#  Deterministic KPZ-type equations with nonlocal “gradient terms”

Boumediene Abdellaoui, Antonio J. Fernández, Tommaso Leonori and Abdelbadie
Younes
Boumediene Abdellaoui
Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Département de
Mathématiques,
Université Abou Bakr Belkaïd, Tlemcen, 13000, Algeria
<EMAIL_ADDRESS>
Antonio J. Fernández
Instituto de Ciencias Matemáticas,
Consejo Superior de Investigaciones Cientf́icas, 28049 Madrid, Spain
<EMAIL_ADDRESS>
Tommaso Leonori
Dipartimento di Scienze di Base e Applicate per l’Ingegneria,
Universitá di Roma “Sapienza”. Via Antonio Scarpa 10, 00161 Roma, Italy
<EMAIL_ADDRESS>
Abdelbadie Younes
Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Département de
Mathématiques,
Université Abou Bakr Belkaïd, Tlemcen, 13000, Algeria
<EMAIL_ADDRESS>
###### Abstract.

The main goal of this paper is to prove existence and non-existence results
for deterministic Kardar–Parisi–Zhang type equations involving non-local
“gradient terms”. More precisely, let $\Omega\subset\mathbb{R}^{N}$,
$N\geqslant 2$, be a bounded domain with boundary $\partial\Omega$ of class
$C^{2}$. For $s\in(0,1)$, we consider problems of the form

$\left\\{\begin{aligned} (-\Delta)^{s}u&=\mu(x)\,|\mathbb{D}(u)|^{q}+\lambda
f(x),\quad&&\textup{ in }\Omega,\\\ u&=0,&&\textup{ in
}\mathbb{R}^{N}\setminus\Omega,\end{aligned}\right.$

where $q>1$ and $\lambda>0$ are real parameters, $f$ belongs to a suitable
Lebesgue space, $\mu\in L^{\infty}(\Omega)$ and $\mathbb{D}$ represents a
nonlocal “gradient term”. Depending on the size of $\lambda>0$, we derive
existence and non-existence results. In particular, we solve several open
problems posed in [4, Section 6] and [2, Section 7].

Keywords: Fractional Laplacian, nonlocal “gradient terms”, deterministic
KPZ–type equations.

2020 MSC: 35R11, 35J60, 26A33.

##  1\. Introduction

In this paper we analyse the existence and non–existence of solutions for
deterministic Kardar–Parisi–Zhang type equations involving non-local “gradient
terms”. For $s,t\in(0,1)$, we consider problems of the form

$\left\\{\begin{aligned}
(-\Delta)^{s}u&=\mu(x)\,|\mathbb{D}_{t}(u)|^{q}+\lambda f(x),\quad&&\textup{
in }\Omega,\\\ u&=0,&&\textup{ in
}\mathbb{R}^{N}\setminus\Omega,\end{aligned}\right.$

depending on a real parameter $\lambda>0$. Here,
$\Omega\subset\mathbb{R}^{N}$, $N\geqslant 2$, is a bounded domain with
boundary $\partial\Omega$ of class $C^{2}$, $f$ belongs to a suitable Lebesgue
space, $\mu\in L^{\infty}(\Omega)$, $q\in(1,+\infty)$ and $\mathbb{D}_{t}$
represents one of the following nonlocal “gradient terms” :

$\displaystyle\circ\quad(-\Delta)^{\frac{t}{2}}u(x):=a_{N,\frac{t}{2}}\,{\rm{P.V.}}\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+t}}dy$
$\displaystyle(\textup{Half $t$--Laplacian}),$ (KPZ1)
$\displaystyle\circ\quad\nabla^{t}u(x):=\mu_{N,t}\int_{\mathbb{R}^{N}}\frac{(x-y)(u(x)-u(y))}{|x-y|^{N+t+1}}dy$
$\displaystyle(\textup{Riesz $t$--Gradient}),$ (KPZ2)
$\displaystyle\circ\quad\mathcal{D}_{t}u(x):=\left(\frac{a_{N,t}}{2}\int_{\mathbb{R}^{N}}\frac{(u(x)-u(y))^{2}}{|x-y|^{N+2t}}dy.\right)^{\frac{1}{2}}$
$\displaystyle(\textup{Stein $t$--Functional}).$ (KPZ3)

Note that the previous definitions make sense for any function $u\in
C_{c}^{\infty}(\mathbb{R}^{N})$. Also, let us point out that

$a_{N,\sigma}:=-\,\frac{2^{2\sigma}\Gamma\left(\frac{N}{2}+\sigma\right)}{\pi^{\frac{N}{2}}\Gamma(-\sigma)}\qquad\textup{
and
}\qquad\mu_{N,\sigma}:=\frac{2^{\sigma}\Gamma(\frac{N+\sigma+1}{2})}{\pi^{\frac{N}{2}}\Gamma(\frac{1-\sigma}{2})},$

are normalization constants and “P.V.” stands for “in the principal value
sense”. Since both these constants and the “principal value sense” will not
play an important role in our work, we will omit them from now on.

Before going further, we would like to emphasize that the three different
nonlocal “gradient terms” that we consider can be traced back many years ago.
Since nowadays the fractional Laplacian does not need any further
presentation, let us focus on the other two terms. As very well explained in
[23, page 3], the origin of the Riesz $t$–Gradient seems to be [19]. Note also
that this operator has been rediscovered several times since [19] and has
received considerable attention in the last few years. See for instance [27,
22, 15, 10]. On the other hand, the Stein $t$–Functional can be at least
traced back to [25]. Moreover, this operator naturally appears as the nonlocal
equivalent to the gradient when considering the minimization of fractional
Harmonic maps into the sphere. See for instance the recent papers [7, 21, 16].

In contrast with the local case

$\left\\{\begin{aligned} -\Delta u&=\mu(x)|\nabla u|^{q}+\lambda
f(x),&&\textup{ in }\Omega,\\\ u&=0,&&\textup{ on
}\partial\Omega,\end{aligned}\right.$ (1.1)

where the literature is very extensive, there exist very few results dealing
with equations of the form (1). We refer to [2], by the first two authors, for
a detailed introduction to the subject. However, since the publication of [2],
some results have been published. For instance, we would like to mention the
recent paper [1], where the authors establish existence and non–existence
results for problems of the form (1) with a local operator (the Laplacian
instead of the fractional Laplacian) and a nonlocal nonlinearity. Also, let us
mention [6], where the authors establish the equivalence between different
notions of solution to problems of the form (1). Finally, let us stress that
solutions to equations with nonlocal diffusion and a nonlocal “gradient term”
with nonsingular kernels have been studied in [13].

In [2], the first two authors analyse the existence and non–existence of
solutions to (1), under the additional assumption $t=s\in(1/2,1)$. The main
goal of this paper is to refine the approach of [2] in order to deal also with
the cases where $s\in(0,1/2]$ and/or $t\neq s$. Depending on the real
parameter $\lambda>0$, we analyse the existence and non–existence of weak
solutions to (1) under the assumptions

$\left\\{\begin{aligned} \ &q\in(1,+\infty),\\\
&0<t<\min\\{1,s(1+(qN)^{-1})\\},\\\ &f\in L^{m}(\Omega)\ \textup{ for some
}m>N/s\ \textup{ and }\ \mu\in L^{\infty}(\Omega).\end{aligned}\right.$ (A1)

Following [9, 8], we introduce the subsequent notion of weak solution to (1) :

######  Definition 1.1.

We say that $u$ is a weak solution to (1) if $u$ and $|\mathbb{D}_{t}(u)|^{q}$
belong to $L^{1}(\Omega)$, $u\equiv 0$ (a.e.) in
$\mathbb{R}^{N}\setminus\Omega$ and

$\int_{\Omega}u(-\Delta)^{s}\phi\,dx=\int_{\Omega}\big{(}\mu(x)|\mathbb{D}_{t}(u)|^{q}+\lambda
f(x)\big{)}\phi\,dx,$ (1.2)

for all $\phi$ belonging to

$\mathbb{X}^{s}(\Omega):=\Big{\\{}\phi\in
C^{s}(\mathbb{R}^{N})\,:\,\phi(x)=0\textup{ for all
}x\in\mathbb{R}^{N}\setminus\Omega\textup{ and }(-\Delta)^{s}\phi\in
L^{\infty}(\Omega)\Big{\\}}.$

######  Remark 1.2.

With a slight abuse of notation we use (KPZ${}_{\,i}$), $i=1,2,3$, to refer to
(1) with $\mathbb{D}_{t}=(-\Delta)^{\frac{t}{2}}$, $\mathbb{D}_{t}=\nabla^{t}$
and $\mathbb{D}_{t}=\mathcal{D}_{t}$ respectively. See (KPZ1), (KPZ2) and
(KPZ3) for the corresponding definitions.

Our main existence result can be informally stated as follows:

######  Theorem 1.3.

Assume that (A1) holds true. Then, there exists $\lambda^{\star}>0$ such that,
for all $0<\lambda\leqslant\lambda^{\star}$, (KPZ${}_{\,i}$), $i=1,2,3$, has a
weak solution $u$. Moreover, $u\in W^{s,p}(\mathbb{R}^{N})\cap
C^{0,s}(\mathbb{R}^{N})$ for all $1<p<+\infty$.

Theorem 1.3 is a particular case of the more general existence results proved
in Section 3. We refer directly to Section 3 for more general statements. In
particular, let us emphasize that, in Section 3, we substantially lower the
regularity imposed on the data $f$. Furthermore, arguing as in the proof of
[2, Theorem 1.3], it is possible to show that the regularity considered in
Section 3 is almost optimal. Note also that our existence results solve
several open problems posed in [2, 4].

The proofs of our existence results rely on the combination of fixed point
arguments in the spirit of [18, 14] with global fractional Calderón–Zygmund
regularity results for the fractional Poisson equation

$\left\\{\begin{aligned} (-\Delta)^{s}u&=h,\quad&&\textup{ in }\Omega,\\\
u&=0,&&\textup{ in }\mathbb{R}^{N}\setminus\Omega.\end{aligned}\right.$ (1.3)

This approach was already implemented in [2] by the first two authors.
However, the required global fractional Calderón–Zygmund regularity theory was
not available for $s\in(0,1/2]$. Furthermore, the Calderón–Zygmund regularity
used in [2] (cf. [2, Section 3]) contains several imprecisions. In the recent
paper [3], we establish sharp global fractional Calderón–Zygmund regularity
results for (1.3) in the full range $s\in(0,1)$ (fixing in particular the
issues of [2, Section 3]). Having at hand these regularity results, the proofs
of our existence results follow from a refinement of the fixed point approach
implemented in [2]. We refer to Section 3 for more details.

Taking into account Theorem 1.3, it is very natural to ask whether the
smallness assumption in $\lambda$ is necessary or not. Note that, in [2,
Theorem 1.2], the first two authors established a non–existence result for
(KPZ3) with $t=s\in(0,1)$. We focus here in proving a non-existence result to
(KPZ1). This was left as an open problem in [2, Section 7].

######  Theorem 1.4.

Assume that (A1) holds true with $t=s$ and $q=2$ and suppose that
$\mu(x)\geqslant\mu_{1}>0$ and $f\gneqq 0$. Then, there exists
$\lambda^{\star\star}>0$ such that, for all $\lambda>\lambda^{\star\star}$,
(KPZ1) has no weak solution in $W^{s,2}(\mathbb{R}^{N})$.

######  Remark 1.5.

The non–existence for $\lambda$ large in the case where
$\mathbb{D}_{t}=\nabla^{t}$ remains completely open and a different approach
is missing.

### Organization of the paper

In the next section we introduce the main function spaces involved in our
results and prove the continuity and compactness of the solution map for the
fractional Poisson equation with $L^{1}$–data. In Section 3, we prove our main
existence results to (1), from which Theorem 1.3 immediately follows. Finally,
in Section 4, we prove non-existence results for (KPZ1) and (KPZ3) when
$\lambda>0$ is large.

### Acknowledgements.

This work has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme through
the Consolidator Grant agreement 862342 (A. J. F.). B. A. is partially
supported by projects MTM2016-80474-P and PID2019-110712GB-I00, MINECO, Spain.
B. A. and A. Y. are partially supported by the DGRSDT, Algeria. Part of this
work was done while A. J. F. was visiting the Università di Roma “La Sapienza”
and the Università di Bologna. He thanks his hosts for the kind hospitality
and the financial support. The authors wish to thank the anonymous referee for
the very useful remarks, which helped to improve the presentation of the
paper.

##  2\. Function spaces and tools

We collect here the definition of the main function spaces involved in our
results and some other tools. First of all, recall that, for all $s\in(0,1)$
and $1\leqslant p<+\infty$, the fractional Sobolev space
$W^{s,p}(\mathbb{R}^{N})$ is defined as

$W^{s,p}(\mathbb{R}^{N}):=\left\\{u\in
L^{p}(\mathbb{R}^{N}):\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy<+\infty\right\\}.$

It is a Banach space endowed with the usual norm

$\|u\|_{W^{s,p}(\mathbb{R}^{N})}=\left(\|u\|_{L^{p}(\mathbb{R}^{N})}^{p}+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{\frac{1}{p}}.$

Also, having at hand $W^{s,p}(\mathbb{R}^{N})$, we define the space
$W_{0}^{s,p}(\Omega)$ as

$W_{0}^{s,p}(\Omega):=\left\\{u\in W^{s,p}(\mathbb{R}^{N}):u\equiv 0\textup{
in }\mathbb{R}^{N}\setminus\Omega\right\\},$

and recall that, thanks to the Sobolev inequality, it is a Banach space
endowed with the norm

$\|u\|_{W^{s,p}_{0}(\Omega)}:=\left(\iint_{D_{\Omega}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{1/p},\quad\textup{
where }\quad
D_{\Omega}:=(\Omega\times\mathbb{R}^{N})\cup((\mathbb{R}^{N}\setminus\Omega)\times\Omega).$

Next, we remind that, for any $s\in(0,1)$ and $1\leqslant p<+\infty$, the
Bessel potential space is defined as

$L^{s,p}(\mathbb{R}^{N})\ :=\ \overline{\big{\\{}u\in
C_{c}^{\infty}(\mathbb{R}^{N})\big{\\}}}^{\
\boldsymbol{|}\cdot\boldsymbol{|}_{L^{s,p}(\mathbb{R}^{N})}},$

where

$\boldsymbol{|}u\boldsymbol{|}_{L^{s,p}(\mathbb{R}^{N})}=\|(1-\Delta)^{\frac{s}{2}}u\|_{L^{p}(\mathbb{R}^{N})}\quad\textup{
and
}\quad(1-\Delta)^{\frac{s}{2}}u=\mathcal{F}^{-1}((1+|\cdot|^{2})^{\frac{s}{2}}\mathcal{F}u),\quad\forall\
u\in C_{c}^{\infty}(\mathbb{R}^{N}).$

Let us stress that, in the case where $s\in(0,1)$ and $1<p<+\infty$,

$\|u\|_{L^{s,p}(\mathbb{R}^{N})}:=\|u\|_{L^{p}(\mathbb{R}^{N})}+\|(-\Delta)^{\frac{s}{2}}u\|_{L^{p}(\mathbb{R}^{N})}$

is an equivalent norm for $L^{s,p}(\mathbb{R}^{N})$ (see e.g. [25, Theorem
2]). By [25, Theorem 1.1], we also know that, if in addition
$2N/(N+2s)<p<+\infty$, then $L^{s,p}(\mathbb{R}^{N})$ can be equipped with the
equivalent norm

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s,p}(\mathbb{R}^{N})}:=\|u\|_{L^{p}(\mathbb{R}^{N})}+\|\mathcal{D}_{s}(u)\|_{L^{p}(\mathbb{R}^{N})}.$

In analogy with $W_{0}^{s,p}(\Omega)$, let us define

$L_{0}^{s,p}(\Omega):=\big{\\{}u\in L^{s,p}(\mathbb{R}^{N}):u\equiv 0\textup{
in }\mathbb{R}^{N}\setminus\Omega\big{\\}},$

and stress that, if $0<s<1$ and $1<p<+\infty$, it is a Banach space endowed
with the norm

$\|u\|_{L^{s,p}_{0}(\Omega)}:=\|(-\Delta)^{\frac{s}{2}}u\|_{L^{p}(\mathbb{R}^{N})}.$

If in addition $2N/(N+2s)<p<+\infty$, then $L_{0}^{s,p}(\Omega)$ can also be
equipped with the equivalent norm

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s,p}_{0}(\Omega)}:=\|\mathcal{D}_{s}(u)\|_{L^{p}(\mathbb{R}^{N})}.$

Let us as well recall that, for all $0<\varepsilon<\sigma<1$ and all
$1<p<+\infty$, by [5, Theorem 7.63, (g)], we have

$L^{\sigma+\varepsilon,p}(\mathbb{R}^{N})\subset
W^{\sigma,p}(\mathbb{R}^{N})\subset L^{\sigma-\varepsilon,p}(\mathbb{R}^{N}).$

It is also well known that, for all $1\leqslant p<+\infty$ and all
$0<\sigma\leqslant\sigma^{\prime}<1$,

$W^{\sigma^{\prime},p}(\mathbb{R}^{N})\subset W^{\sigma,p}(\mathbb{R}^{N}),$

and that (cf. [5, Theorem 7.63 (c)]), if in addition $1<p<+\infty$,

$L^{\sigma^{\prime},p}(\mathbb{R}^{N})\subset L^{\sigma,p}(\mathbb{R}^{N}).$

Since the constants will be useful later on, let us emphasize there exists
$\widetilde{k}:=\widetilde{k}(\sigma,\sigma^{\prime},p)\geqslant 1$ such that

$\|u\|_{L^{\sigma,p}(\mathbb{R}^{N})}\leqslant\widetilde{k}\|u\|_{L^{\sigma^{\prime},p}(\mathbb{R}^{N})},\quad\textup{
for all }u\in L^{\sigma^{\prime},p}(\mathbb{R}^{N}),$ (2.1)

and

$\|u\|_{W^{\sigma,p}(\mathbb{R}^{N})}\leqslant\widetilde{k}\|u\|_{W^{\sigma^{\prime},p}(\mathbb{R}^{N})},\quad\textup{
for all }u\in W^{\sigma^{\prime},p}(\mathbb{R}^{N}),$ (2.2)

Now, for $0<s\leqslant t<\min\\{1,s(1+N^{-1})\\}$, let us set111We use, here
and in the sequel, the convention $1/a^{+}=+\infty$ if $a\leqslant 0$.

$\widetilde{p}\,(m,s,t):=\left\\{\begin{aligned}
&\frac{1}{(t-s)^{+}},\quad&&\textup{ if }m>\frac{N}{2s-t},\\\
&\min\Big{\\{}\frac{mN}{N-ms+mN(t-s)},\frac{1}{(t-s)^{+}}\Big{\\}},\quad&&\textup{
if }1\leqslant m<\frac{N}{2s-t}.\end{aligned}\right.$

In our next result, which will be very useful in the sequel, we describe the
regularity of the (unique) solution to (1.3). Note that such a result is
contained in [3, Theorems 1.3 and 5.2, Corollaries 5.3 – 5.6].

######  Proposition 2.1.

Let $0<s\leqslant t<\min\\{1,s(1+N^{-1})\\}$ and let $u$ be the (unique) weak
solution to (1.3) with $h\in L^{m}(\Omega)$ for some $m\geqslant 1$. Then, for
all $1<p<\widetilde{p}$, there exists $\widetilde{C}(N,s,p,m,\Omega)>0$ such
that

$\|u\|_{L^{t,p}(\mathbb{R}^{N})}\leqslant\widetilde{C}\,\|h\|_{L^{m}(\Omega)}\quad\textup{
and
}\quad\|u\|_{W^{t,p}(\mathbb{R}^{N})}\leqslant\widetilde{C}\,\|h\|_{L^{m}(\Omega)}.$
(2.3)

We also present here a technical but useful lemma proved in [3, Lemma 5.1] and
a classical result from harmonic analysis.

######  Lemma 2.2.

Let $s\in(0,1)$, $s\leqslant t<\min\\{1,2s\\}$ and and let $u$ be the (unique)
weak solution to (1.3) with $h\in L^{1}(\Omega)$. Then, there exists
$C:=C(N,s,t,\Omega)>0$ such that

$\big{|}(-\Delta)^{\frac{t}{2}}u(x)|\leqslant
C\left[g_{1}(x)+|\log\delta(x)|g_{2}(x)+\frac{1}{\delta^{t-s}(x)}g_{3}(x)\right],\quad\mbox{
for a.e. }x\in\Omega.$ (2.4)

Here, $\delta(x):={\mbox{dist}}(x,\partial\Omega)$ and the functions $g_{i}$,
$i=1,2,3$, satisfy:

$\displaystyle\bullet\ \mbox{For all $0<\lambda<2s-t$, there exists
$C:=C(\lambda)>0$ s.t. $\displaystyle g_{1}(x)\leqslant
C\int_{\Omega}\frac{|f(y)|}{|x-y|^{N-(2s-t-\lambda)}}dy,$}$ (2.5)
$\displaystyle\bullet\
g_{2}(x):=\int_{\Omega}\frac{|f(y)|}{|x-y|^{N-(2s-t)}}dy,$ (2.6)
$\displaystyle\bullet\
g_{3}(x):=(t-s)\int_{\Omega}\frac{|f(y)|}{|x-y|^{N-s}}dy.$ (2.7)

Moreover, for any
$R\geqslant\frac{1}{3}+\frac{4}{3}(\operatorname{diam}(\Omega)+{\mbox{dist}}(0,\Omega))$,
it follows that

$\displaystyle\big{|}(-\Delta)^{\frac{t}{2}}u(x)\big{|}\leqslant\int_{\Omega}\frac{|u(y)|}{\delta^{s}(y)}\frac{dy}{|x-y|^{N+(t-s)}},\quad\textup{
for a.e. }x\in B_{R}(0)\setminus\Omega\,,$ (2.8)

and

$\big{|}(-\Delta)^{\frac{t}{2}}u(x)\big{|}\leqslant\frac{4^{N+t}}{(1+|x|)^{N+t}}\int_{\Omega}|u(y)|dy,\quad\textup{
for a.e. }x\in\mathbb{R}^{N}\setminus B_{R}(0).$ (2.9)

######  Lemma 2.3.

Let $\omega\subset\mathbb{R}^{N}$, $N\geqslant 2$, be an open bounded domain
and let $0<\alpha<N$ and $1\leqslant p<\ell<\infty$ be such that
$\dfrac{1}{\ell}=\dfrac{1}{p}-\dfrac{\alpha}{N}$. Moreover, for $g\in
L^{p}(\omega)$, let

$J_{\lambda}(g)(x):=\int_{\omega}\dfrac{g(y)}{|x-y|^{N-\alpha}}dy\,.$

It follows that there exists $C=C(N,\alpha,p,\sigma>0,\ell,\Omega)>0$ such
that

  * $a)$

$J_{\alpha}$ is well defined (in the sense that the integral converges
absolutely for a.e. $x\in\omega$).

  * $b)$

$[J_{\alpha}(g)]_{M^{\ell}(\omega)}\leqslant C\,\|g\|_{L^{1}(\omega)}.$ In
particular, $\|J_{\alpha}(g)\|_{L^{\sigma}(\omega)}\leqslant
C\,\|g\|_{L^{1}(\omega)}$ for all $1\leqslant\sigma<\ell$.

  * $c)$

If $1<p<\frac{N}{\alpha}$, then $\|J_{\alpha}(g)\|_{L^{\ell}(\omega)}\leqslant
C\,\|g\|_{L^{p}(\omega)}$.

  * $d)$

If $p=\frac{N}{\alpha}$, then $\|J_{\alpha}(g)\|_{L^{\sigma}(\omega)}\leqslant
C\,\|g\|_{L^{p}(\omega)}$ for all $1\leqslant\sigma<+\infty$.

  * $e)$

If $p>\frac{N}{\alpha}$, then $\|J_{\alpha}(g)\|_{L^{\infty}(\omega)}\leqslant
C\,\|g\|_{L^{p}(\omega)}.$

###### Proof.

Parts a), b), c) and d) follow from [26, Theorem I, Section 1.2, Chapter V,
page 119]. Part e) is contained in [12, Lemma 7.12] (see also [17, Theorem
2.2]). ∎

We conclude this section proving a compactness result for the fractional
Poisson equation (1.3) that will be key in the proof of our main existence
results. Here, we denote by $G_{s}:\mathbb{R}_{\ast}^{2N}\to\mathbb{R}$ the
Green function associated to $(-\Delta)^{s}$ in $\Omega$ with homogeneous
Dirichlet boundary conditions. Note that
$\mathbb{R}_{\ast}^{2N}:=\\{(x,y)\in\mathbb{R}^{2N}:x\neq y\\}$.

######  Proposition 2.4.

Let $0<t<\min\\{1,s(1+N^{-1})\\}$ and $1<p<N/(N(1+t-s)-s)$. The solution map

$\mathbb{G}_{s}:L^{1}(\Omega)\to L_{0}^{t,p}(\Omega),\qquad
h\mapsto\mathbb{G}_{s}[h]:=\int_{\Omega}G_{s}(x,y)h(y)\,dy,$

is well-defined, continuous and compact.

###### Proof.

First of all, note that without loss of generality we can assume that
$s\leqslant t<\min\\{1,s(1+N^{-1})\\}$. Indeed, once we have the result in
this case, we can infer the result for $t\in(0,s)$ interpolating as in the
proof of [2, Proposition 3.10]. Let us also stress that, by Proposition 2.1,
the solution map $\mathbb{G}_{s}$ is well-defined and continuous for all
$1<p<N/(N(1+t-s)-s)$. Hence, we just have to show that $\mathbb{G}_{s}$ is
compact for the same range of $p$.

Let $(h_{n})_{n}\subset L^{1}(\Omega)$ be a sequence such that
$\|h_{n}\|_{L^{1}(\Omega)}\leqslant 1$ for all $n\in\mathbb{N}$ and let
$u_{n}=\mathbb{G}_{s}(h_{n})$ for all $n\in\mathbb{N}$. By [8, Proposition
2.6] we know that, up to a subsequence, $(u_{n})_{n}$ is strongly convergent
in $L^{q}(\Omega)$ for all $1\leqslant q<N/(N-2s)$. Moreover, combining this
strong convergence with Vitali’s convergence theorem, we deduce that, up to a
subsequence, $(u_{n}/\delta^{s})_{n}$ is strongly convergent in
$L^{r}(\Omega)$ for all $1\leqslant r<N/(N-s)$. Having at hand the (up to a
subsequence) strong convergence of $(u_{n})_{n}$ in $L^{q}(\Omega)$ for all
$1\leqslant q<N/(N-2s)$, to end the proof, we just have to show that, up to a
subsequence, $((-\Delta)^{\frac{t}{2}}u_{n})_{n}$ is strongly convergent in
$L^{p}(\mathbb{R}^{N})$ for all $1<p<N/(N(1+t-s)-s)$.

First, for $0<\alpha<2$, let us consider the integral operator

$T_{\alpha}:L^{1}(\Omega)\to L^{\gamma}(\Omega),\qquad
g\mapsto\int_{\mathbb{R}^{N}}\frac{g(y)\mathds{1}_{\Omega}(y)}{|x-y|^{N-\alpha}}dy.$

Note that, by [12, Lemma 7.12], $T_{\alpha}$ is well-defined and continuous
for all $1\leqslant\gamma<N/(N-\alpha)$. Moreover, following step by step the
proof of [24, Theorem 2.2], one can prove that $T_{\alpha}$ is compact for all
$1\leqslant\gamma<N/(N-\alpha)$. We only sketch the proof. Let
$(g_{n})_{n}\subset L^{1}(\Omega)$ be a sequence such that
$\|g_{n}\|_{L^{1}(\Omega)}\leqslant 1$ for all $n\in\mathbb{N}$,
$v_{n}:=T_{\alpha}(g_{n})$ for all $n\in\mathbb{N}$ and
$1\leqslant\gamma<N/(N-\alpha)$. If we prove the existence of a (not
relabelled) subsequence $(v_{n})_{n}$ that is Cauchy in $L^{\gamma}(\Omega)$,
the compactness immediately follows. Let $\eta_{\varepsilon}$ be a standard
mollifier and let $v_{n}^{\varepsilon}:=v_{n}\star\eta_{\varepsilon}$ for all
$\varepsilon>0$ and all $n\in\mathbb{N}$. Following [24, Theorem 2.2], we
obtain that, for all $1\leqslant\gamma<N/(N-\alpha)$, there exist constants
$C>0$ and $\sigma>0$ (independent of $\varepsilon$ and $n$) such that

$\|v_{n}^{\varepsilon}-v_{n}\|_{L^{\gamma}(\Omega)}\leqslant
C\varepsilon^{\sigma}.$ (2.10)

On the other hand, using the standard properties of the mollifiers
$\eta_{\varepsilon}$, we get that, for any $\varepsilon>0$, the sequence
$(v_{n}^{\varepsilon})_{n}$ is bounded and equicontinuous in
$C(\mathbb{R}^{N})$. Thus, Arzelà-Ascoli Theorem implies the existence of a
subsequence uniformly convergent in $\overline{\Omega}$. Combining (2.10) with
the uniform convergence in $\overline{\Omega}$, a standard diagonal argument
shows the existence of a (not relabelled) Cauchy subsequence $(v_{n})_{n}$ in
$L^{\gamma}(\Omega)$, as desired.

Having at hand the compactness of $T_{\alpha}$ and Lemma 2.2 we now prove
that, up to a subsequence, $((-\Delta)^{\frac{t}{2}}u_{n})_{n}$ is strongly
convergent in $L^{p}(\mathbb{R}^{N})$ for all $1\leqslant
p<N/\big{(}N(1+t-s)-s\big{)}$. To that end, we fix
$R\geqslant\frac{1}{3}+\frac{4}{3}\big{(}\operatorname{diam}(\Omega)+{\mbox{dist}}(0,\Omega)\big{)}$
as in Lemma 2.2 and split $\mathbb{R}^{N}$ in three regions: $\Omega$,
$B_{R}(0)\setminus\Omega$ and $\mathbb{R}^{N}\setminus B_{R}(0)$.

Combining (2.4) with the linearity of the problem (1.3), the compactness of
$T_{\alpha}$ and Hölder inequality, we get that, up to a subsequence,
$((-\Delta)^{\frac{t}{2}}u_{n})_{n}$ is strongly convergent in $L^{p}(\Omega)$
for all $1\leqslant p<N/\big{(}N(1+t-s)-s\big{)}$.

Next, we deal with the strong convergence of
$((-\Delta)^{\frac{t}{2}}u_{n})_{n}$ in $L^{p}(B_{R}(0)\setminus\Omega)$ for
all $1\leqslant p<N/(N(1+t-s)-s)$. Since
$|x-y|\geqslant\max\\{\delta(y),\delta(x)\\}$ for all $y\in\Omega$ and $x\in
B_{R}(0)\setminus\Omega$, by (2.8) we have that, for all $\varepsilon>0$,

$\big{|}(-\Delta)^{\frac{t}{2}}u_{n}(x)\big{|}\leqslant\int_{\Omega}\frac{|u_{n}(y)|}{|x-y|^{N+t}}dy\leqslant\frac{1}{\delta^{t-s+\varepsilon}(x)}\int_{\Omega}\frac{|u_{n}(y)|}{\delta^{s}(y)}\frac{dy}{|x-y|^{N-\varepsilon}}\quad\textup{
for a.e. }x\in\mathbb{R}^{N}\setminus B_{R}(0)\,.$

Combining this inequality with Lemma 2.3, Hölder inequality and the (up to a
subsequence) strong convergence of $(u_{n}/\delta^{s})_{n}$ in $L^{r}(\Omega)$
for all $1\leqslant r<N/(N-s)$, we get the desired convergence in
$L^{p}(B_{R}(0)\setminus\Omega)$.

Finally, the (up to a subsequence) strong convergence of
$((-\Delta)^{\frac{t}{2}}u_{n})_{n}$ in $L^{p}(\mathbb{R}^{N}\setminus
B_{R}(0))$ for all $1\leqslant p<N/(N(1+t-s)-s)$ follows from (2.9). Indeed,
combining (2.9) with the (up to a subsequence) strong convergence of
$(u_{n})_{n}$ in $L^{q}(\Omega)$ for all $1\leqslant q<N/(N-2s)$, we get the
desired convergence in $L^{p}(\mathbb{R}^{N}\setminus B_{R}(0))$. ∎

######  Corollary 2.5.

Let $0<t<\min\\{1,s(1+N^{-1})\\}$ and $1<p<N/(N(1+t-s)-s)$. The solution map

$\mathbb{G}_{s}:L^{1}(\Omega)\to W_{0}^{t,p}(\Omega),\qquad
h\mapsto\mathbb{G}_{s}[h]:=\int_{\Omega}G_{s}(x,y)h(y)\,dy,$

is well-defined, continuous and compact.

###### Proof.

Having at hand Proposition 2.4 the result follows arguing as in the proof of
[3, Corollary 5.6] ∎

######  Remark 2.6.

  * $\circ$

In the case where $1/2<t<\min\\{1,s(1+N^{-1})\\}$, Proposition 2.4 and
Corollary 2.5 can be proved arguing as in the proof of [2, Proposition 3.10].

  * $\circ$

We believe Proposition 2.4 and Corollary 2.5 are of independent interest and
will be useful elsewhere.

##  3\. Existence results

This section is devoted to prove existence results for (1) with the different
choices of $\mathbb{D}_{t}$ present in the introduction. We will analyse in
parallel $\mathbb{D}_{t}=(-\Delta)^{\frac{t}{2}}$ and
$\mathbb{D}_{t}=\nabla^{t}$ and separately $\mathbb{D}_{t}=\mathcal{D}_{t}$.
Let us emphasize that the main existence result stated in the introduction,
namely Theorem 1.3, immediately follows from the results of this section.

We first analyse the case $\mathbb{D}_{t}=(-\Delta)^{\frac{t}{2}}$ (the case
$\mathbb{D}_{t}=\nabla^{t}$ follows arguing on the exact same way, as we will
detail later on). More precisely, for $q\in(1,+\infty)$, we analyse the
existence of weak solution to

$\left\\{\begin{aligned}
(-\Delta)^{s}u&=\mu(x)|(-\Delta)^{\frac{t}{2}}u|^{q}+\lambda
f(x),&&\quad\textup{ in }\Omega,\\\ u&=0,&&\quad\textup{ in
}\mathbb{R}^{N}\setminus\Omega.\\\ \end{aligned}\right.$ (KPZ1)

Let us impose $0<t<\min\\{1,s(1+N^{-1})\\}$ and set

$\overline{q}(m,s,t):=\left\\{\begin{aligned} &+\infty,&&\mbox{ if }t\leqslant
s\ \mbox{ and }\ m\geqslant N/s,\\\ &s/(N(t-s)),\qquad&&\mbox{ if }t>s\ \mbox{
and }\ m>N/s,\\\ &N/(N-ms),&&\mbox{ if }t\leqslant s\ \mbox{ and }1\leqslant
m<N/s,\\\ &N/(N-sm+mN(t-s)),\quad&&\mbox{ if }t>s\ \mbox{ and }1\leqslant
m\leqslant N/s.\end{aligned}\right.$ (3.1)

Having at hand $\overline{q}\in(1,+\infty]$, our main result concerning (KPZ1)
reads as follows:

######  Theorem 3.1.

Assume that $0<t<\min\\{1,s(1+N^{-1})\\}$, $f\in L^{m}(\Omega)$ for some
$m\geqslant 1$ and $\mu\in L^{\infty}(\Omega)$. Then, for all
$1<q<\overline{q}$, there exists $\lambda_{\star}>0$ such that, for all
$0<\lambda\leqslant\lambda_{\star}$, (KPZ1) has a weak solution $u$. Moreover:

  * $\circ$

If $m\geqslant N/s$, then $u\in W^{s,p}(\mathbb{R}^{N})\cap
C^{0,s}(\mathbb{R}^{N})$ for all $1<p<+\infty$.

  * $\circ$

If $1\leqslant m<N/s$, then $u\in W^{s,p}(\mathbb{R}^{N})$ for all
$1<p<mN/(N-ms)$.

###### Proof of Theorem 3.1.

We use some ideas of [2, Sections 4 and 6] and consider separately the cases
$m>N/s$ and $1\leqslant m\leqslant N/s$.

Case 1: $m>N/s$.

First of all, observe that, without loss of generality, we can assume that
$N/s<m<1/(q(t-s)^{+})$. Then, let us fix $r=r(m,s,t,q)>0$ such that
$1<qm<r<\frac{1}{(t-s)^{+}}$ and define

$\lambda_{\star}:=\frac{q-1}{q\|f\|_{L^{m}(\Omega)}}\Big{(}q(\widetilde{C}\widetilde{k})^{q}\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-qm}{mr}}\Big{)}^{-\frac{1}{q-1}},$
(3.2)

with $\widetilde{C}>0$ as in Proposition 2.1 and $\widetilde{k}$ as in (2.1).
Since $q>1$, we know (cf. [2, Lemma 4.1]) there exists a unique
$\ell\in(0,\infty)$ such that

$\widetilde{C}\Big{(}\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-qm}{mr}}\widetilde{k}^{q}\ell+\lambda_{\star}\|f\|_{L^{m}(\Omega)}\Big{)}=\ell^{\frac{1}{q}}.$
(3.3)

Having at hand $\lambda_{\star}$ and $\ell$, we define

$E_{\eta}:=\left\\{v\in
L^{\gamma,1+\eta}_{0}(\Omega):\left\|v\right\|_{L^{\gamma,r}_{0}(\Omega)}\leqslant\ell^{\frac{1}{q}}\right\\},$
(3.4)

with

$\gamma:=\max\\{t,s\\},\quad\textup{ and
}0<\eta<\min\left\\{q-1,\dfrac{s-N(\gamma-s)}{N(1+(\gamma-s))-s}\right\\},$
(3.5)

and point out that $E_{\eta}$ is a closed and convex subset of
$L_{0}^{\gamma,1+\eta}(\Omega)$. Moreover $E_{\eta}$ is also bounded in
$L_{0}^{\gamma,1+\eta}(\Omega)$. Indeed, for any $R>0$, we have that

$\big{\|}(-\Delta)^{\frac{\gamma}{2}}u\big{\|}_{L^{1+\eta}(\mathbb{R}^{N})}\leqslant\big{\|}(-\Delta)^{\frac{\gamma}{2}}u\big{\|}_{L^{1+\eta}(B_{R}(0))}+\big{\|}(-\Delta)^{\frac{\gamma}{2}}u\big{\|}_{L^{1+\eta}(\mathbb{R}^{N}\setminus
B_{R}(0))}\,.$ (3.6)

Then, observe that, since $r>1+\eta$,

$\big{\|}(-\Delta)^{\frac{\gamma}{2}}u\big{\|}_{L^{1+\eta}(B_{R}(0))}\leqslant
C_{R}\big{\|}(-\Delta)^{\frac{\gamma}{2}}u\big{\|}_{L^{r}(B_{R}(0))}\leqslant
C_{R}\big{\|}(-\Delta)^{\frac{\gamma}{2}}u\big{\|}_{L^{r}(\mathbb{R}^{N})}\leqslant
C_{R}\,\ell^{\frac{1}{q}}\,.$ (3.7)

On the other hand, choosing
$R\geqslant\frac{1}{3}+\frac{4}{3}\big{(}\operatorname{diam}(\Omega)+{\mbox{dist}}(0,\Omega)\big{)}$,
we have that $|x-y|\geqslant\frac{1}{4}\big{(}1+|x|\big{)}$ for all
$y\in\Omega$ and all $x\in\mathbb{R}^{N}\setminus B_{R}(0)$ and thus, we get
that

$\displaystyle\big{\|}(-\Delta)^{\frac{\gamma}{2}}u\big{\|}_{L^{1+\eta}(\mathbb{R}^{N}\setminus
B_{R}(0))}^{1+\eta}\leqslant\int_{\mathbb{R}^{N}\setminus
B_{R}(0)}\left|\int_{\Omega}\frac{|u(y)|}{|x-y|^{N+\gamma}}dy\right|^{1+\eta}dx$
(3.8) $\displaystyle\qquad\leqslant
C\|u\|_{L^{1+\eta}(\Omega)}^{1+\eta}\int_{\mathbb{R}^{N}\setminus
B_{R}(0)}\frac{dx}{(1+|x|)^{(N+\gamma)(1+\eta)}}\leqslant\overline{C}\,\|u\|_{L^{1+\eta}(\Omega)}^{1+\eta}\leqslant\widetilde{C}\,\ell^{\frac{1+\eta}{q}}.$

Note that the last inequality follows from the fractional Sobolev inequality
(see for instance [23, Theorem 1.8]) and the definition of $E_{\eta}$.
Gathering (3.6)–(3.8) the boundedness of $E_{\eta}$ in
$L_{0}^{\gamma,1+\eta}(\Omega)$ follows.

To prove the existence of a weak solution to (KPZ1) belonging to $E_{\eta}$,
we use Schauder’s fixed point Theorem. Let us consider

$T_{1}:E_{\eta}\to L_{0}^{\gamma,1+\eta}(\Omega),\quad\varphi\mapsto u,$ (3.9)

where $u$ is the unique weak solution to

$\left\\{\begin{aligned}
(-\Delta)^{s}u&=\mu(x)|(-\Delta)^{\frac{t}{2}}\varphi|^{q}+\lambda
f(x)\,,\quad&&\textup{ in }\Omega,\\\ u&=0,&&\textup{ in
}\mathbb{R}^{N}\setminus\Omega,\end{aligned}\right.$ (3.10)

and observe that, if we prove that $T_{1}$ has a fixed point in $E_{\eta}$,
the existence part immediately follows. Note that, by Proposition 2.4, the
operator $T_{1}$ is well defined. Hence, to end the proof in this case, we
just have to prove that $T_{1}$ is continuous and compact and that
$T_{1}(E_{\eta})\subset E_{\eta}$.

We start proving that $T_{1}(E_{\eta})\subset E_{\eta}$. Let $\varphi\in
E_{\eta}$ and $u=T_{1}(\varphi)$. Using Proposition 2.1, it is immediate to
see that $u\in L^{\gamma,1+\eta}_{0}(\Omega)$ and that

$\displaystyle\|u\|_{L^{\gamma,r}_{0}(\Omega)}$
$\displaystyle\leqslant\widetilde{C}\bigg{(}\lambda_{\star}\|f\|_{L^{m}(\Omega)}+\|\mu\|_{L^{\infty}(\Omega)}\||(-\Delta)^{\frac{t}{2}}\varphi|^{q}\|_{L^{m}(\Omega)}\bigg{)}$
(3.11)
$\displaystyle\leqslant{\widetilde{C}}\bigg{(}\lambda_{\star}\|f\|_{L^{m}(\Omega)}+\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-qm}{rm}}\|(-\Delta)^{\frac{t}{2}}\varphi\|^{q}_{L^{r}(\Omega)}\bigg{)}$
$\displaystyle\leqslant\widetilde{C}\bigg{(}\lambda_{\star}\|f\|_{L^{m}(\Omega)}+\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-qm}{rm}}\|\varphi\|^{q}_{L_{0}^{t,r}(\Omega)}\bigg{)}$
$\displaystyle\leqslant\widetilde{C}\bigg{(}\lambda_{\star}\|f\|_{L^{m}(\Omega)}+\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-qm}{rm}}\widetilde{k}^{q}\|\varphi\|^{q}_{L^{\gamma,r}_{0}(\Omega)}\bigg{)}\leqslant\ell^{\frac{1}{q}}\,.$

Hence, it follows that $T_{1}(E_{\eta})\subset E_{\eta}$.

Next, we prove that $T_{1}$ is compact. Let $(\varphi_{n})_{n}\subset
E_{\eta}$ be such that
$\|\varphi_{n}\|_{L^{\gamma,1+\eta}_{0}(\Omega)}\leqslant 1$ for all
$n\in\mathbb{N}$. Also, let
$h_{n}:=\mu(x)|(-\Delta)^{\frac{t}{2}}\varphi_{n}|^{q}+\lambda f(x)$ for all
$n\in\mathbb{N}$. Arguing as in (3.11), it is immediate to check that
$(h_{n})_{n}$ is bounded in $L^{1}(\Omega)$ and thus, the compactness of
$T_{1}$ immediately follows from Proposition 2.4.

Finally, we prove that $T_{1}$ is continuous. Let $(\varphi_{n})_{n}\subset
E_{\eta}$ be a sequence such that $\varphi_{n}\to\varphi$ in
$L_{0}^{\gamma,1+\eta}(\Omega)$ and let $u_{n}=T_{1}(\varphi_{n})$ for all
$n\in\mathbb{N}$ and $u=T_{1}(\varphi)$. Note that

$\left\\{\begin{aligned}
(-\Delta)^{s}(u_{n}-u)&=\mu(x)\Big{(}|(-\Delta)^{\frac{t}{2}}\varphi_{n}|^{q}-|(-\Delta)^{\frac{t}{2}}\varphi|^{q}\Big{)},\quad&&\textup{
in }\Omega,\\\ u_{n}-u&=0,&&\textup{ in
}\mathbb{R}^{N}\setminus\Omega.\end{aligned}\right.$ (3.12)

If we show that the $L^{1}$–norm of the right hand side in the (3.12) goes to
$0$ as $n\to\infty$, the continuity of $T_{1}$ immediately follows from
Proposition 2.4 and the existence in the case where $m>N/s$ will follow. By
direct computations (using Hölder inequality and the Mean value Theorem), it
follows that

$\Big{\|}\mu(x)\Big{(}|(-\Delta)^{\frac{t}{2}}\varphi_{n}|^{q}-|(-\Delta)^{\frac{t}{2}}\varphi|^{q}\Big{)}\Big{\|}_{L^{1}(\Omega)}\leqslant
C\,\|(-\Delta)^{\frac{t}{2}}(\varphi_{n}-\varphi)\|_{L^{q}(\Omega)},$ (3.13)

for some $C>0$ depending only on $q$, $r$, $\|\mu\|_{L^{\infty}(\Omega)}$,
$|\Omega|$ and $\ell$. On the other hand, using (2.1) and Littlewood’s
inequality (or interpolation in $L^{p}$–spaces), we infer that

$\displaystyle\|(-\Delta)^{\frac{t}{2}}(\varphi_{n}-\varphi)\|_{L^{q}(\Omega)}\leqslant\|\varphi_{n}-\varphi\|_{L_{0}^{t,q}(\Omega)}\leqslant\widetilde{k}\|\varphi_{n}-\varphi\|_{L_{0}^{\gamma,q}(\Omega)}=\widetilde{k}\|(-\Delta)^{\frac{\gamma}{2}}(\varphi_{n}-\varphi)\|_{L^{q}(\mathbb{R}^{N})}$
(3.14)
$\displaystyle\quad\leqslant\widetilde{k}\|(-\Delta)^{\frac{\gamma}{2}}(\varphi_{n}-\varphi)\|_{L^{1+\eta}(\mathbb{R}^{N})}^{\theta}\|(-\Delta)^{\frac{\gamma}{2}}(\varphi_{n}-\varphi)\|_{L^{r}(\mathbb{R}^{N})}^{1-\theta}=\widetilde{k}\|\varphi_{n}-\varphi\|_{L_{0}^{\gamma,1+\eta}(\Omega)}^{\theta}\|\varphi_{n}-\varphi\|_{L_{0}^{\gamma,r}(\Omega)}^{1-\theta}$
$\displaystyle\quad\leqslant(2\ell^{\frac{1}{q}})^{1-\theta}\widetilde{k}\|\varphi_{n}-\varphi\|_{L_{0}^{\gamma,1+\eta}(\Omega)}^{\theta},$

with $\frac{1}{q}=\frac{\theta}{1+\eta}+\frac{1-\theta}{r}$. Since
$\varphi_{n}\to\varphi$ in $L_{0}^{\gamma,1+\eta}(\Omega)$, combining (3.13)
and (3.14), we conclude that

$\lim_{n\to\infty}\Big{\|}\mu\Big{(}|(-\Delta)^{\frac{t}{2}}\varphi_{n}|^{q}-|(-\Delta)^{\frac{t}{2}}\varphi|^{q}\Big{)}\Big{\|}_{L^{1}(\Omega)}=0,$

as desired. The proof of the existence in the case where $m>N/s$ is thus
finished. Once we have the existence of a weak solution $u\in E_{\eta}$, the
claimed regularity immediately follows from the definition of $E_{\eta}$, our
choice of $r$, Proposition 2.1 and [20, Proposition 1.4 (iii)].

Case 2: $1\leqslant m\leqslant N/s$.

First of all, let us fix $r=r(m,s,t,q)>0$ such that
$1<qm<r<\frac{mN}{(N-ms+mN(\gamma-s))^{+}}$ and consider $\lambda_{\star}$ and
$\ell$ as in (3.2) and (3.3) respectively. Then, we define

$\widetilde{E}_{\eta}:=\Big{\\{}v\in
L_{0}^{\gamma,1+\eta}(\Omega):\|v\|_{L_{0}^{\gamma,1+\eta}(\Omega)}\leqslant
M\textup{ and
}\|v\|_{L_{0}^{\gamma,r}(\Omega)}\leqslant\ell^{\frac{1}{q}}\Big{\\}},$ (3.15)

with $\gamma$ and $\eta$ as in (3.5) and

$M:=\tilde{C}_{1}\Big{(}\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-q}{r}}\widetilde{k}^{q}\ell+\lambda_{\star}\|f\|_{L^{1}(\Omega)}\Big{)},$

where $\tilde{C}_{1}>0$ is the constant that appears in Proposition 2.1 for
$m=1$. It is immediate to see that $\widetilde{E}_{\eta}$ is a bounded, closed
and convex set of $L_{0}^{\gamma,1+\eta}(\Omega)$. Moreover, arguing as in the
first case, one can prove that

$\widetilde{T}_{1}:\widetilde{E}_{\eta}\to
L_{0}^{\gamma,1+\eta}(\Omega),\quad\varphi\mapsto u,$ (3.16)

where $u$ is the unique weak solution to (3.10), is well-defined, continuous,
compact and satisfies
$\widetilde{T}_{1}(\widetilde{E}_{\eta})\subset\widetilde{E}_{\eta}$. Hence,
applying again Schauder’s fixed point Theorem, the existence follows also in
this case. Having at hand the existence, the claimed regularity follows again
from Proposition 2.1. ∎

Next, we analyse the existence of weak solution to (1) in the case where
$\mathbb{D}_{t}=\nabla^{t}$. More precisely, we analyse the existence of weak
solution to

$\left\\{\begin{aligned} (-\Delta)^{s}u&=\mu(x)|\nabla^{t}u|^{q}+\lambda
f(x),&&\quad\textup{ in }\Omega,\\\ u&=0,&&\quad\textup{ in
}\mathbb{R}^{N}\setminus\Omega.\\\ \end{aligned}\right.$ (KPZ2)

Our main result concerning (KPZ2) can be formulated as follows:

######  Theorem 3.2.

Assume that $0<t<\min\\{1,s(1+N^{-1})\\}$, $f\in L^{m}(\Omega)$ for some
$m\geqslant 1$ and $\mu\in L^{\infty}(\Omega)$. Then, for all
$1<q<\overline{q}$, there exists $\lambda_{\star}>0$ such that, for all
$0<\lambda\leqslant\lambda_{\star}$, (KPZ2) has a weak solution $u$. Moreover:

  * $\circ$

If $m\geqslant N/s$, then $u\in W^{s,p}(\mathbb{R}^{N})\cap
C^{0,s}(\mathbb{R}^{N})$ for all $1<p<+\infty$.

  * $\circ$

If $1\leqslant m<N/s$, then $u\in W^{s,p}(\mathbb{R}^{N})$ for all
$1<p<mN/(N-ms)$.

######  Remark 3.3.

Let us emphasize that, having at hand [23, Theorem 1.7], the proof of Theorem
3.2 follows arguing exactly as in the proof of Theorem 3.1.

Finally, we analyse the existence of weak solution in the slightly more
involved case where $\mathbb{D}_{t}=\mathcal{D}_{t}$. More precisely, we
analyse the existence of weak solution to

$\left\\{\begin{aligned} (-\Delta)^{s}u&=\mu(x)(\mathcal{D}_{t}u)^{q}+\lambda
f(x),&&\quad\textup{ in }\Omega,\\\ u&=0,&&\quad\textup{ in
}\mathbb{R}^{N}\setminus\Omega.\\\ \end{aligned}\right.$ (KPZ3)

Setting $\gamma:=\max\\{t,s\\}$ and

$\overline{m}(s,t):=\frac{2N}{N+2s-2N(t-s)^{+}}$

our main result concerning (KPZ3) reads as follows:

######  Theorem 3.4.

Assume that $0<t<\min\\{1,s(1+N^{-1})\\}$, $f\in L^{m}(\Omega)$ for some
$m>\overline{m}$ and $\mu\in L^{\infty}(\Omega)$. Then, for all
$1<q<\overline{q}$, there exists $\lambda_{\star}>0$ such that, for all
$0<\lambda\leqslant\lambda_{\star}$, (KPZ3) has a weak solution $u$. Moreover:

  * $\circ$

If $m\geqslant N/s$, then $u\in W^{s,p}(\mathbb{R}^{N})\cap
C^{0,s}(\mathbb{R}^{N})$ for all $1<p<+\infty$.

  * $\circ$

If $\overline{m}<m<N/s$, then $u\in W^{s,p}(\mathbb{R}^{N})$ for all
$1<p<mN/(N-ms)$.

###### Proof.

We consider separately the cases $m>N/s$ and $\overline{m}<m\leqslant N/s$.

Case 1: $m>N/s$.

First of all, observe that, without loss of generality, we can assume that
$N/s<m<1/(q(t-s)^{+})$. Then, let us fix $r=r(m,s,t,q)>0$ such that
$\max\\{qm,2\\}<r<\frac{1}{(t-s)^{+}}$ and define

$\lambda_{\star}:=\frac{q-1}{q\|f\|_{L^{m}(\Omega)}}\Big{(}q(\widetilde{C}\widetilde{k})^{q}\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-qm}{mr}}\Big{)}^{-\frac{1}{q-1}},$
(3.17)

with $\widetilde{C}>0$ as in Proposition 2.1 and $\widetilde{k}$ as in (2.1).
Since $q>1$, we know (cf. [2, Lemma 4.1]) there exists a unique
$\ell\in(0,\infty)$ such that

$\widetilde{C}\Big{(}\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-qm}{mr}}\widetilde{k}^{q}\ell+\lambda_{\star}\|f\|_{L^{m}(\Omega)}\Big{)}=\ell^{\frac{1}{q}}.$
(3.18)

Having at hand $\lambda_{\star}$ and $\ell$, we define

$E_{\eta}:=\left\\{v\in
L^{\gamma,1+\eta}_{0}(\Omega):{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{\gamma,r}_{0}(\Omega)}\leqslant\ell^{\frac{1}{q}}\right\\},$
(3.19)

with

$\gamma=\max\\{t,s\\}\quad\textup{ and }\quad
0<\eta<\min\left\\{q-1,\dfrac{s-N(\gamma-s)}{N(1+(\gamma-s))-s}\right\\},$
(3.20)

and we point out that $E_{\eta}$ is a closed, bounded and convex subset of
$L_{0}^{\gamma,1+\eta}(\Omega)$. To prove the existence of a weak solution to
(KPZ3) belonging to $E_{\eta}$, we use again Schauder’s fixed point Theorem.
Let us define

$T_{2}:E_{\eta}\to L_{0}^{\gamma,1+\eta}(\Omega),\quad\varphi\mapsto u,$
(3.21)

where $u$ the unique weak solution to

$\left\\{\begin{aligned}
(-\Delta)^{s}u&=\mu(x)(\mathcal{D}_{t}\varphi)^{q}+\lambda
f(x)\,,\quad&&\textup{ in }\Omega,\\\ u&=0,&&\textup{ in
}\mathbb{R}^{N}\setminus\Omega.\end{aligned}\right.$ (3.22)

Note that, by Proposition 2.4, the operator $T_{2}$ is well defined. Hence, to
conclude the proof in this case, we just have to prove that $T_{2}$ is
continuous and compact and that $T_{2}(E_{\eta})\subset E_{\eta}$. The
compactness of $T_{2}$ and the fact that $T_{2}(E_{\eta})\subset E_{\eta}$ can
be proved arguing exactly as in the proof of Theorem 3.1. However, to prove
that $T_{2}$ is continuous we have to argue in a different way. Let
$(\varphi_{n})_{n}\subset E_{\eta}$ be a sequence such that
$\varphi_{n}\to\varphi$ in $L_{0}^{\gamma,1+\eta}(\Omega)$ and let
$u_{n}=T_{2}(\varphi_{n})$ for all $n\in\mathbb{N}$ and $u=T_{2}(\varphi)$.
Note that

$\left\\{\begin{aligned}
(-\Delta)^{s}(u_{n}-u)&=\mu(x)\Big{(}(\mathcal{D}_{t}\varphi_{n})^{q}-(\mathcal{D}_{t}\varphi)^{q}\Big{)},\quad&&\textup{
in }\Omega,\\\ u_{n}-u&=0,&&\textup{ in
}\mathbb{R}^{N}\setminus\Omega.\end{aligned}\right.$ (3.23)

If we show that the $L^{1}$–norm of the right hand side in (3.23) goes to $0$
as $n\to\infty$, the continuity of $T_{2}$ immediately follows from
Proposition 2.4. We actually prove something more general from which the
existence part of the result in the case $m>N/s$ immediately follows.

Claim: For all $1<\alpha<r$, it follows that

$\lim_{n\to\infty}\int_{\Omega}\big{|}(\mathcal{D}_{t}\varphi_{n}(x))^{\alpha}-(\mathcal{D}_{t}\varphi(x))^{\alpha}\big{|}dx=0.$
(3.24)

_Proof of the claim._ First of all, let $2\leqslant\beta<r$ be fixed but
arbirary. Using the Mean value Theorem and Hölder and triangular inequalities,
one can easily get that

$\displaystyle\int_{\Omega}|(\mathcal{D}_{t}\varphi_{n}(x))^{\beta}-(\mathcal{D}_{t}\varphi(x))^{\beta}\big{|}dx=\int_{\Omega}\big{|}\big{(}(\mathcal{D}_{t}\varphi_{n}(x))^{2}\big{)}^{\frac{\beta}{2}}-\big{(}(\mathcal{D}_{t}\varphi(x))^{2}\big{)}^{\frac{\beta}{2}}\big{|}dx$
$\displaystyle\quad\leqslant\frac{\beta}{2}\int_{\Omega}\big{|}(\mathcal{D}_{t}\varphi_{n}(x))^{2}-(\mathcal{D}_{t}\varphi(x))^{2}\big{|}\big{(}(\mathcal{D}_{t}\varphi_{n}(x))^{2}+(\mathcal{D}_{t}\varphi(x))^{2}\big{)}^{\frac{\beta-2}{2}}dx$
$\displaystyle\quad\leqslant\frac{\beta}{2}\int_{\Omega}(\mathcal{D}_{t}(\varphi_{n}-\varphi)(x))(\mathcal{D}_{t}(\varphi_{n}+\varphi)(x))\big{(}(\mathcal{D}_{t}\varphi_{n}(x))^{2}+(\mathcal{D}_{t}\varphi(x))^{2}\big{)}^{\frac{\beta-2}{2}}dx$
$\displaystyle\quad\leqslant\frac{\sqrt{2}}{2}\beta\int_{\Omega}(\mathcal{D}_{t}(\varphi_{n}-\varphi)(x))\big{(}\mathcal{D}_{t}\varphi_{n}(x)+\mathcal{D}_{t}\varphi(x)\big{)}^{\beta-1}dx\leqslant\frac{\sqrt{2}}{2}\beta\|\mathcal{D}_{t}(\varphi_{n}-\varphi)\|_{L^{\beta}(\Omega)}\|\mathcal{D}_{t}\varphi_{n}+\mathcal{D}_{t}\varphi\|_{L^{\beta}(\Omega)}^{\beta-1}$
$\displaystyle\quad\leqslant
C\big{(}\|\mathcal{D}_{t}\varphi_{n}\|_{L^{\beta}(\Omega)}^{\beta-1}+\|\mathcal{D}_{t}\varphi\|_{L^{\beta}(\Omega)}^{\beta-1}\big{)}\|\mathcal{D}_{t}(\varphi_{n}-\varphi)\|_{L^{\beta}(\Omega)}\leqslant
C\big{(}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\varphi_{n}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{0}^{t,\beta}(\Omega)}^{\beta-1}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\varphi\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{0}^{t,\beta}(\Omega)}^{\beta-1}\big{)}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\varphi_{n}-\varphi\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{0}^{t,\beta}(\Omega)},$

for some $C>0$ depending only on $\beta$. Then, arguing exactly as in the
proof of (3.14), we conclude that

$\lim_{n\to\infty}\int_{\Omega}\big{|}(\mathcal{D}_{t}\varphi_{n}(x))^{\beta}-(\mathcal{D}_{t}\varphi(x))^{\beta}\big{|}dx=0.$

Since $\beta\in[2,r)$ was fixed but arbitrary, we have proved (3.24) for all
$2\leqslant\alpha<r$.

It remains to deal with the case $1<\alpha<2$. To that end, note that
$(\mathcal{D}_{t}\varphi_{n})_{n}\subset L^{2}(\Omega)$ is a bounded non-
negative sequence such that
$\|\mathcal{D}_{t}\varphi_{n}\|_{L^{2}(\Omega)}\to\|\mathcal{D}_{t}\varphi\|_{L^{2}(\Omega)}$
as $n\to\infty$. Hence, it follows that
$\mathcal{D}_{t}\varphi_{n}\to\mathcal{D}_{t}\varphi$ in $L^{2}(\Omega)$ as
$n\to\infty$. Also, observe that, for any $\alpha\in(1,2)$,

$\displaystyle\int_{\Omega}\big{|}(\mathcal{D}_{t}\varphi_{n}(x))^{\alpha}-(\mathcal{D}_{t}\varphi(x))^{\alpha}\big{|}dx$
$\displaystyle\quad\leqslant
C\,\|\mathcal{D}_{t}(\varphi_{n})-\mathcal{D}_{t}(\varphi)\|_{L^{\alpha}(\Omega)}\big{(}\|\mathcal{D}_{t}\varphi_{n}\|_{L^{\alpha}(\Omega)}^{\alpha-1}+\|\mathcal{D}_{t}\varphi\|_{L^{\alpha}(\Omega)}^{\alpha-1}\big{)}\leqslant\widetilde{C}\,\|\mathcal{D}_{t}\varphi_{n}-\mathcal{D}_{t}\varphi\|_{L^{2}(\Omega)},$

with $C>0$ depending only on $\alpha>0$ and $\widetilde{C}>0$ depending only
on $\alpha$, $r$, $|\Omega|$ and $\ell$. Combining this chain of inequalities
with the fact that $\mathcal{D}_{t}\varphi_{n}\to\mathcal{D}_{t}\varphi$ in
$L^{2}(\Omega)$ as $n\to\infty$ we conclude that

$\lim_{n\to\infty}\int_{\Omega}\big{|}(\mathcal{D}_{t}\varphi_{n}(x))^{\alpha}-(\mathcal{D}_{t}\varphi(x))^{\alpha}\big{|}dx=0.$

Once the claim is proved, to conclude the proof in the case where $m>N/s$, it
just remains to prove the claimed regularity, that follows thanks to
Proposition 2.1, [20, Proposition 1.4 (iii)] and the definition of $E_{\eta}$.

Case 2: $\overline{m}<m\leqslant N/s$.

First of all, let us fix $r=r(m,s,t,q)>0$ such that
$\max\\{2,qm\\}<r<\frac{mN}{(N-ms+mN(\gamma-s))^{+}}$ and consider
$\lambda_{\star}$ and $\ell$ as in (3.17) and (3.18) respectively. Then, let
us define

$\widetilde{E}_{\eta}:=\Big{\\{}v\in
L_{0}^{\gamma,1+\eta}(\Omega):{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{0}^{\gamma,1+\eta}(\Omega)}\leqslant
M\textup{ and
}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{0}^{\gamma,r}(\Omega)}\leqslant\ell^{\frac{1}{q}}\Big{\\}},$
(3.25)

with $\gamma$ and $\eta$ as in (3.20) and

$M:=C\Big{(}\|\mu\|_{L^{\infty}(\Omega)}|\Omega|^{\frac{r-q}{r}}\widetilde{k}^{q}\ell+\lambda_{\star}\|f\|_{L^{1}(\Omega)}\Big{)},$

with $C>0$ depending only on $\Omega$, $N$, $s$ and $t$ (cf. Proposition 2.1).
It is easy to check that $\widetilde{E}_{\eta}$ is a bounded, closed and
convex set of $L_{0}^{\gamma,1+\eta}(\Omega)$. Moreover, arguing as we did in
the first case, one can prove that

$\widetilde{T}_{2}:\widetilde{E}_{\eta}\to
L_{0}^{\gamma,1+\eta}(\Omega),\quad\varphi\mapsto u,$ (3.26)

where $u$ is the unique weak solution to (3.22), is well-defined, continuous,
compact and satisfies
$\widetilde{T}_{2}(\widetilde{E}_{\eta})\subset\widetilde{E}_{\eta}$. Hence,
applying again Schauder’s fixed point Theorem, the existence follows also in
this case. Having at hand the existence, the claimed regularity follows again
from Proposition 2.1. ∎

######  Remark 3.5.

The appearance of the upper bound $\overline{q}\in(1,+\infty]$ in Theorems 3.1
and 3.4 is natural in this kind of existence results. Even in the local case
(1.1), one finds an upper bound on $q$ depending on the regularity of the data
$f$ and on the dimension $N$. In our proofs, the upper bound $\overline{q}$
naturally appears when applying Proposition 2.1 and depends on the regularity
of the datum $f$, on the dimension $N$ and on the order of the “nonlocal
gradient”. The lower bound $\overline{m}$ is again related to the use of
Proposition 2.1 but also to the fact that we need $r>2$ in the proof of
Theorem 3.4.

##  4\. Non–existence results

This section is devoted to prove non-existence results for (KPZ1) and (KPZ3).
Let us start analysing (KPZ1). As already mentioned in the introduction, the
proof of the non-existence result for (KPZ1) is completely different from its
counterpart for (KPZ3). Here, we prove a generalization of Theorem 1.4.

######  Theorem 4.1.

Let $0<t<\min\\{1,2s\\}$, $\mu_{2}\geqslant\mu(x)\geqslant\mu_{1}>0$, $f\in
L^{1}(\Omega)$ with $f\gneqq 0$ and $q>\frac{2(s+1)}{t+2}$. Then, there exists
$\lambda^{\star\star}>0$ such that, for all $\lambda>\lambda^{\star\star}$,
(KPZ1) has no weak solution in $W_{0}^{s,2}(\Omega)$.

###### Proof.

Let $u\in W_{0}^{s,2}(\Omega)$ be a weak solution to (KPZ1) and let $\phi\in
W_{0}^{s,2}(\Omega)\cap C^{s}(\mathbb{R}^{N})$ be the unique (energy) solution
to

$\left\\{\begin{aligned} (-\Delta)^{s}\phi&=1,\quad&&\textup{ in }\Omega,\\\
\phi&=0,&&\textup{ in }\mathbb{R}^{N}\setminus\Omega.\end{aligned}\right.$

First of all, note that $\phi\in\mathbb{X}^{s}(\Omega)$, so that we can use
$\phi$ as test function in (KPZ1) and get that

$\displaystyle\int_{\Omega}u\,dx=\int_{\Omega}\mu(x)|(-\Delta)^{\frac{t}{2}}u|^{q}\phi\,dx+\lambda\int_{\Omega}f\phi\,dx\geqslant\mu_{1}\int_{\Omega}|(-\Delta)^{\frac{t}{2}}u|^{q}\phi\,dx+\lambda\int_{\Omega}f\phi\,dx.$
(4.1)

On the other hand, let $\psi\in W_{0}^{\frac{t}{2},2}(\Omega)\cap
C^{\frac{t}{2}}(\mathbb{R}^{N})$ be the unique (energy) solution to the
problem

$\left\\{\begin{aligned} (-\Delta)^{\frac{t}{2}}\psi&=1,\quad&&\textup{ in
}\Omega,\\\ \psi&=0,&&\textup{ in
}\mathbb{R}^{N}\setminus\Omega.\end{aligned}\right.$ (4.2)

Since $W_{0}^{s,2}(\Omega)\subset W_{0}^{\frac{t}{2},2}(\Omega)$ we can test
(4.2) with $u$ and integrate by parts, so that, using (4.1), we obtain

$\int_{\Omega}\psi(-\Delta)^{\frac{t}{2}}u\,dx=\int_{\Omega}u\,dx\geqslant\mu_{1}\int_{\Omega}|(-\Delta)^{\frac{t}{2}}u|^{q}\phi\,dx+\lambda\int_{\Omega}f\phi\,dx.$
(4.3)

Moreover, using Young’s inequality, we easily see that

$\int_{\Omega}\psi(-\Delta)^{\frac{t}{2}}u\,dx\leqslant\mu_{1}\int_{\Omega}|(-\Delta)^{\frac{t}{2}}u|^{q}\phi\,dx+{C_{q}}\mu_{1}^{-\frac{1}{q-1}}\int_{\Omega}\frac{\psi^{\frac{q}{q-1}}}{\phi^{\frac{1}{q-1}}}dx.$
(4.4)

Thus, combining (4.3) and (4.4), we obtain that

$\lambda\int_{\Omega}f\phi\,dx\leqslant{C_{q}}\mu_{1}^{-\frac{1}{q-1}}\int_{\Omega}\frac{\psi^{\frac{q}{q-1}}}{\phi^{\frac{1}{q-1}}}dx.$
(4.5)

Finally, note that (see e.g. [11, Eq. (1.15)]) there exists $C_{0}>0$
(depending only on $\Omega$, $s,t$ and $N$) such that

$C_{0}^{-1}\delta^{s}(x)\leqslant\phi(x)\leqslant
C_{0}\delta^{s}(x)\quad\textup{ and }\quad
C_{0}^{-1}\delta^{\frac{t}{2}}(x)\leqslant\psi(x)\leqslant
C_{0}\delta^{\frac{t}{2}}(x),\quad\textup{ in }\ \overline{\Omega}.$ (4.6)

Hence, since $q>\frac{2(s+1)}{t+2}$, the right hand side in (4.5) is bounded,
an thus necessarily

$\lambda\leqslant\frac{C_{q}\displaystyle\int_{\Omega}\frac{\psi^{\frac{q}{q-1}}}{\phi^{\frac{1}{q-1}}}dx.}{\displaystyle{\mu_{1}^{\frac{1}{q-1}}}\int_{\Omega}f\phi\,dx}=:\lambda^{\star\star}.$
(4.7)

∎

######  Remark 4.2.

The regularity imposed on $f$ can be slightly weakened. Indeed, to prove our
non–existence result, namely Theorem 4.1, we only need $f\in
L^{1}(\Omega,\delta^{s}(x)dx)$. Moreover, observe that in the proof we only
used that $u$ is a (weak) supersolution to (KPZ1). On the negative side,
observe that the bound appearing on the power $q$ seems to be technical.

We also prove here a generalization of [2, Theorem 1.2] covering the cases
where $q\neq 2$ and/or $t\neq s$.

######  Theorem 4.3.

Let $0<t<\min\\{1,2s\\}$, $\mu_{2}\geqslant\mu(x)\geqslant\mu_{1}>0$, $f\in
L^{1}(\Omega)$ with $f^{+}\not\equiv 0$ and $q>1$. Then, there exists
$\lambda^{\star\star}>0$ such that, for all $\lambda>\lambda^{\star\star}$,
(KPZ3) has no weak solution in $W_{0}^{s,2}(\Omega)$.

###### Proof.

Let $u\in W_{0}^{s,2}(\Omega)$ be a weak solution to (KPZ3) and let $\phi\in
C_{c}^{\infty}(\Omega)$ be an arbitrary non-negative function such that

$\int_{\Omega}f\phi^{\frac{q}{q-1}}dx>0\quad\textup{ and
}\quad\int_{\Omega}\big{(}\mathcal{D}_{2s-t}\phi\big{)}^{\frac{q}{q-1}}dx<+\infty.$
(4.8)

Using $\phi^{\frac{q}{q-1}}$ as test function in (KPZ3), we get

$\int_{\Omega}u(-\Delta)^{s}(\phi^{\frac{q}{q-1}})\,dx\geqslant\mu_{1}\int_{\Omega}(\mathcal{D}_{t}u)^{q}\phi^{\frac{q}{q-1}}dx+\lambda\int_{\Omega}f\phi^{\frac{q}{q-1}}dx.$
(4.9)

On the other hand, using the Mean value Theorem and Hölder’s and Young’s
inequalities, we infer that

$\displaystyle\int_{\Omega}u(-\Delta)^{s}(\phi^{\frac{q}{q-1}})\,dx=\iint_{\mathbb{R}^{2N}}\frac{(u(x)-u(y))(\phi^{\frac{q}{q-1}}(x)-\phi^{\frac{q}{q-1}}(y))}{|x-y|^{N+2s}}\,dydx$
$\displaystyle\quad\leqslant
C_{q}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)||\phi(x)-\phi(y)|}{|x-y|^{N+2s}}(\phi^{\frac{1}{q-1}}(x)+\phi^{\frac{1}{q-1}}(y))\,dydx$
$\displaystyle\quad=2C_{q}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)||\phi(x)-\phi(y)|}{|x-y|^{N+2s}}\phi^{\frac{1}{q-1}}(x)\,dydx\leqslant
2C_{q}\int_{\mathbb{R}^{N}}\big{(}\mathcal{D}_{t}u\big{)}\,\big{(}\mathcal{D}_{2s-t}\phi\big{)}\
\phi^{\frac{1}{q-1}}\,dx$
$\displaystyle\quad=2C_{q}\int_{\Omega}\big{(}\mathcal{D}_{t}u\big{)}\,\big{(}\mathcal{D}_{2s-t}\phi\big{)}\phi^{\frac{1}{q-1}}\,dx\leqslant\mu_{1}\int_{\Omega}(\mathcal{D}_{t}u)^{q}\phi^{\frac{q}{q-1}}dx+\widetilde{C}_{q,\mu_{1}}\int_{\Omega}(\mathcal{D}_{2s-t}\phi)^{\frac{q}{q-1}}dx\,.$

Combining the above chain of inequalities with (4.9), we get that necessarily

$\lambda\int_{\Omega}f\phi^{\frac{q}{q-1}}dx\leqslant\widetilde{C}_{q,\mu_{1}}\int_{\Omega}(\mathcal{D}_{2s-t}\phi)^{\frac{q}{q-1}}dx.$

Hence, defining

$\lambda^{\star\star}:=\inf\left\\{\frac{\displaystyle\widetilde{C}_{q,\mu_{1}}\int_{\Omega}(\mathcal{D}_{2s-t}\phi)^{\frac{q}{q-1}}dx}{\displaystyle\int_{\Omega}f\phi^{\frac{q}{q-1}}dx}:\phi\in
C_{c}^{\infty}(\Omega)\textup{ is non-negative and satisfies
}\eqref{propphi}\right\\},$

the result immediately follows. ∎

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††thanks: Corresponding<EMAIL_ADDRESS>
# Decoherence Effect of Qubits in 1D Transverse Ising Model

Bobin Li  Department of Physics, Lanzhou University of Technology, Lanzhou
730050, China

###### Abstract

With the development of lab technology, the low-order correlation function can
no longer describe the main effect of decoherence in quantum many-body system,
so it is imperative to study the higher-order correlation function of the
system. In this paper, we study the changes of the correlation functions in
the decoherence effect, analytically. And explore when it is possible to
approach the qubit decoherence process only by low-order correlation function,
and when third-order or higher correlation functions are needed in 1D
transverse Ising model. It indicates that, under strong coupling and long
coherence time, the effect of high-order correlation functions can not be
ignored, and the approximation of classical Markov process is limited. But, in
the case of weak coupling and short coherence time, low-order correlation
function can describe well.

Qubit; 1D Transverse Ising Models; Correlation Functions; Decoherence

###### pacs:

03.65.Yz, 64.60.De, 82.20.Sb

††preprint: APS/123-QED

##  I Introduction

The decoherence of a qubit is caused by its bath, which is essentially a
quantum many-body system. Previously, people studied the decoherence of
qubits, and the decoherence effect of the bath is generally approached by the
classical Markov process, which is described by the second-order correlation
function. Under this approximation, the effect of higher-order correlation
function are ignored. In recent years, with the development of quantum control
and dynamic decoupling technology, the coherent time of qubits has been
greatly extended, so the effect of quantum many body increasingly stand out,
then the markov approximation cannot satisfy the needs of the quantum many
body system, and, namely, the second-order correlation function cannot cover
the main effect of baths, and the papers[1] can obtain strict solutions and
comprehensive properties of higher order correlation functions for many-body
systems.

For example, it can be solved strictly to 1D Ising model and 2D Ising model
with zero magnetic field by analytic method. It is available for thosed
systems with not very strong many-body interaction[2,3,4] by using more
advanced numerical methods, such as CCE so on. And the numerical solution[5]
of higher order correlation function is obtained. For systems with strong
many-body interaction but small numbers of particles(less than 30 particles),
they can be solved by the method of numerically strict diagonalization, citing
Haiqing Lin [5]. For systems with strong many-body interaction and large
numbers of particles, it will be available by the methods of extrapolation
that is to extrapolate from the weak interaction strength and less particle
numbers to those cases that is strong interaction and large numbers, and it
could obtain some properties of higher order correlation function
qualitatively.

At the same time, with the update of experimental technology, it becomes
feasible for the experimental measurement of high-order correlation function ,
which has promoted the theoretical improvement for the research of correlation
function, and the study of high-order correlation function is imperative.
Therefore, this work mainly focuses on the role of higher-order correlation
function in the process of decoherence of qubit. By theoretical analysis for
many-body quantum effect in complex baths, it is obtained strictly
analytically for the form of higher order correlation function. The
contribution of the high-order correlation function is proposed by comparing
with the exact solution in 1D transverse Ising model. It is found that in the
process of strong coupling and long time interaction between system and bath,
the contribution of high-order correlation function will increase for to the
decoherence of qubit.

##  II Theory method

###  II.1 Correlation function

The reduced density matrix of the central spin Qubit $\rho^{S}(t)$ is

$\displaystyle\rho^{S}(t)$
$\displaystyle=Tr_{B}\rho(t)=Tr_{B}(\mathcal{T}exp(\int_{0}^{t}d\tau\mathcal{L}(\tau))\rho(0))$
$\displaystyle=\sum_{n}\frac{1}{n!}Tr_{B}(\int_{0}^{t}\mathcal{T}[\mathcal{L}(t_{1})\mathcal{L}(t_{2})\cdots\mathcal{L}(t_{n})\rho(0)]dt_{1}dt_{2}\cdots
dt_{n})$
$\displaystyle=\sum_{n}\frac{2^{n}}{n!}\sum_{\\{\alpha_{n}\\}}\sum_{\\{\eta_{n}\\}}\int_{0}^{t}dt_{1}\cdots
dt_{n}\cdot
C_{\alpha_{n}\cdots\alpha_{1}}^{\eta_{n}\cdots\eta_{1}}\times(\mathcal{T}[\mathcal{S}_{\alpha_{1}}^{\overline{\eta}_{1}}(t_{1})\cdots\mathcal{S}_{\alpha_{n}}^{\overline{\eta}_{n}}(t_{n})]\rho^{S}(0))$

where
$\mathcal{L}(t)=2\sum_{\alpha}(\mathcal{S}_{\alpha}^{+}(t)\mathcal{B}_{\alpha}^{-}(t)+\mathcal{S}_{\alpha}^{-}(t)\mathcal{B}_{\alpha}^{+}(t))$
So
$C_{\alpha_{n}\cdots\alpha_{1}}^{\eta_{n}\cdots\eta_{1}}=Tr_{B}(T\mathcal{B}_{\alpha_{1}}^{\eta_{1}}(t_{1})\cdots\mathcal{B}_{\alpha_{n}}^{\eta_{n}}(t_{n})\rho^{B})$
Is the correlation function, while the irreducible correlation function is
$\tilde{C}_{\alpha_{n}\cdots\alpha_{1}}^{\eta_{n}\cdots\eta_{1}}$[8], The
system dynamics can be described as

$\displaystyle\rho^{S}(t)=\mathcal{T}\exp(\sum_{N=1}^{\infty}\frac{2^{N}}{N!}\int_{0}^{t}dt_{1}\cdots
dt_{n}\cdot\tilde{C}_{\alpha_{n}\cdots\alpha_{1}}^{\eta_{n}\cdots\eta_{1}}\times(\mathcal{S}_{\alpha_{n}}^{\overline{\eta}_{n}}(t_{n})\cdots\mathcal{S}_{\alpha_{1}}^{\overline{\eta}_{1}}(t_{1}))\rho^{S}(0)$
(1)

###  II.2 1D Transverse Ising model

The bath is regarded as a 1D transverse Ising model. In this model, Surrounded
by a circle of bath spins, the qubit is located on the central axis of the
circle. In this bath, the bath spins are coupled in only one direction, and
there is an external magnetic field in the whole environment. The exact
solution of 1D transverse Ising model is shown in Appendix A. Since it is
applied to be periodic boundary conditions, qubit is located in the center of
the circle of 1D transverse Ising model chain, and the system Hamiltonian in
the Schrodinger representation is:

$\displaystyle
H=H_{0}+V=\omega_{0}\sigma^{z}-\sum_{j=1}^{N}\sigma_{j}^{x}\sigma_{j+1}^{x}-\lambda\sum_{j=1}^{N}\sigma_{j}^{z}+\sum_{j=1}^{N}\sigma^{z}\otimes(-g\sigma_{j}^{z})$
(2)

where
$H_{0}=\omega_{0}\sigma^{z}-\sum_{j=1}^{N}\sigma_{j}^{x}\sigma_{j+1}^{x}-\lambda\sum_{j=1}^{N}\sigma_{j}^{z}$and
$V=\sum_{j=1}^{N}\sigma^{z}\otimes(-g\sigma_{j}^{z})$. Use the Jordan–Wigner
transform

$\sigma_{j}^{z}=1-2a_{j}^{\dagger}a_{j},\sigma_{j}^{x}+i\sigma_{j}^{y}=2(\prod_{i<j}\sigma_{i}^{z})a_{j}$

The system Hamiltonian was changed to a Fermi system

$H_{\lambda}=-\sum_{j=1}^{N}[(a_{j}^{\dagger}-a_{j})(a_{j+1}^{\dagger}+a_{j+1})-2\lambda
a_{j}^{\dagger}a_{j}]-\lambda
N+\omega_{0}\sigma^{z}-g\sum_{j=1}^{N}\sigma^{z}\otimes(1-2a_{j}^{\dagger}a_{j})$

By the Fourier transform $a_{j}=\sum_{k}c_{k}exp(-ikj)\sqrt{N}$,The spin
system is mapped to the spin-free Fermi system

$H_{\lambda}=-\sum_{k}[(2cosk-2\lambda)c_{k}^{\dagger}c_{k}+isink(c_{-k}^{\dagger}c_{k}^{\dagger}+c_{-k}c_{k})]-N\lambda+\omega_{0}\sigma^{z}-g\sum_{k}\sigma^{z}\otimes(1-2c_{k}^{\dagger}c_{k})$

where$c_{k}^{\dagger}c_{k}$ is the creation annihilation operator of fermions
with wave vector K. The system Hamiltonian can be diagonally transformed by
the Bogoliubov transformation

$\begin{bmatrix}b_{-k}\\\ b_{k}\\\ b_{-k}^{\dagger}\\\
b_{k}^{\dagger}\end{bmatrix}=\begin{bmatrix}u_{k}&0&0&iv_{k}\\\
0&u_{k}&-iv_{k}&0\\\ 0&-iv_{k}&u_{k}&0\\\
iv_{k}&0&0&u_{k}\end{bmatrix}\begin{bmatrix}c_{-k}\\\ c_{k}\\\
c_{-k}^{\dagger}\\\ c_{k}^{\dagger}\end{bmatrix}$

Here, $u_{k}=cos\theta_{k},v_{k}=sin\theta_{k}$with
$tan(2\theta_{k})=sink/(cosk-\lambda)$. After the transformation, the
diagonalized Fermionic Hamiltonian is

$\displaystyle
H_{\lambda}=\sum_{k}\varepsilon_{k}(b_{k}^{\dagger}b_{k}-1/2)+\omega_{0}\sigma^{z}-g\sum_{k}\sigma^{z}\otimes(cos2\theta_{k}-2b_{k}^{\dagger}b_{k}-isin2\theta_{k}(b_{k}^{\dagger}b_{-k}^{\dagger}-b_{-k}b_{k})))=H+V$
(3)

The interaction representation, the interaction Hamiltonian$V(t)$

$\displaystyle
V(t)=(-g\sigma^{z})\otimes\sum_{k}(cos2\theta_{k}-2b_{k}^{\dagger}b_{k}-isin2\theta_{k}\cdot(b_{k}^{\dagger}b_{-k}^{\dagger}\cdot
exp(-i2\varepsilon_{k}t)-b_{-k}b_{k}\cdot
exp(i2\varepsilon_{k}t)))=S_{1}B_{1}$ (4)

At the same times

$\displaystyle S_{1}$ $\displaystyle=-g\sigma^{z}$ (5) $\displaystyle B_{1}$
$\displaystyle=\sum_{k}(cos2\theta_{k}-2b_{k}^{\dagger}b_{k}-isin2\theta_{k}\cdot(b_{k}^{\dagger}b_{-k}^{\dagger}\cdot
exp(-i2\varepsilon_{k}t)-b_{-k}b_{k}\cdot exp(i2\varepsilon_{k}t)))$ (6)

where $u_{k}=cos\theta_{k},v_{k}=sin\theta_{k}$with
$tan(2\theta_{k})=sink/(cosk-\lambda)$ where
$\varepsilon_{k}=2\sqrt{1-2\lambda cosk+\lambda^{2}}$

##  III Results and Discussion

For the density matrix of the system, there is an initial state relation

$\rho(0)=\rho^{S}(0)\otimes\rho^{B}(0)$

Where bath is $\rho^{B}(0)=\frac{1}{Z}e^{-\beta H_{0}}$,The density matrix of
the qubit

$\rho_{10}^{S}(t)=\rho_{01}^{S*}(t)=\rho_{10}^{S}(0)e^{\Gamma(t)}$

where $\Gamma(t)$ Is the decoherence function. Since 1D transverse Field Ising
model is a pure dephasing model, so

$\displaystyle\rho^{S}(t)$
$\displaystyle=exp[\sum_{N=0}^{+\infty}\frac{2^{N}}{N!}\int_{0}^{t}dt_{N}\cdots
dt_{1}\tilde{C}_{1\cdots
1}^{+\cdots+}\mathcal{S}_{1}^{-}\cdots\mathcal{S}_{1}^{-}]\rho^{S}(0)$
$\displaystyle=\prod_{N=0}^{\infty}\sum_{n=0}^{+\infty}(\frac{1}{n!}[\frac{2^{N}}{N!}\int_{0}^{t}dt_{N}\cdots
dt_{1}\tilde{C}_{1\cdots
1}^{+\cdots+}]^{n}[ig]^{nN})\left[\begin{array}[]{cc}0&c\\\ (-1)^{nN}\cdot
c&0\end{array}\right]$

So

$\displaystyle\rho_{10}^{S}(t)=exp[\sum_{N=0}^{+\infty}(\frac{2^{N}}{N!}\int_{0}^{t}dt_{N}\cdots
dt_{1}\tilde{C}_{1\cdots 1}^{+\cdots+}[ig]^{N})]\cdot c$

So the series representation of the decoherence function is obtained as follow

$\displaystyle\Gamma(t)=\sum_{N=0}^{+\infty}(\frac{2^{N}}{N!}\int_{0}^{t}dt_{N}\cdots
dt_{1}\tilde{C}_{1\cdots 1}^{+\cdots+}[ig]^{N})$ (7)

#### The first-order correlation function

$\displaystyle
C_{1}^{+}=\sum_{k}(cos2\theta_{k}-\frac{2}{e^{\beta\varepsilon_{k}}+1})$

The first-order irreducible correlation function

$\displaystyle
C_{1}^{+}=\tilde{C}_{1}^{+}=\sum_{k}(cos2\theta_{k}-\frac{2}{e^{\beta\varepsilon_{k}}+1})\stackrel{{\scriptstyle\beta\rightarrow\infty}}{{=}}\sum_{k}cos2\theta_{k}$
(8)

#### The second-order correlation function

$\displaystyle C_{11}^{++}$
$\displaystyle=\sum_{k,k^{\prime}}(cos2\theta_{k}(cos2\theta_{k^{\prime}}-\frac{2}{e^{\beta\varepsilon_{k^{\prime}}}+1})-\frac{2}{e^{\beta\varepsilon_{k}}+1}(cos2\theta_{k^{\prime}}-\frac{2}{e^{\beta\varepsilon_{k^{\prime}}}+1}))$
$\displaystyle+\sum_{k}(sin^{2}2\theta_{k}(cos(2\varepsilon_{k}(t_{1}-t_{2}))\frac{1}{(e^{\beta\varepsilon_{k}}+1)^{2}}+cos(2\varepsilon_{k}(t_{1}-t_{2}))(\frac{1}{e^{\beta\varepsilon_{k}}+1}+1)^{2})$
$\displaystyle\stackrel{{\scriptstyle\beta\rightarrow\infty}}{{=}}\sum_{k,k^{\prime}}cos2\theta_{k}cos2\theta_{k^{\prime}}+\sum_{k}cos(2\varepsilon_{k}(t_{1}-t_{2}))$

The second-order irreducible correlation function

$\displaystyle\tilde{C}_{11}^{++}=\sum_{k}cos(2\varepsilon_{k}(t_{1}-t_{2}))(\frac{1}{e^{\beta\varepsilon_{k}}+1}+1)^{2}\stackrel{{\scriptstyle\beta\rightarrow\infty}}{{=}}\sum_{k}cos(2\varepsilon_{k}(t_{1}-t_{2}))$
(9)

#### The third-order correlation function

$\displaystyle\tilde{C}_{111}^{+++}$
$\displaystyle\stackrel{{\scriptstyle\beta\rightarrow\infty}}{{=}}-\sum_{k_{1}}sin^{2}2\theta_{k_{1}}[[1-\theta(t_{3}-t_{1})\theta(t_{1}-t_{2})-\theta(t_{1}-t_{3})\theta(t_{3}-t_{2})]cos(2\varepsilon_{k_{1}}(t_{1}-t_{3}))$
$\displaystyle+[1-\theta(t_{2}-t_{1})\theta(t_{1}-t_{3})-\theta(t_{1}-t_{2})\theta(t_{2}-t_{3})]cos(2\varepsilon_{k_{1}}(t_{1}-t_{2}))$
$\displaystyle+[1-\theta(t_{3}-t_{2})\theta(t_{2}-t_{1})-\theta(t_{2}-t_{3})\theta(t_{3}-t_{1})]cos(2\varepsilon_{k_{1}}(t_{2}-t_{3}))]$
(10)

where $tan(2\theta_{k})=sink/(cosk-\lambda)$ and
$\varepsilon_{k}=2\sqrt{1-2\lambda cosk+\lambda^{2}}$, Applied
$sin(2\theta_{k})=sink\text{/$\sqrt{1-2\lambda cosk+\lambda^{2}}$}$,
$cos(2\theta_{k})=(cosk-\lambda)/\sqrt{1-2\lambda cosk+\lambda^{2}}$. See
Appendix B for the derivation of the third-order correlation function in
detail.

Fig. 1. $N=10000,g=0.01$, Under weak coupling, the first three order
correlation function changes with the external field intensity.
(a)$\lambda=0.0$,(b)$\lambda=0.5$,(c)phase transition
point$\lambda=1.0$,(d)$\lambda=2.0$

Firstly, qubit is approximated to the weak coupling of $G=0.01$in the
thermodynamic limit. With the increase of external magnetic field $\lambda$,
the bath system transfers from ferromagnetic phase$0.0\leqslant\lambda<1.0$ to
paramagnetic phase $\lambda>1.0$. In $\lambda=1.0$,the system underwent a
quantum phase transition. As shown in Figure 1(c), at this phase point, the
total correlation function of the system diverges, because that the
correlation length of the system tends to infinity. Specifically, the odd-
order of the obtained correlation function diverges, and only the second-order
correlation function remains limited. In the weak field condition, the first
two order correlation functions shows the feasibility of Markov approximation
that are always larger than the third-order, with time evolving.

The proportion of the third order correlation function is increasing with
time, but it always contributes less to the total correlation function.It can
be foreseen that, with the evolution time continues, it will become more and
more important for the proportion of the correlation function above the second
order. Therefore, when the time evolution of the decoherence process is
relatively long, The Markov approximation faces the dilemma of too large
accumulation error. And this change is independent of the phase in which the
system is located, as shown in FIG. 1(b) and 1(d).

Fig. 2. $N=10000,g=1.0$,Under strong coupling, the first three order
correlation function changes with the external field intensity.
(a)$\lambda=0.0$,(b)$\lambda=0.5$,(c)phase transition
point$\lambda=1.0$,(d)$\lambda=2.0$ Under the same parameters, the coupling
strength of the system is increased to $g=1.0$ ,becoming a strongly coupled
system. FIG. 2 is consistent with FIG. 1 for the phase transformation process
of the system. However, the difference is that the third-order correlation
function, that is, the non-Markov term, exceeds the second-order correlation
function in a relatively short time. This shows that in the case of strong
coupling, non-Markov terms will become important.

Fig. 3. $N=10000,\lambda=0.97$,Near phase transition point$\lambda=1.0$,The
first three order changes.(a)$g=0.01$,(b)$g=1.0$

Fig. 4. $N=10000$,The three order
changes.(a)$\lambda=0.0$,(b)$\lambda=0.5$,(c)$\lambda=0.97$,(d)$\lambda=2.0$

Since the correlation function diverges at the phase transition point
$\lambda=1.0$, it is difficult to explore the effect of phase transition on
the system. So in FIG. 3, the state of the system immediately adjacent to the
phase transition point $\lambda=0.97$is applied. First, as before, non-Markov
terms become important as the coupling strength increases. Moreover, when the
phase transition point is approached, the quantum interference effect
disappears, showing a smooth relationship, indicating that the phase
transition is a statistical effect rather than a quantum effect.

In FIG. 4, it can be clearly seen that with the increase of coupling strength
$G$of the system, non-Markov term modes become larger and larger, and this
growth almost linearly increases with the increase of coupling strength. When
the system is in different phase states, the non-Markov terms have different
trends with the change of coherent time. In ferromagnetic and paramagnetic
states, far from the phase transition point, the non-Markov term has quantum
interference effect. There is a monotone increase as we approach the phase
transition.

##  IV Conclusion

We study the effect of bath on the decoherence of qubit, which mainly refers
to the contribution of the high-order correlation function of 1D transverse
Ising model.The larger the coupling strength of the quantum many-body system,
the more obvious the effect of the higher order correlation function.The
longer coherent time of the qubit in the quantum many-body bath, the more
obvious the effect of the higher order correlation function.The closer the
quantum many-body system is to the phase transition point, the more obvious
the effect of the higher order correlation function is.At the phase transition
point, the higher-order correlation function should also mutate.

##  Appendix A The exact solution of the transverse field Ising model

$\displaystyle
H=H_{S}+H_{b}+g\sigma_{Z}B=|\uparrow\rangle\langle\uparrow|\otimes
H_{+}+|\downarrow\rangle\langle\downarrow|\otimes H_{-}$ (11)

Where$H_{+}=(\omega_{0}+H_{b}+gB)$ and $H_{-}=(-\omega_{0}+H_{b}-gB)$ At the
same times,

$\displaystyle\rho(0)=\rho^{S}(0)\otimes\rho^{B}(0)and\rho^{B}(0)=\frac{1}{Z}e^{-\beta
H_{0}}$ (12)

Secondly

$\displaystyle\rho^{S}(t)=tr_{b}(\rho(t))=tr_{b}(U(t,0)\rho(0)U^{+}(t,0))$
(13)

Finally

$\displaystyle
U(t,0)=e^{-itH}=|\uparrow\rangle\langle\uparrow|e^{-itH_{+}}+|\downarrow\rangle\langle\downarrow|e^{-itH_{-}}$
(14)

So

$\displaystyle\rho_{\downarrow\uparrow}^{S}(t)=\langle\uparrow|tr_{b}(\rho(t))|\downarrow\rangle=c\cdot
tr_{b}[e^{-itH_{+}}\frac{1}{Z}e^{-\beta H_{b}}e^{itH_{-}}]$ (15)

when temperature is zero $\beta\rightarrow\infty$,we have

$\displaystyle\rho_{\downarrow\uparrow}^{S}(t)$
$\displaystyle=\langle\uparrow|tr_{b}(\rho(t))|\downarrow\rangle$ (16)
$\displaystyle=c\cdot e^{-it2\omega_{0}}\langle
g|e^{-it(H_{b}+gB)}e^{it(H_{b}-gB)}|g\rangle$

The bases of Hilbert space

$\displaystyle\\{\tilde{|0_{-k},0_{k}\rangle},\tilde{|1_{-k},1_{k}\rangle},\tilde{|0_{-k},1_{k}\rangle},\tilde{|1_{-k},0_{k}\rangle}\\}$
(17)

Under those bases

$\displaystyle H_{b}$
$\displaystyle=\left[\begin{array}[]{cccc}-\frac{1}{2}\sum_{k^{\prime}}\varepsilon_{k^{\prime}}&0&0&0\\\
0&-\frac{1}{2}\sum_{k^{\prime}}(-1)^{\delta_{k^{\prime},\pm
k}}\varepsilon_{k^{\prime}}&0&0\\\
0&0&-\frac{1}{2}\sum_{k^{\prime}}(1-\delta_{k^{\prime},\pm
k})\varepsilon_{k^{\prime}}&0\\\
0&0&0&-\frac{1}{2}\sum_{k^{\prime}}(1-\delta_{k^{\prime},\pm
k})\varepsilon_{k^{\prime}}\end{array}\right]$
$\displaystyle=H_{b}^{k}-\frac{1}{2}\sum_{k^{\prime}\neq\pm
k}\varepsilon_{k^{\prime}}$

So

$\displaystyle B$
$\displaystyle=\left[\begin{array}[]{cccc}\sum_{k^{\prime}}cos2\theta_{k^{\prime}}&isin2\theta_{k}\cdot
e^{i2\varepsilon_{k}t}&0&0\\\ -isin2\theta_{k}\cdot
e^{-i2\varepsilon_{k}t}&\sum_{k^{\prime}}cos2\theta_{k^{\prime}}-4&0&0\\\
0&0&\sum_{k^{\prime}}cos2\theta_{k^{\prime}}-2&0\\\
0&0&0&\sum_{k^{\prime}}cos2\theta_{k^{\prime}}-2\end{array}\right]$
$\displaystyle=B^{k}+\sum_{k^{\prime}}cos2\theta_{k^{\prime}}$

So

$\displaystyle\rho_{\downarrow\uparrow}^{S}(t)$
$\displaystyle=\langle\uparrow|tr_{b}(\rho(t))|\downarrow\rangle$
$\displaystyle=c\cdot e^{-it2\omega_{0}}\langle
g|e^{-it(H_{b}+gB)}e^{it(H_{b}-gB)}|g\rangle$ $\displaystyle=c\cdot
e^{-it2(\omega_{0}+g\sum_{k^{\prime}}cos2\theta_{k^{\prime}})}\prod_{k>0}\langle
g|U_{k}^{\dagger}(t)U_{k}(t)|g\rangle$

So

$\displaystyle U_{k}^{\dagger}(t)U_{k}(t)$
$\displaystyle=e^{-it(H_{b}^{k}+gB^{k})}e^{it(H_{b}^{k}-gB^{k})}$
$\displaystyle=\left[\begin{array}[]{cc}e^{-itM_{1}}\cdot e^{itM_{2}}&0\\\
0&e^{i4tg}I\end{array}\right]$

Where

$\displaystyle e^{-itM_{1}}\cdot e^{itM_{2}}=e^{4itg}\cdot
e^{-it\vec{a}\cdot\vec{\sigma}}\cdot e^{it\vec{b}\cdot\vec{\sigma}}$

where
$\vec{a}=(-sin2\theta_{k}sin(2\varepsilon_{k}t)g,-sin2\theta_{k}cos(2\varepsilon_{k}t)g,(2g-\varepsilon_{k})),\\\
\vec{b}=(sin2\theta_{k}sin(2\varepsilon_{k}t)g,sin2\theta_{k}cos(2\varepsilon_{k}t)g,(-2g-\varepsilon_{k}))$and
$\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ is vector of Puli matrix.

$\displaystyle e^{-itM_{1}}\cdot e^{itM_{2}}$ $\displaystyle=e^{4itg}\cdot
e^{-it\vec{a}\cdot\vec{\sigma}}\cdot e^{it\vec{b}\cdot\vec{\sigma}}$
$\displaystyle=e^{4itg}\cdot\left[\begin{array}[]{cc}A&B\\\
C&D\end{array}\right]$

Where

$\displaystyle A$
$\displaystyle=cos(ta)cos(tb)+(\varepsilon_{k}^{2}-sin^{2}2\theta_{k})\frac{sin(ta)sin(tb)}{ab}+i\varepsilon_{k}[\frac{sin(ta)cos(tb)}{a}-\frac{cos(ta)sin(tb)}{b}]$
$\displaystyle B$ $\displaystyle=sin2\theta_{k}\cdot
e^{i2\varepsilon_{k}t}[\frac{cos(ta)sin(tb)}{b}+\frac{sin(ta)cos(tb)}{a}+i2(\varepsilon_{k}+2)\frac{sin(ta)sin(tb)}{ab}]$
$\displaystyle C$ $\displaystyle=-sin2\theta_{k}\cdot
e^{-i2\varepsilon_{k}t}[\frac{sin(ta)cos(tb)}{a}+\frac{cos(ta)sin(tb)}{b}-i2(\varepsilon_{k}-2)\frac{sin(ta)sin(tb)}{ab}]$
$\displaystyle D$
$\displaystyle=cos(ta)cos(tb)+(\varepsilon_{k}^{2}-16-sin^{2}2\theta_{k})\frac{sin(ta)sin(tb)}{ab}+i[(\varepsilon_{k}+4)\frac{sin(tb)cos(ta)}{a}$
$\displaystyle-(\varepsilon_{k}-4)\frac{cos(tb)sin(ta)}{b}]$

Where
$a=\sqrt{(gsin2\theta_{k})^{2}+(2g-\varepsilon_{k})^{2}},b=\sqrt{(gsin2\theta_{k})^{2}+(2g+\varepsilon_{k})^{2}}$
As a same times

$\displaystyle\rho_{\downarrow\uparrow}^{S}(t)=e^{\Gamma(t)}\rho_{\downarrow\uparrow}^{S}(0)$
(18)

So

$\displaystyle
e^{\Gamma(t)}=\prod_{k>0}[cos(ta)cos(tb)+(\varepsilon_{k}^{2}-sin^{2}2\theta_{k})\frac{sin(ta)sin(tb)}{ab}+i\varepsilon_{k}[\frac{sin(ta)cos(tb)}{a}-\frac{cos(ta)sin(tb)}{b}]]$
(19)

So

$\displaystyle\Gamma(t)=\sum_{k>0}In(cos(ta)cos(tb)+(\varepsilon_{k}^{2}-sin^{2}2\theta_{k})\frac{sin(ta)sin(tb)}{ab}+i\varepsilon_{k}[\frac{sin(ta)cos(tb)}{a}-\frac{cos(ta)sin(tb)}{b}]\text{)}$
(20)

Where
$a=\sqrt{(gsin2\theta_{k})^{2}+(2g-\varepsilon_{k})^{2}},b=\sqrt{(gsin2\theta_{k})^{2}+(2g+\varepsilon_{k})^{2}}$and
$\varepsilon_{k}=2\sqrt{1-2\lambda cosk+\lambda^{2}}$and
$sin2\theta_{k}=sink\text{/$\sqrt{1-2\lambda cosk+\lambda^{2}}$}$and
$k=\frac{(2l-1)\pi}{N},l=1,2,\cdots\frac{N}{2}$

##  Appendix B The third order correlation function

$\displaystyle Part$
$\displaystyle=tr(B_{1}(t_{3})B_{1}(t_{2})B_{1}(t_{1})\rho^{B}(0))$
$\displaystyle=\sum_{k_{3}k_{2}k_{1}}(cos2\theta_{k_{3}}-2\frac{1}{e^{\beta\varepsilon_{k_{3}}}+1})(cos2\theta_{k_{2}}-2\frac{1}{e^{\beta\varepsilon_{k_{2}}}+1})(cos2\theta_{k_{1}}-2\frac{1}{e^{\beta\varepsilon_{k_{1}}}+1})$
$\displaystyle+\sum_{k_{3}k_{1}}(cos2\theta_{k_{3}}-2\frac{1}{e^{\beta\varepsilon_{k_{3}}}+1})(sin^{2}2\theta_{k_{1}}\cdot[\frac{1}{e^{\beta\varepsilon_{k_{1}}}+1}\frac{1}{e^{\beta\varepsilon_{-k_{1}}}+1}\cdot$
$\displaystyle
exp(i2\varepsilon_{k_{1}}(t_{1}-t_{2}))+(\frac{1}{e^{\beta\varepsilon_{k_{1}}}+1}+1)(\frac{1}{e^{\beta\varepsilon_{-k_{1}}}+1}+1)\cdot
exp(-i2\varepsilon_{k_{1}}(t_{1}-t_{2}))])$
$\displaystyle+\sum_{k_{2}k_{1}}(sin^{2}2\theta_{k_{1}}cos2\theta_{k_{2}}\cdot\frac{1}{e^{\beta\varepsilon_{k_{1}}}+1}\frac{1}{e^{\beta\varepsilon_{-k_{1}}}+1}\cdot[exp(i2(\varepsilon_{k_{1}}(t_{1}-t_{3}))+exp(-i2\varepsilon_{k_{1}}(t_{1}-t_{3}))])$
$\displaystyle+\sum_{k_{2}k_{1}}(-2sin^{2}2\theta_{k_{1}}\cdot((\frac{1}{e^{\beta\varepsilon_{k_{2}}}+1}-1)\cdot\frac{1}{e^{\beta\varepsilon_{-k_{1}}}+1}\frac{1}{e^{\beta\varepsilon_{k_{1}}}+1})\cdot
exp(i2(\varepsilon_{k_{1}}(t_{1}-t_{3})))$
$\displaystyle+\sum_{k_{2}k_{1}}(sin^{2}2\theta_{k_{2}}\cdot[\frac{1}{e^{\beta\varepsilon_{k_{2}}}+1}\frac{1}{e^{\beta\varepsilon_{-k_{2}}}+1}\cdot
exp(i2\varepsilon_{k_{2}}(t_{2}-t_{3}))+(\frac{1}{e^{\beta\varepsilon_{k_{2}}}+1}+1)(\frac{1}{e^{\beta\varepsilon_{-k_{2}}}+1}+1)\cdot$
$\displaystyle
exp(-i2\varepsilon_{k_{2}}(t_{2}-t_{3}))]\cdot(cos2\theta_{k_{1}}-2\frac{1}{e^{\beta\varepsilon_{k_{1}}}+1})$
$\displaystyle+\sum_{k_{2}k_{1}}(-2sin^{2}2\theta_{k_{1}}\cdot(1+\frac{1}{e^{\beta\varepsilon_{k_{2}}}+1})(1+\frac{1}{e^{\beta\varepsilon_{k_{1}}}+1})(1+\frac{1}{e^{\beta\varepsilon_{-k_{1}}}+1})\cdot
exp(-i2(\varepsilon_{k_{1}}(t_{1}-t_{3})))$
$\displaystyle\stackrel{{\scriptstyle\beta\rightarrow\infty}}{{=}}\sum_{k_{3}k_{2}k_{1}}[cos2\theta_{k_{3}}cos2\theta_{k_{2}}cos2\theta_{k_{1}}+cos2\theta_{k_{3}}sin^{2}2\theta_{k_{1}}\cdot$
$\displaystyle
exp(-i2\varepsilon_{k_{1}}(t_{1}-t_{2}))+cos2\theta_{k_{1}}sin^{2}2\theta_{k_{2}}\cdot
exp(-i2\varepsilon_{k_{2}}(t_{2}-t_{3}))-2sin^{2}2\theta_{k_{1}}\cdot
exp(-i2(\varepsilon_{k_{1}}(t_{1}-t_{3}))]$

$\displaystyle Part\stackrel{{\scriptstyle\beta\rightarrow\infty}}{{=}}$
$\displaystyle\sum_{k_{3}k_{2}k_{1}}[cos2\theta_{k_{3}}cos2\theta_{k_{2}}cos2\theta_{k_{1}}+cos2\theta_{k_{3}}sin^{2}2\theta_{k_{1}}\cdot
exp(-i2\varepsilon_{k_{1}}(t_{1}-t_{2}))$
$\displaystyle+cos2\theta_{k_{1}}sin^{2}2\theta_{k_{2}}\cdot
exp(-i2\varepsilon_{k_{2}}(t_{2}-t_{3}))-2sin^{2}2\theta_{k_{1}}\cdot
exp(-i2(\varepsilon_{k_{1}}(t_{1}-t_{3}))]$ (21)

$\displaystyle C_{111}^{+++}$
$\displaystyle=\frac{1}{2^{2}}[[1-\theta(t_{3}-t_{1})\theta(t_{1}-t_{2})-\theta(t_{1}-t_{3})\theta(t_{3}-t_{2})][Part+Part(t_{1}\leftrightarrow
t_{3})]$
$\displaystyle+[1-\theta(t_{2}-t_{1})\theta(t_{1}-t_{3})-\theta(t_{1}-t_{2})\theta(t_{2}-t_{3})][Part(t_{1}\rightarrow
t_{2},t_{2}\rightarrow t_{3},t_{3}\rightarrow t_{1})+Part(t_{2}\leftrightarrow
t_{3})]$
$\displaystyle+[1-\theta(t_{3}-t_{2})\theta(t_{2}-t_{1})-\theta(t_{2}-t_{3})\theta(t_{3}-t_{1})][Part(t_{2}\leftrightarrow
t_{1})+Part(t_{1}\rightarrow t_{3},t_{2}\rightarrow t_{1},t_{3}\rightarrow
t_{2})]]$
$\displaystyle\overset{\beta\rightarrow\infty}{=}\frac{1}{2^{2}}[[1-\theta(t_{3}-t_{1})\theta(t_{1}-t_{2})-\theta(t_{1}-t_{3})\theta(t_{3}-t_{2})]\times[\sum_{k_{3}k_{2}k_{1}}[2\cdot$
$\displaystyle
cos2\theta_{k_{3}}cos2\theta_{k_{2}}cos2\theta_{k_{1}}+cos2\theta_{k_{3}}sin^{2}2\theta_{k_{1}}\cdot(exp(i2\varepsilon_{k_{1}}(t_{2}-t_{1}))+exp(i2\varepsilon_{k_{1}}(t_{2}-t_{3})))$
$\displaystyle+cos2\theta_{k_{1}}sin^{2}2\theta_{k_{2}}\cdot(exp(i2\varepsilon_{k_{2}}(t_{3}-t_{2}))+exp(i2\varepsilon_{k_{2}}(t_{1}-t_{2})))-2sin^{2}2\theta_{k_{1}}\cdot
2cos(2\varepsilon_{k_{1}}(t_{1}-t_{3}))]$
$\displaystyle+[1-\theta(t_{2}-t_{1})\theta(t_{1}-t_{3})-\theta(t_{1}-t_{2})\theta(t_{2}-t_{3})]$
$\displaystyle\times[\sum_{k_{3}k_{2}k_{1}}[2\cdot
cos2\theta_{k_{3}}cos2\theta_{k_{2}}cos2\theta_{k_{1}}+cos2\theta_{k_{3}}sin^{2}2\theta_{k_{1}}\cdot(exp(i2\varepsilon_{k_{1}}(t_{3}-t_{2}))+exp(i2\varepsilon_{k_{1}}(t_{3}-t_{1})))$
$\displaystyle+cos2\theta_{k_{1}}sin^{2}2\theta_{k_{2}}\cdot(exp(i2\varepsilon_{k_{2}}(t_{1}-t_{3}))+exp(i2\varepsilon_{k_{2}}(t_{2}-t_{3})))-2sin^{2}2\theta_{k_{1}}\cdot
2cos(2\varepsilon_{k_{1}}(t_{1}-t_{2}))]$
$\displaystyle+[1-\theta(t_{3}-t_{2})\theta(t_{2}-t_{1})-\theta(t_{2}-t_{3})\theta(t_{3}-t_{1})]$
$\displaystyle\times[\sum_{k_{3}k_{2}k_{1}}[2\cdot
cos2\theta_{k_{3}}cos2\theta_{k_{2}}cos2\theta_{k_{1}}+cos2\theta_{k_{3}}sin^{2}2\theta_{k_{1}}\cdot(exp(i2\varepsilon_{k_{1}}(t_{1}-t_{2}))+exp(i2\varepsilon_{k_{1}}(t_{1}-t_{3})))$
$\displaystyle+cos2\theta_{k_{1}}sin^{2}2\theta_{k_{2}}\cdot(exp(i2\varepsilon_{k_{2}}(t_{3}-t_{1}))+exp(i2\varepsilon_{k_{2}}(t_{2}-t_{1})))-2sin^{2}2\theta_{k_{1}}\cdot
2cos(2\varepsilon_{k_{1}}(t_{2}-t_{3}))]]$

where $tan(2\theta_{k})=sink/(cosk-\lambda)$ and
$\varepsilon_{k}=2\sqrt{1-2\lambda cosk+\lambda^{2}}$, Applied
$sin(2\theta_{k})=sink\text{/$\sqrt{1-2\lambda cosk+\lambda^{2}}$}$,
$cos(2\theta_{k})=(cosk-\lambda)/\sqrt{1-2\lambda cosk+\lambda^{2}}$.

The irreducible correlation function as follow

$\displaystyle\tilde{C}_{111}^{+++}$
$\displaystyle=\frac{1}{2^{2}}[[1-\theta(t_{3}-t_{1})\theta(t_{1}-t_{2})-\theta(t_{1}-t_{3})\theta(t_{3}-t_{2})][\sum_{k_{1}}-2sin^{2}2\theta_{k_{1}}\cdot
2cos(2\varepsilon_{k_{1}}(t_{1}-t_{3}))]$
$\displaystyle+[1-\theta(t_{2}-t_{1})\theta(t_{1}-t_{3})-\theta(t_{1}-t_{2})\theta(t_{2}-t_{3})][\sum_{k_{1}}-2sin^{2}2\theta_{k_{1}}\cdot
2cos(2\varepsilon_{k_{1}}(t_{1}-t_{2}))]$
$\displaystyle+[1-\theta(t_{3}-t_{2})\theta(t_{2}-t_{1})-\theta(t_{2}-t_{3})\theta(t_{3}-t_{1})][\sum_{k_{1}}-2sin^{2}2\theta_{k_{1}}\cdot
2cos(2\varepsilon_{k_{1}}(t_{2}-t_{3}))]]$
$\displaystyle=-\sum_{k_{1}}sin^{2}2\theta_{k_{1}}[[1-\theta(t_{3}-t_{1})\theta(t_{1}-t_{2})-\theta(t_{1}-t_{3})\theta(t_{3}-t_{2})]cos(2\varepsilon_{k_{1}}(t_{1}-t_{3}))$
$\displaystyle+[1-\theta(t_{2}-t_{1})\theta(t_{1}-t_{3})-\theta(t_{1}-t_{2})\theta(t_{2}-t_{3})]cos(2\varepsilon_{k_{1}}(t_{1}-t_{2}))$
$\displaystyle+[1-\theta(t_{3}-t_{2})\theta(t_{2}-t_{1})-\theta(t_{2}-t_{3})\theta(t_{3}-t_{1})]cos(2\varepsilon_{k_{1}}(t_{2}-t_{3}))]$

So

$\displaystyle\tilde{C}_{111}^{+++}$
$\displaystyle=-\sum_{k_{1}}sin^{2}2\theta_{k_{1}}[[1-\theta(t_{3}-t_{1})\theta(t_{1}-t_{2})-\theta(t_{1}-t_{3})\theta(t_{3}-t_{2})]cos(2\varepsilon_{k_{1}}(t_{1}-t_{3}))$
$\displaystyle+[1-\theta(t_{2}-t_{1})\theta(t_{1}-t_{3})-\theta(t_{1}-t_{2})\theta(t_{2}-t_{3})]cos(2\varepsilon_{k_{1}}(t_{1}-t_{2}))$
$\displaystyle+[1-\theta(t_{3}-t_{2})\theta(t_{2}-t_{1})-\theta(t_{2}-t_{3})\theta(t_{3}-t_{1})]cos(2\varepsilon_{k_{1}}(t_{2}-t_{3}))]$
(22)

## References

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