# Existence and uniqueness of weak solutions to a parabolic nonlocal
1-Laplacian equation

Dingding Li and Chao Zhang∗ Dingding Li School of Mathematics, Harbin
Institute of Technology, Harbin 150001, China<EMAIL_ADDRESS>Chao Zhang
School of Mathematics and Institute for Advanced Study in Mathematics, Harbin
Institute of Technology, Harbin 150001, China<EMAIL_ADDRESS>
###### Abstract.

We consider a class of parabolic nonlocal $1$-Laplacian equation

$\displaystyle u_{t}+(-\Delta)^{s}_{1}u=f\quad\text{ in }\Omega\times(0,T].$

By employing the Rothe time-discretization method, we establish the existence
and uniqueness of weak solutions to the equation above. In particular,
different from the previous results on the local case, we infer that the weak
solution maintains $\frac{1}{2}$-Hölder continuity in time.

###### Key words and phrases:

Weak solutions; existence; nonlocal $1$-Laplace operator; singular parabolic
problem

###### 2020 Mathematics Subject Classification:

35D30, 35R11

∗ Corresponding author.

##  1\. Introduction

Suppose that $\Omega$ is a bounded domain of $\mathbb{R}^{N}(N\geq 2)$ with
smooth boundary $\partial\Omega$ and $T$ is a positive number. Denote
$\Omega_{T}:=\Omega\times(0,T]$. In this paper, we deal with the following
fractional 1-Laplacian problem

$\displaystyle\begin{cases}u_{t}+\left(-\Delta\right)^{s}_{1}u=f&\text{in
}\Omega_{T},\\\ u(x,0)=u_{0}&\text{on }\Omega,\\\ u=0&\text{in
}\mathcal{C}_{\Omega}\times(0,T],\end{cases}$ (1.1)

where $\mathcal{C}_{\Omega}:=\mathbb{R}^{N}\backslash\Omega$. The operator
$(-\Delta)^{s}_{p}$, called the fractional $p$-Laplacian, is given by

$\displaystyle\left(-\Delta\right)^{s}_{p}u(x)=2\text{P.V.}\int_{\mathbb{R}^{N}}\frac{\left|u(x)-u(y)\right|^{p-2}\left(u(x)-u(y)\right)}{|x-y|^{N+sp}}dy,\quad
x\in\Omega$

with $p\in\left[1,+\infty\right)$ and $s\in(0,1)$. P.V. is known as an
abbreviation in the sense of principal value.

Problem (1.1) could be seen as a nonlocal counterpart of the following
1-Laplacian Dirichlet problem

$\displaystyle\begin{cases}\displaystyle u_{t}-\text{div}\left(\frac{\nabla
u}{|\nabla u|}\right)=f&\text{in }\Omega_{T},\\\ u(x,0)=u_{0}&\text{on
}\Omega,\\\ u=0&\text{on }\partial\Omega\times(0,T].\end{cases}$ (1.2)

As to the homogeneous case that $f\equiv 0$ in $\Omega_{T}$, Andrew,
Ballester, Caselles and Mazón in [1] introduced the concept of solutions to
make clear the quotient $\frac{Du}{|Du|}$, even if $Du$ vanishes. Moreover,
they used nonlinear semigroup theory to prove the existence and uniqueness of
solutions and found a vector field $Z\in L^{\infty}(\Omega,\mathbb{R}^{N})$,
satisfying $\|Z\|_{L^{\infty}(\Omega,\mathbb{R}^{N})}\leq 1$ and $(Z,Du)=|Du|$
as measures (see also [2]). For the inhomogeneous case, the nonlinear
semigroup theory was also utilized in [15] when $f\in L^{2}(\Omega_{T})$.
Under the assumption that the source $f$ belongs to
$L^{1}_{\text{loc}}(0,+\infty,L^{2}(\Omega))$, Latorre and Segura de León
demonstrated the existence and uniqueness of solutions in [10], which
retrieved the results of [14] in a correct way. Additional results concerning
problem (1.2) can be found in [3, 5, 4].

Regarding the nonlocal case, it is worth mentioning that the definition of
solutions to (1.1) was originally given by Mazón, Rossi and Toledo in [11],
where they showed that a suitable function $Z\in
L^{\infty}\left(\mathbb{R}^{N}\times\mathbb{R}^{N}\times[0,T]\right)$ should
be found to interpret and describe the fractional ratio
$\frac{u(x,t)-u(y,t)}{|u(x,t)-u(y,t)|}$ and studied the existence of solutions
to homogeneous parabolic fractional 1-Laplacian equation. Different from the
local case, the function $Z$ describes $\frac{u(x,t)-u(y,t)}{|u(x,t)-u(y,t)|}$
in the set
$\left\\{\mathbb{R}^{N}\times\mathbb{R}^{N}\times[0,T]\big{|}u(x,t)\neq
u(y,t)\right\\}$ precisely and $Z\in[-1,1]$ in the set
$\left\\{\mathbb{R}^{N}\times\mathbb{R}^{N}\times[0,T]\big{|}u(x,t)=u(y,t)\right\\}$.
In terms of elliptic equations, the existence of $(s,1)$-harmonic function was
studied in [7]. In addition, by imposing the condition that $f$ belongs to
$L^{\frac{N}{s}}(\Omega)$ and its norm is sufficiently small, Bucur
investigated in [6] the minimizer of the following functional

$\displaystyle\mathcal{F}^{s}_{1}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|}{|x-y|^{N+s}}dxdy-\int_{\Omega}fudx$

and showed that this minimizer corresponds to the weak solution of the
nonlocal 1-Laplacian problem. On the other hand, when
$\|f\|_{L^{\frac{N}{s}}(\Omega)}>\frac{1}{S_{N,s}}$ the energy could be
unbounded and the minimizer may not exist in this case. What’s more, the
flatness results for the minimizer were obtained, which implies that the
expression of $Z$ towards $\frac{u(x)-u(y)}{|u(x)-u(y)|}$ is not precise. We
also refer to [12] for an interesting result, in which Novage and Onoue
studied the minimizers of a nonlocal variational problem concerning the image-
denoising model

$\displaystyle\text{min}\left\\{\mathcal{F}_{K,f}(u)\big{|}u\in
BV_{K}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N})\right\\},$

where $\mathcal{F}_{K,f}$ is given by

$\displaystyle\mathcal{F}_{K,f}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}K(|x-y|)|u(x)-u(y)|dxdy+\frac{1}{2}\int_{\mathbb{R}^{N}}\left(u(x)-f(x)\right)^{2}dx$

with $K$ being a kernel singular at the origin. A typical example is that
$K(x)=\frac{1}{|x|^{N+s}}$. In two dimensions, by assuming the original image
is locally $\beta$-Hölder continuous with $\beta\in(1-s,1]$, they showed the
minimizers have the same local Hölder regularity.

Finally, we would like to turn to the method of Rothe time-discretization,
also called the minimizing movements method or difference method, which is
usually used to prove the existence of solutions to parabolic problems. We
refer to [16, 17, 8, 18] for the application in terms of showing the existence
of weak solutions of nonlinear parabolic problems and refer to [13] for the
application in terms of finding variational solutions.

Motivated by the results mentioned above, our main aim in the present paper is
to study the existence and uniqueness of weak solutions to the nonlocal
parabolic 1-Laplacian problem (1.1). Before we begin our proof of the main
result, it is crucial to demonstrate the equivalence between the minimizers
and the weak solutions to the following problem

$\displaystyle\begin{cases}\dfrac{u-u_{0}}{h}+\left(-\Delta\right)^{s}_{1}u=f&\text{in
}\Omega,\\\ u=0&\text{in }\mathcal{C}_{\Omega},\end{cases}$ (1.3)

where $f\in L^{2}(\Omega)$. However, when dealing with problem (1.1), it is
necessary to assume $f\in L^{\frac{N}{s}}(\Omega_{T})$ to show that the weak
solution belongs to $L^{\infty}(0,T;L^{2}(\Omega))$. Similar to the method of
difference, a function $u$ could be found and proved to be the solution of
(1.1) and a function $Z$ may be found to replace
$\frac{u(x,t)-u(x,y)}{|u(x,t)-u(y,t)|}$ in the weak formulation.

From the purely mathematical point of view, the super-linear growth condition
on the operator seems to be of great significance because a convergence result
is indispensable (see [18, Lemma 2.5] and [8, Lemma 2.6]). In the context of
the $1$-Laplacian equation, diverging from the classic approach, we have to
use an alternative method to derive the convergence result concerning $u$,
provided that the initial value $u_{0}$ possesses finite energy. In other
words, $u_{0}$ not only should belong to $L^{2}(\Omega)$, but also should lie
in the fractional Sobolev space $W^{s,1}_{0}(\Omega)$. Additionally, it should
be emphasized that based on the uniform boundedness result, one can expect $u$
to be $\frac{1}{2}$-Hölder continuous in time.

Consider the set-value sign function sgn given by

$\displaystyle\text{sgn}(s)=\begin{cases}1&\text{if }s>0,\\\ [-1,1]&\text{if
}s=0,\\\ -1&\text{if }s<0.\end{cases}$

We introduce the weak solution to such a singular parabolic problem.

######  Definition 1.1.

A function $u\in C^{0,\frac{1}{2}}\left([0,T];L^{2}(\Omega)\right)\cap
L^{\infty}(0,T;W^{s,1}_{0}(\Omega))$ is said to be a weak solution to problem
(1.1) if there exists a function $Z\in
L^{\infty}(\mathbb{R}^{N}\times\mathbb{R}^{N}\times[0,T])$ satisfying:

  * (1)

For all $\varphi\in C^{1}([0,T];L^{2}(\Omega))\cap
C([0,T];W^{s,1}_{0}(\Omega))$ with $\varphi(\cdot,T)=0$, there holds

$\displaystyle\begin{aligned}
&\quad-\int_{\Omega}u_{0}\varphi(x,0)dx-\int_{0}^{T}\int_{\Omega}u\varphi_{t}dxdt+\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}dxdydt\\\
&=\int_{0}^{T}\int_{\Omega}f\varphi dxdt;\end{aligned}$ (1.4)

  * (2)

$Z(x,y,t)\in\mathrm{sgn}(u(x,t)-u(y,t))$ is an anti-symmetry function with
respect to $(x,y)$.

######  Remark 1.2.

By an approximation argument, the weak solution $u$ could be chosen as a test
function in (1.4).

We are now in a position to state the main result as follows.

######  Theorem 1.3.

Under the assumption that $f\in L^{\frac{N}{s}}(\Omega_{T})$ and $u_{0}\in
L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$, there exists a weak solution to
problem (1.1) in the sense of Definition 1.1.

This paper is organized as follows. In Section 2, we state some notations
regarding the fractional Sobolev spaces and give some crucial lemmas. Section
3 is devoted to proving that the minimizers of a functional are equivalent to
the weak solutions to problem (1.3). We will prove our main result in Section
4.

##  2\. Preliminaries

In this section, we introduce some notations that will be used later. We say
that $\left\\{u_{p}\right\\}$ is a sequence and consider the subsequence of
it. We denote by $|z|$ the Euclidean norm of $z\in\mathbb{R}^{N}$. If not
otherwise specified, we denote by $C$ several constants whose value may change
from line to line even if in the same inequality.

For a function $f\in L^{q}(\Omega_{T})$ with $q\in[1,+\infty)$, we denote by

$\displaystyle[f]_{h}(x,t):=\frac{1}{h}\int_{t}^{t+h}f(x,\mu)d\mu$

the Steklov average of $f$ and we have the following conclusion.

######  Lemma 2.1.

Suppose a function $f$ belongs to $L^{q}(\Omega_{T})$ with $q\in[1,+\infty)$,
there holds

$\displaystyle[f]_{h}\rightarrow f\quad\text{in
}L^{q}(\Omega_{T})\quad(h\rightarrow 0).$

For $0<s<1$ and $1\leq p<+\infty$, the fractional Sobolev space
$W^{s,p}(\mathbb{R}^{N})$ is defined as

$\displaystyle\left\\{u\in
L^{p}(\mathbb{R}^{N})\bigg{|}[u]_{W^{s,p}(\mathbb{R}^{N})}:=\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)^{\frac{1}{p}}<+\infty\right\\}.$

For a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, we denote by
$W^{s,p}_{0}(\Omega)$ a fractional Sobolev space, which is

$\displaystyle W^{s,p}_{0}(\Omega):=\left\\{u\in
W^{s,p}(\mathbb{R}^{N})\big{|}u=0\text{ in
}\mathbb{R}^{N}\backslash\Omega\right\\}.$

We refer to [9] for more information about those fractional Sobolev spaces and
emphasize that the space $L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$ is a closed
and convex set.

Under the fractional framework, we define the following functionals

$\displaystyle\Phi^{s}_{p}(u):=\frac{1}{2p}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy,$

and

$\displaystyle
J^{s}_{p}(u):=\frac{2}{hp}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy+\int_{\Omega}\left(\frac{u-u_{0}}{h}-f\right)^{2}dx,$

where $0<s<1$, $1\leq p<+\infty$, and $h>0$ are fixed constants.

To begin with, we prove the existence and uniqueness of weak solutions to the
following elliptic problem

$\displaystyle\begin{cases}\dfrac{u-u_{0}}{h}+\left(-\Delta\right)^{s}_{p}u=f&\text{in
}\Omega,\\\ u=0&\text{in }\mathcal{C}_{\Omega},\end{cases}$ (2.1)

where $u_{0},f\in L^{2}(\Omega)$, and $h>0,p>1$ are fixed constants.

######  Definition 2.2.

A function $u\in L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$ is said to be a weak
solution to problem (2.1) if for every $\varphi\in L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)$, there holds

$\displaystyle\begin{aligned} &\quad\int_{\Omega}\frac{u-u_{0}}{h}\varphi
dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{N+sp}}dxdy\\\
&=\int_{\Omega}f\varphi dx.\end{aligned}$ (2.2)

######  Lemma 2.3.

There exists a unique weak solution $u\in L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)$ to problem (2.1).

###### Proof.

Consider the variational problem

$\displaystyle\min\left\\{J^{s}_{p}(v)\big{|}v\in L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)\right\\}.$

We will show that $J^{s}_{p}(v)$ has a minimizer $u$ in $L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)$ and then prove that this minimizer is a weak solution to
problem (2.1).

We first show that $J^{s}_{p}$ is coercive on $L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)$. For any $v\in L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$,

$\displaystyle J^{s}_{p}(v)$
$\displaystyle:=\frac{2}{hp}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{N+sp}}dxdy+\int_{\Omega}\left(\frac{v-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle\
\geq\frac{2}{hp}[v]^{p}_{W^{s,p}_{0}(\Omega)}+\left(\bigg{\|}\frac{v}{h}\bigg{\|}_{L^{2}(\Omega)}-\bigg{\|}\frac{u_{0}}{h}\bigg{\|}_{L^{2}(\Omega)}-\|f\|_{L^{2}(\Omega)}\right)^{2},$

by letting $\|v\|_{L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)}$ goes to infinity,
we have $J^{s}_{p}(v)\rightarrow+\infty$.

Next, we will show the existence of minimizers of $J^{s}_{p}$. Because of

$\displaystyle 0\leq\inf_{v\in L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)}J^{s}_{p}(v)\leq
J^{s}_{p}(0)=\int_{\Omega}\left(f+\frac{u_{0}}{h}\right)^{2}dx,$

we could find a minimizing sequence $\left\\{v_{m}\right\\}\subset
L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$, satisfying

$\displaystyle\lim\limits_{m\rightarrow+\infty}J^{s}_{p}(v_{m})=\inf_{v\in
L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)}J^{s}_{p}(v).$

Thus there exists a constant $C$ such that

$\displaystyle\frac{2}{hp}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|v_{m}(x)-v_{m}(y)|^{p}}{|x-y|^{N+sp}}dxdy+\int_{\Omega}\left(\frac{v_{m}-u_{0}}{h}-f\right)^{2}dx\leq
C,$

which implies that

$\displaystyle\|v_{m}\|_{W^{s,p}_{0}(\Omega)}\leq
C[u]_{W^{s,p}_{0}(\Omega)}\leq C(f,u_{0})$

and

$\displaystyle\|v_{m}\|_{L^{2}(\Omega)}\leq C(f,u_{0}).$

Therefore, there exists a function $u\in L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)$ to which $\left\\{u_{m}\right\\}$, up to a subsequence,

$\displaystyle v_{m}\rightharpoonup u\quad\text{weakly in }L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega).$

Combining with the fact that $J^{s}_{p}$ is weakly lower semi-continuous on
$L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$, we obtain

$\displaystyle\inf_{v\in L^{2}(\Omega)\cap
W^{s,p}_{0}(\Omega)}J^{s}_{p}(v)\leq
J^{s}_{p}(u)\leq\liminf_{m\rightarrow+\infty}J^{s}_{p}(v_{m})=\inf_{v\in
L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)}J^{s}_{p}(v).$

It follows that $u\in L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$ is a minimizer of
the functional $J^{s}_{p}$.

To see that the minimizer is a weak solution to problem (2.1), we may perturb
$u$ with test function $\varphi\in L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$ and
use that the first variation of the energy vanishes to deduce $u$ is a weak
solution, i.e.,

$\displaystyle 0$
$\displaystyle=\frac{d}{dt}J^{s}_{p}(u+t\varphi)\bigg{|}_{t=0}$
$\displaystyle=\frac{d}{dt}\left[\frac{2}{hp}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)+t\varphi(x)-u(y)-t\varphi(y)|^{p}}{|x-y|^{N+sp}}dxdy+\int_{\Omega}\left(\frac{u+t\varphi-
u_{0}}{h}-f\right)^{2}dx\right]\Bigg{|}_{t=0}$
$\displaystyle=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{N+sp}}dxdy$
$\displaystyle\quad+\frac{2}{h}\int_{\Omega}\left(\frac{u-u_{0}}{h}-f\right)\varphi
dx.$

Finally, suppose that there is a function $\tilde{u}\neq u$, belonging to $\in
L^{2}(\Omega)\cap W^{s,p}_{0}(\Omega)$ and satisfying the weak formulation
(2.2), we have

$\displaystyle\int_{\Omega}\frac{(u-\tilde{u})^{2}}{h}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{U(x,y)}{|x-y|^{N+sp}}dxdy=0,$
(2.3)

where $U(x,y)$ is defined as follows

$\displaystyle U(x,y):=$
$\displaystyle\left(|u(x)-u(y)|^{p-2}(u(x)-u(y)-|\tilde{u}(x)-\tilde{u}(y)|^{p-2}(\tilde{u}(x)-\tilde{u}(y)))\right)$
$\displaystyle\times(u(x)-\tilde{u}(x)-u(y)+\tilde{u}(y)).$

Since the two terms on the left-hand side of (2.3) are nonnegative, we have
$u=\tilde{u}$ a.e. in $\Omega$. Therefore we obtain the uniqueness of weak
solutions. ∎

Next, we list two lemmas regarding the functional $\Phi^{s}_{p}$ and
fractional Sobolev space. The proofs can be found in [6].

######  Lemma 2.4.

For fixed $0<s<1$, set $s_{p}:=N+s-\frac{N}{p}\in(s,1)$, then for all $u\in
W^{s_{p},p}_{0}(\Omega)$, there holds that

$\displaystyle[u]_{W^{s,1}_{0}(\Omega)}\leq
C^{\frac{1-p}{p}}[u]_{W^{s_{p},p}_{0}(\Omega)},$

where $C$ depends on $N,s,\Omega$.

######  Lemma 2.5.

For fixed $0<s<1$ and
$1<q<\min\left\\{\frac{N}{N+s-1},\frac{N+1}{N+s}\right\\}$, set
$s_{q}:=N+s-\frac{N}{q}\in(s,1)$, if $u\in W^{s_{q},q}_{0}(\Omega)$, it holds
that

$\displaystyle\lim\limits_{p\rightarrow
1}\Phi^{s_{p}}_{p}(u)=\Phi^{s}_{1}(u).$

##  3\. Results on elliptic equations

In this section, we address problem (1.3), that is,

$\displaystyle\begin{cases}\dfrac{u-u_{0}}{h}+\left(-\Delta\right)^{s}_{1}u=f&\text{in
}\Omega,\\\ u=0&\text{in }\mathcal{C}_{\Omega},\end{cases}$

demonstrating the existence of a minimizers for $J^{s}_{1}$ on
$L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$ and establishing the equivalence
between weak solutions and minimizers.

######  Definition 3.1.

A function $u\in L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$ is said to be a weak
solution to problem (1.3) if there exists a function $Z\in
L^{\infty}(\mathbb{R}^{N}\times\mathbb{R}^{N})$, satisfying

  * (1)

For every $\varphi\in L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$, there holds

$\displaystyle\int_{\Omega}\frac{u-u_{0}}{h}\varphi
dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy=\int_{\Omega}f\varphi
dx;$

  * (2)

$Z(x,y)\in\mathrm{sgn}(u(x)-u(y))$ is an anti-symmetry function with respect
to $(x,y)$.

######  Lemma 3.2.

For fixed $s\in(0,1)$, set $s_{p}:=N+s-\frac{N}{p}$. Let $u_{p}$ be the
minimizer of the functional $J^{s_{p}}_{p}$ on $L^{2}(\Omega)\cap
W^{s_{p},p}_{0}(\Omega)$, then there exists a function $u\in L^{2}(\Omega)\cap
W^{s,1}_{0}(\Omega)$ such that, up to a subsequence,

$\displaystyle u_{p}\rightarrow u\quad\text{strongly in }L^{q}(\Omega)\text{
and a.e. in }\mathbb{R}^{N},$

where $1\leq q<2$. Moreover, the function $u$ is the unique minimizer of the
functional $J^{s}_{1}$ in $L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$.

###### Proof.

We prove the consequence by several steps. Some of the reasoning is based on
the ideas developed in [6].

Step 1. To prove the boundedness of $\left\\{u_{p}\right\\}$ in
$L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$.

For fixed $p>1$, by Lemma 2.4 and Definition 2.2, we have

$\displaystyle[u_{p}]^{p}_{W^{s,1}_{0}(\Omega)}$ $\displaystyle\leq
C^{1-p}(N,s,\Omega)[u_{p}]^{p}_{W^{s_{p},p}(\Omega)}$
$\displaystyle=C^{1-p}(N,s,\Omega)\left(\int_{\Omega}fu_{p}dx-\int_{\Omega}\frac{u_{p}-u_{0}}{h}u_{p}dx\right)$
$\displaystyle\leq
C^{1-p}(N,s,\Omega)\int_{\Omega}\left(\frac{f^{2}h}{2}+\frac{u_{p}^{2}}{2h}-\frac{u_{p}^{2}}{h}+\frac{u_{0}^{2}}{2h}+\frac{u_{p}^{2}}{2h}\right)dx$
$\displaystyle=\frac{1}{2}C^{1-p}(N,s,\Omega)\int_{\Omega}\left(f^{2}h+\frac{u_{0}^{2}}{h}\right)dx$
$\displaystyle\leq\left(\frac{C(N,s,\Omega)}{C(N,s,\Omega)+1}\right)^{{1-p}}\int_{\Omega}\left(f^{2}h+\frac{u_{0}^{2}}{h}\right)dx$
$\displaystyle\leq\left(1+\frac{1}{C(N,s,\Omega)}\right)\int_{\Omega}\left(f^{2}h+\frac{u_{0}^{2}}{h}\right)dx,$

where the Young’s inequality was utilized and $p$ was limited in $(1,2)$.

In addition, by taking $u_{p}$ as test function in (2.2) and observing the
following inequality

$\displaystyle\int_{\Omega}u_{p}^{2}dx$ $\displaystyle\leq
h\int_{\Omega}fu_{p}dx+\int_{\Omega}u_{0}u_{p}dx$
$\displaystyle\leq\frac{1}{2}h^{2}\int_{\Omega}f^{2}dx+\frac{1}{2}\int_{\Omega}u_{p}^{2}dx+2\int_{\Omega}u_{0}^{2}dx+\frac{1}{4}\int_{\Omega}u_{p}^{2}dx,$

we know that the sequence $\left\\{u_{p}\right\\}$ is bounded in
$L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$. Thus, by compactness results
involving the fractional Sobolev spaces, up to subsequences,

$\displaystyle u_{p}\to u\quad\text{a.e. in }\mathbb{R}^{N}$

and

$\displaystyle u_{p}\rightarrow u\quad\text{strongly in
}L^{q}(\Omega)\quad\text{for }1\leq q<2.$

Step 2. To prove that $u$ found in Step 1 is a minimizer of $J^{s}_{1}$.

Suppose $u$ is not the minimizer of $J^{s}_{1}$, we may find a function $v$,
such that

$\displaystyle J^{s}_{1}(v)<J^{s}_{1}(u).$ (3.1)

For reason of $C^{\infty}_{0}(\Omega)$ is dense in $L^{2}(\Omega)\cap
W^{s,1}_{0}(\Omega)$, we know that there exists a sequence
$\left\\{v_{m}\right\\}\subset C^{\infty}_{0}(\Omega)$, such that

$\displaystyle v_{m}\rightarrow v\quad\text{in }L^{2}(\Omega)\cap
W^{s,1}_{0}(\Omega)\text{ and a.e. in }\mathbb{R}^{N}.$

Hence

$\displaystyle\lim\limits_{m\rightarrow+\infty}J^{s}_{1}(v_{m})$
$\displaystyle=\lim\limits_{m\rightarrow+\infty}\left[\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|v_{m}(x)-v_{m}(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{v_{m}-u_{0}}{h}-f\right)^{2}dx\right]$
$\displaystyle=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|v(x)-v(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{v-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle=J^{s}_{1}(v).$

According to Lemma 2.5, Fatou’s Lemma and the minimality of $u_{p}$, we derive
that

$\displaystyle J^{s}_{1}(u)$ $\displaystyle\leq\liminf\limits_{p\rightarrow
1}J^{s_{p}}_{p}(u_{p})\leq\limsup\limits_{p\rightarrow
1}J^{s_{p}}_{p}(u_{p})\leq\limsup\limits_{p\rightarrow 1}J^{s_{p}}_{p}(v_{m})$
$\displaystyle=\limsup\limits_{p\rightarrow
1}\left[\frac{4}{h}\Phi^{s}_{p}(v_{m})+\int_{\Omega}\left(\frac{v_{m}-u_{0}}{h}-f\right)^{2}dx\right]$
$\displaystyle=\frac{4}{h}\Phi^{s}_{1}(v_{m})+\int_{\Omega}\left(\frac{v_{m}-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle=J^{s}_{1}(v_{m}).$

By letting $m$ goes to $+\infty$, we have

$\displaystyle J^{s}_{1}(u)\leq J^{s}_{1}(v),$

which is contradict to (3.1).

Step 3. To prove the uniqueness of minimizers.

Suppose there exists a function $\tilde{u}\neq u$, satisfying for any $v\in
L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$, $J^{s}_{1}(u)=J^{s}_{1}(\tilde{u})\leq
J^{s}_{1}(v)$. Thus, by setting $\bar{u}=\frac{1}{2}u+\frac{1}{2}\tilde{u}$,
we have

$\displaystyle J^{s}_{1}(\bar{u})$
$\displaystyle=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\bar{u}(x)-\bar{u}(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{\bar{u}-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\frac{1}{2}u(x)+\frac{1}{2}\tilde{u}(x)-\frac{1}{2}u(y)-\frac{1}{2}\tilde{u}(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{\frac{1}{2}u+\frac{1}{2}\tilde{u}-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle<\frac{1}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|+|\tilde{u}(x)-\tilde{u}(y)|}{|x-y|^{N+s}}dxdy$
$\displaystyle\quad+\frac{1}{2}\int_{\Omega}\left(\frac{u-u_{0}}{h}-f\right)^{2}dx+\frac{1}{2}\int_{\Omega}\left(\frac{\tilde{u}-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle=\frac{1}{2}J^{s}_{1}(u)+\frac{1}{2}J^{s}_{1}(\tilde{u})$
$\displaystyle=J^{s}_{1}(u),$

which is contradict to $u$ being a minimizer of $J^{s}_{1}$. ∎

In the rest part of this section, we shall show the equivalence between the
minimizer of $J^{s}_{1}$ and the weak solution to problem (1.3).

######  Lemma 3.3.

For fixed $s\in(0,1)$, suppose that there exists a weak solution to problem
(1.3), then the function $u$ found in Lemma 3.2 is a weak solution to (1.3).

###### Proof.

Let $\tilde{u}$ be a weak solution to (1.3). From Definition 3.1, we know that
there exists a function $Z$ such that

  * (1)

For every $\varphi\in L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$,

$\displaystyle\int_{\Omega}\frac{\tilde{u}-u_{0}}{h}\varphi
dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy=\int_{\Omega}f\varphi
dx;$

  * (2)

$Z(x,y)\in\mathrm{sgn}(\tilde{u}(x)-\tilde{u}(y))$ is an anti-symmetry
function with respect to $(x,y)$.

Choosing $\varphi=\tilde{u}-u$ and noting that

$\displaystyle\quad\int_{\Omega}\left[\frac{\tilde{u}-u_{0}}{h}(\tilde{u}-u)-f(\tilde{u}-u)\right]dx$
$\displaystyle=\int_{\Omega}\left(\frac{\tilde{u}-u_{0}}{h}-f\right)\left(\tilde{u}-u_{0}-hf-u+u_{0}+hf\right)dx$
$\displaystyle=h\int_{\Omega}\left[\left(\frac{\tilde{u}-u_{0}}{h}-f\right)^{2}-\left(\frac{\tilde{u}-u_{0}}{h}-f\right)\left(\frac{u-u_{0}}{h}-f\right)\right]dx$
$\displaystyle\geq\frac{h}{2}\int_{\Omega}\left[\left(\frac{\tilde{u}-u_{0}}{h}-f\right)^{2}-\left(\frac{u-u_{0}}{h}-f\right)^{2}\right]dx,$

we have

$\displaystyle J^{s}_{1}(u)$
$\displaystyle=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{u-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle\geq\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{u(x)-u(y)}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{u-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle\geq\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{u(x)-u(y)}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{\tilde{u}-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle\quad-\frac{2}{h}\int_{\Omega}\left[\frac{\tilde{u}-u_{0}}{h}(\tilde{u}-u)-f(\tilde{u}-u)\right]dx$
$\displaystyle=\int_{\Omega}\left(\frac{\tilde{u}-u_{0}}{h}-f\right)^{2}dx-\frac{2}{h}\int_{\Omega}\left(\frac{\tilde{u}-u_{0}}{h}\tilde{u}-f\tilde{u}\right)dx$
$\displaystyle=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\tilde{u}(x)-\tilde{u}(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{\tilde{u}-u_{0}}{h}-f\right)^{2}dx$
$\displaystyle=J^{s}_{1}(\tilde{u}).$

Therefore, $u=\tilde{u}$ and the conclusion follows. ∎

######  Remark 3.4.

(1) We ignored the proof of the existence of weak solutions and left the
details to [11, Theorem 3.4] and [6, Theorem 1.6].

(2) The equivalence between the minimizers of $J^{s}_{1}$ and the weak
solutions to problem (1.3) could be proved easily by combining Lemmas 3.2 and
3.3.

##  4\. Proof of the main result

In this section, we will prove the existence and uniqueness of weak solutions
to problem (1.1).

###### Proof of Theorem 1.3.

We divide our proof into several parts.

First, we shall show the asymptotic behaviour of $\left\\{u^{m}\right\\}$ and
$\left\\{Z^{m}\right\\}$, which are defined later. For a fixed number
$m\in\mathbb{N}$, consider the following elliptic problem

$\displaystyle\begin{cases}\displaystyle\frac{u_{k}-u_{k-1}}{h}+(-\Delta)^{s}_{1}u_{k}=[f]_{h}(x,(k-1)h)&\text{in
}\Omega,\\\ u_{k}=0&\text{in }\mathcal{C}_{\Omega},\end{cases}$

where $h:=\frac{T}{m}$, $k\in\left\\{1,2,\dots,m\right\\}$, and $[f]_{h}$ is
the steklov average of $f$.

By Lemma 3.3, for every $k$, there exists a weak solution $u_{k}\in
L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$ and a function
$Z_{k}\in\text{sgn}(u_{k}(x)-u_{k}(y))$, such that for every $\varphi\in
L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$,

$\displaystyle\int_{\Omega}\frac{u_{k}-u_{k-1}}{h}\varphi
dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z_{k}\frac{\varphi(x)-\varphi(y)}{|x-y|^{N+s}}dxdy=\int_{\Omega}[f]_{h}(x,(k-1)h)\varphi
dx.$ (4.1)

We take $\varphi=u_{k}-u_{k-1}$ as test function in (4.1) and use the Young’s
inequality to deduce a estimate for $u_{k}$, i.e.,

$\displaystyle\quad\int_{\Omega}\frac{\left(u_{k}-u_{k-1}\right)^{2}}{h}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdy$
$\displaystyle=\int_{\Omega}[f]_{h}(x,(k-1)h)(u_{k}-u_{k-1})dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z_{k}\frac{u_{k-1}(x)-u_{k-1}(y)}{|x-y|^{N+s}}dxdy$
$\displaystyle\leq\frac{h}{4}\int_{\Omega}[f]_{h}^{2}(x,(k-1)h)dx+\int_{\Omega}\frac{\left(u_{k}-u_{k-1}\right)^{2}}{h}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z_{k}\frac{u_{k-1}(x)-u_{k-1}(y)}{|x-y|^{N+s}}dxdy$
$\displaystyle\leq\frac{h}{4}\int_{\Omega}[f]_{h}^{2}(x,(k-1)h)dx+\int_{\Omega}\frac{\left(u_{k}-u_{k-1}\right)^{2}}{h}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy,$

which implies that

$\displaystyle\begin{aligned}
&\quad\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdy\\\
&\leq\frac{h}{4}\int_{\Omega}[f]_{h}^{2}(x,(k-1)h)dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy\\\
&=\frac{h}{4}\int_{\Omega}\left(\frac{1}{h}\int_{(k-1)h}^{kh}f(x,t)dt\right)^{2}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy\\\
&\leq\frac{1}{4h}\left[\int_{(k-1)h}^{kh}\left(\int_{\Omega}f^{2}(x,t)dx\right)^{\frac{1}{2}}dt\right]^{2}+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy\\\
&\leq\frac{1}{4}\int_{(k-1)h}^{kh}\left(\int_{\Omega}f^{2}(x,t)dx\right)dt+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy.\\\
\end{aligned}$ (4.2)

We add all inequalities (4.2) for $k=1,2,\dots,i$ to get

$\displaystyle\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{i}(x)-u_{i}(y)|}{|x-y|^{N+s}}dxdy\leq\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{0}(x)-u_{0}(y)|}{|x-y|^{N+s}}dxdy+C,$
(4.3)

where $C$ depends on $f$.

Similarly, by taking $\varphi=u_{k}$ in (4.1), we have

$\displaystyle\quad\int_{\Omega}\frac{u_{k}-u_{k-1}}{h}u_{k}dx$
$\displaystyle\leq\int_{\Omega}[f]_{h}(x,(k-1)h)u_{k}dx$
$\displaystyle\leq\int_{\Omega}\left(\varepsilon[f]^{\frac{N}{s}}_{h}(x,(k-1)h)+C(\varepsilon)u_{k}^{\frac{N}{N-s}}\right)dx$
$\displaystyle\leq\varepsilon\int_{\Omega}\left(\frac{1}{h}\int_{(k-1)h}^{kh}f(x,t)dt\right)^{\frac{N}{s}}dx+C(\varepsilon)\int_{\Omega}u_{k}^{\frac{N}{N-s}}dx$
$\displaystyle\leq\frac{\varepsilon}{h^{\frac{N}{s}}}\left(\int_{(k-1)h}^{kh}\left(\int_{\Omega}|f(x,t)|^{\frac{N}{s}}dx\right)^{\frac{s}{N}}dt\right)^{\frac{N}{s}}+C(\varepsilon,N,s,\Omega)\left([u_{k}]_{W^{s,1}_{0}(\Omega)}\right)^{\frac{N}{N-S}}$
$\displaystyle\leq\frac{\varepsilon}{h^{\frac{N}{s}}}h^{\frac{N-s}{s}}\int_{(k-1)h}^{kh}\int_{\Omega}|f(x,t)|^{\frac{N}{s}}dxdt+C(\varepsilon,N,s,\Omega,u_{0})$
$\displaystyle\leq\frac{\varepsilon}{h}\int_{(k-1)h}^{kh}\int_{\Omega}|f(x,t)|^{\frac{N}{s}}dxdt+C(\varepsilon,N,s,\Omega,u_{0}),$

where a Sobolev-type inequality involving the fractional norm
$\|\cdot\|_{W^{s,1}_{0}(\Omega)}$, the Young’s inequality, Hölder’s inequality
and (4.3) were utilized. Hence, we have

$\displaystyle\frac{1}{2}\int_{\Omega}u_{k}^{2}dx\leq
C(f,\varepsilon,N,s,\Omega,u_{0})h+\frac{1}{2}\int_{\Omega}u_{k-1}^{2}dx.$
(4.4)

By adding all inequalities (4.4) for $k=1,2,\dots,i$, we obtain

$\displaystyle\frac{1}{2}\int_{\Omega}u_{i}^{2}dx\leq
C(f,\varepsilon,N,s,\Omega,u_{0})T+\frac{1}{2}\int_{\Omega}u_{0}^{2}dx.$

Now we take a function $Z_{0}\in\text{sgn}(u_{0}(x)-u_{0}(y))$ and define

$\displaystyle u^{m}(x,t):=\begin{cases}u_{0}(x)&\text{if }t=0,\\\ \\\
\dfrac{t}{h}u_{1}(x)+\dfrac{h-t}{h}u_{0}(x)&\text{if }0<t\leq h,\\\ \\\
\dfrac{t-h}{h}u_{2}(x)+\dfrac{2h-t}{h}u_{1}(x)&\text{if }h<t\leq 2h,\\\ \\\
\cdots\cdots&\cdots\cdots,\\\ \\\
\dfrac{t-(m-1)h}{h}u_{m}(x)+\dfrac{mh-t}{h}u_{m-1}(x)&\text{if }(m-1)h<t\leq
mh=T,\end{cases}$

and

$\displaystyle Z^{m}(x,y,t)=\begin{cases}Z_{0}(x,y)&\text{if }t=0,\\\ \\\
\dfrac{t}{h}Z_{1}(x,y)+\dfrac{h-t}{h}Z_{0}(x,y)&\text{if }0<t\leq h,\\\ \\\
\dfrac{t-h}{h}Z_{2}(x,y)+\dfrac{2h-t}{h}Z_{1}(x,y)&\text{if }h<t\leq 2h,\\\
\\\ \cdots\cdots&\cdots\cdots,\\\ \\\
\dfrac{t-(m-1)h}{h}Z_{m}(x,y)+\dfrac{mh-t}{h}Z_{m-1}(x,y)&\text{if
}(m-1)h<t\leq mh=T.\end{cases}$

Thus we have

$\displaystyle\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u^{m}(x,t)-u^{m}(y,t)|}{|x-y|^{N+s}}dxdy\leq
C+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{0}(x)-u_{0}(y)|}{|x-y|^{N+s}}dxdy$

and

$\displaystyle\int_{\Omega}\left(u^{m}(x,t)\right)^{2}dx\leq
C+\int_{\Omega}\left(u_{0}(x)\right)^{2}dx.$

After taking the supermum over $[0,T]$, we deduce

$\displaystyle\sup\limits_{0\leq t\leq
T}\left\\{\int_{\Omega}\left(u^{m}(x,t)\right)^{2}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u^{m}(x,t)-u^{m}(y,t)|}{|x-y|^{N+s}}dxdy\right\\}\leq
C,$ (4.5)

where $C$ is independent of $m$.

Therefore, we could choose a subsequence, still denoted by
$\left\\{u^{m}\right\\}$, such that

$\displaystyle\begin{cases}u^{m}\rightharpoonup u\quad\text{weakly-}^{*}\text{
in }L^{\infty}(0,T;L^{2}(\Omega)),\\\ u\in L^{\infty}(0,T;L^{2}(\Omega)\cap
W^{s,1}_{0}(\Omega)).\end{cases}$

Moreover, we may find a function $Z\in
L^{\infty}(\mathbb{R}^{N}\times\mathbb{R}^{N}\times[0,T])$, such that

$\displaystyle Z^{m}\rightharpoonup Z\quad\text{weakly-}^{*}\text{ in
}L^{\infty}(\mathbb{R}^{N}\times\mathbb{R}^{N}\times[0,T]).$

Next, we will prove that $u$ belongs to the space
$C^{0,\frac{1}{2}}\left([0,T];L^{2}(\Omega)\right)$. Define $J^{s,1}_{k}$ on
$L^{2}(\Omega)\cap W^{s,1}_{0}(\Omega)$ as follows

$\displaystyle
J^{s,1}_{k}(u):=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{u-u_{k-1}}{h}-[f]_{h}(x,(k-1)h)\right)^{2}dx.$

From the equivalence between weak solutions and minimizers, noting that the
function $u_{k-1}$ is an admissible comparison map for the minimizer $u_{k}$,
we have

$\displaystyle\quad\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{u_{k}-u_{k-1}}{h}-[f]_{h}(x,(k-1)h)\right)^{2}dx$
$\displaystyle\leq J^{s,1}_{k}(u_{k})\leq J^{s,1}_{k}(u_{k-1})$
$\displaystyle=\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left([f]_{h}(x,(k-1)h)\right)^{2}dx.$

Thus, the following inequality holds

$\displaystyle\quad\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdy+\frac{1}{2}\int_{\Omega}\left(\frac{u_{k}-u_{k-1}}{h}\right)^{2}dx$
$\displaystyle\leq\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdy+\int_{\Omega}\left(\frac{u_{k}-u_{k-1}}{h}-[f]_{h}(x,(k-1)h)\right)^{2}dx$
$\displaystyle\quad+\int_{\Omega}\left([f]_{h}(x,(k-1)h)\right)^{2}dx$
$\displaystyle\leq\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy+2\int_{\Omega}\left([f]_{h}(x,(k-1)h)\right)^{2}dx.$

Adding the previous inequalities for $k=1,2,\dots,m$, one can show that

$\displaystyle\quad\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{m}(x)-u_{m}(y)|}{|x-y|^{N+s}}dxdy+\frac{1}{2}\sum_{k=1}^{m}\int_{\Omega}\left(\frac{u_{k}-u_{k-1}}{h}\right)^{2}dx$
$\displaystyle\leq\frac{2}{h}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{0}(x)-u_{0}(y)|}{|x-y|^{N+s}}dxdy+2\sum_{k=1}^{m}\int_{\Omega}\left([f]_{h}(x,(k-1)h)\right)^{2}dx.$

By the uniformly boundedness of energy, we have

$\displaystyle\quad\int_{0}^{T}\int_{\Omega}\left|\partial_{t}u^{m}\right|^{2}dxdt$
$\displaystyle=\sum_{k=1}^{m}\int_{(k-1)h}^{kh}\int_{\Omega}\frac{1}{h^{2}}|u_{k}-u_{k-1}|^{2}dxdt$
$\displaystyle=\frac{1}{h}\sum_{k=1}^{m}\int_{\Omega}|u_{k}-u_{k-1}|^{2}dxdt$
$\displaystyle\leq
4\left([u^{m}(0)]_{W^{s,1}_{0}(\Omega)}+[u^{m}(T)]_{W^{s,1}_{0}(\Omega)}\right)+4h\sum_{k=1}^{m}\int_{\Omega}\left(\frac{1}{h}\int_{(k-1)h}^{kh}f(x,t)dt\right)^{2}dx$
$\displaystyle\leq
C(N,s,f,u_{0},\Omega)+\frac{4}{h}\sum_{k=1}^{m}\int_{\Omega}\left(\int_{(k-1)h}^{kh}f(x,t)dt\right)^{2}dx$
$\displaystyle\leq
C(N,s,f,u_{0},\Omega)+\frac{4}{h}\sum_{k=1}^{m}\left(\int_{(k-1)h}^{kh}\left(\int_{\Omega}f^{2}(x,t)dx\right)^{\frac{1}{2}}dt\right)^{2}$
$\displaystyle\leq
C(N,s,f,u_{0},\Omega)+\frac{4}{h}\sum_{k=1}^{m}\left(\left(\int_{(k-1)h}^{kh}\int_{\Omega}f^{2}(x,t)dxdt\right)^{\frac{1}{2}}h^{\frac{1}{2}}\right)^{2}$
$\displaystyle=C(N,s,f,u_{0},\Omega)+4\|f\|^{2}_{L^{2}(\Omega_{T})}.$

Hence, $\left\\{u^{m}\right\\}$ is bounded in $W^{1,2}(0,T;L^{2}(\Omega))$. By
Rellich’s theorem, we conclude that there exists a subsequence, without loss
of generality, still denoted by $u$ the limit of this subsequence,

$\displaystyle\begin{cases}u^{m}\rightarrow u\quad\text{strongly in
}L^{2}(\Omega_{T}),\\\
\partial_{t}u^{m}\rightharpoonup\partial_{t}u\quad\text{weakly}\text{ in
}L^{2}(\Omega_{T}).\end{cases}$

In particular, noting that for any $t_{1},t_{2}\in[0,T]$, there holds

$\displaystyle\displaystyle\frac{\|u(t_{1})-u(t_{2})\|_{L^{2}(\Omega)}}{(t_{1}-t_{2})^{\frac{1}{2}}}$
$\displaystyle=\frac{\left(\int_{\Omega}\left(u(t_{1})-u(t_{2})\right)^{2}dx\right)^{\frac{1}{2}}}{(t_{1}-t_{2})^{\frac{1}{2}}}$
$\displaystyle\leq\frac{\left(\int_{\Omega}\left(\int_{t_{2}}^{t_{1}}\partial_{t}udt\right)^{2}dx\right)^{\frac{1}{2}}}{(t_{1}-t_{2})^{\frac{1}{2}}}$
$\displaystyle\leq\frac{\int_{t_{1}}^{t_{2}}\left(\int_{\Omega}|\partial_{t}u|^{2}dx\right)^{\frac{1}{2}}dt}{(t_{1}-t_{2})^{\frac{1}{2}}}$
$\displaystyle\leq\left(\int_{t_{1}}^{t_{2}}\int_{\Omega}|\partial_{t}u|^{2}dxdt\right)^{\frac{1}{2}},$

which means that $u\in C^{0,\frac{1}{2}}\left([0,T];L^{2}(\Omega)\right)$.

Next, we will show that $u$ and $Z$ satisfy (1) in Definition 1.1.

For each $\varphi\in C^{1}\left([0,T];L^{2}(\Omega)\right)\cap
C([0,T];W^{s,1}_{0}(\Omega))$ with $\varphi(\cdot,T)=0$, we take
$\varphi(x,kh)$ as a test function in (4.1) for every
$k\in\left\\{1,2,\dots,m\right\\}$ to get

$\displaystyle\begin{aligned}
&\quad\int_{\Omega}\frac{u_{k}(x)-u_{k-1}(x)}{h}\varphi(x,kh)dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z_{k}\frac{\varphi(x,kh)-\varphi(y,kh)}{|x-y|^{N+s}}dxdy\\\
&=\int_{\Omega}[f]_{h}(x,(k-1)h)\varphi(x,kh)dx.\end{aligned}$ (4.6)

For any $t\in(h,T)$, there exists $k\in\left\\{2,3,\dots,m\right\\}$, such
that $(k-1)h<t\leq kh$. Thus, we have

$\displaystyle\begin{aligned}
&\quad\int_{\Omega}\frac{kh-t}{h}\frac{u_{k-1}(x)-u_{k-2}(x)}{h}\varphi(x,(k-1)h)dx\\\
&\quad+\int_{\Omega}\frac{t-(k-1)h}{h}\frac{u_{k}(x)-u_{k-1}(x)}{h}\varphi(x,kh)dx\\\
&\quad+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{kh-t}{h}Z_{k-1}\frac{\varphi(x,(k-1)h)-\varphi(y,(k-1)h)}{|x-y|^{N+s}}dxdy\\\
&\quad+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{t-(k-1)h}{h}Z_{k}\frac{\varphi(x,kh)-\varphi(y,kh)}{|x-y|^{N+s}}dxdy\\\
&=\int_{\Omega}\frac{kh-t}{h}[f]_{h}(x,(k-2)h)\varphi(x,(k-1)h)dx\\\
&\quad+\int_{\Omega}\frac{t-(k-1)h}{h}[f]_{h}(x,(k-1)h)\varphi(x,kh)dx.\end{aligned}$
(4.7)

That is, by definitions of $u^{m}$ and $Z^{m}$,

$\displaystyle\begin{aligned}
&\quad\int_{\Omega}\frac{u^{m}(x,t)-u^{m}(x,(t-h))}{h}\varphi(x,(k-1)h)dx\\\
&\quad+\int_{\Omega}\frac{kh-t}{h}\frac{u_{k}-u_{k-1}}{h}\left(\varphi(x,kh)-\varphi(x,(k-1)h)\right)dx\\\
&\quad+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z^{m}\frac{\varphi(x,(k-1)h)-\varphi(y,(k-1)h)}{|x-y|^{N+s}}dxdy\\\
&\quad+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{kh-t}{h}Z_{k}\frac{\varphi(x,kh)-\varphi(y,kh)}{|x-y|^{N+s}}dxdy\\\
&\quad-\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{kh-t}{h}Z_{k}\frac{\varphi(x,(k-1)h)-\varphi(y,(k-1)h)}{|x-y|^{N+s}}dxdy\\\
&=\int_{\Omega}\frac{t-(k-1)h}{h}[f]_{h}(x,(k-2)h)\varphi(x,(k-1)h)dx\\\
&\quad+\int_{\Omega}\frac{kh-t}{h}[f]_{h}(x,(k-1)h)\varphi(x,kh)dx,\end{aligned}$
(4.8)

where $k$ depends on $t$. By integrating (4.8) with respect to $t$ from $h$ to
$T$, for the first term on the left-hand side of (4.8), we get

$\displaystyle\begin{aligned}
&\quad\int_{h}^{T}\int_{\Omega}\frac{u^{m}(x,t)-u^{m}(x,(t-h))}{h}\varphi(x,(k-1)h)dxdt\\\
&=\int_{h}^{T}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,(k-1)h)dxdt-\int_{0}^{T-h}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,kh)dxdt\\\
&=\int_{h}^{T-h}\int_{\Omega}u^{m}(x,t)\frac{\varphi(x,(k-1)h)-\varphi(x,kh)}{h}dxdt\\\
&\quad+\int_{T-h}^{T}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,(m-1)h)dxdt-\int_{0}^{h}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,0)dxdt.\end{aligned}$
(4.9)

Noting that

$\displaystyle\quad\int_{T-h}^{T}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,(m-1)h)dxdt$
$\displaystyle=\int_{T-h}^{T}\int_{\Omega}\left(\dfrac{t-(m-1)h}{h}u_{m}(x)+\dfrac{mh-t}{h}u_{m-1}(x)\right)\dfrac{\varphi(x,(m-1)h)}{h}dxdt$
$\displaystyle=\frac{1}{2}\int_{\Omega}u_{m}(x)\varphi(x,(m-1)h)dx+\frac{1}{2}\int_{\Omega}u_{m-1}(x)\varphi(x,(m-1)h)dx$
$\displaystyle=\frac{1}{2}\int_{\Omega}u^{m}(x,T)\varphi(x,(m-1)h)dx+\frac{1}{2}\int_{\Omega}u^{m}(x,(m-1)h)\varphi(x,(m-1)h)dx$

and

$\displaystyle\quad\int_{0}^{h}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,0)dxdt$
$\displaystyle=\int_{0}^{h}\int_{\Omega}\left(\dfrac{t}{h}u_{1}(x)+\dfrac{h-t}{h}u_{0}(x)\right)\frac{\varphi(x,0)}{h}dxdt$
$\displaystyle=\frac{1}{2}\int_{\Omega}u_{0}(x)\varphi(x,0)dx+\frac{1}{2}\int_{\Omega}u_{1}(x)\varphi(x,0)dx,$

we have

$\displaystyle\begin{aligned}
&\quad\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{T-h}^{T}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,(m-1)h)dxdt\bigg{|}\\\
&=\lim\limits_{m\rightarrow+\infty}\bigg{|}\frac{1}{2}\int_{\Omega}\left(u^{m}(x,T)+u^{m}(x,(m-1)h)\right)\varphi(x,(m-1)h)dx\bigg{|}\\\
&\leq\lim\limits_{m\rightarrow+\infty}\frac{1}{2}\left(\|u^{m}(x,T)\|_{L^{2}(\Omega)}+\|u^{m}(x,T-h)\|_{L^{2}(\Omega)}\right)\|\varphi(x,(m-1)h)\|_{L^{2}(\Omega)}\\\
&\leq\lim\limits_{m\rightarrow+\infty}C(u_{0},f,N,s,\Omega)\|\varphi(x,(m-1)h)-\varphi(x,T)\|_{L^{2}(\Omega)}\\\
&\leq\lim\limits_{m\rightarrow+\infty}C(u_{0},f,N,s,\Omega)\max\limits_{T-h\leq
s\leq T}\|\varphi_{t}(x,s)\|_{L^{2}(\Omega)}\frac{T}{m}\\\ &=0\end{aligned}$
(4.10)

and

$\displaystyle\begin{aligned}
&\quad\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{0}^{h}\int_{\Omega}\frac{u^{m}(x,t)}{h}\varphi(x,0)dxdt-\int_{\Omega}u_{0}(x)\varphi(x,0)dx\bigg{|}\\\
&=\lim\limits_{m\rightarrow+\infty}\bigg{|}\frac{1}{2}\int_{\Omega}u_{1}(x)\varphi(x,0)dxdt-\frac{1}{2}\int_{\Omega}u_{0}(x)\varphi(x,0)dx\bigg{|}\\\
&\leq\lim\limits_{m\rightarrow+\infty}\frac{1}{2}\|u_{1}(x)-u_{0}(x)\|_{L^{2}(\Omega)}\|\varphi(x,0)\|_{L^{2}(\Omega)}\\\
&\leq\lim\limits_{m\rightarrow+\infty}C(u_{0},f,s,N,\Omega)h^{\frac{1}{2}}\|\varphi(x,0)\|_{L^{2}(\Omega)}\\\
&=0.\end{aligned}$ (4.11)

Observing that

$\displaystyle\begin{aligned}
&\quad\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T-h}\int_{\Omega}u^{m}(x,t)\frac{\varphi(x,(k-1)h)-\varphi(x,kh)}{h}dxdt-\int_{0}^{T}\int_{\Omega}u(x,t)\varphi_{t}(x,t)dxdt\bigg{|}\\\
&\leq\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T-h}\int_{\Omega}\left(u^{m}(x,t)\frac{\varphi(x,(k-1)h)-\varphi(x,kh)}{h}-u^{m}(x,t)\varphi_{t}(x,t)\right)dxdt\bigg{|}\\\
&\quad+\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T-h}\int_{\Omega}u^{m}(x,t)\varphi_{t}(x,t)dxdt-\int_{0}^{T}\int_{\Omega}u^{m}(x,t)\varphi_{t}(x,t)dxdt\bigg{|}\\\
&\quad+\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{0}^{T}\int_{\Omega}\left(u^{m}(x,t)-u(x,t)\right)\varphi_{t}(x,t)dxdt\bigg{|}\\\
&\leq\lim\limits_{m\rightarrow+\infty}\int_{h}^{T-h}\int_{\Omega}|u^{m}(x,t)|\bigg{|}\frac{\varphi(x,(k-1)h)-\varphi(x,kh)}{h}-\varphi_{t}(x,t)\bigg{|}dxdt\\\
&\quad+\lim\limits_{m\rightarrow+\infty}\left(\int_{0}^{h}\int_{\Omega}|u^{m}(x,t)\varphi_{t}(x,t)|dxdt+\int_{T-h}^{T}\int_{\Omega}|u^{m}(x,t)\varphi_{t}(x,t)|dxdt\right)\\\
&\leq\lim\limits_{m\rightarrow+\infty}\int_{0}^{T}\|u^{m}(x,t)\|_{L^{2}(\Omega)}\max\limits_{(k-1)h\leq\tau\leq
kh}\|\varphi_{t}(x,\tau)-\varphi_{t}(x,t)\|_{L^{2}(\Omega)}dt\\\
&\quad+\lim\limits_{m\rightarrow+\infty}2\|u^{m}\|_{L^{\infty}(0,T;L^{2}(\Omega))}\|\varphi_{t}\|_{L^{\infty}(0,T;L^{2}(\Omega))}h\\\
&=0,\end{aligned}$ (4.12)

by letting $m$ goes to $+\infty$ and bringing (4.10)–(4.12) into (4.9) one has

$\displaystyle\begin{aligned}
&\quad\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}\int_{\Omega}\frac{u^{m}(x,t)-u^{m}(x,(t-h))}{h}\varphi(x,(k-1)h)dxdt\\\
&=-\int_{0}^{T}\int_{\Omega}u(x,t)\varphi_{t}(x,t)dxdt-\int_{\Omega}u_{0}(x)\varphi(x,0)dx.\end{aligned}$
(4.13)

For the second term on the left-hand side of (4.8), from the boundedness of
$\left\\{u^{m}\right\\}$ in the space
$C^{0,\frac{1}{2}}\left([0,T];L^{2}(\Omega)\right)$, we have

$\displaystyle\begin{aligned}
&\quad\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T}\int_{\Omega}\frac{kh-t}{h}\frac{u_{k}-u_{k-1}}{h}\left(\varphi(x,kh)-\varphi(x,(k-1)h)\right)dxdt\bigg{|}\\\
&\leq\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T}\int_{\Omega}\frac{u_{k}-u_{k-1}}{h}\left(\varphi(x,kh)-\varphi(x,(k-1)h)\right)dxdt\bigg{|}\\\
&\leq\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}\bigg{\|}\frac{u_{k}-u_{k-1}}{h}\bigg{\|}_{L^{2}(\Omega)}\|\varphi(x,kh)-\varphi(x,(k-1)h)\|_{L^{2}(\Omega)}dt\\\
&\leq\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}C(u_{0},f,s,N,\Omega)h^{\frac{1}{2}}\max\limits_{h\leq\tau\leq
T}\|\varphi_{t}(x,\tau)\|_{L^{2}(\Omega)}dt\\\
&\leq\lim\limits_{m\rightarrow+\infty}C(u_{0},f,s,N,\Omega,\varphi,T)h^{\frac{1}{2}}\\\
&=0.\end{aligned}$ (4.14)

Similarly, for the third term on the left-hand side of (4.8), we deduce that

$\displaystyle\quad\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z^{m}\frac{\varphi(x,(k-1)h)-\varphi(y,(k-1)h)}{|x-y|^{N+s}}dxdydt$
$\displaystyle\qquad\qquad\quad-\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}dxdydt\bigg{|}$
$\displaystyle\leq\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z^{m}\frac{\varphi(x,(k-1)h)-\varphi(y,(k-1)h)}{|x-y|^{N+s}}dxdydt$
$\displaystyle\qquad\qquad\quad-\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z^{m}\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}dxdydt\bigg{|}$
$\displaystyle\quad+\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\left(Z^{m}-Z\right)\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}dxdydt\bigg{|}$
$\displaystyle\leq\lim\limits_{m\rightarrow+\infty}\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\bigg{|}\frac{\varphi(x,(k-1)h)-\varphi(y,(k-1)h)}{|x-y|^{N+s}}-\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}\bigg{|}dxdydt$
$\displaystyle=0$

and for the right-hand side of (4.8), we find

$\displaystyle\quad\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}\int_{\Omega}\frac{t-(k-1)h}{h}[f]_{h}(x,(k-2)h)\varphi(x,(k-1)h)dxdt$
$\displaystyle\quad+\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}\int_{\Omega}\frac{kh-t}{h}[f]_{h}(x,(k-1)h)\varphi(x,kh)dxdt$
$\displaystyle=\lim\limits_{m\rightarrow+\infty}\int_{h}^{T-h}\int_{\Omega}[f]_{h}(x,(k-1)h)\varphi(x,kh)dxdt+\lim\limits_{m\rightarrow+\infty}\int_{0}^{h}\int_{\Omega}\frac{t}{h}[f]_{h}(x,0)\varphi(x,h)dxdt$
$\displaystyle\quad+\lim\limits_{m\rightarrow+\infty}\int_{T-h}^{T}\int_{\Omega}\frac{T-t}{h}[f]_{h}(x,T-h)\varphi(x,T)dxdt.$

Due to

$\displaystyle\quad\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T-h}\int_{\Omega}[f]_{h}(x,(k-1)h)\varphi(x,kh)dxdt-\int_{0}^{T}\int_{\Omega}f\varphi
dxdt\bigg{|}$
$\displaystyle\leq\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{h}^{T-h}\int_{\Omega}[f]_{h}(x,(k-1)h)\varphi(x,kh)dxdt-\int_{0}^{T}\int_{\Omega}f\varphi(x,kh)dxdt\bigg{|}$
$\displaystyle\quad+\lim\limits_{m\rightarrow+\infty}\bigg{|}\int_{0}^{T}\int_{\Omega}f\varphi(x,kh)\varphi
dxdt-\int_{0}^{T}\int_{\Omega}f\varphi dxdt\bigg{|}$ $\displaystyle=0$

and

$\displaystyle\quad\lim\limits_{m\rightarrow+\infty}\int_{0}^{h}\int_{\Omega}\frac{t}{h}[f]_{h}(x,0)\varphi(x,h)dxdt+\lim\limits_{m\rightarrow+\infty}\int_{T-h}^{T}\int_{\Omega}\frac{T-t}{h}[f]_{h}(x,T-h)\varphi(x,T)dxdt$
$\displaystyle=\lim\limits_{m\rightarrow+\infty}\frac{h}{2}\int_{\Omega}\frac{1}{h}\int_{0}^{h}f(x,\tau)d\tau\varphi(x,h)dx+\lim\limits_{m\rightarrow+\infty}\frac{h}{2}\int_{\Omega}\frac{1}{h}\int_{T-h}^{T}f(x,\tau)d\tau\varphi(x,T)dx$
$\displaystyle=\frac{1}{2}\lim\limits_{m\rightarrow+\infty}\int_{0}^{h}\int_{\Omega}f(x,t)\varphi(x,h)dxdt$
$\displaystyle=0,$

one has

$\displaystyle\begin{aligned}
&\quad\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z^{m}\frac{\varphi(x,(k-1)h)-\varphi(y,(k-1)h)}{|x-y|^{N+s}}dxdydt\\\
&=\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}dxdydt\end{aligned}$
(4.15)

and

$\displaystyle\begin{aligned}
&\quad\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}\int_{\Omega}\frac{t-(k-1)h}{h}[f]_{h}(x,(k-2)h)\varphi(x,(k-1)h)dxdt\\\
&\quad+\lim\limits_{m\rightarrow+\infty}\int_{h}^{T}\int_{\Omega}\frac{kh-t}{h}[f]_{h}(x,(k-1)h)\varphi(x,kh)dxdt\\\
&=\int_{0}^{T}\int_{\Omega}f\varphi dxdt.\end{aligned}$ (4.16)

Combining (4.13)–(4.16), we get

$\displaystyle\begin{aligned}
&\quad-\int_{\Omega}u_{0}\varphi(x,0)dx-\int_{0}^{T}\int_{\Omega}u\varphi_{t}dxdt+\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}dxdydt\\\
&=\int_{0}^{T}\int_{\Omega}f\varphi dxdt.\end{aligned}$

To conclude the proof, it remains to show that
$Z(x,y,t)\in\text{sgn}\left(u(x,t)-u(y,t)\right)$, i.e., for almost all
$t\in[0,T]$ and $(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$,
$Z(x,y,t)=\frac{u(x,t)-u(y,t)}{|u(x,t)-u(y,t)|}$ in the domain
$\left\\{u(x,t)\neq u(y,t)\right\\}$ and
$\|Z\|_{L^{\infty}(\mathbb{R}^{N}\times\mathbb{R}^{N}\times[0,T])}\leq 1$.

Choosing $u$ as a test function in (1.4) to deduce that

$\displaystyle\begin{aligned}
&\quad\frac{1}{2}\|u(T)\|^{2}_{L^{2}(\Omega)}-\frac{1}{2}\|u_{0}\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{u(x,t)-u(y,t)}{|x-y|^{N+s}}dxdydt\\\
&=\int_{0}^{T}\int_{\Omega}fudxdt.\end{aligned}$ (4.17)

Moreover, by choosing $\varphi=u_{k}$ in (4.1), we obtain

$\displaystyle\int_{\Omega}\frac{u_{k}-u_{k-1}}{h}u_{k}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdy=\int_{\Omega}[f]_{h}(x,(k-1)h)u_{k}dx.$

Thus, for any $t\in[(k-1)h,kh]$, we have

$\displaystyle\begin{aligned}
&\quad\frac{t-(k-1)h}{h}\left[\int_{\Omega}\frac{u_{k}-u_{k-1}}{h}u_{k}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdy\right]\\\
&\quad+\frac{kh-t}{h}\left[\int_{\Omega}\frac{u_{k-1}-u_{k-2}}{h}u_{k-1}dx+\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdy\right]\\\
&=\int_{\Omega}\left[\frac{t-(k-1)h}{h}[f]_{h}(x,(k-2)h)u_{k-1}+\frac{kh-t}{h}[f]_{h}(x,(k-1)h)u_{k}\right]dx.\end{aligned}$
(4.18)

Integrating (4.18) with respect to $t$ from $(k-1)h$ to $kh$ to obtain

$\displaystyle\begin{aligned}
&\quad\frac{1}{2}\int_{\Omega}(u_{k}-u_{k-1})u_{k}dx+\frac{t-(k-1)h}{h}\int_{(k-1)h}^{kh}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k}(x)-u_{k}(y)|}{|x-y|^{N+s}}dxdydt\\\
&\quad+\frac{1}{2}\int_{\Omega}(u_{k-1}-u_{k-2})u_{k-1}dx\\\
&\quad+\frac{kh-t}{h}\int_{(k-1)h}^{kh}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{k-1}(x)-u_{k-1}(y)|}{|x-y|^{N+s}}dxdydt\\\
&=\int_{(k-1)}^{kh}\int_{\Omega}\left[\frac{t-(k-1)h}{h}[f]_{h}(x,(k-2)h)u_{k-1}+\frac{kh-t}{h}[f]_{h}(x,(k-1)h)u_{k}\right]dxdt.\end{aligned}$
(4.19)

By adding all the equation (4.19) for $k\in\left\\{2,3,\dots,m\right\\}$, from
Young’s inequality and definition of $u^{m}$, we have

$\displaystyle\quad\frac{1}{4}\int_{\Omega}(u_{m}^{2}+u_{m-1}^{2})dx+\int_{h}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u^{m}(x,t)-u^{m}(y,t)|}{|x-y|^{N+s}}dxdydt$
$\displaystyle\leq\sum_{k=2}^{m}\int_{(k-1)}^{kh}\int_{\Omega}\left[\frac{t-(k-1)h}{h}[f]_{h}(x,(k-2)h)u_{k-1}+\frac{kh-t}{h}[f]_{h}(x,(k-1)h)u_{k}\right]dxdt$
$\displaystyle\quad+\frac{1}{4}\int_{\Omega}(u_{1}^{2}+u_{0}^{2})dx.$

Letting $m$ goes to $+\infty$ and taking the fact that $u^{m}\rightarrow u$ in
$C\left([0,T];L^{2}(\Omega)\right)$ into consideration, we arrive at

$\displaystyle\begin{aligned}
&\quad\frac{1}{2}\int_{\Omega}u(T)^{2}dx+\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x,t)-u(y,t)|}{|x-y|^{N+s}}dxdydt\\\
&\leq\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}fudx+\frac{1}{2}\int_{\Omega}u_{0}^{2}dx.\end{aligned}$
(4.20)

Combining (4.20) and (4.17), there holds

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x,t)-u(y,t)|}{|x-y|^{N+s}}dxdydt\leq\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}Z\frac{u(x,t)-u(y,t)}{|x-y|^{N+s}}dxdydt,$

which implies $Z(x,y,t)\in\text{sgn}\left(u(x,t)-u(y,t)\right)$. ∎

######  Remark 4.1.

By an approximation argument, we may take $\varphi=u-v$ in the formulation

$\displaystyle\quad-\int_{0}^{T}\int_{\Omega}(u-v)\varphi_{t}dxdt+\int_{\Omega}\left(u(T)-v(T)\right)\varphi(x,T)dxdt$
$\displaystyle=-\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\left(Z_{u}-Z_{v}\right)\frac{\varphi(x,t)-\varphi(y,t)}{|x-y|^{N+s}}dxdydt$

to deduce

$\displaystyle 0$
$\displaystyle\leq\frac{1}{2}\int_{\Omega}\left(u(T)-v(T)\right)^{2}dx$
$\displaystyle=-\int_{0}^{T}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\left(Z_{u}-Z_{v}\right)\frac{u(x,t)-v(x,t)-u(y,t)+v(y,t)}{|x-y|^{N+s}}dxdydt\leq
0,$

where $Z_{u}\in\mathrm{sgn}\left(u(x,t)-u(y,t)\right)$ and
$Z_{v}\in\mathrm{sgn}\left(v(x,t)-v(y,t)\right)$. The uniqueness of weak
solutions to problem (1.1) follows immediately.

### Data availability

No data was used for the research described in the article.

### Conflict of interest

The authors have no non-financial Conflict of interest to declare that are
relevant to the consent of this article.

### Acknowledgments

This work was supported by the National Natural Science Foundation of China
(No. 12071098) and the Fundamental Research Funds for the Central Universities
(No. 2022FRFK060022).

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# Pose2RGBD. Generating Depth and RGB images from absolute positions.

Pîrvu Mihai Cristian
University "Politehnica" of Bucharest & MorphL
<EMAIL_ADDRESS>
###### Abstract

We propose a method at the intersection of Computer Vision and Computer
Graphics fields, which automatically generates RGBD images using neural
networks, based on previously seen and synchronized video, depth and pose
signals. Since the models must be able to reconstruct both texture (RGB) and
structure (Depth), it creates an implicit representation of the scene, as
opposed to explicit ones, such as meshes or point clouds. The process can be
thought of as neural rendering, where we obtain a function $f:Pose\rightarrow
RGBD$, which we can use to navigate through the generated scene, similarly to
graphics simulations. We introduce two new datasets, one based on synthetic
data with full ground truth information, while the other one being recorded
from a drone flight in an university campus, using only video and GPS signals.
Finally, we propose a fully unsupervised method of generating datasets from
videos alone, in order to train the Pose2RGBD networks. Code and datasets are
available at: https://gitlab.com/mihaicristianpirvu/pose2rgbd.

##  1 Introduction

The field of Computer Vision has been mostly concerned with inverse graphics
problems, which takes one or more pictures and tries to explain them using
different levels of representations, such as classification [2], depth
estimation [4, 15], object detection [20] or semantic segmentation [1]. On the
other hand, the field of Computer Graphics has mostly focused on generating
fast and high fidelity sceneries using handcrafted models. Their main focus is
either realism, with advances both in level of detail for the used models and
algorithmic, such as ray-tracing [21] or speed, using various engineering and
hardware solutions that enables the solutions for consumer level PCs.

Classical rendering pipelines involves processing hard-coded meshes, texture
mapping, applying lighting, shadows and other effects, while simultaneously
being conscious about clipping and level of detail of far or occluded views in
order to minimize the amount of work that needs to be done on GPU. Then, the
system must also map 3D coordinates to a 2D viewport, and this process is
repeated at every frame, with very little room for parallelism ahead of time.

Recently, there has been some traction in the domain of differentiable
rendering [12, 17, 8], which proposes end-to-end differentiable methods of
simulating the traditional rendering pipelines, processing explicit
representations, such as meshes, to produce 2D projections and texture
mappings. Inverse problems have also been tackled, like generating meshes from
RGB images [23]. We come to complete the picture by generating novel views
from absolute positions directly, bypassing any sort of dense and redundant
representations of a view.

Classical Computer Vision pipelines, based on the idea of Structure from
Motion [11] and Visual Odoemtry [16] try to solve the same problem, which is
representing a 3D model acquired using 2D imagery that are tied together using
various algorithms. A typical pipeline involves gathering some images of a
scene, using various algorithms to extract keypoints in each image [14],
merging them together, by predicting a relative pose between the images, and
then completing the 3D model image by image. Usually the model is stored
explicitly as point clouds, which may also impose memory restrictions. This
process is very error prone, but has been a main research topic and has led to
very impressive results. Having an explicit 3D model is not always practical,
and thus, we try to solve this issue by using a neural network to implicitly
memorize the 3D structure and directly infer the 2D projections when required
at any given pose inside some predefined boundaries.

Other closely tied problems are the tasks of inferring the absolute pose from
imagery [9], navigating through novel scenes [7], predicting dense maps depth
maps from RGB inputs in supervised [4] or unsupervised fashion [25], as well
as generating novel RGB images from noise using generative models [10, 6].

Finally, the closest article related to our work is [5], which uses RGB images
and absolute poses in order to encode information about a specified scene.
Their Generative Query Network is then able to implicitly render novel views
in their simulated environment. Unlike their work, we show that the model can
be trained using only absolute pose and can infer both RGB and Depth images
simultaneously. We also use much more complicated datasets, that simulates a
flying UAV in an outdoor scene.

While this work is not directly aimed at completely replacing the traditional
rendering mechanism, it aims to offer new way of producing 2D projections from
an implicit 3D model, using a neural network that regresses both textures and
structure in the forms of RGB and Depth images, from an absolute position. The
stored 3D scene is just an approximation of the original scene, so the level
of detail will most likely be reduced, however, in most cases, not every
intricate detail is needed to make high level inferences, like avoiding
obstacles or detecting objects. The main difference between the classical
rendering pipeline and this work is that the details of the world are
compressed in a high level representation directly from observed data. The
network incorporates in its weights a compressed implicit 3D model, which can
then be referenced and used to produce novel viewpoints.

In the following section, we’ll discuss about our proposed method and network
architecture. Then, we’ll introduce the synthetic and real world datasets that
were used followed by a general technique to create a new dataset from scratch
for this problem using only a video footage. Finally, we’ll provide a series
of results and studies we ran, followed by discussing possible future
directions and improvements.

##  2 Proposed method

The proposed model, succinctly called Pose2RGBD, aims to compute a function
$f:(P,w)\rightarrow RGBD$, represented as a neural network, that takes as
input an absolute pose (P) and makes a dense estimation of a 4 channel map
(RGBD), by adjusting the trainable weights (w) such that a pixel-wise cost
function is minimized between the predictions and the ground truth values of
the training set. The absolute pose is represented as 6DoF (translation +
rotation) restricted to a space of $[-1:1]$, by applying min-max normalization
over its extremes. The translation is represented simply as the offset from
the center of the surveilled space. The rotation can represented both as Euler
angles ($\phi,\theta,\psi$) or quaternions ($a,b,c,d$), however the later case
gave us better results, and thus was prefered.

The model must then project this low dimensional and non-redundant information
into a high dimensional dense representation. The output RGBD map is also
normalized to $[-1:1]$ and then reprojected into the original space for
computing metrics, such as Depth error in meters (for the synthetic dataset)
or Absolute Pixel error (for both datasets). The proposed architecture is
inspired by the generator of DCGAN [19] and is described in Figure 1.

There are two versions of models here. The first one, which contains only
orange boxes, is simply the generator of the DCGAN, starting from a 6DoF
representation, using a dense layer to project it to a higher dimensional
space and then reshaping to 4x4x1024, in order to be processed by 2D
transposed Convolution operations, until reaching a 512x512x4 output. The loss
function is computed using standard pixel-wise MSE. The second version, titled
Model Slice, has a few additional quirks. First, the introduction of a
Bottleneck layer, which aims process the high level feature maps. All the
operations done in this layer are done using traditional 2D convolutional
operations, with a kernel size of 3. Then, we are also using a secondary
output map, where we slice the depth map in multiple progressive layers.
Basically, if our depth map has an output range of $[-1:1]$, then, for $S=10$
slices, we’ll get 10 binary channels with positive values at the locations
where the depth map is inside each slice’s interval. The slices are divided
uniformly, so the first one captures the depth in range $[-1,-0.8)$, the
second one has the range $[-0.8,1)$ and so on until the last one that includes
the range $[0.8:1]$. The loss function on this layer is computed using
standard binary cross entropy for each slice independently. Also, the slice
are concatenated to the feature maps of the penultimate layer of the
bottleneck. The reasoning for this is simply to combine the high level feature
maps with the hand-crafted features caused by the layering. We’ve varied the
number of slices, which offers a trade-off between speed and performance, as
we’ll present in the experiments section. Finally, these slices can be
interpreted as confidence maps by summing them. Batch Normalization is used
between all layers, except the output ones, where we use hyperbolic tangent
for the predicted RGBD map and sigmoid for the depth slices.

Figure 1: Pose2RGBD Model Architecture. Inputs are represented as 6DoF poses,
while outputs are dense 4-channeled RGBD maps.

###  2.1 Creating a dataset from scratch

In order to train the Pose2RGBD network, one needs to have all 3 sources of
information synchronized: RGB, Absolute Pose and Depth for each frame of a
video. The network can be trained on multiple videos as well, however they
should all be in the same spacial environment. Having two identical poses
pointing to two different images will simply make no sense.

The first case will assume that we have collected a set of videos and
localization data, however they are unsynchronized and recorded at different
frequencies.

Algorithm 1 Synchronizing RGB and Absolute Pose

1:$GPS^{\prime}\leftarrow\textsc{interpolate}(GPS,RGB-Freq)$

2:$Flow\leftarrow\textsc{Optical Flow}(RGB)$

3:$Flow-Mag\leftarrow\sqrt{Flow(u)^{2}+Flow(v)^{2}}$

4:$GPS-Sync\leftarrow\textsc{Match}(Flow-Mag,GPS^{\prime})$

5:$\textbf{return}{}RGB,GPS-Sync$

This pseudocode is divided in 4 steps. Firstly, we assume that the absolute
pose comes from a GPS source, however this is not the only option. Usually,
the camera sensor records at 24 or 30 FPS, however the GPS can record anywhere
between 5 to 100 updates per second. We need to synchronize the data by
extrapolating or interpolating them missing points according to the RGB
frequency. Then, we’ll compute the Optical Flow of the video, using any off
the shelf algorithm, such as [22]. Plotting the mean magnitude of the optical
flow per frame over all frames, alongside with the speed of the vehicle from
GPS data results in a figure similar to Figure 2.

Figure 2: GPS Speed vs Optical Flow magnitude in a video. Manual
synchronization was done in critical points.

What the Figure shows is that there is a strong correlation between optical
flow and the speed of the vehicle. Most of the times when the magnitude of the
flow is constant, the speed is also constant. When abrupt changes happen, such
as acceleration or direction change, both signals will have spikes, but not
necessarily in the same direction of the magnitude. We have manually annotated
"hot" paths, where the synchroization should be done. We can see the strong
correlation between the red and the green lines the signals were denoised
using a median filter or similar signal processing techniques. Having an
algorithm that matches these correlated matches will give us an offset.
Finally, we will keep only the maximum intersection between the two newly
synchronized data sources.

The next algorithm focuses on computing dense Depth maps from video sources
and scaling them to match the absolute pose coordinate system.

Algorithm 2 Computing and scaling Depth from RGB + Pose

1:$Disparity,RP\leftarrow\textsc{DepthFromRGB}(RGB)$

2:$ScaledRP\leftarrow\textsc{RelativeFromAbsolute}(Pose)$

3:$Scale\leftarrow\textsc{FindScale}(RGB,ScaledRP,RP)$

4:$ScaledDepth\leftarrow\textsc{ApplyScaling}(Scale,Disparity)$

5:$\textbf{return}{}ScaledDepth$

We can pick any off the shelf unsupervised depth estimation network, such as
[25]. This method is able to take as input, once trained, any frame of a video
and return an unscaled disparity map. It can also take any pair of two frames
and return an unscaled relative pose between the two of them. From the set of
synchronized absolute poses, having followed Algorithm 1, we can compute
another set of scaled relative poses between each frame. Having access to
these two sets of relative poses, we can compute the per frame scaling factor
that can be applied to each disparity map. One way to do this is to apply the
view synthesis operator with both sets of relative poses such that we match
and minimize the difference between them. It should be noted that we don’t
really need to apply the last step to train the Pose2RGBD network, since we
already have our 3 ingredients: RGB, Absolute Pose and (Unscaled) Depth. This
is indeed the case for our real world dataset, where we synchronized the GPS
with the video for all the flights, however, we only computed the unscaled
depth maps.

Finally, this process can be done in a completly unsupervised fashion by
composing multiple unscaled relative poses together in order to create a
pseudo absolute pose, relative to a key frame. This is a well studied problem
and the key challenges is that drifts will appear caused by accumulated
errors, however these errors will be consistent with the predicted unscaled
depth maps, therefore the dataset is synchronized. Bundle adjustment
techniques can be applied to fix this as well as using more robust methods,
like Structure from Motion and explicit 3D models.

###  2.2 Proposed Datasets

For the purpose of this paper, we have created two datasets, one based on the
CARLA simulator [3] and one taken by flying an UAV over the campus of an
university and then applying the steps previously described in Algorithms 1
and 2.

####  2.2.1 Synthetic Dataset

The synthetic dataset contains 11,085 RGBD images and Absolute Poses over a
surface of about 300x300 meters. The images are taken at a resolution of
854x854. The altitude is kept at a constant of about 55m, while all the 3
rotation angles of the camera are varied with a random factor, however it
always looks towards the city, similarly to how an UAV would fly. The purpose
of this dataset is to show that, given perfect information, the network is
able to learn sufficient information in order to compute an internal 3D
representation over a relatively large surface. In Figure 3 we can observe the
flight pattern used to export the dataset.

Figure 3: Synthetic Dataset flight path

####  2.2.2 Real World Dataset

The real world dataset is constructed by doing an UAV flight over the campus
of an university over a range of 260x150m. The recording is 12 minutes long
and is recorded at 4K@24FPS. The dataset is sampled such that the altitude is
constant at about 50m above the ground, as this is the desired setup, thus
removing starting/ending frames where the UAV takes off and lands. In total,
we get 15,605 usable RGB frames. The flight is done such that the gimbal is
kept at almost the same angle, pointing towards the ground while the drone
surveys in a U-shaped path, as can be seen in Figure 4. The RGB frames are
then synchronized with the raw GPS log, as described in Algorithm 1. Then, we
train an off-the-shelf unsupervised depth estimation network on these frames
and export the depths map for each RGB frame, as described in Algorithm 2,
however we only keep the unscaled version of the maps, so we have no
correspondence between meters and the resulted values.

Figure 4: Real Dataset flight path

While the pattern is not as nice as the synthetic case, we can see that the
flights try to maintain a preset route. In the following section, where we
present the results of the Pose2RGBD network over these datasets, the first
dataset will be called Synthetic, while the real dataset will be called Real.

Figure 5: Samples from the two datasets alongside with a slicing of 4. Top
row: Real dataset. Bottom row: Synthetic dataset.

In Figure 5 we present two samples from the real and synthetic datasets using
a depth slicing of 4. We observe how the slicing divides the scene in multiple
regions, obtaining a soft segmentation over the scene directly from the depth
maps.

##  3 Experiments and results

The two networks, which we’ll call Base and Slice, as described in Figure 1,
were trained on the two datasets, Real and Synthetic. Most of the experiments
were run on the Synthetic dataset and only a small subset of them were also
redone on the Real dataset. Two input types were used, Translation + Euler
angles, which will be called 6DoF and Translation + Quaternions, which will be
called 6DoF-Quat. Depth errors are computed as mean pixel in meters for the
Synthetic dataset. However, for the Real dataset, since the depth is not
normalized, it has no real world significance. For the RGB predictions, we use
the mean absolute pixel error in the range $[0:255]$. Since these two are
strongly correlated, as we’ll see, we can use this metric as a proxy for the
quality of the depth map as well. All the inputs and outputs were normalized
in the range of $[-1:1]$ and all the outputs had hyperbolic tangent as the
final activation function. We also tried changing the normalization to $[0:1]$
or removing the activation function for the final layer, however the results
were very similar ( 98% relative errors), so, for consistency, we kept the
same setup everywhere. The train/validation splits are done by randomizing the
frames’ order and doing a 0.8/0.2 split. All outputs have a resolution of
512x512, regardless of the Dataset, in order to use the same models. This
means that, even if the Real dataset has a 9:16 aspect ratio, we resize it to
1:1, which loses some realism, but in this way we can compare the results to
the Synthetic dataset directly. All the models were trained using the PyTorch
1.3 [18] framework using the NVIDIA Tesla P100, up to 100 epochs and optimized
using the AdamW optimizer [13], a fixed learning rate of 0.01 and no
hyperparameter tuning.

Input | Output | RGB (px) | Depth (m)
---|---|---|---
6DoF | Depth | n/a | 8.2
6DoF | RGB | 20.96 | n/a
6DoF | RGBD | 21.45 | 8.85
6DoF-Quat | Depth | n/a | 8.02
6DoF-Quat | RGB | 20.78 | n/a
6DoF-Quat | RGBD | 21.13 | 8.41
Table 1: 6DoF vs 6DoF-Quat results for the Base Model on the Synthetic
Dataset.

What we observe in Table 1 is that the Quaternion representation outperforms
the regular Euler angles representation in all the experiments, which is why
they are used for all subsequent experiments. We also observe that, while
there is a small drop in performance to predict both RGB and Depth, the
relative performance loss is just of 1.65%, respectively 4.64%.

We then move to predicting both RGBD as well as Depth Slices, as described in
Section 2.

Model | Output | RGB (px) | Depth (m)
---|---|---|---
Base | Depth | n/a | 8.02
Base | RGBD | 21.13 | 8.41
Slice-10 | Depth | n/a | 7.23
Slice-10 | RGBD | 20.71 | 7.01
Slice-32 | Depth | n/a | 6.31
Slice-32 | RGBD | 20.39 | 6.97
Slice-64 | Depth | n/a | 6.64
Slice-64 | RGBD | 20.37 | 7.1
Table 2: Base vs Slice models on the Synthetic dataset. The input is 6Dof-
Quat, so it is omitted.

In Table 2 we observe how using the updated model with depth slices improves
the quality of the results significantly. We observe, however, that blindly
increasing the number of slices can actually decrease the performance, thus
this is a hyperparameter which must be tuned accordingly. Some sceneries might
perform better with a small number (e.g. indoor), while others might benefit
with a larger number (e.g. aerial images).

Model | Output | RGB (px) | Depth*
---|---|---|---
Base | Depth | n/a | 0.01874
Base | RGBD | 24.91 | 0.01915
Slice-10 | Depth | n/a | 0.0147
Slice-10 | RGBD | 24.82 | 0.01449
Table 3: Base vs Slice models on the Real dataset. The input is 6Dof-Quat, so
it is omitted. *Note: Depth is unscaled.

Table 3 presents the results for the Real dataset. The Depth error is based on
the internal scale of the method that was used to generate the labels, so it
has no real world interpretation. However, we can correlate it to the pixel
error of the RGB results, which can act as a proxy for fidelity. Using the
ratio between RGB and Depth error in Table 2 can also give us a rough
estimation of the actual depth error.

Model | parameters | Batch Size | RAM (MB) | FPS
---|---|---|---|---
CPU | GPU1 | GPU2
Base | 2,928,673 | 1 | 31.34 | 32.85 | 245.75 | 357.74
5 | 86.13 | 41.04 | 322.84 | 1080.73
10 | 124.14 | 37.16 | 336.15 | 1314.48
Slice-10 | 2,975,633 | 1 | 128.78 | 7.85 | 63.67 | 122.38
5 | 320.39 | 7.58 | 78.64 | 330.89
10 | 924.69 | 6.92 | 80.41 | 410.23
Slice-32 | 2,987,601 | 1 | 151.63 | 7.63 | 60.80 | 120.67
5 | 579.22 | 6.74 | 74.43 | 302.26
10 | 1258.21 | 6.67 | 76.66 | 365.44
Slice-64 | 3,005,009 | 1 | 164.69 | 7.04 | 56.86 | 117.71
5 | 655.28 | 6.49 | 68.87 | 264.56
10 | 1446.07 | 6.50 | 65.24 | 167.27
Table 4: Inference performance between multiple models. Experiments ran on
Intel i7-6700HQ, NVIDIA GTX 960m and NVIDIA V100.

Table 4 presents a study about performance in space and time on 3 different
processing units. Inputs to the networks are represented as 6Dof-Quat and
outputs are of shape 512x512x1 (Depth). The FPS is computed by multiplying the
inference time to the batch size. In theory, we could compute, using a batch
of 10, the current frame and 9 frames ahead, given a known trajectory, so the
term FPS makes sense here. The experiments for each configuration were run 100
times and the results were averaged. The observed RAM (or VRAM for GPUs) was
taken by computing the difference of memory usage before and after a forward
pass. Since the models are so shallow, we observe very fast inference times,
while still using little memory. Pipelining multiple consecutive frames also
brings a benefit, since the model can make future predictions ahead of time
and then just feed them with no computation cost. The number of slices should
be treated as a hyperparameter, that is dependant on the type of data since it
imposes a penalty hit if increased too much.

We end this section by providing a few qualitative results from the Synthetic
and Real datasets, from the best performing models in Figure 6. The first and
third columns are ground truth RGB and Depths, while the second and forth
columns are predicted maps. The fifth column is the confidence map, computed
from summing each channel of the predicted depth slices, as described in
Section 2. Black pixels represent low confidence, white pixels are pixels
predicted by multiple maps and gray pixels are confident pixels, predicted by
only one map. Ideally, the whole picture should look gray.

##  4 Conclusions and further work

We have proposed a technique that implicitly learns a high level
representation of a scene, using the correlation between RGB, Depth and
Absolute Pose signals. Upon training, these models can be used as a neural
renderer to produce novel RGBD images for a given pose input, even in unseen
scenes. While the results are not of a very high quality, it should be noted
that the input itself has no redundancy and is very low dimensional, so the
network has to learn a direct mapping between position and the high
dimensional visual space.

These networks resemble very much generative networks, such as GANs or VAEs,
so adding an adversarial training or trying to minimize the divergence between
the input distribution and the output reconstruction are very obvious next
steps that should improve the results.

We could also add other helpful cost functions, such as consistency losses
between nearby poses, as done by [24] should add a significant benefit to the
results.

Figure 6: Qualitative results from the Synthetic (top 4 rows) and Real (bottom
4 rows) datasets.

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