# Time-dependent Rabi frequencies to protect quantum operations on an atomic
qutrit by continuous dynamical decoupling

Adonai Hilario da Silva  São Carlos Institute of Physics, University of São
Paulo, PO Box 369, 13560-970, São Carlos, SP, Brazil  Reginaldo de Jesus
Napolitano  São Carlos Institute of Physics, University of São Paulo, PO Box
369, 13560-970, São Carlos, SP, Brazil  Felipe Fernandes Fanchini  Faculdade
de Ciências, UNESP - Universidade Estadual Paulista, 17033-360 Bauru, São
Paulo, Brazil  Bruno Bellomo  Institut UTINAM, CNRS UMR 6213, Université
Bourgogne Franche-Comté, Observatoire des Sciences de l’Univers THETA, 41 bis
avenue de l’Observatoire, F-25010 Besançon, France

###### Abstract

We investigate the form required for the time-dependent Rabi frequencies
involved in a procedure capable to protect the action of quantum gates on an
atomic qutrit by means of external fields continuously decoupling the system
from the environmental noise. Several simulations are considered to protect
the action of quantum-gate models, including randomly chosen ones. We argue
that the requirements for the Rabi frequencies could be nowadays
experimentally met. We also investigate the transition from one gate operation
to another, including protecting a qutrit memory state. We finally apply our
methodology to protect from noise the application of an algorithm capable of
distinguishing the parity of permutations of three elements.

##  I Introduction

Quantum coherence is the valuable resource that physically distinguishes
between quantum and classical computers. Thus, preserving coherence or
recovering it from decoherence is fundamental for high-fidelity processing of
quantum information. The first specific paper about dynamically suppressing
decoherence of a quantum state involved a single qubit [1]. However, at least
two qubits are necessary for a single entangling gate [2] and useful quantum-
information processing requires more than only a few universal quantum gates
[3]. As quantum algorithms become more sophisticated, the complexity of the
quantum circuits for running them inevitably increases, as does the dimension
of the required Hilbert spaces. Recent advances have introduced photonic qudit
systems [4, 5, 6], with their quantum states belonging to high-dimensional
Hilbert spaces. As an impressive example of how high a dimension recent
technology can reach, we cite the realization of two entangled photonic qudits
in 100 dimensions [6]. As examples of useful applications of quantum-
information processing using qudits we mention quantum-key distribution [7, 8,
9, 10, 11], quantum computation [12, 13, 14, 15, 16], and even basic
investigations on the nature of quantum mechanics [17]. Therefore, approaches
to avoid or recover from errors introduced by decoherence are bound to attract
ever-increasing attention [18, 19, 20, 21]. Evidence has been accumulating
that such higher-dimensional states and the corresponding quantum gates acting
on them provide additional support against decoherence, besides improving the
efficiency of quantum circuits [22, 23, 24]. Proposals to implement qudit
devices abound, not only using optical systems, but also superconducting [24,
25] and atomic-spin [26, 27] devices.

We have recently presented a generalized procedure for continuously and
dynamically decoupling an ensamble of qudits from general environmental
perturbations [28]. Using generalized continuous dynamical decoupling (GCDD)
we can protect any qudit system from general noise even during the operation
of an arbitrary qudit gate. In Ref. [28] we illustrate a practical
implementation of the GCDD method with a very detailed qutrit model,
consisting of the three magnetic hyperfine states of the ground energy level
of ${}^{87}\mathrm{Rb},$ during the action of a qutrit Hadamard gate. The
quantum gate and the control are implemented using laser beams whose
intensities and phases are modulated according to our prescription. Our model
relies on the control over the Rabi frequencies of nine distinguishable laser
beams, in an orchestrated scheme that, in principle, would allow us to
generate the dynamical decoupling from noise, simultaneously executing the
intended one-qudit gate operation. However, this situation raises the question
whether such prescribed modulation of the control fields could be implemented
using current technology, since the required time-varying Rabi frequencies
could prove themselves unrealizable.

Here we explicitly study the form of the time-dependent Rabi frequencies
required to decouple an atomic qutrit from the environmental noise while also
executing quantum operations, also validating the approximations involved in
the protection scheme. Most of the previous investigations used pulsed
dynamical decoupling of qubits [29, 30, 31, 32]. In the case of laser fields
driving two-photon transitions in qutrits, gaussian-shaped time-dependent Rabi
frequencies [32] have been studied in detail. The GCDD method does not use
pulses, but in the case of qudits ($d$-level systems) [28], it prescribes
time-dependent Hamiltonians to generate any $\mathrm{SU}\left(d\right)$
operation continuously. Here we consider our illustrative model of atomic
qutrits, addressing the possibility to tailor the necessary Rabi frequencies
using currently available technological capabilities. We estimate the validity
of approximations and suggest values of the parameters that are currently
achievable. We simulate the protection scheme under quite general
environmental perturbations, during the operation of different kinds of
quantum gates that act on the qutrit. We make use of continuous external
fields, whose period $t_{0}$ is assumed shorter than the correlation time of
the noise, to emulate an appropriate periodic time-dependent unitary
evolution. It is worth emphasizing that, although here we address the qutrit
model using a laser-atom system, the same analysis could be extended to qudit
systems involving higher dimensions, as for example, another set of hyperfine
levels of an atomic system, or to a treatment of more than a single qudit. We
do not expect any qualitatively different impediments in such an extension to
the general conclusions about our analysis presented here.

This article is organized as follows. Section II briefly resumes the GCDD
method. In Sec. III we review our qutrit model and we give the explicit
expression of the external fields required for implementing the required GCDD
Hamiltonian. In Sec. IV we study the required Rabi frequencies for various
one-qutrit quantum operations, we verify the validity of our approximations,
and we argue that the requirements for the Rabi frequencies could be currently
met. We also show that the GCDD procedure can be used to protect the execution
of a quite simple quantum algorithm [33], where a single qutrit is used to
determine the parity of the permutations of a set of three objects. Finally,
Sec. V presents our conclusions.

##  II A resume of the GCDD procedure

The GCDD prescription detailed in Ref. [28], is a generalization of the
continuous dynamical decoupling of qubits [34, 35, 36, 37, 38] and consists of
a continuously-varying control Hamiltonian and a modification of the intended
quantum operation that, together, contrast the negative effects of the
environmental noise. Indeed, the final state results to be closer to the state
which would have been obtained if the ideal noise-free version of the intended
operation had taken place, with respect to the case when the operation is
performed in the presence of the noise but without protection. The GCDD
procedure can be extended to the case of an arbitrary number of qudits,
identical or not [28]. In this section, we briefly resume our previous work
regarding the GCDD prescription, by focusing on the case of a single qudit.
The main point concerns the introduction of a basic unitary evolution that
satisfies a weaker condition of the general dynamical decoupling condition
[39], which is sufficient to effectively decouple the system from its
environment [28].

According to the GCDD prescription, the total Hamiltonian is written as

$H_{\mathrm{tot}}(t)=\left[H_{c}(t)+H_{\mathrm{gate}}(t)\right]\otimes\mathbb{I}_{E}+\mathbb{I}_{d}\otimes
H_{E}+H_{\mathrm{int}},$ (1)

where $\mathbb{I}_{E}$ and $\mathbb{I}_{d}$ are the identity operators of,
respectively, the environmental and the qudit Hilbert spaces. Here
$H_{\mathrm{gate}}(t)$ is the Hamiltonian associated to the ideal gate
Hamiltonian $H_{G}$ one wants to apply to the qudit. The quantum-gate
operation starts at $t=0$ and stops at $t=\tau$. According to our protection
scheme, the Hamiltonian $H_{\mathrm{gate}}(t)$ is defined by

$H_{\mathrm{gate}}(t)\equiv U_{c}(t)H_{G}U_{c}^{\dagger}(t),$ (2)

where $U_{c}(t)$ is the control unitary transformation associated to external
control fields whose action is described by $H_{c}(t)$. $H_{E}$ is the free
Hamiltonian of the environment, and $H_{\mathrm{int}}$ describes the
interaction between the qudit and the environment. The noise effects are
assumed to be resulting from the interaction between the qudit and the
environment, described by the very general Hamiltonian

$\displaystyle H_{\mathrm{int}}$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{d-1}\sum_{s=0}^{d-1}\outerproduct{r}{s}\otimes
B_{rs},$ (3)

where $B_{rs}$, for $r,s=0,1,\dots,d-1$, are operators that act on the
environmental states and $\ket{k}$, for $k=0,1,\dots,d-1$, are $d$ normalized
states composing a logical basis set for the qudit Hilbert space of dimension
$d$.

We remark that the free Hamiltonian of the qudit can be thought as included in
$H_{G}$ or assumed to be eliminated before the application of the GCDD scheme.
The protection protocol is then applied to a system which, in the absence of
$H_{G}$, is degenerate since all qudit levels have the same energy.

The procedure also involves the Hermitian operator $H_{L}$ whose action on the
logical basis states is given by

$H_{L}\ket{k}\equiv k\hbar\omega_{d}\ket{k},$ (4)

for $k=0,1,\dots,d-1$, where $\omega_{d}\equiv\omega_{0}d$ and $\omega_{0}$ is
the control frequency corresponding to the dynamical-decoupling period
$t_{0}\equiv 2\pi/\omega_{0}$. We also make use of the Hermitian operator
$H_{F}$ whose action on the quantum-Fourier transformed basis is defined as

$H_{F}\ket{\psi_{n}}\equiv n\hbar\omega_{0}\ket{\psi_{n}},$ (5)

where

$\ket{\psi_{n}}\equiv\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}\exp(\frac{2\pi
i}{d}jn)\ket{j},$ (6)

for $n=0,1,\dots,d-1.$ The control unitary transformation is then given in
terms of $H_{L}$ and $H_{F}$ as

$U_{c}(t)\equiv\exp(-i\omega_{r}t)\exp(-i\frac{H_{L}}{\hbar}t)\exp(-i\frac{H_{F}}{\hbar}t),$
(7)

where, for convenience, we have introduced a real constant $\omega_{r}$ as

$\omega_{r}\equiv-\frac{\mathrm{Tr}\left(H_{L}\right)+\mathrm{Tr}\left(H_{F}\right)}{\hbar
d}.$ (8)

We stress that the operator $U_{c}(t)$ is built to be periodic and its period
is $t_{0}$. Using the definition of $U_{c}(t)$ in Eq. (7), we can calculate
the control Hamiltonian as

$\displaystyle H_{c}(t)$ $\displaystyle=$ $\displaystyle
i\hbar\frac{\mathrm{d}U_{c}(t)}{\mathrm{d}t}U_{c}^{\dagger}(t)$ (9)
$\displaystyle=$
$\displaystyle\hbar\omega_{r}\mathbb{I}_{d}+H_{L}+\exp(-i\frac{H_{L}}{\hbar}t)H_{F}\exp(i\frac{H_{L}}{\hbar}t).\quad\>$

It is now possible to show that the following sufficient condition for
dynamically decoupling [39] for the interaction defined by Eq. (3) is
satisfied:

$\int_{0}^{t_{0}}\mathrm{d}t\,\left[U_{c}^{\dagger}(t)\otimes\mathbb{I}_{E}\right]H_{\mathrm{int}}\left[U_{c}(t)\otimes\mathbb{I}_{E}\right]=\frac{t_{0}}{d}\mathbb{I}_{d}\otimes\sum_{r=0}^{d-1}B_{rr}.$
(10)

Here the period of $U_{c}(t)$, $t_{0}$, is to be chosen so that it is short
enough compared with the bath correlation time, and so that the gate time
$\tau$ is an integer multiple of $t_{0}$.

In the laboratory we need to generate external fields such that they interact
with the qudit according to the Hamiltonian
$H_{\mathrm{lab}}(t)=H_{c}(t)+H_{\mathrm{gate}}(t)$. We assume that the ideal
gate Hamiltonian $H_{G}$ can be expanded in the computational basis as

$H_{G}=\hbar\sum_{r=0}^{d-1}\sum_{s=0}^{d-1}g_{r,s}\outerproduct{r}{s},$ (11)

with $g_{s,r}^{\ast}=g_{r,s}$ since $H_{G}$ is Hermitian. As explained in Ref.
[28], we have taken $H_{\mathrm{gate}}(t)$ as defined in Eq. (2) so that, in
the presence of the control fields, the intended quantum operation can be
realized even when there is noise with high fidelity, without the control
fields also decoupling the gate action. We define $\hbar G=\hbar
g_{0}\mathbb{I}_{d}-H_{G}$, where $g_{0}$ is the largest eigenvalue of
$H_{G},$ so that $G$ is non negative. We also define the following non-
negative operators:
$H_{L}^{\prime}\equiv\hbar\left(d-1\right)\omega_{d}\mathbb{I}_{d}-H_{L}$ and
$H_{F}^{\prime}\equiv\hbar\left(d-1\right)\omega_{0}\mathbb{I}_{d}-H_{F}$. We
thus obtain

$H_{\mathrm{lab}}(t)=\hbar\omega_{l}\mathbb{I}_{d}-\hbar\Upsilon(t)\Upsilon(t),$
(12)

with $\omega_{l}\equiv g_{0}+\omega_{r}+\left(d^{2}-1\right)\omega_{0}$ and

$\Upsilon(t)\equiv\sqrt{\frac{V}{\hbar}},$ (13)

where

$V\\!=\\!H_{L}^{\prime}+\exp\\!\left(-i\frac{H_{L}}{\hbar}t\right)\\!H_{F}^{\prime}\exp\\!\left(i\frac{H_{L}}{\hbar}t\right)\\!+\hbar
U_{c}(t)GU_{c}^{\dagger}(t),$ (14)

which is a well-defined square root, since $\Upsilon(t)\Upsilon(t)$ is a non-
negative operator.

As one can deduce from Eq. (9), the protection procedure does not depend on
the characteristics of the environment and on the coupling constants
regulating the interaction between the system and the environment. However,
the effectiveness of the protection depends on the ratio between the bath
correlation time and the control period.

##  III An atomic qutrit

Now we briefly review the qutrit model we have introduced in Ref. [28], which
may be consulted for the mathematical derivations and other details. The
qutrit comprises the hyperfine ground manifold of magnetic states of a
${}^{87}\mathrm{Rb}$ atom, as shown in Fig. 1. These three ground states are
degenerate in the absence of magnetic fields and we denote them by $\ket{m}$,
for $m=-1,0,1$. We name $\hbar\omega_{g}$ the energy of the ground states and
$\hbar\omega_{e}$ the energy of the excited state $\ket{e}$.

The effective control fields involved in the GCDD procedure consist of a set
of polarized laser beams. We only use photon frequencies $\omega_{s}$ that are
reduned (this means that the detunings $\Delta_{s}$, defined as
$\Delta_{s}\equiv\omega_{s}-\omega_{e}+\omega_{g}$, are negative), from the
$F=1\leftrightarrow F^{\prime}=0$ transition. Thus we can approximate the
relevant set of atomic states as $\left\\{\ket{m}\text{, for
}m=-1,0,1\right\\}$, together with the excited state $\ket{e}$, forming a
restricted Hilbert space that we denote by $\mathscr{H}_{4}$. If the photons
are detuned far enough to the red of the transitions
$\ket{m}\leftrightarrow\ket{e}$, for $m=-1,0,1$ (virtual transitions), then
the excited state is not going to be effectively populated, avoiding spurious
transitions to the $F=2$ ground states through spontaneous emission from
$\ket{e}$. Accordingly, the effective qutrit consists of the states $\ket{m}$,
with $m=-1,0,1$, whose Hilbert space is denoted by $\mathscr{H}_{3}$. The GCDD
control over the states in $\mathscr{H}_{3}$ is accomplished through two-
photon transitions by means of nine laser beams as comprehensively explained
in Ref. [28]. We make use of frequency and intensity modulation and we choose
the frequencies and two-photon transitions so that a suitable form of the
rotating-wave approximation is valid. The three states $\left\\{\ket{m}\text{,
for }m=-1,0,1\right\\}$ form a qutrit computation basis. In the following we
use the notation $\left\\{\ket{m}\text{, for }m=-1,0,1\right\\}$ instead of
explicitly associating them to computational states $\left\\{\ket{d}\text{,
for }d=0,1,2\right\\}$.

Figure 1: The hyperfine states of the $\mathrm{D}_{2}$ transition of
${}^{87}\mathrm{Rb}$ that we need in the present model (not to scale). We use
two-photon transitions with three different detunings $\Delta_{s}$, for
$s=1,2,3,$ and, for each of these detunings, we use $\sigma^{\pm}$\- and
$\pi$-polarized laser light, as described in full detail in Ref. [28].

Following these parameter guidelines, now we must follow the prescription laid
out in Ref. [28] to connect nine independent time-varying Rabi frequencies
$\Omega_{s,m}(t)$ ($s=1,2,3$ and $m=-1,0,1$), associated to the various
physical transitions involved in the dynamics, with the Hamiltonians
$H_{\mathrm{gate}}(t)$ and $H_{c}(t)$ of Eq. (1). In Ref. [28] we have shown
that the Rabi frequencies we need to generate the dynamics evolving according
to a Hamiltonian equivalent to that of Eq. (12) up to a term proportional to
the identity, which only introduces an immaterial global phase factor (this
aspect is not even present if the condition $\omega_{g}=\omega_{l}$ is
satisfied), are given by the following equations,

$H_{\mathrm{eff}}(t)=\hbar\omega_{g}\mathbb{I}_{3}-\hbar\Theta^{\dagger}(t)\Theta(t),$
(15)

with

$\displaystyle\Theta(t)$ $\displaystyle\equiv$
$\displaystyle\left[\begin{array}[]{ccc}\frac{\Omega_{1,1}(t)}{\sqrt{\Delta_{1}}}&\frac{\Omega_{1,0}(t)}{\sqrt{\Delta_{1}}}&\frac{\Omega_{1,-1}(t)}{\sqrt{\Delta_{1}}}\\\
\frac{\Omega_{1,0}^{\ast}(t)}{\left(\sqrt{\Delta_{1}}\right)^{\ast}}&\frac{\Omega_{2,0}(t)}{\sqrt{\Delta_{2}}}&\frac{\Omega_{2,-1}(t)}{\sqrt{\Delta_{2}}}\\\
\frac{\Omega_{1,-1}^{\ast}(t)}{\left(\sqrt{\Delta_{1}}\right)^{\ast}}&\frac{\Omega_{2,-1}^{\ast}(t)}{\left(\sqrt{\Delta_{2}}\right)^{\ast}}&\frac{\Omega_{3,-1}(t)}{\sqrt{\Delta_{3}}}\end{array}\right],$
(19)

where we use the convention that $\sqrt{\Delta_{s}}=i\sqrt{-\Delta_{s}},$
since all detunings are negative, and choose the parameters so that

$\displaystyle\Theta(t)$ $\displaystyle=$
$\displaystyle\Theta^{\dagger}(t)=\Upsilon(t).$ (20)

Thus, for a given gate Hamiltonian as in Eq. (11), we calculate the elements
of the operator $\Upsilon(t)$ according to Eq. (13), and then we must generate
fields in the laboratory resulting in Rabi frequencies such that Eq. (20) is
satisfied. Notice that the various detunings involved in the procedure are
constant and independent.

In the next section we study several one-qudit quantum gates and show
simulations of the GCDD process.

##  IV Results

Here we explore several applications of the GCDD procedure in the case of a
qutrit, by focusing mainly on the required time-dependence for the various
Rabi frequencies involved in the decoupling protocol. In all the applications
the noise involves both damping and dephasing effects. In subsection IV.1 we
briefly describe the noise model considered in our numerical simulations. In
subsection IV.2 we calculate the Rabi frequencies required to generate and
protect the action of a qutrit generalized Hadamard gate. In particular, the
effect of noise is studied both at zero and nonzero temperature. In subsection
IV.3 we consider two additional gates, which can be thought as a
generalization of the usual $X$ and $Z$ qubit gates, and simulate their
protection using the GCDD method. In subsection IV.4 we verify that it is
possible to emulate any quantum gate represented by a generic $\mathrm{SU}(3)$
matrix without the need of a universal set of gates. In subsection IV.5 we
study how the Rabi frequencies behave when we finish applying a quantum gate
and start applying another one in sequence. Finally, in subsection IV.6 we
show an example of how the GCDD can be used to execute a very simple one-
qutrit algorithm, described in Ref. [33].

###  IV.1 Noise model and fidelity

In our model the dissipative effects are assumed to derive from the
interaction between our qutrit system and two identical bosonic thermal baths,
respectively responsible for amplitude damping and dephasing noise. A complete
description of how to treat the dissipative dynamics can be found in the main
text of Ref. [28] and in its Appendix C. The bosonic fields are characterized
by the same spectral density
$J\left(\omega\right)=\alpha^{2}\omega\exp\left(-\omega/\omega_{c}\right)$,
where $\alpha$ is a dimensionless constant linked to the noise strength and
$\omega_{c}$ is the cut-off frequency which is also connected to the
correlation time of the boson fields $\tau_{c}$, as $\tau_{c}\equiv
2\pi/\omega_{c}$.

Using the state nomenclature of Fig. 1, the amplitude damping noise is due to
emission and absorption processes corresponding to atomic transitions between
the states $\left|\pm 1\right\rangle$ and $\left|0\right\rangle$, while the
dephasing results in destroying the phases between $\left|\pm 1\right\rangle$
and $\left|0\right\rangle$. In fact, in Ref. [28] the dephasing noise does not
couple the states $\ket{1}$ and $\ket{-1}$ directly. In this work, we have
added a direct coupling between these states in the dephasing noise, that is,
we have included a term proportional to
$\outerproduct{-1}{-1}-\outerproduct{1}{1}$ in the interaction Hamiltonian
with the dephasing bath (see Appendix C of Ref. [28]).

As in Ref. [28], in our simulations we assume for the bath correlation time
$\tau_{c}=\tau/4$ and we take all the coupling constants appearing in the
interactions terms with the environments all equal between them, here
including the new coupling constant appearing in the added term for the
dephasing noise. We name $\lambda$ this common coupling constant. In order to
clarify the notation adopted here with respect to Ref. [28] we remark the
following. In Ref. [28] we named as effective coupling parameter the quantity
$\bar{\lambda}=\lambda\alpha$. However, in the master equation the final
quantity giving the effective strength of the noises is
$\lambda_{\mathrm{me}}=\bar{\lambda}\alpha=\lambda\alpha^{2}$. In this sense,
in the figures of Ref. [28] giving the value of $\bar{\lambda}$ was not
sufficient to obtain the value of $\lambda_{\mathrm{me}}$ effectively used in
the master equation. In order to do that, it is enough to consider that it was
like if we assumed $\lambda=1$. In this way the value $\bar{\lambda}=0.1$ used
in Ref. [28] corresponds here to $\lambda_{\mathrm{me}}=0.01$.

The impact of our protection scheme is here quantified by means of the
fidelity measuring how close are the states obtained, respectively, by the
dissipative dynamics associated to the total Hamiltonian of Eq. (1) (treated
by means of a Redfield master equation [28]) and by the noise-free dynamics
governed by the Hamiltonian $H_{\mathrm{lab}}(t)$ of Eq. (12). The latter
dynamics gives after a time $\tau$ the same output state of the ideal noise-
free dynamics governed by the gate Hamiltonian $H_{G}$ of Eq. (11) (i.e., the
dynamics in the absence of the protective scheme). In our study we also
consider the dissipative dynamics in the absence of the protective scheme,
which is obtained by turning off the control Hamiltonian $H_{c}(t)$.
Concerning the fidelity, it is defined for two arbitrary states $\rho$ and
$\sigma$ as $[\mathrm{Tr}\\{\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\\}]^{2}$.
Since the two baths are always assumed to be identical we refer to them using
the term bath at singular.

###  IV.2 The Hadamard gate

Let us consider the single-qudit gate known as the Hadamard gate for $d=3$,
which we have also studied in Ref. [28]. This gate maps a state of the
computational basis into a superposition of all states and its unitary matrix
representation can be written as (here and in the following the matrices are
given in the ground-state subspace basis $\\{\ket{-1}$, $\ket{0}$,
$\ket{1}\\}$)

$\displaystyle U_{H}$ $\displaystyle=$
$\displaystyle-\frac{i}{\sqrt{3}}\left[\begin{array}[]{ccc}1&1&1\\\
1&e^{i\frac{2\pi}{3}}&e^{-i\frac{2\pi}{3}}\\\
1&e^{-i\frac{2\pi}{3}}&e^{i\frac{2\pi}{3}}\end{array}\right].$ (24)

This unitary satisfies

$U_{H}=\exp(-i\frac{H}{\hbar}\tau),$ (25)

where $H$ is the Hamiltonian associated to the Hadamard gate ($H$ gate). We
can invert this relation by using the matrix logarithm in both sides of the
equality to obtain $H$. Then, taking the gate Hamiltonian of Eq. (11), and
combining it with the definition of the $H_{L}^{\prime}$ and $H_{F}^{\prime}$
operators, we get the matrix which corresponds to the square of $\Upsilon(t)$,
using Eq. (13). Unfortunately the resulting matrix is, in general, not
analytically diagonalizable for a generic time instant $t$. The numerical
diagonalization however is straightforward. Therefore in order to calculate
$\Upsilon$ for a specific time $t_{i}$, we first need to evaluate
$\Upsilon^{2}(t_{i})$ and then compute $\Upsilon(t_{i})$.

Figure 2: Numerical solutions for the six independent Rabi frequencies, as
functions of time, used to protect the action of the Hadamard gate on a single
qutrit. The following parameters and the styles of the curves are the same as
in Figs. 5, 6, 7, 8, 9, 10, and 11. The chosen detunings are: $\Delta_{1}=1$
GHz, $\Delta_{2}=2$ GHz, and $\Delta_{3}=3$ GHz. The real parts of the Rabi
frequencies are shown with solid lines, while the imaginary parts with dashed
lines. Due to the fact that $U_{c}(t)$ is periodic, the plots consist of only
one period $t_{0}$, in units of the gate time $\tau$. The control period is
chosen to be $t_{0}=\tau/10$.

The numerical solutions for the Rabi frequencies necessary to implement the
Hadamard gate are shown in Fig. 2. As one can see, the shape of the numerical
solutions resemble to smooth trigonometric functions. It is relevant to
mention that these shapes do not change for different values of $t_{0}$. The
only difference would be the numerical value of the period. This leads us to
conclude that it should be possible to effectively implement the required time
dependency of the Rabi frequencies in a laboratory through the use of electro-
optic modulators, for phase control, combined with Mach-Zehnder
interferometers. To be more specific, the control over the electric field
phase can be realized by controlling the difference of electric potential
applied inside the electro-optical modulator. This creates a time dependency
on the refraction index, which in turn, will create a time dependency on the
electric field phase. If the laser beam is split in two and only one of the
beams goes through the modulator, we create a pair of beams with a difference
in its relative phase that can be completely controlled. Finally, by combining
both beams again, we can create interference, meaning we can translate the
time dependency of the relative phase into control of the amplitude of the
recombined beams [40]. Since the time-dependent Rabi frequencies are
proportional to the laser amplitude, then it should be possible to generate
them through the process described.

To show that with the use of these Rabi frequencies one can protect the qutrit
from noise while also applying the desired Hadamard gate, we have performed
simulations of different kinds of dynamics for several values of the control
period $t_{0}$.

Figure 3:  Numerical solutions to estimate how much the GCDD procedure
protects the amplitude damping and dephasing noises during the action of the
Hadamard gate (the time $t$ is given in units of the gate time $\tau$), with
$\ket{0}$ as initial state, for different control periods $t_{0}$. The
reference case in which there is no protection is also shown. The temperature
is $T=8\pi\hbar/(k_{B}\tau)$. The following parameters are equal to the ones
used in Figs. 4, 12, and 15. The bath correlation time $\tau_{c}$ is set equal
to $\tau/4$, the coupling constants between the system and the baths are all
equal and the effective strength of the noises is
$\lambda_{\mathrm{me}}=0.01$, so that the Born approximation is valid [28].
The quantity $m$ in the legend is defined as $m\equiv\tau/t_{0}$. It follows
that $t_{0}=4\tau_{c}/m$.

The fidelity as a function of time is shown in Fig. 3 in the case of a nonzero
temperature for the bath. Differently from Ref. [28], as in Fig. 2 and in the
subsequent figures, $t_{0}$ is given in units of the gate time $\tau$ (we use
the quantity $m\equiv\tau/t_{0}$), instead of the bath correlation time
$\tau_{c}$, because the protection scheme does not depend on the
characteristics of the noise. This choice is also due to the fact that from an
experimental point of view, the reference time one has access to is the gate
time. One can notice that the gate fidelity becomes higher with respect to the
case without protection the shorter the control period used is. Therefore, the
solutions giving the Rabi frequencies presented in Fig. 2 can indeed provide
protection from noise while also executing the Hadamard gate to our qutrit.

Figure 4:  Impact of the GCDD procedure for different values of $t_{0}$ (we
use the quantity $m=\tau/t_{0}$), for the Hadamard gate at zero temperature
(with both amplitude damping and dephasing noises). As in Fig. 3, we use
$\ket{0}$ as initial state, $\tau_{c}=\tau/4$, and
$\lambda_{\mathrm{me}}=0.01$.

We have also considered the case in which the bath can be assumed to be at
zero temperature. In this limit we have only vacuum-noise effects in terms of
dephasing and amplitude damping. The fidelity as a function of time, for
different control periods, in the case of vacuum noise is shown in Fig. 4.
Notice that, even though the fidelity in the absence of protection drops only
to around $0.735$, we still need $m$ of the same order of those used in the
case where there is also thermal noise (see Fig. 3) in order to achieve
fidelities above $0.99$. This shows that, for this noise model, vacuum noise
is, at least, as difficult to overcome as thermal noise. We notice that the
same Rabi frequencies that arise from the GCDD method are used to protect the
qutrit against both vacuum and thermal noise. This aspect has been already
treated in the final comment of Sec. II.

###  IV.3 Two additional gates

We have performed the same procedure for two other one-qutrit gates, which we
name $U_{X}$ and $U_{Z}$. Their unitary matrix representations are written as
follows:

$\displaystyle U_{X}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\
0&1&0\end{array}\right],$ (29) $\displaystyle U_{Z}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}1&0&0\\\ 0&e^{i\frac{2\pi}{3}}&0\\\
0&0&e^{-i\frac{2\pi}{3}}\end{array}\right].$ (33)

These two gates can be thought as extensions of the usual $X$ and $Z$ one-
qubit gates to the three-dimensional case. $U_{X}$ flips the quantum states
cyclically while $U_{Z}$ applies phases of $\pm 2\pi/3$ to the states
$\ket{0}$ and $\ket{1}$, respectively, and does nothing to the state
$\ket{-1}$.

Figure 5:  Same plots and parameters as in Fig. 2, but for the $X$ gate.
Figure 6:  Same plots and parameters as in Fig. 2, but for the $Z$ gate.

Similarly to the case of the Hadamard gate, we present the Rabi frequencies
for the $U_{X}$ and $U_{Z}$ gates in Figs. 5 and 6, respectively. We are
always assuming that the control period $t_{0}$ is chosen to be sufficiently
shorter than the bath correlation time $\tau_{c}$. The GCDD simulations of the
above three distinct one-qutrit gates (not shown for the $X$ and $Z$ gates)
strongly suggest that this method might be used to generate any one-qutrit
gate while also protecting it from external noise. In particular, as far as
the gate dynamics studied here indicate, by choosing $t_{0}=\tau/36$. we can
achieve fidelity values above $0.99$.

###  IV.4 Beyond an universal set of quantum gates

It is a well known result that a finite set of one-qudit gates combined with
an entangling gate can be used to approximate any multi-qudit quantum gate
within arbitrary precision [41]. This is a useful concept since there are
quantum gates easier to implement than others. In our present atomic qutrit
model, we have shown that in order to apply a protected quantum gate all is
needed is to generate specific time-dependent Rabi frequencies. Therefore,
depending on the precision the Rabi frequencies can be controlled in
laboratory, it may be possible to generate arbitrarily complex gates without
the need of decomposing them into simpler gates.

To illustrate this possibility, we consider three different quantum gates that
are not particularly useful in any direct algorithm, but instead, illustrate
what a general single-qutrit gate looks like. It is possible to write any
$3\times 3$ unitary matrix using a set of eight parameters given by
$p=\\{\theta_{1},\theta_{2},\theta_{3},\phi_{1},\phi_{2},\phi_{3},\phi_{4},\phi_{5}\\}$,
that is

$\displaystyle U=\begin{bmatrix}u_{00}(p)&u_{01}(p)&u_{02}(p)\\\
u_{10}(p)&u_{11}(p)&u_{12}(p)\\\ u_{20}(p)&u_{21}(p)&u_{22}(p)\end{bmatrix}.$
(34)

The parameters are such that $\theta_{j}\in[0,\pi]$ and $\phi_{j}\in[0,2\pi[$.
The explicit form of the entries $u_{jk}(p)$ are given by [42]

$\displaystyle\begin{split}u_{00}(p)&=\cos{\theta_{1}}\cos{\theta_{2}}e^{i\phi_{1}},\\\
u_{01}(p)&=\sin{\theta_{1}}e^{i\phi_{3}},\\\
u_{02}(p)&=\cos{\theta_{1}}\sin{\theta_{2}}e^{i\phi_{4}},\\\
u_{10}(p)&=\sin{\theta_{2}}\sin{\theta_{3}}e^{-i\phi_{4}-i\phi_{5}}\\\
&\quad\,-\sin{\theta_{1}}\cos{\theta_{2}}\cos{\theta_{3}}e^{i\phi_{1}+i\phi_{2}-i\phi_{3}},\\\
u_{11}(p)&=\cos{\theta_{1}}\cos{\theta_{3}}e^{i\phi_{2}},\\\
u_{12}(p)&=-\cos{\theta_{2}}\sin{\theta_{3}}e^{-i\phi_{1}-i\phi_{5}},\\\
&\quad\,-\sin{\theta_{1}}\sin{\theta_{2}}\cos{\theta_{3}}e^{i\phi_{2}-i\phi_{3}+i\phi_{4}},\\\
u_{20}(p)&=-\sin{\theta_{1}}\cos{\theta_{2}}\sin{\theta_{3}}e^{i\phi_{1}-i\phi_{3}+i\phi_{5}}\\\
&\quad\,-\sin{\theta_{2}}\cos{\theta_{3}}e^{-i\phi_{2}-i\phi_{4}},\\\
u_{21}(p)&=\cos{\theta_{1}}\sin{\theta_{3}}e^{i\phi_{5}},\\\
u_{22}(p)&=\cos{\theta_{2}}\cos{\theta_{3}}e^{-i\phi_{1}-i\phi_{2}}\\\
&\quad\,-\sin{\theta_{1}}\sin{\theta_{2}}\sin{\theta_{3}}e^{-i\phi_{3}+i\phi_{4}+i\phi_{5}}.\end{split}$
(35)

This parametrization might seem confusing and dense, but, in the context of
numerical simulation, it is simpler than the usual parametrization using the
Lie algebra generators [43].

We have generated pseudo-random values for the eight parameters to build up
three different one-qutrit gates. The three sets generated are:

$\displaystyle\begin{split}\theta_{1}^{(\alpha)}=0.924939\times\pi,\;\;\;\phi_{1}^{(\alpha)}&=0.773065\times\pi,\\\
\theta_{2}^{(\alpha)}=0.109418\times\pi,\;\;\;\phi_{2}^{(\alpha)}&=1.623579\times\pi,\\\
\theta_{3}^{(\alpha)}=0.424271\times\pi,\;\;\;\phi_{3}^{(\alpha)}&=0.562634\times\pi,\\\
\phi_{4}^{(\alpha)}&=0.966021\times\pi,\\\
\phi_{5}^{(\alpha)}&=0.632344\times\pi,\end{split}$ (36)
$\displaystyle\begin{split}\theta_{1}^{(\beta)}=0.363719\times\pi,\;\;\;\phi_{1}^{(\beta)}&=0.254886\times\pi,\\\
\theta_{2}^{(\beta)}=0.098480\times\pi,\;\;\;\phi_{2}^{(\beta)}&=1.419032\times\pi,\\\
\theta_{3}^{(\beta)}=0.420599\times\pi,\;\;\;\phi_{3}^{(\beta)}&=1.336089\times\pi,\\\
\phi_{4}^{(\beta)}&=1.284848\times\pi,\\\
\phi_{5}^{(\beta)}&=0.837395\times\pi,\end{split}$ (37)
$\displaystyle\begin{split}\theta_{1}^{(\gamma)}=0.539910\times\pi,\;\;\;\phi_{1}^{(\gamma)}&=1.491107\times\pi,\\\
\theta_{2}^{(\gamma)}=0.947045\times\pi,\;\;\;\phi_{2}^{(\gamma)}&=0.950144\times\pi,\\\
\theta_{3}^{(\gamma)}=0.179982\times\pi,\;\;\;\phi_{3}^{(\gamma)}&=1.780184\times\pi,\\\
\phi_{4}^{(\gamma)}&=0.919544\times\pi,\\\
\phi_{5}^{(\gamma)}&=1.573260\times\pi.\end{split}$ (38)

Plugging these values into Eq. (34) yields three one-qutrit gates which we
name $U_{\alpha}$, $U_{\beta}$, and $U_{\gamma}$, respectively. Using the same
procedure described for the gates $H$, $X$, and $Z$, we calculate the time
dependence of the Rabi frequencies as the ones shown in Figs. 7-9. As for the
gates $H$, $X$, and $Z$, also these three general gates require a time
dependency for the Rabi frequencies that resemble simple trigonometric
functions. This suggests that, depending on the precision at which the Rabi
frequencies can be controlled in the laboratory, it might be possible to
implement general one-qutrit gates without the need of decomposing them into
simpler, universal gates.

Figure 7: Same plots and parameters as in Fig. 2, but for the $U_{\alpha}$
gate. Figure 8: Same plots and parameters as in Fig. 2, but for the
$U_{\beta}$ gate. Figure 9: Same plots and parameters as in Fig. 2, but for
the $U_{\gamma}$ gate.

Of course, this is just a speculative conclusion over the numerical results.
However, if in a practical implementation attempt of these gates one could
verify that it is not necessary to decompose them into simpler gates, this
could mean a reasonable improvement in the complexity in algorithms involving
complicated one-qutrit (or qudit) gates.

###  IV.5 Rabi frequencies in gate transitions

It is pertinent to ask how the Rabi frequencies change when different gates
are applied in sequence from one to the next (let us say from $G_{1}$ to
$G_{2}$). The only quantity that must change in the $\Upsilon(t)$ matrix is
the operator $G$. We must then only make the substitution

$\displaystyle G$ $\displaystyle\rightarrow$ $\displaystyle
G_{1}\theta(\tau-t)+G_{2}\theta(t-\tau),$ (39)

where $\theta$ is the Heaviside step function. Obviously this would mean that
the Rabi frequencies would need to be instantaneously discontinuous at the
transition of the gates, which is not implementable in a laboratory. To
overcome this problem we can replace the Heaviside step function with a smooth
function defined as

$\displaystyle\theta_{s}(t)$ $\displaystyle=$ $\displaystyle\frac{1}{\exp(-\mu
t)+1},$ (40)

that becomes the Heaviside function in the limit $\mu\rightarrow\infty$.

As a first example, we consider the case of a transition between the Hadamard
gate and a period where the qutrit state is maintained at its state (this is
done by turning on the GCDD protection associated to the case $H_{G}=0$). The
time dependence of the diagonal elements of the operator $\Upsilon(t)$ are
shown in Fig. 10 using the functions $\theta$ and $\theta_{s}$. By setting the
parameter $\mu=100$ the loss in the final fidelity is of the order $10^{-6}$,
a negligible value compared to the natural losses due to the external noise.
As a second example, we also consider the transition between the Hadamard and
$Z$ gates in Fig. 11. Notice how, in both examples, with just the addition of
the $\theta_{s}$ function, it is possible to make the Rabi frequencies and
their first derivative continuous at the time the transition between two
quantum gates happens.

We have also simulated the noisy evolution of three operations in sequence:
Hadamard gate, followed by a period where the memory is kept (identity
operator), followed by the application of the $X$ gate given by Eq. (29). The
results are shown in Fig. 12. By choosing $m=36$ we can achieve a fidelity of
around $0.995$ at the end of the process. We finally remark that if we start
from the initial state $\ket{0}$, as in Figs. 3 and 4, the protection scheme
is globally less efficient, even if good results can be in any case obtained
by increasing the value of $m$, with a gate fidelity of around 0.992 for
$m=36$.

Figure 10:  Diagonal elements of the operator $\Upsilon(t)$, as functions of
time, with a transition between the application of the Hadamard gate and the
period where the memory is kept, using the Heaviside $\theta$ function (left
column) and the smooth function $\theta_{s}$ (right column) with $\mu=100$.
Same parameters as in Fig. 2. Figure 11:  Same plots and parameters as in Fig.
10, with a transition between the Hadamard gate and the $Z$ gate. Figure 12:
Impact of the GCDD procedure for different values of $t_{0}$ (we use the
quantity $m=\tau/t_{0}$), with two gates ($H$ and $X$) being applied in
sequence, separated by an interval where the memory of the state is supposed
to be kept. As in Fig. 3, we use $T=8\pi\hbar/(k_{B}\tau)$, $\tau_{c}=\tau/4$,
and $\lambda_{\mathrm{me}}=0.01$. However, the initial state is different and
here it is $\ket{-1}$.

###  IV.6 An illustrative example

As a further application of our GCDD protocol, to illustrate the efficiency of
gate transitions in a more complex situation, we show how to protect a quite
simple quantum algorithm. To this purpose, we consider the algorithm developed
in Ref. [33], where it was shown that one can determine the parity in a
permutation of three different elements using a single qutrit (it can be
extended to determine the parity of $n$ different elements using a single
$n$-dimensional quantum system [33]).

If we represent the six possible permutations as

$\displaystyle\begin{split}&f_{1}=\begin{pmatrix}-1&0&1\\\
-1&0&1\end{pmatrix},\;f_{4}=\begin{pmatrix}-1&0&1\\\ 1&0&-1\end{pmatrix},\\\
&f_{2}=\begin{pmatrix}-1&0&1\\\
0&1&-1\end{pmatrix},\;f_{5}=\begin{pmatrix}-1&0&1\\\ 0&-1&1\end{pmatrix},\\\
&f_{3}=\begin{pmatrix}-1&0&1\\\
1&-1&0\end{pmatrix},\;f_{6}=\begin{pmatrix}-1&0&1\\\
-1&1&0\end{pmatrix},\end{split}$ (41)

we have that $f_{1}$, $f_{2}$, and $f_{3}$ are even permutations while
$f_{4}$, $f_{5}$, and $f_{6}$ are odd ones. These permutation operations can
be easily written as unitary operators. In fact, notice that the unitary
defined in Eq. (29) corresponds to the permutation $f_{3}$.

It is straightforward to verify that, if we adopt $\ket{0}$ as our initial
state, then applying the Hadamard gate $U_{H}$, defined in Eq. (24), followed
by one of the permutations, followed by $U_{H}^{\dagger}$, the final state
will be, up to a global phase, $\ket{0}$ if the permutation is even and
$\ket{1}$ if the permutation is odd. Therefore, a single execution of the
circuit involving these three gates can tell us, at the time we measure the
final state, the parity of the permutation.

Figure 13:  Same plots and parameters as in Fig. 2, but for the $f_{2}$
permutation gate.

In Figs. 13 and 14 we show the time dependent Rabi frequencies required to
perform the $f_{2}$ (even) and $f_{5}$ (odd) unitary operations respectively.

Figure 14:  Same plots and parameters as in Fig. 2, but for the $f_{5}$
permutation gate. Figure 15:  Impact of the GCDD procedure on the fidelity at
$\tau$ (gate fidelity) for different values of $m=\tau/t_{0}$, for the
permutation parity determining circuit for the six different permutations,
using $\ket{0}$ as initial state and the global gate
$U_{H}^{\dagger}f_{k}U_{H}$. The gate fidelity is computed using as expected
final state either $\ket{0}$ or $\ket{1}$, according to the parity of the
permutation. The absence of protection is represented by $m=0$ in the graph.
As in Fig. 3, we use $T=8\pi\hbar/(k_{B}\tau)$, $\tau_{c}=\tau/4$, and
$\lambda_{\mathrm{me}}=0.01$.

Using the strategy presented in Sec. IV.4 we also simulate the execution of
the quantum circuit under the action of both, vacuum and thermal noise, with
and without the presence of protection provided by the GCDD method. The
simulations are done for the six permutations in Eq. (41), and the results for
the fidelity of the final measurement, which reveals the parity of the
permutation, are all shown (superposed) in Fig. 15. Notice that for the case
where the protection is active with large $m$, the circuit can be executed
with almost unitary fidelity for every permutation. For the case where the
protection is turned off, the fidelity tends to $1/3$ for all cases (this
point is represented by $m=0$ in the graph), corresponding to an evolution to
the maximally mixed state.

The results of the simulation with active protection and large $m$ show that,
if one of the six permutations $f_{k}$ is treated as a black box operation
inside the circuit, one could infer the parity of $f_{k}$ with a fidelity
arbitrarily close to $1$ by measuring the final state. If we are allowed to
use ancilla qudits, it is possible to even guess the permutation $f_{k}$
instead of just its parity. This could be done by adapting the phase
estimation algorithm to qudits [44], and applying it right after the described
circuit. By measuring the eigenvalue associated with the operation
$U_{H}^{\dagger}f_{k}U_{H}$ one could infer the value of $k$, and hence, the
permutation.

##  V Conclusions

The present study is based on the GCDD protection scheme introduced in Ref.
[28], that provides a method to compute the time-dependent control
Hamiltonians required to decouple the system under study from its environment.
The major question tackled in this article is whether it is possible to tailor
the time-dependent Rabi frequencies required by the GCDD procedure, using the
available nowadays technological capabilities.

We have presented time-dependent Rabi frequencies to decouple an atomic qutrit
from both dephasing and amplitude-damping noises occurring together, while
also executing intrinsically different quantum operations, including the case
of random gates. We notice that the detunings involved in the protection
scheme are constant and independent. Within the validity of the approximations
of our original calculations [28] we have presented an illustration of how the
GCDD control could be realized in the laboratory. We have suggested for the
various involved parameters orders of magnitudes within ranges which might be
achievable today in a realistic experiment. Indeed, in the case of
manipulation of trapped ions for quantum computing, it is already usual in the
laboratory to follow several procedures that, combined, illustrate more
technological prowess than the GCDD method requires [45]. We have also
considered the protection of a sequence of quantum gates on the system,
showing how the required rabi-frequencies should vary at the transitions.
Finally, we have explicitly shown that the protection scheme can be used to
implement efficiently a simple quantum algorithm allowing one to determine the
parity of the permutations of a set of three elements.

The analysis we present here shows that, at least in principle, it is possible
to realize an experiment to test our protection scheme if we are able to
emulate the noise effects involved in our model, either by finding a naturally
noisy system similar to our setup, or by artificially introducing the same
kinds of perturbations. If such an experiment is accomplished, the principles
used in the present model could be translated to other systems with the same
logical capabilities and even for qudits with $d>3$. The latter condition
could be also realised using atomic qudits based on an extension of our simple
qutrit model. It is always important to have thought experiments illustrating
how future realistic implementations should work, and it really matters when
the illustrative models show possible drawbacks and obstacles one should face
when implementing theoretical ideas in the laboratory. We strongly recommend a
serious analysis by experimentalists about how to implement a version of such
an experiment, at least as a proof-of-principle, to assess possible unforeseen
new questions that might further improve the technological aspects of quantum-
information processing and quantum computing.

###### Acknowledgements.

R.d.J.N. and F.F.F. acknowledge support from Fundação de Amparo à Pesquisa do
Estado de São Paulo (FAPESP), Projects No. 2018/00796-3 and 2021/04655-8, and
also from the National Institute of Science and Technology for Quantum
Information (CNPq INCT-IQ 465469/2014-0) and the National Council for
Scientific and Technological Development (CNPq).

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# Non-autonomous Overdetermined Problems for the Normalized $p\,$-Laplacian

Lucio Cadeddu, Antonio Greco and Benyam Mebrate  Lucio Cadeddu and Antonio
Greco
Department of Mathematics and Computer Science, University of Cagliari
Postal address: via Ospedale 72, 09124 Cagliari, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>Benyam Mebrate
Department of Mathematics, Wollo University, Dessie, Ethiopia
Postal address: 1145<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We present existence and nonexistence results on the solution of an
overdetermined problem for the normalized $p$-Laplacian in a bounded open set,
with $p$ ranging from $1$ to infinity. More precisely we consider a non-
constant Neumann condition at the boundary. The definitions and statements
needed to understand the main results are recalled in detail.

###### Key words and phrases:

Overdetermined problems, Viscosity solutions, Normalized infinity-Laplacian

MSC 2020: 35B06, 35N25, 35R35

##  1\. Introduction

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with a differentiable
boundary $\partial\Omega$, whose outer normal we denote by $\nu$. Choose
$\bar{x}\in\Omega$ and define

$R_{1}=\min_{x\in\partial\Omega}|x-\bar{x}|,\quad
R_{2}=\max_{x\in\partial\Omega}|x-\bar{x}|.$ (1.1)

In this paper we study overdetermined problems ruled by the normalized
$p$-Laplacian for $p\in[1,\infty]$: more precisely, we study the problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}^{N}u=1&\mbox{in $\Omega$;}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr u=0,\;-\dfrac{\partial
u}{\partial\nu}=q(|x-\bar{x}|)&\mbox{on $\partial\Omega$,}\end{array}\right.$
(1.2)

where $q(r)$ is a real-valued function defined on $[R_{1},R_{2}]$. For a
smooth function $u$ with nonvanishing gradient $Du$, the operator
$\Delta_{p}^{N}u$ is given by

$\Delta_{p}^{N}u=\begin{cases}\frac{1}{p}\,|Du|^{2-p}\,\mathop{\rm
div}\big{(}|Du|^{p-2}\,Du\big{)},&p\in[1,\infty);\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr|Du|^{-2}\,\langle D^{2}u\,Du,Du\rangle&p=\infty.\end{cases}.$
(1.3)

The relationship between the case when $p$ is finite and the case $p=\infty$
is put into evidence by the equality

$\Delta_{p}^{N}u=\frac{\,p-1\,}{p}\,\Delta_{\infty}^{N}\,u+\frac{1}{\,p\,}\,\Delta_{1}^{N}u\quad\mbox{for
$p\in[1,\infty)$}$

(see [2, (1.2)] or [12, (1.6)]): for a given function $u$ and at a fixed point
$x$ such that $Du(x)\neq 0$, it follows that
$\Delta_{p}^{N}u(x)\to\Delta_{\infty}u(x)$ when $p\to\infty$. The term
“normalized” is used to make a distinction from the classical $p$-Laplace
operator $\Delta_{p\,}u$ given by

$\Delta_{p\,}u=\begin{cases}\mathop{\rm
div}\big{(}|Du|^{p-2}\,Du\big{)},&p\in[1,\infty);\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\langle D^{2}u\,Du,Du\rangle,&p=\infty.\end{cases}$ (1.4)

Solutions to (1.2) are intended in the viscosity sense for $p\in[1,\infty]$:
the definition of a viscosity solution is recalled in Section 2, together with
the meaning of $\partial u/\partial\nu$ in (1.2). In the special case when
$p=1$, we also consider solutions which are smooth near the boundary (see
below). Concerning the usual Laplace operator $\Delta=\Delta_{2}$, in the
fundamental work [18] Serrin showed, in particular, that if $\Omega$ is
sufficiently smooth and there exists a solution $u\in
C^{2}(\overline{\Omega})$ to the problem

$\left\\{\begin{array}[]{ll}-\Delta u=1&\mbox{in $\Omega$;}\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr u=0,\;-\dfrac{\partial u}{\partial\nu}=q&\mbox{on
$\partial\Omega$,}\end{array}\right.$ (1.5)

where $q$ is a positive constant, then $\Omega$ is a ball centered at some
$x_{1}\in\mathbb{R}^{n}$ and $u$ is given by
$u(x)=\dfrac{\,r^{2}-|x-x_{1}|^{2}}{2n}$, where $r$ is the radius of $\Omega$.
Notice that problem (1.5) is invariant under translations, hence the point
$x_{1}$ is arbitrary, while the radius $r$ depends on the value of $q$.
Buttazzo and Kawohl in [3] studied the corresponding overdetermined problem
both for the infinity-Laplacian:

$\left\\{\begin{array}[]{ll}-\Delta_{\infty}\,u=1&\mbox{in $\Omega$;}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr u=0,\;-\dfrac{\partial
u}{\partial\nu}=q&\mbox{on $\partial\Omega$}\end{array}\right.$ (1.6)

and the normalized one:

$\left\\{\begin{array}[]{ll}-\Delta_{\infty}^{N}\,u=1&\mbox{in $\Omega$;}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr u=0,\;-\dfrac{\partial
u}{\partial\nu}=q&\mbox{on $\partial\Omega$.}\end{array}\right.$

Banerjee and Kawohl [2], instead, considered the corresponding problem for the
normalized $p$-Laplacian with $p\in(1,\infty)$. They proved that if $u\in
C(\overline{\Omega})$ is a viscosity solution to

$\left\\{\begin{array}[]{ll}-\Delta_{p}^{N}u=1&\mbox{in $\Omega$;}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr u=0,\;-\dfrac{\partial
u}{\partial\nu}=q&\mbox{on $\partial\Omega$},\end{array}\right.$

then $\Omega$ is a ball. The result is also true if $p=1$ provided that $u$ is
smooth near the boundary [12, Remark 4.3]. However, for $p=\infty$ it is
generally false [3, Theorem 2]. Problem (1.6), where $\Omega$ contains the
origin and $u$ satisfies a non-constant Neumann condition at the boundary
$\partial\Omega$, given by

$-\dfrac{\partial u}{\partial\nu}=q(|x|),$ (1.7)

has been studied in [8]. The corresponding problem for the classical
$p$-Laplacian with finite $p>1$, namely

$\left\\{\begin{array}[]{ll}-\Delta_{p\,}u=1&\mbox{in $\Omega$;}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr u=0,\;-\dfrac{\partial
u}{\partial\nu}=q(|x|)&\mbox{on $\partial\Omega$},\end{array}\right.$

was considered in [6, 7, 9]. The main results were basically focused on the
geometry of $\Omega$. This paper deals with a similar problem related to the
normalized $p$-Laplacian, $p\in[1,\infty]$. Most of the notations we use are
standard. By $c_{p}$ we denote the constant

$c_{p}=\begin{cases}\frac{p}{\,p+n-2\,},&p\in[1,\infty);\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr 1,&p=\infty.\end{cases}$ (1.8)

Our first result is the following:

######  Theorem 1.1.

Let $p\in[1,\infty]$ and let $c_{p}$ be as above.

  1. (1)

Suppose that the equation $q(r)-c_{p}\,r=0$ possesses a unique solution
$R\in[R_{1},R_{2}],$ and $(q(r)-c_{p}\,r)(r-R)>0$ for all
$r\in[R_{1},R_{2}]\setminus\\{R\\}$. Then the viscosity solution to problem
(1.2) exists only in the special case when $R_{1}=R=R_{2}$ (i.e.,
$\Omega=B(\bar{x},R)$). If, instead, $R_{1}<R_{2},$ then the viscosity
solution to (1.2) does not exist.

  2. (2)

Suppose that the function $\rho(r)=\frac{q(r)}{r}$ is strictly increasing.
Then the viscosity solution to problem (1.2) exists only if $\Omega$ is a ball
centered at $\bar{x}$.

  3. (3)

If $q$ is continuous, and if the equation $q(r)-c_{p}\,r=0$ does not possess
any solution, then the viscosity solution to problem (1.2) does not exist.

The theorem is proved in Section 4 by means of a comparison argument. The
result shows that the behavior of the normalized $p$-Laplacian with respect to
the overdetermined problem (1.2) enjoys a continuity property at infinity:
more precisely, since $c_{\infty}=1=\lim\limits_{p\to\infty}c_{p}$, the
statement for $p=\infty$ is readily obtained from the case when $p$ is finite
by just letting $p\to\infty$. By contrast, the classical (not normalized)
$p$-Laplacian (1.4) exhibits a different behavior: indeed, a result similar to
Theorem 1.1 for $p\in(1,\infty)$ has been proved in [6, Theorem 1.2]. There,
the ratio $q(r)/r^{p-1}$ (which is obtained by letting $\varepsilon_{0}=p-1$
in (5)) is required to be strictly increasing: such a ratio makes no sense
when $p\to\infty$. Nevertheless, a corresponding result for the infinity-
Laplacian is found in [8, Theorem 1.1]: surprisingly, the relevant ratio in
this case is $q(r)/r^{1/3}$.

Unlike [6, Theorem 1.2], our Theorem 1.1 also applies to the special case when
$p=1$. In such a case, as mentioned before, we focus on solutions $u$ which
are smooth near the boundary. To this purpose, we adopt the following
definition:

######  Definition 1.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set of class $C^{2}$, and
for $\varepsilon>0$ define
$\Omega_{\varepsilon}=\\{\,x\in\Omega\mid\mathop{\rm
dist}(x,\partial\Omega)<\varepsilon\,\\}$. We say that a viscosity solution to
(1.2) is a smooth solution near the boundary if:

  1. (1)

$u\in C^{2}(\overline{\Omega}_{\varepsilon})$ for some $\varepsilon>0$;

  2. (2)

$Du\neq 0$ in $\overline{\Omega}_{\varepsilon}$;

  3. (3)

the equation $-\Delta_{p}^{N}u=1$ is satisfied in the classical sense in
$\overline{\Omega}_{\varepsilon}$;

  4. (4)

the boundary conditions in (1.2) hold pointwise.

A solution $u$ which is smooth near the boundary satisfies the equation

$(n-1)\,|Du|\,H(x)=1\quad\mbox{in $\overline{\Omega}_{\varepsilon}$,}$ (1.9)

where $H(x)$ is the mean curvature of the level surface $u={}$constant passing
through the point $x$ (see [5, (14.102)] and [12, Remark 4.3]). Note that if
(1.2) has a solution $u$ wich is smooth near the boundary, then the surface
$\partial\Omega$ must have a positive mean curvature $H(x)$ as a consequence
of (1.9). In such a case we may prescribe the Neumann condition by means of a
function $q(|x-\bar{x}|,\,H(x))$ that not only depends on the distance from
$x\in\partial\Omega$ to some fixed point $\bar{x}\in\Omega$, but is also
allowed to depend on the mean curvature $H(x)$ of $\partial\Omega$. More
precisely, we consider the overdetermined problem

$\left\\{\begin{array}[]{ll}-\Delta_{1}^{N}u=1&\mbox{in $\Omega$;}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr u=0,\;-\dfrac{\partial
u}{\partial\nu}=q(|x-\bar{x}|,\,H(x))&\mbox{on
$\partial\Omega$,}\end{array}\right.$ (1.10)

where $q\colon[R_{1},R_{2}]\times(0,\infty)\to(0,\infty)$ is a prescribed,
positive function. In the case when $q(r,h)$ is independent of $h$, problem
(1.10) clearly reduces to (1.2). We have:

######  Theorem 1.3.

Let $\Omega$ be a bounded open set of class $C^{2}$. Choose $\bar{x}\in\Omega$
and define $R_{1},R_{2}$ as in (1.1). Consider a positive function $q(r,h)$
such that

  1. (1)

$q(r,h)$ is monotone non-decreasing in $h$ for every $r\in[R_{1},R_{2}]$;

  2. (2)

the ratio $\frac{q(r,\,1/r)}{r}$ is strictly increasing.

If problem (1.10) has a solution $u$ which is smooth near the boundary, then
$R_{1}=R_{2}$ (i.e., $\Omega=B(\bar{x},R_{1})$).

######  Example 1.4.

If the function $q$ has the special form $q(r,h)=r^{\alpha}\,h^{\beta}$, then
$q(r,\,1/r)\allowbreak=r^{\alpha-\beta}$ and the assumptions in the theorem
are satisfied provided that $\alpha-1>\beta\geq 0$.

######  Remark 1.5.

Several symmetry results were obtained in [8] for overdetermined problems
related to the equation $\Delta_{\infty}u=0$. As mentioned in [1, Remark 2.2,
p. 599], “there is no difference between the two resulting equations (in the
viscosity sense) when the right-hand side $f\equiv 0$”. Hence all results in
[8] concerning solutions to $\Delta_{\infty}u=0$ also hold for normalized
infinity-harmonic functions, i.e., for solutions of $\Delta_{\infty}^{N}u=0$.

The paper is organized as follows: in Section 2 we define viscosity solutions,
and in Section 3 we recall some preliminary lemmas which will be used to prove
our main results in Section 4.

##  2\. Viscosity Solutions

Let $p\in[1,\infty]$. As usual (see for instance [2, 13, 16]), if $u\in
C(\Omega)$ is twice differentiable at $x_{0}\in\Omega$ and if $Du(x_{0})\neq
0$, we define the upper and lower normalized $p$-Laplacian of $u$ at $x_{0}$,
respectively by
$\Delta_{p}^{+}\,u(x_{0})=\Delta_{p}^{-}\,u(x_{0})=\Delta_{p\,}u(x_{0})$,
where $\Delta_{p\,}u$ is given by (1.3). If, instead, $Du(x_{0})=0$, we denote
by $\lambda_{\rm min}=\lambda_{1}\leq\ldots\leq\lambda_{n}=\lambda_{\rm max}$
the eigenvalues of the Hessian matrix $D^{2}u(x_{0})$ and define

$\Delta_{p}^{+}\,u(x_{0})=\begin{cases}\frac{\,p-1\,}{p}\,\lambda_{1}+\frac{1}{\,p\,}\,\sum\limits_{i=2}^{n}\lambda_{i},&p\in[1,2];\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\frac{\,p-1\,}{p}\,\lambda_{n}+\frac{1}{\,p\,}\,\sum\limits_{i=1}^{n-1}\lambda_{i},&p\in(2,\infty);\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\lambda_{n},&p=\infty,\end{cases}$

and

$\Delta_{p}^{-}\,u(x_{0})=\begin{cases}\frac{\,p-1\,}{p}\,\lambda_{n}+\frac{1}{\,p\,}\,\sum\limits_{i=1}^{n-1}\lambda_{i},&p\in[1,2];\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\frac{\,p-1\,}{p}\,\lambda_{1}+\frac{1}{\,p\,}\,\sum\limits_{i=2}^{n}\lambda_{i},&p\in(2,\infty);\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\lambda_{1},&p=\infty.\end{cases}$

In the case when $p<\infty$, the definitions above may equivalently be
rewritten as follows (cf. [13, p. 177]):

$\Delta_{p}^{+}\,u(x_{0})=\begin{cases}\frac{\,p-2\,}{p}\,\lambda_{\rm
min}+\frac{1}{\,p\,}\,\Delta u,&p\in[1,2];\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\frac{\,p-2\,}{p}\,\lambda_{\rm max}+\frac{1}{\,p\,}\,\Delta
u,&p\in(2,\infty).\end{cases}$

and

$\Delta_{p}^{-}\,u(x_{0})=\begin{cases}\frac{\,p-2\,}{p}\,\lambda_{\rm
max}+\frac{1}{\,p\,}\,\Delta u,&p\in[1,2];\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\frac{\,p-2\,}{p}\,\lambda_{\rm min}+\frac{1}{\,p\,}\,\Delta
u,&p\in(2,\infty).\end{cases}$

For $f\colon\Omega\to\mathbb{R}$ we will give the definition of viscosity
solution to the PDE

$-\Delta_{p}^{N}u(x)=f(x)\;\text{in}\;\Omega$ (2.1)

We denote by $USC(\Omega)$ and $LSC(\Omega)$, respectively, the spaces of
upper semicontinuous and lower semicontinuous real-valued functions on
$\Omega$. Furthermore, for any $\varphi\in C^{2}(\Omega)$ we write
$u\prec_{x_{0}}\varphi$ (respectively, $u\succ_{x_{0}}\varphi$) if the
difference $u-\varphi$ has a local maximum (minimum) at $x_{0}$.

######  Definition 2.1.

  1. (1)

$u\in USC(\Omega)$ is called a viscosity subsolution (or simply subsolution)
of the PDE (2.1) in $\Omega$ if for every $x_{0}\in\Omega$, and for every
$\varphi\in C^{2}(\Omega)$ satisfying $u\prec_{x_{0}}\varphi$, we have

$-\Delta_{p}^{+}\varphi(x_{0})\leq f(x_{0}).$

In this case we write $-\Delta_{p}^{N}u(x)\leq f(x)$ in $\Omega$.

  2. (2)

$u\in LSC(\Omega)$ is called a viscosity supersolution (or simply
supersolution) of the PDE (2.1) in $\Omega$ if for every $x_{0}\in\Omega$, and
for every $\varphi\in C^{2}(\Omega)$ satisfying $u\succ_{x_{0}}\varphi$, we
have

$-\Delta_{p}^{-}\varphi(x_{0})\geq f(x_{0}).$

In this case we write $-\Delta_{p}^{N}u(x)\geq f(x)$ in $\Omega.$

  3. (3)

$u\in C(\Omega)$ is called viscosity solution (or simply solution) of the PDE
(2.1) in $\Omega,$ if u is both a subsolution and a supersolution of the PDE
(2.1).

We now consider a boundary datum $g(x)$ and define a viscosity solution of the
Dirichlet problem

$\left\\{\begin{array}[]{rcll}-\Delta_{p}^{N}u&=&f,&\mbox{in $\Omega$}\\\
u&=&g,&\mbox{on $\partial\Omega$}\end{array}\right.$ (2.2)

as follows. We also give a meaning to the boundary condition (1.7) (see [2,
Remark 1.2]).

######  Definition 2.2.

  1. (1)

$u\in USC(\overline{\Omega})$ is a subsolution of (2.2) if $u$ is a
subsolution of $-\Delta_{p}^{N}u=f$ in $\Omega$ and $u\leq g$ on
$\partial\Omega.$

  2. (2)

$u\in LSC(\overline{\Omega})$ is a supersolution of (2.2) if $u$ is a
supersolution of $-\Delta_{p}^{N}u=f$ in $\Omega$ and $u\geq g$ on
$\partial\Omega.$

  3. (3)

$u\in C(\overline{\Omega})$ is a solution of (2.2) if $u$ is both a
subsolution and supersolution of (2.2).

  4. (4)

A solution $u$ of (2.2) satisfies the boundary condition (1.7) if for every
$x_{0}\in\partial\Omega$, and for every $\varphi\in C^{2}$ in a neighborhood
of $x_{0}$ satisfying $u\prec_{x_{0}}\varphi$, we have
$-\frac{\partial\varphi}{\partial\nu}\geq q(|x_{0}|)$. If, instead, we take
$\varphi\in C^{2}$ in a neighborhood of $x_{0}$ satisfying
$u\succ_{x_{0}}\varphi$, we have $-\frac{\partial\varphi}{\partial\nu}\leq
q(|x_{0}|)$.

A function $u\in C(\Omega)$ satisfying $-\Delta_{p}^{N}u=0$ in $\Omega$ is
called a normalized $p$-harmonic function.

######  Remark 2.3.

(i) A smooth function $u$ with $Du\neq 0$ in $\Omega$ satisfying (2.2) in the
classical sense is also a viscosity solution. (ii) The normalized
$p$-Laplacian is a nonlinear operator for $p\neq 2$. Nevertheless, if $u\in
USC(\Omega)$ is a subsolution of (2.2), then $v=-u\in LSC(\Omega)$ and $v$ is
a supersolution of the Dirichlet problem

$\left\\{\begin{array}[]{rcll}-\Delta_{p}^{N}v&=&-f,&\mbox{in $\Omega$}\\\
v&=&-g,&\mbox{on $\partial\Omega$}\end{array}\right.$ (2.3)

Similarly, if $u\in LSC(\Omega)$ is a supersolution of (2.2), then $v=-u\in
USC(\Omega)$ and $v$ is a subsolution of (2.3).

##  3\. Well-posedness, comparison principle, radial solutions

The proof of Theorem 1.1, which holds for $p>1$, is based on the comparison
principle and the explicit expression of the radial solutions which are
recalled in this section.

######  Lemma 3.1 (Comparison principle).

Let $p\in[1,\infty]$, let $\Omega\subset\mathbb{R}^{n}$ be a bounded (possibly
disconnected) open set, and $f\in C(\Omega)$. We assume that $f\neq 0$ in
$\Omega$ and does not change sign. Let $u,v\in C(\overline{\Omega})$ satisfy

$-\Delta_{p}^{N}u\leq f(x)\quad\mbox{and}\quad-\Delta_{p}^{N}v\geq f(x),\quad
x\in\Omega.$

If $u\leq v$ on $\partial\Omega,$ then $u\leq v$ in $\Omega$. The result also
holds if $p=\infty$ and $f\equiv 0$ in $\Omega$.

###### Proof.

The first claim follows from [13, Theorem 5], taking Remark 2.3 $(ii)$ into
account. The case when $p=\infty$ and $f\equiv 0$ follows from Jensen’s
fundamental result [10, Theorem 3.11] by virtue of the equivalence between
infinity-harmonicity and normalized infinity-harmonicity (Remark 1.5). ∎

Uniqueness for problem (2.2) is a consequence of the comparison principle
stated above. Recall that uniqueness lacks in the case when $p=1$ and $f\equiv
0$, and a famous example was given in [19, Section 3.6] (see Fig. 1 below and
[12, Fig. 2]).

Figure 1. Sternberg & Ziemer’s example: a least-gradient function (left) and
another solution of $\Delta_{1}^{N}u=0$ (right)

######  Lemma 3.2 (Radial solution).

Let $p\in[1,\infty]$ and let $c_{p}$ be as in (1.8). For every $R>0$ the
function

$u_{R}(x)=\dfrac{\,c_{p}\,}{2}\,\left(R^{2}-|x-\bar{x}|^{2}\right)$ (3.1)

is the unique solution of the problem

$\left\\{\begin{array}[]{rcll}-\Delta_{p}^{N}u&=&1,&x\in B(\bar{x},R)\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr u&=&0,&x\in\partial
B(\bar{x},R)\end{array}\right.$ (3.2)

Moreover,

$-\dfrac{\partial u_{R}}{\partial\nu}=c_{p}\,R,$

where $\nu=R^{-1}\,(x-\bar{x})$ is the outer normal at $x\in\partial
B(\bar{x},R)$.

###### Proof.

The solution is unique by Lemma 3.1. For $p\in(1,\infty)$, the representation
(3.1) is found in [12, p. 20]. If $p=\infty$, the result follows from [3, pag.
243] with the observation that $d(x,\partial\Omega)=R-|x-\bar{x}|$ and by
letting $a=R$. We give details for the case when $p=1$, with reference to
Definition 2.2 and Remark 2.3 $(i)$. By differentiation of (3.1),
$Du_{R}(x)=-\frac{1}{n-1}(x-\bar{x})$ and $D^{2}u_{R}(x)=-\frac{1}{n-1}\,I$,
where $I$ denotes the identity matrix. Hence $\lambda_{\rm min}=\lambda_{\rm
max}=-\frac{1}{n-1}$. If $x\neq\bar{x}$, using (1.3) we see that the equation
is satisfied in the classical sense. Otherwise $Du_{R}(\bar{x})=0$ and we have
$\Delta_{1}^{+}\,u_{R}(\bar{x})=\Delta_{1}^{-}\,u_{R}(\bar{x})=\Delta
u_{R}-\lambda_{\rm min}=-1$. Thus, $-\Delta_{1}^{N}\,u_{R}(x)=1$ in all of
$B(\bar{x},R)$ and, of course, $u_{R}(x)=0$ on $\partial B(\bar{x},R)$. ∎

In order to prove Theorem 1.1, we also need to establish the positivity of the
solution to the following Dirichlet problem:

$\left\\{\begin{array}[]{rcll}-\Delta_{p}^{N}u&=&1,&x\in\Omega\\\
u&=&0,&x\in\partial\Omega\\\ \end{array}\right.$ (3.3)

######  Lemma 3.3.

Let $p\in[1,\infty]$. Any solution of (3.3) is positive in $\Omega$.

###### Proof.

Notice that $v=0$ satisfies $-\Delta_{p}^{N}v\leq 1$ in $\Omega$ and $v=0$ on
$\partial\Omega$. As $v$ is a subsolution, by the comparison principle (Lemma
3.1) we have $0\leq u$ in $\Omega$. Consider $B(x_{0},R)$ contained in
$\Omega$, and consider the solution $u_{R}$ of problem (3.2) in $B(x_{0},R)$.
Since $u_{R}(x)=0$ on $\partial B(x_{0},R)$ and $u\geq 0$ on $\partial
B(x_{0},R)$, Lemma 3.1 implies $0<u_{R}(x)\leq u(x)$ in $B(x_{0},R)$. Since
$x_{0}$ is arbitrary in $\Omega,$ we have $u>0$ in $\Omega$. ∎

By choosing $B(x_{0},R)\subset\Omega$ so that $\partial
B(x_{0},R)\cap\partial\Omega\neq\emptyset$, we immediately obtain the
following boundary-point lemma (see also [2, Lemma 2.3]):

######  Lemma 3.4 (Hopf).

Let $p\in[1,\infty]$. Suppose $\Omega$ satisfies an interior sphere condition
at every boundary point, and let $u\in C(\overline{\Omega})$ be a viscosity
solution of (3.3). Then for all $x\in\partial\Omega$ we have

$\limsup_{t\rightarrow 0^{+}}\dfrac{u(x)-u(x-\nu t)}{t}<0.$

We conclude this section by quoting some existence and regularity results.

######  Lemma 3.5 (Existence).

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded (possibly disconnected) open
set, and let $f\in C(\Omega)$ and $g\in C(\partial\Omega)$. The Dirichlet
problem (2.2) has a viscosity solution provided that one of the following
conditions hold:

  1. (1)

$p\in(n,\infty]$, and $f$ does not vanish and does not change sign in
$\Omega$.

  2. (2)

$p=\infty$ and $f$ is bounded in $\Omega$.

###### Proof.

If $p=\infty$, Claim (1) follows from [15, Theorem 1.8] using Remark 2.3 (ii).
The claim was extended to $p>n$ in [14, Corollary 4.5]: indeed, assumption
(3.19) of [14] reduces to $p>n$. Claim (2) is a special case of [16, Theorem
6.1] corresponding to $F(x)=|x|$. ∎

######  Remark 3.6.

Concerning regularity, global $C^{1,\beta}$-regularity is proved in [2,
Theorem 4.2] for $p\in(1,\infty)$. In the case when $p=\infty$ it is known
that the viscosity solution to (3.3) is locally Lipschitz continuous in
$\Omega$: see, for instance, [20, Lemma 5.3] with $F(x)=|x|$. See also [4] for
further details.

Let us point out that the existence, uniqueness and regularity results
recalled above allow to construct the following counterexample, which mimics
the one in [8, p. 242]. The example is valid for $p\in(n,\infty)$ and shows
that if we let the function $q$ in (1.2) be arbitrary, i.e., if we drop every
assumption on $q$, then problem (1.2) may well be solvable even though the
domain $\Omega$ is not a ball.

######  Example 3.7.

Let $\Omega\subset\mathbb{R}^{2}$ be an ellipse in canonical position, with
semi-axes $a<b$, and let $p\in(n,\infty)$. Thus, there exists a unique
solution $u=u_{0}$ of the Dirichlet problem (3.3). Note that the problem is
invariant under reflection with respect to each axis: i.e., if we define
$v(x_{1},x_{2})=u_{0}(\pm x_{1},\pm x_{2})$ for whatever choice of the signs
$\pm$, we always find $\Delta_{p}^{N}v=\Delta_{p}^{N}u_{0}$. But since the
solution of problem (3.3) is unique, we must have $u_{0}\equiv v$, hence
$u_{0}(x_{1},x_{2})=u_{0}(\pm x_{1},\pm x_{2})$. Furthermore, since $u_{0}$ is
differentiable up to the boundary, the last equality implies that $|\nabla
u_{0}(x_{1},x_{2})|=|\nabla u_{0}(\pm x_{1},\pm x_{2})|$ for every
$(x_{1},x_{2})\in\partial\Omega$. Now observe that for every $r\in[a,b]$ the
set $F_{r}$ of all $x\in\partial\Omega$ such that $|x|=r$ is invariant under
reflection with respect to each axis, and therefore it is legitimate to define
$q(r)=|\nabla u_{0}(x)|$ by choosing any $x=(x_{1},x_{2})\in F_{r}$ (because
the value of $q(r)$ is independent of the choice of $x\in F_{r}$). Then, with
this particular function $q$, problem (3.3) is solvable (and has the solution
$u_{0}$) altough $\Omega$ is not a disc.

##  4\. Existence and nonexistence of solutions

In this section we prove our main results.

###### Proof of Theorem 1.1.

We follow the same guidelines as in [8]. (1) Let $\Omega=B(\bar{x},R)$, where
$R$ is the solution of $q(r)-c_{p}\,r=0$. By Lemma 3.2,
$u_{R}(x)=\frac{\,c_{p}\,}{2}\,(R^{2}-|x-\bar{x}|^{2})$ is the solution to the
problem (1.2) in $B(\bar{x},R)$. Therefore, the solution to (1.2) exists in
$B(\bar{x},R)$.
On the other hand, assume $u$ is the solution to (1.2). Define
$u_{i}(x)=\frac{\,c_{p}\,}{2}\,(R_{i}^{2}-|x-\bar{x}|^{2})$ for $i=1,2$. Then
$u_{i}$ is the solution to (3.3) in the ball $B(\bar{x},R_{i})$. Since $u\geq
0$ on $\partial B(\bar{x},R_{1})$ (see Lemma 3.3) and $u_{1}=0$ on $\partial
B(\bar{x},R_{1}),$ we have $u_{1}\leq u$ on $\partial B(\bar{x},R_{1})$. By
Lemma 3.1, $u_{1}\leq u$ in $B(\bar{x},R_{1})$. Since $u_{2}\geq 0$ on
$\partial\Omega$ and $u=0$ on $\partial\Omega,$ we have $u=0\leq u_{2}$ on
$\partial\Omega$ and hence $u\leq u_{2}$ in $\Omega$ by Lemma 3.1. Let
$P_{1}\in\partial B(\bar{x},R_{1})\cap\partial\Omega$. Then the outer normal
$\nu$ to $\partial\Omega$ at $P_{1}$ equals $\dfrac{P_{1}-\bar{x}}{R_{1}}$,
the outer normal to $B(\bar{x},R_{1})$. Since $u_{1}$ is a smooth function
satisfying $u\succ_{P_{1}}u_{1}$, by Definition 2.2 (4) we may write

$c_{p}\,R_{1}=-\dfrac{\partial u_{1}}{\partial\nu}(P_{1})\leq q(R_{1}).$ (4.1)

Let $P_{2}\in\partial B(\bar{x},R_{2})\cap\partial\Omega$. Then the outer
normal $\nu$ to $\partial\Omega$ at $P_{2}$ equals
$\dfrac{P_{2}-\bar{x}}{R_{2}}$, the outer normal to $B(\bar{x},R_{2})$.
Furthermore $u\prec_{P_{2}}u_{2}$. Hence

$q(R_{2})\leq-\dfrac{\partial u_{2}}{\partial\nu}(P_{2})=c_{p}\,R_{2}.$ (4.2)

Inequalities (4.1) and (4.2) may be rephrased as

$q(R_{1})-c_{p}\,R_{1}\geq 0$ and $q(R_{2})-c_{p}\,R_{2}\leq 0$. (4.3)

Since the equation $q(r)-c_{p}\,r=0$ has the unique solution $R$ in
$[R_{1},R_{2}]$ and $q(r)-c_{p}\,r<0$ for $r<R,$ we have $R=R_{1}$ and again
since $q(r)-c_{p}\,r>0$ for $r>R,$ we have $R=R_{2}$. Therefore,
$R_{1}=R=R_{2},$ which is $\Omega=B(\bar{x},R).$

(2) If (1.2) has a solution, then we obtain (4.1) and (4.2), hence $\rho(r)$
satisfies

$\rho(R_{2})=\dfrac{q(R_{2})}{R_{2}}\leq
c_{p}\leq\dfrac{q(R_{1})}{R_{1}}=\rho(R_{1}).$

Since $\rho$ is strictly increasing, we must have $R_{1}=R_{2}$ and the result
follows.

(3) Suppose that (1.2) has a solution. Then, by (4.3), and since $q$ is
continuous, we have $q(R)-c_{p}\,R=0$ at some point $R\in[R_{1},R_{2}]$,
contradicting the assumption. Therefore problem (1.2) must be unsolvable. ∎

###### Proof of Theorem 1.3.

The result follows by exploiting (1.9). As mentioned in the Intoduction, we
have $H(x)>0$ for every $x\in\partial\Omega$. Take $P_{i}\in\partial
B(\bar{x},R_{i})\cap\partial\Omega$, $i=1,2$ as in the proof of Theorem 1.1,
and recall that the mean curvature of the sphere $\partial B(\bar{x},R_{i})$
is $1/R_{i}$. Hence we may write $H(P_{1})\leq 1/R_{1}$ and $H(P_{2})\geq
1/R_{2}$. This and (1.9) imply

$\displaystyle q(R_{1},\,1/R_{1})\geq
q(R_{1},H(P_{1}))=|Du(P_{1})|=\frac{1}{\,(n-1)\,H(P_{1})\,}\geq\frac{R_{1}}{\,n-1\,}=c_{1}\,R_{1},$
$\displaystyle q(R_{2},\,1/R_{2})\leq
q(R_{2},H(P_{2}))=|Du(P_{2})|=\frac{1}{\,(n-1)\,H(P_{2})\,}\leq\frac{R_{2}}{\,n-1\,}=c_{1}\,R_{2}$

because $q(r,h)$ is monotone non-decreasing in $h$. The inequalities above
imply

$\frac{\,q(R_{1},\,1/R_{1})\,}{R_{1}}\geq
c_{1}\geq\frac{\,q(R_{2},\,1/R_{2})\,}{R_{2}}.$

Then, since the ratio $\frac{q(r,\,1/r)}{r}$ is strictly increasing in $r$, we
must have $R_{1}=R_{2}$ as claimed. ∎

## Funding

L. Cadeddu and A. Greco are members of the Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto
Nazionale di Alta Matematica (INdAM) and are partially supported by the
research project Evolutive and stationary Partial Differential Equations with
a focus on bio-mathematics, funded by Fondazione di Sardegna (2019). L.
Cadeddu’s research has been accomplished within the UMI Group TAA
Approximation Theory and Applications.

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