\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx, subfigure}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 145, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/145\hfil 
 The Poisson equation from non-local to local]
{The Poisson equation from non-local to local}

\author[U. Biccari, V. Hern\'andez-Santamar\'ia \hfil EJDE-2018/145\hfilneg]
{Umberto Biccari, V\'ictor Hern\'andez-Santamar\'ia}

\address{Umberto Biccari \newline
DeustoTech, University of Deusto, 48007 Bilbao,
Basque Country, Spain. \newline
Facultad de Ingenier\'ia, Universidad de Deusto,
Avenida de las Universidades 24, 48007 Bilbao,
Basque Country, Spain, +34 944139003 - 3282}
\email{umberto.biccari@deusto.es, u.biccari@gmail.com}

\address{V\'ictor Hern\'andez-Santamar\'ia \newline
DeustoTech, University of Deusto, 48007 Bilbao,
Basque Country, Spain. \newline
Facultad de Ingenier\'ia, Universidad de Deusto,
Avenida de las Universidades 24, 48007 Bilbao,
Basque Country, Spain, +34 944139003 - 3282}
\email{victor.santamaria@deusto.es}

\dedicatory{Communicated by Raffaella Servadei}

\thanks{Submitted January 22, 2018. Published July 17, 2018.}
\subjclass[2010]{35B30, 35R11, 35S05}
\keywords{Fractional Laplacian; elliptic equations; weak solutions}

\begin{abstract}
 We analyze the limiting behavior as $s\to 1^-$ of the solution to the
 fractional Poisson equation $(-\Delta)^s{u_s}=f_s$, $x\in\Omega$ with
 homogeneous Dirichlet boundary conditions $u_s\equiv 0$, $x\in\Omega^c$.
 We show that $\lim_{s\to 1^-} u_s =u$, with $-\Delta u =f$, $x\in\Omega$
 and $u=0$, $x\in\partial\Omega$. Our results are complemented by a discussion
 on the rate of convergence and on extensions to the parabolic setting.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction and statement of main results}\label{intro}

Let $0<s<1$ and let $\Omega\subset\mathbb{R}^N$ be a bounded and regular domain.
Let us consider the following elliptic problem
\begin{equation}\label{PE}
 \begin{gathered}
 (-\Delta)^s u = f, \quad x\in\Omega \\
 u\equiv 0, \quad x\in\Omega^c.
 \end{gathered}
\end{equation}
Here $(-\Delta)^s $ indicates the fractional Laplace operator,
defined for any function $u$ regular enough as the  singular integral
\begin{equation}\label{frac_lapl}
\begin{gathered}
 (-\Delta)^s u(x):=C_{N,s}\operatorname{P.V.}\int_{\mathbb{R}^N}
\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy\,,
\end{gathered}
\end{equation}
where $C_{N,s}$ is a normalization constant
\begin{equation}\label{cns}
 C_{N,s}:= \frac{s2^{2s}\Gamma(\frac{N+2s}{2})}{\pi^{N/2}\Gamma(1-s)},
\end{equation}
where $\Gamma$ is the usual Gamma function.
Moreover, we have to mention that, for having a completely rigorous definition
of the fractional Laplace operator, it is necessary to introduce also
the class of functions $u$ for which computing $(-\Delta)^s u$ makes sense.
We postpone this discussion to the next section.

Models involving the fractional Laplacian or other types of non-local operators
 have been widely used in the description of several complex phenomena
for which the classical local approach turns up to be inappropriate or limited.
Among others, we mention applications in turbulence \cite{bakunin2008turbulence},
elasticity \cite{dipierro2015dislocation}, image processing \cite{gilboa2008nonlocal},
 laser beams design \cite{longhi2015fractional}, anomalous transport and diffusion
\cite{meerschaert2012fractional}, porous media flow \cite{vazquez2012nonlinear}.
Also, it is well known that the fractional Laplacian is the generator of s-stable
 processes, and it is often used in stochastic models with applications,
for instance, in mathematical finance \cite{levendorskii2004pricing}.

One of the main differences between these non-local models and  classical
partial differential equations is  that the fulfillment of a non-local equation
at a point involves the values of the function far away from that point.

The Poisson problem \eqref{PE} is one of the most classical models involving
the Fractional Laplacian, and it has been extensively studied in the past.
Nowadays, there are many contributions concerning, but not limited to, existence
and regularity of solutions, both local and global
\cite{biccari2017local,cozzi2017interior,grubb2015fractional,
leonori2015basic,ros2016boundary,ros2014dirichlet,servadei2014weak},
unique continuation properties \cite{fall2014unique}, Pohozaev identities
\cite{ros2014pohozaev}, spectral analysis \cite{frank2016refined}
 and numerics \cite{acosta2017fractional}.

In this article, we are interested in analyzing the behavior of the solutions
to \eqref{PE} under the limit $s\to 1^-$. Indeed, it is well-known
(see, e.g., \cite{dihitchhiker,stinga2010extension}) that, at least
for regular enough functions, it holds
\begin{itemize}
 \item  $\lim_{s\to 0^+}(-\Delta)^s u = u$.

 \item  $\lim_{s\to 1^-}(-\Delta)^s u = -\Delta u$.
\end{itemize}
In view of this, it is interesting to investigate whether, when $s\to 1^-$,
a solution $u_s$ to \eqref{PE} converges to a solution to the classical
Poisson equation
\begin{equation}\label{poisson}
 \begin{gathered}
 -\Delta u = f, \quad x\in\Omega \\
 u = 0, \quad x\in\partial\Omega.
 \end{gathered}
\end{equation}
In our opinion, this is a very natural issue which, to the best of our knowledge,
 has never been fully addressed in the literature in the setting of weak
solutions with minimal assumptions. As we will see, the answer to this
question is positive.

Before introducing our main result, let us recall that we have the following
definition of weak solutions.

\begin{definition}\label{weak_sol_def} \rm
 Let $f\in H^{-s}(\Omega)$. A function $u\in H_0^s(\overline{\Omega})$
is said to be a weak solution of the Dirichlet problem \eqref{PE} if
 \begin{equation}\label{weak-sol}
 \frac{C_{N,s}}{2}\int_{\mathbb{R}^N}
\int_{\mathbb{R}^N}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\,dx\,dy
 = \int_\Omega  fv\,dx
 \end{equation}
 holds for every $v\in\mathcal{D}(\Omega)$.
\end{definition}

Here $H_0^s(\overline{\Omega})$ denotes the fractional order Sobolev space
which consists of all functions $u\in H^s(\Omega)$ which are zero on $\Omega^c$,
while $H^{-s}(\Omega)$ is its dual. We will give a more exhaustive description
of these spaces in Section \ref{prel}.
The main result of our work is the following.

\begin{theorem}\label{limit_thm}
 Let $\mathcal{F}_s=\{f_s\}_{0<s<1}\subset H^{-s}(\Omega)$ be a sequence
satisfying the following assumptions:
 \begin{itemize}
  \item[(H1)] $\|f_s\|_{H^{-s}(\Omega)}\leq C$, for all $0<s<1$ and uniformly
with respect to $s$;

  \item[(H2)] $f_s\rightharpoonup f$ weakly in $H^{-1}(\Omega)$ as $s\to 1^-$.
 \end{itemize}
For all $f_s\in\mathcal{F}_s$, let $u_s\in H_0^s(\overline{\Omega})$ be the unique
weak solution to the Dirichlet problem \eqref{PE}, in the sense of
Definition \ref{weak_sol_def}. Then, as $s\to 1^-$, $u_s\to u$ strongly
in $H^{1-\delta}_0(\Omega)$ for all $0<\delta\leq 1$.
Moreover, $u\in H^1_0(\Omega)$ and satisfies
\[
 \int_{\Omega} \nabla u\cdot\nabla v\,dx
= \int_{\Omega}fv\,dx, \quad \forall v\in\mathcal{D}(\Omega),
\]
i.e. it is the unique weak solution to \eqref{poisson}.
\end{theorem}

The proof of Theorem \ref{limit_thm} will be based on classical PDEs techniques.
Moreover, the result will follow from the limit behavior as $s\to 1^-$ of
the operator $(-\Delta)^s $ (\cite{dihitchhiker,stinga2010extension})
and of the norm $\|\cdot\|_{H^s(\Omega)}$ \cite{bourgain2001another}.

Furthermore, notice that Theorem \ref{limit_thm} requires the existence of
a sequence $\mathcal{F}_s$ satisfying the assumptions (H1) and (H2).
We point out that such sequence indeed exists, and that it is possible to
 construct it systematically. We will give a proof of this fact in
Section \ref{prel}.

This paper will be organized as follows: Section \ref{prel} will be devoted
to introduce some preliminary definitions and results that will be needed
in our analysis. In Section \ref{weak_sec}, instead, we will present the proof
 of Theorem \ref{limit_thm}, concerning the limit behavior of the solutions
to \eqref{PE}. Finally, in Section \ref{rem_sec}, we will present an additional
result of convergence under weaker assumptions, a discussion on the rate of
approximation and an extension to the the parabolic setting.

\section{Preliminaries}\label{prel}

In this section, we introduce some preliminary results that will be useful
for the proof of our main theorem.

We start by giving a more rigorous definition of the fractional Laplace operator,
as we have anticipated in Section \ref{intro}. Define
\[
 \mathcal L_s^{1}(\mathbb{R}^N):=\big\{u:\mathbb{R}^N\to\mathbb{R}\text{ measurable},\;
\int_{\mathbb{R}^N}\frac{|u(x)|}{(1+|x|)^{N+2s}}\,dx<\infty\big\}.
\]
For $u\in \mathcal L_s^{1}(\mathbb{R}^N)$ and $\varepsilon>0$ we set
\[
 (-\Delta)_\varepsilon^s u(x)
:= C_{N,s}\int_{\{y\in\mathbb{R}^N:\;|x-y|>\varepsilon\}}
\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy,\quad x\in\mathbb{R}^N.
\]

The fractional Laplace operator $(-\Delta)^s $ is then defined by the
singular integral
\begin{equation}\label{fl_def}
 (-\Delta)^s u(x)=C_{N,s}\operatorname{P.V.}
\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy
=\lim_{\varepsilon\downarrow 0}(-\Delta)_\varepsilon^s u(x),\quad x\in\mathbb{R}^N,
\end{equation}
provided that the limit exists.

We notice that if $0<s<1/2$ and $u$ is smooth, for example bounded
and Lipschitz continuous on $\mathbb{R}^N$, then the integral in \eqref{fl_def}
is in fact not really singular near $x$ (see e.g.\ \cite[Remark 3.1]{dihitchhiker}).
Moreover, $\mathcal L_s^{1}(\mathbb{R}^N)$ is the right space for which
$ v:=(-\Delta)_\varepsilon^s u$ exists for every $\varepsilon>0$, $v$ being
also continuous at the continuity points of $u$.

It is by now well-known (see, e.g., \cite{dihitchhiker}) that the natural
functional setting for problems involving the Fractional Laplacian is the one
of the fractional Sobolev spaces. Since these spaces are not so familiar as
 the classical integral order ones, for the sake of completeness,
we recall here their definition.

Given $\Omega\subset\mathbb{R}^N$ regular enough and $s\in(0,1)$, the
fractional Sobolev space $H^s({\Omega})$ is defined as
\[
 H^s(\Omega):= \big\{u\in L^2(\Omega):
 \frac{|u(x)-u(y)|}{|x-y|^{\frac N2+s}}\in L^2(\Omega\times\Omega)\big\}.
\]

It is known that this is a Hilbert space, endowed with the norm
(derived from the scalar product)
\[
 \|u\|_{H^s(\Omega)} := \Big(\|u\|_{L^2(\Omega)}^2
+ |u|_{H^s(\Omega)}^2\Big)^{1/2},
\]
where
\[
 |u|_{H^s(\Omega)}:= \Big(\int_\Omega\int_\Omega \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}
\,dx\,dy\Big)^{1/2}
\]
is the so-called Gagliardo seminorm of $u$. For all $0<s<1$, we set
\[
 H_0^s(\overline{\Omega}):= \{u\in H^s(\mathbb{R}^N): u=0\textrm{ in }
\Omega^c\},
\]
and we indicate with $H^{-s}(\Omega)=(H^s(\overline{\Omega}))'$ its dual
 with respect to the pivot space $L^2(\Omega)$. Moreover, if $s>1/2$,
according to \cite[Theorem 6]{fiscella2015density} we have the identity
\[
 H_0^s(\overline{\Omega})= H_0^s(\Omega),
\]
where
\[
 H_0^s(\overline{\Omega})= H_0^s(\Omega)
:= \overline{C_0^\infty(\Omega)}^{\,H^s(\Omega)}
\]
is the closure of the continuous infinitely differentiable functions
compactly supported in $\Omega$ with respect to the $H^s(\Omega)$-norm.

A more exhaustive description of fractional Sobolev spaces and of their
properties can be found in several classical references
(see, e.g., \cite{adams2003sobolev,dihitchhiker,jllions1972non}).

Coming back to our problem, let us recall that the existence and uniqueness
of weak solutions to \eqref{PE} is guaranteed by the following result
(see, e.g., \cite[Proposition 1.2.23]{peradottolaplaciano}).

\begin{proposition}\label{prop-ex}
Let $\Omega\subset\mathbb{R}^N$ be an arbitrary bounded open set and $0<s<1$.
 Then for every $f\in H^{-s}(\Omega)$, the Dirichlet problem \eqref{PE}
has a unique weak solution $u\in H_0^s(\overline{\Omega})$.
Moreover, there exists a constant $C>0$ such that
\begin{equation}\label{est-sol}
 \|u\|_{H_0^s(\overline{\Omega})}\le C\|f\|_{H^{-s}(\Omega)}.
\end{equation}
In addition, we can take $C=\sqrt{2/C_{N,s}}$.
\end{proposition}

We remind that our main interest in the present work is the analysis
of the behavior of the solutions of \eqref{PE} when $s\to 1^-$.
The proof of Theorem \ref{limit_thm} is obtained employing classical
techniques in functional analysis, as well as the following results.

\begin{proposition}[{\cite[Proposition 4.4]{dihitchhiker}}]\label{limit_prop}
For any $u\in C_0^\infty(\mathbb{R}^N)$ the following statements hold:
\begin{itemize}
 \item[(i)] $\lim_{s\to 0^+}(-\Delta)^s u = u$.
 \item[(ii)] $\lim_{s\to 1^-} (-\Delta)^s u = -\Delta u$.
\end{itemize}
\end{proposition}

\begin{proposition}[{\cite[Corollary 7]{bourgain2001another}}]\label{brezis_prop}
For any $\varepsilon>0$, let $g_\varepsilon \in H^{1-\varepsilon}(\Omega)$.
Assume that
\[
 \varepsilon\|g_\varepsilon\|_{H^{1-\varepsilon}(\Omega)}^2\leq C_0,
\]
where $C_0$ is a positive constant not depending on $\varepsilon$.
Then, up to a subsequence, $\{g_\varepsilon\}_{\varepsilon>0}$ converges
in $L^2(\Omega)$ (and, in fact, in $H^{1-\delta}(\Omega)$, for all
$\delta > 0$) to some $g\in H^1(\Omega)$.
\end{proposition}

Finally, as we pointed out in Section \ref{intro}, our main result requires
a sequence $\mathcal{F}_s$ satisfying the assumptions (H1) and (H2).
The existence of such a sequence is guaranteed by the following result.

\begin{proposition}\label{limit_f}
 For any $f\in H^{-1}(\Omega)$ there exists a sequence
$\mathcal{F}_s=\{f_s\}_{0<s<1}\subset H^{-s}(\Omega)$ satisfying the assumptions
{\rm (H1)} and
 \begin{itemize}
  \item[(H2')] $f_s\to f$ strongly in $H^{-1}(\Omega)$ as $s\to 1^-$.
 \end{itemize}
\end{proposition}

\begin{proof}
Recall that every $f\in H^{-1}(\Omega)$ can be written as
$f=\operatorname{div}(g)$ with $g\in L^2(\Omega)$. Furthermore, let us
 introduce a standard mollifier $\rho_\varepsilon$ defined as
\[
 \rho_\varepsilon(x):= \begin{cases}
 C\varepsilon^{-N}\exp\big(\frac{\varepsilon^2}{|x|^2-\varepsilon^2}\big),
& \text{if } |x|<\varepsilon  \\
 0, \quad \textrm{if } |x|\geq\varepsilon
 \end{cases}
\]
and set $g_\varepsilon:=g\star\rho_\varepsilon$. It is knwon that:
\begin{itemize}
  \item[(i)] $g_\varepsilon$ is well defined, since $g\in L^2(\Omega)$,
 hence it is locally integrable.
  \item[(ii)] $g_\varepsilon\in C_0^\infty(\Omega_\varepsilon)$, with
$\Omega_\varepsilon:=\{x\in\Omega: \textrm{dist}(x,\partial\Omega)>\varepsilon\}$.
  \item[(iii)] $\partial_{x_i}g_\varepsilon$ is bounded uniformly with respect to
$\varepsilon$ for all $i=1,\ldots,N$.
  \item[(iv)] $\lim_{\varepsilon\to 0^+} g_\varepsilon = g$, strongly in
$L^2(\Omega)$.
\end{itemize}

Thus we can take $f_\varepsilon:=\operatorname{div}(g_\varepsilon)$ and,
from Property (iii) above, we immediately have that
$\|f_\varepsilon\|_{H^{-1+\varepsilon}(\Omega)}$ is bounded uniformly with
respect to $\varepsilon$. In addition, using Properties (ii) and (iv),
it is straightforward that, for all $i=1,\ldots,N$,
$\partial_{x_i} g_\varepsilon = \rho_\varepsilon\star g_{x_i} \to g_{x_i}$ as
$\varepsilon\to 0^+$. Hence,
\[
 \lim_{\varepsilon\to 0^+}f_\varepsilon
= \lim_{\varepsilon\to 0^+}\operatorname{div}(g_\varepsilon)
= \operatorname{div}(g) = f,
\]
where the convergence is strong in $H^{-1}(\Omega)$.
Therefore, by choosing $\varepsilon=1-s$, following the above argument
 we can construct a sequence $\{f_s\}_{0<s<1}\subset H^{-s}(\Omega)$
satisfying (H1) and (H2').
\end{proof}

\begin{remark} \rm
 Notice that (H2') is a property of strong convergence in $H^{-1}(\Omega)$
 which, clearly, implies the weak convergence in the same functional setting
 (property (H2)). Therefore, Proposition \ref{limit_f} provides a sequence
$\mathcal{F}_s$ which is within the hypotheses of Theorem \ref{limit_thm}.
\end{remark}

\section{Elliptic case: proof of Theorem \ref{limit_thm}}\label{weak_sec}

In this Section, we give the proof of Theorem \ref{limit_thm} employing
the definition of weak solution that we gave in Section \ref{prel}.

\begin{proof}[Proof of Theorem \ref{limit_thm}]
First of all, since we are interested in the behavior for $s\to 1^-$,
until the end of the proof we will assume $s> 1/2$.
Moreover, from (H2) and the definition of weak convergence we get
\begin{equation}\label{limit-rhs}
 \lim_{s\to 1^-}\int_{\Omega} f_sv\,dx
= \int_{\Omega} fv\,dx, \quad \forall v\in\mathcal{D}(\Omega).
\end{equation}

For all $0<s<1$, let $u_s\in H_0^s(\overline{\Omega})$ be the solution
to \eqref{PE} corresponding to the right-hand side $f_s$.
According to Proposition \ref{prop-ex}, for $s$ sufficiently close to one
we have the estimate
\begin{equation}\label{norm_est}
 \sqrt{1-s}\|u_s\|_{H^s(\Omega)}\leq\mathcal{C}(s,N)\|f_s\|_{H^{-s}(\Omega)}\,,
\end{equation}
with
\[
 \mathcal{C}(s,N) := \sqrt{\frac{2-2s}{C_{N,s}}}
\]

Moreover, for all $N$ fixed, the constant $\mathcal{C}(s,N)$ is decreasing
as a function of $s$ (see Figure \ref{fig1}). This of course implies
\[
 \mathcal{C}(s,N) < \mathcal{C}\Big(\frac 12,N\Big)
= \sqrt{\frac{\pi}{\Gamma(\frac{N+1}{2})}}.
\]

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} % cns
\end{center}
 \caption{Behavior of $\mathcal{C}(s,N)$ as a function of
 $s\in[1/2,1]$ for different fixed values of $N$.}\label{fig1}
\end{figure}

Therefore, from \eqref{norm_est} and the uniform boundedness of
$\|f_s\|_{H^{-s}(\Omega)}$ we deduce that
\[
 \sqrt{1-s}\|u_s\|_{H^s(\Omega)}\leq C
\]
with $C$ depending only on $N$ and $\Omega$. This, thanks to Proposition
\ref{brezis_prop}, allows us to conclude that $u_s\to u$ strongly in
$H^{1-\delta}_0(\Omega)$ for any $0<\delta\leq 1$, and that $u\in H_0^1(\Omega)$.

Notice that, according to \cite[Section 6]{warma2015fractional}, for all
$\phi\in H_0^s(\overline{\Omega})$ and $\psi\in\mathcal{D}(\Omega)$
we have the identity
\begin{align*}
 \big\langle (-\Delta)^s{\phi},\psi\big\rangle_{L^2(\Omega)}
&=\frac{C_{N,s}}{2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{(\phi(x)-\phi(y))(\psi(x)-\psi(y))}{|x-y|^{N+2s}}\,dx\,dy \\
 &= \big\langle \phi,(-\Delta)^s{\psi}\big\rangle_{L^2(\Omega)}.
\end{align*}

This can be applied to the variational formulation \eqref{weak-sol},
 which can thus be rewritten as
\begin{equation}\label{weak-def-s}
 \big\langle u_s,(-\Delta)^s{v}\big\rangle_{L^2(\Omega)} = \int_\Omega f_sv\,dx.
\end{equation}

Now, since $u_s\to u$ strongly in $H^{1-\delta}_0(\Omega)$ for any
$0<\delta\leq 1$ and $v\in\mathcal{D}(\Omega)$, Proposition \ref{limit_prop}
and Cauchy-Schwarz inequality imply that
\begin{align*}
&\big|\langle u_s,(-\Delta)^s{v}\rangle_{L^2(\Omega)}-\langle u,
-\Delta v\rangle_{L^2(\Omega)}\big|  \\
& = \big|\langle u_s,(-\Delta)^s{v}-(-\Delta v)\rangle_{L^2(\Omega)}
 +\langle u_s-u,-\Delta v\rangle_{L^2(\Omega)}\big| \\
&\leq\|u_s\|_{L^2(\Omega)}\|(-\Delta)^s{v}-(-\Delta v)\|_{L^2(\Omega)}
 + \|-\Delta v\|_{L^2(\Omega)}\|u_s-u\|_{L^2(\Omega)}\to 0,
\end{align*}
as $s\to 1^-$. Consequently,
\[
 \lim_{s\to 1^-} \big\langle u_s,(-\Delta)^s{v}\big\rangle_{L^2(\Omega)}
= \big\langle u,-\Delta v\big\rangle_{L^2(\Omega)}
= -\int_\Omega u\Delta v\,dx = \int_\Omega \nabla u\cdot\nabla v\,dx.
\]
This, together with \eqref{limit-rhs} and \eqref{weak-def-s} implies that
$u$ satisfies
\[
 \int_\Omega \nabla u\cdot\nabla v\,dx
= \int_{\Omega} fv\,dx, \quad \forall v\in\mathcal{D}(\Omega),
\]
i.e. it is a weak solution to \eqref{poisson}.
\end{proof}

\begin{remark} \rm
The result that we just proved is to some extent not surprising, due to
the limit behavior of the fractional Laplacian as $s\to 1^-$.
In fact, a hint that Theorem \ref{limit_thm} had to be true comes from
the very classical example
\begin{gather*}
  (-\Delta)^s{u_s} =1, \quad x\in B(0,1)  \\
  u_s\equiv 0, \quad x\in B(0,1)^c,
\end{gather*}
whose solution is given explicitly by
\[
 u_s(x) = \frac{2^{-2s}\Gamma(\frac N2)}{\Gamma(\frac{N+2s}{2})
\Gamma(1+s)}(1-|x|^2)^s\chi_{B(0,1)}.
\]
Indeed, it can be readily checked that, for $x\in B(0,1)$,
\[
 \lim_{s\to 1^-} u_s(x) = \frac{1}{2N}(1-|x|^2):=u(x),
\]
which is the unique solution to the limit problem
 \begin{gather*}
  -\Delta u =1 , \quad x\in B(0,1)  \\
  u= 0, \quad x\in \partial B(0,1).
 \end{gather*}
Of course, the above fact does not tell anything about the general case
of problem \eqref{PE}. To the best of our knowledge, this is an issue that,
 although natural and probably expected, has not yet been fully addressed
in the literature (at least, not in the setting of weak solutions with
minimal assumptions) and our contribution helps to fill in this gap.
\end{remark}

\section{Additional results and further comments}\label{rem_sec}

\subsection{Weakening the assumptions of Theorem \ref{limit_thm}}

Scope of this section is to show that a convergence result in the
spirit of Theorem \ref{limit_thm} can be obtained under weaker assumption
on the sequence $\mathcal F_s$ of the right-hand sides of \eqref{PE}.
 In particular, we are going to prove the following result.

\begin{theorem}\label{limit_thm_weak}
Let $\mathcal F_s=\{f_s\}_{0<s<1}\subset H^{-1}(\Omega)$ be a sequence
such that $f_s\rightharpoonup f$ weakly in $H^{-1}(\Omega)$.
For all $f_s\in\mathcal F_s$, let $u_s$ be the corresponding solution to \eqref{PE}.
 Then, as $s\to 1^-$, $u_s\rightharpoonup u$ weakly in $L^2(\Omega)$, with
$u$ solution to \eqref{poisson} in the transposition sense.
\end{theorem}

\begin{proof}
First of all, since we are interested in analyzing the behavior of $u_s$
as $s\to 1^-$, until the end of this proof we will always assume $s>1/2$.
Moreover, observe that, the right-hand side $f_s$ belongs to $H^{-1}(\Omega)$,
which is strictly greater than $H^{-s}(\Omega)$. Therefore, we cannot apply
Lax-Milgram Theorem. Instead, we shall define the solution to \eqref{PE}
 in a different way.

For all $\phi\in L^2(\Omega)$, let $y$ be solution of the elliptic problem
\begin{equation}\label{PE_transp}
 \begin{gathered}
  (-\Delta)^s{y}=\phi, \quad x\in\Omega  \\
  y \equiv 0, \quad x\in\Omega^c.
 \end{gathered}
\end{equation}

Recall that, from the regularity of $\phi$ and the results in
 \cite{biccari2017local,cozzi2017interior}, for all $\varepsilon>0$ we
 have $y\in H^{2s-\varepsilon}_0(\Omega)\hookrightarrow H^1_0(\Omega)$,
with continuous and compact embedding.

Moreover, the map $\Lambda: \phi\mapsto y$ is linear and continuous
from $L^2(\Omega)$ into $H^{2s-\varepsilon}_0(\Omega)$.
Thus, $\Lambda$ is compact from $L^2(\Omega)$ into $H^1_0(\Omega)$ and
its adjoint $\Lambda^*$ is a compact operator from $H^{-1}(\Omega)$
into $L^2(\Omega)$. In addition,
\[
 \langle f_s,y\rangle_{H^{-1}(\Omega),H^1_0(\Omega)}
= \langle f_s,\Lambda \phi\rangle_{H^{-1}(\Omega),H^1_0(\Omega)}
 = (\Lambda^* f_s,\phi)_{L^2(\Omega)}.
\]
Therefore, $u_s:=\Lambda^*f_s\in L^2(\Omega)$ is a solution defined by
transposition to \eqref{PE}, i.e.\ it satisfies
\begin{equation}\label{transp_def_s}
 \int_\Omega u_s\phi\,dx = \langle f_s,y\rangle_{H^{-1}(\Omega),H^1_0(\Omega)}.
\end{equation}
Moreover, we have
\begin{equation}\label{us_norm}
 \|u_s\|_{L^2(\Omega)} \leq C\|f_s\|_{H^{-1}(\Omega)}\leq C',
\end{equation}
with $C'$ independent of $s$ since, $f_s$ being in $H^{-1}(\Omega)$,
it is uniformly bounded in that space.

In particular, $\{u_s\}_{0<s<1}$ is a bounded sequence in $L^2(\Omega)$,
which implies that $u_s\rightharpoonup u$ weakly in $L^2(\Omega)$.

Notice that \eqref{transp_def_s} is obtained multiplying \eqref{PE} for
$y$ and integrating over $\Omega$. Observe also that in this expression
the functional spaces involved (namely $L^2(\Omega)$, $H_0^1(\Omega)$ and
$H^{-1}(\Omega)$) do not depend on $s$. Then, using the definition of weak
limit and \eqref{transp_def_s} we have
\[
 \int_\Omega u\phi\,dx = \lim_{s\to 1^-}\int_\Omega u_s\phi\,dx
= \lim_{s\to 1^-} \langle f_s,y\rangle_{H^{-1}(\Omega),H_0^1(\Omega)}
= \langle f,y\rangle_{H^{-1}(\Omega),H_0^1(\Omega)},
\]
i.e.\ $u$ is a solution by transposition to \eqref{poisson}.
Moreover, since the $L^2(\Omega)$-regularity of $u_s$ cannot be improved,
its convergence to a solution to \eqref{poisson} can be expected only
in the weak sense.
\end{proof}

\subsection{Remarks on the convergence rate}

Our interest in the subject of this paper is motivated by previous results
 concerning the numerical approximation of the fractional Laplacian.
In more detail, the issue that we addressed came from the observation that
for the stiffness matrix $\mathcal A_h^s$ derived in
\cite{biccari2017controllability} from the FE discretization of \eqref{frac_lapl}
in dimension $N=1$  the following holds:
\begin{itemize}
 \item[(i)] $\lim_{s\to 0^+}\mathcal A_h^s = h\operatorname{Tridiag}(1/6,2/3,1/6):=\mathcal I_h$, an approximation of the identity;

 \item[(ii)] $\lim_{s\to 1^-}\mathcal A_h^s = h^{-1}
\operatorname{Tridiag}(-1,2,-1):=\mathcal A_h$, the classical tridiagonal
matrix for the FE approximation of the one-dimensional Laplacian.
\end{itemize}

The second property in particular implies that also the numerical solution
$u_h^s$ associated to $\mathcal A_h^s$ converges to the one corresponding
to $\mathcal A_h$. Therefore, investigating whether this still holds in the
continuous case was a question that arose naturally.

While we answered to this question in Theorem \ref{limit_thm},
there we did not specify under which rate this convergence occurs.
In what follow, we present an informal discussion on this particular point.

During the proof of Theorem \eqref{limit_thm}, we showed that the sequence
$\{u_s\}_{0<s<1}$ of solutions to \eqref{PE} is bounded in
$H_0^s(\overline{\Omega})$, with the  estimate
\begin{equation}\label{us_norm_est}
 \sqrt{1-s}\|u_s\|_{H^s(\Omega)}\leq C,
\end{equation}
with $C$ a constant uniform with respect to $s$. This last inequality,
in turn, was obtained as a consequence of Proposition \ref{prop-ex} and
of the assumption (H1) on the sequence $\{f_s\}_{0<s<1}$ of the
right-hand sides.

Moreover, the factor $\sqrt{1-s}$ in \eqref{us_norm_est} already appears
in \cite{bourgain2001another} to correct the well-known defect
of the seminorm $|\cdot|_{H^s(\Omega)}$ which, as $s\to 1^-$, does not
converge to $|\cdot|_{H^1(\Omega)}$.

In fact, if $\zeta$ is any smooth non-constant function, then for all $1< p<\infty$
we have $|\zeta|_{W^{s,p}(\Omega)}\to +\infty$ as $s\to 1^-$. This situation
 may be rectified by multiplying by $(1 -s)^{1/p}$ in front of
 $|\zeta|_{W^{s,p}(\Omega)}\to +\infty$. In particular, we have
\begin{equation}\label{conv_weighted}
 \lim_{s\to 1^-} (1-s)^{1/p}|\zeta|_{W^{s,p}(\Omega)}
= K(N,p,\Omega)\Big(\int_\Omega |\nabla\zeta|^p\,dx\Big)^{1/p}.
\end{equation}

Also notice that the constant $K$ in the expression above is uniform in $s$.
In view of these observations, we claim that the convergence that we obtained
in Theorem \ref{limit_thm} satisfies the rate
\[
 \lim_{s\to 1^-}\|u_s-u\|_{H^s(\Omega)} \sim \mathcal O(\sqrt{1-s}).
\]
Indeed, if this convergence were slower, then we would still have blow-up
phenomena in the $H^s(\Omega)$-seminorm. On the other hand, if the
convergence were faster, then for some $\alpha>1/2$
\[
 \lim_{s\to 1^-}(1-s)^{\alpha}|\cdot|_{H^s(\Omega)}
= \lim_{s\to 1^-}\underbrace{(1-s)^{\alpha-\frac 12}}_{\to 0}
\underbrace{\sqrt{1-s}\,|\cdot|_{H^s(\Omega)}}_{\to |\cdot|_{H^1(\Omega)}}  = 0.
\]

Clearly, the discussion that we just presented is not a rigorous proof of our claim.
 Nevertheless, we believe that our statement is true, and a further confirmation
is given by the  numerical simulations
in Fugures \ref{behavior1_fig} and \ref{behavior2_fig},
where we compared the solution
to \eqref{PE} and \eqref{poisson} for different values of $s$ and we computed
the approximation error in the $H^s(\Omega)$-norm. As expected, we observe
a convergence of $u_s$ to $u$, with a rate of $\sqrt{1-s}$.

\begin{figure}[ht]
\begin{center}
\subfigure[Solutions to $(-\Delta)^s{u_s}=\sin(\pi x^2)$ for different 
values of $s\in{[1/2,1]}$]{
 \includegraphics[width=0.7\textwidth]{fig2a} % convergence1
}
\subfigure[Decay of $\|u_s-u\|_{H^s(-1,1)}$ with respect to $s\in{[1/2,1]}$]{
 \includegraphics[width=0.5\textwidth]{fig2b} % convergence2
 }
\end{center}
 \caption{Convergence of the solutions to $(-\Delta)^s{u_s}=\sin(\pi x^2)$
with Dirichlet homogeneous boundary conditions as $s\to 1^-$, and
its corresponding error in the $H^s(-1,1)$-norm.}
 \label{behavior1_fig}
\end{figure}

\begin{figure}[ht]
\begin{center}
 \subfigure[Solutions to $(-\Delta)^s{u_s}=f$ with $f$ piecewise constant 
for different values of $s\in{[1/2,1]}$]{
\includegraphics[width=0.7\textwidth]{fig3a} % convergence3
}
\subfigure[Decay of $\|u_s-u\|_{H^s(-1,1)}$ with respect to $s\in{[1/2,1]}$]{
\includegraphics[width=0.5\textwidth]{fig3b} % convergence4
}
\end{center}
 \caption{Convergence of the solutions to $(-\Delta)^s{u_s}=f$ with $f$
piecewise constant and Dirichlet homogeneous boundary conditions as $s\to 1^-$,
and its corresponding error in the $H^s(-1,1)$-norm.}
 \label{behavior2_fig}
\end{figure}

\subsection{Parabolic case}

As it most often happens, the properties of the solutions to elliptic problems
can be naturally transferred into the parabolic setting. In our case,
this translates in the fact that the solution $\phi_s$ to the fractional
heat equation
\begin{equation}\label{FE}
 \begin{gathered}
  \partial_t\phi_s + (-\Delta)^s{\phi_s} = g_s, \quad
(x,t)\in\Omega\times(0,T)  \\
  \phi_s\equiv 0, \quad (x,t)\in\Omega^c\times(0,T)   \\
  \phi_s(x,0) = 0, \quad x\in\Omega,
 \end{gathered}
\end{equation}
converges as $s\to 1^-$ to the one to the local problem
\begin{equation}\label{HE}
 \begin{gathered}
  \partial_t\phi -\Delta\phi = g, \quad(x,t)\in\Omega\times(0,T)\\
  \phi= 0, \quad (x,t)\in\partial\Omega\times(0,T)  \\
  \phi(x,0) = 0, \quad x\in\Omega.
 \end{gathered}
\end{equation}

First of all, let us recall that we have the following definition of weak
solution for the parabolic problem \eqref{FE} (see, e.g., \cite{leonori2015basic}).

\begin{definition}\label{weak_sol_def_parabolic} \rm
Let $g_s\in L^2(0,T;H^{-s}(\Omega))$. A function
$\phi_s\in L^2(0,T;H_0^s(\overline{\Omega}))\cap C([0,T];L^2(\Omega))$ with
$\partial_t\phi_s\in L^2(0,T;H^{-s}(\Omega))$ is said to be a weak solution
to the parabolic problem \eqref{FE} if for every
$\psi\in\mathcal{D}(\Omega\times(0,T))$, it holds the equality
\begin{equation}\label{weak-sol-par}
\begin{aligned}
&\int_0^T \int_\Omega\partial_t\phi_s\psi\,dx\,dt \\
& + \frac{C_{N,s}}{2}\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
 \frac{(\phi_s(x)-\phi_s(y))(\psi(x)-\psi(y))}{|x-y|^{N+2s}}\,dx\,dy\,dt  \\
&= \int_0^T\int_\Omega g_s\psi\,dx\,dt.
\end{aligned}
\end{equation}
\end{definition}

Moreover, thanks to \cite[Theorem 26]{leonori2015basic}, 
existence and uniqueness of solutions is guaranteed. 

\begin{proposition}
Assume that $f_s\in L^2(0,T;H^{-s}(\Omega))$. Then problem \eqref{FE} 
has a unique finite energy solution, defined according to 
\eqref{weak_sol_def_parabolic}.
\end{proposition}

Then, adapting the methodology for the proof of Theorem \eqref{limit_thm}, 
the following result is immediate.

\begin{theorem}\label{limit_thm_parabolic}
 Let $\mathcal{G}_s:=\{g_s\}_{0<s<1}\subset L^2(0,T;H^{-s}(\Omega))$ 
be a sequence satisfying the following assumptions for all $0<t<T$:
 \begin{itemize}
  \item[(H3)] $\|g_s(t)\|_{H^{-s}(\Omega)}\leq C$, for all 
$0<s<1$ and uniformly with respect to $s$.

  \item[(H4)] $g_s(t)\rightharpoonup g(t)$ weakly in $H^{-1}(\Omega)$ as $s\to 1^-$.
 \end{itemize}
For any $f_s\in\mathcal{G}_s$, let $\phi_s\in L^2(0,T;H_0^s(\overline{\Omega}))$ 
be the unique weak solution to the corresponding parabolic problem \eqref{FE} 
in the sense of Definition \ref{weak_sol_def_parabolic}. 
Then, as $s\to 1^-$, $(\phi_s,\partial_t\phi_s)\to(\phi,\partial_t\phi)$ 
strongly in $L^2(0,T;H^{1-\delta}_0(\Omega))\times L^2(0,T;H^{-1}(\Omega))$ 
for any $0<\delta\leq 1$. Moreover, 
$\phi\in L^2(0,T;H^1_0(\Omega))\times L^2(0,T;H^{-1}(\Omega))$ and satisfies
\[
 \int_0^T \int_\Omega\partial_t\phi\psi\,dx\,dt 
+ \int_0^T\int_\Omega \nabla\phi\cdot\nabla\psi\,dx\,dt 
= \int_0^T\int_\Omega g\psi\,dx\,dt, \quad 
\forall\psi\in\mathcal{D}(\Omega\times(0,T)),
\]
i.e. it is the unique weak solution to \eqref{HE}.
\end{theorem}

\begin{proof}
First of all, notice that a sequence $\mathcal{G}_s$ satisfying (H3) and 
(H4) exists. In fact, it can be constructed following the methodology of 
Proposition \ref{limit_f}, since both properties are independent of the 
time variable. Moreover, it is evident that we shall only analyze  the 
first term on the left-hand side of \eqref{weak-sol-par},
because of the following two facts:
\begin{itemize}
 \item The functional space in which the integration in time is carried out 
is fixed and does not depend on $s$. Therefore, the limit process does not 
affect the regularity in the time variable.

 \item For the remaining two terms in \eqref{weak-sol-par}, the limit as 
$s\to 1^-$ can be addressed in an analogous way as in the proof of 
Theorem \ref{limit_thm}.
\end{itemize}

Moreover, multiplying \eqref{FE} by $\phi_s$, integrating by parts, and using 
a classical methodology for heat-like equations, it is not difficult to obtain
 the  energy estimate
\begin{equation}\label{energy_est}
 \|\phi_s\|_{L^2(0,T;H_0^s(\overline{\Omega}))} 
+ \|\partial_t\phi_s\|_{L^2(0,T;H^{-s}(\Omega))}
\leq C\|g_s\|_{L^2(0,T;H^{-s}(\Omega))}.
\end{equation}

From here, an analogous argument as the one presented in the proof of
 Theorem \ref{limit_thm} can be developed to show that, 
as $s\to 1^-$, $\phi_s\to \phi$ strongly in $L^2(0,T;H^{1-\delta}_0(\Omega))$ 
for all $0<\delta\leq 1$. Moreover, from \eqref{energy_est} we have that
 $\{\partial_t \phi_s\}_{0<s<1}$ is bounded in $L^2(0,T;H^{-s}(\Omega))$, 
which is compactly embedded in $L^2(0,T;H^{-1}(\Omega))$. 
Thus, as $s\to 1^-$, $\partial_t\phi_s\to \partial_t\phi$ strongly in 
$L^2(0,T;H^{-1}(\Omega))$, and we can conclude that 
$(\phi_s,\partial_t\phi_s)\to (\phi,\partial_t\phi)$ strongly in 
$L^2(0,T;H^{1-\delta}_0(\Omega))\times L^2(0,T;H^{-1}(\Omega))$ for all 
$0<\delta\leq 1$. In particular,
\[
 \lim_{s\to 1^-}\int_0^T\int_{\Omega} \partial_t\phi_s\psi\,dx\,dt 
= \int_0^T\int_{\Omega} \partial_t\phi\psi\,dx\,dt.
\]
This, together with the above remarks, implies that the function $\phi$ satisfies
\[
 \int_0^T \int_\Omega\partial_t\phi\psi\,dx\,dt 
 + \int_0^T\int_\Omega \nabla\phi\cdot\nabla\psi\,dx\,dt
 = \int_0^T\int_\Omega g\psi\,dx\,dt, \quad \forall\psi\in\mathcal{D}
(\Omega\times(0,T)),
\]
i.e. it is the unique weak solution to \eqref{HE}.
\end{proof}

\subsection*{Acknowledgments}
The authors wish to acknowledge Enrique Zuazua (Universidad Aut\'onoma de Madrid, 
DeustoTech and Laboratoire Jacques-Louis Lions) for having suggested the 
topic of this work. Moreover, a special thank goes to Xavier Ros-Oton 
(Universit\"at Z\"urich) and Enrico Valdinoci (University of Melbourne) 
for interesting and clarifying discussions.

 This project received funding from the European Research Council (ERC)
 under the European Union's Horizon 2020 research and innovation programme
 (grant agreement No. 694126-DyCon), and from the MTM2017-92996-C2-1-R
 grant of MINECO (Spain).
 U. Biccari was partially  supported by the Grants MTM2014-52347 of
 MINECO (Spain) and FA9550-18-1-0242 of AFOSR (USA).

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\end{document}