformable fractional
derivative to express the fractional radial SE in the N-dimensional space for
the extended Cornall potential by using generalized extended Nikiforov-Uvarov
(ENU) method to the fractional domain. In Ref. $\left[23\right],$ the
fractional form of the NU method is applicable in order to solve fractional
radial SE with its applications on variety of potentials such as the
oscillator potential, Woods-Saxon potential, and Hulthen potential. In Ref.
$\left[24\right]$, the Caputo fractional derivative is applied on a fractional
Schrödinger wave equation with using quantization of the classical
nonrelativistic Hamiltonian. The free particle solutions are obtained, which
are confined to a certain region of space.

Thus, The fractional calculus focuses on vacuum space without considering the
effects of medium. Thus, the present study is to study the effect of
fractional-order on quarkonium masses in a hot medium which are not considered
in recent works such as in Refs. $\left[18-24\right]$, The conformable
fractional of the Nikiforov-Uvarov (CF-NU) method is applied to obtain the
analytic solutions of the $N$-dimensional radial SE.

The paper is organized as follows: In Sec. 2, the CF method is briefly
explained. In Sec. 3, The energy eigenvalue and wave function are calculated
in the $N$-dimensional space using CF-NU method. In Sec. 4, the results are
discussed. In Sec. 5, the summary and conclusion are presented.

##  II 2\. Theoretical Tools

###  II.1 2.1. Conformable fractional derivative

Fractional derivative plays an important role in applied science.
Riemann–Liouville and Riesz and Caputo give a good formula that allows to
apply boundary and initial conditions as in Ref. $\left[19\right]$

$D_{t}^{\alpha}f\left(t\right)=\int_{t_{0}}^{t}K_{\alpha}\left(t-s\right)f^{\left(n\right)}\left(s\right)ds,\
\ \ \ t>t_{0}$ (1)

with

$K_{\alpha}\left(t-s\right)=\frac{\left(t-s\right)^{n-\alpha-1}}{\Gamma\left(n-\alpha\right)},$
(2)

where $f^{\left(n\right)}$ is the $n$ th derivative of the function
$f\left(t\right)$, and $K_{\alpha}\left(t-s\right)$ is the kernel, which is
fixed for a given real number $\alpha$. The kernel
$K_{\alpha}\left(t-s\right)$ has singularity at $t=s$. Caputo and Fabrizio
$\left[25\right]$ suggested a new formula of the fractional derivative with
smooth exponential kernel of the form to avoid the difficulties that found in
Eq. $\left(1\right)$

$D_{t}^{\alpha}f\left(t\right)=\frac{M\left(\alpha\right)}{1-\alpha}\int_{t_{0}}^{t}\exp\left(\frac{\alpha\left(t-s\right)}{1-\alpha}\right)\dot{y}\left(s\right)ds,$
(3)

where $M\left(\alpha\right)$ is a normalization function with $M(0)=M(1)=1$. A
new formula of fractional derivative, called conformable fractional derivative
(CFD) is proposed by Khalil et al. $\left[26\right]$

$D_{t}^{\alpha}f\left(t\right)=\lim_{\epsilon\rightarrow
0}\frac{f\left(t+\epsilon t^{1-\alpha}\right)-f\left(t\right)}{\epsilon}\ \ \
t>0$ (4)

$f\left(0\right)=\lim_{\epsilon\rightarrow 0}f\left(t\right)$ (5)

with $0<\alpha\leq 1$. This new definition is simple and provides a natural
extension of differentiation with integer order $n\in Z$ to fractional order
$\alpha\in C$. Moreover, the CFD operator is linear and satisfies the
interesting properties that traditional fractional derivatives do not, such as
the formula of the derivative of the product or quotient of two functions and
the chain rule $\left[23\right].$ The concept of CFD has successfully applied
to the CF-NU method to obtain the eigenvalues and eigenfunctions of SE as in
Refs. $\left[22,23\right].$

##  III Conformable Fractional NU method

In this section, the CF-NU method is briefly given to solve the conformable
fractional of differential equation which takes the following form {see Ref.
[23], for details}

$D^{\alpha}\left[D^{\alpha}\Psi(s)\right]+\frac{\bar{\tau}(s)}{\sigma(s)}D^{\alpha}\Psi(s)+\frac{\tilde{\sigma}(s)}{\sigma^{2}(s)}\Psi(s)=0,$
(6)

where $\sigma(s)$ and $\tilde{\sigma}(s)$ are polynomials of maximum second
degree and $\bar{\tau}(s)$ is a polynomial of maximum first degree with an
appropriate $s=s(r)$ coordinate transformation.

$D^{\alpha}\Psi(s)=s^{1-\alpha}\Psi^{\prime}(s),$ (7)

$D^{\alpha}\left[D^{\alpha}\Psi(s)\right]=\left(1-\alpha\right)s^{1-2\alpha}\Psi^{\prime}(s)+s^{2-2\alpha}\Psi^{"}(s).$
(8)

Substituting by Eqs. (7) and (8) into (6), we obtain

$\Psi^{\prime\prime}(s)+\frac{\bar{\tau}_{f}(s)}{\sigma_{f}(s)}\Psi^{\prime}(s)+\frac{\tilde{\sigma}_{f}(s)}{\sigma_{f}^{2}(s)}\Psi(s)=0,$
(9)

To find particular solution of Eq. (9) by separation of variables, if one
deals with the transformation

$\Psi(s)=\Phi(s)\chi(s),$ (10)

it reduces to an equation of hypergeometric type as follows

$\sigma_{f}(s)\chi^{\prime\prime}(s)+\tau_{f}(s)\chi^{\prime}(s)+\lambda\chi(s)=0,$
(11)

where

$\sigma_{f}(s)=\pi_{f}(s)\frac{\Phi(s)}{\Phi^{\prime}(s)},$ (12)
$\tau_{f}(s)=\bar{\tau}_{f}(s)+2\pi_{f}(s);~{}\ \ \tau_{f}^{\prime}(s)<0,$
(13)

and

$\lambda=\lambda_{n}=-n\tau_{f}^{\prime}(s)-\frac{n(n-1)}{2}\sigma_{f}^{\prime\prime}(s),n=0,1,2,...$
(14)

$\chi(s)=\chi_{n}(s)$ which is a polynomial of $n$ degree which satisfies the
hypergeometric equation, taking the following form

$\chi_{n}(s)=\frac{B_{n}}{\rho_{n}}\frac{d^{n}}{ds^{n}}(\sigma_{f}^{\prime\prime}(s)\rho(s)),$
(15)

where $B_{n}$ is a normalization constant and $\rho(s)$ is a weight function
which satisfies the following equation

$\frac{d}{ds}\omega(s)=\frac{\tau(s)}{\sigma_{f}(s)}\omega(s);\ \ \
\omega(s)=\sigma_{f}(s)\rho(s),$ (16)
$\pi_{f}(s)=\frac{\sigma_{f}^{\prime}(s)-\bar{\tau}_{f}(s)}{2}\pm\sqrt{(\frac{\sigma_{f}^{\prime}(s)-\bar{\tau}_{f}(s)}{2})^{2}-\tilde{\sigma}_{f}(s)+K\sigma_{f}(s),}$
(17)

and

$\lambda=K+\pi_{f}^{\prime}(s),$ (18)

the $\pi_{f}(s)$ is a polynomial of first degree. The values of $K$ in the
square-root of Eq. (17) is possible to calculate if the expressions under the
square root are square of expressions. This is possible if its discriminate is
zero.

##  IV Conformable Fractional of the Dependent Temperature Schrödinger
Equation

The SE for two particles interacting via a spherically symmetric (central)
potential $V(r)$ in the N-dimensional space, where $r$ is inter-particle
distance, is given by $\left[11-13\right]$

$[\frac{d^{2}}{dr^{2}}+\frac{(N-1)}{r}\frac{d}{dr}-\frac{L(L+N-2)}{r^{2}}+2\mu(E-V(r))]\Psi(r)=0,$
(19)

where $L,N,$ and $\mu$ are the angular momentum quantum number, the
dimensionality number and the reduced mass for the quarkonium particle (for
charmonium $\mu=\frac{m_{c}}{2}$ and for bottomonium $\mu=\frac{m_{b}}{2}$),
respectively. Setting the wave function $\Psi(r)=R(r)r^{\frac{1-N}{2}}$, the
following radial SE is obtained

$[\frac{d^{2}}{dr^{2}}+2\mu(E-V(r,T)-\frac{(L+\frac{(N-2)}{2})^{2}-\frac{1}{4}}{2\mu
r^{2}})]R(r)=0.$ (20)

where $V(r,T)$ is the Cornell potential at the finite temperature as in Ref.
$\left[15\right]$ and references therein which take following as follows

$V(r,T)=a\left(T,r\right)r-\frac{b\left(T,r\right)}{r},$ (21)

where $a\left(T,r\right)=\frac{a}{m_{D}(T)r}(1-e^{-m_{D}\left(T\right)r})$ and
$b\left(T,r\right)=be^{-m_{D}\left(T\right)r}$ where $m_{D}\left(T\right)$ is
the Debye mass that vanishes at $T\rightarrow 0$. $a=0.184$ GeV2 and $b$ will
determine later. By substituting Eq. (21) into Eq. (20) and using
approximation
$e^{-m_{D}\left(T\right)r}={\displaystyle\sum\limits_{j=0}^{\infty}}\frac{\left(-m_{D}\left(T\right)r\right)^{j}}{j!}$
up to second-order, which gives good accuracy when $m_{D}r\ll 1$. One obtains

$[\frac{d^{2}}{dr^{2}}+2\mu(E-A+\frac{b}{r}-Cr+Dr^{2}-\frac{(L+\frac{(N-2)}{2})^{2}-\frac{1}{4}}{2\mu
r^{2}})]R(r)=0.$ (22)

where,

$A=b~{}m_{D}\left(T\right),C=a-\frac{1}{2}bm_{D}^{2}\left(T\right),\text{and \
}D=\frac{1}{2}a~{}m_{D}\left(T\right).$ (23)

By taking $r=\frac{1}{x}$, Eq. (23) takes the following form

$[\frac{d^{2}}{dx^{2}}+\frac{2x}{x^{2}}\frac{d}{dx}+\frac{2\mu}{x^{4}}(E-A+bx-\frac{C}{x}+\frac{D}{x^{2}}-\frac{(L+\frac{(N-2)}{2})^{2}-\frac{1}{4}}{2\mu}x^{2})]R(x)=0.$
(24)

The expansion of $\frac{C}{x}$ and $\frac{D}{x^{2}}$ in a power series around
the characteristic radius $r_{0}$ of meson up to the second order is given
Ref. $\left[15\right]$. The following equation is obtained

$[\frac{d^{2}}{dx^{2}}+\frac{2x}{x^{2}}\frac{d}{dx}+\frac{2}{x^{4}}(-D_{1}+D_{2}x-D_{3}x^{2})]R(x)=0,$
(25)

where,

$D_{1}=-\mu(E-A-\frac{3C}{\delta}+\frac{6D}{\delta^{2}}),D_{2}=\mu(\frac{3C}{\delta^{2}}-\frac{8D}{\delta^{3}}+b),\text{and
}D_{3}=\mu(\frac{C}{\delta^{3}}-\frac{3D}{\delta^{4}}+\frac{(L+\frac{(N-2)}{2})^{2}-\frac{1}{4}}{2\mu}).$
(26)

The transition of the conformable fractional of Eq. (25) is obtained as in
Ref. $\left[23\right]$

$[D^{\alpha}\left[D^{\alpha}R(x)\right]+\frac{2x^{\alpha}}{x^{2\alpha}}D^{\alpha}R(x)+\frac{2}{x^{4\alpha}}(-D_{1}+D_{2}x^{\alpha}-D_{3}x^{2\alpha})]R(x)=0,$
(27)

substituting by Eqs. (7) and (8) into (27), we obtain

$R^{\prime\prime}(x)+\frac{\bar{\tau}_{f}(x)}{\sigma_{f}(x)}R^{\prime}(x)+\frac{\tilde{\sigma}_{f}(x)}{\sigma_{f}^{2}(x)}R(x)=0,$
(28)

where

$\bar{\tau}_{f}(s)=3x^{\alpha}-\alpha
x^{\alpha},\sigma_{f}(s)=x^{\alpha+1},\text{and
}\tilde{\sigma}(s)=2(-D_{1}+D_{2}x^{\alpha}-D_{3}x^{2\alpha}).$ (29)

Hence, the Eq. (28) satisfies Eq. (9). Therefore, Eq. (17) takes the following
form after substituting by Eq. (29)

$\pi_{f}=-x^{\alpha}+\alpha x^{\alpha}\pm\sqrt{\left(-x^{\alpha}+\alpha
x^{\alpha}\right)^{2}-2(-D_{1}+D_{2}x^{\alpha}-D_{3}x^{2\alpha})+Kx^{1+\alpha}}.$
(30)

The constant $K$ is chosen such as the function under the square root has a
double zero, i.e. its discriminant equals zero. Hence,

$K=\left(\frac{D_{2}^{2}}{2D_{1}}-(1-2\alpha+\alpha^{2}+2D_{3})\right)x^{\alpha-1}\text{.}$
(31)

Substituting by Eq. (31) into Eq. (30), we obtain

$\pi_{f}\left(x\right)=-x^{\alpha}+\alpha
x^{\alpha}+\frac{D_{2}}{\sqrt{2D_{1}}}x^{\alpha}-\sqrt{2D_{1}}$ (30)

The positive sign in Eq. (30) is determined as in Ref. $\left[15\right]$. By
using Eq. (13), we obtain

$\tau_{f}(x)=3x^{\alpha}+\alpha
x^{\alpha}-2\left(\frac{D_{2}}{\sqrt{2D_{1}}}x^{\alpha}-\sqrt{2D_{1}}\right),$
(24)

and using Eq. (14), we obtain

$\lambda_{n}=\left(-n\left(3\alpha-\alpha^{2}\right)-\frac{2D_{2}\alpha}{\sqrt{2D_{1}}}-\frac{n\left(n-1\right)\alpha\left(\alpha+1\right)}{2}\right)x^{\alpha-1}\text{.}$
(25)

From Eqs. (14 and 18 ); $\lambda=\lambda_{n}$. The energy eigenvalues of Eq.
(22) in the $N$-dimensional space is given

$\displaystyle E_{nL}^{N}$
$\displaystyle=A+\frac{3C}{\delta}-\frac{6D}{\delta^{2}}-$
$\displaystyle\frac{2\mu(\frac{3C}{\delta^{2}}+b-\frac{8~{}D}{\delta^{3}})^{2}}{[(2n+1)\pm\sqrt{W+\frac{8\mu
C}{\delta^{3}}+4((L+\frac{N-2}{2})^{2}-\frac{1}{4})-\frac{24\mu
D}{\delta^{4}}}]^{2}}.$ (26)

with

$W=\left(2n\alpha+\alpha\right)^{2}-4\left(n\left(3\alpha-\alpha^{2}\right)+\frac{1}{2}n\left(n-1\right)\alpha\left(\alpha+1\right)+\alpha-1\right).$
(27)

The radial of wave function takes the following form

$R_{nL}\left(r^{\alpha}\right)=C_{nL}~{}r^{\left(-\frac{D_{2}}{\sqrt{2D_{1}}}-1\right)\alpha}e^{\sqrt{2D_{1}}r^{\alpha}}(-r^{2\alpha}D^{\alpha})^{n}(r^{\left(-2n+\frac{D_{2}}{\sqrt{2D_{1}}}\right)\alpha}e^{-2\sqrt{2D_{1}}r^{\alpha}}).$
(28)

$C_{nL}$ is the normalization constant that is determined by
$\int\left|R_{nL}\left(r^{\alpha}\right)\right|^{2}dr=1$. We note that the
radial wave function in Eq. (28) does not explicitly depend on the number of
dimensions. Hence, $\int\left|R_{nL}\left(r\right)\right|^{2}dr=1$ remains
unchanged.

##  V Discussion of Results

In this section, the above results are applied on quarkonium masses. the
following relation is used as in Refs. $\left[12,27\right]$

$M=2m+E_{nL}^{N},$ (29)

where $m$ is quarkonium bare mass for the charmonium or bottomonium mesons. By
using Eq. (26), we write Eq. (29) as follows:

$\displaystyle M_{Q}$
$\displaystyle=2m+A+\frac{3C}{\delta}-\frac{6D}{\delta^{2}}-$
$\displaystyle\frac{2\mu(\frac{3C}{\delta^{2}}+b-\frac{8~{}D}{\delta^{3}})^{2}}{[(2n+1)\pm\sqrt{W+\frac{8\mu
C}{\delta^{3}}+4((L+\frac{N-2}{2})^{2}-\frac{1}{4})-\frac{24\mu
D}{\delta^{4}}}]^{2}}.$ (30)

Eq. (30) represents quarkonium mass that is calculated in the N-dimensional
space with considering fraction-order, Number of dimensionality, and finite
temperature.. One can obtain the quarkonium masses at zero temperature by
taking $T=0$ leads to $A=D=0$ and $C=a$ and $\alpha=1$. Therefore, Eq. (30)
takes the following form

$M_{Q}=2m+\frac{3a}{\delta}-\frac{2\mu(\frac{3a}{\delta^{2}}+b)^{2}}{[(2n+1)\pm\sqrt{1+\frac{8\mu
a}{\delta^{3}}+4L(L+1)}]^{2}}.$ (31)

Eq. (31) coincides with Ref. $\left[28\right]$, in which the authors obtained
the quarkonium mass at zero temperature and $\alpha=1$ in 3D. At $\alpha=1,$
the following equation is obtained

$M_{Q}=2m+A+\frac{3C}{\delta}-\frac{6D}{\delta^{2}}-\frac{2\mu(\frac{3C}{\delta^{2}}+b-\frac{8~{}D}{\delta^{3}})^{2}}{[(2n+1)\pm\sqrt{1+\frac{8\mu
C}{\delta^{3}}+4((L+\frac{N-2}{2})^{2}-\frac{1}{4})-\frac{24\mu
D}{\delta^{4}}}]^{2}}.$ (32)

Eq. (32) is compatible with Ref. $\left[15\right]$, in which the author
obtained the quarkonium mass at finite temperature in the N-dimensional space.
To calculate quarkonium mass accordance to Eq. (30), the Debye mass is defined
as in Ref. $\left[29\right]$

$m_{D}\left(T\right)=g(T)T\sqrt{\frac{N_{c}}{3}+\frac{N_{f}}{6}},$ (34)

and

$g^{2}(T)=\frac{24\pi^{2}}{\left(33-2N_{f}\right)\ln\left(\frac{2\pi
T}{\Lambda_{MS}}\right)},$ (35)

with $N_{f}$ and $N_{c}$ as the number of flavours and colors, respectively.
$\Lambda_{MS}=0.1$ GeV and $b=\frac{g^{2}(T)}{3\pi}$. In Figs. (1, 2), the
bottomonium mass is plotted as a function of temperature ratio
$\frac{T}{T_{c}}$ where $T_{c}$ is the critical temperature equals 170 MeV as
in Ref. $\left[30\right]$. In Fig. (1), the bottomonium mass is plotted versus
temperature ratio $\frac{T}{T_{c}}$ in the three-dimensional space. At
$\alpha=1$ which represents the normal case for normal calculus. One note that
the bottomonium mass increases up $T=1.36T_{c}$ then the curve of bottomonium
decreases with increasing temperature. This indicates that the bottomonium is
stable up to $T=1.36T_{c}$ then the bottomonium melts in hot QCD medium. In
Refs. $\left[6,29,31\right],$ the authors found that the bottomonium melts
above the critical temperature. In Ref. $\left[6\right],$ the dissociation of
bottomonium $T_{D}\succ$ $Tc$ , $T_{D}=3.2$ $Tc$ in Ref. $\left[29\right]$,
and $T_{D}=1.11$ $Tc$ in Ref. $\left[31\right]$. In these works, the SE is
solved for quarkonium by using different methods. In addition, the
interpretation is an agreement with Ref. $\left[27\right]$, in which the SE is
solved for a nucleon in hot QCD medium. Hence, the present result is a
qualitative agreement with Refs. $\left[6.29,31\right]$. To investigate the
effect of fractional parameter, one takes the $\alpha=0.4,0.6,0.8$. One notes
that the curve of bottomonium strongly increases and then decreases in higher
temperatures. In addition, the bottomonium mass is shifted to higher values by
decreasing fractional parameter $\alpha$ which indicates the binding energy is
more bound. Therefore, the dissociation temperature increases with increasing
binding energy. Hence the effect of fraction parameter supports the bound
state of bottomonium. The effect is not considered in other works.

In this work, one interests to study the effect on the dimensional number on
quarkonium mass. The motivation for this as a natural consequence of the
unification of the two modern theories of quantum mechanics and relativity and
the emergence of the string theory, the investigation of the Standard Model
particles in extra or higher-dimensional space is a hot topic of interest.
From the experimental point of view, the investigation of the existence of
extra dimensions is one of the primary goals of the LHC. The search for extra
dimensions with the ATLAS and CMS detectors is discussed in Ref.
$\left[32\right]$. In Fig. 2, the bottomonium mass is plotted as a function of
temperature ratio for different of fractional parameter at $N=5$. In Fig. 2,
one note that the bottomonium mass decreases with increasing temperature. This
indicates that the bottomonium melts around the critical temperature when the
higher dimensional space is considered and also the quarkonium mass slightly
shifts with increasing fractional parameter. Therefore, the fractional
parameter is a little effect in the higher-dimensional space. In addition, one
notes that the bottomonium mass increases with increasing dimensionality
number. This finding is in agreement with Ref. $\left[22\right]$

Fig. 1: Mass Specrum 1S of bottomonium is plotted as a function of ratio
temperature $\frac{T}{T_{c}}$ for parameters $m_{b}=4.823$ GeV and $a=0.184$
GeV2,at different values of fractional parameter at 3D

Fig. 2: Mass Specrum 1S of bottomonium is plotted as a function of ratio
temperature $\frac{T}{T_{c}}$ for parameters $m_{b}=4.823$ GeV and $a=0.184$
GeV2at different values of fractional parameter at 5D

In Figs. (3, 4), the charmonium is plotted as a function of temperature ratio
for different values of fractional parameter in 3D and 5D spaces,
respectively. In Fig. (3), one notes that the charmonium mass decreases with
increasing temperature and notes the charmonium melts in given interval of
temperature. This indicates that the charmonium melts under critical
temperature. In Ref. $\left[31\right]$, the authors found the 1S state of
charmonuim is melts around 0.99 $T_{c}$. By decreasing fractional parameter,
one notes the charmonium mass increases. In Fig. (4), in 5D, the charmonium
mass decreases with increasing temperature and increases with increasing
dimensionality number. The effect of fractional parameter is a little
sensitive on charmonium when the dimensionality number increases. Thus, the
charmonium mass increases with decreasing fractional parameter and increasing
dimensionality number.

Fig. 3: Mass Specrum 1S of charmonium is plotted as a function of ratio
temperature $\frac{T}{T_{c}}$ for parameters $m_{c}=1.209$ GeV and $a=0.184$
GeV2at different values of fractional parameter at 3D

Fig. 4: Mass Specrum 1S of charmonium is plotted as a function of ratio
temperature $\frac{T}{T_{c}}$ for parameters $m_{c}=1.209$ GeV and $a=0.184$
GeV2at different values of fractional parameter at 5D

##  VI Summary and Conclusion

The CF-NU method is applied to solve N-fractional radial SE. The eigenvalues
of energy and corresponding wave functions are obtained, in which they depend
on the fractional parameter $0<\alpha\leq 1$, finite temperature, and
dimensionality number. Particular cases are obtained at $\alpha=1$, $T=0$ and
$T\neq 0$ that compatible with recent works. The present results are applied
on the quarkonium system such as charmonium and bottomonium masses in the hot
QCD medium. The present results show that the fractional parameter plays an
important role in the hot QCD medium since the binding energy of charmonium
and bottomonium is more bound by decreasing fractional parameter. In addition,
the effect of fractional parameter is studied in hot medium when the higher
dimensional space is considered $\left(N=5\right)$. The results show that the
effect of fractional parameter is a little sensitive on quarkonium masses at
$N=5.$ The dissociation of charmonium and bottomonium is investigated at
$\alpha=1$ which represents a normal case and the agreement is noted in
comparison with other works. One concludes that the fraction calculus plays an
important role to give more information about quark-gluon plasma and expects a
key for solving many theoretical problems in high energy physics.

##  VII References

## References

  * (1)

  1. 1.

B. K. Patra and D. K. Srivastava, Phys. Lett. B 505, 113 (2001).

  2. 2.

D. Pal, B. K. Patra, and D. K. Srivastava, Eur. J. Phys. C 17, 179 (2000).

  3. 3.

V. Agotiya, V. Chandra, and B. K. Patra1, Phys. Rev. C 80, 025210 (2009).

  4. 4.

T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986).

  5. 5.

G. P. Malik, R. K. Jha, and V. S. Varma, Astrophysical J. 503, 446 (1998).

  6. 6.

C.-Y. Wong, Phys. Rev. C 65, 034902 (2002).

  7. 7.

F. Arleo, J. Cugnon, and Y. Kalinovsky, Phys. Lett. B 614, 44 (2005).

  8. 8.

C.-Y. Wong, Phys. Rev. C 76, 014902 (2007).

  9. 9.

W. M. Alberico, A. Beraudo, A. D. Pace, and A. Molinari, Phys. Rev. D 75,
074009 (2007).

  10. 10.

X. Y. Wu, B. Zhang, and X. Liu, Inter. J. Theor. Phys. 50, 2546 (2011).

  11. 11.

M. Abu-Shady and A. N. Ikot, Euro. Phys. J. Plus 134, 321, (2019).

  12. 12.

M. Abu-Shady, T. A. Abdel-Karim, and S. Y. Ezz-Alarab, J. Egyp. Math. Soci.
27, 14, (2019).

  13. 13.

M. Abu-Shady, T. A. Abdel-Karim, E. M. Khokha, Adv. High Ener. Phys. 2018,
7356843, (2018).

  14. 14.

M. Abu-Shady, E. M. Khokha, Adv. High Ener. Phys. 2018, 7032041, (2018).

  15. 15.

M. Abu-Shady, J. Egypt Math. Society 25, 86 (2017).

  16. 16.

E. M. Khokha, M. Abu-Shady, T. A. Abdel-Karim, Inter. J. Theor. App. Math. 2,
86, (2016).

  17. 17.

M. Abu-Shady, Adv. Math. Phys. 2016, 1, (2016).

  18. 18.

K. B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New
York, 1974).

  19. 19.

I. Podlubny, Fractional Differential Equations (Academic Press, 1999).

  20. 20.

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus
Models and Numerical Methods (World Scientific, 2017).

  21. 21.

15\. J. L. Wang and H. F. Li, Comput. Math. Appl. 62, 1562 (2011).

  22. 22.

A. Al-Jamel, Int. J. Mod. Phys. A 34, 1950054 (2019).

  23. 23.

H. Karayer, D. Demirhan and F. Büyükkılıç, Commun. Theor. Phys. 66, 12 (2018).

  24. 24.

R. Herrmann, arXiv:math-ph/0510099v4

  25. 25.

M. Caputo and M. Fabrizio, Prog. Fract. Differ. Appl. 1, 73 (2015).

  26. 26.

R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, J. Comput. Appl. Math.
264, 65 (2014).

  27. 27.

P. Cheng, Q. Meng, Y. Xia, J. Ping, and H. Zong, Phys. Rev. D 98, 116010
(2018).

  28. 28.

S. M. Kuchin and N. V. Maksimenko, Univ. J. Phys. Appl. 1, 295 (2013).

  29. 29.

P. K. Srivastava, O. S. K. Chaturvedi, L. Thakur, Eur. Phys. J. C 78, 440
(2018).

  30. 30.

M. Y. Jamal, S. Mitra, and V. Chandra, Phys. Rev. D 97, 094033 (2018).

  31. 31.

C.-Y. Wong, J. Phys. Part. Phys. 28, 2349 (2002).

  32. 32.

ATLAS and CMS Collabs. (J. Kretzschmar), Nucl. Part. Phys. Proc. 273275, 541
(2016).

## References

  * (1)


# Hausdorff measure estimates for the degenerate quenching problem

Damião J. Araújo  Department of Mathematics, Universidade Federal da Paraíba,
58059-900, João Pessoa-PB, Brazil<EMAIL_ADDRESS>,  Rafayel Teymurazyan
Applied Mathematics and Computational Sciences Program (AMCS), Computer,
Electrical and Mathematical Sciences and Engineering Division (CEMSE), King
Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900,
Kingdom of Saudi Arabia and CMUC, Department of Mathematics, University of
Coimbra, 3000-143 Coimbra, Portugal<EMAIL_ADDRESS>and
José Miguel Urbano  Applied Mathematics and Computational Sciences Program
(AMCS), Computer, Electrical and Mathematical Sciences and Engineering
Division (CEMSE), King Abdullah University of Science and Technology (KAUST),
Thuwal, 23955 -6900, Kingdom of Saudi Arabia and CMUC, Department of
Mathematics, University of Coimbra, 3000-143 Coimbra, Portugal
<EMAIL_ADDRESS>
###### Abstract.

We study analytical and geometric properties of minimizers of non-
differentiable functionals epitomizing the degenerate quenching problem. Our
main finding unveils finite $(n-1)-$Hausdorff measure estimates for the
pertaining free boundaries. The approach hinges upon deriving optimal gradient
decay estimates, coupled with a fine analysis of an intrinsic auxiliary
equation stripped of the singularity.

###### Key words and phrases:

Free boundary problems; Hausdorff estimates; degenerate equations

###### 2020 Mathematics Subject Classification:

Primary 35R35. Secondary 35A21, 35J70

##  1\. Introduction

In this work, we are concerned with minimizers of $p-$energy functionals of
the type

$v\longmapsto\int_{\Omega}\frac{|Dv|^{p}}{p}+F(x,v)\,dx,$

for non-differentiable potentials $F(x,s)\geq 0$.

This class of problems is well-understood in the case $p=2$, particularly in
the context of the obstacle problem, for which $F(x,s)=s_{+}$ (see [6, 8] and
the book [26]). Another emblematic potential is the discontinuous
$F(x,s)=\chi_{\\{s>0\\}}$, leading to the cavity or Bernoulli free boundary
problem, investigated by Alt and Caffarelli in [1]. An intermediary scenario
emerges by interpolating between the two previous cases and considering
potentials exhibiting $\gamma-$growth, namely with

$F(x,s)=s_{+}^{\gamma},\qquad\gamma\in(0,1).$ (1.1)

Often known as Alt-Phillips potentials (cf. [2, 27, 28]), they bring about the
so-called quenching problem, which models phenomena characterized by abrupt
changes in certain quantities along unknown interfaces. In all these cases,
the absence of differentiability for $F$ strongly influences the efficiency of
the regularity mechanisms for minimizers compared to the classical scenarios.
Still, for $p=2$, it has been shown that the optimal regularity class for
minimizers is $C_{\rm loc}^{1,\alpha}$, with

$\alpha=\frac{\gamma}{2-\gamma},\qquad\gamma\in(0,1].$

For the cavity problem, corresponding to $\gamma=0$, minimizers are locally
Lipschitz. Concerning the free boundary, in the seminal works [7] for the
obstacle problem, and [28] for the case $\gamma\in(0,1)$, analytical and
geometric measure theory methods were applied to establish that interfaces
have finite $(n-1)-$dimensional Hausdorff measure. For the two-phase problem
$(p=2)$, the free boundaries are known to be $C^{1}$ in dimension two (see
[23]).

For the nonlinear case $p\neq 2$ much less is known as the challenges posed by
the non-quadratic growth in the functional muddle the understanding of the
regularity and geometric properties of the minimizes and the free boundary.
Obstacle problems, corresponding to $\gamma=1$, were treated in [3, 14] for
non-zero obstacles $\varphi$. In particular, in [3], the authors show that a
minimizer $u$ is of class $C^{1,p^{\prime}-1}$ at the free boundary
$\partial\\{u>\varphi\\}$, for $p^{\prime}=p/(p-1)$. For the sign-changing
case, the intermediate scenario $\gamma\in(0,1)$ was treated in [22]. In [5],
the authors considered potentials with varying non-differentiability levels

$F(x,s)\sim(s-\varphi(x))_{+}^{\gamma},$

obtaining improved regularity estimates at contact points in
$\partial\\{u>\varphi\\}$. For the particular case of an obstacle
$\varphi\equiv 0$, it is shown therein that minimizers are of class
$C^{1,\alpha}$ at the quenching interface $\partial\\{u>0\\}$, for

$\alpha=\frac{\gamma}{p-\gamma},$

revealing the precise interplay between the singularity parameter $\gamma$ and
the exponent $p$ in determining the regularity of minimizers. Related problems
were studied in [20, 11], establishing the Lipschitz regularity of minimizers
using a singular perturbation. These efforts notwithstanding, the study of the
free boundary for problems involving the $p-$Laplace operator remains
virtually virgin ground, the only significant contribution being the results
in [17, 21, 14, 10] for the $p-$obstacle problem, and in [12] for the cavity
problem, where it was shown that near flat points, the free boundary is of
class $C^{1,\beta}$, for a $\beta\in(0,1)$.

In this paper, we advance the theory one step forward, carrying the analysis
in the degenerate case $p>2$ and for the quenching scenario (1.1), for which
the corresponding Euler-Lagrange equation is singular since the right-hand
side blows up at the free boundary

$\mathcal{F}(u)=\partial\\{u>0\\}.$

We consider non-negative minimizers and chiefly obtain the finiteness of the
$(n-1)-$dimensional Hausdorff measure of the free boundary, a result that
plays a significant role in the context of trying to universally characterize
the “size” or “dimension” of the interface. These estimates are closely
related to the (lack of) regularity and the geometric properties of the free
boundary. Our approach relies on a pointwise gradient control based on an
intrinsic Harnack-type inequality and a suitable application of regularity
estimates at points relatively close to the free boundary available in [5].
This gradient control is already crucial in [28] for treating the case $p=2$,
but its proof heavily relies on the linearity of the operator and fails to
work in the degenerate case. After establishing a non-degeneracy estimate and
porosity of the free boundary, we remove the singularity, thereby obtaining a
problem with a positive and bounded right-hand side. This strategy,
complemented by ensuring that the gradient of the right-hand side of a
transformed problem belongs to a suitable Morrey space, allows us to analyze
the free boundary regularity as that of a refined obstacle problem.

The paper is organized as follows. In section 2, we introduce some notation,
formulate the problem rigorously and gather several preliminary results.
Section 3 brings the optimal gradient decay, and, in Section 4, we establish a
non-degeneracy estimate and the porosity of the free boundary. In Section 5,
we remove the singularity in the problem and in Section 6, we prove the main
result.

##  2\. Problem setting and preliminary results

Let $\Omega$ be an $n-$dimensional bounded domain and $p>2$. For a given non-
negative boundary data $g\in W^{1,p}(\Omega)$, we consider minimizers of the
functional

$J(v):=\int_{\Omega}\left(\frac{|Dv(x)|^{p}}{p}+\left[v_{+}(x)\right]^{\gamma}\right)dx,$
(2.1)

among competitors in the set

$\mathbb{K}:=\left\\{v\in W^{1,p}(\Omega);\,\,v\geq 0,\,\,v-g\in
W^{1,p}_{0}(\Omega)\right\\}.$ (2.2)

The corresponding Euler-Lagrange equation is the singular PDE, holding in the
non-coincidence set $\\{u>0\\}\cap\Omega$,

$\Delta_{p}u:=\mathrm{div}\left(|Du|^{p-2}Du\right)=\gamma u^{\gamma-1},$
(2.3)

whose right-hand side blows up at free boundary points since $\gamma$ is
assumed to be in $(0,1)$.

Most of the results in the paper are valid for local minimizers defined as
follows. Set

$J_{r}(v):=\int_{B_{r}}\left(\frac{|Dv(x)|^{p}}{p}+\left[v_{+}(x)\right]^{\gamma}\right)dx.$

######  Definition 2.1.

A non-negative function $u\in W^{1,p}(\Omega)$ is called a local minimizer
with respect to a ball $B_{r}\subset\Omega$ if

$J_{r}(u)\leq J_{r}(v),$

for all $v\geq 0$ such that $u-v\in W_{0}^{1,p}(B_{r})$.

######  Remark 2.1.

Observe that if $u$ is a local minimizer with respect to a ball $B_{r}$, then

$u_{s}(x):=\frac{u(sx)}{s^{\frac{p}{p-\gamma}}}$

is a local minimizer with respect to the ball $B_{\frac{r}{s}}$. More
precisely,

$J_{r}(u)=s^{\frac{p\gamma}{p-\gamma}}J_{\frac{r}{s}}(u_{s}).$

The existence of a minimizer for (2.1) is obtained in [5, Theorem 2.1] (see
also [22], for the particular case of a zero obstacle), together with the
bound

$\|u\|_{\infty}\leq\|g\|_{\infty},$

for bounded boundary data. The local $C^{1,\alpha}-$regularity is the object
of [5, Theorem 3.1], where the existence of a constant $C>0$, depending only
on $\mathrm{dist}(\Omega^{\prime},\partial\Omega)$,
$\|u\|_{L^{\infty}(\Omega)}$, $n$, $p$ and $\gamma$, is obtained such that

$\|u\|_{C^{1,\alpha}(\Omega^{\prime})}\leq C,$ (2.4)

for

$\alpha:=\min\left\\{\sigma^{-},\frac{\gamma}{p-\gamma}\right\\},$ (2.5)

where $\sigma>0$ is the Hölder regularity exponent for the gradient of
$p-$harmonic functions. It follows from the results in [16] (see also [4])
that, for $n=2$, one has $\alpha=\frac{\gamma}{p-\gamma}$. Note also that when
$\gamma=0$, the regularity result above recovers the Lipschitz regularity of
solutions obtained in [11, Theorem 3.3]. Moreover, in [5, Theorem 4.1 and
Theorem 6.1], the growth of the minimizer away from free boundary points is
revealed to be

$u\leq C\,r^{\frac{p}{p-\gamma}},\quad\textrm{in}\ B_{r}(y)$ (2.6)

for any $r\in(0,r_{0})$ and any $y\in\partial\\{u>0\\}\cap\Omega^{\prime}$,
where $\Omega^{\prime}\subset\subset\Omega$, for universal constants $C>0$ and
$r_{0}>0$. The approach is based on geometric tangential analysis and a fine
perturbation combined with an adjusted scaling argument, the idea of which is
to ensure that at the limit, one gets a linear elliptic equation without the
zero-order term. The intuition behind the proof is that the problem behaves
essentially as an obstacle problem for a uniformly elliptic operator.

We conclude this section by showing that any minimizer $u$ of (2.1) is
$p-$subharmonic and solves the corresponding Euler-Lagrange equation. As $u\in
W^{1,p}(\Omega)$, $p-$subharmonicity is equivalent to proving that $u$ stays
below any $p-$harmonic replacement, according to [24, Chapter 5]. The proof is
known and can be obtained by combining tools used in [1, 11, 22, 27]. We
include it below for the reader’s convenience.

######  Lemma 2.1.

If $u$ is a minimizer of (2.1) and $v$ is a $p-$harmonic function in
$B\subset\Omega$ that agrees with $u$ in $\Omega\setminus B$, then $u\leq v$.

###### Proof.

Set $w:=\min(u,v)$. We aim to show that, in fact, $w=u$. As $w-u\in
W^{1,p}(B)$, one has $D(w-u)=0$ a.e. on $\\{w=u\\}$, [18, Lemma A.4]. Hence,

$\int_{B}|Dw|^{p-2}Dw\cdot D(w-u)=\int_{B}|Dv|^{p-2}Dv\cdot D(w-u)=0,$ (2.7)

where the last equality follows from the fact that $v$ is $p-$harmonic and
agrees with $u$ on $\partial B$. On the other hand, defining for $0\leq s\leq
1$,

$u_{s}(x):=su(x)+(1-s)w(x),$

recalling (2.7) and noting that $u_{s}-w=s(u-w)$, we have

$\begin{split}&\int_{B}\left(|Du|^{p}-|Dw|^{p}\right)\\\
&=\int_{0}^{1}\frac{d}{ds}\left(\int_{B}|Du_{s}|^{p}\right)\,ds\\\
&=p\int_{0}^{1}ds\int_{B}|Du_{s}|^{p-2}Du_{s}\cdot D(u-w)\\\
&=p\int_{0}^{1}ds\int_{B}\left(|Du_{s}|^{p-2}Du_{s}-|Dw|^{p-2}Dw\right)\cdot
D(u-w)\\\
&=p\int_{0}^{1}\frac{ds}{s}\int_{B}\left(|Du_{s}|^{p-2}Du_{s}-|Dw|^{p-2}Dw\right)\cdot
D(u_{s}-w).\end{split}$

Combining this with the well-known inequality

$\left(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta\right)\cdot(\xi-\eta)\geq
c\,|\xi-\eta|^{p},$

where $c>0$ is a constant depending only on $n$ and $p$, we reach

$\displaystyle\int_{B}\left(|Du|^{p}-|Dw|^{p}\right)$ $\displaystyle\geq$
$\displaystyle cp\int_{0}^{1}\frac{ds}{s}\int_{B}|D(u_{s}-w)|^{p}$
$\displaystyle=$ $\displaystyle cp\int_{0}^{1}s^{p-1}\,ds\int_{B}|D(u-w)|^{p}$
$\displaystyle=$ $\displaystyle c\int_{B}|D(u-w)|^{p}\geq 0.$

Using this and the fact that $w\leq u$, we obtain

$J(w)-J(u)\leq\int_{B}(w_{+}^{\gamma}-u_{+}^{\gamma})\leq 0.$

By the minimality of $u$, we conclude that $J(w)=J(u)$, which only holds if
$w=u$. ∎

######  Proposition 2.1.

If $u$ is a minimizer of (2.1), then

$\Delta_{p}u=\gamma u^{\gamma-1}\textrm{ in }\,\\{u>0\\}$

in the sense of distributions.

###### Proof.

Let $y\in\Omega$ be such that $u(y)>0$. Since $u$ is continuous
(particularly), 2.4, then for $r>0$ small, one has $u\geq u(y)/2$ in
$B_{r}(y)$. Therefore, if $\xi\in C_{0}^{\infty}(\Omega)$ and $|\varepsilon|$
is small enough, then $u+\varepsilon\xi\geq u(y)/4$, i.e.,
$u+\varepsilon\xi\in\mathbb{K}$. As $u$ is a minimizer of (2.1), then
$\varepsilon=0$ is a minimizer of $\varepsilon\mapsto J(u+\varepsilon\xi)$. In
other words, one should have
$J^{\prime}(u+\varepsilon\xi)|_{\varepsilon=0}=0$, i.e.,

$\int_{B_{r}(y)}\left(|Du|^{p-2}Du\cdot D\xi+\gamma
u^{\gamma-1}\xi\right)\,dx=0,$

which is the desired result. ∎

##  3\. Pointwise gradient estimate

In this section, we prove a pointwise gradient estimate, which plays an
essential role in the analysis of the free boundary. We first establish an
intrinsic Harnack-type inequality for small radii.

######  Lemma 3.1.

If $u$ is a local minimizer of (2.1) in $B_{1}$, then there exists a constant
$C>0$, depending only on $n$, $p$, $\gamma$ and $\|u\|_{\infty}$, such that

$\sup_{B_{r}(x)}u\leq C\,u(x),$

for each $x\in B_{1/2}$ such that
$r:=\left[u(x)\right]^{\frac{p-\gamma}{p}}\leq 1$.

###### Proof.

Suppose the conclusion fails. Then there exist a sequence of points $x_{k}\in
B_{1/2}$ and local minimizers $u_{k}$ in $B_{1}$, such that

$s_{k}:=\sup_{B_{r_{k}}(x_{k})}u_{k}>ku_{k}(x_{k})=kr_{k}^{\frac{p}{p-\gamma}}.$
(3.1)

where $r_{k}:=\left[u_{k}(x_{k})\right]^{\frac{p-\gamma}{p}}$. Set

$v_{k}(y):=s_{k}^{-1}u_{k}\left(x_{k}+r_{k}y\right)\quad\mathrm{in}\ B_{1}.$

Note that $v_{k}(y)\in[0,1]$, and, using (3.1),

$v_{k}(0)=s_{k}^{-1}u_{k}(x_{k})=s_{k}^{-1}r_{k}^{\frac{p}{p-\gamma}}<\frac{1}{k}.$

Additionally,

$\sup_{B_{1}}v_{k}=1.$

On the other hand, $v_{k}$ minimizes

$\int_{B_{1}}\left(\frac{|Dv|^{p}}{p}+\frac{r_{k}^{p}}{s_{k}^{p-\gamma}}\,v^{\gamma}\right)\,dx,$

and, recalling (3.1) once more,

$\frac{r_{k}^{p}}{s_{k}^{p-\gamma}}<\frac{1}{k^{p-\gamma}}.$ (3.2)

Hence, from 2.4, up to a subsequence, $v_{k}$ converges locally uniformly to a
function $v_{\infty}$ defined in $B_{1}$. Obviously, $v_{\infty}(0)=0$ and

$\sup_{B_{1}}v_{\infty}=1.$ (3.3)

Moreover, from (3.2), it follows that $v_{\infty}$ minimizes

$\int_{B_{1}}\frac{|Dv|^{p}}{p}\,dx,$

so $v_{\infty}$ is $p-$harmonic in $B_{1}$. Hence, as $v_{\infty}(0)=0$, we
conclude that $v_{\infty}\equiv 0$, which contradicts (3.3). ∎

As a consequence, we obtain the following pointwise gradient estimate.

######  Theorem 3.1.

If $u$ is a local minimizer of (2.1) in $B_{1}$, then there exists a constant
$C>0$, depending only on $n$, $p$, $\gamma$ and $\|u\|_{\infty}$, such that

$|Du(x)|^{p}\leq C\,u^{\gamma}(x),$

for each $x\in B_{1/2}$.

###### Proof.

Let $x\in B_{1/2}$. We divide the proof into two steps.

Step 1. If $\|u\|_{\infty}\leq 1$, then setting

$v(y):=r^{-\frac{p}{p-\gamma}}\,u(x+ry),\quad y\in B_{1},$

for $r:=\left[u(x)\right]^{\frac{p-\gamma}{p}}\leq 1$ and applying Lemma 3.1,
we conclude that there exists a universal constant $C>0$ such that

$\sup_{B_{1}}v\leq C.$

Local $C^{1,\alpha}$ regularity of $u$ implies, in particular,

$|Dv(0)|\leq C^{\prime}C,$

for a universal constant $C^{\prime}>0$. Observe now that

$|Dv(0)|=r^{-\frac{\gamma}{p-\gamma}}|Du(x)|=|Du(x)|u^{-\frac{\gamma}{p}}(x),$

and the result follows.

Step 2. If $\|u\|_{\infty}>1$, set
$r:=\left[u(x)/\|u\|_{\infty}\right]^{\frac{p-\gamma}{p}}\leq 1$. Applying
Lemma 3.1 for $w(x):=\left[u(x)/\|u\|_{\infty}\right]$, we obtain

$\sup_{B_{r}(x)}w\leq Cw(x).$

Therefore, arguing as in Step 1, we get

$|Dw(x)|^{p}\leq Cw^{\gamma}(x),$

i.e.,

$|Du(x)|^{p}\leq C\|u\|_{\infty}^{p-\gamma}u^{\gamma}(x).$

∎

##  4\. Non-degeneracy and porosity

In this section, we prove non-degeneracy and positive density results for
minimizers of (2.1), obtaining, as a consequence, the porosity of the free
boundary.

######  Lemma 4.1.

If $u$ is a minimizer of (2.1) and $x_{0}\in\overline{\\{u>0\\}}$, then there
exists a constant $c>0$, depending only on $n$, $p$ and $\gamma$, such that

$\sup_{\partial B_{r}(x_{0})}u\geq cr^{\frac{p}{p-\gamma}},$ (4.1)

for any $r<\mathrm{dist}(x_{0},\Omega)$.

###### Proof.

By continuity, it is enough to prove the result for $x_{0}\in\\{u>0\\}$. Set

$\lambda:=\frac{p-\gamma}{p-1}$

and define

$v(x):=u^{\lambda}(x)-c|x-x_{0}|^{\frac{p}{p-1}}\quad\textrm{in}\quad\\{u>0\\}\cap
B_{r}(x_{0}),$

for a constant $c>0$. A direct calculation shows that

$\Delta_{p}\left(u^{\lambda}\right)=\lambda^{p-1}u^{-\gamma}\left[(1-\gamma)|Du|^{p}+u\Delta_{p}u\right],$

which, recalling Proposition 2.1, leads to

$\Delta_{p}\left(u^{\lambda}\right)=\lambda^{p-1}u^{-\gamma}\left[(1-\gamma)|Du|^{p}+\gamma
u^{\gamma}\right]\geq\gamma\lambda^{p-1}>0.$

Moreover,

$\Delta_{p}(c|x-x_{0}|^{\frac{p}{p-1}})=c^{p-1}n\left(\frac{p}{p-1}\right)^{p}.$

Since $\Delta_{p}v$ is continuous with respect to $c$, we can choose $c>0$ so
small that $\Delta_{p}v\geq 0$ in $\\{u>0\\}\cap B_{r}(x_{0})$. As
$v(x_{0})>0$, the maximum principle implies the existence of
$y\in\partial\left(\\{u>0\\}\cap B_{r}(x_{0})\right)$ such that $v(y)>0$. But
$v\leq 0$ on $\partial\\{u>0\\}$, hence $y\in\partial B_{r}(x_{0})$, and
therefore $u(y)\geq cr^{\frac{p}{p-\gamma}}$. ∎

######  Remark 4.1.

Note that from 2.6, we already knew that in $B_{r}(x_{0})$, with
$x_{0}\in\partial\\{u>0\\}$, the supremum of a minimizer grows at most like
$r^{\frac{p}{p-\gamma}}$. Lemma 4.1 reveals that it grows exactly as
$r^{\frac{p}{p-\gamma}}$.

As a consequence of the lemma, we get the following positive density result.

######  Corollary 4.1.

If $u$ is a minimizer of (2.1), then, for every
$\Omega^{\prime}\subset\subset\Omega$, there exists a constant
$\delta\in(0,1)$, depending only on $\Omega^{\prime}$, $n$, $p$ and $\gamma$,
such that, for any small ball $B_{r}(x_{0})\subset\Omega^{\prime}$, with
$x_{0}\in\partial\\{u>0\\}$, one has

$\frac{|\\{u>0\\}\cap B_{r}(x_{0})|}{|B_{r}(x_{0})|}\geq\delta.$

###### Proof.

Indeed, Lemma 4.1 guarantees the existence of $y\in\partial B_{r/2}(x_{0})$
such that

$u(y)\geq cr^{\frac{p}{p-\gamma}}>0.$

Therefore, $u>0$ in $B_{\rho r}(y)\subset B_{r}(x_{0})$, for $\rho>0$ small.
In other words, $B_{\rho r}(y)\subset\\{u>0\\}\cap B_{r}(x_{0})$. Thus,

$\frac{|\\{u>0\\}\cap B_{r}(x_{0})|}{|B_{r}(x_{0})|}\geq\frac{|B_{\rho
r}(y)|}{|B_{r}(x_{0})|}=\rho^{n}=:\delta.$

∎

We recall the definition of a porous set to state another consequence of the
non-degeneracy estimate.

######  Definition 4.1.

A set $E\subset\mathbb{R}^{n}$ is called porous, with porosity constant
$\delta>0$, if there exists a constant $\rho>0$ such that, for each $x\in E$
and $r\in(0,\rho)$, there exists a $y\in\mathbb{R}^{n}$ such that

$B_{\delta r}(y)\subset B_{r}(x)\setminus E.$

The Hausdorff dimension of a porous set does not exceed $n-C\delta^{n}$, where
$C>0$ is a constant depending only on $n$ (see, for example, [25]). Hence, the
Lebesgue measure of a porous set is zero.

######  Corollary 4.2.

Let $u$ be a minimizer of (2.1). If $x_{0}\in\Omega$ and $r>0$ are such that
$B_{2r}(x_{0})\subset\Omega$, then the set

$E:=\partial\\{u>0\\}\cap\overline{B_{r}(x_{0})}$

is porous. Hence, the free boundary $\partial\\{u>0\\}$ is a set of Lebesgue
measure zero.

###### Proof.

Let $x\in E$. We have $B_{r/2}(x)\subset B_{2r}(x_{0})\subset\Omega$. From
Lemma 4.1, there exists $y\in\partial B_{r/2}(x)$ such that

$u(y)\geq cr^{\frac{p}{p-\gamma}},$

for a constant $c>0$ depending only on $n$, $p$ and $\gamma$. Hence,

$y\in B_{2r}(x_{0})\cap\\{u>0\\}.$

Set
$d(y):=\mathrm{dist}\left(y,\overline{B_{2r}(x_{0})}\setminus\\{u>0\\}\right)$,
then 2.6 provides

$u(y)\leq C\left[d(y)\right]^{\frac{p}{p-\gamma}},$

for a constant $C>0$ depending only on $n$, $p$ and $\gamma$. Therefore,
setting

$\delta:=\min\left\\{\frac{1}{2},\left[cC^{-1}\right]^{\frac{p-\gamma}{p}}\right\\}<1,$

we have

$d(y)\geq\delta r.$

Hence, $B_{\delta r}(y)\subset B_{d(y)}(y)\subset\\{u>0\\}$. In particular,

$B_{\delta r}(y)\cap B_{r}(x)\subset\\{u>0\\}.$

On the other hand, if $z\in[x,y]$ is such that $|z-y|=\delta r/2$, then

$B_{(\delta/2)r}(z)\subset B_{\delta r}(y)\cap B_{r}(x).$

Indeed, if $z^{*}\in B_{(\delta/2)r}(z)$, then

$|z^{*}-y|\leq|z^{*}-z|+|z-y|<\frac{\delta r}{2}+\frac{\delta r}{2}=\delta r$

and, since $|x-y|=|x-z|+|z-y|$,

$|z^{*}-x|\leq|z^{*}-z|+|x-y|-|z-y|<\frac{\delta r}{2}+r-\frac{\delta
r}{2}=r.$

Thus,

$B_{(\delta/2)r}(z)\subset B_{\delta r}(y)\cap B_{r}(x)\subset
B_{r}(x)\setminus\partial\\{u>0\\}\subset B_{r}(x)\setminus E,$

i.e., $E$ is porous with porosity constant $\delta/2$. ∎

##  5\. Removing the singularity

One of the main difficulties in analyzing equation (2.3) is the fact that the
right-hand side blows up across the free boundary $\partial\\{u>0\\}$.
However, Proposition 2.1 combined with Theorem 3.1 allow us to transform the
problem into one with a bounded right-hand side. That is the gist of this
section.

Let $u$ be a minimizer of (2.1) and set

$v(x):=u^{\theta}(x),\,x\in\Omega,$ (5.1)

where $\theta>1$ is defined by

$\theta:=\max\left\\{\frac{p-1}{p-2},\frac{p+1}{p-1}\right\\}=\left\\{\begin{array}[]{ccl}\frac{p-1}{p-2}&\mathrm{if}&2<p\leq
3\\\ \\\ \frac{p+1}{p-1}&\mathrm{if}&p\geq 3.\end{array}\right.$ (5.2)

Recalling Proposition 2.1, in $\\{u>0\\}=\\{v>0\\}$, one has

$\Delta_{p}v=f,$

where

$f:=\theta^{p-1}u^{(\theta-1)(p-1)-1}\left[(\theta-1)(p-1)|Du|^{p}+\gamma
u^{\gamma}\right].$ (5.3)

Observe that $(\theta-1)(p-1)-1\geq 0$, and therefore, in $\\{u>0\\}$, one has

$0<f\leq C,$ (5.4)

where the upper bound is a consequence of Theorem 3.1, with $C>0$ a constant
depending only on $n$, $p$, $\gamma$ and $\|u\|_{\infty}$. Thus, $v$ is a
(weak) solution of the obstacle problem

$\begin{cases}\Delta_{p}v=f\,\textrm{ in }\,\\{v>0\\},\\\ v\geq 0\,\textrm{ in
}\,\Omega,\\\ v-g^{\theta}\in W_{0}^{1,p}(\Omega),\end{cases}$ (5.5)

where $f$ is defined by (5.3).

######  Remark 5.1.

Note that the free boundaries for $v$ and $u$ coincide. Hence, the conclusions
of Corollary 4.1 and Corollary 4.2 are valid for $\partial\\{v>0\\}$.

######  Remark 5.2.

Although the obstacle problem (5.5) has a unique solution (see, for example,
[9, 10, 26]), every minimizer $u$ of (2.1) generates a right-hand side $f$
defined by (5.3). Hence, there are as many $v$ solutions as minimizers of
(2.1).

Since $v$ is the weak solution of (5.5), then we have

$f\chi_{\\{v>0\\}}\leq\Delta_{p}v\leq f,$

almost everywhere in $\Omega$ (cf. [9, Proposition 1.2]). Therefore, as a
direct consequence of the porosity of the free boundary, we obtain

$\Delta_{p}v=f\,\textrm{ a.e. in }\Omega.$

Moreover, $v\in W^{2,2}_{\mathrm{loc}}(\Omega)$ thanks to (5.4) (see [19,
29]), and, by (5.1), one has

$|D^{2}u|\leq\theta^{2}\left(\frac{|Du|^{2}}{u}+u^{1-\theta}|D^{2}v|\right),$
(5.6)

in $\\{u>0\\}$.

##  6\. Hausdorff measure estimates

From Corollary 4.2, we already know that the $n-$dimensional Lebesgue measure
of the free boundary is zero. In this section, we show that the
$(n-1)-$dimensional Hausdorff measure of the free boundary is finite. The case
$p=2$ is treated in [28], but the nonlinear setting is considerably more
demanding.

Recall that the $s-$dimensional Hausdorff measure of a set $E$ is defined by

$\mathcal{H}^{s}(E):=\lim_{\delta\to 0}\mathcal{H}_{\delta}^{s}(E),$

where

$\mathcal{H}_{\delta}^{s}(E):=\inf\left\\{\sum_{j=1}^{\infty}\mu(s)\left(\frac{\mathrm{diam}\,E_{j}}{2}\right)^{s}\right\\}.$

Here, the infimum is taken over all countable coverings $\\{E_{j}\\}$ of $E$
such that $\mathrm{diam}\,E_{j}\leq\delta$,
$\mu(s):=\frac{\pi^{s/2}}{\Gamma\left(\frac{s}{2}+1\right)}$ and
$\Gamma(s):=\int_{0}^{\infty}e^{-t}t^{s-1}\,dt$, for $s>0$, is the usual Gamma
function.

Since the free boundaries of $u$ and $v$ coincide, we study the free boundary
regularity for $v$, as the latter solves (5.5) with a bounded right-hand side
$f$ defined by (5.3).

######  Remark 6.1.

A Hausdorff measure estimate for problem (5.5) is obtained in [21, Theorem
3.3], provided $f>0$ is Lipschitz. As observed in [10, Section 4] (see also
[9]), the Lipschitz continuity assumption can be relaxed, assuming instead
that $Df$ is locally in a Morrey space. More precisely, it can be replaced in
our context by the assumption that

$\int_{B_{r}}|Df|\,dx\leq Cr^{n-1},\quad r\in(0,1).$ (6.1)

######  Proposition 6.1.

The function $f$ defined by (5.3) satisfies (6.1).

###### Proof.

Recalling (5.2) and (5.6), a direct calculation and the use of Young’s
inequality reveal

$\displaystyle|Df|$ $\displaystyle\leq$ $\displaystyle
Cu^{(\theta-1)(p-1)-2}|Du|^{p+1}$
$\displaystyle+C^{\prime}u^{(\theta-1)(p-1)-2+\gamma}|Du|$
$\displaystyle+C^{\prime\prime}u^{(\theta-1)(p-1)-\theta}\left(|Du|^{2(p-1)}+|D^{2}v|^{2}\right)$
$\displaystyle\leq$ $\displaystyle C\left(1+|D^{2}v|^{2}\right),$

where in the last inequality we used the fact that $(\theta-1)(p-1)-2\geq 0$,
$(\theta-1)(p-1)-\theta\geq 0$ and that $u$ and its gradient are bounded. The
constant $C>0$ depends only on $n$, $p$ and $\gamma$.

Then, we obtain, for $r\in(0,1)$,

$\displaystyle\int_{B_{r}}|Df|\,dx$ $\displaystyle\leq$ $\displaystyle
Cr^{n}+C\int_{B_{r}}|D^{2}v|^{2}\,dx$ (6.2) $\displaystyle\leq$ $\displaystyle
Cr^{n}+Cr^{n}\|D^{2}v\|^{2}_{L^{2}(B_{r})}$ $\displaystyle\leq$ $\displaystyle
Cr^{n},$

since $v\in W^{2,2}_{\mathrm{loc}}(\Omega)$. The result follows because
$r^{n}\leq r^{n-1}$. ∎

We next define a class of functions on the unit ball.

######  Definition 6.1.

We say that $v\in\mathcal{F}$ if, in $B_{1}$, it satisfies
$\Delta_{p}v=f\chi_{\\{v>0\\}}$, $0\leq v\leq 1$ and $0\in\partial\\{v>0\\}$.

Since $f$ is continuous and positive, one has the following estimates, deduced
in [10, Lemma 4.3], [21, Lemma 2.4] and [21, Lemma 2.5].

######  Lemma 6.1.

If $v\in\mathcal{F}$ and $x_{0}\in\partial\\{v>0\\}\cap B_{1/2}$, then there
exist positive constants $M=M(p,n,\gamma)$ and $0<\sigma=\sigma(f)$ such that

$\fint_{B_{r}(x_{0})}\left[|Dv(x)|^{p-2}|D^{2}v(x)|\right]^{2}\,dx\leq
M,\quad\forall r\leq 1/2$

and

$\frac{\sigma}{(p-1)^{2}}\leq\left[|Dv(x)|^{p-2}|D^{2}v(x)|\right]^{2},\
\textrm{ a.e. in }\,\\{v>0\\}.$

We now define the sets

$Q_{\varepsilon}:=\left\\{|Dv|\leq\varepsilon^{\frac{1}{p-1}}\right\\}\quad\textrm{
and }\quad
Q_{\varepsilon}^{i}:=\left\\{|v_{x_{i}}|\leq\varepsilon^{\frac{1}{p-1}}\right\\}.$

The next result is the main step towards proving the finiteness of the
$(n-1)-$dimensional Hausdorff measure of the free boundary. Its proof is
essentially the same as in [21, Theorem 3.3] (see also [10, Section 4]). We
sketch it here for the reader’s convenience.

######  Lemma 6.2.

If $v\in\mathcal{F}$ and $x_{0}\in\partial\\{v>0\\}\cap B_{1/2}$, then

$|Q_{\varepsilon}\cap B_{r}(x_{0})\cap\\{v>0\\}|\leq C\varepsilon
r^{n-1},\quad\forall r<1/4,$

where $C>0$ is a constant depending only on $n$, $p$ and $\gamma$.

###### Proof.

Note that it is enough to prove that

$\int_{0}^{1}\left|Q_{\varepsilon}\cap
B_{rs}(x_{0})\cap\\{v>0\\}\right|\,ds\leq C\varepsilon r^{n},$ (6.3)

for any $B_{r}(x_{0})\subset B_{1}$, $r<1/2$. Indeed, if (6.3) holds, then the
lemma follows, since otherwise there exists $B_{r}(x_{0})$ such that

$|Q_{\varepsilon}\cap B_{r}(x_{0})\cap\\{v>0\\}|\geq C_{0}\varepsilon
r^{n-1},$

with $C_{0}>0$ arbitrarily large, and we would have

$\displaystyle C_{0}\varepsilon r^{n-1}$ $\displaystyle\leq$
$\displaystyle|Q_{\varepsilon}\cap B_{r}(x_{0})\cap\\{v>0\\}|$
$\displaystyle\leq$ $\displaystyle\int_{0}^{1}|Q_{\varepsilon}\cap
B_{rs}(x_{0})\cap\\{v>0\\}|\,ds$ $\displaystyle\leq$ $\displaystyle
C\varepsilon r^{n},$

which leads to a contradiction for $C_{0}$ large enough. Thus, it remains to
prove (6.3). The idea is to differentiate the PDE satisfied by $v$ on its
positivity set, multiply the outcome by the truncated function

$G(\eta):=\left\\{\begin{array}[]{ccl}\varepsilon&\mathrm{if}&\eta>\varepsilon^{\frac{1}{p-1}}\\\
|\eta|^{p-1}\mathrm{sign}(\eta)&\mathrm{if}&|\eta|\leq\varepsilon^{\frac{1}{p-1}}\\\
-\varepsilon&\mathrm{if}&\eta<-\varepsilon^{\frac{1}{p-1}}\end{array}\right.$

and integrate over $B_{rs}(x_{0})$.

From (5.5), in $\\{v>0\\}$ we have

$D_{e}\Delta_{p}v=D_{e}f,$

where $e$ is any unit vector and $D_{e}$ is the directional derivative. Thus,
putting $v_{e}:=D_{e}v$,

$\mathrm{div}\left(|Dv|^{p-2}Dv_{e}+(p-2)|Dv|^{p-4}DvDv\cdot
Dv_{e}\right)=D_{e}f,$

in the weak sense. Multiplication by $G(v_{e})$ and integration over
$B_{rs}(x_{0})$, $s\in(0,1)$, leads to

$\begin{split}&\int_{B_{rs}(x_{0})}\left(|Dv|^{p-2}Dv_{e}+(p-2)|Dv|^{p-4}DvDv\cdot
Dv_{e}\right)\cdot DG(v_{e})\,dx\\\ &=\int_{\partial
B_{rs}(x_{0})}\left(|Dv|^{p-2}D_{\nu}v_{e}+(p-2)|Dv|^{p-4}D_{\nu}vDv\cdot
Dv_{e}\right)G(v_{e})\,dx\\\
&\quad-\int_{B_{rs}(x_{0})}D_{e}fG(v_{e})\,dx,\end{split}$ (6.4)

where $D_{\nu}$ denotes the outward normal derivative.

Integrating the first term of the right-hand side and using Hölder’s
inequality combined with the first estimate in Lemma 6.1, we obtain

$\int_{0}^{1}\int_{\partial
B_{rs}(x_{0})}\left(|Dv|^{p-2}D_{\nu}v_{e}+(p-2)|Dv|^{p-4}D_{\nu}vDv\cdot
Dv_{e}\right)G(v_{e})\,dx\,ds$
$\begin{split}&\leq(p-1)\int_{B_{r}(x_{0})}|Dv|^{p-2}|D^{2}v||G(v_{e})|\,dx\\\
&\leq(p-1)\varepsilon
r^{n/2}\left(\int_{B_{r}(x_{0})}\left[|Dv|^{p-2}|D^{2}v|\right]^{2}\right)^{\frac{1}{2}}\\\
&\leq C\varepsilon r^{n}.\end{split}$ (6.5)

To estimate the second term of the right-hand side, we use (6.2), obtaining

$\int_{B_{rs}(x_{0})}D_{e}fG(v_{e})\,dx\leq\int_{B_{r}(x_{0})}|Df||G(v_{e})|\,dx\leq
C\varepsilon r^{n}.$ (6.6)

The constants $C>0$ in (6.5) and (6.6) depend only on $n$, $p$ and $\gamma$.

Now, to estimate below the left-hand side of (6.4), we take $e=e_{i}$, where
$\\{e_{i}\\}_{i=1}^{n}$ is the standard basis in $\mathbb{R}^{n}$. Note that

$G^{\prime}(\eta)=(p-1)|\eta|^{p-2}\chi_{\left\\{|\eta|<\varepsilon^{\frac{1}{p-1}}\right\\}}$

and $Q_{\varepsilon}\subset Q_{\varepsilon}^{i}$. Recalling the second
estimate in Lemma 6.1, we obtain

$\begin{split}&\sum_{i=1}^{n}\left[\int_{B_{rs}(x_{0})}|Dv|^{p-2}Dv_{x_{i}}+(p-2)|Dv|^{p-4}DvDv\cdot
Dv_{x_{i}}\right]\cdot DG(v_{x_{i}})\\\
&\geq(p-1)\int_{Q_{\varepsilon}^{i}\cap
B_{rs}(x_{0})\cap\\{v>0\\}}|Dv|^{2(p-2)}|D^{2}v|^{2}\\\
&\quad+(p-1)(p-2)\int_{Q_{\varepsilon}^{i}\cap
B_{rs}(x_{0})\cap\\{v>0\\}}|Dv|^{2(p-2)}\sum_{i=1}^{n}\left(\frac{Dv}{|Dv|}\cdot
Dv_{x_{i}}\right)^{2}\\\ &\geq(p-1)\int_{Q_{\varepsilon}^{i}\cap
B_{rs}(x_{0})\cap\\{v>0\\}}\left[|Dv|^{p-2}|D^{2}v|\right]^{2}\\\
&\geq\sigma\frac{|Q_{\varepsilon}^{i}\cap B_{rs}(x_{0})\cap\\{v>0\\}|}{p-1}\\\
&\geq\sigma\frac{|Q_{\varepsilon}\cap
B_{rs}(x_{0})\cap\\{v>0\\}|}{p-1}.\end{split}$

Combining the last inequality with (6.4), (6.5), and (6.6), we obtain (6.3). ∎

We are now ready to prove the main result of this section.

######  Theorem 6.1.

If $u$ is a minimizer of (2.1), and $x_{0}\in\partial\\{u>0\\}\cap B_{1/2}$,
then

$\mathcal{H}^{n-1}(\partial\\{u>0\\}\cap B_{r}(x_{0}))<Cr^{n-1},$

for any $r\in(0,1/4)$, where the constant $C>0$ depends only on $n$, $p$ and
$\gamma$.

###### Proof.

It is enough to prove the result for the solution $v$ of the obstacle problem
(5.5), since $\partial\\{v>0\\}=\partial\\{u>0\\}$.

Recalling Besicovitch’s covering lemma, let
$\\{B_{\varepsilon}(x_{i})\\}_{i\in I}$ be a finite covering of
$\partial\\{v>0\\}\cap B_{r}(x_{0})$, with $x_{i}$ at the free boundary and at
most $L=L(n)$ overlapping at each point. From the proof of Corollary 4.2,
there exists a constant $c$ such that, for any $i\in I$, we can find $y_{i}\in
B_{\varepsilon}(x_{i})$ for which

$B_{c\varepsilon}(y_{i})\subset Q_{\varepsilon}\cap
B_{\varepsilon}(x_{i})\cap\\{v>0\\}.$

Using Lemma 6.2, we deduce

$\displaystyle|B_{1}|c^{n}\varepsilon^{n}\\#I$ $\displaystyle=$
$\displaystyle\sum_{i\in I}|B_{c\varepsilon}(y_{i})|$ $\displaystyle\leq$
$\displaystyle\sum_{i\in I}|Q_{\varepsilon}\cap
B_{\varepsilon}(x_{i})\cap\\{v>0\\}|$ $\displaystyle\leq$ $\displaystyle
L(n)|Q_{\varepsilon}\cap B_{\varepsilon}(x_{0})\cap\\{v>0\\}|$
$\displaystyle\leq$ $\displaystyle L(n)C\varepsilon r^{n-1}.$

Thus,

$\mathcal{H}_{\delta}^{n-1}(\partial\\{v>0\\}\cap B_{r}(x_{0}))\leq Cr^{n-1},$

and letting $\delta\to 0$, we arrive at

$\mathcal{H}^{n-1}(\partial\\{u>0\\}\cap B_{r}(x_{0}))\leq Cr^{n-1}.$

∎

We conclude with two remarks.

######  Remark 6.2.

Since the free boundary has locally finite $\mathcal{H}^{n-1}-$measure, the
set $\\{u>0\\}$ has locally finite perimeter in $\Omega$. Thus,
$D(\chi_{\\{u>0\\}})$ is, in the sense of distributions, a vector-valued Borel
measure supported on the free boundary. Moreover, its total variation is a
Radon measure (see [13]).

######  Remark 6.3.

Up to a negligible set of null perimeter, the free boundary is a union of, at
most, a countable family of $C^{1}$ hypersurfaces (see [15]).

Acknowledgments. DJA is partially supported by Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq) grant 311138/2019-5 and
Paraíba State Research Foundation (FAPESQ) grant 2019/0014. RT is partially
supported by FCT, DOI 10.54499/2022.02357.CEECIND/CP1714/CT0001. RT and JMU
are partially supported by the King Abdullah University of Science and
Technology (KAUST) and by the Centre for Mathematics of the University of
Coimbra (CMUC, funded by the Portuguese Government through FCT/MCTES, DOI
10.54499/UIDB/00324/2020).

## References

  * [1] H.W. Alt and L. Caffarelli, Existence and regularity for a minimum problem with a free boundary, J. Reine Angew. Math. 325 (1981), 105–144.
  * [2] H.W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63–107.
  * [3] J. Andersson, E. Lindgren and H. Shahgholian, Optimal regularity for the obstacle problem for the $p-$Laplacian, J. Differential Equations 259 (2015), 2167–2179.
  * [4] D.J. Araújo, E. Teixeira and J.M. Urbano, A proof of the $C^{p^{\prime}}-$regularity conjecture in the plane, Adv. Math. 316 (2017), 541–553.
  * [5] D.J. Araújo, R. Teymurazyan and V. Voskanyan, Sharp regularity for singular obstacle problems, Math. Ann. 387 (2023), 1367–1401.
  * [6] L. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427–448.
  * [7] L. Caffarelli, A remark on the Hausdorff measure of a free boundary, and the convergence of the coincidence sets, Boll. Un. Mat. Ital. A(5) 18 (1981), 109–113.
  * [8] L. Caffarelli and D. Kinderlehrer, Potential methods in variational inequalities, J. Anal. Math. 37 (1980), 285–295.
  * [9] S. Challal, A. Lyaghfouri and J.F. Rodrigues, On the $A-$obstacle problem and the Hausdorff measure of its free boundary, Ann. Mat. Pura Appl. 191 (2012), 113–165.
  * [10] S. Challal, A. Lyaghfouri, J.F. Rodrigues and R. Teymurazyan, On the regularity of the free boundary for quasilinear obstacle problems, Interfaces Free Bound. 16 (2014), 359–394.
  * [11] D. Danielli and A. Petrosyan, A minimum problem with free boundary for a degenerate quasilinear operator, Calc. Var. Partial Differential Equations 23 (2005), 97–124.
  * [12] D.Danielli, A.Petrosyan and H.Shahgholian, A singular perturbation problem for the p-Laplace operator, Indiana Univ. Math. J. 52 (2003), 457–476.
  * [13] L.C. Evans and R.L. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, 2015. xiv+299 pp.
  * [14] A. Figalli, B. Krummel and X. Ros-Oton, On the regularity of the free boundary in the $p-$Laplacian obstacle problem, J. Differential Equations 263 (2017), 1931–1945.
  * [15] E. Giusti, Minimal surfaces and functions of bounded variations, Birkhäuser Verlag, Basel, 1984. xii+240 pp.
  * [16] T. Iwaniec and J.J. Manfredi, Regularity of $p-$harmonic functions on the plane, Rev. Mat. Iberoamericana 5 (1989), 1–19.
  * [17] L. Karp, T. Kilpeläinen, A. Petrosyan and H. Shahgholian, On the porosity of free boundaries in degenerate variational inequalities, J. Differential Equations 164 (2000), 110–117.
  * [18] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. xx+313 pp.
  * [19] O.A. Ladyzhenskaya and N.N. Uraltseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. xviii+495 pp.
  * [20] C. Lederman and D. Oelz, A quasilinear parabolic singular perturbation problem, Interfaces Free Bound. 10 (2008), 447–482.
  * [21] K. Lee and H. Shahgholian, Hausdorff measure and stability for the $p-$obstacle problem $(2<p<\infty)$, J. Differential Equations 195 (2003), 14–24.
  * [22] R. Leitão, O. de Queiroz and E. Teixeira, Regularity for degenerate two-phase free boundary problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), 741–762.
  * [23] E. Lindgren and A. Petrosyan, Regularity of the free boundary in a two-phase semilinear problem in two dimensions, Indiana Univ. Math. J. 57 (2008), 3397–3417.
  * [24] P. Lindqvist, Notes on the stationary $p-$Laplace equation, SpringerBriefs Math., Springer, Cham, 2019. xi+104 pp.
  * [25] O. Martio and M. Vuorinen, Whitney cubes, $p-$capacity, and Minkowski content, Exposition. Math. 5 (1987), 17–40.
  * [26] A. Petrosyan, H. Shahgholian and N. Uraltseva, Regularity of free boundaries in obstacle-type problems, Grad. Stud. Math. 136, American Mathematical Society, Providence, RI, 2012. x+221 pp.
  * [27] D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J. 32 (1983), 1–17.
  * [28] D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Partial Differential Equations 8 (1983), 1409–1454.
  * [29] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.