# On the limiting problems for two eigenvalue systems and variations

H. Bueno  Departmento de Matemática, Universidade Federal de Minas Gerais,
31270-901 - Belo Horizonte - MG, Brazil<EMAIL_ADDRESS>and  Aldo H. S.
Medeiros  Departamento de Matemática, Universidade Federal de Viçosa,
36570-900 - Viçosa - MG, Brazil<EMAIL_ADDRESS>
###### Abstract.

Let $\Omega$ be a bounded, smooth domain. Supposing that
$\alpha(p)+\beta(p)=p$, $\forall\,p\in\left(\frac{N}{s},\infty\right)$ and
$\displaystyle\lim_{p\to\infty}\alpha(p)/{p}=\theta\in(0,1)$, we consider two
systems for the fractional $p$-Laplacian and a variation on the first system.
The first system is the following.

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u|^{\alpha(p)-2}u|v(x_{0})|^{\beta(p)}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{t}v(x)=\lambda\beta(p)\left(\displaystyle\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)|v(x_{0})|^{\beta(p)-2}v(x_{0})\delta_{x_{0}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
\end{array}\right.$

where $x_{0}$ is a point in $\overline{\Omega}$, $\lambda$ is a parameter,
$0<s\leq t<1$, $\delta_{x}$ denotes the Dirac delta distribution centered at
$x$ and $p>N/s$.

A variation on this system is obtained by considering $x_{0}$ to be a point
where the function $v$ attains its maximum. In this case, we denote
$x_{0}=x_{v}$.

The second one is the system

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u(x_{1})|^{\alpha(p)-2}u(x_{1})|v(x_{2})|^{\beta(p)}\delta_{x_{1}}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{t}v(x)=\lambda\beta(p)|u(x_{1})|^{\alpha(p)}|v(x_{2})|^{\beta(p)-2}v(x_{2})\delta_{x_{2}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $x_{1},x_{2}\in\Omega$ are arbitrary, $x_{1}\neq x_{2}$. Although we not
consider here, a variation similar to that on the first system can be solved
by practically the same method we apply.

We obtain solutions for the systems (including the variation on the first
system) and consider the asymptotic behavior of these solutions as
$p\to\infty$. We prove that they converge, in the viscosity sense, to
solutions of problems on $u$ and $v$.

###### Key words and phrases:

fractional systems, variational methods, viscosity solutions

###### 1991 Mathematics Subject Classification:

35R11, 35A15, 35D40

##  1\. Introduction

In this paper we deal with different systems for the fractional $p$-Laplacian
and study the behavior of their solutions $(u_{p},v_{p})$ as $p$ goes to
infinity: we prove that these solutions converge, in the viscosity sense, to
solutions $(u_{\infty},v_{\infty})$ of related systems.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded, smooth domain and, for each
$x\in\Omega$, let $\delta_{x}$ be the Dirac mass concentrated at $x$. Consider
also functions
$\alpha,\beta\colon\left(\frac{N}{s},\infty\right)\to(1,\infty)$ satisfying

  1. $(h_{1})$

$\alpha(p)+\beta(p)=p$, $\forall\,p\in\left(\frac{N}{s},\infty\right)$;

  2. $(h_{2})$

$\displaystyle\lim_{p\to\infty}\frac{\alpha(p)}{p}=\theta\in(0,1)$.

For each $p>\frac{N}{s}$, we consider the system

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u|^{\alpha(p)-2}u|v(x_{0})|^{\beta(p)}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{t}v(x)=\lambda\beta(p)\left(\displaystyle\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)|v(x_{0})|^{\beta(p)-2}v(x_{0})\delta_{x_{0}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
\end{array}\right.$ ($P^{1}_{p}$)

where $x_{0}$ is a point in $\overline{\Omega}$, $\lambda$ is a parameter,
$0<s\leq t<1$ and $(-\Delta_{p})^{r}$ denotes the $r$-fractional $p$-Laplacian
operator, which is defined, for any $p>1$, by

$(-\Delta_{p})^{r}\phi(x)=\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}\frac{|\phi(x)-\phi(y)|^{p-2}(\phi(x)-\phi(y))}{|x-y|^{N+rp}}\mathrm{d}x\mathrm{d}y$
(1)

for any $\phi\in C^{\infty}_{0}(\Omega)$, which is a dense subespace of
$W^{r,p}_{0}(\Omega)$. We also recall that

$\big{\langle}(-\Delta_{p})^{r}u,\varphi\big{\rangle}:=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{N+rp}}\mathrm{d}x\mathrm{d}y$

is the expression of $(-\Delta_{p})^{r}$ as an operator from
$W^{r,p}_{0}(\Omega)$ into its dual. (The definition of the space
$W^{r,p}_{0}(\Omega)$ will be given in the sequence.)

We first prove that, for each $p>N/s$, this system has a unique solution. Then
we consider the behavior of a sequence of these solutions as $p\to\infty$ and
prove that they converge uniformly to $(u_{\infty},v_{\infty})$, which are
viscosity solutions of a related system. (Precise statements are given in the
sequence.)

As a variation on system ($P^{1}_{p}$), we consider the system

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u|^{\alpha(p)-2}u|v(x_{v})|^{\beta(p)}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{t}v(x)=\lambda\beta(p)\left(\displaystyle\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)|v(x_{v})|^{\beta(p)-2}v(x_{v})\delta_{x_{v}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ ($P^{1}_{\infty}$)

where $x_{v}$ is a maximum point of $v$ in $\overline{\Omega}$. Observe that
the first equation in ($P^{1}_{\infty}$) can be replaced by
$(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u|^{\alpha(p)-2}u\|v\|_{\infty}^{\beta(p)}$
in $\Omega$. To solve the above system we apply the same method used to handle
problem ($P^{1}_{p}$), see Remark 8.

We also handle the system

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u(x_{1})|^{\alpha(p)-2}u(x_{1})|v(x_{2})|^{\beta(p)}\delta_{x_{1}}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{t}v(x)=\lambda\beta(p)|u(x_{1})|^{\alpha(p)}|v(x_{2})|^{\beta(p)-2}v(x_{2})\delta_{x_{2}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$ ($P^{2}_{p}$)

where $x_{1},x_{2}\in\Omega$ are arbitrary points, $x_{1}\neq x_{2}$.

Of course, we could also consider the case where $x_{u}$ and $x_{v}$ are
points of maxima of $u$ and $v$, respectively, since our reasoning also solves
this case.

In Section 2–5 we handle system ($P^{1}_{p}$), while system ($P^{1}_{\infty}$)
is considered in Remark 8. Finally, in Section 6 we deal with problem
($P^{2}_{p}$).

##  2\. Background, setting and description of results

Due to the appropriate Sobolev embedding, the solutions $(u,v)$ of both
problems ($P^{1}_{p}$) and ($P^{2}_{p}$) must be continuous.

Since both equations in the system have the same homogeneity, ($P^{1}_{p}$)
and ($P^{2}_{p}$) are actually eigenvalue problems. The eigenvalue problem for
the $s$-fractional $p$-Laplacian operator was studied by Lindgren and
Lindqvist in the pioneering paper [9]. Precisely, they studied the problem

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u=\lambda_{1}(s,p)|u|^{p-2}u(x)&{\rm
in}\ \ \Omega,\\\ u=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega.\\\
\end{array}\right.$ (2)

The authors proved that the minimum of the Rayleigh quotient associated with
(2), that is,

$\lambda_{1}(s,p)=\inf_{u\in
W^{s,p}_{0}(\Omega)\setminus\\{0\\}}\frac{[u]_{s,p}^{p}}{\|u\|_{p}^{p}}=\frac{[\phi_{p}]_{s,p}^{p}}{\|\phi_{p}\|_{p}^{p}}.$

is attained by a function that does not change sign in $\Omega$.

In the case $p=\infty$ of the same paper, Lindgren and Lindqvist denoted

$\lambda_{1}(s,\infty)=\inf\left\\{\frac{\left\|\frac{u(x)-u(y)}{|x-y|^{s}}\right\|_{\infty}}{\|u\|_{\infty}}\,:\,u\in
W^{s,\infty}_{0}(\Omega)\setminus\\{0\\}\right\\}$

and showed that

$\lambda_{1}(s,\infty)=\frac{1}{R^{s}}\qquad\text{and}\qquad\lim_{p\to\infty}\sqrt[p]{\lambda_{1}(s,p)}=\lambda_{1}(s,\infty),$

where $R=\underset{x\in\Omega}{\mathrm{max\
}}\textup{dist}(x,\mathbb{R}^{N}\setminus\Omega)=\|\textup{dist}(\cdot,\mathbb{R}^{N}\setminus\Omega)\|_{\infty}$.

The results obtained in relation with Eq. (2) were extended by Del Pezzo and
Rossi in [3] to the case of systems of the form

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{r}u(x)=\lambda\alpha(p)|u(x)|^{\alpha(p)-2}u(x)|v(x)|^{\beta(p)}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{s}v(x)=\lambda\beta(p)|u(x)|^{\alpha(p)}|v(x)|^{\beta(p)-2}v(x)&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
\end{array}\right.$ (3)

when assumptions $(h_{1})$ and $(h_{2})$ are fulfilled. If for each
$p\in(\frac{N}{s},\infty)$ we denote

$\lambda_{1,p}=\inf\left\\{\frac{\frac{1}{p}[u]_{r,p}^{p}+\frac{1}{p}[v]_{s,p}^{p}}{\displaystyle\int_{\Omega}|u|^{\alpha(p)}|v|^{\beta(p)}\,\mathrm{d}x}\,:\,(u,v)\in
W^{s,p}(\Omega),\ \ uv\neq 0\right\\}$

the authors showed that $\lambda_{1,p}$ is _principal eigenvalue_ (that is, an
eigenvalue associated with an eigenfunction that does not change its sign) and

$\lambda_{s,p}^{\frac{1}{p}}\to\Lambda_{1,\infty}=\left[\frac{1}{R}\right]^{\theta
r+(1-\theta)s}\ \ \text{as}\ \ p\to\infty.$ (4)

More recently, Mihǎilescu, Rossi and Stancu-Dumitru [11] studied the system

$\left\\{\begin{array}[]{ll}-\Delta_{p}u(x)=\lambda\alpha(p)|u(x_{1})|^{\alpha(p)-2}u(x_{1})|v(x_{2})|^{\beta(p)}\delta_{x_{1}}&{\rm
in}\ \ \Omega,\\\
-\Delta_{p}v(x)=\lambda\beta(p)|u(x_{1})|^{\alpha(p)}|v(x_{2})|^{\beta(p)-2}v(x_{2})\delta_{x_{2}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm on}\ \partial\Omega,\\\ \end{array}\right.$ (5)

where $x_{1},x_{2}\in\Omega$ are arbitrary points, $x_{1}\neq x_{2}$. If
$x_{1}$ and $x_{2}$ are points of maxima of $u$ and $v$, respectively, using
arguments like those in [1, 5, 7], it can be proved that ($P^{2}_{p}$) is the
limit, as $r\to\infty$, of the problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}u=\lambda\alpha(p)\|u\|_{r}^{\alpha(p)-r}|u|^{r}\|v\|_{r}^{\beta(p)}&{\rm
in}\ \ \Omega,\\\
-\Delta_{p}v=\lambda\beta(p)\|u\|_{r}^{\alpha(p)}\|v\|_{r}^{\beta(p)-r}|v|^{r}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm on}\ \partial\Omega,\end{array}\right.$ (6)

which can be solved by classical minimization procedures.

As in [3], they proved that system (5) has a principal eigenvalue and studied
the asymptotic behavior of the principal eigenvalues and corresponding
positive eigenfunctions $u_{p}$ and $v_{p}$ as $p$ goes to infinity.
Mihǎilescu, Rossi and Stancu-Dumitru proved that the converge to $u_{\infty}$
and $v_{\infty}$, both viscosity solutions of the equation
$-\Delta_{\infty}w=0$ in .

The main goal of this work is to study system ($P^{1}_{p}$). Note that this
system is related to both systems (3) and (5). In the last section of this
article, we make clear that the method used to solve system ($P^{1}_{p}$) also
applies to system ($P^{2}_{p}$), thus generalizing system (5) from [11] to the
fractional $p$-Laplacian operator.

Due to the presence of the Dirac mass $\delta_{x}$, it is more natural to
compare the present work with [11]. We note that the integral form of the
fractional $p$-Laplacian is more difficult to handle than that of the
$p$-Laplacian. Also, in [11], it is valid the convergence

$\|\nabla u\|_{L^{p}(\Omega)}\to\||\nabla u|\|_{L^{\infty}(\Omega)},\ \
\text{for all}\ \ u\in W_{0}^{1,p}(\Omega)$

in the $p$-Laplacian case, what does not happen when we are dealing with the
Gagliardo semi-norm. Furthermore, a direct calculation with the distance
function $\text{dist}(x,\mathbb{R}^{N}\setminus\Omega)$ shows that
$|\nabla\text{dist}(x,\mathbb{R}^{N}\setminus\Omega)|=1$, but this is not
valid in our case, making more difficult to estimate the solutions of system
($P^{2}_{p}$). Furthermore, the presence of the integral term in ($P^{1}_{p}$)
changes the equation that the viscosity solutions $u_{\infty}$ and
$v_{\infty}$ satisfy, see Theorem 4.

On its turn, we will show that the eigenvalues of ($P^{1}_{p}$) converge, as
$p\to\infty$ to the same value $\Lambda_{1,\infty}$ given by (4), a result
obtained in [3].

We introduce the notation used while handling problem ($P^{1}_{p}$). In the
last section of this article, we consider problem ($P^{2}_{p}$) and make the
necessary adjustments.

For each $0<r<1$ and $p\in[1,\infty]$, we consider the Sobolev spaces
$W^{r,p}(\Omega)$

$W^{r,p}(\Omega)=\left\\{u\in
L^{p}(\Omega)\,:\,\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+rp}}\mathrm{d}x\mathrm{d}y<\infty\right\\},$

and also the spaces

$W^{r,p}_{0}(\Omega)=\left\\{u\in L^{p}(\mathbb{R}^{N})\,:\,u=0\ \text{in}\ \
\mathbb{R}^{N}\setminus\Omega\ \text{and}\ [u]_{r,p}<\infty\right\\},$

where

$[u]_{r,p}^{p}=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+rp}}\mathrm{d}x\mathrm{d}y.$

We recall that, for $0<s\leq t<1$ and $1<p<\infty$, there exists q constant
$C>0$ depending only on $s$, $N$ and $p$ such that

$\|f\|_{W^{s,p}(\Omega)}\leq C\|f\|_{W^{t,p}(\Omega)},\ \ \text{for all}\ \
f\in W^{t,p}(\Omega).$

In particular, $W_{0}^{t,p}(\Omega)\hookrightarrow W_{0}^{s,p}(\Omega)$, for
more details see [4]. So, we can consider only the space
$W_{0}^{s,p}(\Omega)$.

For each $0<s\leq t<1$, $x_{0}\in\Omega$ fixed and $p\in[1,\infty]$, we denote
$X_{s,t,p}(\Omega)=W_{0}^{s,p}(\Omega)\times W_{0}^{t,p}(\Omega)$ and

$X^{*}_{s,t,p}(\Omega)=\left\\{(u,v)\in
X_{s,t,p}(\Omega)\,:\,\left(\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)v(x_{0})\neq
0\right\\}.$

If $C_{0}(\overline{\Omega})$ stands for the space $\left\\{u\in
C(\Omega)\,:\,u=0\ \text{in}\ \mathbb{R}^{N}\setminus\Omega\right\\}$, it is
well-known that the immersion $W^{s,p}_{0}(\Omega)\hookrightarrow
C_{0}(\overline{\Omega})$ is compact for any
$p\in\left(\frac{N}{s},\infty\right)$. The compactness of this immersion is
consequence of the following Morrey’s type inequality (see [4])

$\sup_{y\neq x}\frac{|u(x)-u(y)|}{|x-y|^{s-\frac{N}{p}}}\leq C[u]_{s,p},\ \
\forall u\in W_{0}^{s,p}(\Omega),$ (7)

which holds whenever $p>\frac{N}{s}$. If $p$ is sufficiently large, the
positive constant $C$ in (7) can be chosen uniformly with respect to $p$ (see
[8], Remark 2.2).

Thus, denoting

$X_{0}(\Omega)=C_{0}(\overline{\Omega})\times C_{0}(\overline{\Omega}),$

we have the compact immersion

$X_{s,t,p}(\Omega)\hookrightarrow X_{0}(\Omega)$

for any $p\in\left(\frac{N}{s},\infty\right)$.

For $p\in\left(\frac{N}{s},\infty\right)$ and $u,v\in X^{*}_{s,t,p}$, we
define

$Q_{s,t,p}(u,v)=\frac{\frac{1}{p}[u]_{s,p}^{p}+\frac{1}{p}[v]_{t,p}^{p}}{\left(\displaystyle\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)|v(x_{0})|^{\beta(p)}}$

and

$\Lambda_{1}(p)=\inf_{(u,v)\in X^{*}_{s,t,p}(\Omega)}Q_{,s,t,p}(u,v).$

Straightforward calculations show that

$\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\left(\frac{1}{p}[u+t\varphi]_{r,p}^{p}\right)=\big{\langle}(-\Delta_{p})^{r}u,\varphi\big{\rangle},\
\ \forall\varphi\in W_{0}^{r,p}(\Omega).$ (8)

If $0<m<\infty$, then

$\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}|(u+t\varphi)(x)|^{m}=m|u(x)|^{m-2}u(x)\varphi(x),\
\ \forall\,\varphi\in L^{m}(\Omega).$ (9)

We also have, for all $1<\alpha<\infty$ and $\varphi\in L^{\alpha}(\Omega)$,

$\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\left(\int_{\Omega}|(u+t\varphi)(x)|^{\alpha}\mathrm{d}x\right)|v(x_{0})|^{\beta}=\alpha\left(\int_{\Omega}|u(x)|^{\alpha-2}u(x)\varphi(x)\mathrm{d}x\right)|v(x_{0})|^{\beta}.$
(10)

######  Definition 1.

A pair $(u,v)\in X_{s,t,p}(\Omega)$ is a weak solution to ($P^{1}_{p}$) if

$\displaystyle\left\langle(-\Delta_{p})^{s}u,\varphi\right\rangle+\left\langle(-\Delta_{p})^{t}v,\psi\right\rangle=$
$\displaystyle\lambda\left[\alpha(p)|u|^{\alpha(p)-2}u(x)|v(x_{0})|^{\beta(p)}\varphi(x)\right.$
(11) $\displaystyle\left.+\
\beta(p)\left(\int_{\Omega}|u(x)|^{\alpha(p)}\mathrm{d}x\right)|v(x_{0})|^{\beta(p)-2}v(x_{0})\psi(x_{0})\right]$

for all $(\varphi,\psi)\in X_{s,t,p}(\Omega)$.

The functional at the left-hand side of (11) is the Gâteaux derivative of the
Fréchet differentiable functional
$(u,v)\mapsto\displaystyle\frac{1}{p}[u]_{s,p}^{p}+\displaystyle\frac{1}{p}[v]_{t,p}^{p}$.
However, the functional at the right-hand side of (11) is merely related to
the right-hand Gâteaux-derivative of the functional
$(u,v)\mapsto\lambda\left(\displaystyle\int_{\Omega}|u(x)|^{\alpha(p)}\mathrm{d}x\right)|v(x_{0})|^{\beta(p)}$,
thus motivating the definition of $Q_{p}$ and $\Lambda_{1}(p)$. It is
noteworthy that minimizing that integral term is enough to minimize the whole
system.

By applying minimization methods, our first result shows that the problem
($P^{1}_{p}$) has a principal eigenvalue – and therefore, a weak solution –
for each $p\in\left(\frac{N}{s},\infty\right)$. Its proof simply adapts
Theorem 1 in [11]. We sketch the proof for the convenience of the reader in
Section 3.

######  Theorem 1.

For each $p\in\left(\frac{N}{s},\infty\right)$ we have

  1. $(i)$

$\Lambda_{1}(p)>0$;

  2. $(ii)$

there exists $(u_{p},v_{p})\in X^{*}_{s,t,p}(\Omega)$ such that

$\Lambda_{1}(p)=Q_{s,t,p}(u_{p},v_{p}),$

with $u_{p},v_{p}>0$ and
$\left(\displaystyle\int_{\Omega}|u_{p}|^{\alpha(p)}\mathrm{d}x\right)|v_{p}(x_{0})|^{\beta(p)}=1$.

The next step is to look for an operator that will motivate the study of the
problem ($P^{1}_{p}$) as $p\to\infty$. So, for each $0<s\leq t<1$ and
$p\in\left(\frac{N}{s},\infty\right)$ we denote

$\displaystyle S_{p}$ $\displaystyle=\left\\{(u,v)\in
X_{s,t,p}(\Omega)\,:\,\left(\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)|v(x_{0})|^{\beta(p)}=1\right\\}$
$\displaystyle S_{\infty}$ $\displaystyle=\left\\{(u,v)\in
X_{s,t,\infty}(\Omega)\,:\,\|u\|_{\infty}^{\theta}|v(x_{0})|^{1-\theta}=1\right\\},$

where $\theta$ was defined in ($h_{2}$).

Furthermore, for each $0<s\leq t<1$ and $p\in\left(\frac{N}{s},\infty\right]$,
we define the functions $\chi_{S_{p}}\colon X_{0}(\Omega)\to[0,\infty]$ and
$F_{p}\colon X_{0}(\Omega)\to[0,\infty]$ by

$\chi_{S_{p}}(u,v)=\left\\{\begin{array}[]{ll}0,&\text{if}\quad(u,v)\in
S_{p};\\\ \infty,&\text{otherwise}\\\ \end{array}\right.$ (12)

and

$F_{p}(u,v)=\left\\{\begin{array}[]{ll}G_{p}(u,v)+\chi_{S_{p}}(u,v),&\text{if}\quad(u,v)\in
X^{*}_{s,t,p}(\Omega);\\\ \infty,&\text{otherwise},\end{array}\right.$ (13)

with $G_{p}$ defined by

$G_{p}(u,v)=\left\\{\begin{array}[]{ll}Q_{s,t,p}(u,v)^{\frac{1}{p}},&\text{if}\quad
p\in(\frac{N}{s},\infty),\vspace*{.1cm}\\\
\displaystyle\frac{\max\left\\{|u|_{s},|v|_{t}\right\\}}{\|u\|_{\infty}^{\theta}|v(x_{0})|^{1-\theta}},&\text{if}\quad
p=\infty,\\\ \end{array}\right.$ (14)

where, for $0<\sigma<1$,

$|u|_{\sigma}=\sup_{y\neq x}\frac{|u(x)-u(y)|}{|x-y|^{\sigma}}.$

The method we apply is known as $\Gamma$-convergence, but everything we use
are the properties listed in Theorem 2. Once again, the next result follows
from a straightforward adaptation of the proof of [11, Theorem 2].

######  Theorem 2.

The function $F_{\infty}$ satisfy the following properties.

  1. $(i)$

If $\\{(u_{p},v_{p})\\}$ is a sequence such that $(u_{p},v_{p})\to(u,v)$ in
$X_{0}(\Omega)$, then

$F_{\infty}(u,v)\leq\lim_{p\to\infty}\inf F_{p}(u_{p},v_{p}).$

  2. $(ii)$

For each $(u,v)\in X_{0}(\Omega)$, there exists a sequence
$\\{(U_{p},V_{p})\\}\subset X_{0}(\Omega)$ such that $(U_{p},V_{p})\to(u,v)$
in $X_{0}(\Omega)$ and

$F_{\infty}(u,v)\geq\lim_{p\to\infty}\sup F_{p}(U_{p},V_{p}).$

Thus, as a consequence of Theorem 2-($i$), we have

$F_{\infty}(u,v)\leq\lim_{p\to\infty}\inf F_{p}(u_{p},v_{p}).$

Applying this inequality to the solutions $(u_{p},v_{p})$ given by Theorem 1,
we obtain the estimate

$F_{\infty}(u,v)\leq\lim_{p\to\infty}\inf\Lambda_{1}(p)^{\frac{1}{p}}=\frac{1}{R^{s\theta+(1-\theta)t}}=\max\\{|u_{\infty}|_{s},|v_{\infty}|_{t}\\},$
(15)

where the last equality will be shown in the proof of Theorem 3. As a
consequence of Theorem 2-($ii$) and (15), we can analyze problem ($P^{1}_{p}$)
as $p\to\infty$.

Therefore, considering Theorems 1 and 2, we study the behavior of the
eigenvalues and eigenfunctions of problem ($P^{1}_{p}$) as $p\to\infty$.

######  Theorem 3.

Let $\\{p_{n}\\}$ be a sequence converging to $\infty$ and
$(u_{p_{n}},v_{p_{n}})$ the solution of ($P^{1}_{p}$) given in Theorem 1.
Passing to a subsequence if necessary,
$\\{(u_{p_{n}},v_{p_{n}})\\}_{n\in\mathbb{N}}$ converges uniformly to
$(u_{\infty},v_{\infty})\in C_{0}^{0,s}(\overline{\Omega})\times
C_{0}^{0,t}(\overline{\Omega})$. Furthermore

  1. $(i)$

$u_{\infty}\geq 0$, $v_{\infty}\geq 0$ and
$\|u_{\infty}\|^{\theta}_{\infty}|v_{\infty}(x_{0})|^{1-\theta}=1$;

  2. $(ii)$

$\displaystyle\lim_{n\to\infty}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}=\Lambda_{1,\infty}=\frac{1}{R^{s\theta+(1-\theta)t}}$;

  3. $(iii)$

$\max\left\\{|u_{\infty}|_{s},|v_{\infty}|_{t}\right\\}=\displaystyle\frac{1}{R^{s\theta+(1-\theta)t}}.$

As we will see in the sequence, the functions $u_{\infty}$ and $v_{\infty}$
are solutions, in the viscosity sense, of regular boundary value problems. In
order to distinguish between the cases (and also to avoid a double minus
sign), we change notation: for each $1<p<\infty$ we denote the
$\sigma$-fractional $p$-Laplacian by
$(-\Delta_{p})^{\sigma}=-\mathcal{L}_{\sigma,p}$, where, if $1<p<\infty$ and
$0<\sigma<1$,

$(\mathcal{L}_{\sigma,p}u)(x):=2\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+\sigma
p}}\mathrm{d}y.$

As argued in [9], this expression appears formally as follows

$\displaystyle\left\langle(-\Delta_{p})^{\sigma}u,\varphi\right\rangle$
$\displaystyle=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{N+\sigma
p}}\mathrm{d}x\mathrm{d}y$
$\displaystyle=\int_{\mathbb{R}^{N}}\varphi(x)\left(\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+\sigma
p}}\mathrm{d}y\right)\mathrm{d}x$
$\displaystyle\quad-\int_{\mathbb{R}^{N}}\varphi(y)\left(\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+\sigma
p}}\mathrm{d}x\right)\mathrm{d}y$
$\displaystyle=\int_{\mathbb{R}^{N}}\varphi(x)(\mathcal{L}_{\sigma,p}u)(x)\mathrm{d}x,\
\ \ \forall\varphi\in W_{0}^{\sigma,p}(\Omega).$

If $p=\infty$, we define

$\mathcal{L}_{\sigma,\infty}=\mathcal{L}^{+}_{\sigma,\infty}+\mathcal{L}^{-}_{\sigma,\infty},$

where

$(\mathcal{L}^{+}_{\sigma,\infty}u)(x)=\sup_{y\in\mathbb{R}^{N}\setminus\\{x\\}}\frac{u(x)-u(y)}{|x-y|^{\sigma}}\quad\text{and}\quad(\mathcal{L}^{-}_{\sigma,\infty}u)(x)=\inf_{y\in\mathbb{R}^{N}\setminus\\{x\\}}\frac{u(x)-u(y)}{|x-y|^{\sigma}},$

see Chambolle, Lindgren and Monneau [2], where the concept was introduced, but
also [9]. Observe that, since $\mathcal{L}_{\sigma,\infty}$ is not
sufficiently smooth, its solutions must be interpreted in the viscosity sense.

We recall the definition of a solution in the viscosity sense by considering
the problem

$\left\\{\begin{array}[]{ll}\mathcal{L}_{\sigma,p}u=0&{\rm in}\ \ \Omega,\\\
u=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\ \end{array}\right.$ (16)

for all $p\in(1,\infty]$.

######  Definition 2.

Let $u\in C(\mathbb{R}^{N})$ satisfy $u=0$ in $\mathbb{R}^{N}\setminus\Omega$.
The function $u$ is a viscosity supersolution of (16) if

$(\mathcal{L}_{\sigma,p}\varphi)(x_{0})\leq 0$

for each pair $(x_{0},\varphi)\in\Omega\times C_{0}^{1}(\mathbb{R}^{N})$ such
that

$\varphi(x_{0})=u(x_{0})\qquad\text{and}\qquad\varphi(x)\leq u(x)\ \ \forall
x\in\mathbb{R}^{N}.$

On its turn, $u$ is a viscosity subsolution of (16) if

$(\mathcal{L}_{\sigma,p}\varphi)(x_{0})\geq 0$

for all pair $(x_{0},\varphi)\in\Omega\times C_{0}^{1}(\mathbb{R}^{N})$ such
that

$\varphi(x_{0})=u(x_{0})\ \ \text{e}\ \ \varphi(x)\geq u(x)\ \ \forall
x\in\mathbb{R}^{N}.$

The function $u$ is a viscosity solution to the problem (16) if $u$ is both a
viscosity super- and subsolution to problem (16).

Finally, in Section 5, we prove that the solutions $u_{\infty}$ and
$v_{\infty}$ given by Theorem 3 are viscosity solutions.

######  Theorem 4.

Let $1<s\leq t<1$. Then, the functions $u_{\infty}$ and $v_{\infty}$, given by
Theorem 3, are viscosity solutions of the system

$\left\\{\begin{array}[]{llll}\max\left\\{\mathcal{L}_{s,\infty}u,\mathcal{L}^{-}_{s,\infty}u-\Lambda_{1,\infty}|u(x)|^{\theta}|v_{\infty}(x_{0})|^{1-\theta}\right\\}=0&{\rm
in}\ \ \Omega,\\\ \mathcal{L}_{t,\infty}v=0&{\rm in}\ \
\Omega\setminus\\{x_{0}\\},\\\ u=v=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega,\\\
v(x_{0})=v_{\infty}(x_{0}).\end{array}\right.$ (17)

##  3\. Some remarks on the proofs of Theorems 1 and 2

Since the proofs of Theorems 1 and 2 are simple adaptations of that one given
in [11], we only sketch them for the convenience of the reader. For details,
see [11, Theorem 1 and Theorem 2].

Sketch of proof of Theorem 1. Estimating the denominator in the definition of
$Q_{s,t,p}$, the inequalities of Young and Sobolev imply that $\Lambda_{1}>0$.
By defining

$\displaystyle U_{n}(x)$
$\displaystyle=\frac{u_{n}(x)}{\left(\displaystyle\int_{\Omega}|u_{n}|^{\alpha(p)}\mathrm{d}x\right)^{\frac{1}{p}}|v_{n}(x_{0})|^{\frac{\beta(p)}{p}}}$
and $\displaystyle V_{n}(x)$
$\displaystyle=\frac{v_{n}(x)}{\left(\displaystyle\int_{\Omega}|u_{n}|^{\alpha(p)}\mathrm{d}x\right)^{\frac{1}{p}}|v_{n}(x_{0})|^{\frac{\beta(p)}{p}}},$

we have $(U_{n},V_{n})\in X_{s,p}(\Omega)$ satisfy
$\left(\displaystyle\int_{\Omega}|U_{n}(x)|^{\alpha(p)}\mathrm{d}x\right)|V_{n}(x_{0})|^{\beta(p)}=1$.
Furthermore,

$\lim_{n\to\infty}Q_{s,t,p}(U_{n},V_{n})=\lim_{n\to\infty}Q_{s,t,p}(u_{n},v_{n})=\Lambda_{1}(s,p),$

guaranteeing the existence of $u_{p},v_{p}\in W^{s,p}(\Omega)$ such that

$\left(\int_{\Omega}|u_{p}|^{\alpha(p)}\mathrm{d}x\right)|v_{p}(x_{0})|^{\beta(p)}=1.$

and

$Q_{s,t,p}(u_{p},u_{p})=\Lambda_{1}(p).$

For any $(\phi,\psi)\in X_{s,t,p}(\Omega)$, considering

$g(t)=Q_{s,t,p}(u_{p}+t\phi,v_{p}+t\psi),$

it follows the existence of $t_{0}>0$ such that $g(t)>g(0)=\Lambda_{1}(p)$.
Since $g\in C^{1}((-t_{0},t_{0}),\mathbb{R})$m we have $g^{\prime}(0)=0$, from
what follows that $(u_{p},v_{p})$ is a weak solution to system ($P^{1}_{p}$).
An argument similar [9, Lemma 22] proves that $u_{p}>0$ and $v_{p}>0$ in
$\Omega$, showing that $\Lambda_{1}(s,p)$ is a principal eigenvalue to system
($P^{1}_{p}$). $\Box$

Sketch of proof of Theorem 2. In order to prove ($i$), suppose that
$(u_{p},v_{p})\to(u,v)\in X_{0}(\Omega)$. Passing to a subsequence, we assume
that
$\displaystyle\lim_{p\to\infty}F_{p}(u_{p},v_{p})=\displaystyle\liminf_{p\to\infty}F_{p}(u_{p},v_{p})$.
It is not difficult to discard the case $(u,v)\notin
X^{*}_{s,t,\infty}(\Omega)\cap S_{\infty}$. So, we consider the case $(u,v)\in
X^{*}_{s,t,\infty}(\Omega)\cap S_{\infty}$, which implies
$\|u\|_{\infty}^{\theta}|v(x_{0})|^{1-\theta}=1$. We can assume that
$F_{p}(u_{p},v_{p})\leq C<\infty$, since otherwise ($i$) is valid. So, for $p$
large enough, we have $(u_{p},v_{p})\in S_{p}$ and, if $k>\frac{N}{s}$, then

$\displaystyle\left(\int_{\Omega}\int_{\Omega}\frac{|u_{p}(x)-u_{p}(y)|^{k}}{|x-y|^{\left(\frac{N}{p}+s\right)k}}+\frac{|v_{p}(x)-v_{p}(y)|^{k}}{|x-y|^{\left(\frac{N}{p}+t\right)k}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{k}}$
$\leq
2^{\frac{1}{k}}|\Omega|^{2\left(\frac{1}{k}-\frac{1}{p}\right)}p^{\frac{1}{p}}\left[\frac{1}{p}[u_{p}]_{s,p}^{p}+\frac{1}{p}[v_{p}]_{t,p}^{p}\right]^{\frac{1}{p}}.$

Thus,

$\displaystyle F_{p}(u_{p},v_{p})$
$\displaystyle=Q_{s,t,p}(u_{p},v_{p})=\left[\frac{1}{p}[u_{p}]_{s,p}^{p}+\frac{1}{p}[v_{p}]_{t,p}^{p}\right]^{\frac{1}{p}}$
$\displaystyle\geq
2^{-\frac{1}{k}}|\Omega|^{2\left(\frac{1}{p}-\frac{1}{k}\right)}p^{-\frac{1}{p}}\left(\int_{\Omega}\int_{\Omega}\frac{|u_{p}(x)-u_{p}(y)|^{k}}{|x-y|^{\left(\frac{N}{p}+s\right)k}}+\frac{|v_{p}(x)-v_{p}(y)|^{k}}{|x-y|^{\left(\frac{N}{p}+t\right)k}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{k}}.$

As $p\to\infty$, results from the uniform convergence and Fatou’s Lemma that

$\displaystyle\liminf_{p\to\infty}F_{p}(u_{p},v_{p})$ $\displaystyle\geq
2^{-\frac{1}{k}}|\Omega|^{-\frac{2}{k}}\left(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{k}}{|x-y|^{sk}}+\frac{|v(x)-v(y)|^{k}}{|x-y|^{tk}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{k}}.$

Making $k\to\infty$, we obtain

$\displaystyle\liminf_{p\to\infty}F_{p}(u_{p},v_{p})$
$\displaystyle\geq\max\left\\{|u|_{s},|v|_{t}\right\\}=F_{\infty}(u,v),$ (18)

concluding the proof of ($i$).

Now we deal with the second claim. Take any $(u,v)\in X_{0}(\Omega)$ and
initially suppose that $(u,v)\notin X^{*}_{s,t,\infty}(\Omega)\cap
S_{\infty}$. Then $F_{s,\infty}(u,v)=\infty$. Consider then a sequence of
values $p\to\infty$ and, for any $p\in\left(\frac{N}{s},\infty\right)$ in the
sequence, define $u_{p}:=u$ and $v_{p}:=v$. Of course we have
$(u_{p},v_{p})\to(u,v)$ as $p\to\infty$ in $X_{0}(\Omega)$. It is not
difficult to discard the cases
$\left(\displaystyle\int_{\Omega}|u_{p}|^{\alpha(p)}\mathrm{d}x\right)|v_{p}(x_{0})|^{\beta(p)}\neq
1$. If, however, $(u,v)\in X^{*}_{s,t,\infty}(\Omega)\cap S_{\infty}$,
consider then a sequence of values $p\to\infty$ and, for any
$p\in\left(\frac{N}{s},\infty\right)$ in the sequence, define

$U_{p}(x)=\frac{u(x)}{\left(\displaystyle\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)^{\frac{1}{p}}|v(x_{0})|^{\frac{1}{p}}}\qquad\text{and}\qquad
V_{p}(x)=\frac{v(x)}{\left(\displaystyle\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)^{\frac{1}{p}}|v(x_{0})|^{\frac{\beta(p)}{p}}}.$

Then $(U_{p},V_{p})\in S_{p}$ and

$\displaystyle\limsup_{p\to\infty}F_{p}(U_{p},V_{p})=\max\bigg{\\{}|u|_{s},|v|_{t}\bigg{\\}}=F_{\infty}(u,v),$

completing the proof of ($ii$). $\hfill\Box$

##  4\. Proof of Theorem 3

Let us denote

$R=\max_{x\in\overline{\Omega}}\text{dist}(x,\mathbb{R}^{N}\setminus\Omega)=\|\text{dist}(.,\mathbb{R}^{N}\setminus\Omega)\|_{L^{\infty}(\Omega)}.$

For a fixed $x_{1}\in\Omega$ we consider the functions
$\phi_{R}\colon\overline{B_{R}(x_{1})}\rightarrow[0,R]$ and
$\psi_{R}\colon\overline{B_{R}(x_{0})}\rightarrow[0,R]$ given by

$\phi_{R}(x)=R^{(\theta-1)t-s\theta}\left(R-|x-x_{1}|\right)^{s}_{+}\quad\text{and}\quad\psi_{R}(x)=R^{(\theta-1)t-s\theta}\left(R-|x-x_{0}|\right)^{t}_{+}.$

Of course we have $\phi_{R}\in C_{0}^{0,s}(\overline{B_{R}(x_{1})}$ and
$\psi_{R}\in C_{0}^{0,s}(\overline{B_{R}(x_{0})}$. Furthermore,

$\|\phi_{R}\|_{\infty}=R^{(\theta-1)(t-s)},\quad|\psi_{R}(x_{0})|=R^{\theta(t-s)}\quad\text{and}\quad|\phi_{R}|_{s}=|\psi_{R}|_{s}=R^{(\theta-1)t-s\theta}.$

We can extend $\phi_{R}$ and $\psi_{R}$ to $\overline{\Omega}$ by putting
$\phi_{R}=0$ in $\mathbb{R}^{N}\setminus\overline{B_{R}(x_{1})}$ and
$\psi_{R}=0$ in $\mathbb{R}^{N}\setminus\overline{B_{R}(x_{0})}$ to that
$\phi_{R},\psi_{R}\in C_{0}^{0,s}(\overline{\Omega})$, maintaining its
$s$-Hölder norm. Additionally, we still have $\phi_{R},\psi_{R}\in
W_{0}^{1,m}(\Omega)\hookrightarrow W_{0}^{s,m}(\Omega)$ for all $s\in(0,1)$
and $m\geq 1$. For details, see [7, 9].

######  Lemma 5.

For any fixed $0<s\leq t<1$ we have

$\Lambda_{1,\infty}=\inf_{(u,v)\in
X_{s,t,\infty}^{*}(\Omega)}\frac{\max\big{\\{}|u|_{s},|v|_{t}\big{\\}}}{\|u\|_{\infty}^{\theta}|v(x_{0})|_{\infty}^{1-\theta}}=\frac{1}{R^{s\theta+(1-\theta)t}}.$

###### Proof.

We note that we have

$\|\phi_{R}\|_{\infty}^{\theta}|\psi_{R}(x_{0})|^{1-\theta}=R^{\theta(\theta-1)(t-s)+\theta(1-\theta)(t-s)}=1$

and therefore

$\Lambda_{1,\infty}=\inf_{(u,v)\in
X_{s,t,\infty}^{*}(\Omega)}\frac{\max\big{\\{}|u|_{s},|v|_{t}\big{\\}}}{\|u\|_{\infty}^{\theta}|v(x_{0})|_{\infty}^{1-\theta}}\leq\frac{\max\big{\\{}|\phi_{R}|_{s},|\psi_{R}|_{t}\big{\\}}}{\|\phi_{R}\|_{\infty}^{\theta}|\psi_{R}(x_{0})|_{\infty}^{1-\theta}}=\frac{1}{R^{s\theta+(1-\theta)t}}.$

Also note that, given $(u,v)\in X_{s,t,p}^{*}(\Omega)$, then $u=0=v$ in
$\overline{\Omega}$. Since $u$ is continuous, there exists
$x_{1}\in\overline{\Omega}$ such that

$\|u\|_{\infty}=|u(x_{1})|.$

The compactness of $\overline{\Omega}$ guarantees the existence of
$y_{x_{0}},y_{x_{1}}\in\partial\Omega$ such that

$|x_{0}-y_{x_{0}}|=\text{dist}(x_{0},\mathbb{R}^{N}\setminus\Omega)\quad\text{and}\quad|x_{1}-y_{x_{1}}|=\text{dist}(x_{1},\mathbb{R}^{N}\setminus\Omega).$

Thus, since $u(y_{x_{1}})=v(y_{x_{0}})=0$, it follows

$\|u\|_{\infty}^{\theta}=|u(x_{1})-u(y_{x_{1}})|^{\theta}\leq|u|_{s}^{\theta}|x_{1}-y_{x_{1}}|^{s\theta}\leq|u|_{s}^{\theta}\,R^{s\theta}.$

On the other hand,

$|v(x_{0})|^{1-\theta}=|v(x_{0})-v(y_{x_{0}})|^{1-\theta}\leq|v|_{t}^{1-\theta}|x_{0}-y_{x_{0}}|^{t(1-\theta)}\leq|v|_{t}^{1-\theta}\,R^{t(1-\theta)}.$

So, for any $(u,v)\in X^{*}_{s,t,p}(\Omega)$, we have

$\displaystyle\frac{1}{R^{s\theta+t(1-\theta)}}=\frac{1}{R^{s\theta}\,R^{(1-\theta)t}}$
$\displaystyle\leq\frac{|u|_{s}^{\theta}|v|_{t}^{1-\theta}}{\|u\|_{\infty}^{\theta}|v(x_{0})|^{1-\theta}}\leq\frac{\left(\max\big{\\{}|u|_{s},|v|_{t}\big{\\}}\right)^{\theta}\left(\max\big{\\{}|u|_{s},|v|_{t}\big{\\}}\right)^{1-\theta}}{\|u\|_{\infty}^{\theta}|v(x_{0})|^{1-\theta}}$
$\displaystyle=\frac{\max\big{\\{}|u|_{s},|v|_{t}\big{\\}}}{\|u\|_{\infty}^{\theta}|v(x_{0})|^{1-\theta}}.$

Therefore,

$\Lambda_{1,\infty}=\inf_{(u,v)\in
X_{s,t,\infty}^{*}(\Omega)}\frac{\max\big{\\{}|u|_{s},|v|_{t}\big{\\}}}{\|u\|_{\infty}^{\theta}|v(x_{0})|_{\infty}^{1-\theta}}\geq\frac{1}{R^{s\theta+(1-\theta)t}},$

concluding the proof. ∎

The next result is pivotal in our analysis of the asymptotic behavior of
solutions in problems driven by the fractional $p$-Laplacian.

######  Lemma 6.

Let $u\in C_{0}^{0,\sigma}(\overline{\Omega})$ be extended as zero outside
$\Omega$. If $u\in W^{\sigma,q}(\Omega)$ for some $q>1$, then $u\in
W_{0}^{\sigma,p}(\Omega)$ for all $p\geq q$ and

$\lim_{p\to\infty}[u]_{\sigma,p}=|u|_{\sigma}.$

The proof of Lemma 6 can be found in [6, Lemma 7].

Proof of Theorem 3. Of course we have

$\Lambda_{1}(p_{n})\leq\frac{\frac{1}{p_{n}}[\phi_{R}]_{s,p_{n}}^{p_{n}}+\frac{1}{p_{n}}[\psi_{R}]_{t,p_{n}}^{p_{n}}}{\displaystyle\int_{\Omega}\left(|\phi_{R}|^{\alpha(p_{n})}\mathrm{d}x\right)|\psi_{R}(x_{0})|^{\beta(p_{n})}}.$

Thus,

$\displaystyle\limsup_{n\to\infty}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}$
$\displaystyle\leq\limsup_{n\to\infty}\left(\frac{1}{p_{n}}\frac{[\phi_{R}]_{s,p_{n}}^{p_{n}}+[\psi_{R}]_{t,p_{n}}^{p_{n}}}{\displaystyle\int_{\Omega}\left(|\phi_{R}|^{\alpha(p_{n})}\mathrm{d}x\right)|\psi_{R}(x_{0})|^{\beta(p_{n})}}\right)^{\frac{1}{p_{n}}}$
$\displaystyle\leq\limsup_{n\to\infty}\left(\left(\frac{2}{p_{n}}\right)^{\frac{1}{p_{n}}}\frac{\max\big{\\{}[\phi_{R}]_{s,p_{n}},[\psi_{R}]_{t,p_{n}}\big{\\}}}{\displaystyle\int_{\Omega}\left(|\phi_{R}|^{\alpha(p_{n})}\mathrm{d}x\right)|\psi_{R}(x_{0})|^{\beta(p_{n})}}\right)$
$\displaystyle=\frac{\max\big{\\{}|\phi_{R}|_{s},|\psi_{R}|_{t}\big{\\}}}{\|\phi_{R}\|_{\infty}^{\theta}|\psi_{R}(x_{0})|^{1-\theta}}\leq\frac{1}{R^{s\theta+(1-\theta)t}},$

proving that the sequence
$\left\\{\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\right\\}_{n\in\mathbb{N}}$ is
bounded in $\mathbb{R}$, that is, there exists $M_{0}>0$ such that

$\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\leq M_{0}\quad\ \text{for all}\ \
n\in\mathbb{N}.$ (19)

Theorem 1 guarantees that we can take $(u_{p_{n}},v_{p_{n}})$ so that

$u_{p_{n}}>0,\
v_{p_{n}}>0\quad\text{and}\quad\left(\int_{\Omega}|u_{p_{n}}|^{\alpha(p_{n})}\mathrm{d}x\right)|v_{p_{n}}(x_{0})|^{\beta(p_{n})}=1.$

Therefore

$\Lambda_{1}(p_{n})=\frac{1}{p_{n}}[u_{p_{n}}]_{s,p_{n}}^{p_{n}}+\frac{1}{p_{n}}[v_{p_{n}}]_{t,p_{n}}^{p_{n}}\geq\frac{1}{p_{n}}\max\bigg{\\{}[u_{p_{n}}]_{s,p_{n}}^{p_{n}},[v_{p_{n}}]_{s,p_{n}}^{p_{n}}\bigg{\\}},$

what yields

$\displaystyle[u_{p_{n}}]_{s,p_{n}}\leq
p_{n}^{\frac{1}{p_{n}}}\sqrt[p_{n}]{\Lambda_{1}(s,p_{n})}.$ (20)

For a fixed $m_{0}>\frac{N}{s}$, denoting the diameter of $\Omega$ by
$\text{diam}(\Omega)$, it follows from (19) and (20) that

$\displaystyle|u_{p_{n}}|_{s-\frac{N}{m_{0}}}$ $\displaystyle=\sup_{x\neq
y}\frac{|u_{p_{n}}(x)-u_{p_{n}}(y)|}{|x-y|^{s-\frac{N}{m_{0}}}}=\sup_{x\neq
y}\frac{|u_{p_{n}}(x)-u_{p_{n}}(y)|}{|x-y|^{s-\frac{N}{p_{n}}}}\,|x-y|^{\frac{N}{m_{0}}-\frac{N}{p_{n}}}$
$\displaystyle\leq\left(\text{diam}(\Omega)\right)^{\frac{N}{m_{0}}-\frac{N}{p_{n}}}\sup_{x\neq
y}\frac{|u_{p_{n}}(x)-u_{p_{n}}(y)|}{|x-y|^{s-\frac{N}{p_{n}}}}$
$\displaystyle\leq
C\left(\text{diam}(\Omega)\right)^{\frac{N}{m_{0}}-\frac{N}{p_{n}}}\,[u_{p_{n}}]_{s,p_{n}}$
$\displaystyle\leq
C\left(\text{diam}(\Omega)\right)^{\frac{N}{m_{0}}-\frac{N}{p_{n}}}\,p_{n}^{\frac{1}{p_{n}}}\,\sqrt[p_{n}]{\Lambda_{1}(s,p_{n})}$

the constant $C$ not depending on $p_{n}$. We conclude that the sequence
$\\{u_{p_{n}}\\}$ is uniformly bounded in
$C_{0}^{0,s-\frac{N}{m_{0}}}(\overline{\Omega})$ and the same reasoning is
valid for $\\{v_{p_{n}}\\}$, showing that $\\{v_{p_{n}}\\}_{n\in\mathbb{N}}$
is uniformly bounded in $C_{0}^{0,t-\frac{N}{m_{0}}}(\overline{\Omega})$.

Passing to subsequences if necessary, there exist $u_{\infty}\in
C_{0}^{0,s-\frac{N}{m_{0}}}(\overline{\Omega})$ and $v_{\infty}\in
C_{0}^{0,t-\frac{N}{m_{0}}}(\overline{\Omega})$ such that

$u_{p_{n}}\to u_{\infty}\quad\text{and}\quad v_{p_{n}}\to v_{\infty}\ \
\text{uniformly in}\ \ \Omega.$

We also observe that

$\|u_{\infty}\|_{\infty}^{\theta}|v_{\infty}(x_{0})|^{1-\theta}=\lim_{n\to\infty}\left(\left(\int_{\Omega}|u_{p_{n}}|^{\alpha(p_{n})}\mathrm{d}x\right)|v_{p_{n}}(x_{0})|^{\beta(p_{n})}\right)^{\frac{1}{p_{n}}}=1.$

Fix $k>\frac{N}{s}$. By applying Fatou’s, Hölder’s inequality and (20), we
obtain

$\displaystyle\int_{\Omega}\int_{\Omega}\frac{|u_{\infty}(x)-u_{\infty}(y)|^{k}}{|x-y|^{sk}}\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq\liminf_{n\to\infty}\int_{\Omega}\int_{\Omega}\frac{|u_{p_{n}}(x)-u_{p_{n}}(y)|^{k}}{|x-y|^{\left(\frac{N}{p_{n}}+s\right)k}}\mathrm{d}x\mathrm{d}y$
$\displaystyle\leq\liminf_{n\to\infty}|\Omega|^{2\left(\frac{p_{n}-k}{p_{n}}\right)}\left(\int_{\Omega}\int_{\Omega}\frac{|u_{p_{n}}(x)-u_{p_{n}}(y)|^{p_{n}}}{|x-y|^{N+sp_{n}}}\mathrm{d}x\mathrm{d}y\right)^{\frac{k}{p_{n}}}$
$\displaystyle\leq|\Omega|^{2}\liminf_{n\to\infty}[u_{p_{n}}]_{s,p_{n}}^{k}$
(21)
$\displaystyle\leq|\Omega|^{2}\liminf_{n\to\infty}\left(p_{n}^{\frac{1}{p_{n}}}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\right)^{k}$
$\displaystyle\leq|\Omega|^{2}\left(\frac{1}{R^{s\theta+(1-\theta)t}}\right)^{k}.$

Thus,

$|u_{\infty}|_{s}=\lim_{k\to\infty}\left(\int_{\Omega}\int_{\Omega}\frac{|u_{\infty}(x)-u_{\infty}(y)|^{k}}{|x-y|^{sk}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{k}}\leq\lim_{n\to\infty}|\Omega|^{\frac{2}{k}}\,\frac{1}{R^{s\theta+(1-\theta)t}}=\frac{1}{R^{s\theta+(1-\theta)t}}.$

Analagously,

$|v_{\infty}|_{t}=\lim_{k\to\infty}\left(\int_{\Omega}\int_{\Omega}\frac{|v_{\infty}(x)-v_{\infty}(y)|^{k}}{|x-y|^{tk}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{k}}\leq\lim_{n\to\infty}|\Omega|^{\frac{2}{k}}\,\frac{1}{R^{s\theta+(1-\theta)t}}=\frac{1}{R^{s\theta+(1-\theta)t}}$

and therefore

$\max\big{\\{}|u_{\infty}|_{s},|v_{\infty}|_{t}\big{\\}}\leq\frac{1}{R^{s\theta+(1-\theta)t}}.$

It follows from Lemma 5 that

$\frac{1}{R^{s\theta+(1-\theta)t}}=\inf_{(u,v)\in
X_{s,t,\infty}^{*}(\Omega)}\frac{\max\big{\\{}|u|_{s},|v|_{t}\big{\\}}}{\|u\|_{\infty}^{\theta}|v(x_{0})|^{1-\theta}}\leq\max\big{\\{}|u_{\infty}|_{s},|v_{\infty}|_{t}\big{\\}}\leq\frac{1}{R^{s\theta+(1-\theta)t}},$

thus producing

$\max\big{\\{}|u_{\infty}|_{s},|v_{\infty}|_{t}\big{\\}}=\frac{1}{R^{s\theta+(1-\theta)t}}.$

On its turn, inequality (4) yields

$\displaystyle\max\left\\{\left(\int_{\Omega}\int_{\Omega}\frac{|u_{\infty}(x)-u_{\infty}(y)|^{k}}{|x-y|^{sk}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{k}},\left(\int_{\Omega}\int_{\Omega}\frac{|v_{\infty}(x)-v_{\infty}(y)|^{k}}{|x-y|^{tk}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{k}}\right\\}$
$\displaystyle\leq|\Omega|^{\frac{2}{k}}\liminf_{n\to\infty}\left(p_{n}^{\frac{1}{p_{n}}}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\right).$

Thus, as $k\to\infty$ we obtain

$\displaystyle\frac{1}{R^{s\theta+(1-\theta)t}}=\max\big{\\{}|u_{\infty}|_{s},|v_{\infty}|_{s}\big{\\}}$
$\displaystyle\leq\liminf_{n\to\infty}\left(p_{n}^{\frac{1}{p_{n}}}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\right)$
$\displaystyle\leq\limsup_{n\to\infty}\left(p_{n}^{\frac{1}{p_{n}}}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\right)\leq\frac{1}{R^{s\theta+(1-\theta)t}},$

from what follows

$\lim_{n\to\infty}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}=\lim_{n\to\infty}\left(p_{n}^{\frac{1}{p_{n}}}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\right)=\frac{1}{R^{s\theta+(1-\theta)t}}=\Lambda_{1,\infty}.$

$\hfill\Box$

##  5\. Proof of Theorem 4

The next result only shows that solutions in the weak sense are viscosity
solutions. Its proof can be achieved by adapting the arguments given by
Lindgren and Lindqvist in [9, Proposition 1].

######  Proposition 7.

The function $u_{p}$ e $v_{p}$ given by Theorem 1 are viscosity solutions to
the problems

$\left\\{\begin{array}[]{ll}\mathcal{L}_{s,p}u=\Lambda_{1}(p)\alpha(p)|u|^{\alpha(p)-1}v(x_{0})&{\rm
in}\ \ \Omega,\\\ u=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

and

$\left\\{\begin{array}[]{ll}\mathcal{L}_{t,p}v=0&{\rm in}\
\Omega\setminus\\{x_{0}\\},\\\ v=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
v(x_{0})=v_{p}(x_{0}),\end{array}\right.$

respectively.

Proof of Theorem 4. We start showing that $v_{\infty}$ is a viscosity solution
to the problem

$\left\\{\begin{array}[]{ll}\mathcal{L}_{t,\infty}v=0&{\rm in}\ \
\Omega\setminus\\{x_{0}\\},\\\ v=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
v(x_{0})=v_{\infty}(x_{0}).\end{array}\right.$ (22)

According to Theorem 3 we have $v_{\infty}=0$ in
$\mathbb{R}^{N}\setminus\Omega$ and $v_{\infty}(x_{0})=v_{\infty}(x_{0})$. So,
we need only show that $v_{\infty}$ is a viscosity solution. Fix
$(z_{0},\varphi)\in\left(\Omega\setminus\\{x_{0}\\}\right)\times
C_{0}^{1}(\mathbb{R}^{N}\setminus\\{x_{0}\\})$ satisfying

$\varphi(z_{0})=v_{\infty}(z_{0})\qquad\text{and}\qquad\varphi(x)\leq
v_{\infty}(x),\ \ \forall x\in\mathbb{R}^{N}\setminus\\{x_{0},z_{0}\\}.$

Theorem 3 also guarantees the existence of a sequence
$\\{(u_{p_{n}},v_{p_{n}})\\}_{n\in\mathbb{N}}\in
C_{0}^{0,s}(\overline{\Omega})\times C_{0}^{0,t}(\overline{\Omega})$ such that
$u_{p_{n}}\to u_{\infty}$ and $v_{p_{n}}\to v_{\infty}$ uniformly in $\Omega$.
Thus, there exists a sequence $\\{x_{p_{n}}\\}_{n\in\mathbb{N}}$ so that
$x_{p_{n}}\to z_{0}$ and $v_{p_{n}}(x_{p_{n}})=\varphi(x_{p_{n}})$. Since
$x_{0}\neq z_{0}$, we can assume the existence of $n_{0}\geq 0$ and a ball
$B_{\rho}(z_{0})$ such that

$x_{p_{n}}\notin B_{\rho}(z_{0})\subset\Omega\setminus\\{z_{0}\\},\quad\forall
n\geq n_{0}.$

Since $v_{p_{n}}$ weakly satisfies

$(-\Delta_{p_{n}})^{t}v_{p_{n}}(x)=\Lambda_{1}(p_{n})\alpha(p_{n})\left(\int_{\Omega}|u_{p_{n}}|^{\alpha(p_{n})}\mathrm{d}x\right)|v_{p_{n}}(x_{0})|^{\beta(p_{n})}v_{p_{n}}(x_{0})\delta_{x_{0}}$

in $\Omega$, then also in $\Omega\setminus\\{x_{0}\\}$, Proposition 7 yields
that $v_{p_{n}}$ is a viscosity solution to the problem

$\left\\{\begin{array}[]{llll}\mathcal{L}_{t,p_{n}}v=0&{\rm in}\ \
\Omega\setminus\\{x_{0}\\},\\\ v=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
v(x_{0})=v_{p_{n}}(x_{0}).\end{array}\right.$ (23)

By standard arguments, we obtain a sequence
$\\{z_{n}\\}_{n\in\mathbb{N}}\subset B_{\rho}(x_{0})$ such that $z_{n}\to
z_{0}$ and

$\sigma_{n}:=\min_{B_{\rho}(x_{0})}\left(v_{p_{n}}-\varphi\right)=v_{p_{n}}(z_{n})-\varphi(z_{n})<v_{p_{n}}(x)-\varphi(x),\
\ \forall x\neq x_{p_{n}}.$

Now, define $\Psi_{n}:=\varphi+\sigma_{n}$. We have

$\Psi_{n}(z_{n})=\varphi(z_{n})+\sigma_{n}=v_{p_{n}}(z_{n})\qquad\text{and}\qquad\Psi_{n}(x)=\varphi(x)+\sigma_{n}<v_{p_{n}}(x),\
\ \forall x\in B_{\rho}(x_{0}).$

Since $v_{p_{n}}$ satisfies (23) in $\Omega\setminus\\{x_{0}\\}$,

$(\mathcal{L}_{t,\infty}\Psi_{n})(z_{n})\leq 0,\qquad\forall n\geq n_{0}.$

Thus, defining

$\left(A_{p_{n},t}(\varphi(z_{n}))\right)^{p_{n}-1}:=2\int_{\mathbb{R}^{N}}\frac{|\varphi(z_{n})-\varphi(y)|^{p_{n}-2}(\varphi(z_{n})-\varphi(y))^{+}}{|z_{n}-y|^{N+tp_{n}}}\mathrm{d}y$

and

$\left(B_{p_{n},t}(\varphi(z_{n}))\right)^{p_{n}-1}:=2\int_{\mathbb{R}^{N}}\frac{|\varphi(z_{n})-\varphi(y)|^{p_{n}-2}(\varphi(z_{n})-\varphi(y))^{-}}{|z_{n}-y|^{N+tp_{n}}}\mathrm{d}y,$

we have

$\displaystyle\left(A_{p_{n},t}(\varphi(z_{n}))\right)^{p_{n}-1}-\left(B_{p_{n},t}(\varphi(z_{n}))\right)^{p_{n}-1}$
$\displaystyle=2\int_{\mathbb{R}^{N}}\frac{|\varphi(z_{n})-\varphi(y)|^{p_{n}-2}(\varphi(z_{n})-\varphi(y))}{|z_{n}-y|^{N+sp_{n}}}\mathrm{d}y$
$\displaystyle\leq 0,\quad\forall n\geq n_{0}.$ (24)

Applying [7, Lemma 3.9] (see also [8, Lemma 6.1]), we obtain

$\lim_{n\to\infty}A_{p_{n},t}(\varphi(z_{n}))=\left(\mathcal{L}_{t,\infty}^{+}\varphi\right)(z_{0})\qquad\text{and}\qquad\lim_{n\to\infty}B_{p_{n},t}(\varphi(z_{n}))=\left(-\mathcal{L}_{t,\infty}^{-}\varphi\right)(z_{0}).$

As $n\to\infty$ in $\eqref{Lim}$ we get

$\left(\mathcal{L}_{t,\infty}\varphi\right)(x_{0})=\left(\mathcal{L}_{t,\infty}^{+}\varphi\right)(x_{0})+\left(\mathcal{L}_{t,\infty}^{-}\varphi\right)(x_{0})\leq
0,$

showing that $v_{\infty}$ is a viscosity supersolution of (22). Analogously,
we obtain that $v_{\infty}$ is a viscosity subsolution of the same equation,
and thus a viscosity solution of (22).

Now we show that $u_{\infty}$ is a viscosity solution to the problem

$\left\\{\begin{array}[]{ll}\max\bigg{\\{}\mathcal{L}_{s,\infty}u,\mathcal{L}^{-}_{s,\infty}u+\Lambda_{1,\infty}|u(x)|^{\theta}|v_{\infty}(x_{0})|^{1-\theta}\bigg{\\}}=0&{\rm
in}\ \ \Omega,\\\ u=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$ (25)

The same reasoning used before imply that, for given
$(z_{0},\varphi)\in\Omega\times C_{0}^{1}(\mathbb{R}^{N})$, we find a sequence
$\\{u_{p_{n}}\\}_{n\in\mathbb{N}}$ in $C_{0}^{0,s}(\overline{\Omega})$ such
that $u_{p_{n}}\to u_{\infty}$ uniformly in $\Omega$ and a sequence
$\\{x_{p_{n}}\\}_{n\in\mathbb{N}}$ satisfying $x_{p_{n}}\to z_{0}$ and
$u_{p_{n}}(x_{p_{n}})=\varphi(x_{p_{n}})$. Thus, there exist $n_{0}\geq 0$ and
a ball $B_{\rho}(z_{0})$ so that

$x_{p_{n}}\notin B_{\rho}(z_{0})\subset\Omega\setminus\\{z_{0}\\},\ \ \forall
n\geq n_{0}.$

As before, we obtain that $u_{p_{n}}$ is a viscosity solution to the problem

$\left\\{\begin{array}[]{ll}\mathcal{L}_{s,p_{n}}u_{p_{n}}=\Lambda_{1}(p_{n})\alpha(p_{n})|u_{p_{n}}|^{\alpha(p_{n})-1}v_{p_{n}}(x_{0})&{\rm
in}\ \ \Omega,\\\ u=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$

Considering, as before, a sequence $\\{z_{n}\\}_{n\in\mathbb{N}}\subset
B_{\rho}(z_{0})$ such that $z_{n}\to z_{0}$ and defining $\Psi_{n}$ as in the
previous proof, we obtain

$(\mathcal{L}_{s,p_{n}}\Psi_{n})(z_{n})\leq\Lambda_{1}(p_{n})\alpha(p_{n})|\Psi_{n}(z_{n})|^{\alpha(p_{n})-1}v_{p_{n}}(x_{0})\
\ \forall n\geq n_{0},$

which is equivalent to the inequality

$\left(A_{p_{n},s}(\varphi(z_{n}))\right)^{p_{n}-1}-\left(B_{p_{n},s}(\varphi(z_{n}))\right)^{p_{n}-1}\leq\left(C_{p_{n}}(\varphi(z_{n}))\right)^{p_{n}-1}\
\ \forall n\geq n_{0},$

where

$\bigg{(}C_{p_{n}}(\varphi(z_{n}))\bigg{)}^{p_{n}-1}:=\Lambda_{1}(p_{n})\alpha(p_{n})|\varphi+\sigma_{n}|^{\alpha(p_{n})-1}v_{p_{n}}(x_{0})$

and the other terms are analogous to that of the previous case, just changing
$t$ for $s$.

Observe that a direct calculation yields

$\displaystyle\lim_{n\to\infty}C_{p_{n}}(\varphi(z_{n}))$
$\displaystyle=\lim_{n\to\infty}\left(\sqrt[p_{n}]{\Lambda_{1}(p_{n})}\sqrt[p_{n}]{\alpha(p_{n})}|\varphi(z_{n})+\sigma_{n}|^{\frac{\alpha(p_{n})}{p_{n}-1}}v_{p_{n}}(x_{0})^{\frac{\beta(p_{n})}{p_{n}-1}}\right)$
$\displaystyle=\Lambda_{1,\infty}|\varphi(z_{0})|^{\theta}v_{\infty}(x_{0})^{1-\theta}$

So, as $n\to\infty$ em $\eqref{Lim}$ we obtain

$\left(\mathcal{L}_{s,\infty}\varphi\right)(x_{0})=\left(\mathcal{L}_{s,\infty}^{+}\varphi\right)(z_{0})+\left(\mathcal{L}_{s,\infty}^{-}\varphi\right)(z_{0})\leq\Lambda_{1,\infty}|\varphi(z_{0})|^{\theta}v_{\infty}(x_{0})^{1-\theta}$

and therefore

$\max\left\\{\mathcal{L}_{s,\infty}u,\mathcal{L}^{-}_{s,\infty}u-\Lambda_{1,\infty}|u(x)|^{\theta}|v_{\infty}(x_{0})|^{1-\theta}\right\\}\leq
0\ \ {\rm in}\ \ \Omega,$

that is, $u_{\infty}$ is a viscosity supersolution to problem (22).
Analogously, $u_{\infty}$ is a viscosity subsolution to the same problem. We
are done. $\hfill\Box$

######  Remark 8.

We observe that the system

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u|^{\alpha(p)-2}u|v(x_{v})|^{\beta(p)}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{t}v(x)=\lambda\beta(p)\left(\displaystyle\int_{\Omega}|u|^{\alpha(p)}\mathrm{d}x\right)|v(x_{v})|^{\beta(p)-2}v(x_{v})\delta_{x_{v}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
\end{array}\right.$ ($P^{1}_{\infty}$)

where $x_{v}$ is a maximum point of $v$ in $\overline{\Omega}$ can be treated
in the same setting given in Section 2, applying the same procedure used to
solve system ($P^{1}_{p}$).

##  6\. On the system($P^{2}_{p}$)

In this section we consider the functional system ($P^{2}_{p}$).

$\left\\{\begin{array}[]{ll}(-\Delta_{p})^{s}u(x)=\lambda\alpha(p)|u(x_{1})|^{\alpha(p)-2}u(x_{1})|v(x_{2})|^{\beta(p)}\delta_{x_{1}}&{\rm
in}\ \ \Omega,\\\
(-\Delta_{p})^{t}v(x)=\lambda\beta(p)|u(x_{1})|^{\alpha(p)}|v(x_{2})|^{\beta(p)-2}v(x_{2})\delta_{x_{2}}&{\rm
in}\ \ \Omega,\\\ u=v=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$

where $x_{1},x_{2}\in\Omega$ are arbitrary points, $x_{1}\neq x_{2}$. Observe
that both equations are functional, so their treatment recall that used to
deal with the second equation in system ($P^{1}_{p}$).

######  Definition 3.

A pair $(u,v)\in X_{s,t,p}(\Omega)$ is a weak solution to ($P^{2}_{p}$) if

$\displaystyle\left\langle(-\Delta_{p})^{s}u,\varphi\right\rangle+\left\langle(-\Delta_{p})^{s}v,\psi\right\rangle=\lambda$
$\displaystyle\left[\alpha(p)|u(x_{1})|^{\alpha(p)-2}u(x_{1})|v(x_{2})|^{\beta(p)}\varphi(x_{1})\right.$
(26) $\displaystyle\left.\
+\beta(p)|u(x_{1})|^{\alpha(p)}|v(x_{2})|^{\beta(p)-2}v(x_{2})\psi(x_{2})\right]$

for all $(\varphi,\psi)\in X_{s,t,p}(\Omega)$.

The denominator in the definition of $Q_{s,t,p}$ should be changed into
$|u(x_{1})|^{\alpha(p)}\,|v(x_{2})|^{\beta(p)}$, maintaining the definition of
$\Lambda_{1}(p)$. The first result, which is similar to Theorem 1 is the
following.

######  Theorem 9.

For each $p\in\left(\frac{N}{s},\infty\right)$ we have

  1. $(i)$

$\Lambda_{1}(p)>0$;

  2. $(ii)$

there exist $(u_{p},v_{p})\in X^{*}_{s,t,p}(\Omega)$ such that $u_{p}>0$,
$v_{p}>0$ and

$|u_{p}(x_{1})|^{\alpha(p)}|v_{p}(x_{2})|^{\beta(p)}=1\qquad\text{and}\qquad\Lambda_{1}(s,p)=Q_{s,t,p}(u_{p},v_{p}).$

Its proof is also similar to that of Theorem 1. For details, see the proof
sketched in Section 3 or [11, Theorem 1].

The next step is to prove a result similar to Theorem 2. Changing the
definition of $S_{p}$ and $S{\infty}$ into

$\displaystyle S_{p}$ $\displaystyle=\left\\{(u,v)\in
X_{s,t,p}(\Omega)\,:\,|u(x_{1})|^{\alpha(p)}|v(x_{2})|^{\beta(p)}=1\right\\}$
and $\displaystyle S_{\infty}$ $\displaystyle=\left\\{(u,v)\in
X_{s,t,p}\,:\,|u(x_{1})|^{\theta}|v(x_{2})|^{1-\theta}=1\right\\}$

and also the denominator in $G_{p}$ into
$|u(x_{1})|^{\theta}|v(x_{2})|^{1-\theta}$, we obtain the version of Theorem 2
with the same statement.

Up to this point, the points $x_{1},x_{2}\in\Omega$ were taken arbitrarily.
Now, we consider sequences $u_{n}:=u_{p_{n}}$ and $v_{n}:=u_{p_{n}}$ given by
Theorem 1. Since $u_{n},v_{n}>0$, we can take $x_{1}$ as a maximum $x_{n}$ of
$u_{n}$ and $x_{2}$ as a maximum $y_{n}$ of $v_{n}$. Observe that we do not
suppose that the maxima $x_{n}$ and $y_{n}$ are unique. However, we will prove
that the sequence $(x_{n},y_{n})$ has a subsequence that converges to
$(x_{\infty},y_{\infty})$ and the equality
$|u_{\infty}(x_{\infty})|^{\theta}|v_{\infty}(y_{\infty})|^{1-\theta}=1$ still
holds true.

######  Theorem 10.

Let $\\{p_{n}\\}$ be a sequence converging to $\infty$ and
$(u_{p_{n}},v_{p_{n}})$ the solution of ($P^{1}_{p}$) given in Theorem 9.
Denote $x_{n}:=x_{u_{p_{n}}}$ and $y_{n}:=x_{v_{p_{n}}}$ a sequence of maxima
to $u_{p_{n}}$ and $v_{p_{n}}$, respectively. Passing to a subsequence if
necessary, $\\{(u_{p_{n}},v_{p_{n}})\\}_{n\in\mathbb{N}}$ converges uniformly
to $(u_{\infty},v_{\infty})\in C_{0}^{0,s}(\overline{\Omega})\times
C_{0}^{0,s}(\overline{\Omega})$, while the sequences $\\{x_{n}\\}$ and
$\\{y_{n}\\}$ converge to $x_{\infty}\in\Omega$ and $y_{\infty}\in\Omega$,
respectively, which are the maxima of $u_{\infty}$ and $v_{\infty}$.
Furthermore

  1. $(i)$

$u_{\infty}\geq 0$, $v_{\infty}\geq 0$ and
$|u_{\infty}(x_{\infty})|^{\theta}|v_{\infty}(y_{\infty})|^{1-\theta}=1$;

  2. $(ii)$

$\displaystyle\lim_{n\to\infty}\sqrt[p_{n}]{\Lambda_{1}(p_{n})}=\frac{1}{R^{s\theta+(1-\theta)t}}$

  3. $(iii)$

$\max\big{\\{}|u_{\infty}|_{s},|v_{\infty}|_{t}\big{\\}}=\displaystyle\frac{1}{R^{s\theta+(1-\theta)t}}$;

  4. $(iv)$

If $s=t$, then

$0\leq
u_{\infty}(x)\leq\displaystyle\frac{\left(\textup{dist}(x,\mathbb{R}^{N}\setminus\Omega)\right)^{s}}{R^{s}}\quad\text{and}\quad
0\leq
v_{\infty}(x)\leq\frac{\left(\textup{dist}(x,\mathbb{R}^{N}\setminus\Omega)\right)^{s}}{R^{s}}.$

Its proof can be obtained by mimicking the method used to prove Theorem 3.
Comparing this result with the one in [11], we first note that our result
brings information about the sequence of maxima of $u_{p_{n}}$ and
$v_{p_{n}}$, which are absent in that paper.

Finally, the analogue to Theorem 4 is the following. Once again, its proof is
obtained by adapting that of the Theorem 4.

######  Theorem 11.

The functions $u_{\infty}$ and $v_{\infty}$, given by Theorem 10, are
viscosity solutions of the problems

$\left\\{\begin{array}[]{ll}\mathcal{L}_{s,\infty}u=0&{\rm in}\ \
\Omega\setminus\\{x_{1}\\},\\\ u=0&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\\
u(x_{1})=u_{\infty}(x_{1})\end{array}\right.\qquad\text{and}\qquad\left\\{\begin{array}[]{ll}\mathcal{L}_{t,\infty}v=0&{\rm
in}\ \ \Omega\setminus\\{x_{2}\\},\\\ v=0&{\rm in}\
\mathbb{R}^{N}\setminus\Omega,\\\
v(x_{2})=v_{\infty}(x_{2}),\end{array}\right.$

respectively.

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# Scalable underwater assembly with reconfigurable visual fiducials

Samuel Lensgraf, Ankita Sarkar, Adithya Pediredla, Devin Balkcom, Alberto
Quattrini Li This project was partially supported by the NSF GRFP,
CNS-1919647, 2024541, 2144624. Dartmouth College

###### Abstract

We present a scalable combined localization infrastructure deployment and task
planning algorithm for underwater assembly. Infrastructure is autonomously
modified to suit the needs of manipulation tasks based on an uncertainty model
on the infrastructure’s positional accuracy. Our uncertainty model can be
combined with the noise characteristics from multiple devices. For the task
planning problem, we propose a layer-based clustering approach that completes
the manipulation tasks one cluster at a time. We employ movable visual
fiducial markers as infrastructure and an autonomous underwater vehicle (AUV)
for manipulation tasks. The proposed task planning algorithm is
computationally simple, and we implement it on AUV without any offline
computation requirements. Combined hardware experiments and simulations over
large datasets show that the proposed technique is scalable to large areas.

##  I Introduction

Autonomous assembly of structures using drones or free-floating robots is a
promising direction for creating rapidly deployable, flexibly designed
structures [1]. In most real-world systems, localization relative to a
reference is achieved using calibrated and fixed positional infrastructure
such as motion capture systems or visual fiducials [2, 3, 4, 5].
Unfortunately, these systems are not scalable as the coverage area is fixed
and scaling beyond the coverage area requires redesigning the positioning
technology.

To overcome the limited coverage area of the positioning technologies, we
propose to design the positioning infrastructure as a dynamic component of the
construction plan. Our method allows localizing against large structures with
minimal modification to the area around them and can also be integrated with
existing underwater construction structures to make them scalable – see Fig. 1
to show our robot in action while moving a fiducial marker.

Figure 1: AUV placing a reconfigurable fiducial marker on a foundation while
localizing using another marker.

Localization infrastructure is often considered to have a constant noise
distribution, allowing coverage algorithms to plan based on the noise
properties of the fixed infrastructure. However, the properties of
localization infrastructure often depend on environmental factors: distance
from the infrastructure, reflections, or water temperature gradients influence
the accuracy of the positioning systems [6]. As the accuracy depends on
relative positioning between the infrastructure and the robot, high-accuracy
positioning can only be provided in a small fixed area resulting in either
large infrastructure requirements, or small structures. Instead, by
understanding the noise properties of the infrastructure components and
modeling them accurately as a function of environmental factors, we show that
it is possible to dynamically reconfigure the infrastructure to maximize the
positioning accuracy of any region. Our noise formulation model is also
conducive to sensor fusion techniques and, hence, can be extended for the case
of multiple sensors.

For a given construction structure manipulation task, we have to plan the
movement of infrastructure such that the repositioned infrastructure
guarantees the accuracy of the manipulation task. This planning is challenging
and results in the combined deployment and sequencing problem.

To solve this problem, we group the manipulation tasks using a clustering
algorithm that guarantees that the radius of any cluster is within the high
localization accuracy achievable with the dynamic markers. We reposition the
markers to have high accuracy for the cluster and execute the manipulation
tasks, one cluster at a time.

For implementation, we use visual fiducial markers mounted on plates that are
movable on our previously developed error-correcting construction foundations
[7]. The AUV views these markers with a downward-facing wide-angle camera and
can compute the position accuracy using our noise model. The implementation of
our clustering-based planning algorithm is computationally simple, and we
implemented it on the AUV itself without any external cloud computing
requirements. We validated the proposed technique both experimentally and on
large simulation datasets.

Our technique is particularly interesting for the case where deploying large
amounts of infrastructure is impractical or expensive. In real-world
scenarios, it is often the case as the sparse area of interest is often near a
protected region that we cannot permanently mar. For example, in the
construction of artificial reefs, the surrounding areas are critically
important to protect. In hard to reach places such as caves, or very deep
waters, transporting large amounts of infrastructure can be impossible. Our
technique is a foundational component towards a scalable solution for these
cases. A number of practical challenges, such as flexibility in terms of
material of connected components per layer, will be subject of future studies
as discussed at the end of the paper.

##  II Related work

Uncertainty-based planning. Infrastructure placement for small scale scenes is
a well-studied problem. In particular, [8] define a landmark placement
algorithm that computes a static landmark placement based on certainty
requirements. We consider the problem of dynamically altering landmark
placements. We also develop a direct model of positional certainty based on
visual fiducial measurements.

[9] develop a swarm robot foraging algorithm that dynamically deploys and re-
deploys sensor motes. While the sensors are moved to provide certainty
implicitly, they do not model localization quality during their deployment.

Sensor coverage problems also directly relate to our work. In particular,
dynamic sensor coverage problems consider moving sensors. [10] consider
randomly moving mobile sensors and analyze coverage properties. Clustering has
been applied to dynamic coverage problems [11], but the sensors themselves
were considered as mobile units rather than beacons, which are deployed and
re-deployed by a moving robot. Sensor deployments via actuators are also
similar to our problem [12]. However, the problems of deployments of actuators
often focus on coverage and do not discuss the coupling of localization and
deployment.

In sensor coverage problems, some work incorporate a continuous sensor field
intensity [13, 14]. In the attenuated disk model, sensing quality decays with
the distance to the sensor. This modeling is similar in spirit to our modeling
of visual fiducials, but our model focuses as much on directionality as on
distance in determining the noise model.

Deployment planning algorithms for heterogeneous robot teams also relate to
our work. Such algorithms consider both cases where rewards are known apriori
and those in which rewards are randomly distributed [15, 16, 17]. Our problem
statement is similar in that there is competition for resources and
dependencies imposed by deployment decisions, but we judge the quality of the
assignment by the time required to execute the deployment plan rather than as
rewards accumulated at each assignment.

Localization. Mainstream underwater localization relies acoustic sensors, such
as Doppler Velocity Log (DVL), long/short/ultrashort baseline acoustic
positioning systems, and multibeam or sidescan sonars [6, 18, 19]. While such
sensors allow the robot to navigate in large areas, their accuracy depends on
a number of external factors, including multipath effect and their overall
resolution, making acoustic sensors not best suited to support manipulation
tasks. Vision-based perception is ubiquitously adopted for many robotics tasks
[20], including underwater [21], given camera’s low cost and ability to
capture rich information of the surrounding. The literature classified state
estimation methods according to different axes, with one being on whether they
minimize reprojection errors of tracked features – indirect methods, such as
ORB-SLAM [22] – or the alignment error considering image intensity values –
direct methods, e.g., DSO [23]. Adding IMUs [24] can improve state estimation
and including loop closure will allow the odometry estimate to be corrected.
Underwater, however, vision-based perception still remains a challenge mainly
due to the haze, color loss, and featureless environments [25, 26]. Given the
precision required by the underwater construction task, we rely on visual
fiducial markers and extend the operation area of the robot by allowing the
robot to move them.

Fisheye cameras are often used to localize mobile robots [27, 28], but there
are no techniques for modeling the quality of features detected using a visual
fiducial marker.

Visual fiducial markers have been developed specifically for fisheye cameras
[29] with the purpose of providing better position information. Other visual
fiducial markers have been developed to reduce positioning noise [30]. To our
knowledge, no attempt has been made to directly model the uncertainty of
detecting visual fiducials. An exploration of the noise properties of visual
fiducials is presented by Kalaitzakis et al. [31], but the geometry of the
noise distribution is unexplored.

Free-floating construction systems. We are inspired by the limitations of our
autonomous underwater construction system [7]. This work builds on and extends
our autonomous underwater construction robot. Previously, it localized using a
single visual fiducial that provided a limited coverage area.

Existing aerial free-floating construction systems commonly make use of fixed-
place motion capture systems [32, 1, 2]. These motion capture systems provide
precise, low latency position information but require numerous precisely
calibrated cameras with limited coverage area. We want to provide coverage to
large areas with limited need for complex fixturing.

##  III Problem Model

We consider the problem of deploying and moving infrastructure dynamically to
provide high quality localization information for a set of tasks at known
positions in global coordinates. The robot localizes using $m$ beacons which
can be placed and moved throughout the mission. The quality of information
coming from the beacons depends on how the robot is positioned relative to the
beacons. Information from each of the beacons can be combined to increase the
localization accuracy. Our goal is to find a mission plan, $\mathcal{A}$,
which consists of an ordered set of actions $a_{i}$. Each $a_{i}$ can
correspond to picking up a beacon, placing a beacon, or completing a task.

Each of the $n$ tasks, located at $t_{i}\in\mathbb{R}^{3}$, requires a high
enough precision of localization information. We model the quality of
information coming from a beacon $b_{i}$ using a function
$\Sigma(r_{i})\mapsto\Sigma_{r_{i}}\in\mathbb{R}^{3\times 3}$ which maps
relative positions ($r_{i}$) into covariance matrices that describe the noise
distribution of the information coming from the beacon. We assume zero mean
error. Information from multiple sensors can be combined by using sensor
fusion equations. We use the equations described by [33]. Algorithm 1 shows
how we combine noise distributions from multiple sensors.

Algorithm 1 Procedure to fuse covariance matrices of uncertain positions [33].

Covariance matrices $\Sigma_{1},\dots,\Sigma_{n}$ Fused covariance matrix
$\Sigma$

1:$\Sigma\leftarrow\Sigma_{1}$ $i\in 1\dots,n$

2:$K\leftarrow\Sigma(\Sigma+\Sigma_{i})^{-1}$

3:$\Sigma\leftarrow\Sigma-K\Sigma$

4:$\Sigma$

Each task $t_{i}$ requires a high enough precision to be completed. We model
the precision requirements using a scalar $C_{i}$ which is obtained by
applying a certainty approximation function $C(\Sigma)\mapsto
C_{i}\in\mathbb{R}$ to the fused covariance matrix $\Sigma$. We define
$C(\Sigma)$ as a function which approximates the probability that location
readings are inside of a given error range. Receiving a location reading
outside of the error range could cause the robot to fail at its task.

To move a beacon $b_{i}$, the robot must have precise enough location
information for both pickup and placement. This means that beacons must be
clustered to provide coverage of one another. For simplicity in our initial
exploration, we assume perfect placement of the beacons. This assumption is
reasonable for moving beacons on error-correcting foundations. In future work,
we plan to extend our method to model a decay in the quality of information of
moved beacons because of small placement errors.

For simplicity, we also assume a known relative orientation $R$ of the AUV. In
practice, the orientation of a free-floating robot can be sensed with a high
accuracy using out-of-the-box AHRS boards. An initial calibration step can be
used to measure the relative rotation for the set of beacons.

1D example.

Figure 2: Series of steps to cover a task at $t_{1}$ with the red and blue
beacons.

Consider a robot operating in a 1D world with no collisions. Two beacons, red
and blue in Figure 2, provide coverage with precision $\Sigma_{r}(r)=r^{2}$,
that is the quality of information decays quadratically with the distance to
the beacon. We set $C(\Sigma)=1-\Sigma$ because, in 1D, $\Sigma$ is a scalar
and can be used directly. The beacons start at positions $b_{1}=-0.1$ and
$b_{2}=0.1$. The robot is given one task to complete at position $t_{1}=.7$.
Our fusion function in Algorithm 1 becomes
$\Sigma=r_{1}^{2}-\frac{r_{1}^{2}}{r_{1}^{2}+r_{2}^{2}}$. The task has a
requirement $1-\Sigma\geq 0.95$. Moving a beacon requires the same certainty.
Figure 2 shows an example of the problem.

We can compute a coverage area for a single beacon and a pair of beacons to
guide our creation of a simple mission plan: $[b_{i}-0.224,b_{i}+0.224]$. To
reach and cover our task at position $0.7$, we need to move one beacon to
position $.7-0.224=0.476$. Moving the beacons will require multiple hops due
to their limited coverage area. Our final plan $\mathcal{A}$ is then
$\mathcal{A}=$ $\textsc{MoveBeacon}(b_{1},0.324)$,
$\textsc{MoveBeacon}(b_{2},0.548)$, $\textsc{complete}(t_{1})$.

In the specific case of assembly a task $t_{i}$ will represent placing a block
at $t_{i}$’s location. For this application, we write
$\textsc{PlaceBlock}(t_{i})$ to mean placing a block at location $t_{i}$. We
also replace MoveBeacon with MoveMarker when we are dealing with a
reconfigurable visual fiducial marker.

##  IV Noise characterization of visual fiducials

(a)

(b)

Figure 3: (a) Predicted and measured largest eigenvector directions in real
world experiment. The arrows extend from the marker’s position. (b) Results
from simulated corner noise. In both cases the predicted and measured
directions closely match.

The first step to implementing our assembly planning and localization method
is to accurately model the noise distribution of visual fiducial markers. To
understand how the noise distribution varies based on the relative position
between the fisheye camera and a marker, we built a simulator. The simulator
applies Gaussian distributed noise to the distorted corner positions of the
visual fiducial, then undistorts them using the Kannala-Brandt Camera model
[34] – typically used for fisheye lenses – and solves the Perspective-n-point
problem. This simulation captures the important sources of noise: sensor noise
and barrel distortion. We experimentally validate our simulator and noise
model in Section VI-C.

Figure 3 shows outputs from a real world experiment (a) and from our
simulation (b). We found that the noise distribution was highly structured and
could be predicted using only two values: the largest eigenvector and
eigenvalue. Further, the eigenvalue is parallel to the position vector. In the
remainder of this section, we discuss how we predict the two components.

###  IV-A Scale noise

Figure 4: Position readings from a static fisheye lens camera for a static
visual fiducial marker.

In our experiments on fiducial marker noise when viewed through a fisheye
lens, we found that the axis of largest noise very consistently pointed
towards the camera. Figure 4 shows an example of a set of relative position
measurements for a single visual fiducial measured by a static camera. The
position vector is marked in black and aligns closely with the largest
eigenvector of the noise distribution (green). We noticed this phenomenon to
be consistent across various positions. The largest eigenvector dominates the
noise distribution, and the other two are an order of magnitude smaller. We
provide empirical evidence for this observation in Section VI-C.

###  IV-B A definition of $\Sigma(p)$

Algorithm 2 Find covariance matrix $\Sigma_{p}$ for a relative position $p$.

$p$, $\beta$, $\lambda_{i}$ $\Sigma_{p}$

1:$\lambda_{*}\leftarrow\beta(p)$

2:$v_{1}\leftarrow\frac{p}{\|p\|}$

3:$v_{2}\leftarrow\textsc{orthogonal}(p)$

4:$v_{3}\leftarrow\textsc{orthogonal}(p,v_{2})$

5:$\Lambda\leftarrow\textsc{hstack}(v_{1},v_{2},v_{3})$

6:$S\leftarrow\textsc{diag}(\lambda_{*},\lambda_{i},\lambda_{i})$

7:$\Lambda S\Lambda^{-1}$

Algorithm 2 shows our covariance matrix prediction procedure. It accepts as
arguments $p$, the relative position, $\beta$, a predictor of the largest
eigenvalue and $\lambda_{i}$, an upper bound on the two smaller eigenvalues.

The largest eigenvalue, $\lambda_{*}$, is predicted using a spline which is
calibrated on experimental data. $v_{1}$, $v_{2}$, and $v_{3}$ are our
predictions of the eigenvectors. The largest eigenvector, $v_{1}$, is
predicted to be the normalized position vector. The other two eigenvectors,
$v_{1}$, and $v_{2}$ are predicted to be orthogonal to $p$ and one another.
Line 6 accumulates the eigenvalues into a $3\times 3$ diagonal matrix. The
matrices $\Lambda$ and $S$ are multiplied together to produce a predicted
covariance matrix which has the predicted eigenvalues and eigenvectors.

##  V Finding feasible assembly plans in practice

We model the assembly process as a set of manipulation tasks located at points
in 3D space. The probability of completing a task $C(\Sigma_{r})$ is the
probability that a block dropped will be inside of the acceptance area of the
slot it is aimed at. As a structure is built, it can occlude the view of
markers, therefore, the markers must be continually moved as the structure is
erected to increase $C(\Sigma_{r})$.

###  V-A Computing $C(\Sigma_{r})$

To enable planning during the construction process, we define
$C:\Sigma_{r}\mapsto\mathcal{R}$, which takes as input a covariance matrix and
outputs the probability of successfully dropping a block. A block or marker is
successfully placed if the robot decides to initiate the placement action
within an acceptable range of the ideal position. This acceptable range
($\alpha$) is dictated by the design of the error-correcting construction
foundation. Assuming that the uncertainty in position is well approximated
with a multivariate Gaussian, $C(\Sigma_{r})$ is the probability that a random
sample position drawn from $\mathcal{N}(0,\Sigma_{r})$ is within the sphere of
radius $\alpha$.

Analytically computing this probability is challenging. Instead, we use a
conservative approximation. The largest eigenvalue ($\lambda_{*}$) is a
conservative estimate of the standard deviation of noise in any direction. So,
we assume that the noise distribution along all three coordinate axes is
independent and is equal to $\lambda_{*}$. This approximation results in the
closed form estimate $C^{*}(\Sigma_{r})$:

$C(\Sigma_{r})\geq\operatorname{erf}{\left(\frac{\alpha}{\sqrt{\lambda_{*}}\sqrt{2}}\right)}^{3}=C^{*}(\Sigma_{r}).$
(1)

###  V-B A layer-based approach for assembly

To assemble a structure, the robot must manipulate markers which can rest on
top of blocks in order to localize while blocks are placed. When placed, the
blocks can obscure sight of the markers. Planning around obscured markers
introduces difficult nonlinearities into the constraints for any solver hoping
to find feasible solutions.

To construct feasible plans efficiently without needing to model occlusions
between the structure as it is erected and the markers, we propose a layer-by-
layer algorithm. Algorithm 3 shows our strategy for generating feasible plans
for a structure with $n$ slots, using $m$ markers.

Our algorithm works by dividing the blocks into layers $l_{1},\dots,l_{h}$,
where $l_{1}$ is the bottom layer of blocks, $l_{2}$ is the layer to be placed
above $l_{1}$, and so on with $l_{h}$ being the topmost layer of blocks. For
each $i\in\\{1,2,\dots,h\\}$, we first cluster $l_{i}$ to obtain clusters of
width at most $r$, where $r$ is an empirical determination of a marker’s
coverage radius based on the bound in Equation 1. In our implementation, this
is achieved using a subroutine ClusterUntilRadius$(l_{i},r)$. This subroutine
performs $k$-means clustering on $l_{i}$ using a value of $k$ that is tuned,
via binary search, to be the minimum possible such that all cluster widths are
at most $r$.

The sub-procedure ExtractCenters finds the center point of each cluster. The
cluster centers are then passed into FindTour which computes a tour of the
cluster centers $O$ which has elements that index the clusters.

In each cluster produced by ClusterUntilRadius$(l_{i},r)$, we select $m$
points to serve as marker destinations. We choose the marker destinations to
be the $m$ farthest points from each other in the cluster. We then use the
sub-procedure WalkToCoverage to transition the markers between destinations.
We achieve this via a simple hopping strategy, like the one discussed in
Section III. This strategy repeatedly hops one marker to the outside of the
other marker’s coverage area, resembling a “gait” if one imagines the markers
to be a robot’s feet. In this way, we position the markers at the $m$
destinations within a cluster, place the blocks within that cluster, and
repeat for successive clusters in that layer. Since the maximum cluster width
is the coverage radius of a marker, we can ensure that each marker is always
covered by another marker, enabling us to continue the gait after the blocks
in a cluster have been placed.

After blocks have been placed around each of the markers in the cluster, the
markers must be moved to make room for the remaining blocks. To do this, the
loop in 8 to 12 iterates over each marker, moves it on top of the nearest
block to it, and then places a block where the marker used to be. If the
cluster has fewer than $m$ points, two markers might share the same nearest
neighbor and cause the markers to be placed in the same location. We avoid
this edge case by requiring a minimum cluster size of $m$ in
ClusterUntilRadius$(l_{i},r)$; this is feasible via a reasonable assumption
that $m$ markers can be placed within radius $r$.

Algorithm 3 Layer-by-layer traversal.

Structure with slots $S=\\{s_{1},\dots,s_{n}\\}$, marker positions
$M=\\{b_{1},b_{2},\dots,b_{m}\\}$ Assembly plan $\mathcal{A}$

1:Divide $S$ into layers $l_{1},\dots,l_{h}$

2:$\mathcal{A}\leftarrow[]$ $i\in\\{1,\dots,h\\}$

3:$\mathcal{C}\leftarrow\textsc{ClusterUntilRadius}(l_{i},r)$

4:$C\leftarrow\textsc{ExtractCenters}(\mathcal{C})$

5:$c_{1},c_{2},\dots,c_{k}\leftarrow\textsc{FindTour}(C)$ $\triangleright$
efficient tour on cluster centers $j=1$ to $k$ $\triangleright$ process
clusters in tour order

6:$C_{j}\leftarrow\textsc{ClusterOf}(c_{j})$

7:$\mathcal{A}\leftarrow\mathcal{A}$.extend(WalkToCoverage$(M,c_{j})$)
$\triangleright$ move markers into cluster $s\in C_{j}\setminus M$
$\triangleright$ place blocks in slots unoccupied by markers

8:$\mathcal{A}\leftarrow\mathcal{A}.\textit{append}\big{(}\textsc{PlaceBlock}(s)\big{)}$
$s\in M$ $\triangleright$ move markers to place remaining blocks

9:$p\leftarrow\textsc{NearestNeighbor}(s,C_{j})$

10:$p\leftarrow p+(0,0,1)$

11:$\mathcal{A}\leftarrow\mathcal{A}.\textit{append}\big{(}\textsc{MoveMarker}(s,p)\big{)}$

12:$\mathcal{A}\leftarrow\mathcal{A}.\textit{append}\big{(}\textsc{PlaceBlock}(s)\big{)}$

13:$\mathcal{A}$

##  VI Experiments

We implement our reconfigurable visual fiducial localization system in both
hardware and simulation, and the results are described in the following.

###  VI-A Experimental setup

Our hardware implementation is deployed on the Droplet AUV system [7, 5]. A
ROS node holds the known global position of the visual fiducial markers. As
the markers are moved, the node is notified and then marker readings are
offset accordingly before performing the sensor fusion steps described in
Algorithms 1 and 2.

For viewing visual fiducial markers, the robot is equipped with FLIR Blackfly
camera with a Senko fisheye lens mounted facing downards. The camera is
mounted in a 3 inch Blue Robotics acrylic enclosure with a dome. Figures 6, 1
show the camera mounted on the robot and Figure 5 (b) shows how the scene
looks through the camera.

###  VI-B Hardware testing

(a)

(b)

Figure 5: (a) Reconfigurable visual fiducial design. (b) Robot’s eye view
during assembly with the reconfigurable fiducials. Figure 6: The AUV
performing the last placement of the two hop maneuver.

To validate the concept of reconfigurable visual fiducials in practice, we
mounted visual fiducial markers on our error correcting connector geometry
[7]. We show the design in Figure 5. The fiducials provide error correction
during both pickup via the top handle (red in Figure 5) and placement.

We tested the concept of walking behaviors by implementing a 1D hopping
strategy. The robot was able to successfully perform manipulation tasks in a
large area with a side of 2.8 meters using cement blocks placed by hand. This
side is 57% longer than the widest side usable by our previous implementation.
Note that the side of the line covered was bounded by the physical width of
the pool and not the robot’s ability to stack blocks successfully. Figure 6
shows the AUV completing the maneuver.

To verify that the visual fiducial marker fusion procedure reduces the
localization noise, we recorded the measured positions and the number of valid
marker readings when the robot was still. When only one marker was determined
as valid, the standard deviation of position measurements on the X axis was
$4.3\text{\,}\mathrm{c}\mathrm{m}$, but when two markers were valid, the
standard deviation was reduced to $0.69\text{\,}\mathrm{c}\mathrm{m}$; around
$6\times$ improvement. On the Y axis, we observe similar improvements: from
$6.0\text{\,}\mathrm{c}\mathrm{m}$ to $1.5\text{\,}\mathrm{c}\mathrm{m}$;
around $4\times$ improvement.

###  VI-C Validation of noise model

To validate our prediction model of covariance matrices given in Algorithm 2,
we conducted both real world and simulation experiments. Our simulator
projects a marker at a given relative position into the camera and offsets the
corners according to a Gaussian distribution. This noise addition simulates
the effects of pixel flicker on corner detection.

Figure 3 (a) shows the results of our physical testing. We arrayed visual
fiducials in an area spanning about 1.4 meters on the positive X and Y axes.
This setup mimics the distances between markers used in practice. We predict
the direction of the largest eigenvector as the median of the measured
relative positions. We found that the predicted direction is accurate to
within $4$ degrees. This result shows that computing the direction of the
largest eigenvector as the position vector is effective in practice.

To more extensively test our prediction algorithm, we conducted a test using a
spline predictor of the largest eigenvalue trained on 6250 noise distributions
generated at known relative positions in our simulator. We set the upper bound
for the smallest two eigenvalues to $10^{-4}$. We set $\alpha$ to
$2\text{\,}\mathrm{c}\mathrm{m}$ for Equation 1. We tested the trained
predictions from Algorithm 2 using 5184 test relative positions not present in
the training data.

To evaluate the quality of Algorithm 2, we checked whether its predictions
were more or less conservative than measured covariance matrices in the test
dataset. We evaluated whether Algorithm 2 produced more or less conservative
results by comparing the bound in Equation 1 for the predicted covariance
matrix $\Sigma_{r}^{*}$ against the measured covariance matrix $\Sigma_{r}$.
We found that in 98.3% of cases $C^{*}(\Sigma_{r}^{*})\leq C^{*}(\Sigma_{r})$.
In the other 1.7% of cases, $C^{*}(\Sigma_{r})$ was only smaller than
$C^{*}(\Sigma_{r}^{*})$ by at most 3.4%. To avoid this case, we configure our
planning algorithm so two markers are always visible.

When combining the two worst predictions using Algorithm 1, we find that the
fused covariance matrix is a conservative estimate. We combined the covariance
matrices of the two worst over-predictions of $C^{*}(\Sigma_{r})$ for both
measured and predicted covariances. We found that the predicted
$C^{*}(\Sigma_{r})$ is 40% lower (59%) than the measured bound (99%).

Our estimate of the noise is often very conservative, but we will show in the
following section that large structures still have a high predicted success
probability when planned using Algorithm 3.

###  VI-D Assembly algorithm

(a)

(b)

Figure 7: (a) Visualization of plan checker. Green blocks represent markers.
(b) Robot moving markers in the DAVE simulator.

To test Algorithm 3 we use both a hand crafted plan checker and a full
underwater robot 3D simulator called DAVE [35]. Our plan checker checks lines
of sight to visual fiducials and computes the predicted certainty of every
step. We use the underwater robot simulator to determine whether a plan which
is feasible according to the plan check is also feasible under realistic
control noise. We ran a construction process with 74 build steps. Figure 7 (b)
shows the partially completed structure in the DAVE simulator. With realistic
control noise, the robot kept at least two markers in view 100% of the time.

Table I: Effect of increasing the cluster radius on plan efficiency and probability of success. After a certain cluster size, occlusions make construction impossible. $r$ | Predicted $P(\text{success})$ | # steps
---|---|---
2.5 | 0 | 226
2.0 | 0.82 | 240
1.5 | 0.96 | 259
1.0 | 0.99 | 345

As the number of manipulation steps increases, the time to build a structure
increases. Our layer-based construction (Algorithm 3) takes an input a
parameter $r$ which describes the radius of clusters used for marker
placement. Between each cluster, the markers are moved in an expensive walking
procedure, so increasing $r$ could improve the efficiency of the construction
process but at the cost of reducing the reliability. Table I shows how
changing $r$ affects the number of steps required to build a 200 block
structure and the certainty afforded during construction with three markers.
We measure the predicted $P(\text{success})$ as the product of the probability
of success of every state in the construction process. We also tested a
pyramidal structure containing 1800 blocks. The assembly planning algorithm
took about 5 minutes to plan the structure and the predicted probability of
success for $r=1.5$ is $91\%$.

##  VII Conclusions & future work

This paper proposes a novel strategy for localizing relative to error
correcting structures while planning the construction process. Our method is
shown to work in practice at small scale and at large scale in simulation. We
also show the robot being able to reliably complete the “hopping” strategy for
moving the markers, extending the area for assembly.

We plan to improve the scale and quality of our hardware implementation.
Planning for large scale construction with heterogeneous materials will
require adaptive and flexible clustering strategies. Materials which are not
well described by a bounding box may require more sophisticated strategies for
avoiding occlusions.

Our assembly process is currently limited to structures which have only a
single connected component per layer. If there is more than one connected
component in a layer, the markers can become stranded as they are lifted up
the structure. In the future, we plan to explore ways to increase the
flexibility of our method.

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