# Parabolic $\alpha$-Riesz flows and limit cases $\alpha\to 0^{+}$, $\alpha\to
d^{-}$

L. De Luca  Istituto per le Applicazioni del Calcolo “M. Picone” IAC-CNR, Via
dei Taurini 19, I-00185 Roma, Italy<EMAIL_ADDRESS>,  M. Morini
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli
Studi di Parma, Parma, Italy<EMAIL_ADDRESS>,  M. Ponsiglione
Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università di Roma,
Piazzale Aldo Moro 2, I-00185 Roma, Italy<EMAIL_ADDRESS>and  E.
Spadaro  Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università
di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce the notion of parabolic $\alpha$-Riesz flow, for
$\alpha\in(0,d)$, extending the notion of $s$-fractional heat flows to
negative values of the parameter $s=-\frac{\alpha}{2}$. Then, we determine the
limit behaviour of these gradient flows as $\alpha\to 0^{+}$ and $\alpha\to
d^{-}$.

To this end we provide a preliminary $\Gamma$-convergence expansion for the
Riesz interaction energy functionals. Then we apply abstract stability results
for uniformly $\lambda$-convex functionals which guarantee that
$\Gamma$-convergence commutes with the gradient flow structure.

Keywords: Gagliardo seminorms; $\Gamma$-convergence; Fractional heat flow AMS
subject classifications:  49J45 35R11 35K20

###### Contents

  1. 1 Description of the problem
  2. 2 $\Gamma$-convergence for the functionals $\mathcal{J}^{\alpha}$ and $\widehat{\mathcal{J}}^{\alpha}$ as $\alpha\to 0^{+}$
  3. 3 $\Gamma$-convergence for the functionals $\mathcal{J}^{\alpha}$ as $\alpha\to d^{-}$
  4. 4 Convergence of the $\alpha$-Riesz flows

## Introduction

Fractional versions of local operators attracted much attention in the last
decades. For $s\in(0,1)$ the so-called $s$-fractional Laplacians and
$s$-fractional perimeters represent natural and interesting nonlocal
counterparts of the classical Laplacian operator and Euclidean perimeter. It
is also well known [17, 26] that for negative values of the parameter $s$, and
specifically for $s=-\frac{\alpha}{2}$ with $\alpha\in(0,d)$ ($d$ being the
dimension of the ambient space) , the $\alpha$-Riesz potentials represent the
natural continuation of the one-parameter family of $s$-fractional Laplacians
(as well as of $s$-fractional perimeters and curvatures [7]) for negative
values of $s$.

Much analysis has been devoted to studying the limit cases of $s$-fractional
perimeters and Laplacians as $s\to 0^{+}$ and $s\to 1^{-}$ ; remarkably, as
$s\to 1^{-}$ one recovers the local canonical objects, while the limit $s\to
0^{+}$ is somehow more degenerate. The analogous analysis for Riesz type
operators seems to be less investigated, with recent results only for what
concerns geometric flows [7]. This paper gives a further contribution in this
direction, introducing the notion of parabolic $\alpha$-Riesz flows and
studying the limit cases as $\alpha\to 0^{+}$ and $\alpha\to d^{-}$.

We will now revisit the literature concerning the limit analysis of
$s$-fractional operators as $s\to 0^{+}$ (referring the interested reader to
[1, 3, 4, 5, 9, 10, 18, 24, 19] for the case $s\to 1^{-}$).

In [20] the authors show that, as $s\to 0^{+}$ , the squared $s$-fractional
Gagliardo seminorms multiplied by $s$ pointwise converge to (a multiple of)
the squared $L^{2}$ norm. A $\Gamma$-convergence expansion (as $s\to 0^{+}$)
with respect to the $L^{2}$ topology has been derived in [9]. The first term
of such an expansion is, according with [20], the squared $L^{2}$ norm divided
by $s$ , whereas the next order term can be understood as a kind of
$0$-fractional Gagliardo seminorm. The first variation of the zero-order
$\Gamma$-limit is trivial, while the first variation of the first order
$\Gamma$-limit gives an interesting elliptic operator, referred to as
$0$-fractional Laplacian, which is a slight variant of the logarithmic
Laplacian introduced in [8]. Still in [9], the minimizing movements approach
was used as a bridge between the $\Gamma$-convergence analysis and the
stability of the corresponding fractional parabolic flows: After scaling the
time by $s$ , the $s$-fractional parabolic flows converge, as $s\to 0^{+}$ ,
to the degenerate parabolic flow of the squared $L^{2}$ norm (with respect to
the $L^{2}$ metric, itself); this is nothing but a trivial ODE corresponding
to the exponential decay of the initial datum. If, instead of scaling in time,
we renormalize the parabolic flow by subtracting the forcing leading term
(corresponding to the first variation of the squared $L^{2}$ norm, scaled by
$s$), we end up with a much more interesting dynamics: the heat flow governed
by the $0$-fractional Laplacian, namely, the parabolic flow of the first order
$\Gamma$-limit of the $s$-fractional Gagliardo seminorms.

A first task of this paper is to provide a similar analysis as $s\to 0^{-}$.
As already explained, for negative values of $s$ the analogue of fractional
Laplacians is given by $\alpha$-Riesz potentials [17, 26], with $\alpha=-2s$ .
The first step in our analysis is to formalize such an analogy introducing,
for negative values of $s$, explicit counterparts of $s$-fractional Gagliardo
seminorms, Laplacians, and the corresponding parabolic flows, referred to as
parabolic $\alpha$-Riesz flows. Such a track has already been pursued in the
geometric framework [7]; there, suitable notions of $\alpha$-Riesz perimeters,
curvatures and geometric flows have been introduced and analyzed as $\alpha\to
0^{+}$, leading to the geometric flow of the $0$-fractional perimeter
introduced in [13]. (In fact, such a flow also coincides with the limit as
$s\to 0^{+}$ of the $s$-fractional mean curvature flows [7].)

As in [9, 7], our approach is purely variational, in the sense that our
stability analysis relies on the gradient flow structure of the parabolic
$\alpha$-Riesz flows. We consider an open bounded set
$\Omega\subset\mathbb{R}^{d}$ and, for every $\alpha\in(0,d)$ , we define the
$\alpha$-Riesz functionals $\mathcal{J}^{\alpha}$ on $L^{2}(\Omega)$ as

(0.1)
$\mathcal{J}^{\alpha}(u):=-\int_{\Omega}\int_{\Omega}\frac{u(x)u(y)}{|x-y|^{d-\alpha}}\,\mathrm{d}y\,\mathrm{d}x\,.$

We stress that the first variation of $\mathcal{J}^{\alpha}$ gives back the
opposite of the $\alpha$-Riesz potentials $(-\Delta)^{-\frac{\alpha}{2}}$ [26,
17]. In terms of a more genuine variational viewpoint, $\mathcal{J}^{\alpha}$
can be understood as a continuous version of the $XY$ energy functional with
long-range interactions and real-valued spin variable. Oversimplifying, in
classical $XY$ scalar systems the spin variable $u$ takes values $\pm 1$, and,
for nearest neighbor interactions, the energy can be seen as a discrete
perimeter of the phase $\\{u=1\\}$ , as well as a discrete Dirichlet energy
for functions defined on a lattice and taking only two values. Clearly, long-
range variants of the $XY$ model are naturally related with nonlocal
perimeters; specifically, with fractional type perimeters, when the
interaction kernel is non-integrable, and with Riesz type perimeters, when the
kernel is locally integrable as in (0.1). Although such analogies are part of
the motivations for our study, here we will not push further considerations
within such a statistical mechanics framework; in fact, we focus our analysis
on real-valued functions, adopting tools and terminology of fractional
seminorms and operators.

Now we describe in detail our results, starting from the asymptotic behavior
of $\mathcal{J}^{\alpha}$ as $\alpha\to 0^{+}$ . In Theorem 2.1 we prove that
the functionals $-\alpha\mathcal{J}^{\alpha}$ $\Gamma$-converge as $\alpha\to
0^{+}$ to a multiple of $\|\cdot\|^{2}_{L^{2}(\Omega)}$ with respect to the
weak $L^{2}$ topology. Some considerations about the signs of the energy
functionals are in order: the functionals $-\mathcal{J}^{\alpha}$ penalize
more and more, as $\alpha\to 0^{+}$ , functions with large $L^{2}$ norm,
which, instead, are favored by the functionals $\mathcal{J}^{\alpha}$ .
Indeed, also after the natural scaling, the functionals
$\alpha\mathcal{J}^{\alpha}$ would $\Gamma$-converge to the relaxation (with
respect to the weak $L^{2}$ topology) of $-\|\cdot\|_{L^{2}(\Omega)}^{2}$,
which is constantly equal to $-\infty$ (see Remark 2.2). On the other hand,
renormalizing the (otherwise asymptotically ill-posed) family of functionals
$\mathcal{J}^{\alpha}$ , by setting
$\widehat{\mathcal{J}}^{\alpha}(u):=\mathcal{J}^{\alpha}(u)+\frac{d\omega_{d}}{\alpha}\|u\|_{L^{2}(\Omega)}^{2}$
, it turns out that the energies $\widehat{\mathcal{J}}^{\alpha}$ penalize
oscillations. In fact, in Theorem 2.3 we prove that the $\Gamma$-limit of
$\widehat{\mathcal{J}}^{\alpha}$ (as $\alpha\to 0^{+}$), with respect to the
strong $L^{2}$ topology, is a nonlocal coercive Dirichlet type energy
functional $\widehat{\mathcal{J}}^{0}$ , which is nothing but the
$\Gamma$-limit derived in [9] for the renormalized $s$-fractional Gagliardo
seminorms (as $s\to 0^{+}$). Similarly to what observed above, here there is
only one option in choosing the signs in order to deal with coercive
functionals: reversing all the signs, one would end up with a kind of
relaxation of $-\widehat{\mathcal{J}}^{0}$ , which would give back once again
$-\infty$ (Remark 2.5). Similar considerations can be done in the setting of
nonlocal geometric flows, but, trying to draw such a parallelism, a relevant
difference emerges: in the geometrical context $u$ is a characteristic
function, so that oscillations relaxing the energy to $-\infty$ are
automatically prevented. A relevant consequence is that, contrarily to the
case discussed above, in the geometric setting, the right sign is the negative
one, since the functionals $\alpha\mathcal{J}^{\alpha}$ computed on
characteristic functions, together with their limit, provide a true perimeter,
whose first variation is a nonlocal curvature satisfying natural comparison
principles [7]. These different paths of geometrical and linear framework
rejoin when looking at the next order, where we fix the negative sign for the
$\alpha$-fractional Laplacians.

In this framework, we introduce the parabolic $\alpha$-Riesz flows as the
$L^{2}$ gradient flows of the functionals $\mathcal{J}^{\alpha}$ , scaled or
renormalized according with the $\Gamma$-convergence analysis. As mentioned
above, the first variation of the functional $-\mathcal{J}^{\alpha}$ (for
$\alpha\in(0,d)$) is already known in the literature [26, 17] under the name
of $\alpha$-Riesz potential $(-\Delta)^{-\frac{\alpha}{2}}$ . Therefore, the
fractional parabolic flows considered here are nothing but fractional heat
flows governed by $s$-fractional Laplacians for negative values of the
fractional parameter $s\in(-\frac{d}{2},0)$.

As one may expect from the static asymptotic analysis, scaling the time by
$\alpha$ , the $\alpha$-parabolic flows governed by the operator
$(-\Delta)^{-\frac{\alpha}{2}}$ converge (Theorem 4.2) as $\alpha\to 0^{+}$,
to a degenerate ODE describing the exponential decay of the initial datum,
namely, to the gradient flow of the (up to a prefactor) $\Gamma$-limit
$\|\cdot\|^{2}_{L^{2}(\Omega)}$; this result agrees with the analysis of
$s$-fractional heat flows as $s\to 0^{+}$ done in [9]. In analogy with the
geometric case of fractional curvature flows [7] and following the discussions
on the signs done above, one could wonder what happens reversing all the
signs. In fact, the parabolic flows governed by
$-\alpha(-\Delta)^{-\frac{\alpha}{2}}$ is actually well defined; however, as
explained in Remark 4.3, our variational methods based on $\Gamma$-convergence
do not provide its convergence (as $\alpha\to 0^{+}$) to the parabolic flow
$u_{t}=2u$. The latter describes the exponential growth of the initial datum,
and is consistent with the corresponding geometric flows, where balls tend to
expand (instead of shrinking) faster and faster as $\alpha\to 0^{+}$.

Following the first order $\Gamma$-convergence analysis, the gradient flow of
the functional $\widehat{\mathcal{J}}^{\alpha}$ is a parabolic flow governed
by $-(-\Delta)^{-\frac{\alpha}{2}}$ plus the forcing leading linear term
(diverging as $\alpha\to 0^{+}$). This flow converges (Theorem 4.4), as
$\alpha\to 0^{+}$, to the gradient flow of $\widehat{\mathcal{J}}^{0}$ ,
namely, to the parabolic flow governed by the $0$-fractional Laplacian. Again,
this result agrees with the analogous analysis done for $s\to 0^{+}$ in [9].
In fact, the proofs of the stability results above are obtained as an
application of the abstract approach developed in [9]; loosely speaking, the
latter consists in showing that uniform $\lambda$-convexity and
$\lambda$-positivity on an Hilbert setting is a sufficient condition to
guarantee that $\Gamma$-convergence of the underlying functionals commutes
with the gradient flow structure. We refer to Subsection 4.2 for a short recap
on such an abstract setting.

The same approach, namely, first analyzing the asymptotic behavior of the
functionals $\mathcal{J}^{\alpha}$ as $\alpha\to d^{-}$ and then using the
abstract result in [9] to derive the convergence of the corresponding
parabolic flows, allows to study also the limit as $\alpha\to d^{-}$ for the
parabolic $\alpha$-Riesz flows. In such a case, the analysis is simpler.
Indeed, the interaction kernel in (0.1) converges to $1$ (strongly in
$L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$), whence, for every $u\in
L^{2}(\Omega)$ , we deduce that $\mathcal{J}^{\alpha}(u)$ converges (as
$\alpha\to d^{-}$) to $\mathcal{J}^{d}(u):=-\big{(}\int_{\Omega}u\big{)}^{2}$
. Actually, in Theorem 3.1, we prove that
$\mathcal{J}^{\alpha_{n}}(u^{n})\to\mathcal{J}^{d}(u)$ for every
$\alpha_{n}\to d^{-}$ and for every $u^{n}\rightharpoonup u$ in
$L^{2}(\Omega)$ , which provides in particular the $\Gamma$-convergence of the
functionals $\pm\mathcal{J}^{\alpha}$ to $\pm\mathcal{J}^{d}$ . As a
consequence, in Theorem 4.5 we prove that the $\alpha$-parabolic flows
governed by the operator $\pm(-\Delta)^{-\frac{\alpha}{2}}$ converge (Theorem
4.2) , as $\alpha\to d^{-}$, to a degenerate ODE describing now the
exponential growth/decay of the average of the initial datum, namely, to the
gradient flow of the $\Gamma$-limit $\mp\big{(}\int_{\Omega}u\big{)}^{2}$.
Moreover, for every $u^{n}\rightharpoonup u$ in $L^{2}(\Omega)$ and for every
$\alpha_{n}\to d^{-}$, the renormalized functionals
$\widetilde{\mathcal{J}}^{\alpha_{n}}(u^{n}):=\frac{\mathcal{J}^{\alpha_{n}}(u^{n})-\mathcal{J}^{d}(u^{n})}{d-\alpha_{n}}\,$
converge to the Riesz energy functional with logarithmic kernel

$\widetilde{\mathcal{J}}^{d}(u):=-\int_{\Omega}\int_{\Omega}u(x)u(y)\log\frac{1}{|x-y|}\,\mathrm{d}y\,\mathrm{d}x\,.$

In particular, as $\alpha\to d^{-}$ , the functionals
$\pm\widetilde{\mathcal{J}}^{\alpha}$ $\Gamma$-converge to the functional
$\pm\widetilde{\mathcal{J}}^{d}$ with respect to the weak $L^{2}$ convergence
(see Theorem 3.4). Finally, Theorem 4.6 establishes that the gradient flows of
$\pm\widetilde{\mathcal{J}}^{\alpha}$ converge, as $\alpha\to d^{-}$, to the
parabolic flow governed by the operator $\pm(-\Delta)^{-\frac{d}{2}}$, namely,
by the first variation of $\pm\widetilde{\mathcal{J}}^{d}$.

Summarizing, we have introduced the parabolic $\alpha$-Riesz flows, i.e., the
flows governed by the $\alpha$-Riesz potential $(-\Delta)^{-\frac{\alpha}{2}}$
, for $\alpha\in(0,d)$, and we have analyzed their convergence as $\alpha\to
0^{+}$ and $\alpha\to d^{-}$ . Along the way, we have focused on the
interesting class of integral functionals

(0.2)
$-\int_{\Omega}\int_{\Omega}\frac{u(x)u(y)}{|x-y|^{p}}\,\mathrm{d}y\,\mathrm{d}x\,,$

representing, for $p\in(0,d)$, natural extensions of Gagliardo-type energies.
In particular, through rigorous renormalization procedures, we have given a
meaning to the critical case $p=d$. It may be of some interest to extend our
methods to $p\geq d$, adopting suitable renormalization procedures; in this
direction, a popular method is the so-called core radius approach, already
used in [12] in the geometric framework. The general case $p\in\mathbb{R}$,
accounting also for negative values of $p$, could deserve further
investigations. Ultimately, our analysis invites to a methodical comparison
between the one-parameter family (0.2) of generalized Riesz
functionals/potentials and the (generalized) Gagliardo seminorms/fractional
Laplacians.

Acknoledgments: LDL and MM are members of the Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto
Nazionale di Alta Matematica (INdAM). The authors have been partially funded
by the ERC-STG Grant n. 759229 HiCoS “Higher Co-dimension Singularities:
Minimal Surfaces and the Thin Obstacle Problem”.

##  1\. Description of the problem

Let $d\in\mathbb{N}$ and let $\Omega\subset\mathbb{R}^{d}$ be a bounded and
open set with Lipschitz continuous boundary. For every $\alpha\in(0,d)$ , we
define the functional $\mathcal{J}^{\alpha}:L^{2}(\Omega)\to(-\infty,+\infty)$
as

(1.1)
$\mathcal{J}^{\alpha}(u):=-\int_{\Omega}\int_{\Omega}\frac{u(x)u(y)}{|x-y|^{d-\alpha}}\,\mathrm{d}y\,\mathrm{d}x\,.$

We first show that the functionals $\mathcal{J}^{\alpha}$ are well defined and
actually finite in $L^{2}(\Omega)$. Let
$k^{\alpha}:\mathbb{R}^{d}\to[0,+\infty)$ be the function defined by

(1.2) $k^{\alpha}(z):=\frac{1}{|z|^{d-\alpha}}\,.$

Then, $k^{\alpha}\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$ . Moreover, for
every $R>0$ we set

(1.3) $k^{\alpha}_{R}:=\chi_{B_{R}(0)}k^{\alpha}\,;$

it is immediate to check that

(1.4)
$\|k^{\alpha}_{R}\|_{L^{1}(\mathbb{R}^{d})}=\frac{d\omega_{d}}{\alpha}R^{\alpha}\,.$

Now, for any $u\in L^{2}(\Omega)$ and for every $R>\mathrm{diam}(\Omega)$, we
have

(1.5) $\mathcal{J}^{\alpha}(u)=-\langle\tilde{u},\tilde{u}\ast
k^{\alpha}\rangle_{L^{2}(\mathbb{R}^{d})}=-\langle\tilde{u},\tilde{u}\ast
k^{\alpha}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}\,,$

where, here and throughout the paper, $\tilde{u}\in L^{2}(\mathbb{R}^{d})$ is
defined by $\tilde{u}=u$ in $\Omega$ and $\tilde{u}=0$ in
$\mathbb{R}^{d}\setminus\overline{\Omega}$ .

Therefore, by (1.5) and (1.4), using Hölder and convolution’s Young
inequalities, we obtain that for every $u\in L^{2}(\Omega)$

(1.6)
$|\mathcal{J}^{\alpha}(u)|\leq\|k^{\alpha}_{R}\|_{L^{1}(\mathbb{R}^{d})}\|\tilde{u}\|^{2}_{L^{2}(\mathbb{R}^{d})}\leq\frac{d\omega_{d}}{\alpha}R^{\alpha}\|u\|^{2}_{L^{2}(\Omega)}\,,$

which shows that $\mathcal{J}^{\alpha}$ takes values in $(-\infty,+\infty)$ .

We highlight that for any $\alpha\in(0,d)$ the Fourier transform
$\mathscr{F}[k^{\alpha}]$ of $k^{\alpha}$ is given by

(1.7)
$\mathscr{F}[k^{\alpha}](\xi)=2^{\alpha}\pi^{\frac{d}{2}}\frac{\Gamma(\frac{\alpha}{2})}{\Gamma(\frac{d-\alpha}{2})}k^{d-\alpha}(\xi)\,,$

where $\Gamma$ is the Euler Gamma function, defined by
$\Gamma(\beta):=\int_{0}^{+\infty}t^{\beta-1}e^{-t}\,\mathrm{d}t$ , for
$\beta>0$ . Indeed, let $\varphi:\mathbb{R}^{d}\to\mathbb{R}$ belong to the
Schwarz class. Then, setting $\beta:=d-\alpha$ , by the very definition of
Euler Gamma function and by Fubini Theorem, we have that

(1.8)
$\int_{\mathbb{R}^{d}}\varphi(x)k^{\alpha}(x)\,\mathrm{d}x=\frac{1}{\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}\\!\,\mathrm{d}t\,t^{\frac{\beta}{2}-1}\int_{\mathbb{R}^{d}}\\!\,\mathrm{d}x\,\varphi(x)e^{-t|x|^{2}}\,.$

Moreover, since

$\mathscr{F}[e^{-t|\cdot|^{2}}](\xi):=\int_{\mathbb{R}^{d}}e^{-t|x|^{2}}e^{-ix\cdot\xi}\,\mathrm{d}x=\pi^{\frac{d}{2}}\frac{e^{-\frac{|\xi|^{2}}{4t}}}{t^{\frac{d}{2}}},$

by Parseval identity we obtain

$\int_{\mathbb{R}^{d}}\varphi(x)e^{-t|x|^{2}}\,\mathrm{d}x=\frac{\pi^{\frac{d}{2}}}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\mathscr{F}[\varphi](\xi)\frac{e^{-\frac{|\xi|^{2}}{4t}}}{t^{\frac{d}{2}}}\,\mathrm{d}\xi\,,$

which, plugged into (1.8), yields

$\displaystyle\int_{\mathbb{R}^{d}}\varphi(x)k^{\alpha}(x)\,\mathrm{d}x=$
$\displaystyle\,\frac{\pi^{\frac{d}{2}}}{(2\pi)^{d}}\frac{1}{\Gamma(\frac{\beta}{2})}\int_{0}^{+\infty}\\!\,\mathrm{d}t\,t^{\frac{\beta-d}{2}-1}\int_{\mathbb{R}^{d}}\mathscr{F}[\varphi](\xi)e^{-\frac{|\xi|^{2}}{4t}}\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\,\frac{\pi^{\frac{d}{2}}}{(2\pi)^{d}}2^{\alpha}\frac{\Gamma\big{(}\frac{\alpha}{2}\big{)}}{\Gamma(\frac{d-\alpha}{2})}\int_{\mathbb{R}^{d}}{\mathcal{F}[\varphi](\xi)}k^{d-\alpha}(\xi)\,\mathrm{d}\xi\,,$

where we have used first the change of variable $t=s|\xi|^{2}$ and then
$u=\frac{1}{4s}$ to get

$\displaystyle\int_{0}^{+\infty}t^{\frac{\beta-d}{2}-1}e^{-\frac{|\xi|^{2}}{4t}}\,\mathrm{d}t=$
$\displaystyle\,|\xi|^{-\alpha}\int_{0}^{+\infty}s^{\frac{\beta-d}{2}-1}e^{-\frac{1}{4s}}\,\mathrm{d}s$
$\displaystyle=$
$\displaystyle\,|\xi|^{-\alpha}2^{d-\beta}\int_{0}^{+\infty}u^{\frac{d-\beta}{2}-1}e^{-u}\,\mathrm{d}u$
$\displaystyle=$
$\displaystyle\,|\xi|^{-\alpha}2^{\alpha}\Gamma\big{(}\frac{\alpha}{2}\big{)}\,.$

Therefore, (1.7) holds true and hence the kernels $k^{\alpha}$ are positive
definite, i.e.,

(1.9) $\langle v,v\ast k^{\alpha}\rangle_{L^{2}(\mathbb{R}^{d})}\geq
0\qquad\textrm{for any }v\in L^{2}(\mathbb{R}^{d})\,,$

which, by (1.5), yields

(1.10) $\mathcal{J}^{\alpha}(u)\leq 0\qquad\textrm{for any }u\in
L^{2}(\Omega)\,.$

Moreover, by (1.5), using the symmetry of $k^{\alpha}_{R}$, for any $u,v\in
L^{2}(\Omega)$ we have that

(1.11) $\displaystyle|\mathcal{J}^{\alpha}(u)-\mathcal{J}^{\alpha}(v)|=$
$\displaystyle\Big{|}\langle u-v,(u+v)\ast
k^{\alpha}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}\Big{|}$ $\displaystyle\leq$
$\displaystyle\frac{d\omega_{d}}{\alpha}R^{\alpha}\big{(}\|u\|_{L^{2}(\Omega)}+\|v\|_{L^{2}(\Omega)}\big{)}\|u-v\|_{L^{2}(\Omega)}\,,$

whence we deduce the continuity of $\mathcal{J}^{\alpha}$ with respect to the
strong convergence in $L^{2}(\Omega)$ .

Now, for every $\alpha\in(0,d)$ and for every $u\in L^{2}(\Omega)$ we set

(1.12)
$\widehat{\mathcal{J}}^{\alpha}(u):=\mathcal{J}^{\alpha}(u)+\frac{d\omega_{d}}{\alpha}\|u\|^{2}_{L^{2}(\Omega)}\,.$

Let
$\mathcal{D}_{1}:=\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\,:\,|x-y|\leq
1\\}$ ; for every $\alpha\in[0,d)$ we define the functionals
$\mathcal{G}^{\alpha}_{1}:L^{2}(\Omega)\to[0,+\infty]$ and
$\mathcal{J}^{\alpha}_{1}:L^{2}(\Omega)\to\mathbb{R}$ as

$\mathcal{G}^{\alpha}_{1}(u):=\frac{1}{2}\iint_{\mathcal{D}_{1}}\frac{|\tilde{u}(x)-\tilde{u}(y)|^{2}}{|x-y|^{d-\alpha}}\,\mathrm{d}y\,\mathrm{d}x\,,\qquad\mathcal{J}^{\alpha}_{1}(u):=-\iint_{(\Omega\times\Omega)\setminus\mathcal{D}_{1}}\frac{u(x)u(y)}{|x-y|^{d-\alpha}}\,\mathrm{d}y\,\mathrm{d}x\,.$

We highlight that for every $\alpha\in(0,d)$ and for every $u\in
L^{2}(\Omega)$ it holds

(1.13)
$\widehat{\mathcal{J}}^{\alpha}(u)=\mathcal{G}^{\alpha}_{1}(u)+\mathcal{J}^{\alpha}_{1}(u)\,,$

Notice that, by Hölder inequality, for every $\alpha\in[0,d)$ and for every
$u\in L^{2}(\Omega)$

(1.14)
$|\mathcal{J}^{\alpha}_{1}(u)|\leq\iint_{\Omega\times\Omega}|u(x)u(y)|\,\mathrm{d}y\,\mathrm{d}x\leq\|u\|^{2}_{L^{1}(\Omega)}\leq|\Omega|\|u\|^{2}_{L^{2}(\Omega)}\,,$

and hence $\mathcal{J}^{\alpha}_{1}$ is finite on $L^{2}(\Omega)$ also for
$\alpha=0$ . In full analogy with (1.13) we can define the functional
$\widehat{\mathcal{J}}^{\alpha}$ also for $\alpha=0$ by setting

(1.15)
$\widehat{\mathcal{J}}^{0}(u):=\mathcal{G}^{0}_{1}(u)+\mathcal{J}^{0}_{1}(u)\,,$

for every $u\in L^{2}(\Omega)$ . In view of (1.14), the functional
$\widehat{\mathcal{J}}^{0}:L^{2}(\Omega)\to(-\infty,+\infty]$ is well defined.
Notice that $\widehat{\mathcal{J}}^{0}$ in (1.15) coincides with the
functional $\hat{F}^{0}$ defined in [9, formula (1.9)].

######  Remark 1.1.

In view of its very definition (1.12) and of (1.11), the functionals
$\widehat{\mathcal{J}}^{\alpha}$ with $\alpha\in(0,d)$ are continuous with
respect to the strong convergence in $L^{2}(\Omega)$ . Moreover, for every
$\alpha\in[0,d)$ and for all $u,\,v\in L^{2}(\Omega)$ we have

(1.16)
$\displaystyle|\mathcal{J}^{\alpha}_{1}(u)-\mathcal{J}^{\alpha}_{1}(v)|\leq$
$\displaystyle(\|u\|_{L^{1}(\Omega)}+\|v\|_{L^{1}(\Omega)})\|u-v\|_{L^{1}(\Omega)}$
$\displaystyle\leq$
$\displaystyle|\Omega|(\|u\|_{L^{2}(\Omega)}+\|v\|_{L^{2}(\Omega)})\|u-v\|_{L^{2}(\Omega)}\,.$

which shows, in particular, that the functionals $\widehat{\mathcal{J}}^{0}$
are lower semicontinuous with respect to the strong convergence in
$L^{2}(\Omega)$ .

The following result will be used throughout the paper.

######  Lemma 1.2.

Let $k\in L^{2}(\mathbb{R}^{d})$ with $\mathrm{supp}\;k\subset B_{R}(0)$ for
some $R>0$ . Let moreover $\\{v^{n}\\}_{n\in\mathbb{N}}\subset
L^{2}(\mathbb{R}^{d})$ be such that $\mathrm{supp}\;v^{n}\subset B_{R}(0)$ and
$v^{n}\rightharpoonup v$ for some $v\in L^{2}(\mathbb{R}^{d})$ . Then,
$v^{n}\ast k\to v\ast k$ (strongly) in $L^{2}(\mathbb{R}^{d})$ .

###### Proof.

We set $g^{n}:=v^{n}-v$ , so that $g^{n}\rightharpoonup 0$ in
$L^{2}(B_{R}(0))$ . For every $x\in\mathbb{R}^{d}$ , $[g^{n}\ast k](x)\to 0$
as $n\to+\infty$ . Moreover, let $h\in\mathbb{R}^{d}$ ; by Young’s convolution
inequality, we have

(1.17) $\Big{\|}[g^{n}\ast k](\cdot+h)-[g^{n}\ast
k](\cdot)\Big{\|}_{L^{2}(\mathbb{R}^{d})}\leq\|g^{n}\|_{L^{2}(\mathbb{R}^{d})}\|k(\cdot+h)-k(\cdot)\|_{L^{1}(\mathbb{R}^{d})}\,;$

since the right-hand side of (1.17) tends to $0$ as $h\to 0$ , uniformly with
respect to $n$ , by the Fréchet-Kolmogorov criterion we get the claim. ∎

##  2\. $\Gamma$-convergence for the functionals $\mathcal{J}^{\alpha}$ and
$\widehat{\mathcal{J}}^{\alpha}$ as $\alpha\to 0^{+}$

We start by studying the asymptotic behavior of the functionals
$-\alpha\mathcal{J}^{\alpha}$ .

######  Theorem 2.1.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to 0^{+}$ as $n\to+\infty$.

  * (i)

($\Gamma$-liminf inequality) For every $u\in L^{2}(\Omega)$ and for every
$\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ with $u^{n}\rightharpoonup
u$ in $L^{2}(\Omega)$, it holds

${d\omega_{d}}\|u\|^{2}_{L^{2}(\Omega)}\leq\liminf_{n\to+\infty}-\alpha_{n}\mathcal{J}^{\alpha_{n}}(u^{n}).$

  * (ii)

($\Gamma$-limsup inequality) For every $u\in L^{2}(\Omega)$

$\lim_{n\to+\infty}-\alpha_{n}\mathcal{J}^{\alpha_{n}}(u)={d\omega_{d}}\|u\|^{2}_{L^{2}(\Omega)}\,.$

###### Proof.

For every $n\in\mathbb{N}$ we set
$\bar{k}^{\alpha_{n}}:=\alpha_{n}k^{\alpha_{n}}$ with $k^{\alpha}$ defined in
(1.2) and, for every $R>\mathrm{diam}(\Omega)$ we set
$\bar{k}^{\alpha_{n}}_{R}:=\bar{k}^{\alpha_{n}}\chi_{B_{R}(0)}=\alpha_{n}k^{\alpha_{n}}_{R}$
.

Proof of (i). Since
$\bar{k}^{\alpha_{n}}_{R}\,\mathrm{d}z\overset{\ast}{\rightharpoonup}{d\omega_{d}}\delta_{0}$
we have that $\tilde{u}\ast\bar{k}^{\alpha_{n}}_{R}\to d\omega_{d}\tilde{u}$
(strongly) in $L^{2}(\Omega)$ , and hence

(2.1)
$\displaystyle\lim_{n\to+\infty}\langle\tilde{u},(\tilde{u}^{n}-\tilde{u})\ast\bar{k}^{\alpha_{n}}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle=$
$\displaystyle\lim_{n\to+\infty}\langle\tilde{u}\ast\bar{k}^{\alpha_{n}}_{R},\tilde{u}^{n}-\tilde{u}\rangle_{L^{2}(\mathbb{R}^{d})}=0\,,$
(2.2)
$\displaystyle\lim_{n\to+\infty}\langle\tilde{u}^{n},\tilde{u}\ast\bar{k}^{\alpha_{n}}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle=$ $\displaystyle d\omega_{d}\|u\|^{2}_{L^{2}(\Omega)}\,.$

Moreover, by (1.5), we have

(2.3) $\displaystyle-\alpha_{n}\mathcal{J}^{\alpha_{n}}(u^{n})=$
$\displaystyle\langle\tilde{u}^{n}-\tilde{u},(\tilde{u}^{n}-\tilde{u})\ast\bar{k}^{\alpha_{n}}\rangle_{L^{2}(\mathbb{R}^{d})}+\langle\tilde{u},(\tilde{u}^{n}-\tilde{u})\ast\bar{k}^{\alpha_{n}}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle+\langle\tilde{u}^{n},\tilde{u}\ast\bar{k}^{\alpha_{n}}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}\,,$

and hence, by (1.10), (2.1), and (2.2), we obtain

(2.4)
$\displaystyle\liminf_{n\to+\infty}-\alpha_{n}\mathcal{J}^{\alpha_{n}}(u^{n})\geq$
$\displaystyle\liminf_{n\to+\infty}\langle\tilde{u}^{n}-\tilde{u},(\tilde{u}^{n}-\tilde{u})\ast\bar{k}^{\alpha_{n}}\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle+\liminf_{n\to+\infty}\langle\tilde{u},(\tilde{u}^{n}-\tilde{u})\ast\bar{k}^{\alpha_{n}}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}+\liminf_{n\to+\infty}\langle\tilde{u}^{n},\tilde{u}\ast\bar{k}^{\alpha_{n}}_{R}\rangle_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\geq$ $\displaystyle d\omega_{d}\|u\|^{2}_{L^{2}(\Omega)}\,.$

Proof of (ii). The claim follows directly by (2.2) and by the first equality
in (1.5). ∎

######  Remark 2.2.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ with $\alpha_{n}\to 0^{+}$
as $n\to+\infty$ . By (2.1)-(2.3), for any $u\in L^{2}(\Omega)$ and for any
sequence $\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ with $u^{n}\to u$
(strongly) in $L^{2}(\Omega)$ , we have that

$\lim_{n\to+\infty}\alpha_{n}\mathcal{J}^{\alpha_{n}}(u^{n})=-d\omega_{d}\|u\|^{2}_{L^{2}(\Omega)}\,,$

which shows, in particular, that the $\Gamma$-limit of the functionals
$\alpha\mathcal{J}^{\alpha}$ (as $\alpha\to 0^{+}$) with respect to the strong
convergence in $L^{2}(\Omega)$ is the functional
$-d\omega_{d}\|\cdot\|^{2}_{L^{2}(\Omega)}$ . This fact implies that the
$\Gamma$-limit of $\alpha\mathcal{J}^{\alpha}$ (as $\alpha\to 0^{+}$) with
respect to the weak convergence in $L^{2}(\Omega)$ equals to $-\infty$ .

In Theorem 2.3 below we show that the functional
$\widehat{\mathcal{J}}^{\alpha}$ $\Gamma$-converges to the functional
$\widehat{\mathcal{J}}^{0}$ . To this end, following [9, Remark 1.3], we
introduce the space

$H^{0}_{0}(\Omega):=\\{u\in
L^{2}(\Omega)\,:\,\mathcal{G}^{0}_{1}(u)<+\infty\\}\,.$

Such a functional space could be endowed with a $0$-Gagliardo type norm
defined as

$\|u\|_{H_{0}^{0}(\Omega)}:=\|u\|_{L^{2}(\Omega)}+\big{(}2\mathcal{G}^{0}_{1}(u)\big{)}^{\frac{1}{2}}\,.$

######  Theorem 2.3.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to 0^{+}$ as $n\to+\infty$ . The following $\Gamma$-convergence
result holds true.

  * (i)

(Compactness) Let $\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such
that

(2.5)
$\widehat{\mathcal{J}}^{\alpha_{n}}(u^{n})+(|\Omega|+m)\|u^{n}\|^{2}_{L^{2}(\Omega)}\leq
M,$

for some constants $m,M>0$ independent of $n$ . Then, up to a subsequence,
$u^{n}\to u$ strongly in $L^{2}(\Omega)$ for some $u\in H_{0}^{0}(\Omega)$ .

  * (ii)

($\Gamma$-liminf inequality) For every $u\in H^{0}_{0}(\Omega)$ and for every
$\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ with $u^{n}\to u$ in
$L^{2}(\Omega)$ , it holds

$\widehat{\mathcal{J}}^{0}(u)\leq\liminf_{n\to+\infty}\widehat{\mathcal{J}}^{\alpha_{n}}(u^{n})\,.$

  * (iii)

($\Gamma$-limsup inequality) For every $u\in H_{0}^{0}(\Omega)$ there exists
$\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ with $u^{n}\to u$ in
$L^{2}(\Omega)$ such that

$\widehat{\mathcal{J}}^{0}(u)=\lim_{n\to+\infty}\widehat{\mathcal{J}}^{\alpha_{n}}(u^{n})\,.$

In order to prove (i) of Theorem 2.3 we adopt a strategy similar to that used
in the proof of [16, Theorem 1.1] . To this end we prove the following result.

######  Lemma 2.4.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to 0^{+}$ as $n\to+\infty$ . Let moreover
$\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ with

(2.6) $\|u^{n}\|_{L^{2}(\Omega)}+\mathcal{G}^{\alpha_{n}}_{1}(u^{n})\leq C\,,$

for some constant $C>0$ (independent of $n$) . Then, up to a subsequence,
$u^{n}\to u$ in $L^{2}(\Omega)$ for some $u\in H^{0}_{0}(\Omega)$ .

###### Proof.

By (2.6), there exists $u\in L^{2}(\Omega)$ such that, up to a (not-relabeled)
subsequence, $u^{n}\rightharpoonup u$ in $L^{2}(\Omega)$ .

For every $\alpha\in[0,d)$ and for every $0<\delta<1$ , we define
$j_{\delta}^{\alpha}:\mathbb{R}^{d}\to(0,+\infty)$ as

$j_{\delta}^{\alpha}(z):=\frac{\chi_{A_{\delta,1}(0)}(z)}{|z|^{d-\alpha}}\,,$

where we have set $A_{r,R}(\xi):=B_{R}(\xi)\setminus\overline{B}_{r}(\xi)$ for
every $0<r<R$ and for every $\xi\in\mathbb{R}^{d}$ . Furthermore, for every
$\alpha\in[0,d)$ and for every $0<\delta<1$ we set
$w_{\delta}^{\alpha}:=\frac{j_{\delta}^{\alpha}}{\|j_{\delta}^{\alpha}\|_{L^{1}(\mathbb{R}^{d})}}$
.

Notice that

(2.7)
$\|j_{\delta}^{\alpha_{n}}\|_{L^{1}(\mathbb{R}^{d})}=d\omega_{d}\frac{1-\delta^{\alpha_{n}}}{\alpha_{n}}\quad\textrm{and}\quad\sup_{n\in\mathbb{N}}\|j_{\delta}^{\alpha_{n}}\|_{L^{2}(\mathbb{R}^{d})}\leq\sup_{n\in\mathbb{N}}\frac{\omega_{d}^{\frac{1}{2}}}{\delta^{d-\alpha_{n}}}=\frac{\omega_{d}^{\frac{1}{2}}}{\delta^{d}}\,.$

We observe that for every $0<\delta<1$

(2.8) $j^{\alpha_{n}}_{\delta}\to j^{0}_{\delta}\textrm{ in
}C^{k}(\overline{A_{\delta,1}(0)})\text{ for all }k\in\mathbb{N}\textrm{ and
}\|j^{\alpha_{n}}_{\delta}\|_{L^{1}(\mathbb{R}^{d})}\to
d\omega_{d}|\log\delta|=\|j^{0}_{\delta}\|_{L^{1}(\mathbb{R}^{d})}\,,$

and, by Lemma 1.2,

(2.9) $\|w^{0}_{\delta}\ast(u^{n}-u)\|_{L^{2}(\mathbb{R}^{d})}\to
0\qquad\textrm{as }n\to+\infty.$

In turn, by triangular and convolution’s Young inequalities, using (2.8) and
(2.9), we get

(2.10) $\|w^{\alpha_{n}}_{\delta}\ast u^{n}-w^{0}_{\delta}\ast
u\|_{L^{2}(\mathbb{R}^{d})}\\\
\leq\|w^{\alpha_{n}}_{\delta}-w^{0}_{\delta}\|_{L^{1}(\mathbb{R}^{d})}\|u^{n}\|_{L^{2}(\mathbb{R}^{d})}+\|w^{0}_{\delta}\ast(u^{n}-u)\|_{L^{2}(\mathbb{R}^{d})}\to
0\qquad\textrm{ as }n\to+\infty\,.$

Now, following [16, Lemma 2.2], using Jensen’s inequality, we get

(2.11) $\displaystyle\|u^{n}-w_{\delta}^{\alpha_{n}}\ast
u^{n}\|^{2}_{L^{2}(\mathbb{R}^{d})}=$
$\displaystyle\int_{\mathbb{R}^{d}}\,\mathrm{d}x\bigg{(}\int_{\mathbb{R}^{d}}(u^{n}(x+z)-u^{n}(x))w_{\delta}^{\alpha_{n}}(z)\,\mathrm{d}z\bigg{)}^{2}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{d}}\,\mathrm{d}x\int_{\mathbb{R}^{d}}|u^{n}(x+z)-u^{n}(x)|^{2}w_{\delta}^{\alpha_{n}}(z)\,\mathrm{d}z$
$\displaystyle\leq$
$\displaystyle\frac{2}{\|j^{\alpha_{n}}_{\delta}\|_{L^{1}(\mathbb{R}^{d})}}\mathcal{G}^{\alpha_{n}}_{1}(u^{n})\leq\frac{C}{|\log\delta|}\,,$

where in the last inequality we have used (2.6) and (2.8). Moreover, since
$w^{0}_{\delta}\,\mathrm{d}z\overset{\ast}{\rightharpoonup}\delta_{0}$ as
$\delta\to 0$ , we have that

(2.12) $\|w^{0}_{\delta}\ast u-u\|_{L^{2}(\mathbb{R}^{d})}\to
0\qquad\textrm{as }\delta\to 0\,.$

Therefore, by triangular inequality, we obtain that, for every $0<\delta<1$ ,

$\displaystyle\|u^{n}-u\|_{L^{2}(\mathbb{R}^{d})}\leq\|u^{n}-w_{\delta}^{\alpha_{n}}\ast
u^{n}\|_{L^{2}(\mathbb{R}^{d})}+\|w_{\delta}^{\alpha_{n}}\ast
u^{n}-w_{\delta}^{0}\ast u\|_{L^{2}(\mathbb{R}^{d})}+\|w^{0}_{\delta}\ast
u-u\|_{L^{2}(\mathbb{R}^{d})}\,,$

whence the strong convergence of $u^{n}$ to $u$ follows by and by (2.10),
(2.11), and (2.12), sending first $n\to+\infty$ and then $\delta\to 0$ .
Finally, by Fatou Lemma and by (2.6),

(2.13)
$\mathcal{G}_{1}^{0}(u)\leq\liminf_{n\to+\infty}\mathcal{G}_{1}^{\alpha_{n}}(u_{n})\leq
C\,,$

whence we deduce that $u\in H^{0}_{0}(\Omega)$ , thus concluding the proof of
the lemma. ∎

###### Proof of Theorem 2.3.

We start by proving compactness. By (2.5), (1.13) and (1.14), we have

$M\geq\mathcal{G}^{\alpha_{n}}_{1}(u^{n})+m\|u^{n}\|^{2}_{L^{2}(\Omega)}\,,$

which, in view of Lemma 2.4, yields (i). Moreover, in view of (1.13), the
proof of (ii) follows by (2.13) and by (1.16). Finally, the proof of (iii)
follows by using standard density arguments in $\Gamma$-convergence (see for
instance the proof of [9, Theorem 1.4 (iii)]). ∎

######  Remark 2.5.

Analogously to what already observed in Remark 2.2, one can easily prove that
the functionals $-\widehat{\mathcal{J}}^{\alpha}$ $\Gamma$-converge to
$-\infty$ (as $\alpha\to 0^{+}$) with respect to the strong $L^{2}$
convergence. For example, assume that $0\in\Omega$ and consider
$v_{n}:=\frac{n^{\frac{d}{2}}}{\log^{\frac{1}{4}}n}\chi_{B_{\frac{1}{n}}(0)}$
. Then, one can easily check that $v_{n}\to 0$ strongly in $L^{2}(\Omega)$ .
Moreover, for $n$ large enough, $\mathcal{G}^{\alpha}_{1}(v_{n})\geq
C\frac{1-n^{-\alpha}}{\alpha}\frac{1}{\log^{\frac{1}{2}}n}$ ; since

$\lim_{n\to+\infty}\lim_{\alpha\to
0}\frac{1-n^{-\alpha}}{\alpha}\frac{1}{\log^{\frac{1}{2}}n}=\lim_{n\to+\infty}\log^{\frac{1}{2}}n=+\infty\,,$

by a standard diagonal argument there exists a sequence
$\\{v_{\alpha}\\}_{\alpha}=\\{v_{n(\alpha)}\\}$ with
$\|v_{\alpha}\|_{L^{2}(\Omega)}\to 0$ and such that
$\mathcal{G}^{\alpha}_{1}(v_{\alpha})\to+\infty$ (as $\alpha\to 0^{+}$). As a
consequence, for every $u\in L^{2}(\Omega)$ , $u+v_{\alpha}\to u$ (strongly)
in $L^{2}(\Omega)$ and $\mathcal{G}^{\alpha}_{1}(u+v_{\alpha})\to+\infty$ .
Recalling (1.13) and recalling Remark 1.1, we conclude that
$-\widehat{\mathcal{J}}^{\alpha}(u+v_{\alpha})\to-\infty$ for every $u\in
L^{2}(\Omega)$ .

##  3\. $\Gamma$-convergence for the functionals $\mathcal{J}^{\alpha}$ as
$\alpha\to d^{-}$

We define the functional $\mathcal{J}^{d}:L^{2}(\Omega)\to(-\infty,0]$ as
$\mathcal{J}^{d}(u):=-\Big{(}\int_{\Omega}u(x)\,\mathrm{d}x\Big{)}^{2}$ .

######  Theorem 3.1.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to{d}^{-}$ as $n\to+\infty$ . For every $u\in L^{2}(\Omega)$ and
for every $\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ with
$u^{n}\rightharpoonup u$ in $L^{2}(\Omega)$ it holds

(3.1)
$\lim_{n\to+\infty}\mathcal{J}^{\alpha_{n}}(u^{n})=\mathcal{J}^{d}(u)\,.$

In particular, as $\alpha\to d^{-}$ , the functionals
$\pm\mathcal{J}^{\alpha}$ $\Gamma$-converge to the functional
$\pm\mathcal{J}^{d}$ with respect to the weak $L^{2}$ convergence in
$L^{2}(\Omega)$ .

###### Proof.

We first notice that for any compact set $K\subset\mathbb{R}^{d}$

(3.2) $\lim_{n\to+\infty}\|k^{\alpha_{n}}-1\|_{L^{1}(K)}=0\,,$

where $k^{\alpha_{n}}$ is defined in (1.2) . By (1.5), we thus have that

(3.3)
$\mathcal{J}^{\alpha_{n}}(u^{n})=-\langle\tilde{u}^{n},\tilde{u}^{n}\ast(k^{\alpha_{n}}-1)\rangle_{L^{2}(\mathbb{R}^{d})}-\langle\tilde{u}^{n},\tilde{u}^{n}\ast
1\rangle_{L^{2}(\mathbb{R}^{d})}\,.$

Now, taking $R>\mathrm{diam}(\Omega)$ , by Young’s convolution inequality and
by (3.2), we have

(3.4)
$\limsup_{n\to+\infty}\langle\tilde{u}^{n},\tilde{u}^{n}\ast(k^{\alpha_{n}}-1)\rangle_{L^{2}(\mathbb{R}^{d})}\leq\limsup_{n\to+\infty}\|u^{n}\|^{2}_{L^{2}(\Omega)}\|k^{\alpha_{n}}-1\|_{L^{1}(B_{R}(0))}=0\,,$

and, since $u^{n}\rightharpoonup u$ in $L^{2}(\Omega)$ ,

(3.5) $\lim_{n\to+\infty}\langle\tilde{u}^{n},\tilde{u}^{n}\ast
1\rangle_{L^{2}(\mathbb{R}^{d})}=\langle\tilde{u},\tilde{u}\ast
1\rangle_{L^{2}(\mathbb{R}^{d})}=-\mathcal{J}^{d}(u)\,.$

By (3.3), (3.4) and (3.5), we obtain (3.1).

∎

For every $\alpha\in(0,d)$ we define the functional
$\widetilde{\mathcal{J}}^{\alpha}:L^{2}(\Omega)\to\mathbb{R}$ as

$\widetilde{\mathcal{J}}^{\alpha}(u):=\frac{\mathcal{J}^{\alpha}(u)-\mathcal{J}^{d}(u)}{d-\alpha}\,;$

moreover, and for every $u\in L^{2}(\Omega)$ we define

(3.6)
$\widetilde{\mathcal{J}}^{d}(u):=-\int_{\Omega}\int_{\Omega}u(x)u(y)\tilde{k}^{d}(|x-y|)\,\mathrm{d}y\,\mathrm{d}x\,,$

where we have set $\tilde{k}^{d}(z):=\log\frac{1}{|z|}$ . Notice that

(3.7) $\tilde{k}^{d}\in L^{p}_{\mathrm{loc}}(\mathbb{R}^{d})\qquad\textrm{for
any }p\in[1,+\infty)\,;$

therefore, for every $R>\mathrm{diam}(\Omega)$ , by Young’s convolution
inequality, we get

(3.8)
$|\widetilde{\mathcal{J}}^{d}(u)|\leq\|\tilde{k}^{d}\|_{L^{1}(B_{R}(0))}\|u\|^{2}_{L^{2}(\Omega)}\qquad\textrm{for
every }u\in L^{2}(\Omega)\,,$

which shows, in particular, that the functional $\widetilde{\mathcal{J}}^{d}$
defined by (3.6) takes values in $\mathbb{R}$ .

######  Remark 3.2.

Let $k\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$ and let $\mathscr{J}_{k}$ be
the functional defined on $L^{2}(\Omega)$ by

$\mathscr{J}_{k}(u):=\int_{\Omega}\int_{\Omega}u(x)u(y)k(x-y)\,\mathrm{d}y\,\mathrm{d}x\qquad\textrm{for
every }u\in L^{2}(\Omega)\,.$

Let $R\geq\mathrm{diam}(\Omega)$ . Then, by Hölder and Young’s convolution
inequalities,

$|\mathscr{J}_{k}(u)|\leq\|u\|_{L^{2}(\Omega)}^{2}\|k\|_{L^{1}(B_{R}(0))}\,.$

######  Remark 3.3.

Let $\\{k^{n}\\}_{n\in\mathbb{N}}\subset L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$
with $\sup_{n\in\mathbb{N}}\|k^{n}\|_{L^{1}(B_{R}(0))}\leq C$ , for some $C>0$
and $R>\mathrm{diam}(\Omega)$ . Then, by Remark 3.2 the functionals
$\mathscr{J}_{k^{n}}$ are $\lambda$-positive, for any $\lambda>2C$ .
Therefore, for any sequence $\\{u_{n}\\}_{n\in\mathbb{N}}\subset
L^{2}(\Omega)$ with

$\sup_{n\in\mathbb{N}}\mathscr{J}_{k^{n}}(u^{n})+\lambda\|u^{n}\|^{2}_{L^{2}(\Omega)}<+\infty\,,$

we have that, up to a subsequence, $u^{n}\rightharpoonup u$ in $L^{2}(\Omega)$
, for some $u\in L^{2}(\Omega)$ .

In particular, this applies to any sequence of functionals
$\\{\mathcal{J}^{\alpha_{n}}\\}_{n}$ with $\alpha_{n}\to d^{-}$ .

######  Theorem 3.4.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to d^{-}$ as $n\to+\infty$ .

For every $u\in L^{2}(\Omega)$ and for every
$\\{u^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ with $u^{n}\rightharpoonup
u$ in $L^{2}(\Omega)$ it holds

(3.9)
$\lim_{n\to+\infty}\widetilde{\mathcal{J}}^{\alpha_{n}}(u^{n})=\widetilde{\mathcal{J}}^{d}(u)\,.$

In particular, as $\alpha\to d^{-}$ , the functionals
$\pm\mathcal{J}^{\alpha}$ $\Gamma$-converge to the functional
$\pm\mathcal{J}^{d}$ with respect to the weak $L^{2}$ convergence in
$L^{2}(\Omega)$ .

###### Proof.

For any $n\in\mathbb{N}$ we set
$\tilde{k}^{\alpha_{n}}:=\frac{k^{\alpha_{n}}-1}{d-\alpha_{n}}$ , with
$k^{\alpha_{n}}$ defined in (1.2). For any $v\in L^{2}(\Omega)$ , it holds

(3.10)
$\widetilde{\mathcal{J}}^{\alpha_{n}}(v)=-\langle\tilde{v},\tilde{v}\ast\tilde{k}^{\alpha_{n}}\rangle_{L^{2}(\mathbb{R}^{d})}\,,\qquad\widetilde{\mathcal{J}}^{d}(v)=-\langle\tilde{v},\tilde{v}\ast\tilde{k}^{d}\rangle_{L^{2}(\mathbb{R}^{d})}\,,$

and hence

(3.11)
$\widetilde{\mathcal{J}}^{\alpha_{n}}(v)=-\langle\tilde{v},\tilde{v}\ast(\tilde{k}^{\alpha_{n}}-\tilde{k}^{d})\rangle_{L^{2}(\mathbb{R}^{d})}+\widetilde{\mathcal{J}}^{d}(v)\\\
$

Moreover, by Taylor expansion, for any $z\in\mathbb{R}^{d}\setminus\\{0\\}$ ,
we have

(3.12)
$\tilde{k}^{\alpha_{n}}(z)=\tilde{k}^{d}(z)+\mathrm{O}\Big{(}(d-\alpha_{n})\log^{2}\frac{1}{|z|}\Big{)}\,,$

with $\lim_{t\to 0}\frac{\mathrm{O}(t)}{t}\in\mathbb{R}$ . Therefore, for any
compact set $K\subset\mathbb{R}^{d}$ , it holds

(3.13)
$\lim_{n\to+\infty}\|\tilde{k}^{\alpha_{n}}-\tilde{k}^{d}\|_{L^{p}(K)}=0\qquad\textrm{for
any }p\in[1,+\infty)\,.$

Let $R>\mathrm{diam}(\Omega)$ .

We start by proving (a). By Young’s convolution inequality and by (3.13), we
have

(3.14)
$\limsup_{n\to+\infty}\langle\tilde{u}^{n},\tilde{u}^{n}\ast(\tilde{k}^{\alpha_{n}}-\tilde{k}^{d})\rangle_{L^{2}(\mathbb{R}^{d})}\leq\limsup_{n\to+\infty}\|u^{n}\|^{2}_{L^{2}(\Omega)}\|\tilde{k}^{\alpha_{n}}-\tilde{k}^{d}\|_{L^{1}(B_{R}(0))}=0\,,$

and, by Lemma 1.2,

(3.15)
$\lim_{n\to+\infty}\widetilde{\mathcal{J}}^{d}(u^{n})=\widetilde{\mathcal{J}}^{d}(u)\,.$

By (3.11), (3.14), and (3.15), we thus obtain

$\lim_{n\to+\infty}\widetilde{\mathcal{J}}^{\alpha_{n}}(u^{n})=\widetilde{\mathcal{J}}^{d}(u)\,,$

i.e., (3.9).

∎

######  Remark 3.5.

By Remark 3.3 and by (3.11), (3.13) and (3.8), it immediately follows that any
sequence $\\{\widetilde{\mathcal{J}}^{\alpha_{n}}\\}_{n}$ with $\alpha_{n}\to
d^{-}$ is uniformly $\lambda$-positive and hence weakly equi-coercive in
$L^{2}(\Omega)$ (in the sense of Remark 3.3).

##  4\. Convergence of the $\alpha$-Riesz flows

###  4.1. The $\alpha$-Riesz potential for $\alpha\in(0,d)$ and for $\alpha=0$
and $\alpha=d$

For every $\alpha\in(0,d)$ and for every $v\in L^{2}(\mathbb{R}^{d})$ the
$\alpha$-Riesz potential is formally defined by

$(-\Delta)^{-\frac{\alpha}{2}}v(x):=2\int_{\mathbb{R}^{d}}\frac{v(y)}{|x-y|^{d-\alpha}}\,\mathrm{d}y\,,\qquad\qquad\textrm{for
a.e. }x\in\mathbb{R}^{d}\,.$

Notice that such a definition coincides, up to multiplicative constants
depending on $\alpha$ and $d$ , with [26, formula (4), p. 117]. By [26,
Theorem 1, p. 119] the above definition is well posed, and there exists a
constant $C(d,\alpha)$ such that

$\|(-\Delta)^{-\frac{\alpha}{2}}v\|_{L^{q}(\mathbb{R}^{d})}\leq
C(d,\alpha)\|v\|_{L^{2}(\mathbb{R}^{d})}\qquad\qquad\textrm{for all }v\in
L^{2}(\mathbb{R}^{d})\,,$

where $q=\frac{2d}{d-\alpha}$. Since $q\geq 2$, by Hölder inequality, we
deduce that

(4.1) $\|(-\Delta)^{-\frac{\alpha}{2}}v\|_{L^{2}(\Omega)}\leq
C(d,\alpha,\Omega)\|v\|_{L^{2}(\mathbb{R}^{d})};$

we also notice that, if $v\in L^{2}(\Omega)$ with $v\equiv 0$ on
$\mathbb{R}^{d}\setminus\Omega$, then (4.1) is an easy consequence of
convolution’s Young inequality. With a little abuse of notation, for every
$\alpha\in(0,d)$ and for every $u\in L^{2}(\Omega)$ we set
$(-\Delta)^{-\frac{\alpha}{2}}u:=(-\Delta)^{-\frac{\alpha}{2}}\tilde{u}$ .

One can easily check that the $\alpha$-Riesz potential is nothing but the
first variation of the $\alpha$-Riesz functionals $-\mathcal{J}^{\alpha}$,
i.e., for every $u,\varphi\in L^{2}(\Omega)$ we have

(4.2) $\lim_{t\to
0}\frac{\mathcal{J}^{\alpha}(u+t\varphi)-\mathcal{J}^{\alpha}(u)}{t}=-\langle(-\Delta)^{-\frac{\alpha}{2}}u,\varphi\rangle_{L^{2}(\Omega)}.$

In analogy with [9, Subsection 4.1], for every
$\psi\in\mathrm{C}^{\infty}_{\mathrm{c}}(\mathbb{R}^{d})$ we define the
$0$-fractional laplacian of $\psi$ as

$(-\Delta)^{0}\psi(x):=\int_{B_{1}}\frac{2\psi(x)-\psi(x+z)-\psi(x-z)}{|z|^{d}}\,\mathrm{d}z-2\int_{\mathbb{R}^{d}\setminus\overline{B}_{1}}\frac{\psi(x+z)}{|z|^{d}}\,\mathrm{d}z\,,\quad
x\in\mathbb{R}^{d}\,.$

Furthermore, for every $u\in H^{0}_{0}(\Omega)$ we define $0$-fractional
laplacian of $u$ by duality as

(4.3) $\langle(-\Delta)^{0}u,\varphi\rangle:=\langle
u,(-\Delta)^{0}\tilde{\varphi}\rangle_{L^{2}(\Omega)}\,,\qquad\textrm{for all
}\varphi\in\mathrm{C}^{\infty}_{\mathrm{c}}(\Omega)\,.$

Clearly, the $0$-fractional laplacian is the first variation of the functional
$\widehat{\mathcal{J}}^{0}$ , as shown in [9, Proposition 3.2].

Finally, for every $v\in L^{2}(\Omega)$ we set

$(-\Delta)^{-\frac{d}{2}}v(x):=2\int_{\Omega}v(y)\log\frac{1}{|x-y|}\,\mathrm{d}y\,,\qquad
x\in\mathbb{R}^{d}\,,$

and we notice that
$(-\Delta)^{-\frac{d}{2}}:L^{2}(\Omega)\to\mathrm{C}(\mathbb{R}^{d})$ .

It is easy to see that $(-\Delta)^{-\frac{d}{2}}$ is the first variation of
the functional $-\widetilde{\mathcal{J}}^{d}$ defined in (3.6) , i.e., that
for any $u,\varphi\in L^{2}(\Omega)$

$\lim_{t\to
0}\frac{\widetilde{\mathcal{J}}^{d}(u+t\varphi)-\widetilde{\mathcal{J}}^{d}(u)}{t}=-\langle(-\Delta)^{-\frac{d}{2}}u,\varphi\rangle_{L^{2}(\Omega)}.$

###  4.2. Abstract stability results for parabolic flows

It is well known that, under suitable uniform convexity assumptions,
$\Gamma$-convergence commutes with gradient flows. This fact, first exploited
in [6], has been generalized in many respects, such as for instance in metric
spaces [11, 2, 25]. For the specific framework of uniform $\lambda$-convex
functionals we refer for instance to [21, 22, 23, 9]. Here, we recall the
stability result as appeared in [9, Theorem 3.8], that best suits our
purposes. We start by recalling the notion of $\lambda$-positive and
$\lambda$-convex functions. Let $\mathscr{H}$ be a Hilbert space and let
$\lambda>0$ . We say that a function
$\mathcal{F}:\mathscr{H}\rightarrow(-\infty,+\infty]$ is $\lambda$-convex if
the function $\mathcal{F}(\cdot)+\frac{\lambda}{2}|\cdot|_{\mathscr{H}}^{2}$
is convex. Moreover, we say that $\mathcal{F}$ is $\lambda$-positive if
$\mathcal{F}(x)+\frac{\lambda}{2}|x|_{\mathscr{H}}^{2}\geq 0$ for every
$x\in\mathscr{H}$ .

######  Theorem 4.1.

Let $\mathscr{H}$ be a Hilbert space. Let
$\\{\mathcal{F}^{n}\\}_{n\in\mathbb{N}}$ with
$\mathcal{F}^{n}:\mathscr{H}\to(-\infty,+\infty]$ for every $n\in\mathbb{N}$
be a sequence of proper, strongly lower semicontinuous functions which are
$\lambda$-convex and $\lambda$-positive, for some $\lambda>0$ independent of
$n$ . Let $\\{x_{0}^{n}\\}_{n\in\mathbb{N}}\subset\mathscr{H}$ be such that
$\sup_{n\in\mathbb{N}}\mathcal{F}^{n}(x_{0}^{n})<+\infty$ and $x_{0}^{n}\to
x^{\infty}_{0}$ for some $x^{\infty}_{0}\in\mathscr{H}$ . Assume that one of
the following statements is satisfied:

  * (a)

The functions $\mathcal{F}^{n}$ $\Gamma$-converge to some proper function
$\mathcal{F}^{\infty}$ with respect to the weak $\mathscr{H}$-convergence.
Moreover, the $\Gamma$-limsup inequality is satisfied with respect to the
strong $\mathscr{H}$-convergence, i.e., for every $y\in\mathscr{H}$ there
exists a sequence $\\{y^{n}\\}_{n\in\mathbb{N}}$ with
$y^{n}\overset{\mathscr{H}}{\to}y$ such that
$\mathcal{F}^{n}(y^{n})\to\mathcal{F}^{\infty}(y)$ as $n\to+\infty$ .

  * (b)

The functions $\mathcal{F}^{n}$ $\Gamma$-converge to some proper function
$\mathcal{F}^{\infty}$ with respect to the strong $\mathscr{H}$-convergence
(as $n\to+\infty$) and every sequence
$\\{y^{n}\\}_{n\in\mathbb{N}}\subset\mathscr{H}$ with

$\sup_{n\in\mathbb{N}}\mathcal{F}^{n}(y^{n})+\frac{\lambda}{2}|y^{n}|_{\mathscr{H}}^{2}<+\infty$

admits a strongly convergent subsequence.

Let $T>0$ . Then, $\mathcal{F}^{\infty}(x_{0}^{\infty})<+\infty$ and the
unique solutions $x^{n}\in H^{1}([0,T];\mathscr{H})$ of the Cauchy problem

(4.4) $\begin{cases}\dot{x}(t)\in-\partial\mathcal{F}^{n}(x(t))\qquad\text{for
a.e.\ }t\in(0,T)\,,\\\ x(0)=x_{0}^{n}\end{cases}$

weakly converge, as $n\to+\infty$ , in $H^{1}([0,T];\mathscr{H})$ to the
unique solution $x^{\infty}$ of the problem

(4.5)
$\begin{cases}\dot{x}(t)\in-\partial\mathcal{F}^{\infty}(x(t))\qquad\text{for
a.e.\ }t\in(0,T)\,,\\\ x(0)=x^{\infty}_{0}\,.\end{cases}$

Furthermore, if

(4.6)
$\lim_{n\to+\infty}\mathcal{F}^{n}(x^{n}_{0})=\mathcal{F}^{\infty}(x^{\infty}_{0})\,,$

then, we have that

(4.7) $x^{n}\to x^{\infty}\qquad\textrm{(strongly) in
}H^{1}([0,T];\mathscr{H})\,,$ (4.8)
$x^{n}(t)\overset{\mathscr{H}}{\to}x^{\infty}(t)\quad\textrm{and}\quad\mathcal{F}^{n}(x^{n}(t))\to\mathcal{F}^{\infty}(x^{\infty}(t))\qquad\textrm{for
every }t\in[0,T]\,.$

###  4.3. Convergence of the $\alpha$-parabolic flows as $\alpha\to 0^{+}$

Here we state and prove the convergence results for the parabolic flows
corresponding to the (either scaled or renormalized) $\alpha$-Riesz
functionals as $\alpha\to 0^{+}$ . These results follow by applying our
$\Gamma$-convergence analysis together with the stability result Theorem 4.1.

######  Theorem 4.2.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to 0^{+}$ as $n\to+\infty$ . Let $u^{\infty}_{0}\in L^{2}(\Omega)$
and $\\{u_{0}^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such that
$u^{n}_{0}\to u^{\infty}_{0}$ in $L^{2}(\Omega)$ . Let $T>0$; then, for every
$n\in\mathbb{N}$ there exists a unique solution $u^{n}\in
H^{1}([0,T];L^{2}(\Omega))$ of

(4.9)
$\begin{dcases}u_{t}(t)=-\alpha_{n}(-\Delta)^{-\frac{\alpha_{n}}{2}}u(t)\qquad\textrm{for
a.e.\ }t\in(0,T)\\\ u(0)=u^{n}_{0}\,.\end{dcases}$

Moreover, $u^{n}\to u^{\infty}$ in $H^{1}([0,T];L^{2}(\Omega))$ as
$n\to+\infty$ , where $u^{\infty}$ is the unique solution of

(4.10) $\begin{dcases}u_{t}(t)=-2d\omega_{d}u(t)\qquad\textrm{for a.e.\
}t\in(0,T)\,,\\\ u(0)=u^{\infty}_{0}\,,\end{dcases}$

and

$\|u^{n}(t)-u^{\infty}(t)\|_{L^{2}(\Omega)}\to
0\quad\textrm{and}\quad-\alpha_{n}\mathcal{J}^{\alpha_{n}}(u^{n}(t))\to
d\omega_{d}\|u^{\infty}(t)\|^{2}_{L^{2}(\Omega)}\qquad\textrm{for every
}t\in[0,T]\,.$

###### Proof.

Recall that, by (1.6), the functionals $-\alpha_{n}\mathcal{J}^{\alpha_{n}}$
are finite on $L^{2}(\Omega)$ and, by (1.10) and Remark 3.2, they are also
positive and convex. Furthermore, by (1.11), the functionals
$-\alpha_{n}\mathcal{J}^{\alpha_{n}}$ are continuous with respect to the
strong convergence in $L^{2}(\Omega)$ . Moreover, for every $u,\varphi\in
L^{2}(\Omega)$ ,

(4.11) $\lim_{t\to
0}\frac{d\omega_{d}\|u+t\varphi\|^{2}_{L^{2}(\Omega)}-d\omega_{d}\|u\|^{2}_{L^{2}(\Omega)}}{t}=2\langle
d\omega_{d}u,\varphi\rangle_{L^{2}(\Omega)}\,.$

Finally, by Remark 2.2,
$\lim_{n\to+\infty}-\alpha_{n}\mathcal{J}^{\alpha_{n}}(u^{n}_{0})=d\omega_{d}\|u^{\infty}_{0}\|^{2}_{L^{2}(\Omega)}$
. Hence, the conclusion follows by applying Theorem 4.1 with
$\mathscr{H}=L^{2}(\Omega)$ ,
$\mathcal{F}^{n}=-\alpha_{n}\mathcal{J}^{\alpha_{n}}$ and
$\mathcal{F}^{\infty}(\cdot)=d\omega_{d}\|\cdot\|^{2}_{L^{2}(\Omega)}$ , once
noticed that, in view of Theorem 2.1, assumption (a) is satisfied. ∎

######  Remark 4.3.

For every $n\in\mathbb{N}$ let $u^{n}$ be the unique solution to the Cauchy
problem (4.9) and let $u$ be the solution to (4.10) with $-2d\omega_{d}$
replaced by $2d\omega_{d}$ . Existence and uniqueness of the solutions for the
problems above are consequence of the classical gradient flow theory for
$\lambda$-convex functionals (see, for instance, [2, 21] or [9, Theorem 3.2]).
It is a natural question if $u^{n}$ converge to $u$ . In the geometric context
of characteristic functions this is the case for the corresponding curvature
flows. In the present setting, unfortunately, Theorem 4.1 does not apply. In
fact, on the one hand, compactness can be achieved only in the weak $L^{2}$
topology, so that assumption (b) does not hold true. On the other hand, in
view of Remark 2.2, the $\Gamma$-limit of the functionals
$\alpha\mathcal{J}^{\alpha}$ (as $\alpha\to 0^{+}$) is
$-d\omega_{d}\|\cdot\|^{2}_{L^{2}}$ only with respect to the strong $L^{2}$
topology, so that neither assumption (a) is satisfied. More in general, it
would be interesting to detect natural assumptions weaker than (a) and (b)
that guarantee convergence for similar parabolic flows, when the
$\Gamma$-convergence property holds true with respect to a topology that is
stronger than that for which equicoercivity can be provided.

######  Theorem 4.4.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to 0^{+}$ as $n\to+\infty$ . Let $u^{\infty}_{0}\in L^{2}(\Omega)$
and let $\\{u^{n}_{0}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such that
$sup_{n\in\mathbb{N}}\widehat{\mathcal{J}}^{\alpha_{n}}(u^{n}_{0})<+\infty$
and $u^{n}_{0}\to u^{\infty}_{0}$ in $L^{2}(\Omega)$ . Let $T>0$; then, for
every $n\in\mathbb{N}$ there exists a unique solution $u^{n}\in
H^{1}([0,T];L^{2}(\Omega))$ of

(4.12)
$\begin{dcases}u_{t}(t)=-\Big{[}-(-\Delta)^{-\frac{\alpha_{n}}{2}}u(t)+2\frac{d\omega_{d}}{\alpha_{n}}u(t)\Big{]}\qquad\textrm{for
a.e.\ }t\in(0,T)\,,\\\ u(0)=u^{n}_{0}\,.\end{dcases}$

Moreover, $u^{\infty}_{0}\in H^{0}_{0}(\Omega)$ and $u^{n}\rightharpoonup
u^{\infty}$ in $H^{1}([0,T];L^{2}(\Omega))$ as $n\to+\infty$ , where
$u^{\infty}$ is the unique solution of

(4.13) $\begin{dcases}u_{t}(t)=-(-\Delta)^{0}u(t)\qquad\textrm{for a.e.\
}t\in(0,T)\\\ u(0)=u^{\infty}_{0}\,.\end{dcases}$

Furthermore, if

$\lim_{n\to+\infty}\widehat{\mathcal{J}}^{\alpha_{n}}(u^{n}_{0})=\widehat{\mathcal{J}}^{0}(u^{\infty}_{0})\,,$

then, $u^{n}\to u^{\infty}$ (strongly) in $H^{1}([0,T];L^{2}(\Omega))$ and

$\|u^{n}(t)-u^{\infty}(t)\|_{L^{2}(\Omega)}\to
0\quad\textrm{and}\quad\widehat{\mathcal{J}}^{\alpha_{n}}(u^{n}(t))\to\widehat{\mathcal{J}}^{0}(u^{\infty}(t))\qquad\textrm{for
every }t\in[0,T]\,.$

###### Proof.

By (1.13) and (1.14), the functionals $\widehat{\mathcal{J}}^{\alpha_{n}}$ are
$\lambda$-convex and $\lambda$-positive for every $\lambda>2|\Omega|$ .
Moreover, by Remark 1.1, they are lower semicontinuous with respect to the
strong convergence in $L^{2}(\Omega)$ . Now, by (4.2) and (4.11), using the
very definition of $\widehat{\mathcal{J}}^{\alpha_{n}}$ in (1.12), we have
that, for every $u,\varphi\in L^{2}(\Omega)$

$\lim_{t\to
0}\frac{\widehat{\mathcal{J}}^{\alpha_{n}}(u+t\varphi)-\widehat{\mathcal{J}}^{\alpha_{n}}(u)}{t}=\Big{\langle}-(-\Delta)^{-\frac{\alpha_{n}}{2}}u+2\frac{d\omega_{d}}{\alpha_{n}}u,\varphi\Big{\rangle}_{L^{2}(\Omega)}\,.$

Hence, the stability claim follows by applying Theorem 4.1 with
$\mathscr{H}=L^{2}(\Omega)$ ,
$\mathcal{F}^{n}=\widehat{\mathcal{J}}^{\alpha_{n}}$ and
$\mathcal{F}^{\infty}=\widehat{\mathcal{J}}^{0}$ , once noticed that, in view
of Theorem 2.3, assumption (b) is satisfied. ∎

###  4.4. Convergence of the $\alpha$-parabolic flows as $\alpha\to d^{-}$

Here we state and prove the convergence results for the parabolic flows
corresponding to the (either scaled or renormalized) $\alpha$-Riesz
functionals as $\alpha\to d^{-}$ . Such results follow by Theorem 4.1.

######  Theorem 4.5.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to d^{-}$ as $n\to+\infty$ . Let $u^{\infty}_{0}\in L^{2}(\Omega)$
and $\\{u_{0}^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such that and
$u^{n}_{0}\to u^{\infty}_{0}$ in $L^{2}(\Omega)$ . Let $T>0$ ; then, for every
$n\in\mathbb{N}$ there exists a unique solution $u^{n}\in
H^{1}([0,T];L^{2}(\Omega))$ of

(4.14)
$\begin{dcases}u_{t}(t)=\pm(-\Delta)^{-\frac{\alpha_{n}}{2}}u(t)\qquad\textrm{for
a.e.\ }t\in(0,T)\\\ u(0)=u^{n}_{0}\,.\end{dcases}$

Moreover, $u^{n}\to u^{\infty}$ in $H^{1}([0,T];L^{2}(\Omega))$ as
$n\to+\infty$ , where $u^{\infty}$ is the unique solution of

(4.15) $\begin{dcases}u_{t}(t)=\pm
2\int_{\Omega}u(t)\,\mathrm{d}y\qquad\textrm{for a.e.\ }t\in(0,T)\,,\\\
u(0)=u^{\infty}_{0}\,,\end{dcases}$

and

$\|u^{n}(t)-u^{\infty}(t)\|_{L^{2}(\Omega)}\to
0\quad\textrm{and}\quad\mathcal{J}^{\alpha_{n}}(u^{n}(t))\to\mathcal{J}^{d}(u^{\infty}(t))\qquad\textrm{for
every }t\in[0,T]\,.$

###### Proof.

By (1.6) the functionals $\pm\mathcal{J}^{\alpha_{n}}$ are finite on
$L^{2}(\Omega)$ and, by Remark 3.2, they are also $\lambda$-positive and
$\lambda$-convex for every
$\lambda>2\omega_{d}\big{(}\mathrm{diam}(\Omega)\big{)}^{d}$ . Furthermore, by
(1.11), the functionals $\mathcal{J}^{\alpha_{n}}$ are continuous with respect
to the strong convergence in $L^{2}(\Omega)$ . Moreover, for every
$u,\varphi\in L^{2}(\Omega)$ ,

(4.16) $\lim_{t\to
0}\frac{\mathcal{J}^{d}(u+t\varphi)-\mathcal{J}^{d}(u)}{t}=-\big{\langle}2\int_{\Omega}u\,\mathrm{d}y,\varphi\big{\rangle}_{L^{2}(\Omega)}\,.$

Finally, by Theorem 3.1, we have that
$\lim_{n\to+\infty}\mathcal{J}^{\alpha}(u^{n}_{0})=\mathcal{J}^{\infty}(u^{\infty}_{0})$
. Hence, the conclusion follow by applying Theorem 4.1 with
$\mathcal{F}^{n}=\pm\mathcal{J}^{\alpha_{n}}$ and
$\mathcal{F}^{\infty}=\pm\mathcal{J}^{d}$, once noticed that, in view of
Theorem 3.1(a), assumption (a) is satisfied. ∎

######  Theorem 4.6.

Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset(0,d)$ be such that
$\alpha_{n}\to d^{-}$ as $n\to+\infty$ . Let $u^{\infty}_{0}\in L^{2}(\Omega)$
and let $\\{u^{n}_{0}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such that
$u^{n}_{0}\to u^{\infty}_{0}$ in $L^{2}(\Omega)$ . Let $T>0$ ; then, for every
$n\in\mathbb{N}$ there exists a unique solution $u^{n}\in
H^{1}([0,T];L^{2}(\Omega))$ to

(4.17)
$\begin{dcases}u_{t}(t)=\pm\frac{1}{d-\alpha_{n}}\Big{[}(-\Delta)^{-\frac{\alpha_{n}}{2}}u(t)-2\int_{\Omega}u(t)\,\mathrm{d}y\Big{]}\qquad\textrm{for
a.e.\ }t\in(0,T)\,,\\\ u(0)=u^{n}_{0}\,.\end{dcases}$

Moreover, $u^{n}\to u^{\infty}$ in $H^{1}([0,T];L^{2}(\Omega))$ as
$n\to+\infty$ , where $u^{\infty}\in H^{1}([0,T];L^{2}(\Omega))$ is the unique
solution to

(4.18)
$\begin{dcases}u_{t}(t)=\pm(-\Delta)^{-\frac{d}{2}}u(t)\qquad\textrm{for a.e.\
}t\in(0,T)\\\ u(0)=u^{\infty}_{0}\,,\end{dcases}$

and

$\|u^{n}(t)-u^{\infty}(t)\|_{L^{2}(\Omega)}\to
0\quad\textrm{and}\quad\widetilde{\mathcal{J}}^{\alpha_{n}}(u^{n}(t))\to\widetilde{\mathcal{J}}^{d}(u^{\infty}(t))\qquad\textrm{for
every }t\in[0,T]\,.$

###### Proof.

By (3.7), (3.13), and (3.11), using Remark 3.2, we have that the functionals
$\pm\widetilde{\mathcal{J}}^{d}$ and $\pm\widetilde{\mathcal{J}}^{\alpha_{n}}$
are $\lambda$-convex and $\lambda$-positive for every
$\lambda>4\|\tilde{k}^{d}\|_{L^{1}(B_{\mathrm{diam}(\Omega)}(0))}$ . Moreover,
by (1.11), the functionals $\widetilde{\mathcal{J}}^{\alpha_{n}}$ are
continuous with respect to the strong convergence in $L^{2}(\Omega)$ , and, by
arguing as in (1.11) one can show that also the functional
$\widetilde{\mathcal{J}}^{d}$ is continuous with respect to the strong
convergence in $L^{2}(\Omega)$ . Now, by (4.2) and (4.16), we have that, for
every $u,\varphi\in L^{2}(\Omega)$

$\lim_{t\to
0}\frac{\widetilde{\mathcal{J}}^{\alpha_{n}}(u+t\varphi)-\widetilde{\mathcal{J}}^{\alpha_{n}}(u)}{t}=\frac{1}{d-\alpha_{n}}\Big{\langle}-(-\Delta)^{-\frac{\alpha_{n}}{2}}u+2\int_{\Omega}u(t)\,\mathrm{d}y,\varphi\Big{\rangle}_{L^{2}(\Omega)}\,.$

Finally, by Theorem 3.4,
$\lim_{n\to+\infty}\widetilde{\mathcal{J}}^{\alpha_{n}}(u^{n}_{0})=\widetilde{\mathcal{J}}^{d}(u^{\infty}_{0})$
. Hence, the conclusion follows by applying Theorem 4.1 with
$\mathcal{F}^{n}=\pm\widetilde{\mathcal{J}}^{\alpha_{n}}$ and
$\mathcal{F}^{\infty}=\pm\widetilde{\mathcal{J}}^{d}$ , once noticed that, in
view of Theorem 3.4, assumption (a) is satisfied. ∎

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# ‘SOS TUTORÍA UC’: A Diversity-Aware Application for Tutor Recommendation
Based on Competence and Personality

Laura Achón  DEI - Universidad Católica Nuestra Señora de la Asunción -
Paraguay  Aalborg University - Denmark  Ana De Souza  Alethia Hume  Ronald
Chenu-Abente  Amalia De Götzen  Luca Cernuzzi

(2023)

###### Abstract

‘SOS TUTORÍA UC’ is a student connection application aimed at facilitating
academic assistance between students through external tutoring outside of the
application. To achieve this, a responsive web application was designed and
implemented, integrated with the WeNet platform, which provides various
services for user management and user recommendation algorithms. This study
presents the development and validation of the experience in the application
by evaluating the importance of incorporating the dimension of personality
traits, according to the Big Five model, in the process of recommending
students for academic tutoring. The goal is to provide support for students to
find others with greater knowledge and with a personality that is “different”,
“similar” or “indifferent” to their own preferences for receiving academic
assistance on a specific topic. The integration with the WeNet platform was
successful in terms of components, and the results of the recommendation
system testing were positive but have room for improvement.

###### keywords:

Diversity Tutorship Big Five Academic Competence Recommendation System

##  1 Introduction

This paper presents ‘SOS TUTORÍA UC’, an application developed by Universidad
Católica Nuestra Señora de la Asunción (UC) in Paraguay and integrated into
the WeNet platform, which aims to facilitate connections between students for
tutoring based on academic competencies and personality, utilizing the Big
Five model [1] to measure personality traits. Additionally, the application
uses hybrid machine artificial intelligence, leveraging the information of the
personality traits and the level of academic competencies of potential tutors
to make recommendations. This allows users to choose tutors with “different”
or “similar” personalities to their own, and they can also choose to disregard
this parameter if they feel “indifferent” towards it. The study explores how
these personality traits can influence tutoring effectiveness, in addition to
academic knowledge in the specific subject to be taught. This is in line with
various previous studies that have examined the impact of peer-to-peer tutor
recommendation systems [2, 3, 4, 5] and personality-based tutor recommendation
systems [6, 7] on the educational process. Furthermore, the current research
builds upon previous studies [8, 9] that have shown a correlation between
personality traits and interaction levels in WeNet project pilots,
highlighting the importance of diversity in social interactions.

##  2 ‘SOS TUTORÍA UC’

The ‘SOS TUTORÍA UC’ web application was designed and implemented, providing
an interface for students to request tutoring sessions and find suitable
tutors based on their academic expertise and personality traits. To
incorporate personality into the matching process, we adopted the Big Five
model [1], that uses the traits, namely Extraversion, Agreeableness,
Conscientiousness, Emotional Stability/Neuroticism, and Openness to
Experience, to provide a comprehensive framework for describing and
understanding individual differences in personality. The ‘SOS TUTORÍA UC’
application was integrated with the WeNet platform [10], ensuring good user
management and leveraging the platform’s recommendation algorithms.

Figure 1: Communication between the modules of ’SOS TUTORÍA UC’ and the WeNet
platform

Figure 1 showcases the integration process, which involved analyzing the
components and APIs of the WeNet platform. The goal was to create and adapt
the logic for tutoring requests to align with the existing WeNet model. The
Service REST API manages model entities and communicates with modules such as
the Profile Manager (user management) and the Task Manager (handling task and
message creation as transactions).

In ’SOS TUTORÍA UC,’ the creation of a task corresponds to a new tutoring
request. The WeNet platform provides five recommended users as potential
tutors, and they receive a notification to review the request details. Based
on the information, they can accept or reject the request for tutorship. The
responses to these requests are recorded as task response transactions, which
can be approval or rejection responses determined by an attribute value. Once
a request is approved by at least one student, the requester can select the
desired tutor through a transaction type indicating the best response. This
informs the platform which response sent by the tutors has been ultimately
chosen by the requesting student.

As part of the integration, the application logic had to be adapted on the
rules side as well. The relevant dimensions that contribute to diversity in
the algorithm are academic competence, personality, and physical proximity.
Finding individuals based on physical proximity is determined by their
distance from the person who posted the question. A distance greater than 500
meters is considered far and out of reach, while a shorter distance is
classified as closer. Competence is also considered a strict constraint. The
system only selects individuals who are “better than me” in terms of academic
competence, if they are available. Finding individuals with “similar” or
“different” personality profiles is based on whether they share a “similar”
personality traits with the person requesting tutoring or not. Competence and
physical proximity requirements are considered essential and given more
importance than other requirements (such as personality). However, it is
acceptable to “diversify” the list of potential tutors by selecting someone
who is not physically close in order to diversify based on gender.

##  3 Analysis of results

To evaluate the system, a pilot study was conducted at the UC. Participants
were invited to use the ‘SOS TUTORÍA UC’ application and provide feedback on
their experience. Since this work corresponds to the integration of a new type
of technology (i.e., a responsive web application) into the WeNet platform,
differentiating it from other consortium pilots, a component-level integration
analysis was conducted. It involved evaluating the communication through the
interfaces of the WeNet platform; that is, the Service REST APIs in the
management of transactions and users, as well as the integration with the
authentication and authorization module. The tests were performed with a test
user in production environment to the three main modules (Auth Controller,
User Controller and Task Controller). All calls to the WeNet APIs during the
tests returned a satisfactory response, in addition to executing the expected
results. Therefore, the component-level integration was successful.

Additionally, we also discuss a preliminary experience with the recommendation
process. For this purpose, a testing scheme was established to analyze whether
the user profiles suggested by WeNet to fulfill a tutoring request adequately
match the parameters indicated by the requester. In the tests performed for
applications with personality “different”, “similar” or “indifferent” to the
applicant, the algorithm prioritized students with high scores in the
requested competency and the specified personality preference. However, it is
worth noting that for some of the tests, the algorithm should have recommend
other tutors that better met the applicant’s requirements. Indeed, despite the
recommendation algorithm yielded positive results, further improvements are
needed to better align with user preferences and selection filters specified.
This could be attributed to low student participation during the pilot
experience or the platform’s gender diversification approach.

In the pilot experience participated 43 students from different campuses and
careers at the UC. After completing the pilot of ‘SOS TUTORÍA UC’ three
primary instruments were used to analyze the experience and evaluate the
relevance of incorporating personality as a search parameter for potential
tutors, thereby introducing an additional element of diversity: i)
questionnaires for tutors and requesters after their connection; ii) an exit
questionnaire for the pilot; and iii) a focus group.

The pilot study revealed valuable insights from the participants. Mainly of
the suggestions arose from the focus group in which participated 7 students in
a hybrid meeting (3 in person and 4 virtually). They expressed the importance
of being able to choose tutors based on personality traits, as this can
significantly impact the dynamics and effectiveness of the tutoring
relationship. Participants suggested the inclusion of additional information,
such as the “level of compatibility” or “level of personality traits”, to aid
in the selection process and provide a more informed decision-making
framework.

One notable challenge observed during the pilot study was the low usage of the
‘SOS TUTORÍA UC’ application. This was attributed to the limited availability
of participants during the exam period, which resulted in a reduced number of
tutors available to provide support. Consequently, there was a decreased
incentive for students to request tutoring sessions through the application.
To address this issue, participants recommended extending the usage of the
application to the entire academic semester, allowing for more opportunities
for tutoring and increasing overall engagement.

Based on the results obtained and the feedback from participants during the
focus group, several priority areas for future work were identified. Firstly,
there is a need to fine-tune the recommendation algorithms to better align
with user preferences and selection filters. This could involve incorporating
additional parameters related to personality traits, compatibility levels or
other diversification elements. Secondly, developing a mobile application with
push notifications for events could enhance the user experience and
interaction, making it more convenient for students to engage with the system.
Lastly, conducting a longer-term study that covers an entire academic semester
would provide more data and insights into the student’s experience with the
personality filtering feature.

##  4 Discussion and conclusions

This paper presented the incorporation of personality traits based on the Big
Five model into a peer tutoring system, ‘SOS TUTORÍA UC’, aimed at providing
academic support. The integration with the WeNet platform successfully enabled
the matching of students with tutors based on “compatible”, “complementary” or
“indifferent” personalities. The pilot study yielded positive results in terms
of system integration, and the recommendation algorithm showed favorable
outcomes by prioritizing the academic level in making recommendations.
However, there is still potential for improvement concerning the consideration
of potential tutor’s personality traits.

The feedback from participants highlighted the importance of considering
personality traits in the tutor-tutee matching process. They expressed the
desire for additional information and parameters to facilitate the selection
of tutors, such as compatibility levels and more detailed personality trait
profiles. The limited usage during the exam period underscored the need to
extend the application’s availability throughout the academic semester.

###### Acknowledgements.

This research received funding from the proactive FET Horizon 2020 project of
the European Union, WeNet: Internet of Us, Grant Agreement No. 823783. We
would also like to express our gratitude to the various institutions
associated with the WeNet: Internet of Us project for their valuable
cooperation.

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