# The coherent cross section of vector mesons in ultraperipheral PbPb
collisions at the LHC

Ya-ping Xie<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  Department of Physics, Lanzhou
University, Lanzhou 730000, Chnia  Key Laboratory of Quark and Lepton Physics
(MOE) and Institute of Particle Physics, Central China Normal University,
Wuhan 430079, China  Xurong Chen<EMAIL_ADDRESS>Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China

###### Abstract

The coherent cross section of $J/\psi$, $\rho$, $\phi$ are computed in dipole
model in the ultraperipheral PbPb collisions, the IP-Sat and IIM model are
applied in the calculation of the differential cross section of the dipole
scattering off the nucleon, three kinds of forward vector meson wave functions
are used in the overlap. The prediction of $J/\psi$ and $\rho$ is compared
with the experimental data of the ALICE Collaboration, and the prediction of
$\phi$ is also given in this paper.

###### pacs:

24.85.+p, 12.38.Bx, 12.39.St, 13.88.+e

##  I Introduction

The production of vector mesons through a virtual photon-hadron scattering had
been studied in Deep Inelastic Scattering at HERA Ivanov:2004ax , the
ultraperipheral collisions at the LHC offer an interesting way to study the
photonproduction of vector mesons through a real photon scattering off a
hadron at the high energy limit Bertulani:2005ru ; Baltz:2007kq , recently,
the ALICE Collaboration have measured vector mesons production in PbPb
ultraperipheral collisions Aaij:2013jxj ; Abbas:2013oua ; TheALICE:2014dwa ;
Adam:2015gsa ; Abelev:2012ba . On the theoretical front, the photon production
of vector mesons have been studied through various approaches, including
perturbative quantum chromodynamics(pQCD), $k_{T}$-factorization and color
dipole model(CDM) Klein:1999qj ; Frankfurt:2002sv ; Goncalves:2005yr ;
Rebyakova:2011vf ; Adeluyi:2012ph ; Cisek:2012yt ; Ducati:2013bya ;
Lappi:2013am ; Guzey:2013qza , in this work, we use the dipole model to
predict the vector mesons production in PbPb ultraperipheral collisions at
$\sqrt{s_{NN}}$=2.76TeV, the goal of this work is to update the prediction of
vector mesons with new fit of IP-Sat model and several wave function models.
According to the dipole model, the process of the photon-hadron scattering can
be viewed as three steps, at the first step, the virtual or real photon splits
into a dipole with quark and antiquark, at the second step, the dipole
scatters off the hadrons, at the last step, the dipole becomes a vector meson.
The amplitude of the photon-hadrons scattering contains three portions, the
light-cone wave function of the photon splitting into dipole, the cross
section of the dipole scattering off the proton, and the forward wave function
for vector mesons. The calculations of cross section of the dipole scattering
off a proton is relative to the gluon distribution in the small-x region, in
the literature, various parameterization models have successfully implement to
calculate the cross section of the dipole scatters off a proton, such GBW
model GolecBiernat:1998js ; GolecBiernat:1999qd , IP-Sat model Bartels:2002cj
; Kowalski:2003hm ; Kowalski:2006hc ; Rezaeian:2012ji  and IIM model
Iancu:2003ge ; Soyez:2007kg ; Watt:2007nr ; Rezaeian:2013tka . The light-cone
wave function of the photon splitting into the dipole can be calculated in
QED, but the forward wave function of the vector meson can not be calculated
analytically, the forward wave functions of vector meson almost are modeled
the light-cone wave function of the photon, the models of the vector mesons
include the Gaus-LC Kowalski:2006hc , DGKP Dosch:1996ss ; Goncalves:2004bp ,
Boosted Gaussian Nemchik:1996cw ; Forshaw:2003ki  and so on.
This paper is organized as follow, the brief review of the dipole model and
wave function will be presented in Sec II, the numerical results and some
discussion will be presented in Sec III.

##  II The coherent vector meson cross section

###  II.1 Formulas in the ultraperipheral collision

This work, we consider the coherent cross section of the vector meson in the
PbPb ultraperipheral collisions, in hadronic collisions, when the impact
parameter is larger, the two hadrons almost don’t touch each other, but the
real photon can be emitted from the hadrons in the high energy limit,
therefore, the real photon can scatter off the hadrons, in this process, the
rapidity distribution can be factorized into the equivalent photon flux and
the cross section of the photon-hadrons scattering, the formula is

$\displaystyle\frac{d\sigma^{h_{1}h_{2}}}{dy}=\bigg{[}n^{h_{1}}(\omega)\sigma^{\gamma
h_{2}}(\omega)\bigg{]}_{\omega_{left}}+\bigg{[}n^{h_{2}}(\omega)\sigma^{\gamma
h_{1}}(\omega)\bigg{]}_{\omega_{right}},$ (1)

where $y$ is the rapidity of the vector meson, the $\sigma^{\gamma A}(\omega)$
is the cross section of the photon-hadrons scattering, the $n(\omega)$ is the
equivalent photon flux in the hadrons, with
$\omega_{left}=\frac{M_{v}}{2}\exp(-y)$, and
$\omega_{right}=\frac{M_{v}}{2}\exp(y)$, whtere $M_{V}$ is the mass the vector
meson. In proton-proton scattering, the equivalent photon flux is Drees:1988pp

$\displaystyle
n(\omega)=\frac{\alpha_{em}}{2\pi}\bigg{[}1+\bigg{(}1-\frac{2\omega}{\sqrt{s_{NN}}}\bigg{)}\bigg{]}\bigg{[}\ln
D-\frac{11}{6}+\frac{3}{D}-\frac{3}{2D^{2}}+\frac{1}{3D^{3}}\bigg{]},$ (2)

where $\sqrt{s_{NN}}$ is the nucleon-nucleon center energy,
$D=1+\frac{0.71\mathrm{GeV}^{2}}{Q^{2}_{min}}$, with
$Q^{2}_{min}=\omega^{2}/\gamma_{L}^{2}$, $\gamma_{L}$ is the lorentz factor,
with $\gamma_{L}=\sqrt{s_{NN}}/2m_{p}$. In the nucleus-nucleus scattering, the
equivalent photon flux is Klein:1999qj

$\displaystyle n(\omega)=\frac{2Z^{2}\alpha_{em}}{\pi}\big{[}\xi
K_{1}(\xi)K_{0}(\xi)-\frac{\xi^{2}}{2}[K_{1}^{2}(\xi)-K_{0}^{2}(\xi)]\big{]},$
(3)

where $\xi=2\omega R_{A}/\gamma_{L}$, with $R_{A}$ is the radius of the
nucleus, $K_{0}(x)$ and $K_{1}(x)$ are the second kind of Bessel functions.
The $\sigma(\omega)$ is the cross section of the photon-hadrons scattering, it
can be integrated from the differential cross section, the coherent
differential cross section of the photon-proton scattering is calculated as
Kowalski:2003hm ; Kowalski:2006hc

$\displaystyle\frac{d\sigma^{\gamma p\to
Vp}}{dt}=\frac{R_{g}^{2}(1+\beta^{2})}{16\pi}\left|\mathcal{A}^{\gamma p\to
Vp}(x_{p},Q^{2},\Delta)\right|^{2},$ (4)

where the amplitude is computed as

$\displaystyle\mathcal{A}^{\gamma p\to Vp}(x_{A},Q^{2},\Delta)=i\int
d^{2}r\int_{0}^{1}\frac{dz}{4\pi}\int
d^{2}b(\Psi_{V}^{*}\Psi_{\gamma})_{T}(z,r,Q^{2})e^{-i(b-(1-z)r)\cdot\Delta}\frac{d\sigma_{q\bar{q}}}{d^{2}b}.$
(5)

Where $t=-\Delta^{2}$, the relationship between the $x_{p}$ amd $y$ is
$x_{p}=M_{v}\exp(-y)/\sqrt{s_{NN}}$ and $T$denotes the transverse overlap of
the wave functions of the photon and vector meson, the $\mathcal{N}(x,r,b)$ is
the amplitude of the dipole scattering off the nucleon, which will be
considered in the next subsection. The factor $\beta$ is the ratio of the real
part to the imaginary part of amplitude, it is computed as

$\beta=\tan(\frac{\pi}{2}\delta),$ (6)

where $\delta$ is calculated as

$\delta=\frac{\partial\ln(\mathrm{Im}\mathcal{A}(x))}{\partial\ln 1/x}.$ (7)

The factor $R_{g}^{2}$ reflects the skewdness, it gives Shuvaev:1999ce

$R_{g}=\frac{2^{2\delta+3}}{\sqrt{\pi}}\frac{\Gamma(\delta+5/2)}{\Gamma(\delta+4)}.$
(8)

The differential cross section of $\gamma A\to VA$ is written as

$\displaystyle\frac{d\sigma^{\gamma A\to
VA}}{dt}=\frac{R_{g}^{2}(1+\beta^{2})}{16\pi}\left|\langle\mathcal{A}^{\gamma
A\to VA}(x_{p},Q^{2},\Delta)\rangle_{N}\right|^{2},$ (9)

where the average amplitude is calculated as Kowalski:2008sa ;
Caldwell:2010zza ; Lappi:2010dd

$\displaystyle\langle\mathcal{A}^{\gamma A\to
VA}(x_{p},Q^{2},\Delta)\rangle_{N}$ $\displaystyle=$ $\displaystyle i\int
d^{2}r\int_{0}^{1}\frac{dz}{4\pi}\int
d^{2}b(\Psi_{V}^{*}\Psi_{\gamma})_{T}(z,r,Q^{2})e^{-i(b-(1-z)r)\cdot\Delta}$
(10) $\displaystyle\times 2(1-\exp(-2\pi B_{p}AT_{A}(b)\mathcal{N}(x_{p},r)).$

The shape function is defined as

$\displaystyle
T_{A}(b)=\int_{-\infty}^{\infty}dz\rho_{A}(\sqrt{b^{2}+z^{2}}),$ (11)

with Wood-Saxon distribution

$\rho_{A}(r)=\frac{N}{\exp(\frac{r-R_{A}}{\delta_{0}})+1},$ (12)

where $\delta_{0}=0.54$fm,
$R_{A}=(1,12\mathrm{fm})A^{1/3}-(0,86\mathrm{fm})A^{-1/3}$, A is the number of
nucleus.

###  II.2 The IP-Sat and IIM model

There are various approaches to calculate the cross section of dipole
scattering off the proton, GBW model was proposed by Golec-Biernat and
Wüsthoff GolecBiernat:1998js ; GolecBiernat:1999qd , but the GBW model has a
shortcoming that it do not match DGLAP evolution equation at large $Q^{2}$.
Then, the Impact Parameter saturation (IP-Sat) model was proposed according to
the DGLAP evolution equation, the amplitude of the IP-Sat model reads
Kowalski:2008sa ; Caldwell:2010zza ; Lappi:2010dd

$\frac{d\sigma_{q\bar{q}}}{d^{2}b}=2[1-\exp(-\frac{1}{2\pi
B_{p}}\frac{\pi^{2}}{2N_{c}}r^{2}\alpha_{s}(\mu^{2})xg(x,\mu^{2})T_{p}(b)],$
(13)

where the $T_{p}(b)$ is defined

$\displaystyle T_{p}(b)=\exp(-\frac{b^{2}}{2B_{p}}).$ (14)

The scale $\mu^{2}$ has relationship with the dipole size $r$,

$\mu^{2}=\mu^{2}_{0}+\frac{C}{r^{2}},$ (15)

with $C=4$, where the $xg(x,\mu^{2})$ is the gluon distribution in the proton,
the initial gluon distribution is

$xg(x,\mu^{2}_{0})=A_{g}x^{-\lambda_{g}}(1-x)^{5.6}.$ (16)

The parameters $A_{g}$, $\mu^{2}_{0}$, $\lambda_{g}$, $B_{p}$ are determined
from the fit to experimental data $F_{2}$, we take the values of parameters
according to Ref. Rezaeian:2012ji , There are two sets of the parameters,
which are different from the Ref. Kowalski:2003hm ; Kowalski:2006hc ,
especially for the mass of the light quarks, in the fit of Kowalski et al, the
mass of the light quark is $m_{q}$=0.14 GeV, in the fit of Rezaeian et al, the
mass of the light quarks is $m_{q}\approx$0 GeV.

|  $B_{p}$ |  $m_{u,d,s}$ |  $m_{c}$ |  $\mu_{0}^{2}$ |  $A_{g}$ |  $\lambda_{g}$
---|---|---|---|---|---|---
Para 1 |  $4.0$ $\mathrm{GeV}^{2}$ |  $\approx 0$ GeV |  1.27 GeV |  1.51 GeV2 |  2.308 |  0.058
Para 2 |  $4.0$ $\mathrm{GeV}^{2}$ |  $\approx 0$ GeV |  1.4 GeV |  1.428 GeV2 |  2.373 |  0.052
Table 1: The parameters of IP-Sat model Rezaeian:2012ji

In Ref. Bartels:2002cj ; Lappi:2010dd , the authors used a factorized impact
parameter saturation model, it reads

$\displaystyle\frac{d\sigma_{q\bar{q}}}{d^{2}b}$ $\displaystyle\approx$
$\displaystyle 2T_{p}(b)\mathcal{N}(x,r)$ (17) $\displaystyle=$ $\displaystyle
2T_{p}(b)[1-\exp(-\frac{1}{2\pi
B_{p}}\frac{\pi^{2}}{2N_{c}}r^{2}\alpha_{s}(\mu^{2})xg(x,\mu^{2})].$

we use fIP-Sat model to calculate the vector meson cross section in photon-
nucleus scattering.
On the other side, Iancu, Itakura and Munier proposed a saturation model based
on the solution to BK evolution equation Iancu:2003ge , we use the impact
parameter dependent saturation model, we take the form as the same as Refs.
Soyez:2007kg ; Lappi:2010dd

$\displaystyle\frac{d\sigma_{q\bar{q}}}{d^{2}b}=2T_{b}(b)\mathcal{N}(x,r),$
(18)

the amplitude is written as

$\displaystyle\mathcal{N}(x,r)=\begin{cases}\mathcal{N}_{0}(\frac{rQ_{s}}{2})^{2(\gamma_{s}+(1/\kappa\lambda
Y)\ln(2/rQ_{s}))},\quad rQ_{s}\leq 2,\\\
1-\exp\big{(}-a\ln^{2}(brQs)\big{)},\quad\quad\quad\\!\\!\\!rQ_{s}>2.\end{cases}$
(19)

with $Y=\ln(1/x)$ and $\kappa=9.9$, where $Qs(x,b)=(x_{0}/x)^{\lambda/2}$ GeV,
$a$ and $b$ are

$\begin{split}&a=-\frac{\mathcal{N}^{2}_{0}\gamma_{s}^{2}}{(1-\mathcal{N}_{0})^{2}\ln(1-\mathcal{N}_{0})},\\\
&b=\frac{1}{2}(1-\mathcal{N}_{0})^{-(1-\mathcal{N}_{0})/(2\mathcal{N}_{0}\gamma_{s})}.\end{split}$
(20)

The parameters $B_{p},\mathcal{N}_{0},\gamma_{c},\lambda,x_{0}$ are need to be
determined from the fit to the experimental data $F_{2}$, we take the
parameter as the same as Ref. Soyez:2007kg , they are presented in the
following Table:

$B_{p}$ |  $m_{u,d,s}$ |  $m_{c}$ |  $m_{b}$ |  $\mathcal{N}_{0}$ |  $\gamma_{c}$ |  $\lambda$ |  $x_{0}$
---|---|---|---|---|---|---|---
$5.59$ GeV-2 |  $0.14$ GeV |  1.4 GeV |  4.5 GeV |  0.7 |  0.7376 |  0.2197 |  $1.632\times 10^{-4}$
Table 2: The parameters of the IIM model Soyez:2007kg .

###  II.3 The forward vector meson wave functions

The $(\Psi_{V}^{*}\Psi_{\gamma})_{T}(r,z)$ is the transverse overlap of the
functions of vector meson and the photon, there are various models for the
forward vector meson wave function in the literature, in this work, we take
the three kinds of model for the vector meson wave functions. At first, we
consider the Boosted Gaussian and Gaus-LC model, the overlap takes following
form in Boosted Gaussian and Gaus-LC model Kowalski:2006hc ,

$\displaystyle(\Psi_{V}^{*}\Psi_{\gamma})_{T}(r,z)=e_{f}e\frac{N_{c}}{\pi
z(1-z)}\\{m_{f}^{2}K_{0}(\epsilon r)\phi_{T}(r,z)-(z^{2}+(1-z)^{2})\epsilon
K_{1}(\epsilon r)\partial_{r}\phi_{T}(r,z)\\},$

where $e=\sqrt{4\pi\alpha_{em}}$, $m_{f}$ is the mass of quarks, $e_{f}$ is
the electric charge of the quarks, $\epsilon=\sqrt{z(1-z)Q^{2}+m_{f}^{2}}$,
$N_{c}$ is the number of the colors. The scalar function $\phi_{T}(r,z)$ of
Gaus-LC model Kowalski:2006hc  reads

$\displaystyle\phi_{T}(r,z)=N_{T}(z(1-z))^{2}\exp(-\frac{r^{2}}{2R_{T}^{2}}),$
(22)

The Boosted Gaussian model is simplified from NNPZ model Nemchik:1996cw ;
Forshaw:2003ki , the scalar function of Boosted Gaussian reads

$\displaystyle\phi_{T}(z,r)=N_{T}z(1-z)\exp\big{(}-\frac{m_{f}^{2}\mathcal{R}^{2}}{8z(1-z)}-\frac{2z(1-z)r^{2}}{\mathcal{R}^{2}}+\frac{m_{f}^{2}\mathcal{R}^{2}}{2}\big{)}.$
(23)

DGKP model is another famous model for the forward vector meson wave function
Dosch:1996ss ; Goncalves:2004bp ; Forshaw:2003ki , in this work, we also
consider the contribution of DGKP model, the overlap is different from the
above models, in the DGKP model, the overlap reads Forshaw:2003ki

$\displaystyle(\Psi_{V}^{*}\Psi_{\gamma})_{T}(r,z)=\frac{ef_{v}}{M_{V}}f_{T}(z)\exp(-\frac{\omega_{T}^{2}r^{2}}{2})\\{(\omega_{T}^{2}\epsilon
r[(z^{2}+(1-z)^{2})]K_{1}(\epsilon r)+m_{f}^{2}K_{0}(\epsilon r)\\},$

where $f_{v}$ is the decay constant of the vector meson, the $f(z)$ reads

$f_{T}(z)=N_{T}\sqrt{z(1-z)}\exp(-\frac{M_{V}^{2}(z-1/2)^{2}}{2\omega^{2}_{T}}).$
(25)

The parameters of the vector meson functions are determined by the
normalization condition and the decay constant, we present the parameters in
Table. 3, some parameters are taken from the Ref. Kowalski:2006hc ;
Rezaeian:2013tka .

meson |  $e_{f}$ |  mass |  $f_{v}$ |  $m_{f}$ |  $N_{T}$ |  $R_{T}^{2}$ |  $N_{T}$ |  $\mathcal{R}^{2}$ |  $N_{T}$ |  $\omega_{T}$
---|---|---|---|---|---|---|---|---|---|---
|  |  GeV |  GeV |  GeV |  |  GeV2 |  |  GeV2 |  |  GeV
$J/\psi$ |  $2/3$ |  3.097 |  0.274 |  1.4 |  1.23 |  6.5 |  0.578 |  2.3 |  8.264 |  0.56
$J/\psi$ |  $2/3$ |  3.097 |  0.274 |  1.27 |  1.45 |  5.57 |  0.60 |  2.36 |  9.18 |  0.568
$\phi$ |  $1/3$ |  1.019 |  0.076 |  0.14 |  4.75 |  21.9 |  0.919 |  11.2 |  12.12 |  0.269
$\phi$ |  $1/3$ |  1.019 |  0.076 |  0.01 |  5.91 |  16.45 |  1.021 |  11.4 |  14.81 |  0.268
$\rho$ |  $1/\sqrt{2}$ |  0.776 |  0.156 |  0.14 |  4.47 |  21.9 |  0.911 |  12.9 |  8.62 |  0.223
$\rho$ |  $1/\sqrt{2}$ |  0.776 |  0.156 |  0.01 |  5.89 |  21.68 |  1.004 |  13.3 |  11.27 |  0.222
Table 3: The parameter of the wave functions, the column 6, 7 are the
parameters of Gaus-LC, the column 8,9 are parameters of Boosted Gaussian
model, the column 10,11 are parameters of DGKP model.

##  III results and discussion

In this section, we shall give our prediction using the fIP-Sat and IIM model
with different kinds of wave functions, and compare the prediction to the
experimental data. In the calculation using IIM model, we take the mass of
charm quark as $m_{c}=$1.4 GeV, and the mass of the light quarks as
$m_{q}$=0.14 GeV. In the calculation using fIP-Sat model, we take the mass of
quarks as two parameters sets, in parameter set 1, the quark mass is $m_{c}=$
1.27 GeV and $m_{q}=$0.01 GeV, in parameter set 2, the quark mass is
$m_{c}=$1.4 GeV and $m_{q}=$0.01 GeV Armesto:2014sma . The parameters of the
wave functions are taken according to the quark mass in fIP-Sat or IIM model,
which are presented in Table. 3, the $Q^{2}=$0 $\mathrm{GeV}^{2}$ in all
calculation in this work because the photon is real photon.
In Fig. 1, we present the prediction of rapidity distribution of $J/\psi$ in
PbPb at $\sqrt{s_{NN}}=$2.76 TeV, the prediction was also calculated using the
parameters in Ref. Kowalski:2006hc . We can see that the result of this work
is lower than the result of Ref. Lappi:2013am , because the newer fit of IP-
Sat model in Ref. Rezaeian:2012ji  is more accurate than the older fit in the
Ref. Kowalski:2006hc .

Figure 1: (Color online) The coherent $J/\psi$ rapidity distribution in PbPb
collision at $\sqrt{s_{NN}}$=2.76TeV computed using fIP-Sat model and compared
to the experimental data of ALICE Abelev:2012ba ; Abbas:2013oua , the black
thick lines are using the parameters of fIP-Sat model parameter set 2 in
Table. 1, the green thin curve are the results from Ref. Lappi:2013am  .

The prediction using IIM model is also calculated in this work, which is
compared with two sets parameters of fIP-Sat model, the results are presented
in Fig. 2, we can see that result of parameter set 2 is closer to the
experimental data than the result of IIM model and parameter set 1, the result
of Gaus-LC wave function is closer than the Boosted Gaussian and DGKP model.

Figure 2: The coherent $J/\psi$ rapidity distribution in PbPb collision at
$\sqrt{s_{NN}}$=2.76TeV computed using IIM and fIP-Sat model with Gaus-LC
(solid), Boosted Gaussian(dashed), DGKP(dot-dashed) and compared to the
experimental data of ALICE Abelev:2012ba ; Abbas:2013oua .

The rapidity distribution of $\rho$ meson was also measured at
ALICEAaij:2013jxj , the prediction had been presented in Ref.
Frankfurt:2015cwa , we also compute the prediction of $\rho$ meson, which is
showed in Fig. 3, we compute the result of $\rho$ meson using IIM model and
fIP-Sat with two sets of parameters, we take the light quarks mass $m_{q}$=
0.14 GeV in the IIM model and take the light quark mass $m_{q}$= 0.01 GeV in
fIP-Sat model.

Figure 3: The coherent $\rho$ meson rapidity distribution in PbPb collision at
$\sqrt{s_{NN}}$=2.76 TeV computed using IIM and fIP-Sat model with Gaus-LC
(solid), Boosted Gaussian(dashed), DGKP(dot-dashed) and compared to the
experimental data of ALICE Aaij:2013jxj .

We also give the prediction of the rapidity distribution of $\phi$ meson in
PbPb collision at $\sqrt{s_{NN}}=$2.76 TeV, there is no experimental data for
the $\phi$ meson, we expect the experimental data of the $\phi$ meson at the
LHC in the furture.

Figure 4: The coherent $\phi$ rapidity distribution in PbPb collision at
$\sqrt{s_{NN}}=$2.76 TeV computed using IIM and fIP-Sat model with Gaus-LC
(solid), Boosted Gaussian(dashed), DGKP(dot-dashed).

.

Summary, we calculate the coherent cross section of vector mesons in PbPb
ultraperipheral collisions with fIP-Sat and IIM model, the parameters of this
work are determined in fit of HERA data. We can find that the newer fit of IP-
Sat model is more accurate than the older fit in the calculation of $J/\psi$
production. The IIM model is little upper than the fIP-Sat model in $J/\psi$
and $\rho$ calculations, the production of $\phi$ is also calculated in this
work and we hope in the future the production will be measured at the LHC.

##  IV Acknowledgements

One of the authors, Y. P. Xie, thanks communication with Bo-Wen Xiao, T.
Lappi, H. Mantysaari and A. H. Rezaenian. This work is supported in part by
the National Natural Science Foundation of China (Grant No. 11175220), the One
Hundred Person Project (Grant No. Y101020BR0), and the Key Laboratory of Quark
and Lepton Physics (MOE), Central China Normal University (Grant No.
QLPL201414).

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# Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

Xiaoqiang Sun111 School of Mathematics, Sun Yat-sen University, Guangzhou,
510275, China. ⊠ Jiguang Bao222 School of Mathematical Sciences, Beijing
Normal University, Laboratory of Mathematics and Complex Systems, Ministry of
Education, Beijing 100875, China. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>$\boxtimes$ Corresponding author

###### Abstract

In this paper, we apply blow-up analysis to study pointwise a priori estimates
for some p-Laplace equations based on Liouville type theorems. With newly
developed analysis techniques, we first extend the classical results of
interior gradient estimates for the harmonic function to that for the
p-harmonic function, i.e., the solution of $\Delta_{p}u=0,\ x\in\Omega$. We
then obtain singularity and decay estimates of the sign-changing solution of
Lane-Emden-Fowler type p-Laplace equation $-\Delta_{p}u=|u|^{\lambda-1}u,\
x\in\Omega$, which are then extended to the equation with general right hand
term $f(x,u)$ with certain asymptotic properties. In addition, pointwise
estimates for higher order derivatives of the solution to Lane-Emden type
p-Laplace equation, in a case of $p=2$, are also discussed.

##  1 Introduction

p-Laplacian equation arises in many studies of non-linear phenomena, for
instance, in mathematical modeling of non-Newtonian fluids in physics [1, 2],
and in the theory of quasi-regular, quasi-conformal mappings in geometry [3,
4, 5]. The p-Laplacian equation represents an important type of quasi-linear
equations. It has the following form

$-\Delta_{p}u=f(x,u),\qquad x\in\Omega$ (1.1)

where $\Omega$ is a domain in $\mathbb{R}^{N}$, and

$\Delta_{p}u:=div(|\nabla u|^{p-2}\nabla u),\qquad 1<p<\infty$

is p-Laplace operator. When $p=2$, $\Delta_{p}u$ is Laplace operator.
$\Delta_{p}u$ is degenerate elliptic when $p>2$ and has singularity when
$p<2$.

We call $u\in C^{1}(\Omega)$ as a weak solution of (1.1), if

$\int_{\Omega}|\nabla u|^{p-2}(\nabla
u,\nabla\varphi)dx=\int_{\Omega}f(x,u)\varphi dx,\qquad\forall\varphi\in
C_{c}^{\infty}(\Omega).$ (1.2)

In this paper we assume $f\in C^{1}(\mathbb{R}^{N})$. In the following text, a
weak solution always refers to that defined above with $C^{1,\alpha}_{loc}$
regularity.

######  Remark 1.1.

Although weak solutions in other weaker forms could be defined as well, for
example $u\in W^{1,m}_{loc}(\Omega)\cap L^{\infty}_{loc}(\Omega)$ [1], or
$C^{1,\alpha}_{loc}$ weak solution [6], however, we can deduce that these
definitions are equivalent to each other due to the following established
conclusion on regularity [7, 8, 1, 9]: _If $u\in C^{1}$ is a weak solution of
(1.1), then there exists $\beta\in(0,1)$ such that $u\in
W^{2,2}_{loc}(\Omega)\cap C^{1,\beta}_{loc}(\Omega).$ _

When $f$=0, Eq. (1.1) becomes

$-\Delta_{p}u=0,\qquad x\in\Omega.$ (1.3)

The solution $u$ of (1.3) is called p-harmonic function. In the case of $p=2$,
it is well-known harmonic function, satisfying the following Laplace equation

$-\Delta u=0,\qquad x\in\Omega.$ (1.4)

For harmonic function, the following pointwise interior estimate of its
gradient [10] can be derived via Poisson integrative formula or mean vale
theorem:

$|\nabla u(x)|\leq N\cdot\sup_{\Omega}|u|\cdot
dist^{-1}(x,\partial\Omega),\qquad\forall x\in\Omega.$ (1.5)

A natural question is that whether the above gradient estimate for harmonic
function can be extended to that for p-harmonic function? We will seek to
answer it in this paper.

On the other hand, when $f$ has the form of $u^{\lambda}$, then Eq. (1.1)
becomes the following Lane-Emden type equation

$-\Delta_{p}u=u^{\lambda},\qquad x\in\Omega.$ (1.6)

J. Serrin and H. Zou [1] proved the Liouville type theorems of (1.6), i.e.
nonexistence of the nonnegative nontrivial solution on the whole space
$\mathbb{R}^{N}$. Based on the Liouville type theorems, P. Polacik, P.
Quittner and P. Souplet [11] obtained the following singularity and decay
estimates for the nonnegative weak solution of (1.6) by applying the method of
blow-up analysis:

_Let $0<p-1<\lambda<\Lambda_{s}(p),$ and let $\Omega\neq\mathbb{R}^{N}$ be a
domain, then there exists a constant $C=C(p,N,\lambda)>0$ $($independent of
$\Omega$ and $u),$ such that any $($ nonnegative weak $)$ solution $u$ of
(1.6) satisfies_

$u(x)+|\nabla u(x)|^{\frac{p}{\lambda+1}}\leq
Cdist^{-\frac{p}{\lambda+1-p}}(x,\partial\Omega),\qquad\forall x\in\Omega$
(1.7)

where,

$\Lambda_{s}(p):=\begin{cases}\frac{N(p-1)+p}{n-p},\qquad\qquad ifN>p\\\
\infty,\qquad\qquad\qquad ifN\leq p\end{cases}$ (1.8)

is the corresponding Sobolev critical exponent ([12, 13]).

Again, we are curious about that whether the decay estimates (1.7) could be
obtained for the following Lane-Emden-Fowler type p-Laplacian equation [8]

$-\Delta_{p}u=|u|^{\lambda-1}u,\qquad x\in\Omega.$ (1.9)

To answer the above two questions, we here develop new analytic techniques to
get pointwise a priori estimates for the solution and/or gradient of some
types of quasilinear equations by coupling blow up analysis with Liouville-
type theorems. In this paper, we first extend the interior estimates for the
gradient of harmonic function to p-harmonic function. Subsequently, we study
the singularity and decay estimates of the sign-changing solutions of the
Lane-Emden-Fowler type p-Laplacian equation, $-\Delta_{p}u=|u|^{\lambda-1}u$.
Furthermore, we also discuss the pointwise a priori estimates for higher order
derivatives of solution to $-\Delta u=u^{\lambda},x\in\Omega$, i.e., the Lane-
Emden type p-Laplacian equation in the case of $p=2$. Different from the
estimates in the form of $L^{p}$ norm or $C^{\alpha}$ seminorm, the pointwise
a priori estimates deduced in this paper have some interesting features: (1)
$dist^{-1}(x,\partial\Omega)$ is explicitly expressed in the right hand of the
estimates. (2) The constant $C$ in the right hand of the estimates depends
only on the exponents in the equation ($p$, $N$ and/or $\lambda$), and is
independent of domain $\Omega$ and solution $u$. In this sense the pointwise
estimates are also universal.

##  2 Main Results

###  2.1 Interior estimates of gradient of p-harmonic function

######  Theorem 2.1.

Let $p\in(1,\infty)$, and let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$
$(N\geq 2)$. Then there exists a constant $C=C(p,N)>0$, such that any $C^{1}$
weak solution $u$ of

$\Delta_{p}u=0,\qquad x\in\Omega,$

satisfies

$|\nabla u(x)|\leq C\sup_{\Omega}|u|\cdot
dist^{-1}(x,\partial\Omega),\qquad\forall x\in\Omega.$ (2.1)

######  Remark 2.1.

This result extends the classical interior estimates (1.5) of gradient of the
harmonic function [10].

###  2.2 Singularity estimates for Lane-Emden-Fowler type equation

For Lane-Emden-Fowler type equation

$-\Delta_{p}u=|u|^{\lambda-1}u,\qquad x\in\Omega,$ (2.2)

according to (1.2), we say $u\in C^{1}(\Omega)$ is a weak solution of (2.2),
if

$\int_{\Omega}|\nabla u|^{p-2}(\nabla
u,\nabla\varphi)dx=\int_{\Omega}|u|^{\lambda-1}u\varphi
dx,\qquad\forall\varphi\in C_{c}^{\infty}(\Omega).$ (2.3)

A weak solution is called "stable" if the second variation of energy function
of (2.2) is nonnegative. More specifically, we have the following definition
[8].

######  Definition 2.1.

Denote

$L_{u}(v,\varphi)=\int_{\Omega}|\nabla u|^{p-2}(\nabla
v,\nabla\varphi)dx+\int_{\Omega}(p-2)|\nabla u|^{p-4}(\nabla u,\nabla
v)\cdot(\nabla v,\nabla\varphi)-\lambda|u|^{\lambda-1}v\varphi dx.$

The weak solution $u\in C^{1}$ of (2.2) is called stable, if

$L_{u}(\varphi,\varphi)\geq 0,\qquad\forall\varphi\in C_{c}^{1}(\Omega).$
(2.4)

Accordingly, the weak solution $u\in C^{1}$ of (2.2) is called stable outside
a compact set $K\subset\Omega$, if

$L_{u}(\varphi,\varphi)\geq 0,\qquad\forall\varphi\in
C_{c}^{1}(\Omega\setminus K).$ (2.5)

######  Remark 2.2.

If $u$ has finite Morse index, then $u$ is stable outside a compact set.

######  Theorem 2.2.

Let $p>2$, and let $\Omega\neq\mathbb{R}^{N}$ be an arbitrary domain of
$\mathbb{R}^{N}$. If

$\begin{cases}p-1<\lambda<\infty,\qquad\qquad\ \ if\
N\leq\frac{p(p+3)}{p-1},\\\ p-1<\lambda<\Lambda_{c}(N,p),\qquad if\
N>\frac{p(p+3)}{p-1},\\\ \end{cases}$ (2.6)

then there exists $C=C(p,N,\lambda)>0$ $($independent of $\Omega$ and $u$ $)$,
such that any stable weak solution $u\in C^{1}(\Omega)$ of (2.2) satisfies

$|u(x)|+|\nabla u(x)|^{\frac{p}{\lambda+1}}\leq
Cdist^{-\frac{p}{\lambda+1-p}}(x,\partial\Omega),\qquad\forall x\in\Omega.$
(2.7)

######  Remark 2.3.

This theorem expands the estimates of singularity and decay for nonnegative
solution of Lane-Emden type equation in [11] to that for sign-changing
solution of Lane-Emden-Fowler type equation, under the condition of
"stability" of the weak solution.

######  Theorem 2.3.

Let $p>2$, and let $\Omega\neq\mathbb{R}^{N}$ be an arbitrary domain of
$\mathbb{R}^{N}$, if

$\begin{cases}p-1<\lambda<\infty,\qquad\qquad\ ifN\leq p,\\\
p-1<\lambda<\frac{N(p-1)+p}{N-p},\qquad ifN>p,\\\ \end{cases}$ (2.8)

then there exists $C=C(p,N,\lambda)>0$ (independent of $\Omega$ and $u$), such
that any weak solution $u\in C^{1}(\Omega)$ of (2.2) which is stable outside a
compact set $K\subset\mathbb{R}^{N}$ satisfies

$|u(x)|+|\nabla u(x)|^{\frac{p}{\lambda+1}}\leq
Cdist^{-\frac{p}{\lambda+1-p}}(x,\partial\Omega),\qquad\forall x\in\Omega.$
(2.9)

The conclusions in the above two theorems can be extended for the following
equation:

$-\Delta_{p}u=f(u),\qquad x\in\Omega.$ (2.10)

######  Corollary 2.4.

Assume that the relations between $N,p,\lambda$ are the same as that in
Theorem 2.2. Let $\Omega\neq\mathbb{R}^{N}$ be an arbitrary domain of
$\mathbb{R}^{N}$, and let $f$ satisfy

$\lim_{u\rightarrow\infty}|u|^{1-\lambda}u^{-1}f(u)=l\in(0,+\infty).$ (2.11)

If $u\in C^{1}(\Omega)$ is a weak solution of (2.10) and satisfies (2.4), then
there exists $C=C(p,N,\lambda,f)>0$ (independent of $\Omega$ and $u$), such
that

$|u(x)|+|\nabla u(x)|^{\frac{p}{\lambda+1}}\leq
C(1+dist^{-\frac{p}{\lambda+1-p}}(x,\partial\Omega)),\qquad\forall
x\in\Omega.$ (2.12)

######  Corollary 2.5.

Assume that the relations between $N,p,\lambda$ are the same as that in
Theorem 2.3. Let $\Omega\neq\mathbb{R}^{N}$ be an arbitrary domain of
$\mathbb{R}^{N}$. Assume that $f$ satisfies

$\lim_{u\rightarrow\infty}|u|^{1-\lambda}u^{-1}f(u)=l\in(0,+\infty).$ (2.13)

If $u\in C^{1}(\Omega)$ is a weak solution of (2.10) and satisfies (2.5), then
there exists $C=C(p,N,\lambda,f)>0$ $($independent of $\Omega$ and $u)$, such
that

$|u(x)|+|\nabla u(x)|^{\frac{p}{\lambda+1}}\leq
C(1+dist^{-\frac{p}{\lambda+1-p}}(x,\partial\Omega)),\qquad\forall
x\in\Omega.$ (2.14)

###  2.3 Estimates of higher order derivatives

######  Theorem 2.6.

Let $n\geq 2$, $1<\lambda<\lambda_{s}$ , here,

$\lambda_{s}:=\begin{cases}\frac{n+2}{n-2},\quad if\ N>2,\\\ \infty,\quad if\
N=2,\\\ \end{cases}$ (2.15)

and let $\Omega$ be an arbitrary domain in $\mathbb{R}^{N}$,
$\Omega\neq\mathbb{R}^{n}$. Then for any nonnegative integer $s$, there exists
$C=C(p,N,\lambda,s)>0$ (independent of $\Omega$ and $u$) such that any
nonnegative smooth solution $u$ of

$-\Delta u=u^{\lambda},\qquad x\in\Omega,$ (2.16)

satisfies

$\sum_{r=0}^{s}|\nabla^{r}u(x)|^{\frac{\lambda-1}{(\lambda-1)r+2}}\leq
Cdist^{-1}(x,\partial\Omega),\qquad\forall x\in\Omega.$ (2.17)

######  Remark 2.4.

The requirements for the regularity of solution actually only need $u\in
C^{s}(\Omega)$. What’s more, in fact, the requirements for the regularity of
solution can be relaxed by applying the regularity theory of elliptic
equation.

######  Remark 2.5.

For the equation with general right hand term $f$ that satisfies corresponding
asymptotic conditions, by using the same method one can obtain the estimates
of higher order derivatives of the solutions. The details are omitted here.

######  Remark 2.6.

Compared to the singularity and decay estimates of the solutions to elliptic
equations that have been obtained in previous studies (e.g., [14]), our result
presented here gives a more precise estimate for higher order derivatives of
the solution to p-Laplacian equation.

##  3 Lemmas

In this section , we first introduce Liouville theorem for p-harmonic function
[1].

######  Lemma 3.1.

Let $u$ be a nonnegative weak solution of

$\Delta_{p}u=0,\qquad x\in\mathbb{R}^{N}\qquad(n>p)$

or

$\Delta_{p}u\leq 0,\qquad x\in\mathbb{R}^{N}\qquad(n\geq 2)$

then $u$ is constant.

For Lane-Emden-Fowler type equation (2.2), the following Liouville type
theorems have been established in [8].

######  Lemma 3.2.

Let $u\in C^{1}(\mathbb{R}^{N})$ be a stable weak solution of (2.2), $p>2$, if

$\begin{cases}p-1<\lambda<\infty,\qquad\qquad ifN\leq\frac{p(p+3)}{p-1},\\\
p-1<\lambda<\Lambda_{c}(N,p),\qquad ifN>\frac{p(p+3)}{p-1},\\\ \end{cases}$
(3.1)

here,

$\Lambda_{c}(N,p)=\frac{[(p-1)N-p]^{2}+p^{2}(p-2)-p^{2}(p-1)N+2p^{2}\sqrt{(p-1)(N-1)}}{(N-p)[(p-1)N-p(p+3)]}.$
(3.2)

then $u\equiv 0.$

######  Lemma 3.3.

Let $u\in C^{1}(\mathbb{R}^{N})$ be a weak solution of (2.2) which is stable
outside of a compact set of $K$, $p>2$, if

$\begin{cases}p-1<\lambda<\infty,\qquad\qquad ifN\leq p,\\\
p-1<\lambda<\frac{N(p-1)+p}{N-p},\qquad ifN>p,\\\ \end{cases}$ (3.3)

then $u\equiv 0.$

Along with the above Liouville type theorems, the following "Doubling lemma"
proposed in [11] also plays an important role in blow-up analysis for the
proof in this paper.

######  Lemma 3.4.

Let $(X,d)$ be a compact metric space and let $D\subset\Sigma\subset X$, where
$D$ is non-empty and $\Sigma$ is closed. Set $\Gamma:=\Sigma\setminus D$. Let
$M:D\rightarrow(0,\infty)$ be bounded on compact subset of $D$, and fix a real
$k>0$. If $y\in D$ satisfies

$M(y)dist(y,\Gamma)>2k,$

then there exists $x\in D$ such that

$M(x)dist(x,\Gamma)>2k,\qquad M(x)\geq M(y),$

and

$M(z)\leq 2M(x),\qquad\forall z\in D\cap\overline{B}(x,kM^{-1}(x)).$

In addition, we present here the following result regarding the convergence of
weak stable solutions of the p-Laplacian equation, which will be used in
proofs of Theorems 2.2 and 2.3 .

######  Lemma 3.5.

Let $\Omega_{k}\upuparrows\mathbb{R}^{N},K$ be a compact subset in
$\mathbb{R}^{N}$. Assume that $u_{k}$ are a set of weak solutions to (2.2) on
$\Omega_{k}$, $k=1,2,\cdots$, respectively and that $($as $k$ large enough$)$
$u_{k}$ is stable outside of $K$. If

$u_{k}\rightarrow u\qquad in\ C_{loc}^{1,\alpha}(\mathbb{R}^{N}),$

then $u$ is a weak solution of (2.2) and is stable outside of $K$.

##  4 Proofs

In this section we apply the method of blow-up analysis to derive pointwise
estimates. The method of blow-up is an efficient approach to deduce estimates
in partial differential equations, which has been applied by many scholars in,
for example, [15, 11, 16]. The corresponding Liouville type theorems ( e.g.
[17, 10, 8, 15, 1, 14]) are important basis of this method. This method can be
briefly described as follows: by proof of contradiction, assuming that an
estimate (in terms of the distance to $\partial\Omega$) fails, we could
construct an appropriate auxiliary function and use "doubling" property, then
the sequence of violating solutions $u_{k}$ will be increasingly large along a
sequence of points $x_{k}$, such that each $x_{k}$ has a suitable neighborhood
where the relative growth of $u_{k}$ remains controlled. After appropriate
rescaling, we can blow up the sequence of neighborhoods and pass to the limit
to obtain a bounded solution of a limiting problem in the whole of
$\mathbb{R}^{N}$ based on $L^{p}$ estimates or $C^{\alpha}$ estimates. By
applying Liouville type theorems to the limiting equation we finally derive a
contradiction.

###  4.1 The proof of Theorem 2.1

###### Proof of Theorem2.1.

Under the same condition with that in Theorem 2.1, we just need to prove the
following estimate: $\forall\epsilon\in(0,1)$, there exists a constant
$C=C(p,N)$, such that any $C^{1}$ weak solution $u$ of

$\Delta_{p}u=0,\qquad x\in\Omega,$

satisfies

$|\nabla u(x)|\leq C\sup_{\Omega}|u|\cdot diam^{\epsilon}(\Omega)\cdot
dist^{-(1+\epsilon)}(x,\partial\Omega),\qquad\forall x\in\Omega.$ (4.1)

Assume $u\geq 0$, otherwise let

$w=u+sup_{\Omega}|u|,$

then $w\geq 0$, and

$\Delta_{p}w=0,\qquad x\in\Omega.$

If for $w,$ (4.1) holds, then for $u$ and $\forall x\in\Omega$ we have

$\displaystyle|\nabla u(x)|$ $\displaystyle=|\nabla w(x)|\leq
C\mathop{\sup_{\Omega}}|w|\cdot diam^{\epsilon}(\Omega)\cdot
dist^{-1+\epsilon}(x,\partial\Omega)$ $\displaystyle\leq
C\mathop{\sup_{\Omega}}|u|\cdot diam^{\epsilon}(\Omega)\cdot
dist^{-(1+\epsilon)}(x,\partial\Omega).$

Moreover, we assume $\mathop{\sup_{\Omega}}|u|\cdot
diam^{\epsilon}(\Omega)=1$. Otherwise, let

$\widetilde{w}=\frac{u}{\mathop{\sup_{\Omega}}|u|\cdot
diam^{\epsilon}(\Omega)},$

then $\mathop{\sup_{\Omega}}|\widetilde{w}|\cdot diam^{\epsilon}(\Omega)=1$,
and

$\Delta_{p}\widetilde{w}=0,\qquad x\in\Omega.$

If for $\widetilde{w}$, (4.1) holds, then for $u$, by

$\frac{|\nabla u(x)|}{\mathop{\sup_{\Omega}}|u|\cdot
diam^{\epsilon}(\Omega)}=|\nabla\widetilde{w}(x)|\leq
Cdist^{-(1+\epsilon)}(x,\partial\Omega),$

we know that

$|\nabla u(x)|\leq C\sup_{\Omega}|u|\cdot diam^{\epsilon}(\Omega)\cdot
dist^{-1+\epsilon}(x,\partial\Omega),\qquad\forall x\in\Omega.\\\ $ (4.2)

Therefore, we only need to prove that if $u$ satisfies $\Delta_{p}u=0$ on
$\Omega$, and $u\geq 0,\sup_{\Omega}|u|\cdot diam(\Omega)=1$, then there holds

$|\nabla u(x)|\leq C\cdot dist^{-(1+\epsilon)}(x,\partial\Omega),\qquad\forall
x\in\Omega.$ (4.3)

If (4.3) fails, then for $k=1,2,\cdots$, there exist sequences
$\Omega_{k},u_{k},y_{k}\in\Omega_{k}$ and $\epsilon_{k}\in(0,1)$, such that
$u_{k}$ is nonnegative $C^{1}$ weak solution of $\Delta_{p}u=0$ on
$\Omega_{k}$, with $\displaystyle{\sup_{\Omega_{k}}u_{k}\cdot
diam^{\epsilon_{k}}(\Omega_{k})=1}$, and the function

$M_{k}:=|\nabla u_{k}|^{\frac{1}{1+\epsilon_{k}}}$

satisfies

$M_{k}(y_{k})>2kdist^{-1}(y_{k},\partial\Omega_{k}).$

By Lemma 3.4, it follows that $\exists x_{k}\in\Omega_{k}$, such that

$M_{k}(x_{k})>2kdist^{-1}(x_{k},\partial\Omega_{k}),$

and

$M_{k}(z)\leq 2M_{k}(x_{k}),\quad\forall z:|z-x_{k}|\leq kM_{k}^{-1}(x_{k}).$

Let $\lambda_{k}:=M_{k}^{-1}(x_{k})$, and we rescale $u_{k}$ by setting

$v_{k}(y):=k^{\epsilon_{k}}\lambda_{k}^{\epsilon_{k}}\cdot
u_{k}(x_{k}+\frac{1}{k^{\epsilon_{k}}}\lambda_{k}y),\qquad\forall y:|y|\leq
k.$

Note that as $|y|\leq k$,

$|\frac{1}{k^{\epsilon_{k}}}\cdot\lambda_{k}y|\leq\frac{1}{k^{\epsilon_{k}}}\cdot\frac{1}{2k}dist(x_{k},\partial\Omega_{k})\cdot
k^{1+\epsilon_{k}}=\frac{1}{2}dist(x_{k},\partial\Omega_{k}),$

thus $x_{k}+\frac{1}{k^{\epsilon_{k}}}\cdot\lambda_{k}y\in\Omega_{k}$. As
such, $v_{k}$ is well defined. Meanwhile we have

$\displaystyle|v_{k}(y)|$
$\displaystyle=k^{\epsilon_{k}}\lambda_{k}^{\epsilon_{k}}\cdot
u_{k}(x_{k}+\frac{1}{k^{\epsilon_{k}}}\lambda_{k}y)$ $\displaystyle\leq
k^{\epsilon_{k}}\cdot(\frac{1}{2k})^{\epsilon_{k}}\cdot
dist^{\epsilon_{k}}(x_{k},\partial\Omega_{k})\cdot\sup_{\Omega_{k}}u_{k}$
$\displaystyle\leq(\frac{1}{2})^{\epsilon_{k}}\cdot\sup_{\Omega_{k}}u_{k}\cdot
diam^{\epsilon_{k}}(\Omega_{k})$
$\displaystyle=(\frac{1}{2})^{\epsilon_{k}}\leq 1,$ $\displaystyle|\nabla
v_{k}(y)|$ $\displaystyle=\lambda_{k}^{1+\epsilon_{k}}\cdot|\nabla
u_{k}(x_{k}+\frac{1}{k^{\epsilon_{k}}}\lambda_{k}y)|$
$\displaystyle=M_{k}(x_{k})^{-(1+\epsilon_{k})}\cdot
M_{k}(x_{k}+\frac{1}{k^{\epsilon_{k}}}\lambda_{k}y)^{1+\epsilon_{k}}\leq
4,\qquad\forall y:|y|\leq k$ $\displaystyle|\nabla v_{k}(0)|$
$\displaystyle=1,$

and

$\Delta_{p}v_{k}(y)=div(|\nabla v_{k}|^{p-2}\cdot\nabla
v_{k})=(\lambda_{k}^{1+\epsilon_{k}})^{p-1}\cdot\Delta_{p}u_{k}=0,$ (4.4)

that is, $v_{k}$ solves

$\Delta_{p}v_{k}=0,\qquad x\in B_{k}(0).$

By using $C^{\alpha}$ estimates, we deduce that there exist $\beta\in(0,1)$
and constant $C$ (independent of $k$) such that

$\parallel v_{k}\parallel_{C_{loc}^{1+\beta}}\leq C.$

Therefore there exist a subsequence of ${v_{k}}$, still denoted by ${v_{k}}$,
such that

$v_{k}\rightarrow v\qquad in\ C_{loc}^{1}(\mathbb{R}^{N}),$ (4.5)

and at point $0$, $v$ satisfies

$|\nabla v(0)|=1.$ (4.6)

By (4.4) it follows that $v$ satisfies

$\Delta_{p}v=0,\qquad x\in\mathbb{R}^{N}.$

By Liouville theorem (Lemma 3.1) we deduce that

$v=const,\qquad x\in\mathbb{R}^{N}.$

So $|\nabla v|\equiv 0$ in $\mathbb{R}^{N}$, this contradicts with (4.6).
Therefore (4.3) holds and we have proved Theorem 2.1.

∎

######  Remark 4.1.

In the above proof, we used an approximation method to get the estimate by
first relaxing the inequality through introducing $\epsilon$ in (4.1) and then
taking limit, which is a critical step since we found that it was hard to
apply blow up analysis for the estimate (2.1) directly.

###  4.2 The proof of Theorem 2.2

###### Proof of Lemma 3.5.

$\forall\varphi\in C_{c}^{1}(\mathbb{R}^{N}\setminus K)$, let
$V=supp\varphi\subset\mathbb{R}^{N}\setminus K$, $V\subset\Omega_{k}$ as $k$
large enough. We first prove that $u$ satisfies (2.3). In fact, as $k$ large
enough,

$\displaystyle|\int_{V}|\nabla u_{k}|^{p-2}(\nabla
u_{k},\nabla\varphi)dx-\int_{V}|\nabla u|^{p-2}(\nabla u,\nabla\varphi)dx|$
$\displaystyle\leq\int_{V}(|\nabla u_{k}|-|\nabla u|)^{p-2}(\nabla
u_{k},\nabla\varphi)dx+\int_{V}|\nabla u|^{p-2}(\nabla u_{k}-\nabla
u,\nabla\varphi)dx\rightarrow 0,\qquad(k\rightarrow\infty).$

We next prove

$L_{u_{k}}(\varphi,\varphi)\rightarrow
L_{u}(\varphi,\varphi)\qquad(k\rightarrow\infty).$ (4.7)

In fact, as $k$ large enough,

$\displaystyle|\int_{V}|\nabla
u_{k}|^{p-2}|\nabla\varphi|^{2}dx-\int_{V}|\nabla
u|^{p-2}|\nabla\varphi|^{2}dx|$
$\displaystyle\leq\displaystyle\max_{V}(|\nabla u_{k}|^{p-2}|-|\nabla
u|^{p-2}|)\int_{V}|\nabla\varphi|^{2}dx\rightarrow
0,\qquad(k\rightarrow\infty).$

Similarly,

$\int_{V}|u_{k}|^{\lambda-1}\varphi^{2}dx\rightarrow\int_{V}|u|^{\lambda-1}\varphi^{2}dx,\qquad(k\rightarrow\infty).$

Moreover,

$\displaystyle|\int_{V}|\nabla u_{k}|^{p-4}|(\nabla
u_{k},\nabla\varphi)|^{2}dx-\int_{V}|\nabla u|^{p-4}|(\nabla
u,\nabla\varphi)|^{2}dx|$ $\displaystyle\leq\int_{V}(|\nabla u_{k}|-|\nabla
u|)^{p-4}|(\nabla u_{k},\nabla\varphi)|^{2}dx+\int_{V}|\nabla u|^{p-4}(\nabla
u_{k}+\nabla u,\nabla\varphi)(\nabla u_{k}-\nabla u,\nabla\varphi)dx$
$\displaystyle\rightarrow 0\qquad(k\rightarrow\infty).$

Therefore,

$L_{u_{k}}(\varphi,\varphi)\rightarrow L_{u}(\varphi,\varphi)\geq
0\qquad(k\rightarrow\infty).$ (4.8)

So $u$ is a weak solution to (2.2) and is stable outside of $K$ according to
the Definition 2.1. ∎

###### Proof of Theorem 2.2.

Let $\alpha=\frac{p}{\lambda+1-p}$. Assume that estimate (2.9) fails, then
there exist sequences $\Omega_{k},u_{k},y_{k}\in\Omega_{k}$ $(k=1,2,\cdots)$,
such that each $u_{k}$ is a stable weak solution of (2.2) on $\Omega_{k}$ and
functions

$M_{k}:=|u_{k}|^{\frac{1}{\alpha}}+|\nabla
u_{k}|^{\frac{1}{(\alpha+1)}},\qquad k=1,2,\cdots,$

satisfy

$M_{k}(y_{k})>2kdist^{-1}(y_{k},\partial\Omega_{k}).$

By Lemma 3.4 it follows that $\exists x_{k}\in\Omega_{k}$, such that

$M_{k}(x_{k})>2kdist^{-1}(x_{k},\partial\Omega_{k}),$ $M_{k}(z)\leq
2M_{k}(x_{k}),\quad\forall z:|z-x_{k}|\leq kM_{k}^{-1}(x_{k}).$

Let $\lambda_{k}:=M_{k}^{-1}(x_{k})$, and rescale $u_{k}$ by setting

$v_{k}(y):=\lambda_{k}^{\alpha}u_{k}(x_{k}+\lambda_{k}y),\qquad\forall
y:|y|\leq k.$

Note that $(p-1)(\alpha+1)+1=\lambda\alpha$. It’s easy to verify that $v_{k}$
is a stable weak solution of

$-\Delta_{p}v_{k}=|v_{k}|^{\lambda-1}v_{k},\qquad y\in B_{k}(0)$ (4.9)

and

$[|v_{k}|^{\frac{1}{\alpha}}+|\nabla
v_{k}|^{\frac{1}{(\alpha+1)}}](0)=1,\qquad\qquad$ (4.10)

$[|v_{k}|^{\frac{1}{\alpha}}+|\nabla v_{k}|^{\frac{1}{(\alpha+1)}}](y)\leq
2,\qquad|y|\leq k.$ (4.11)

By using $C_{loc}^{1,\beta}$ estimates of p-Laplacian equation, we know
$\exists\beta\in(0,1)$ and $C$ (independent of $k$$)$ such that

$\parallel v_{k}\parallel_{C_{loc}^{1+\beta}}\leq C,\qquad\forall
k=1,2,\cdots.$

By Arzela-Ascolli Theorem, it follows that there exist a subsequence of
${v_{k}}$, still denoted by ${v_{k}}$, such that

$v_{k}\rightarrow v\qquad in\ C_{loc}^{1}(\mathbb{R}^{N}).$ (4.12)

and by (4.9) we deduce that $v$ satisfies

$\Delta_{p}v=|v|^{\lambda-1}v,\qquad x\in\mathbb{R}^{N}.$ (4.13)

By Lemma 3.5, $v$ is a stable weak solution of $\eqref{v22}$. By Lemma 3.2 we
know that

$v=0,\qquad x\in\mathbb{R}^{N}.$

But by (4.10) it follows that

$[|v|^{\frac{1}{\alpha}}+|\nabla v|^{\frac{1}{(\alpha+1)}}](0)=1.$ (4.14)

Contradiction is derived. As such we have proved Theorem 2.2. ∎

###### Proof of Theorem 2.3.

By applying the same procedure in proof of Theorem 2.2, it is easy to deduce
the conclusion by Lemma 3.3 and Lemma 3.5. ∎

###### Proofs of Corollary 2.4 and Corollary 2.5.

Similar to the proofs of Theorem 2.2 and Theorem 2.3, the difference lies in
that

$M_{k}(y_{k})>2k(1+dist^{-1}(y_{k},\partial\Omega_{k}))>2kdist^{-1}(y_{k},\partial\Omega_{k}),$

and

$\lambda_{k}\rightarrow 0\quad(k\rightarrow\infty).$

$v_{k}$ solves

$-\Delta_{p}v_{k}(y)=f_{k}(v_{k}(y)):=\lambda_{k}^{(\alpha+1)(p-1)+1}f(\lambda_{k}^{-\alpha
v_{k}(y)}),\qquad|y|\leq k.$ (4.15)

By using $C^{\alpha}$ estimates, we deduce that there exist $\beta\in(0,1)$
and constant $C($independent of $k$$)$ such that

$\parallel v_{k}\parallel_{C_{loc}^{1+\beta}}\leq C,\qquad\forall
k=1,2,\cdots.$

Therefore there exist a subsequence of ${v_{k}}$, still denoted by ${v_{k}}$,
such that

$v_{k}\rightarrow v\qquad in\ C_{loc}^{1}(\mathbb{R}^{N}),$ (4.16)

and by (4.15) we deduce that $v$ satisfies (4.10)-(4.11) and solves

$\Delta_{p}v=l\cdot|v|^{\lambda-1}v,\qquad x\in\mathbb{R}^{N}.$ (4.17)

By Liouville theorems $($Lemma 3.2 and Lemma 3.3$)$ we know

$v=0,\qquad x\in\mathbb{R}^{N}.$

The contradiction is also derived. ∎

###### Proofs of Theorem 2.6.

Assume that (2.17) fails, then, there exist sequences
$\Omega_{k},u_{k},y_{k}\in\Omega_{k}$ $(k=1,2,\cdots)$, such that each $u_{k}$
is a smooth solution of $-\Delta u=u^{\lambda}$ on $\Omega_{k}$, and functions

$M_{k}:=\sum_{r=0}^{s}|\nabla^{r}u_{k}|^{\frac{1}{\alpha+r}},\qquad
k=1,2,\cdots,$ (4.18)

satisfy

$M_{k}(y_{k})>2kdist^{-1}(y_{k},\partial\Omega_{k}),$

where $\alpha=\frac{2}{\lambda-1}$.
By Lemma 3.4 it follows that there exists $x_{k}\in\Omega_{k}$, such that

$M_{k}(x_{k})>2kdist^{-1}(x_{k},\partial\Omega_{k}),$ $M_{k}(z)\leq
2M_{k}(x_{k}),\quad\forall z:|z-x_{k}|\leq kM_{k}^{-1}(x_{k}).$

Let $\lambda_{k}:=M_{k}^{-1}(x_{k})$, and rescale $u_{k}$ by setting

$v_{k}(y):=\lambda_{k}^{\alpha}u_{k}(x_{k}+\lambda_{k}y),\qquad\forall
y:|y|\leq k.$

Note that as $|y|\leq k$ we have

$|\lambda_{k}y|\leq\frac{1}{2k}dist(x_{k},\partial\Omega_{k})\cdot
k=\frac{1}{2}dist(x_{k},\partial\Omega_{k}).$

This implies that

$x_{k}+\lambda_{k}y\in\Omega_{k},$

Therefore $v_{k}$ is well defined. We further deduce that

$\displaystyle-\Delta v_{k}(y)$
$\displaystyle=-\lambda_{k}^{\alpha}\cdot\Delta_{y}u_{k}(x_{k}+\lambda_{k}y)$
$\displaystyle=-\lambda_{k}^{\alpha+2}\Delta u_{k}$
$\displaystyle=\lambda_{k}^{\alpha+2}u_{k}^{\lambda}=(\lambda_{k}^{\alpha}u_{k})^{\lambda}$
$\displaystyle=v_{k}^{\lambda}(y).$

So $v_{k}$ solves

$-\Delta v_{k}=v_{k}^{\lambda},\qquad y\in B_{k}(0),$

and

$\displaystyle\sum_{r=0}^{s}|\nabla^{r}v_{k}(y)|^{\frac{1}{\alpha+r}}$
$\displaystyle=\sum_{r=0}^{s}|\lambda_{k}^{\alpha+r}\nabla^{r}u_{k}|^{\frac{1}{\alpha+r}}$
$\displaystyle=\lambda_{k}M_{k}(x_{k}+\lambda_{k}y)$ $\displaystyle\leq
2,\qquad\qquad\qquad|y|\leq k;$
$\sum_{r=0}^{s}|\nabla^{r}v_{k}|^{\frac{1}{\alpha+r}}(0)=\lambda_{k}M_{k}(x_{k})=1.$
(4.19)

By using $C^{\alpha}$ estimates, we deduce that there exist $\beta\in(0,1)$
and constant $C$ (independent of $k$) such that

$\parallel v_{k}\parallel_{C_{loc}^{s+\beta}}\leq C.$

Therefore there exist a subsequence of ${v_{k}}$, still denoted by ${v_{k}}$,
such that

$v_{k}\rightarrow v\qquad in\ C_{loc}^{s}(\mathbb{R}^{N}),$ (4.20)

and $v$ solves

$-\Delta v=v^{\lambda},\qquad x\in\mathbb{R}^{N}.$ (4.21)

By Lemma 3.1 we deduce

$v=const,\qquad x\in\mathbb{R}^{N}.$

This contradicts with (4.19), thus we have proved Theorem 2.6. ∎

Acknowledgements. This study was supported by grants from the National Natural
Science Foundation of China (11871070), the Guangdong Basic and Applied Basic
Research Foundation (2020B151502120), the Fundamental Research Funds for the
Central Universities (20ykzd20).

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# Agile Software Engineering and Systems Engineering at SKA Scale

Juande Santander-Vela1 1SKA Organisation, Lower Withington, Cheshire, UK;
<EMAIL_ADDRESS>
###### Abstract

Systems Engineering (SE) is the set of processes and documentation required
for successfully realising large-scale engineering projects, but the classical
approach is not a good fit for software-intensive projects, especially when
the needs of the different stakeholders are not fully known from the
beginning, and requirement priorities might change. The SKA is the ultimate
software-enabled telescope, with enormous amounts of computing hardware and
software required to perform its data reduction. We give an overview of the
system and software engineering processes in the SKA1 development, and the
tension between classical and agile SE.

##  1 Introduction

The Square Kilometre Array (SKA) is an international project with the aim of
building multi-purpose radio telescopes, with an equivalent collecting area of
at least one square kilometre. It will provide unprecedented sensitivity and
survey speed, so that key questions in modern astrophysics and cosmology can
be answered. Currently, Phase 1 of the SKA (SKA1) is under design, nearing
Critical Design Review (CDR) status.

The data rates generated by each of the SKA1 telescopes is in the order of 1
terabyte/s at the interface with the Central Signal Processor (CSP). The
amount of processing required by the CSP to deal with that traffic is 50% of
the current most powerful supercomputer in the world,11198 PFLOPS as of June
2017, see https://top500.org even if they are simple Multiply-Accumulate
operations, and the Science Data Processor (SDP) will require in the order of
2.5 times the capability of the current most powerful computer in the world.

Eleven international Consortia, whose members span 17 timezones,222From
Vancouver (GMT-9) to Sydney (GMT+12); if we include New Zealand, and the
difference between daylight savings in northern and southern hemispheres, we
can reach 19-21 hours; the sun always shines over at least one SKA
contributor.A are designing each of the subsystems. However, the SKA
Organisation (SKAO) is the design authority, with Integrated Engineering Teams
(IETs) per Element with Systems Engineering, Project Management, Operations,
and Science considerations always represented. The External IETs include the
counterparts from Consortia, and we have Telescope Teams which can spawn
Resolution Teams when issues need to be discussed and resolved across the
whole project. SKAO retains ownership of the L1 requirements, System
Architecture, System Baseline Design, and Functional Analysis, and after CDR
we will have ownership of all system and subsystem requirements, and the SE
and design artefacts from Consortia. More information of SKA System
Engineering practices and challenges can be found in (Cremonini et al. 2016).

##  2 Systems Engineering vs Agile development principles

Systems Engineering (SE) is an interdisciplinary approach and means to enable
the full life cycle of successful systems, including problem formulation,
solution development, and operational sustainment and use (INCOSE 2015).
During all of those phases, understanding of the system increases, and changes
are inevitable. _Managing change_ , and making sure change happens while
keeping a working system, is central to SE.

However, in most of the typical SE processes there is the underlying
assumption that _change will decrease with time_ , which is typically true for
hardware/mechanical systems, but _tends not to be true for software heavy
systems_.

The main principles from the Agile Manifesto333http://agilemanifesto.org are:

  * •

_Individuals and interactions_ over processes and tools;

  * •

_Working software_ over comprehensive documentation;

  * •

_Customer collaboration_ over contract negotiation;

  * •

_Responding to change_ over following a plan

Following those principles, the traditional Software Engineering Management
Plan becomes more a framework for the best practices that need to be followed
—pull requests between branches/forks versus commits to master, code reviews,
Test Driven Design—, and not a final blueprint on how the software will be
managed.

It also means that traditional Work-Breakdown-Structure oriented approaches
don’t fully work, and that instead, _what is finally built at each stage is a
result of agreements with stakeholders_, as we will:

  * •

Keep _communication with all stakeholders_ involved,with speedy and clear
dissemination of agreements;

  * •

To have _continuous integration of engineering artifacts_ , and software;

  * •

To _hire software developers (or development entities) that are invested in
the system_ , not merely contracted; but also sharing gains with contractors
if something is done faster than expected, or some desirable metrics are
exceeded

  * •

To _have systems and processes that accommodate system (and software) change_
and can still _prove requirements are satisfied_ ; this means _not using a
document-centric requirements management system_ , but being able to
_integrate requirements, development, testing, and results_ with a quick
turnaround.

And all of this is made on a _fixed cadence_ —with the fastest period being
the _sprint_ —, trying to keep momentum, allowing teams to self-organise their
prioritised tasks.

We must not forget that the reason to become agile is to embed quality in the
system being developed, and that the system is useful to the different
stakeholders, which is also a goal of SE. Rees (2018) expands on quality and
software.

##  3 Marrying SE with Agile

Successfully marrying Agile principles and SE principles means that, instead
of making sure the SE processes are correct so that we remove the human factor
as a source of mistakes, and that we can prove that the system, as designed,
will behave as intended when built, _we focus on delivering a minimum viable
system as soon as possible, constructed from minimum viable subsystems, and we
keep adding functionality_ , making sure we keep the intended behaviours that
we have already built, and that we incorporate the still not incorporated
behaviours as development progresses.

That means that we need metrics to keep track of the progress towards design
completion during the design phase —the downward part of the classical V—, and
then to record validation of the design artefacts —upward part of the V—. But
you also want to be able to support those processes even when the design phase
does not go through a traditional V, but it is made of multiple, indented Vs,
or can be evolved, and the processes followed for each potential delivery.

If the killer function of Agile development is Continuous Integration, and be
always available to deliver working software, _the killer function for Agile
SE is Continuous Inspection_ , and be always able to deliver status of
compliance, and of progress, almost at any moment, and on cadence.

##  4 SE functions during SKA pre-construction and construction

During pre-construction, the main SE function from SKAO is the supervision of
Interface Control Documents (ICDs), but also assessing design compliance
against L2 requirements, through compliance matrices, and evidence for
compliance. Many statements can be easily justified through functional
allocations, while others require architectural support in order for them to
be met.

Given the different levels of design maturity at system and elevent level, and
of adherence to SE principles across Consortia, and the relatively small
amounts of SE personell at SKAO, tradicional processes with full flown-down
from Design Reference Mission, to Operational Concept, to Functional Analysis,
to System Requierements have not been fully possible. Rather, iterations over
the available detail of those SE artefacts have been steadily increasing
definition, with the drawing of Element and System PDR baselines, and future
Element and System CDR baselines.

In all these processes, we have required submissions of some design and SE
artifacts linked with individual milestones. SKAO has a unified system for
Configuration Management,
eBentley444https://www.bentley.com/en/products/product-line/asset-
performance/assetwise-alim (eB), that is also used by all Consortia. However,
in spite of SKAO offers to Consortia for usage of our own Requirements
Management system (Jama555http://jamasoftware.com/), it has not been
standardised, and requirement submissions require manual processes from the
SKAO SEs. There is still a large amount of engineering understanding required,
and the process enables a less mechanistic, more engineering-centric review of
requirements, but it is also more prone to pitfalls.

For SKA1 construction, the Scaled Agile
Framework666http://scaledagileframework.com (SAFe) has been selected as the
way software development will be performed. SAFe allows for recursive
groupings of Agile teams (using Scrum) into Agile Release Trains (ARTs) —that
also use Scrum for managing the Train backlog, that then gets assigned to the
ART backlog(s)—, and can group multiple ART into a Large Solution. As
previously indicated, the SE function will be organised from the SKA Inter-
Governmental Organisation (IGO) that will exist before triggering
construction. However, it needs to trickle down to the Agile Release Trains,
and to the individual agile teams. Part of it has to go through the figure of
the product owners, and consumer representatives, but there must be a strong
presence of the system’s mission and behaviour, and the Operations Concept
Document, so that it is really implemented. We inted to define additional
metrics from the inspection of the backlogs, and the processing of the
outcomes of the Program Increment demos.

##  5 Conclusions

Applying Systems Engineering _by the book_ is never a good idea. It is even
less ideal for projects that are optimising a multi-valued function in the
cost, science capability, and time-to-construction, among other dimensions.
Processes need to be tailored, and they need to be kept responsive, while at
the same time keeping track of the SE deliverables that are required as design
artefacts, and managing the risk of the uncertainties that are still carried
forward to the construction phase. When software is involved, design needs to
focus on documenting the system architecture, instead of designing down to the
application or module level, as agile methods will require continuous
adaptation of the modules to be developed. It also requires definition and
constant evaluation of the key performance metrics that will be used during
the construction phase, and be able to translate between SE artefacts such as
ICDs, the software architecture, functional analysis, and the needs for the
different releases. We have been applying evidence-based SE artefacts by
different means, and found the shortcomings of the approach, specially when
dealing with a world-wide distributed team with no traditional line of
command.

## References

  * Cremonini et al. (2016) Cremonini, A., Caiazzo, M., Hayden, D., Labate, M. G., Olgin, R., & Santander-Vela, J. 2016, in Modeling, Systems Engineering, and Project Management for Astronomy VI, vol. 9911 of Proc. SPIE, 99110W
  * INCOSE (2015) INCOSE 2015, Systems Engineering Handbook (Wiley)
  * Rees (2018) Rees, N. 2018, in ADASS XXVII, edited by TBD (San Francisco: ASP), vol. TBD of ASP Conf. Ser., TBD