# Observation of Microcanonical Atom Number Fluctuations in a Bose-Einstein
Condensate

M. B. Christensen  T. Vibel  A. J. Hilliard  Center for Complex Quantum
Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade
120, DK-8000 Aarhus C, Denmark.  M. B. Kruk  K. Pawłowski  Center for
Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668
Warsaw, Poland.  D. Hryniuk  Center for Theoretical Physics, Polish Academy of
Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland.  Institute of Physics,
Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland  K.
Rzążewski  Center for Theoretical Physics, Polish Academy of Sciences, Al.
Lotników 32/46, 02-668 Warsaw, Poland.  M. A. Kristensen  J. J. Arlt  Center
for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark.

###### Abstract

Quantum systems are typically characterized by the inherent fluctuation of
their physical observables. Despite this fundamental importance, the
investigation of the fluctuations in interacting quantum systems at finite
temperature continues to pose considerable theoretical and experimental
challenges. Here we report the characterization of atom number fluctuations in
weakly interacting Bose-Einstein condensates. Technical fluctuations are
mitigated through a combination of nondestructive detection and active
stabilization of the cooling sequence. We observe fluctuations reduced by
$27\text{\,}\mathrm{\char 37\relax}$ below the canonical expectation for a
noninteracting gas, revealing the microcanonical nature of our system. The
peak fluctuations have near linear scaling with atom number $\Delta
N_{0,\mathrm{p}}^{2}\propto N^{1.134}$ in an experimentally accessible
transition region outside the thermodynamic limit. Our experimental results
thus set a benchmark for theoretical calculations under typical experimental
conditions.

Bose-Einstein condensates (BECs) have become a cornerstone of current
developments in quantum simulation Bloch2012, and one would expect a complete
description of their properties to be available. However, in spite of
considerable theoretical effort, the atom number fluctuations between the
thermal and condensed component of an interacting Bose gas at relevant atom
densities are still not fully understood. In principle, such a quantum system
can be described through all moments of its probability distribution, which
have indeed not been obtained for large interacting BECs Politzer1996;
Navez1997; Giorgini1998; Meier1999; Kocharovsky2006. To date, experiments have
been limited to the first moment corresponding to the condensate fraction
Gerbier2004; Meppelink2010; Tammuz2011 as well as the critical temperature
indicating the onset of condensation Gerbier2004a; Smith2011. Only recently
BEC atom number fluctuations corresponding to the second moment have become
experimentally accessible Kristensen2019.

These fluctuations pose a subtle problem with surprising challenges and rich
physics Kocharovsky2006. Historically, a description of the noninteracting
Bose gas was developed within the grand canonical ensemble, which shows
unphysically large fluctuations below the critical temperature Ziff1977. This
is referred to as the grand canonical catastrophe Grossmann1996; Holthaus1998,
and is one of the few examples where the predictions of different statistical
ensembles differ dramatically. Thus, any description of the fluctuations must
be based on canonical or microcanonical approaches.

Figure 1: Atom number fluctuations of a BEC. (a) Calculated variance of BEC
atom number as a function of relative temperature in the canonical (red
diamonds) and microcanonical (blue circles) ensembles, with (empty symbols)
and without (filled symbols) interactions at a scattering length
$a=$100\text{\,}\mathrm{a_{0}}$$ for $N=1000$ 87Rb atoms in a trap with
$\omega_{a}=2\pi\times$20.0\text{\,}\mathrm{Hz}$$ and $\lambda=10$. (b)
Experimental variance of BEC atom number as a function of temperature (blue
points). The error bars represent statistical uncertainties of the variance
(vertical) and 1-$\sigma$ spread in temperature (horizontal). The dashed line
is a fit (see text), where the blue band represents the uncertainty of the
fit. The gray area indicates the offset due to technical fluctuations. The
aspect ratio was $\lambda=$7.28\pm 0.02$$ and the atom number at peak
fluctuations was $N_{\mathrm{p}}=$(1.49\pm 0.06)\text{\times}{10}^{5}$$. The
results are plotted as a function of the temperature rescaled with the
temperature at peak fluctuations. The solid black lines in both panels are
exact canonical calculations for a noninteracting gas Weiss1997.

In the canonical ensemble, the asymptotic behavior of the condensate atom
number variance in the noninteracting Bose gas was first obtained by Politzer
Politzer1996. Later, exact numerical recursion relations for this
noninteracting case were found Weiss1997, which recently allowed for an
initial comparison with experimental results Kristensen2019. However, the
canonical ensemble assumes the existence of an external reservoir exchanging
energy with the system under investigation. This assumption is not met in the
case of an ultracold gas isolated from the environment. Therefore, the
microcanonical ensemble needs to be invoked, which is numerically demanding
and currently does not allow for a full comparison at typical experimental
parameters.

In the microcanonical ensemble the key quantity is the partition function
$\Gamma(E,\,N)$, i.e. the number of ways to distribute the energy $E$ between
$N$ atoms. Calculation of $\Gamma(E,\,N)$ for a noninteracting gas in a
harmonic trap is closely related to the classic mathematical problem of
partitions. Discussions between Leibniz and Bernoulli inspired intensive work
on this topic Dickson1971, with breakthroughs achieved a century ago by
Ramanujan and Hardy Hardy1918. In particular it was shown that the canonical
and microcanonical fluctuations agree for large atom number in the
noninteracting 1D gas Grossmann1997.

The results for the 3D case in the microcanonical ensemble were not known
until 1997 Navez1997, when a fourth statistical ensemble, called the Maxwell’s
demon ensemble, was proposed. In this ensemble the system exchanges particles
with the reservoir, without changing its energy. It is applied to the thermal
cloud at sufficiently low temperatures, where the BEC comprises the reservoir.
The ensemble yields an asymptotic expression for the microcanonical
fluctuations in the noninteracting gas 111The Maxwell’s demon ensemble yields
the variance of the number of atoms in the thermal cloud. Due to conservation
of the total number of atoms, this also exactly corresponds to the variance of
the number of atoms in the condensate.

$\Delta
N_{0}^{2}=\left(\frac{\zeta(2)}{\zeta(3)}-\frac{3\zeta(3)}{4\zeta(4)}\right)N\left(\frac{T}{T_{c}^{0}}\right)^{3}.$
(1)

Here $T_{c}^{0}$ is the critical temperature of a noninteracting gas in a
harmonic trap and $\zeta(x)$ is the Riemann zeta function. In particular, the
second term corresponds to the reduction in the number fluctuations in the
microcanonical ensemble with respect to the canonical result given by the
first term. Thus, the ratio between the canonical and microcanonical
fluctuations in the 3D isotropic case is 0.39, corresponding to a reduction by
$61\text{\,}\mathrm{\char 37\relax}$.

Importantly, interactions are expected to modify the fluctuations of the
condensate atom number $\Delta N_{0}^{2}$. In the homogeneous case, initial
results showed that interactions suppress the fluctuations by a factor of two
at very low temperatures due to a strong pair correlation of the atoms
reducing the number of degrees of freedom Giorgini1998; Meier1999;
Zwerger2004. Moreover, interactions have been included using numerous methods
such as the aforementioned Maxwell’s demon ensemble Idziaszek1999, number-
conserving quasiparticle methods Kocharovsky2000a; Kocharovsky2000, master-
equation and hybrid quasiparticle approaches Svidzinsky2006; Svidzinsky2010,
and a correlated many-body approach Bhattacharyya2016. However, only some of
these methods are based on a microcanonical ensemble and none of them allow
for the analysis of harmonically trapped interacting BECs at typical
experimental atom numbers. Moreover, most approaches are only valid at low
temperature and do not capture the peak fluctuations near
$T_{\mathrm{c}}^{0}$.

Additionally, the fluctuations are expected to scale anomalously with atom
number: $\Delta N_{0}^{2}\propto N^{1+\gamma}$, with $\gamma\neq 0$
Giorgini1998; Meier1999. This is in stark contrast to most classical systems
where the central limit theorem ensures normal scaling of the fluctuations
$\Delta N^{2}\propto N$ due to the existence of a finite microscopic coherence
length scale Zwerger2004; Yukalov2005a. The characterization of BEC atom
number fluctuations is thus an important outstanding challenge that has been
hindered by technical fluctuations of BEC experiments until recently. This
technical limitation was mitigated in our recent experiments, enabling the
current experimental characterization of BEC fluctuations.

In this Letter, we characterize the atom number fluctuations in a weakly
interacting BEC as a function of atom number and trapping geometry. To avoid
technical fluctuations, BECs with a well-controlled atom number and
temperature are prepared through a combination of nondestructive detection and
active stabilization of the cooling sequence. We quantitatively analyze the
fluctuations in two ways. The fluctuations are compared to an exact canonical
calculation for the noninteracting Bose gas, which shows fluctuations reduced
by $27\text{\,}\mathrm{\char 37\relax}$ with respect to the canonical
expectation, indicating the microcanonical nature of the system. Second, the
fluctuations are fitted with a phenomenological model showing anomalous atom
number scaling with an exponent $\gamma=$0.134\pm 0.005$$, which we compare to
the expectation in the thermodynamic limit.

Figure 2: Ratio, $S$, between the calculated peak variance of the
noninteracting BEC in the microcanonical and canonical ensembles as a function
of the trap aspect ratio $\lambda$ for $N=10^{2}$ (blue points), $N=10^{3}$
(green diamonds), $N=10^{4}$ (red triangles) and $N=10^{5}$ (purple square)
atoms. The black dashed line corresponds to the asymptotic value estimated
within the Maxwell’s demon ensemble (see Eq.(1)). The three experimental
points with the lowest atom number
$N_{\mathrm{p}}=$8.8\text{\times}{10}^{4}$-$10.4\text{\times}{10}^{4}$$ are
shown for comparison (open points). The typical error bar shown indicates the
$68.2\text{\,}\mathrm{\char 37\relax}$ confidence interval on the experimental
value. (Inset) $S$ as a function of the number of atoms for fixed aspect ratio
$\lambda=6$.

In a first step, we theoretically estimate the importance of using an
interacting microcanonical ensemble with respect to the more readily available
canonical ensemble calculations. This is achieved for ensembles of up to
$10^{5}$ atoms using a new computational method. In brief, we apply the usual
Metropolis algorithm but perform a random walk in the space of Fock states,
accounting for Bose enhancement with perturbatively included interactions. The
random walk results in the set of visited Fock states obeying canonical
ensemble statistics. By post-selecting samples according to their energy, we
obtain representative “microstates” of the microcanonical ensemble
SupplementaryMaterial. Figure 1(a) show the results for $N=1000$ atoms for
trap aspect ratio $\lambda=10$. Importantly, the microcanonical result lies
clearly below the canonical expectation as indicated by Eq. (1). In the
elongated case shown here, the ratio, $S$, between the peak variance in the
microcanonical and canonical ensembles is $S=0.72$, corresponding to a
reduction by $28\text{\,}\mathrm{\char 37\relax}$ for this atom number.
Overall, it is evident from Fig. 1(a) that the effect of interactions is small
and leads to a minor increase of the variance in both ensembles.

Given the small effect of interactions, we calculate $S$ for a range of aspect
ratios and atom numbers for the noninteracting case as shown in Fig. 2. For
low aspect ratios $\lambda$ and large atom numbers the ratio $S$ tends towards
the limiting value 0.39 given by Eq. (1). In highly elongated traps with large
aspect ratios, $S$ tends towards 1, since microcanonical and canonical
fluctuations are identical in the noninteracting 1D case. The dependence of
$S$ on the atom number is shown in Fig. 2 (inset) for $\lambda=6$. This
indicates that the microcanonical and canonical results scale differently with
atom number. In particular we observe a roughly logarithmic dependence of $S$
on $N$ for atom numbers $10^{2}<N<10^{4}$. These results show that a clear
reduction of the fluctuations below the canonical result is expected under
realistic conditions and motivates the following experimental investigation.

The experimental apparatus used to produce BECs was previously described in
detail in Ref. Kristensen2017. Briefly, around $10^{9}$ ${}^{\text{87}}$Rb
atoms are first captured and cooled in a magneto-optical trap. The cloud is
then optically pumped to the $\left|F=2,m_{F}=2\right\rangle$ state. Then it
is transported into a cigar-shaped quadrupole-Ioffe-configuration magnetic
trap with axial trapping frequency $\omega_{a}$ and radial frequency
$\omega_{\rho}$ SupplementaryMaterial, where it is cooled by radio-frequency
(rf) forced evaporation.

The sequence outlined above is typically subject to technical fluctuations
that prevent the observation of atom number fluctuations. To overcome this
challenge, a stabilization procedure is initiated when the cloud contains
$\sim 4\times 10^{6}$ atoms at a temperature
$14\text{\,}\mathrm{\SIUnitSymbolMicro K}$. The cloud is probed using
minimally destructive Faraday imaging, and the atom number is corrected using
a weak rf pulse of controllable duration. The pulse removes excess atoms, and
the outcome of the stabilization procedure is verified by a second Faraday
measurement. This allows us to prepare a well-controlled number of atoms with
a relative stability at the $10^{-4}$ level Gajdacz2016; Kristensen2017.

For efficient cooling towards BEC, a tight magnetic trap and consequently
large collision rate is desirable. In its most compressed configuration, our
trap has an aspect ratio $\lambda=17$. This can lead to significant phase
fluctuations across the spatial extent of the cloud Hellweg2003, which evolve
into density modulations during time of flight (TOF) and thus hinder the
precise determination of atom number and temperature. We therefore decompress
the magnetic trap for our measurements, limiting the aspect ratio to
$4.5<\lambda<10$ where the phase coherence length is larger than the
condensate length in the long direction.

In the final step of the experimental sequence, BECs are produced by forced
evaporation ending at an rf-frequency corresponding to the desired BEC
temperature. To ensure proper thermal equilibrium, the BEC is first held at
the final rf frequency for $800\text{\,}\mathrm{ms}$ and a further
$400\text{\,}\mathrm{ms}$ without rf radiation before the trap is turned off.

The clouds are probed after $35\text{\,}\mathrm{ms}$ TOF expansion using
resonant absorption imaging on the $F=2\rightarrow F^{\prime}=3$ cycling
transition. Fringes in the processed image due to vibrations of optical
components are mitigated by minimizing the time-delay between the absorption
and reference image. Optical pumping is applied between these images to
transfer the atoms to the transparent $F=1$ state. Thus, the atom and beam
images can be taken only $340\text{\,}\mathrm{\SIUnitSymbolMicro s}$ apart,
limited only by the camera shift speed.

For each experimental configuration corresponding to a chosen aspect ratio and
atom number, the fluctuations were measured as a function of the temperature
$T$. At every temperature (corresponding to an rf end frequency) a set of 60
measurements according to the experimental sequence outline above was taken
222Measurements for which the stabilization process failed were excluded, such
that each set includes at least 45 measurements..

Figure 3: Peak fluctuations of a BEC. (Data points) Observed peak variance of
the BEC atom number as a function of the atom number at peak variance
$N_{\mathrm{p}}$ and trap aspect ratio $\lambda$. The thick black bars
indicate the $68.2\text{\,}\mathrm{\char 37\relax}$ confidence interval.
Narrow stems connect the data points to the base to guide the eye. (Surface)
Theoretical expectation for a noninteraction canonical ensemble scaled
according to Eq. (6).

To determine the BEC atom number in each image, the wings of the cloud are fit
with a Bose-enhanced thermal distribution from which the temperature is
obtained. The fitted distribution is subtracted from the image, and the BEC
atom number $N_{0}$ is obtained by integration of the remaining density.
However, the variance cannot be determined directly from $N_{0}$, since small
remaining drifts of the magnetic offset field lead to minor temperature
variations with a median standard deviation of $\sim$3\text{\,}\mathrm{nK}$$.
We eliminate this drift by subtracting a linear fit of the condensate number
as a function of the total atom number Kristensen2019 and determine the
residuals $\eta_{i}$, where $i$ indicates the order in time. The BEC atom
number variance is then given by a two-sample variance of the residuals

$\Delta
N_{0}^{2}=\frac{1}{2}\left\langle\left(\eta_{i+1}-\eta_{i}\right)^{2}\right\rangle.$
(2)

Thus, this two-sample variance contains the BEC fluctuations and detection
noise, but excludes slow technical drifts. Figure 1(b) shows this variance of
the measurements as a function of $T$ for a given configuration of the
experiment.

The peak fluctuations are determined by a fit to $\Delta N_{0}^{2}$ also
illustrated in Fig. 1(b). The fit model is inspired by the asymptotic behavior
of the fluctuations in a noninteracting gas $\Delta
N_{0}^{2}=\zeta(2)\left(\frac{k_{\mathrm{B}}}{\hbar\overline{\omega}}\right)^{3}T^{3}$,
where $\overline{\omega}$ is the geometric mean of the trapping frequencies
Politzer1996. Moreover, the fluctuations decay in near steplike manner close
to the critical temperature, which we model with a Heaviside step function
$\Theta(T_{p}-T)$, where $T_{\mathrm{p}}$ is the temperature at the peak
fluctuations. To account for small temperature drifts the expression is
furthermore convolved with a normal distribution $\mathcal{N}(T,\sigma_{T})$
centered on the temperature $T$ with a standard deviation $\sigma_{T}$ given
by the median of the measured temperature variation. Thus, we fit the data
with the model

$\Delta N_{0}^{2}(T)=\left(f\ast g\right)(T)+\mathcal{O}$ (3)

where $f$ and $g$ are given by

$\displaystyle f(T)$ $\displaystyle=\Delta
N_{\mathrm{0,p}}^{2}\left(\frac{T}{T_{\mathrm{p}}}\right)^{3}\Theta(T_{\mathrm{p}}-T),$
(4) $\displaystyle g(T)$ $\displaystyle=\mathcal{N}(T,\sigma_{T}).$ (5)

The three fit parameters are the peak atom number variance $\Delta
N_{\mathrm{0,p}}^{2}$, $T_{\mathrm{p}}$ and an offset $\mathcal{O}$ which
accounts for experimental noise Kristensen2019. Since the atom number varies
as a consequence of the evaporation, we determine the atom number at peak
fluctuations $N_{\mathrm{p}}$ from a spline interpolation of the atom number
as a function of the temperature for all measurements in a given
configuration, evaluated at $T_{\mathrm{p}}$. Figure 3 shows the measured peak
atom number variance for all 13 experimental configurations of aspect ratio
and atom number (for a tabular presentation of the data see Supplemental
Material SupplementaryMaterial). The three points at lowest atom number are
also included in Fig. 2.

We quantitatively evaluate the peak fluctuations using two approaches. First,
we compare the measured fluctuations to the expected fluctuations in a
canonical noninteracting gas, which can be calculated under conditions
comparable to the experiment. Second, the fluctuations are investigated using
a phenomenological model to probe their dependence on atom number and trapping
geometry.

Figure 4: Scaling model for peak atom number fluctuations of a BEC. (Data
points) Normalized peak variance of the BEC atom number as a function of
$N_{\mathrm{p}}$ and aspect ratio $\lambda$. The thick black bars indicate the
$68.2\text{\,}\mathrm{\char 37\relax}$ confidence interval. Narrow stems
connect the data points to the base to guide the eye. (Surface)
Phenomenological model given by Eq. (7) of the fluctuations.

To compare experiment and theory the average ratio $S$ between the observed
fluctuations and the exact theoretical result for the noninteracting canonical
ensemble $\Delta N_{\mathrm{0,p,can}}^{2}$ is evaluated. We obtain $S$ from a
fit of

$\Delta N_{\mathrm{0,p}}^{2}=S(1+a\,\delta\lambda+b\,\delta\\!\ln(N))\Delta
N_{\mathrm{0,p,can}}^{2},$ (6)

inspired by the scalings with atom number and aspect ratio in Fig. 2. The
coefficients $a$ and $b$ allow for a linear scaling with the trap aspect ratio
and the logarithm of the atom number, where $\delta x=x-\langle x\rangle$
denotes that the mean over the set of experimental parameters has been
subtracted. This yields an average ratio $S=$0.73\pm 0.05$$ corresponding to a
$27\text{\,}\mathrm{\char 37\relax}$ reduction of the fluctuations and clearly
reveals the microcanonical nature of our system with reduced fluctuations
below the canonical ensemble expectation. The coefficients $a=$0.04\pm 0.04$$
and $b=$0.24\pm 0.26$$ represent a correction to $S$ which are barely resolved
within our data range, supporting the interpretation that the microcanonical
nature of the system constitutes the most important deviation from the
canonical result.

We use a complementary approach to analyze the scaling of the fluctuations
with atom number and aspect ratio $\lambda$. Inspired by Eq. (1), we use the
phenomenological model

$\Delta N_{0,\mathrm{p}}^{2}/N=a+b\lambda N^{\gamma},$ (7)

where $a$, $b$ and $\gamma$ are fit coefficients allowing for linear aspect
ratio and a nonlinear number dependence. In particular, the exponent $\gamma$
quantifies the degree of anomalous atom number scaling.

We first fit the exact canonical result $\Delta N_{\mathrm{0,p,can}}^{2}$ with
this model which yields $a=$1.327\pm 0.014$$, $b=$1.5\pm 0.5$$ and
$\gamma=$-0.27\pm 0.03$$. Thus, the coefficient $a$ is close to its limiting
value $\zeta(2)/\zeta(3)=1.37$ corresponding to the first term in Eq. 1. More
significantly, the negative sign of $\gamma$ combined with the relatively
large value of $b$ show that even the exact theoretical result scales
anomalously with $N$. This can be attributed to the transition from a system
with reduced dimensionality at low atom numbers to the limiting 3D case
Kristensen2019. Hence, we do not expect our experiment to reflect the
thermodynamic limit, where the scaling has been investigated theoretically
Giorgini1998; Idziaszek1999; Meier1999.

Figure 4 shows the experimental data and the fit according to Eq. (7) using a
maximum likelihood method. Contrary to the noninteracting theoretical case, we
find anomalous scaling with $\gamma=$0.134\pm 0.005$>0$. We interpret our
result as being due to the interplay between the dimensionality effects in a
microcanonical system and interaction effects, which are predicted to yield
$\gamma=1/3$ in the thermodynamic limit Giorgini1998; Meier1999. In
particular, Fig. 2 shows that a microcanonical system generally scales
different from a canonical one in both $\lambda$ and $N$. Since $\gamma>0$,
Eq. (7) diverges and the coefficient $a=$0.55\pm 0.04$$ cannot be interpreted
as a limiting value. Moreover, $b=$0.020\pm 0.012$$ is very small due to the
large value of $N^{\gamma}$. This result thus provides a simple analytical
model as a benchmark for future theoretical investigations.

In conclusion, we have measured the peak fluctuations of a weakly interacting
Bose-Einstein condensate as a function of atom number and trap aspect ratio.
The significant differences between a canonical and microcanonical ensemble
description of these fluctuations were investigated numerically by post-
processing ensembles generated via the quantum Metropolis algorithm.
Experimentally the fluctuations were found to be reduced by
$27\text{\,}\mathrm{\char 37\relax}$ relative to the expectation for a
canonical gas, in qualitative agreement with our theoretical prediction. This
clearly demonstrates the micro-canonical nature of the system and thus guides
future theoretical work. In addition the peak fluctuations were found to
follow a weakly anomalous scaling with the total atom number $\Delta
N_{0p}^{2}\propto N^{1.134}$. We interpret the scaling as an interplay between
dimensional effects and the limiting case in the thermodynamic limit with
$\Delta N_{0p}^{2}\propto N^{4/3}$.

Future experiments will aim to observe the slight increase of the fluctuations
due to interactions, predicted by our theoretical calculation. To date, this
effect is shrouded by measurement uncertainties and the dependence of the
fluctuations on trapping geometry. Moreover, the use of larger atomic clouds
will provide experimental insight in the fluctuations in the thermodynamic
limit Giorgini1998; Meier1999. Finally, technical improvements will allow for
larger data sample sizes and enable the investigation of the time dependence
and higher moments of the atom number distribution Zeiher2020.

###### Acknowledgements.

M.B.C., T.V., A.J.H., M.A.K., and J.J.A. acknowledge support by the Villum
Foundation, the Carlsberg Foundation, and the Danish National Research
Foundation through the Center of Excellence “CCQ” (Grant agreement No.
DNRF156). K.R., M.B.K. and D.H. acknowledge support from the (Polish) National
Science Center Grant No. 2018/29/B/ST2/01308. K.P. acknowledges support from
the (Polish) National Science Center Grant No. 2019/34/E/ST2/00289. This
research was supported in part by PLGrid Infrastructure. Center for
Theoretical Physics of the Polish Academy of Sciences is a member of the
National Laboratory of Atomic, Molecular and Optical Physics (KL FAMO). MBK
acknowledges support from the (Polish) National Science Center Grant No.
2018/31/B/ST2/01871. This article has been supported by the Polish National
Agency for Academic Exchange under Grant No. PPI/ PZA/2019/1/00094/U/00001.


# Quantitative comparison results for first-order Hamilton-Jacobi equations

V. Amato, L. Barbato

###### Abstract

In this paper, we prove a quantitative version of the comparison result for
solutions to first-order Hamilton-Jacobi equations proved in [22]. The key
role is played by quantitative versions of the Pólya-Szegő inequality and of
the Hardy-Littlewood inequality.
Keywords: Hamilton-Jacobi equations, stability, comparison results.
MSC 2020: 35F21, 35B35, 35B51.

##  1 Introduction

In this paper, we will deal with symmetrization in the class of first-order
Hamilton-Jacobi equations.

One of the earliest references on this topic is [22], where the authors treat
the following problems involving the Hamilton-Jacobi equations

Here $\Omega^{\sharp}$ stands for the ball centered at the origin with the
same measure as $\Omega$ (in the sequel just centered ball),
$H:\operatorname{\mathbb{R}}^{n}\to\operatorname{\mathbb{R}}$ and
$K:\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$ are measurable
functions, $f\in L^{p}(\Omega)$ and $f_{\sharp}$ is the increasing
rearrangement of $f$ (see Section 2 for its definition).

In particular, under some suitable conditions on $H$ and $K$, it is proven
([22, Theorem 2.2]) that if $u,u^{\text{G}}$ solve (1a) and (1b) respectively,
it holds

$\big{\lVert}u\big{\rVert}_{L^{1}(\Omega)}\leq\big{\lVert}u^{\text{G}}\big{\rVert}_{L^{1}(\Omega^{\sharp})}.$
(2)

In [7, 29] the authors characterize the case of equality in (2). In
particular, for $H(\cdot)={\left|\cdot\right|}$ they find out that if equality
occurs in (2), then up to a translation $\Omega$ is a ball, and both $u$ and
$f$ are radially symmetric.

In such a case, it is natural to ask if inequality (2) can be improved in a
quantitative version. More precisely, if the quantity

${\big{\lVert}u^{\text{G}}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}-{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}$

is small, can we say that $\Omega$ is close to a ball, and both $u$ and $f$
are almost radially symmetric? While the notion of the closeness of a set to a
ball is now classical and we can refer, for example, to the Fraenkel’s index
of asymmetry (see for example [20]), the notion of almost radiality is not as
common. In many papers, see for instance [3, 11, 12], the authors use an
$L^{p}-$norm of kind

$\lVert g-g^{\sharp}\rVert_{L^{p}(\operatorname{\mathbb{R}}^{n})},$

where $g\in L^{p}(\operatorname{\mathbb{R}}^{n})$ and $g^{\sharp}$ is its
decreasing rearrangement.

The case of our interest is when $H(\cdot)={\left|\cdot\right|}$ since the
rigidity is proved in such a case. Namely, we consider the following Hamilton-
Jacobi equations

Before stating our main theorem, we recall the Fraenkel asymmetry index, i.e.

$\alpha(\Omega):=\min_{x\in\operatorname{\mathbb{R}}^{n}}\bigg{\\{}\dfrac{|\Omega\triangle
B_{r}(x)|}{|B_{r}(x)|}\;,\;|B_{r}(x)|=|\Omega|\bigg{\\}},$

where the symbol $\triangle$ stands for the symmetric difference.

Hence, our main result is the following.

######  Theorem 1.1.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $f\in L^{\infty}(\Omega)$ be a non-negative function with

$0<m\leq f\leq M.$

Suppose that $u$ and $u^{\text{G}}$ are the solutions to (3a) and (3b),
respectively. Then, there exist some positive constants $\theta=\theta(n)$ and
$C_{1}:=C_{1}(n,{\left|\Omega\right|},f^{*}),\,C_{2}:=C_{2}(n,{\left|\Omega\right|},f^{*})$
such that

$C_{1}\alpha^{3}(\Omega)+C_{2}\inf_{x_{0}\in\operatorname{\mathbb{R}}^{n}}{\big{\lVert}u-u^{\sharp}(\cdot+x_{0})\big{\rVert}}_{L^{1}(\operatorname{\mathbb{R}}^{n})}^{\theta}\leq{\big{\lVert}u^{G}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}-{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}.$
(4)

Moreover, the dependencies of $C_{1}$ and $C_{2}$ from ${\left|\Omega\right|}$
and $f^{\ast}$ are explicit.

On the source term, we have much less information. To be more precise, we
consider $\mu(t)={\left|\Set{x\in\Omega\,:\,{\left|u(x)\right|}>t}\right|}$
the distribution function of $u$ and we consider $f_{u}$, i.e. a rearrangement
of $f$ such that each level set agrees with the one of $u$, i.e.

$f_{u}(x)=f^{\ast}(\mu(u(x))).$

Then the following proposition holds true.

######  Proposition 1.2.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $f\in L^{\infty}(\Omega)$ be a non-negative function with

$0<m\leq f\leq M.$

Suppose that $u$ and $u^{\text{G}}$ are the solutions to (3a) and (3b),
respectively. Then, there exist a positive constant
$C_{3}:=C_{3}(n,{\left|\Omega\right|},f^{*}),$ such that

$C_{3}{\big{\lVert}f-f_{u}\big{\rVert}}_{L^{1}(\operatorname{\mathbb{R}}^{n})}^{2}\leq{\big{\lVert}u^{G}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}-{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}.$
(5)

Moreover, the dependence of $C_{3}$ from ${\left|\Omega\right|}$ and
$f^{\ast}$ is explicit.

The main ingredient to prove that both $u$ and $f$ are almost radial is the
following theorem.

######  Theorem 1.3.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $u$ be the solution to (3a). Then it holds

${\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}\leq{\big{\lVert}(u^{\sharp})^{\text{G}}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}\leq{\big{\lVert}u^{\text{G}}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}.$
(6)

Theorem 1.3 can be interpreted as a quantitative result of (2), since we are
able to add a term between ${\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}$ and
${\big{\lVert}u^{\text{G}}\big{\rVert}}_{L^{1}(\Omega)}$.

These kinds of results involving the rearrangement of the gradient are not new
and they have been studied in different contexts. In [23] Giarrusso and
Nunziante study the same result for Hamilton-Jacobi equations with a lower
order term of the kind $f(x)-\lambda u$, comparing its solution to a suitable
symmetric one.

In [1, 15] the authors show the existence of a radially symmetric
rearrangement of the gradient of $f$ which gives the $L^{q}$ comparison as in
(2) for a given $q$.

In addition, Cianchi in [10] also gives a characterization of the above
rearrangement; as it turns out, the rearrangement found by Cianchi is the
increasing rearrangement if $q=1$ and the decreasing one if $q=+\infty$, while
is not so trivial for the case $1<q<+\infty$.

In [4, 5] the authors study firstly the case of a Sobolev non-zero trace
function and later the one of a BV function. In both cases, they find out that
the rearrangement of the gradient is a Sobolev function of $\Omega^{\sharp}$
with non-zero trace.

In [16] the authors deal with the non-stationary case, obtaining the same kind
of estimates.

Lastly, let us recall that in [32] Talenti study the case in which the Lorentz
norm $L^{q,1}$ takes the place of the $L^{1}$ norm in (2).

Moreover, in the last decades, there has been a growing interest in the
quantitative issue, not only in quantifying the distance of the sets to a
given ball but also in giving such information on the functions involved. For
example, we recall the quantitative Pólya-Szegő proved in [11] and the
quantitative Hardy-Littlewood proved in [12]. A similar quantitative result to
Theorem 1.1 concerning second order elliptic equation has been done in [3],
where the authors prove a quantitative version of the Talenti comparison
result (see [31]).

The proof of Theorem 1.1 is divided into two main propositions in which we
show each term on the left-hand side is bounded from above by
${\big{\lVert}u^{G}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}-{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}$.

The paper is organized as follows: in Section 2 we introduce some notations
and preliminary results; Sections 3, 4 and 5 are devoted to proving the
boundedness of each term on the left-hand side.

##  2 Notation and preliminary

In this article, we denote by $|\Omega|$ the Lebesgue measure of an open and
bounded set of $\mathbb{R}^{n}$, with $n\geq 2$. For convenience, we will
denote by $||\cdot||_{p}$ the norm $||\cdot||_{L^{p}(\Omega^{\sharp})}$ or
$||\cdot|_{L^{p}(\Omega)}$; otherwise, we will explicitly write the space.

If $\Omega$ is a bounded open set and $E\subset\Omega$, then we denote with
$P(E,\Omega)$ the perimeter of $E$ inside $\Omega$ (see for instance [6]), the
quantity

$P(E,\Omega):\inf\left\\{\liminf_{n}\int_{E}{\left|\nabla
u_{n}\right|}\,dx:\,u\in\text{Lip}_{\text{loc}}(E),\,u_{n}\to\chi_{E}\in
L^{1}_{\text{loc}}(E)\right\\}.$

If $\Omega=\operatorname{\mathbb{R}}^{n}$, we denote
$\text{Per}(E):=\text{Per}(E,\operatorname{\mathbb{R}}^{n})$.

Moreover, if $\Omega$ is an open set, the following coarea formula applies.
Some references for results relative to the sets of finite perimeter and for
the coarea formula are, for instance, [6, 18, 28].

######  Theorem 2.1 (Coarea formula).

Let $\Omega\subset\mathbb{R}^{n}$ be an open set. Let $f\in
W^{1,1}_{\text{loc}}(\Omega)$ and let $u:\Omega\to\operatorname{\mathbb{R}}$
be a measurable function. Then,

${\displaystyle\int_{\Omega}u(x)|\nabla
f(x)|dx=\int_{\mathbb{R}}dt\int_{\Omega\cap
f^{-1}(t)}u(y)\,d\mathcal{H}^{n-1}(y)}.$ (7)

Moreover, if $f\in\text{BV}(\Omega)$, then it holds the Fleming-Rishel
formula, i.e.

${\left|Df\right|}(\Omega)=\int_{-\infty}^{+\infty}\text{Per}(\Set{u>t},\Omega)\,dt.$
(8)

The rest of this section is devoted to the introduction of the rearrangement
of functions and both some qualitative and quantitative properties. Lastly, in
order to make the proof easier some rescaling properties are remarked.

###  2.1 Rearrangement of functions

A general survey of all the basic results about this topic is, for example,
[26].

######  Definition 2.1.

Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_distribution function_ of $u$ is the function
$\mu:[0,+\infty[\,\to[0,+\infty[$ defined as the measure of the superlevel
sets of $u$, i.e.

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

Also using the Coarea formula (7), is possible to deduce an explicit
expression for $\mu$ in terms of integrals of $u$

$\mu(t)={\left|\\{u>t\\}\cap\\{|\nabla
u|=0\\}\right|}+\int_{t}^{+\infty}\left(\int_{u=s}\frac{1}{{\left|\nabla
u\right|}}\,d\,\mathcal{H}^{n-1}\right)\,ds,$

as a consequence, for almost all $t\in(0,+\infty)$,

$\infty>-\mu^{\prime}(t)\geq\displaystyle\int_{u=t}\dfrac{1}{|\nabla
u|}d\mathcal{H}^{n-1}$ (9)

moreover if $\mu$ is absolutely continuous, equality holds in (9).

######  Definition 2.2.

Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_decreasing rearrangement_ of $u$, denoted by $u^{\ast}(\cdot)$,is defined as

$u^{*}(s)=\inf\\{t\geq 0:\mu(t)<s\\}.$

######  Remark 2.1.

Let us notice that the function $\mu(\cdot)$ is decreasing and right
continuous and thus the function $u^{\ast}(\cdot)$ can be seen as its
generalized inverse.

Also by this remark, one can prove using Definitions 2.1 and 2.2 that

$u^{\ast}(\mu(t))\leq t,\quad\forall t\geq 0,$ $\mu(u^{\ast}(s))\leq
s\quad\forall s\geq 0.$

######  Definition 2.3.

The _Schwartz rearrangement_ of $u$ is the function $u^{\sharp}$ whose
superlevel sets are balls with the same measure as the superlevel sets of $u$.

We emphasize the relation between $u^{\sharp}$ and $u^{*}$:

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

where $\omega_{n}$ stands for the measure of the unit ball in
$\operatorname{\mathbb{R}}^{n}$. By their construction the functions $u$,
$u^{*}$ e $u^{\sharp}$ have the same distribution function, or equi-
distributed, and it holds

$\displaystyle{{\left\|u\right\|}_{p}={\left\|u^{*}\right\|}_{p}=\lVert{u^{\sharp}}\rVert_{p}},\quad\text{for
all }p\geq 1.$

The absolute continuity of $\mu$ is ensured by the following lemma (we refer,
for instance, to [9, 13]).

######  Lemma 2.2.

Let $u\in W^{1,p}(\operatorname{\mathbb{R}}^{n})$, with $p\in(1,+\infty)$. The
distribution function $\mu$ of $u$ is absolutely continuous if and only if

${\left|\\{0<u<||u^{\sharp}||_{\infty}\\}\cap\\{|\nabla
u^{\sharp}|=0\\}\right|}=0.$

The decreasing rearrangement has a lot of useful inequality such as the Hardy-
Littlewood inequality, see [25].

######  Theorem 2.3 (Hardy-Littlewood inequaliy).

Let us consider $h\in L^{p}(\Omega)$ and $g\in L^{p^{\prime}}(\Omega)$. Then

$\int_{\Omega}{\left|h(x)g(x)\right|}\,dx\leq\int_{0}^{{\left|\Omega\right|}}h^{*}(s)g^{*}(s)\,ds.$
(10)

Furthermore, if we assume that the function $u$ is a _Sobolev function_ , i.e.
$u\in W^{1,p}_{0}(\Omega)$, then also $u^{\sharp}$ is a Sobolev function and
its gradient does not increase under symmetrization, as a consequence of the
Pólya-Szegő inequality (see [30]).

######  Theorem 2.4 (Pólya-Szegő inequality).

Let $u\in W^{1,p}_{0}(\Omega)$, then $u^{\sharp}\in
W^{1,p}_{0}(\Omega^{\sharp})$ and

$\lVert\nabla u^{\sharp}\rVert_{p}\leq{\left\|\nabla u\right\|}_{p}.$ (11)

Also, the Pólya-Szegő inequality holds even in the case $p=\infty$ (see [26])
and will be useful for the next results to state this case.

######  Corollary 2.5.

Let $u\in W^{1,\infty}_{0}(\Omega)$, then $u^{\sharp}\in
W^{1,\infty}_{0}(\Omega^{\sharp})$ and

$\lVert\nabla u^{\sharp}\rVert_{\infty}\leq{\left\|\nabla
u\right\|}_{\infty}.$ (12)

Also, we recall the notion of pseudo-rearrangement of a function, introduced
in [2]. Given $u$ solution of (3a), $\forall s\in[0,|\Omega|]$ we can find a
subset $D(s)$ such that:

  1. 1.

$|D(s)|=s$;

  2. 2.

$s_{1}\leq s_{2}\implies D(s_{1})\subset D(s_{2})$;

  3. 3.

$D(s)=\\{x\in\Omega:\,|u(x)|>t\\}$ if $s=\mu(t)$.

Moreover, if $f\in L^{p}(\Omega)$ then

$\int_{D(s)}f(x)\,dx$

is an absolutely continuous function of $s$, so it exists a function $F$ such
that

$\int_{0}^{s}F(t)\,dt=\int_{D(s)}f(x)\,dx.$

In particular, the following lemma [2, Lemma 2.2] holds

######  Lemma 2.6.

Let $f\in L^{p}(\Omega),p>1$. There is a sequence $\\{F_{k}(s)\\}$ of
functions which have the same rearrangement $f^{*}$ of $f$, such that

$F_{k}(s)\rightharpoonup F(s)\quad\text{in }L^{p}(0,|\Omega|)$

If $f\in L^{1}(\Omega)$, it follows that:

$\lim_{k}\int_{0}^{|\Omega|}F_{k}(s)g(s)\,ds=\int_{0}^{|\Omega|}F(s)g(s)\,ds$

for each function $g$ belonging to the space $BV(0,|\Omega|)$.

###  2.2 Rearrangement of the gradient

As we want to obtain a quantitative version of (2), it is useful to recall
their result ([22, Theorem 2.2]).

######  Theorem 2.7.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, let $\Omega^{\sharp}$ be the centered ball, let $p\geq 1$, let
$f\colon\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, let
$H\colon\operatorname{\mathbb{R}}^{n}\to\operatorname{\mathbb{R}}$ be
measurable non-negative functions and let $K\colon[0,+\infty)\to[0,+\infty)$
be a strictly increasing real-valued function such that

$0\leq K({\left|y\right|})\leq H(y)\qquad\forall
y\in\operatorname{\mathbb{R}}^{n}\qquad\text{ and }K^{-1}(f)\in
L^{p}(\Omega).$

Let $v\in W_{0}^{1,p}(\Omega)$ be a function that satisfies

$\begin{cases}H(\nabla v)=f(x)&\text{a.e. in }\Omega\\\ v=0&\text{on
}\partial\Omega\end{cases},$

denoting by $z\in W_{0}^{1,p}(\Omega^{\sharp})$ the unique spherically
decreasing symmetric solution to

$\begin{cases}K({\left|\nabla z\right|})=f_{\sharp}(x)&\text{a.e. in
}\Omega^{\sharp}\\\ z=0&\text{on }\partial\Omega^{\sharp}\end{cases},$

then

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

Moreover, in [7, 29] the following uniqueness result is proved:

######  Theorem 2.8.

Let $\Omega\subset\operatorname{\mathbb{R}}^{n}$ be a bounded open set, let
$v\in W_{0}^{1,1}(\Omega)$ be a non-negative function. Denote by
$f(x)={\left|\nabla v\right|}(x)$ and by $w\in W_{0}^{1,1}(\Omega^{\sharp})$
the decreasing spherically symmetric solution to

${\left|\nabla w\right|}=f_{\sharp}.$

If ${\left\|v\right\|}_{L^{1}}={\left\|w\right\|}_{L^{1}}$ then there exists
$x_{0}\in\operatorname{\mathbb{R}}^{n}$ such that
$\Omega=x_{0}+\Omega^{\sharp}$, $f=f_{\sharp}(\cdot\,+x_{0})$ and
$v=w(\cdot\,+x_{0})$.

###  2.3 Some quantitative inequalities

In order to prove our main theorem we recall some quantitative results that we
need in the following. The starting point is the quantitative isoperimetric
inequality, proved in [21] (see also [24, 14, 19]).

If we denote by $\alpha(\Omega)$ the Fraenkel’s asymmetry index

$\alpha(\Omega):=\min_{x\in\operatorname{\mathbb{R}}^{n}}\bigg{\\{}\dfrac{|\Omega\triangle
B_{r}(x)|}{|B_{r}(x)|}\;,\;|B_{r}(x)|=|\Omega|\bigg{\\}},$

then we have the following Theorem.

######  Theorem 2.9.

There exists a constant $\gamma_{n}$ such that, for any measurable set
$\Omega$ of finite measure

$P(\Omega)\geq
n\omega_{n}^{\frac{1}{n}}{\left|\Omega\right|}^{\frac{n-1}{n}}\left(1+\dfrac{\alpha^{2}(\Omega)}{\gamma_{n}}\right).$
(13)

We recall that the dimensional constant $\gamma_{n}$ is explicitly computed in
[17].

When we study the asymmetry of our domain in section 3, we use the
quantitative isoperimetric inequality (13) on the superlevel set of the
solution to (3a). Hence, the difficulty here is to estimate how the asymmetry
propagates from the whole domain to the superlevel sets and we can use the
following result (see for instance [8, Lemma 2.8]).

######  Lemma 2.10.

Let $\Omega\subset\operatorname{\mathbb{R}}^{n}$ be an open set with finite
measure and $U\subset\Omega$ a subset of positive measure such that

$\frac{|\Omega\backslash U|}{|\Omega|}\leq\frac{1}{4}\alpha(\Omega)$

then holds that

$\alpha(U)\geq\frac{1}{2}\alpha(\Omega)$

Another key ingredient is the quantitative version of Pólya-Szegő inequality
(2.4), proved in [11].

######  Theorem 2.11 (Quantitative Pólya-Szegő inequality).

Let $u\in W^{1,2}(\operatorname{\mathbb{R}}^{n})$, $n\geq 2$. Then, there
exist positive constants $r$, $s$ and $C$, depending only on $n$, such that,
for every $u\in W^{1,2}(\operatorname{\mathbb{R}}^{n})$, it holds

$\inf_{x_{0}\in\operatorname{\mathbb{R}}^{n}}\dfrac{\displaystyle{\int_{\operatorname{\mathbb{R}}^{n}}{\left|u(x)\pm
u^{\sharp}(x+x_{0})\right|}\,dx}}{{\left|\\{u>0\\}\right|}^{\frac{1}{n}+\frac{1}{2}}{\left\|\nabla
u^{\sharp}\right\|}_{2}}\leq
C(n)\left[M_{u^{\sharp}}(E(u)^{r})+E(u)\right]^{s},$ (14)

where

$E(u)=\frac{\displaystyle{\int_{\operatorname{\mathbb{R}}^{n}}{\left|\nabla
u\right|}^{2}}}{\displaystyle{\int_{\operatorname{\mathbb{R}}^{n}}|\nabla
u^{\sharp}|^{2}}}-1\qquad\text{ and }\qquad
M_{u^{\sharp}}(\delta)=\dfrac{{\left|\left\\{|\nabla
u^{\sharp}|<\delta\right\\}\cap\left\\{0<u^{\sharp}<||u||_{\infty}\right\\}\right|}}{{\left|\\{{\left|u\right|}>0\\}\right|}}.$
(15)

Lastly, the main tool to obtain (2) is the Hardy-Littlewood inequality (10),
so we will use a quantitative version (see [12]). Before stating the result,
we need to introduce the following notation:

$\displaystyle\mu_{h}(t)={\left|\\{h>t\\}\right|},$
$\displaystyle\mu_{g}(t)={\left|\\{g>t\\}\right|},$ $\displaystyle
g_{h}(x)=g^{\ast}(\mu_{h}(h(x))).$

Moreover, when $q\in[1,\infty)$, let us introduce the following Lorentz-type
norm

${\big{\lVert}g\big{\rVert}}_{\Lambda^{q}_{p}(\Omega)}=\begin{cases}\displaystyle{\left(\int_{0}^{{\left|\Omega\right|}}g^{\ast}(s)^{q}\theta^{\prime}_{q}(s)\;ds\right)^{1/q}}&\text{
if }p\in(1,\infty)\\\
\displaystyle{\left(\int_{0}^{{\left|\Omega\right|}}g^{\ast}(s)^{q}\,d\theta_{1}(s)+\theta_{1}(0^{+}){\big{\lVert}g\big{\rVert}}_{\infty}^{q}\right)^{\frac{1}{q}}}&\text{
if }p=1.\end{cases}$ (16)

where, for $s\in[0,{\left|\Omega\right|})$,

$\theta_{p}(s)=\begin{cases}\displaystyle{\left(\int_{0}^{s}\left(-(h^{\ast})^{\prime}(\sigma)\right)^{-1/(p-1)}d\sigma\right)^{1/p^{\prime}}}&\text{
if }p\in(1,\infty)\\\
\displaystyle{\text{ess}\sup_{\sigma\in[0,s)}\frac{1}{-(h^{\ast})^{\prime}(\sigma)}}&\text{
if }p=1.\end{cases}$ (17)

######  Theorem 2.12 (Quantitative Hardy Littlewood inequality).

Let $h$ and $g$ be two measurable functions such that $hg\in L^{1}(\Omega)$
and let us assume that there exist $p\in[1,+\infty)$ and $q\in[1,+\infty)$
such that $\theta_{p}(s)<\infty$ for every $0\leq s<|\Omega|$ and
${\left\|g\right\|}_{\Lambda^{q}_{p}(\Omega)}<\infty$. Then, by setting

$m=\dfrac{qp+1}{p+1},$ (18)

it holds

$\int_{\Omega}h(x)g(x)\;dx+\dfrac{1}{2^{p+1}eq}{\left\|g\right\|}^{-qp}_{\Lambda^{q}_{p}(\Omega)}{\left\|g-g_{h}\right\|}^{1+pq}_{L^{m}(\Omega)}\leq\int_{0}^{|\Omega|}h^{\ast}(s)g^{\ast}(s)\;ds.$
(19)

###  2.4 Rescaling properties

To lighten the proofs we will usually add two assumptions:

$\displaystyle{\left|\Omega\right|}=1,\qquad\qquad{\left\|f\right\|}_{\infty}=1.$
(20)

Even in the general case we can recover this assumption by setting

$a=\dfrac{{\left|\Omega\right|}^{-\frac{1}{n}}}{{\left\|f\right\|}_{\infty}},\quad\quad
b={\left|\Omega\right|}^{-\frac{1}{n}}.$

If $u$ and $u^{G}$ are solutions to (3a) and (3b) respectively, we define the
functions $w(x)=au\left(\frac{x}{b}\right)$ and
$z(x)=av\left(\frac{x}{b}\right)$, that are solutions to

$\begin{cases}|\nabla w|=g&\text{in }\tilde{\Omega}\\\ w=0&\text{on
}\partial\tilde{\Omega},\end{cases}\qquad\qquad\begin{cases}|\nabla
z|=g_{\sharp}&\text{in }\tilde{\Omega}^{\sharp}\\\ z=0&\text{on
}\partial\tilde{\Omega}^{\sharp},\end{cases}$

with $|\tilde{\Omega}|=1$ and ${\left\|g\right\|}_{\infty}=1$. Furthermore,
the following holds

$\displaystyle\alpha(\tilde{\Omega})$ $\displaystyle=\alpha(\Omega);$
$\displaystyle{\left\|z\right\|}_{1}-{\left\|w\right\|}_{1}$
$\displaystyle=\dfrac{{\left|\Omega\right|}^{-1-\frac{1}{n}}}{{\left\|f\right\|}_{\infty}}\left(||u^{G}||_{1}-||u||_{1}\right);$
$\displaystyle{\big{\lVert}w-w^{\sharp}\big{\rVert}}_{1}$
$\displaystyle=\dfrac{{\left|\Omega\right|}^{-1-\frac{1}{n}}}{{\left\|f\right\|}_{\infty}}{\big{\lVert}u-u^{\sharp}\big{\rVert}}_{1};$
$\displaystyle\min g$ $\displaystyle=\frac{\min
f}{{\big{\lVert}f\big{\rVert}}_{\infty}}.$

Moreover, if $\sigma(t)$ and $\mu(t)$ are the distribution functions of $w$
and $u$ respectively, then

$\sigma(t)={\left|\Omega\right|}^{-1}\mu\left(\frac{t}{a}\right),$

which yields

${\big{\lVert}g-g_{w}\big{\rVert}}_{1}=\dfrac{1}{{\left|\Omega\right|}{\left\|f\right\|}_{\infty}}{\big{\lVert}f-f_{u}\big{\rVert}}_{1}.$

##  3 Asymmetry of the domain

Our first step in proving Theorem 1.1 is to study the asymmetry of the set
$\Omega$. To this aim we recall the definition of $s_{\Omega}$, given in [8]

$s_{\Omega}=\sup\left\\{t\geq
0:\mu(t)\geq|\Omega|\left(1-\frac{\alpha(\Omega)}{4}\right)\right\\},$

and we prove the following lemma.

######  Lemma 3.1.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $f\in L^{\infty}(\Omega)$ be a non-negative function.

Suppose that $u$ is the solution to (3a). Then, it holds

${\big{\lVert}g\big{\rVert}}_{L^{1}(\Omega^{\sharp})}-{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}\geq
s_{\Omega}\frac{\alpha(\Omega)^{2}}{8\gamma_{n}},$

where $\gamma_{n}$ is the constant appearing in the quantitative isoperimetric
inequality and $g$ is unique decreasing spherically symmetric solution of the
problem

$\begin{cases}|\nabla g|=F(\omega_{n}|x|^{n})\quad\text{in }\Omega^{\\#}\\\
g=0\quad\text{on }\partial\Omega^{\sharp}.\end{cases}$ (21)

###### Proof.

By Fleming-Rishel Formula (8) and the quantitative isoperimetric inequality
(13)

$-\frac{d}{dt}\int_{u>t}|\nabla u|dx=P(u>t)\geq
n\omega_{n}^{\frac{1}{n}}\mu(t)^{1-\frac{1}{n}}\left(1+\frac{\alpha({u>t})^{2}}{\gamma_{n}}\right)$
(22)

Hence, since $u$ is solution to (3a), we have

$-\frac{d}{dt}\int_{u>t}|\nabla
u|dx=-\frac{d}{dt}\int_{u>t}f(x)dx=-\frac{d}{dt}\int_{0}^{\mu(t)}F(\tau)d\tau=F(\mu(t))(-\mu^{\prime}(t)),$
(23)

and, using (22) and (23), it follows

$\frac{\alpha({u>t})^{2}}{\gamma_{n}}\leq\frac{F(\mu(t))(-\mu^{\prime}(t))}{n\omega_{n}^{\frac{1}{n}}\mu(t)^{1-\frac{1}{n}}}-1.$
(24)

Integrating both sides from $0$ to $t$, we have

$\int_{0}^{t}\frac{F(\mu(\tau))(-\mu^{\prime}(\tau))}{n\omega_{n}^{\frac{1}{n}}\mu(\tau)^{1-\frac{1}{n}}}d\tau-t\geq\int_{0}^{t}\frac{\alpha({u>\tau})^{2}}{\gamma_{n}}d\tau,$

which gives, bthe definition of rearrangement

$\int_{s}^{|\Omega|}\frac{F(t)}{t^{1-\frac{1}{n}}}dt-u^{*}(s)\geq\int_{0}^{s}\frac{\alpha({u>t})^{2}}{\gamma_{n}}dt.$
(25)

In order to apply the lemma (2.10) let define the set

$A=\left\\{t\geq
0:\mu(t)\geq|\Omega|\left(1-\frac{\alpha({\Omega})}{4}\right)\right\\}.$

The set $A$ is not empty if $\alpha(\Omega)>0$, and it is an interval.
Moreover, for every $t\in A$ holds

$\frac{|\Omega\backslash\\{u>t\\}|}{|\Omega|}=1-\frac{\mu(t)}{|\Omega|}\leq\frac{\alpha({\Omega})}{4}\Rightarrow\alpha({u>t})\geq\frac{\alpha({\Omega})}{2}.$

Thus, if $s_{\Omega}=\sup A$, the integral of the asymmetry in (25) can be
estimated from below as

$\int_{0}^{s}\frac{\alpha({u>t})^{2}}{\gamma_{n}}dt\geq\min\\{s_{\Omega},u^{*}(s)\\}\frac{\alpha({\Omega})^{2}}{4\gamma_{n}}.$
(26)

The estimate on the asymmetry follows by noticing that $u^{*}$ is decreasing
and by definition

$s_{\Omega}=u^{*}\left(|\Omega|\left(1-\frac{\alpha({\Omega})}{4}\right)\right)$

while for $t>s_{\Omega}$ holds

$\mu(t)<\mu(s_{\omega})=\mu\left(u^{*}\left(|\Omega|\left(1-\frac{\alpha({\Omega})}{4}\right)\right)\right)\leq|\Omega|\left(1-\frac{\alpha({\Omega})}{4}\right).$

Hence, by combining (25) and (26) and integrating in $[0,|\Omega|]$ the first
two terms are respectively the norm of $g$ and the latter is by definition the
norm of $u$. Last, the integral of the minimum can be estimated as the area
below $s_{\Omega}$ before the intersection with $u^{*}$ obtaining

$\|g\|_{1}-\|u\|_{1}\geq
s_{\Omega}|\Omega|\left(1-\frac{\alpha({\Omega})}{4}\right)\frac{\alpha({\Omega})^{2}}{4\gamma_{n}}\geq
s_{\Omega}\frac{\alpha({\Omega})^{2}}{8\gamma_{n}}.$

∎

We are now in position to prove that the excess
$\displaystyle{{\big{\lVert}u^{\text{G}}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}-{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}}$
control the asymmetry index of $\Omega$.

######  Proposition 3.2.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $f\in L^{\infty}(\Omega)$ with $f\geq m>0$ be a non-negative
function. Suppose that $u$ and $u^{\text{G}}$ are the solutions to (3a) and
(3b), respectively.

Then, there exists an explicit constant
$C_{1}=C_{1}({\left|\Omega\right|},n,m)$ such that the following inequality
holds

$C_{1}\alpha(\Omega)^{3}\leq{\big{\lVert}u^{\text{G}}\big{\rVert}}_{L^{1}(\Omega^{\sharp})}-{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}.$

###### Proof.

Let us recall that, by the mean value theorem, it holds that

$F(s)=\lim_{h\to 0}\frac{1}{h}\int_{s}^{s+h}F(t)dt=\lim_{h\to
0}\frac{1}{h}\int_{D(s+h)\backslash D(s)}f(x)dx\geq m\qquad
a.e.\,s\in[0,\lvert\Omega\rvert].$

Therefore let $\nu_{g}$ be the distribution function of $g$, by the equation
(21) immediately follows that the gradient vanishes at most at the origin
therefore $\nu_{g}$ is absolutely continuous. Moreover, due to spherical
symmetry the function verifies the isoperimetric inequality as equality.

Hence, for $\nu_{g}$ we have

$1=\frac{F(\nu_{g}(t))(-\nu_{g}^{\prime}(t))}{n\omega_{n}^{\frac{1}{n}}\nu_{g}(t)^{1-\frac{1}{n}}}\Rightarrow-\nu_{g}^{\prime}(t)=\frac{n\omega_{n}^{\frac{1}{n}}\nu_{g}(t)^{1-\frac{1}{n}}}{F(\nu_{g}(t))}.$

Now, following the approach in [27], let $t_{1}$ be such that

$\nu_{g}(2t_{1})=|\Omega|\left(1-\frac{\alpha(\Omega)}{8}\right),$

and we have

$\displaystyle\frac{\alpha(\Omega)}{8}$
$\displaystyle=|\Omega|-|\Omega|\left(1-\frac{\alpha(\Omega)}{8}\right)=\nu_{g}(0)-\nu_{g}(2t_{1})=$
$\displaystyle=-\int_{0}^{2t_{1}}\nu_{g}^{\prime}(s)ds=\int_{0}^{2t_{1}}\frac{n\omega_{n}^{\frac{1}{n}}\nu_{g}(s)^{1-\frac{1}{n}}}{F(\nu_{g}(s))}ds\leq\frac{n\omega_{n}^{\frac{1}{n}}|\Omega|^{1-\frac{1}{n}}2t_{1}}{m}.$

Lastly, we can obtain that

$t_{1}\geq
m\frac{\alpha(\Omega)}{16n\omega_{n}^{\frac{1}{n}}|\Omega|^{1-\frac{1}{n}}}.$
(27)

Now let us distinguish two cases:

  * •

$s_{\Omega}\geq t_{1}$ then from (27) directly follows that

$\|g\|_{1}-\|u\|_{1}\geq C\alpha(\Omega)^{3}.$

  * •

$s_{\Omega}<t_{1}$ in general holds the pointwise inequality
$u^{\sharp}(x)\leq g(x)$ in $\Omega^{\sharp}$, see [22], therefore it is true
that

$\|g\|_{1}-\|u\|_{1}=\int_{0}^{\infty}\left(\nu_{g}(t)-\mu(t)\right)dt\geq\int_{t_{1}}^{2t_{1}}\left(\nu_{g}(t)-\mu(t)\right)dt.$
(28)

Also by $s_{\Omega}<t_{1}\leq t\leq 2t_{1}$, follows that

$\nu_{g}(t)\geq\nu_{g}(2t_{1})=|\Omega|\left(1-\frac{\alpha(\Omega)}{8}\right)\qquad\mu(t)\leq|\Omega|\left(1-\frac{\alpha(\Omega)}{4}\right).$
(29)

Therefore, by combining (28) and (29)

$\displaystyle\|g\|_{1}-\|u\|_{1}\geq\int_{t_{1}}^{2t_{1}}\left(\nu_{g}(t)-\mu(t)\right)dt\geq$
$\displaystyle\geq\frac{|\Omega|}{8}\alpha(\Omega)\int_{t_{1}}^{2t_{1}}dt\geq
C\alpha(\Omega)^{2}\geq C\alpha(\Omega)^{3}.$

Eventually, the lemma 2.6 and Hardy-Littlewood inequality brings to

$\displaystyle{\big{\lVert}g\big{\rVert}}_{L^{1}(\Omega^{\sharp})}=\int_{0}^{|\Omega|}\int_{s}^{|\Omega|}\frac{F(t)}{t^{1-\frac{1}{n}}}dtds=\int_{0}^{|\Omega|}F(t)t^{\frac{1}{n}}dt=\lim_{k}\int_{0}^{|\Omega|}f_{k}(t)t^{\frac{1}{n}}dt\leq$
$\displaystyle\leq\int_{0}^{|\Omega|}f_{*}(t)t^{\frac{1}{n}}dt={\big{\lVert}u^{\text{G}}\big{\rVert}}_{L^{1}(\Omega^{\sharp})},$

and the thesis follows. ∎

##  4 Almost radiality of the solution

Our first step in proving the almost radiality of the solution is to prove
Theorem 1.3.

###### Proof of Theorem 1.3.

Firstly, let us stress the fact that it is always possible to construct a
function $h$ whose rearrangement is $t^{\frac{1}{n}}$ and such that

$\int_{0}^{|\Omega|}|\nabla
u^{\sharp}|_{*}t^{\frac{1}{n}}\,dt=\int_{\Omega^{\sharp}}|\nabla
u^{\sharp}|h(x)\,dx.$ (30)

Since $|\nabla u^{\sharp}|$ is radial we can build $D(s)$, defined in Section
2, such that it is a ball or an annulus. Now choosing as super level sets of
$h$ the corresponding $D(s)$ of the same measure, we have for every
$t,\tau>0$, either

$\\{|\nabla u^{\sharp}|>t\\}\subset\\{h>\tau\\}\text{ or }\\{|\nabla
u^{\sharp}|>t\\}\supset\\{h>\tau\\}.$

The last one assures us that (30) holds true.

Now, our goal is to show that

$\int_{0}^{|\Omega|}|\nabla
u^{\\#}|_{*}(t)t^{\frac{1}{n}}dt\leq\int_{0}^{|\Omega|}|\nabla
u|_{*}(t)t^{\frac{1}{n}}dt.$

By coarea formula (7), isoperimetric inequality and the properties of
rearrangements, we have

$\displaystyle\int_{0}^{|\Omega|}|\nabla
u^{\sharp}|_{*}t^{\frac{1}{n}}\,dt=\int_{\Omega^{\sharp}}|\nabla
u^{\sharp}|h(x)\,dx=\int_{0}^{\infty}\int_{u^{*}=s}h(t(s))\,d\mathcal{H}^{n-1}\,ds$
(31)
$\displaystyle=\int_{0}^{\infty}\mathcal{H}^{n-1}(\\{u^{*}=s\\})h(t(s))\,ds\leq\int_{0}^{\infty}\mathcal{H}^{n-1}(\\{u=s\\})h(t(s))\,ds$
$\displaystyle=\int_{0}^{\infty}\int_{u=s}h(t(s))\,d\mathcal{H}^{n-1}\,ds=\int_{\Omega}|\nabla
u|h_{u}(x)\,dx\leq\int_{0}^{|\Omega|}|\nabla u|_{*}t^{\frac{1}{n}}\,dt,$

where $h_{u}$ is a function with the same level superlevel sets of $u$ and
whose increasing rearrangement is $t^{\frac{1}{n}}$.

Hence by definition

$\begin{gathered}\|(u^{\\#})^{G}\|_{1}=\frac{1}{n\omega_{n}}\int_{0}^{|\Omega|}\int_{s}^{|\Omega|}|\nabla
u^{\\#}|_{*}(t)t^{\frac{1}{n}-1}dtds=\\\
=\frac{1}{n\omega_{n}}\int_{0}^{|\Omega|}|\nabla
u^{\\#}|_{*}(t)t^{\frac{1}{n}}dt\leq\frac{1}{n\omega_{n}}\int_{0}^{|\Omega|}|\nabla
u|_{*}(t)t^{\frac{1}{n}}dt=\|u^{G}\|_{1}\end{gathered}$ (32)

While the first inequality is a trivial consequence of (2) and
$\|u\|_{1}={\big{\lVert}u^{\sharp}\big{\rVert}}_{1}$. ∎

For all the next lemmas and corollaries we will use as extra assumptions that
$|\Omega|=1$ and $||f||_{\infty}=1$. Moreover we set
$\varepsilon:={\big{\lVert}u^{G}\big{\rVert}}_{1}-{\big{\lVert}u\big{\rVert}}_{1}$.

Let us observe that (32) brings immediately to the following corollary.

######  Corollary 4.1.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $u$ be the solution to (3a). Then

$\int_{0}^{|\Omega|}\left[|\nabla u|_{*}-|\nabla
u^{\\#}|_{*}\right]t^{\frac{1}{n}}dt<\varepsilon.$

######  Lemma 4.2.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $u$ be the solution to (3a). Let $|\Omega|=1$ and
$||f||_{\infty}=1$. For all $\alpha>0$, if $I$ is the set defined as

$I=\left\\{t\in[0,1]:\left[|\nabla u|_{*}(t)-|\nabla
u^{\\#}|_{*}(t)\right]t^{\frac{1}{n}}>\varepsilon^{\alpha}\right\\}$

then $|I|<\varepsilon^{1-\alpha}$.

###### Proof.

By the previous corollary

$\displaystyle|I|=\int_{I}dt=\frac{1}{\varepsilon^{\alpha}}\int_{I}\varepsilon^{\alpha}dt<\frac{1}{\varepsilon^{\alpha}}\int_{I}\left[|\nabla
u|_{*}(t)-|\nabla u^{\\#}|_{*}(t)\right]t^{\frac{1}{n}}dt<$
$\displaystyle<\frac{1}{\varepsilon^{\alpha}}\int_{0}^{|\Omega|}\left[|\nabla
u|_{*}(t)-|\nabla
u^{\\#}|_{*}(t)\right]t^{\frac{1}{n}}dt<\varepsilon^{1-\alpha}$

∎

######  Lemma 4.3.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $u$ be the solution to (3a). Let $|\Omega|=1$ and
$||f||_{\infty}=1$. Then it exist a constant $C=C(m)$ such that

$E(u)=\frac{\displaystyle{\int_{\Omega}|\nabla
u|^{2}dx}}{\displaystyle{\int_{\Omega^{\\#}}|\nabla
u^{\\#}|^{2}dx}}-1\leq\frac{16}{m^{2}}\varepsilon^{\frac{1}{2}}$

###### Proof.

By the properties of the rearrangements and the Pólya-Szegő for the
$L^{\infty}$ case 2.5, we get

$\displaystyle\int_{\Omega}|\nabla u|^{2}dx-\int_{\Omega^{\\#}}|\nabla
u^{\\#}|^{2}dx$ $\displaystyle=\int_{0}^{|\Omega|}|\nabla u|_{*}^{2}-|\nabla
u^{\\#}|_{*}^{2}dx$ $\displaystyle=\int_{0}^{|\Omega|}\left(|\nabla
u|_{*}+|\nabla u^{\\#}|_{*}\right)\left(|\nabla u|_{*}-|\nabla
u^{\\#}|_{*}\right)dx$ $\displaystyle\leq
2\|f\|_{\infty}\int_{0}^{|\Omega|}|\nabla u|_{*}-|\nabla u^{\\#}|_{*}dx.$

Lemma 4.2 allows us to estimate the last term, indeed

$\displaystyle\int_{0}^{|\Omega|}|\nabla u|_{*}-|\nabla u^{\\#}|_{*}$
$\displaystyle=\int_{[0,|\Omega|]\cap I}|\nabla u|_{*}-|\nabla
u^{\\#}|_{*}+\int_{[0,|\Omega|]\backslash I}(|\nabla u|_{*}-|\nabla
u^{\\#}|_{*})t^{\frac{1}{n}}t^{-\frac{1}{n}}$ $\displaystyle\leq
2\|f\|_{\infty}|I|+\varepsilon^{\alpha}\frac{n}{n-1}|\Omega|^{1-\frac{1}{n}}$
$\displaystyle\leq 2\varepsilon^{1-\alpha}+\varepsilon^{\alpha}\frac{n}{n-1}.$

By choosing $\alpha=\frac{1}{2}$ we get

$\int_{\Omega}|\nabla u|^{2}dx-\int_{\Omega^{\\#}}|\nabla u^{\\#}|^{2}dx\leq
8\varepsilon^{\frac{1}{2}}.$

Moreover, if

$8\varepsilon^{\frac{1}{2}}\leq\frac{m^{2}|\Omega|}{2}=\frac{m^{2}}{2}\implies\varepsilon\leq\frac{m^{4}}{256}$

we have

$\int_{\Omega^{\\#}}|\nabla u^{\\#}|^{2}dx\geq\int_{\Omega}|\nabla
u|^{2}dx-8\varepsilon^{\frac{1}{2}}\geq\frac{m^{2}}{2}.$

The last gives us

$\frac{\displaystyle{\int_{\Omega}|\nabla
u|^{2}dx}}{\displaystyle{\int_{\Omega^{\\#}}|\nabla
u^{\\#}|^{2}dx}}-1\leq\frac{8\varepsilon^{\frac{1}{2}}}{\displaystyle{\int_{\Omega^{\\#}}|\nabla
u^{\\#}|^{2}dx}}\leq\frac{16}{m^{2}}\varepsilon^{\frac{1}{2}}.$

∎

We are now in position to prove the stability for $u$.

######  Proposition 4.4.

Let $\Omega$ be a bounded open set of $\operatorname{\mathbb{R}}^{n}$, $n\geq
2$, and let $f\in L^{\infty}(\Omega)$ be a non-negative function with

$0<m\leq f\leq M.$

Suppose that $u$ and $u^{\text{G}}$ are the solutions to (3a) and (3b),
respectively. Then, there exist some positive constants $\theta=\theta(n)$ and
$C_{2}:=C_{2}(n,{\left|\Omega\right|},f^{\sharp})$ such that

$C_{2}\inf_{x_{0}\in\operatorname{\mathbb{R}}^{n}}{\big{\lVert}u-u^{\sharp}(\cdot+x_{0})\big{\rVert}}_{L^{1}(\operatorname{\mathbb{R}}^{n})}^{\theta}\leq{\big{\lVert}u^{G}\big{\rVert}}_{1}-{\big{\lVert}u\big{\rVert}}_{1}.$
(33)

###### Proof.

Firstly, let us assume ${\left|\Omega\right|}=1$, and
${\left\|f\right\|}_{\infty}=1$. Moreover we set
$\varepsilon:={\big{\lVert}u^{G}\big{\rVert}}_{1}-{\big{\lVert}u\big{\rVert}}_{1}$.

Taking into account the quantitative Pólya-Szegő principle in [11], we
immediately get

$C\inf_{x_{0}\in\operatorname{\mathbb{R}}^{n}}{\big{\lVert}u-u^{\sharp}(\cdot+x_{0})\big{\rVert}}_{L^{1}(\operatorname{\mathbb{R}}^{n})}\leq\left[M(E(u)^{r})+E(u)\right]^{s},$

for some positive constants $C,r$ and $s$.

In order to conclude, it is necessary to estimate the term

$M(E(u)^{r})=\frac{|\\{|\nabla u|\leq E(u)^{r}\\}\cap\\{0<u<\sup
u\\}|}{|\\{|u|>0\\}|}.$

To this aim, we will use Lemma 4.3. Indeed, since $|\nabla u|=f\geq m$ a.e.,
if

$m>\left(\frac{16}{m^{2}}\varepsilon^{\frac{1}{2}}\right)^{r}\implies\varepsilon<\frac{1}{256}m^{\frac{4r+2}{r}},$

we have

$|\nabla u|\geq
m>\left(\frac{16}{m^{2}}\varepsilon^{\frac{1}{2}}\right)^{r}\geq
E(u)^{r}\,\,a.e.\text{ in }\Omega.$

Hence $M(E(u)^{r})=0$.

On the other hand, if $\varepsilon>\frac{1}{256}m^{\frac{4r+2}{r}}$ we have

$\displaystyle{\big{\lVert}u-u^{\sharp}\big{\rVert}}_{L^{1}(\operatorname{\mathbb{R}}^{n})}\leq
2{\big{\lVert}u\big{\rVert}}_{L^{1}(\Omega)}\leq
2{\big{\lVert}u^{G}\big{\rVert}}_{L^{1}(\Omega)}=2\int_{0}^{1}f_{\sharp}(t)t^{\frac{1}{n}}dt$
$\displaystyle\leq
2\frac{n}{n+1}\leq\frac{512n}{(n+1)}\frac{1}{m^{\frac{4r+2}{r}}}\varepsilon.$

Finally, we remove the additional assumptions ${\left|\Omega\right|}=1$ and
$||f||_{\infty}=1$ using the rescaling properties in Section 2.4. ∎

Hence, Theorem 1.1 can be recover by putting together Propositions 3.2 and
4.4.

###### Proof of Theorem 1.1.

The proof of the main Theorem is recovered taking into account Propositions
3.2 and 4.4. ∎

##  5 Property of the source term

The last step is to obtain some information on the source term.

###### Proof of Proposition 1.2.

Firstly, let us assume ${\left|\Omega\right|}=1$, and
${\left\|f\right\|}_{\infty}=1$. Moreover we set
$\varepsilon:={\big{\lVert}u^{G}\big{\rVert}}_{1}-{\big{\lVert}u\big{\rVert}}_{1}$.

Theorem 1.3 shows that

$\int_{0}^{|\Omega|}f^{*}(t)(1-t)^{\frac{1}{n}}\,dt-\int_{\Omega}fh_{u}(x)\,dx=\int_{0}^{1}|\nabla
u|_{*}t^{\frac{1}{n}}\,dt-\int_{\Omega}|\nabla u|h_{u}(x)\,dx\leq\varepsilon.$

Applying the quantitative Hardy-Littlewood inequality (see [12]) with $p=q=1$,
we have

${\left\|f-f_{u}\right\|}_{L^{m}(\operatorname{\mathbb{R}}^{n})}\leq\left[4\,e\,{\left\|f\right\|}_{\Lambda_{1}^{1}(\Omega)}\left(\int_{0}^{|\Omega|}f^{*}(t)(1-t)^{\frac{1}{n}}\,dt-\int_{\Omega}fh_{u}(x)\,dx\right)\right]^{\frac{1}{2}}.$

By direct calculations, we have that

${\left\|f\right\|}_{\Lambda_{1}^{1}(\Omega)}\leq\frac{{\big{\lVert}f\big{\rVert}}_{\infty}}{n}=\frac{1}{n}.$

Hence, there exists $C_{3}$ such that

$C_{3}{\left\|f-f_{u}\right\|}_{1}^{2}\leq\varepsilon.$

Finally, we remove the additional assumptions ${\left|\Omega\right|}=1$ and
$||f||_{\infty}=1$ using the rescaling properties in Section 2.4. ∎

### Acknowledgements

The authors were partially supported by Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto
Nazionale di Alta Matematica (INdAM).

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

E-mail address, L. Barbato (corresponding author<EMAIL_ADDRESS>
Mathematical and Physical Sciences for Advanced Materials and Technologies,
Scuola Superiore Meridionale, Largo San Marcellino 10, 80126 Napoli, Italy.