# The variational approach to $s$-fractional heat flows and the limit cases
$s\to 0^{+}$ and $s\to 1^{-}$

V. Crismale  Dipartimento di Matematica “Guido Castelnuovo”, Sapienza
Università di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy
<EMAIL_ADDRESS>,  L. De Luca  IAC-CNR, Via dei Taurini 19, I-00185
Roma, Italy<EMAIL_ADDRESS>,  A. Kubin  Dipartimento di Matematica
“Guido Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo Moro 2,
I-00185 Roma, Italy<EMAIL_ADDRESS>,  A. Ninno  Dipartimento di
Matematica “Guido Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo
Moro 2, I-00185 Roma, Italy<EMAIL_ADDRESS>and  M. Ponsiglione
Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università di Roma,
Piazzale Aldo Moro 2, I-00185 Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

This paper deals with the limit cases for $s$-fractional heat flows in a
cylindrical domain, with homogeneous Dirichlet boundary conditions, as $s\to
0^{+}$ and $s\to 1^{-}$ .

To this purpose, we describe the fractional heat flows as minimizing movements
of the corresponding Gagliardo seminorms, with respect to the $L^{2}$ metric.
First, we provide an abstract stability result for minimizing movements in
Hilbert spaces, with respect to a sequence of $\Gamma$-converging uniformly
$\lambda$-convex energy functionals. Then, we provide the $\Gamma$-convergence
analysis of the $s$-Gagliardo seminorms as $s\to 0^{+}$ and $s\to 1^{-}$ , and
apply the general stability result to such specific cases.

As a consequence, we prove that $s$-fractional heat flows (suitably scaled in
time) converge to the standard heat flow as $s\to 1^{-}$, and to a degenerate
ODE type flow as $s\to 0^{+}$ . Moreover, looking at the next order term in
the asymptotic expansion of the $s$-fractional Gagliardo seminorm, we show
that suitably forced $s$-fractional heat flows converge, as $s\to 0^{+}$ , to
the parabolic flow of an energy functional that can be seen as a sort of
renormalized $0$-Gagliardo seminorm: the resulting parabolic equation involves
the first variation of such an energy, that can be understood as a zero (or
logarithmic) Laplacian.

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

###### Contents

  1. 1 $\Gamma$-convergence of the squared $s$-Gagliardo seminorms as $s\to 0^{+}$
  2. 2 $\Gamma$-convergence of the squared $s$-Gagliardo seminorms as $s\rightarrow 1^{-}$
  3. 3 Minimizing movements for $\lambda$-convex functionals defined on a Hilbert space
  4. 4 Convergence of the $s$-fractional heat flows

## Introduction

In this paper we consider the $s$-fractional heat equation ($s\in(0,1)$)

(0.1) $u_{t}(t)+C(s)(-\Delta)^{s}u(t)=0\,,\qquad t\geq 0$

posed in a bounded set $\Omega\subset\mathbb{R}^{d}$ with homogeneous
Dirichlet conditions, and we study its asymptotic behavior as $s\to 0^{+}$ and
$s\to 1^{-}$. We adopt a purely variational approach, combining the theory of
_minimizing movements_ for _gradient flows_ and $\Gamma$-convergence
properties for the underlying energies.

The fractional heat equation may be seen as the $L^{2}$-gradient flow of the
squared $s$-Gagliardo seminorm

$[u]^{2}_{s}:=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{d+2s}}\,\mathrm{d}y\,\mathrm{d}x\,,$

with the support of $u$ contained in $\Omega$ when the equation is posed in a
bounded domain with homogeneous Dirichlet boundary conditions.

The asymptotic behavior of $s$-Gagliardo seminorms has been studied by several
authors. The case $s\to 1^{-}$ has been first considered in [6], where it is
proven that the pointwise limit of the squared $s$-Gagliardo seminorms
multiplied by $(1-s)$ is given by (a multiple of) the Dirichlet integral. Such
a result is indeed proven for every exponent $1<p<+\infty$ ($[\cdot]_{s}$
corresponds to $p=2$). For $p=1$ only a control of the limit in terms of the
total variation is provided, allowing to characterize the BV space; this has
been extended in several directions: first, in [13], showing that the
pointwise limit is exactly (a multiple of) the total variation, and then in
[26, 21] for more general kernels. The asymptotic behavior of (relative)
fractional perimeters (roughly said, considering characteristic functions in
the framework of [13]) is provided in [3] in terms of $\Gamma$-convergence.

For what concerns the limit as $s\to 0^{+}$ , the literature is much poorer.
In [23] 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 (see also [16] for a similar result in the context of
$s$-fractional perimeters). The corresponding asymptotic analysis in terms of
$\Gamma$-convergence has been developed in [15] in the context of fractional
perimeters (that is, restricting to characteristic functions). A functional
with more interesting properties is obtained in the limit as $s\to 0^{+}$ by
studying the next order term in the asymptotic expansion of the squared
$s$-fractional Gagliardo seminorms: in [15] it is shown, still restricting to
fractional perimeters, that the corresponding $\Gamma$-limit provides a new
nonlocal energy, referred to as $0$-fractional perimeter.

The first step in our analysis is to extend the results in [15] to the
seminorms. In fact, we remove the constraint on the admissible functions to be
characteristic functions and we obtain a $\Gamma$-convergence expansion (as
$s\to 0^{+}$) with respect to the $L^{2}$ topology for functions whose support
is in a bounded set $\Omega$ . The zero-order $\Gamma$-convergence result is
Theorem 1.2, while the next order analysis is done in Theorem 1.4. The
$\Gamma$-convergence analysis of the (scaled) $s$-Gagliardo seminorms as $s\to
1^{-}$ is done in Theorem 2.1, giving back the $\Gamma$-convergence version of
the result in [6].

The above asymptotic results, which are of independent interest, serve as a
tool for the stability of the corresponding parabolic flows. Stability of
gradient flows with respect to the $\Gamma$-convergence of the corresponding
energies is nowadays a classical problem, which has been widely investigated
in recent years in increasing generality (we refer, for instance, to [28, 27,
4]). In the present paper we exploit the fact that the underlying energies are
uniformly (with respect to $s$) $\lambda$-convex.

The gradient flows of $\lambda$-convex energies, namely energies which are
convex up to a quadratic perturbation multiplied by $\lambda$, admits a unique
solution. Moreover, such a solution can be approximated by the discrete-in-
time implicit Euler scheme; namely, the solution coincides with the so called
minimizing movements solution. This well-known fact is the content of Theorem
3.6, which provides also quantitative convergence estimates for the discrete-
in-time solutions to the unique solution of the parabolic flow, the
convergence rates depending essentially on $\lambda$ and on the initial data
(see also [2, 12, 25]). The uniformity of the convergence rates is the key
tool for proving Theorem 3.8, which guarantees stability for gradient flows in
Hilbert spaces with respect to sequences of $\Gamma$-converging uniformly
$\lambda$-convex functionals.

With Theorems 3.6 and 3.8 on hand, we are in a position to provide our
stability results for the $s$-fractional heat flows as $s\to 0^{+}$ and $s\to
1^{-}$ .

First, we apply Theorem 3.6 in order to show that for every $s\in(0,1)$ there
exists a unique solution to the Cauchy problem governed by the $s$-fractional
parabolic flow (0.1) and that such a solution is the limit, as the time step
vanishes, of the discrete-in-time evolution. Actually, the parabolic flow in
the abstract result Theorem 3.6 is written - as usual - in terms of a
differential inclusion of $-u_{t}$ in the subdifferential of the underlying
energy evaluated in $u$, which in general could be multivalued. This is not
our case, since the $s$-Gagliardo seminorms are differentiable in the
fractional Sobolev spaces $\mathcal{H}^{s}_{0}(\Omega)$ , which are dense in
$L^{2}(\Omega)$ ; this implies (see Proposition 3.7) that along the parabolic
flow, the subdifferential of the nonlocal energy reduces to a point for a.e.
$t>0$ . Actually existence, uniqueness and regularity for the fractional heat
flows have been extensively studied; we refer the interested reader to [9, 18]
and the references therein. Moreover, in the framework of nonnegative
solutions for the fractional heat equation in $\mathbb{R}^{d}$, the problem
has been studied in the context of a general Widder theory [30], in e.g. [5,
8, 29], with even not regular (but nonnegative) initial datum.

Finally, since the energies we consider are uniformly $\lambda$-convex, we can
apply the abstract stability result Theorem 3.8 to obtain the asymptotic
behavior of the $s$-fractional heat flows as $s\to 0^{+}$ and $s\to 1^{-}$ .
This is contained in our main results, Theorems 4.4, 4.5, 4.6: the limit
evolutions are an ODE corresponding to an exponential growth for the $0$-th
order as $s\to 0^{+}$, a $0$-fractional heat equation for the first order as
$s\to 0^{+}$, and the classical local heat equation as $s\to 1^{-}$. The
stability consists in a weak $H^{1}$-in-time-convergence, which is proven to
be strong if the initial data are well prepared, namely if the approximating
initial data are a recovery sequence for the limit datum with respect to the
$\Gamma$-converging energies. Furthermore, in this case for every time $t$ the
approximating evolutions are recovery sequences (with respect to the
$\Gamma$-converging energies) for the limit evolution, namely there is
convergence of the corresponding energies for every $t$.

A result similar in spirit to our asymptotic analysis for $s\to 1^{-}$ is
provided in [1], where the authors consider limit of convolution type energies
and of the corresponding parabolic flows as the support of the convolution
kernel concentrates to a point. In that paper the authors focus on homogeneous
Neumann boundary conditions, and are able to deal also with anisotropic
kernels; we believe that, up to minor differences in the setting, their
approach and the one developed in this paper are consistent and that our
analysis as $s\to 1^{-}$ could be derived by their analysis with minor
modifications. As well, we believe that our results (also for $s\to 0^{+}$)
could be generalized to natural variants of the fractional heat flows
considered here, dealing for instance with different boundary conditions as
well as anisotropic variants of the underlying Gagliardo seminorms.

The paper is structured as follows: in Sections 1 and 2 we study the
$\Gamma$-convergence of the functionals as $s\to 0^{+}$ and $s\to 1^{-}$ ,
respectively. Section 3 collects the abstract stability result for gradient
flows of uniformly $\lambda$-convex $\Gamma$-converging energies. In Section 4
we prove the main results of convergence for $s$-fractional parabolic flows as
$s\to 0^{+}$ and $s\to 1^{-}$.

Acknoledgments: The authors 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).

##  1\. $\Gamma$-convergence of the squared $s$-Gagliardo seminorms as $s\to
0^{+}$

In this section we analyze the asymptotic behaviour of the scaled and
renormalized $s$-Gagliardo seminorms as $s\to 0^{+}$.

Let $d\in\mathbb{N}$, $d\geq 1$, and let $s\in(0,1)$ . The Gagliardo
$s$-seminorm of a measurable function $u:\mathbb{R}^{d}\to\mathbb{R}$ is
defined by

$[u]_{s}:=\bigg{[}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{d+2s}}\,\mathrm{d}y\,\mathrm{d}x\bigg{]}^{\frac{1}{2}}\,,$

whenever the double integral above is finite. Let $\Omega$ be an open bounded
subset of $\mathbb{R}^{d}$ with Lipschitz continuous boundary. We denote by
$\mathcal{H}^{s}_{0}(\Omega)$ the completion of
$C^{\infty}_{\mathrm{c}}(\Omega)$ with respect to the Gagliardo $s$-seminorm
defined above. For every measurable function $u:\Omega\to\mathbb{R}$ we denote
by $\tilde{u}$ its extension to $0$ on the whole $\mathbb{R}^{d}$ , i.e.,
defined by $\tilde{u}=u$ in $\Omega$ and $\tilde{u}=0$ in
$\mathbb{R}^{d}\setminus\Omega$ .

In [23, Theorem 2] it has been proven that there exists a constant $C(d)$
depending only on the dimension $d$, such that for $d>2s$

$\int_{\Omega}\frac{|u(x)|^{2}}{|x|^{2s}}\,\mathrm{d}x\leq
C(d)\frac{s(1-s)}{(d-2s)^{2}}[\tilde{u}]^{2}_{s}\,\quad\textrm{ for every
}u\in\mathcal{H}^{s}_{0}(\Omega)\,.$

It follows that $\mathcal{H}^{s}_{0}(\Omega)\subset L^{2}(\Omega)$ for every
$d\geq 1$ and every $s\in(0,1)$: for $2s<d$ this comes from the above
estimate, being $\Omega$ bounded; for $d\leq 2s$ it is enough to pass to
suitable $s^{\prime}<s$ with $2s^{\prime}<d$, recalling that
$[\tilde{u}]_{s_{1}}\leq C(d,s)[\tilde{u}]_{s_{2}}$ for $0<s_{1}\leq s_{2}<1$
(see e.g. [17, Proposition 2.1]).

Along with [19, Theorem 1.4.2.2] (see also [24, Theorem 3.29]), the inclusion
$\mathcal{H}^{s}_{0}(\Omega)\subset L^{2}(\Omega)$ gives that

$\mathcal{H}^{s}_{0}(\Omega)=\left\\{u\in
L^{2}(\Omega)\,:\,[\tilde{u}]_{s}<+\infty\right\\}\,.$

For every $s\in(0,1)$, we define the functional
$F^{s}:L^{2}(\Omega)\to[0,+\infty]$ as

(1.1) $F^{s}(u):=[\tilde{u}]^{2}_{s}\,.$

###  1.1. $0$-th order $\Gamma$-convergence for the functionals $F^{s}$ as
$s\to 0^{+}$

We define the functional $F^{0}:L^{2}(\Omega)\to[0,+\infty)$ as

(1.2) $F^{0}(u):=\frac{d\omega_{d}}{2}\|u\|^{2}_{L^{2}},$

where $\omega_{d}$ is the measure of the unit ball of $\mathbb{R}^{d}$. The
following result is a trivial consequence of [23, formula (9)].

######  Theorem 1.1.

Let $\delta\in(0,1)$ . For every $s\in\big{(}0,\frac{\delta^{2}}{8}\big{)}$
and for every $u\in\mathcal{H}^{s}_{0}(\Omega)$ we have

(1.3)
$\frac{d\omega_{d}}{2}\int_{\Omega}\frac{|u(x)|^{2}}{|x|^{2s}}\,\mathrm{d}x\leq
s\frac{2^{2s}}{(1-\delta)^{2}}F^{s}(u)\,.$

The following theorem follows easily from the above estimate.

######  Theorem 1.2.

Let $\\{s_{n}\\}_{n\in\mathbb{N}}\subset(0,1)$ be such that $s_{n}\to 0^{+}$
as $n\to+\infty$.

  * (i)

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

$\sup_{n\in\mathbb{N}}s_{n}F^{s_{n}}(u^{n})\leq C,$

for some constant $C\in\mathbb{R}$. Then, up to a subsequence,
$u^{n}\rightharpoonup u$ in $L^{2}(\Omega)$ for some $u\in L^{2}(\Omega)$.

  * (ii)

($\Gamma$-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

$F^{0}(u)\leq\liminf_{n\to+\infty}s_{n}F^{s_{n}}(u^{n}).$

  * (iii)

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

$F^{0}(u)=\lim_{n\to+\infty}s_{n}F^{s_{n}}(u^{n}).$

###### Proof.

Since $\Omega$ is bounded, there exists $0<R<+\infty$ such that $\Omega\subset
B_{R}$ . Therefore, in view of (1.3) and of the energy bound, for $n$ large
enough, we have that

(1.4)
$\displaystyle\frac{1}{R^{2s_{n}}}\int_{\Omega}|u^{n}(x)|^{2}\,\mathrm{d}x\leq\int_{\Omega}\frac{|u^{n}(x)|^{2}}{|x|^{2s_{n}}}\,\mathrm{d}x\leq
C(d)\,.$

It follows that $\|u^{n}\|_{L^{2}(\Omega)}$ is uniformly bounded and hence, up
to a subsequence, $u^{n}\rightharpoonup u$ in $L^{2}(\Omega)$ for some $u\in
L^{2}(\Omega)$ , proving (i). Let us pass to the proof of (ii). Let
$\delta\in(0,1)$ be fixed. Using again (1.3), for $n$ large enough, we have

$\displaystyle
s_{n}F^{s_{n}}(u^{n})\geq\frac{(1-\delta)^{2}}{2^{2s_{n}}}\frac{d\omega_{d}}{2}\int_{\Omega}\frac{|u^{n}(x)|^{2}}{|x|^{2s_{n}}}\,\mathrm{d}x\geq\frac{(1-\delta)^{2}}{2^{2s_{n}}R^{2s_{n}}}\frac{d\omega_{d}}{2}\int_{\Omega}|u^{n}(x)|^{2}\,\mathrm{d}x,$

which, passing to the limit as $n\to+\infty$ and using the weak lower
semicontinuity of the $L^{2}$ norm, yields

$\liminf_{n\to+\infty}s_{n}F^{s_{n}}(u^{n})\geq(1-\delta)^{2}\frac{d\omega_{d}}{2}\int_{\Omega}|u(x)|^{2}\,\mathrm{d}x;$

by the arbitrariness of $\delta$, the claim (ii) follows. Now we show that
also (iii) holds true. If $u\in\mathrm{C}^{\infty}_{\mathrm{c}}(\Omega)$ , the
claim is proven in [23, Theorem 3], with $u^{n}\equiv u$ . Since
$\mathrm{C}^{\infty}_{\mathrm{c}}(\Omega)$ is dense in $L^{2}(\Omega)$ , the
general case follows by a standard diagonal argument. ∎

###  1.2. The first order $\Gamma$-limit of the functionals $F^{s}$ as $s\to
0^{+}$

In order to compute the $\Gamma$-limit of the renormalized functionals
$F^{s}-\frac{1}{s}F^{0}$ as $s\to 0^{+}$ we need to rewrite the functional
$F^{s}$ in a different manner. Let $s\in[0,1)$. We define the functional
$G^{s}_{1}:L^{2}(\Omega)\to[0,+\infty]$ as

(1.5) $\displaystyle
G^{s}_{1}(u):=\frac{1}{2}\iint_{\mathcal{B}_{1}}\frac{|\tilde{u}(x)-\tilde{u}(y)|^{2}}{|x-y|^{d+2s}}\,\mathrm{d}y\,\mathrm{d}x\,,$

where
$\mathcal{B}_{1}:=\\{(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\,:\,|x-y|<1\\}$
, and the functional $J^{s}_{1}:L^{2}(\Omega)\to(-\infty,+\infty)$ as

(1.6)
$J^{s}_{1}(u):=-\iint_{\mathbb{R}^{2d}\setminus\overline{\mathcal{B}}_{1}}\frac{\tilde{u}(x)\tilde{u}(y)}{|x-y|^{d+2s}}\,\mathrm{d}y\,\mathrm{d}x\,.$

We notice that the functionals $J^{s}_{1}$ are well-defined in $L^{2}(\Omega)$
since, by Hölder inequality,

(1.7)
$|J^{s}_{1}(u)|\leq\|u\|_{L^{1}(\Omega)}^{2}\leq|\Omega|\|u\|^{2}_{L^{2}(\Omega)}\,.$

It is easy to check that for every $s\in(0,1)$

(1.8)
$\hat{F}^{s}(u):=F^{s}(u)-\frac{1}{s}F^{0}(u)=G^{s}_{1}(u)+J^{s}_{1}(u)\qquad\textrm{for
every }u\in L^{2}(\Omega).$

In analogy with (1.8), we define the functionals
$\hat{F}^{0}:L^{2}(\Omega)\to(-\infty,+\infty]$ as

(1.9) $\hat{F}^{0}(u):=G^{0}_{1}(u)+J^{0}_{1}(u)\,,$

and we introduce the space

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

######  Remark 1.3.

It is natural to endow the space $\mathcal{H}_{0}^{0}(\Omega)$ with a
0-Gagliardo type norm

$[u]_{0}:=(2G^{0}_{1}(u))^{\frac{1}{2}}\,.$

We are now in a position to state our $\Gamma$-convergence result for the
functionals $\hat{F}^{s}$ defined in (1.8) .

######  Theorem 1.4.

Let $\\{s_{n}\\}_{n\in\mathbb{N}}\subset(0,1)$ be such that $s_{n}\to 0^{+}$
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

(1.10) $\hat{F}^{s_{n}}(u^{n})+2|\Omega|\|u^{n}\|^{2}_{L^{2}(\Omega)}\leq M,$

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

  * (ii)

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

$\hat{F}^{0}(u)\leq\liminf_{n\to+\infty}\hat{F}^{s_{n}}(u^{n})\,.$

  * (iii)

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

$\hat{F}^{0}(u)=\lim_{n\to+\infty}\hat{F}^{s_{n}}(u^{n})\,.$

######  Remark 1.5.

Notice that for all $s\in[0,1)$ and for all $u,\,v\in L^{2}(\Omega)$ we have

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

In particular, for all $s\in[0,1)$ the functionals $J^{s}_{1}$ are continuous
with respect to the strong $L^{2}$ convergence.

###  1.3. Compactness and $\Gamma$-liminf inequality

In order to prove (i) of Theorem 1.4, we recall the following result proven in
[20] .

######  Theorem 1.6 (Local compactness [20]).

Let $k:\mathbb{R}^{d}\to[0,+\infty]$ be a radially symmetric kernel such that

$\int_{\mathbb{R}^{d}}k(z)\,\mathrm{d}z=+\infty\qquad\textrm{and}\qquad\int_{\mathbb{R}^{d}}\min\\{1,|z|^{2}\\}k(z)\,\mathrm{d}z<+\infty$

and let

$\mathcal{W}^{k}(\Omega):=\left\\{u\in
L^{2}(\Omega)\,:\,\iint_{\mathbb{R}^{2d}}|\tilde{u}(x)-\tilde{u}(y)|^{2}k(x-y)\,\mathrm{d}y\,\mathrm{d}x<+\infty\right\\}$

be the Banach space endowed with the norm

$\|u\|_{\mathcal{W}^{k}(\Omega)}:=\|u\|_{L^{2}(\Omega)}+\Big{(}\iint_{\mathbb{R}^{2d}}|\tilde{u}(x)-\tilde{u}(y)|^{2}k(x-y)\,\mathrm{d}y\,\mathrm{d}x\Big{)}^{\frac{1}{2}}\,.$

Then, the embedding $\mathcal{W}^{k}(\Omega)\hookrightarrow L^{2}(\Omega)$ is
compact.

With Theorem 1.6 on hand, we are in a position to prove compactness.

###### Proof of Theorem 1.4(i).

By (1.10), (1.8) and (1.7), we have that

$M\geq\hat{F}^{s_{n}}(u^{n})+2|\Omega|\|u^{n}\|^{2}_{L^{2}(\Omega)}\geq-|\Omega|\|u^{n}\|_{L^{2}(\Omega)}^{2}+2|\Omega|\|u^{n}\|^{2}_{L^{2}(\Omega)}=|\Omega|\|u^{n}\|^{2}_{L^{2}(\Omega)}\,,$

i.e., that $\|u^{n}\|_{L^{2}(\Omega)}$ is uniformly bounded. Therefore, by
(1.8) we deduce

(1.12) $\displaystyle G^{0}_{1}(u^{n})\leq$ $\displaystyle
G^{s_{n}}_{1}(u^{n})\leq
M+\iint_{\mathbb{R}^{2d}\setminus\overline{\mathcal{B}}_{1}}\frac{|\tilde{u}^{n}(x)||\tilde{u}^{n}(y)|}{|x-y|^{d+2s_{n}}}\,\mathrm{d}y\,\mathrm{d}x\leq
M+|\Omega|\|u^{n}\|^{2}_{L^{2}(\Omega)}$ $\displaystyle\leq$ $\displaystyle
2M\,,$

whence, by applying Theorem 1.6 with $k(z):=\frac{\chi_{B_{1}}(z)}{|z|^{d}}$ ,
we deduce that, up to a subsequence, $u^{n}\to u$ in $L^{2}(\Omega)$ for some
$u\in L^{2}(\Omega)$ . Finally, by (1.12) and by the lower semicontinuity of
the functional $G^{0}_{1}$ with respect to the strong $L^{2}$ convergence, we
get that $u\in\mathcal{H}_{0}^{0}(\Omega)$ . ∎

Now we prove the $\Gamma$-liminf inequality.

###### Proof of Theorem 1.4(ii).

By Fatou lemma we have

$G^{0}_{1}(u)\leq\liminf_{n\to+\infty}G^{s_{n}}_{1}(u^{n})\,;$

moreover, by (1.11) we easily get

$J^{0}_{1}(u)=\lim_{n\to+\infty}J^{s_{n}}_{1}(u^{n})\,.$

In view of (1.8) and (1.9), we get the claim. ∎

###  1.4. $\Gamma$-limsup inequality

Here we construct the recovery sequence for the functionals $\hat{F}^{s}$ . We
start by showing that, for smooth functions, the pointwise limit of the
functional $\hat{F}^{s}$ as $s\to 0^{+}$ coincides with the functionals
$\hat{F}^{0}$ .

######  Lemma 1.7.

For every $u\in\mathrm{C}^{\infty}_{\mathrm{c}}(\Omega)$ we have that

$\lim_{s\to 0^{+}}\hat{F}^{s}(u)=\hat{F}^{0}(u)\,.$

###### Proof.

In view of the definition of $\hat{F}^{s}$ in (1.8) it is enough to show

(1.13) $\displaystyle\lim_{s\to 0^{+}}G^{s}_{1}(u)=G^{0}_{1}(u)\,,$ (1.14)
$\displaystyle\lim_{s\to 0^{+}}J^{s}_{1}(u)=J^{0}_{1}(u)\,.$

We start by proving (1.13). To this end, we note that, since
$\tilde{u}\in\mathrm{C}^{\infty}(\mathbb{R}^{d})$, for every
$x,\,y\in\mathbb{R}^{d}$ we have

$|\tilde{u}(y)-\tilde{u}(x)|^{2}\leq\|\nabla\tilde{u}\|_{L^{\infty}}^{2}|x-y|^{2}\,.$

Let $U\subset\mathbb{R}^{d}$ be an open set such that
$\mathrm{dist}(\Omega,\mathbb{R}^{d}\setminus U)>1$ and let
$\varepsilon\in(0,1)$; we have

$\displaystyle|G^{s}_{1}(u)-G^{0}_{1}(u)|\leq$
$\displaystyle\frac{1}{2}\int_{U}\,\mathrm{d}x\int_{B_{\varepsilon}(x)}|\tilde{u}(x)-\tilde{u}(y)|^{2}\bigg{|}\frac{1}{|x-y|^{d+2s}}-\frac{1}{|x-y|^{d}}\bigg{|}\,\mathrm{d}y$
$\displaystyle+\frac{1}{2}\iint_{\mathcal{B}_{1}\setminus\overline{\mathcal{B}}_{\varepsilon}}|\tilde{u}(x)-\tilde{u}(y)|^{2}\bigg{|}\frac{1}{|x-y|^{d+2s}}-\frac{1}{|x-y|^{d}}\bigg{|}\,\mathrm{d}y\,\mathrm{d}x.$

By Dominated Convergence Theorem the second addend in the righthand side tends
to zero (for fixed $\varepsilon$) as $s\to 0^{+}$, while the first addend is
bounded from above by
$|U|\|\nabla\tilde{u}\|^{2}_{L^{\infty}}\int_{B_{\varepsilon}}\frac{1}{|z|^{d+2s-2}}\,\mathrm{d}z$,
which tends to zero as $\varepsilon\to 0^{+}$. This clearly yields (1.13).

Finally, (1.14) is a trivial consequence of the Dominated Convergence Theorem,
once noticed that

$J^{s}_{1}(u)=-\int_{\Omega}u(x)\int_{\Omega\setminus\overline{B}_{1}(x)}\frac{u(y)}{|x-y|^{d+2s}}\,\mathrm{d}y\,\mathrm{d}x\,.$

∎

######  Lemma 1.8 (Density of smooth functions).

For every $u\in\mathcal{H}_{0}^{0}(\Omega)$ there exists
$\\{u_{k}\\}_{k\in\mathbb{N}}\subset\mathrm{C}^{\infty}_{\mathrm{c}}(\Omega)$
such that $u_{k}\to u$ (strongly) in $L^{2}$ and

$\lim_{k\to+\infty}J^{0}_{1}(u_{k})=J^{0}_{1}(u)\qquad\textrm{and}\qquad\lim_{k\to+\infty}G^{0}_{1}(u_{k})=G_{1}^{0}(u).$

###### Proof.

This result is proven in [24, Theorem 3.29], for domains with a continuous
boundary. For the reader’s convenience we recall the sketch of the proof: Up
to a partition of the unity argument, one may assume $\Omega$ to be the
subgraph of a continuous function: thus it is enough to approximate first with
$u_{\delta}(x):=u(x^{\prime},x_{n}+\delta)$, for small $\delta$, whose support
is well contained in $\Omega$, and then to take
$u_{\delta}\ast\phi_{\varepsilon}$, for a family of mollifiers
$\\{\phi_{\varepsilon}\\}_{\varepsilon}$ and small $\varepsilon$. ∎

The limsup inequality in Theorem 1.4 follows directly from the density proved
above.

###### Proof of Theorem 1.4(iii).

Let $u\in\mathcal{H}^{0}(\Omega)$ . By Lemma 1.8 there exists a sequence of
functions
$\\{u^{k}\\}_{k\in\mathbb{N}}\subset\mathrm{C}^{\infty}_{\mathrm{c}}(\mathbb{R}^{d})$
such that $u^{k}\to u$ in $L^{2}$ and

$\limsup_{k\to+\infty}\hat{F}^{0}(u^{k})=\hat{F}^{0}(u)\,.$

In view of Lemma 1.7 we have

$\lim_{n\to+\infty}\hat{F}^{s_{n}}(u^{k})=\hat{F}^{0}(u^{k})\qquad\textrm{for
every }k\in\mathbb{N}\,.$

Therefore, by a standard diagonal argument, there exists a sequence
$\\{u^{n}\\}_{n\in\mathbb{N}}\subset\mathrm{C}^{\infty}_{\mathrm{c}}(\mathbb{R}^{d})$
with $u^{n}=u^{k(n)}$ for every $n\in\mathbb{N}$ satisfying the desired
properties. ∎

##  2\. $\Gamma$-convergence of the squared $s$-Gagliardo seminorms as
$s\rightarrow 1^{-}$

Here we study the $\Gamma$-convergence of the functionals $(1-s)F^{s}$ as
$s\rightarrow 1^{-}$, where $F^{s}$ is defined in (1.1). The candidate
$\Gamma$-limit is the functional
$F^{1}:L^{2}(\Omega)\rightarrow\mathbb{R}\cup\\{+\infty\\}$ defined by

$F^{1}(u):=\left\\{\begin{aligned} &\frac{\omega_{d}}{4}\int_{\Omega}|\nabla
u(x)|^{2}\,\mathrm{d}x&\text{ if }u\in H_{0}^{1}(\Omega)\,,\\\
&+\infty&\textrm{elsewhere in }L^{2}(\Omega)\,.\end{aligned}\right.$

######  Theorem 2.1.

Let $\\{s_{n}\\}_{n\in\mathbb{N}}\subset(0,1)$ be such that $s_{n}\rightarrow
1^{-}$ 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

(2.1)
$\sup_{n\in\mathbb{N}}(1-s_{n})F^{s_{n}}(u^{n})+\|u^{n}\|_{L^{2}(\Omega)}^{2}\leq
M,$

for some constant $M$ independent of $n$ . Then, up to a subsequence,
$u^{n}\to u$ strongly in $L^{2}(\Omega)$ for some $u\in H_{0}^{1}(\Omega)$ .

  * (ii)

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

$F^{1}(u)\leq\liminf_{n\to+\infty}(1-s_{n})F^{s_{n}}(u^{n})\,.$

  * (iii)

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

(2.2) $F^{1}(u)=\lim_{n\to+\infty}(1-s_{n})F^{s_{n}}(u^{n})\,.$

###  2.1. Proof of Compactness

This subsection is devoted to the proof of Theorem 2.1(i). To accomplish this
task, we adopt the strategy in [3] adapting it to our case. To this purpose,
for every function $v\in L^{2}(\Omega)$ and for every $h\in\mathbb{R}^{d}$ we
denote by $\tau_{h}v$ the shift of $v$ by $h$, defined by
$\tau_{h}v(\cdot):=v(\cdot+h)$ . We recall the following two classical
results.

######  Theorem 2.2 (Fréchet-Kolmogorov).

Let $\\{v^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\mathbb{R}^{d})$ be such tha
$\sup_{n\in\mathbb{N}}\|v^{n}\|_{L^{2}(\mathbb{R}^{d})}\leq M$ , for some
constant $M$ independent of $n$ . If

$\lim_{|h|\rightarrow
0^{+}}\sup_{n\in\mathbb{N}}\|\tau_{h}v^{n}-v^{n}\|_{L^{2}(\mathbb{R}^{d})}=0\,,$

then $\\{v^{n}\\}_{n\in\mathbb{N}}$ is pre-compact in
$L_{\mathrm{loc}}^{2}(\mathbb{R}^{d})$ .

######  Theorem 2.3.

Let $v\in L^{2}(\mathbb{R}^{d})$. Then $v\in H^{1}(\mathbb{R}^{d})$ if and
only if there exists $C>0$ such that

$\|\tau_{h}v-v\|_{L^{2}(\mathbb{R}^{d})}\leq C|h|\quad\textrm{ for every
}h\in\mathbb{R}^{d}\,.$

For every $A\subset\mathbb{R}^{d}$ and for every $t>0$ we define the set

(2.3) $A_{t}:=\\{x\in\mathbb{R}^{d}:\;\mathrm{dist}(x,A)<t\\}.$

The following result which allows to estimate the $L^{2}$ distance of a
function from its shift has been proven in [3, Proposition 5] in $L^{1}$ ; for
the sake of completeness, we state and prove it also in our case.

######  Proposition 2.4.

There exists a constant $C(d)>0$ such that the following holds true: for every
$v\in L^{2}(\mathbb{R}^{d})$, for every $h\in\mathbb{R}^{d}$ and for every
open bounded set $\Omega^{\prime}\subset\mathbb{R}^{d}$ we have

(2.4) $\|\tau_{h}v-v\|_{L^{2}(\Omega^{\prime})}^{2}\leq
C(d)\frac{|h|^{2}}{\rho^{d+2}}\int_{B_{\rho}}\|\tau_{y}v-v\|_{L^{2}(\Omega^{\prime}_{|h|})}^{2}\,\mathrm{d}y\,\quad\textrm{for
every }\rho\in(0,|h|],$

with $\Omega^{\prime}_{|h|}$ defined as in (2.3).

###### Proof.

The proof closely resembles the one of [3, Proposition 5]. Let
$\varphi\in\mathrm{C}_{\mathrm{c}}^{1}(B_{1})$ be a fixed function with
$\varphi\geq 0$ and $\int_{B_{1}}\varphi(x)\,\mathrm{d}x=1$ . For every
$\rho>0$ we define the functions
$U_{\rho},V_{\rho}:\mathbb{R}^{d}\to\mathbb{R}$ as

$U_{\rho}(x):=\frac{1}{\rho^{d}}\int_{B_{\rho}}v(x+y)\varphi\bigg{(}\frac{y}{\rho}\bigg{)}\,\mathrm{d}y\,,\qquad
V_{\rho}(x):=\frac{1}{\rho^{d}}\int_{B_{\rho}}(v(x)-v(x+y))\varphi\bigg{(}\frac{y}{\rho}\bigg{)}\,\mathrm{d}y\,;$

clearly, for every $\rho>0$ and for every $x\in\mathbb{R}^{d}$

$v(x)=U_{\rho}(x)+V_{\rho}(x)\,,$

and hence

(2.5) $|\tau_{h}v(x)-v(x)|^{2}\leq
3|U_{\rho}(x+h)-U_{\rho}(x)|^{2}+3|V_{\rho}(x)|^{2}+3|V_{\rho}(x+h)|^{2}\,.$

By Jensen inequality, for every $\xi\in\mathbb{R}^{d}$ we have

(2.6)
$|V_{\rho}(\xi)|^{2}\leq\frac{\omega_{d}}{\rho^{d}}\|\varphi\|^{2}_{L^{\infty}(B_{1})}\int_{B_{\rho}}|v(\xi)-\tau_{y}v(\xi)|^{2}\,\mathrm{d}y.$

Moreover, by the change of variable $z=x+y$, we have that

$U_{\rho}(x)=\frac{1}{\rho^{d}}\int_{B_{\rho}(x)}v(z)\varphi\bigg{(}\frac{z-x}{\rho}\bigg{)}\,\mathrm{d}z\,,$

whence we deduce that

$\begin{split}\mathrm{D}\,U_{\rho}(x)=&-\frac{1}{\rho^{d+1}}\int_{B_{\rho}(x)}v(z)\mathrm{D}\varphi\bigg{(}\frac{z-x}{\rho}\bigg{)}\,\mathrm{d}z\\\
=&-\frac{1}{\rho^{d+1}}\int_{B_{\rho}(x)}(v(z)-v(x))\mathrm{D}\varphi\bigg{(}\frac{z-x}{\rho}\bigg{)}\,\mathrm{d}z\\\
=&-\frac{1}{\rho^{d+1}}\int_{B_{\rho}}(v(x+y)-v(x))\mathrm{D}\varphi\bigg{(}\frac{y}{\rho}\bigg{)}\,\mathrm{d}y\,;\end{split}$

therefore, by the fundamental Theorem of Calculus and by Jensen inequality, we
obtain

(2.7)
$\begin{split}&|U_{\rho}(x+h)-U_{\rho}(x)|^{2}\leq|h|^{2}\int_{0}^{1}|\mathrm{D}\,U_{\rho}(x+th)|^{2}\,\mathrm{d}t\\\
&\leq\omega_{d}\frac{|h|^{2}}{\rho^{d+2}}\|\mathrm{D}\varphi\|_{L^{\infty}(B_{1})}^{2}\int_{0}^{1}\int_{B_{\rho}}|\tau_{y}v(x+th)-v(x+th)|^{2}\,\mathrm{d}y\,\mathrm{d}t.\end{split}$

Now, by (2.5), (2.6), and (2.7), taking $\rho<|h|$ , we have

(2.8)
$\begin{split}|\tau_{h}v(x)-v(x)|^{2}\leq&3\omega_{d}\frac{|h|^{2}}{\rho^{d+2}}\|\mathrm{D}\varphi\|_{\infty}^{2}\int_{0}^{1}\int_{B_{\rho}}|\tau_{y}v(x+th)-v(x+th)|^{2}\,\mathrm{d}y\,\mathrm{d}t\\\
&+3\omega_{d}\frac{|h|^{2}}{\rho^{d+2}}\|\varphi\|_{\infty}^{2}\int_{B_{\rho}}|\tau_{y}v(x)-v(x)|^{2}\,\mathrm{d}y\\\
&+3\omega_{d}\frac{|h|^{2}}{\rho^{d+2}}\|\varphi\|_{\infty}^{2}\int_{B_{\rho}}|\tau_{y}v(x+h)-v(x+h)|^{2}\,\mathrm{d}y\,.\end{split}$

Finally, by integrating (2.8) on $\Omega^{\prime}$ , by Fubini theorem, we
obtain (2.4) with
$C(d):=3\omega_{d}(2\|\varphi\|_{L^{\infty}(B_{1})}^{2}+\|\mathrm{D}\varphi\|_{L^{\infty}(B_{1})}^{2})$
. ∎

We recall the following version of Hardy’s inequality, that is proven in [3,
Proposition 6].

######  Lemma 2.5.

Let $g:\mathbb{R}\rightarrow[0,+\infty)$ be a Borel measurable function. Then
for all $l\geq 0$ we have

$\int_{0}^{r}\frac{1}{\rho^{d+l+1}}\int_{0}^{\rho}g(t)\,\mathrm{d}t\,\mathrm{d}\rho\leq\frac{1}{d+l}\int_{0}^{r}\frac{g(t)}{t^{d+l}}\,\mathrm{d}t\quad\textrm{for
every }r\geq 0.$

The following result will be used in the proof of Theorem 2.1(i). It is the
$L^{2}$ analog of [3, Proposition 4].

######  Proposition 2.6.

There exists a constant $\bar{C}(d)>0$ such that for every $v\in
L^{2}(\mathbb{R}^{d})$, for every bounded open set
$\Omega^{\prime}\subset\mathbb{R}^{d}$, for every $s\in(0,1)$ , and for every
$h\in\mathbb{R}^{d}$ , it holds

$\|\tau_{h}v-v\|_{L^{2}(\Omega^{\prime})}^{2}\leq|h|^{2s}\bar{C}(d)(1-s)\int_{B_{|h|}}\frac{\|\tau_{y}v-v\|_{L^{2}(\Omega^{\prime}_{|h|})}^{2}}{|y|^{d+2s}}\,\mathrm{d}y\,.$

###### Proof.

For every fixed $v\in L^{2}(\mathbb{R}^{d})$, we define the function
$g_{v}:[0,|h|]\to\mathbb{R}$ as

$g_{v}(t):=\int_{\partial
B_{t}}\|\tau_{y}v-v\|_{L^{2}(\Omega^{\prime}_{|h|})}^{2}\,\mathrm{d}\mathcal{H}^{d-1}(y)\,.$

By integrating in polar coordinates formula (2.4) we have

(2.9) $\|\tau_{h}v-v\|_{L^{2}(\Omega^{\prime})}^{2}\leq
C(d)\frac{|h|^{2}}{\rho^{d+2}}\int_{0}^{\rho}g_{v}(t)\,\mathrm{d}t\,.$

By multiplying both sides of (2.9) by $\rho^{1-2s}$ and integrating in the
interval $[0,|h|]$ , using Lemma 2.5 and the very definition of $g_{v}$ , we
obtain

$\displaystyle\|\tau_{h}v-v\|_{L^{2}(\Omega^{\prime})}^{2}\leq$ $\displaystyle
2C(d)(1-s)|h|^{2s}\int_{0}^{|h|}\frac{1}{\rho^{d+2s+1}}\int_{0}^{\rho}g_{v}(t)\,\mathrm{d}t\,\mathrm{d}z$
$\displaystyle\leq$ $\displaystyle
C(d)(1-s)|h|^{2s}\int_{0}^{|h|}\frac{g_{v}(t)}{t^{d+2s}}\,\mathrm{d}t$
$\displaystyle=$ $\displaystyle
C(d)(1-s)|h|^{2s}\int_{B_{|h|}}\frac{\|\tau_{y}v-v\|_{L^{2}(\Omega^{\prime}_{|h|})}^{2}}{|y|^{d+2s}}\,\mathrm{d}y\,,$

which concludes the proof. ∎

We are now in a position to prove Theorem 2.1(i).

###### Proof of Theorem 2.1(i).

By Proposition 2.6 and by the upper bound (2.1) we obtain that for every open
bounded set $\Omega^{\prime}\subset\mathbb{R}^{d}$ and for every
$h\in\mathbb{R}^{d}$

(2.10) $\|\tau_{h}\tilde{u}^{n}-\tilde{u}^{n}\|_{L^{2}(\Omega^{\prime})}\leq
C(d,M)|h|^{s_{n}}\,,$

where we recall that $\tilde{u}^{n}$ is the extension of $u^{n}$ to $0$ in
$\mathbb{R}^{d}\setminus\Omega$ . Therefore, the sequence
$\\{\tilde{u}^{n}\\}_{n\in\mathbb{N}}$ satisfies the assumption of Theorem
2.2, and hence there exists a function $v\in L^{2}(\mathbb{R}^{d})$ with $v=0$
in $\mathbb{R}^{d}\setminus\Omega$, such that, up to a subsequence,
$\tilde{u}^{n}\rightarrow v$ in $L_{\mathrm{loc}}^{2}(\mathbb{R}^{d})$. Now,
sending $n\to+\infty$ in (2.10), we obtain that for every open bounded set
$\Omega^{\prime}\subset\mathbb{R}^{d}$

$\|\tau_{h}v-v\|_{L^{2}(\Omega^{\prime})}\leq C(d,M)|h|\quad\textrm{for every
}h\in\mathbb{R}^{d}\,,$

and hence by Theorem 2.3 (choosing $\Omega^{\prime}=\mathbb{R}^{d}$ in the
above inequality) we obtain that $\mathrm{D}v\in L^{2}(\mathbb{R}^{d})$ .
Since $v=0$ in $\mathbb{R}^{d}\setminus\Omega$, by the regularity of
$\partial\Omega$, we have that $v$ is the extension to $0$ in
$\mathbb{R}^{d}\setminus\Omega$ of a function $u\in H_{0}^{1}(\Omega)$ , thus
concluding the proof. ∎

###  2.2. Proof of the $\Gamma$-liminf inequality

Here we prove the $\Gamma$-liminf inequality in Theorem 2.1. The proof of this
result closely resembles the one of [6, Theorem 2].

###### Proof of Theorem 2.1(ii).

We can assume without loss of generality that (2.1) holds true so that the
function $u$ is actually in $H^{1}_{0}(\Omega)$.

Claim 1. Let $\eta\in\mathrm{C}_{\mathrm{c}}^{\infty}(B_{1})$ be a standard
mollifier, i.e., $\eta\geq 0$ and $\int_{B_{1}}\eta(x)\,\mathrm{d}x=1$ . For
every $\varepsilon>0$ , we set
$\eta_{\varepsilon}(\cdot):=\frac{1}{\varepsilon^{d}}\eta(\frac{\cdot}{\varepsilon})$
. Then, for every $s\in(0,1)$

$\frac{1}{2}[\tilde{v}_{\varepsilon}]^{2}_{s}\leq F^{s}(v)\qquad\textrm{for
every }v\in L^{2}(\Omega)\textrm{ and for every }\varepsilon>0\,,$

where $\tilde{v}_{\varepsilon}:=\tilde{v}\ast\eta_{\varepsilon}$ .

Indeed, setting $\Omega_{\varepsilon}:=\\{x\in\mathbb{R}^{d}\ :\
\text{dist}(x,\Omega)\leq\varepsilon\\}$ , we have that
$\tilde{v}_{\varepsilon}=0$ in $\mathbb{R}^{d}\setminus\Omega_{\varepsilon}$ ;
therefore, by applying Jensen inequality to the probability measure
$\eta_{\varepsilon}\,\mathrm{d}z$ , we get

$\displaystyle{}^{2}_{s}\leq$
$\displaystyle\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|\tilde{v}(x-z)-\tilde{v}(y-z)|^{2}}{|x-y|^{d+2s}}\eta_{\varepsilon}(z)\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|\tilde{v}(x-z)-\tilde{v}(y-z)|^{2}}{|x-z-(y-z)|^{d+2s}}\eta_{\varepsilon}(z)\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=$ $\displaystyle 2F^{s}(v)\,.$

Claim 2. For every $\varepsilon>0$ and for every $R>0$, it holds

(2.11)
$\frac{\omega_{d}}{4}\liminf_{n\to+\infty}\,\int_{B_{R}}|\nabla\tilde{u}^{n}_{\varepsilon}|^{2}(\mathrm{dist}(x,\partial
B_{R}))^{2(1-s_{n})}\,\mathrm{d}x\leq\liminf_{n\to+\infty}(1-s_{n})\frac{[\tilde{u}^{n}_{\varepsilon}]^{2}_{s_{n}}}{2}\,,$

where $\tilde{u}_{\varepsilon}^{n}:=\tilde{u}^{n}\ast\eta_{\varepsilon}$ ,
with $\eta_{\varepsilon}$ as in Claim 1.

Indeed, by Taylor expansion, using that
$\sup_{n\in\mathbb{N}}\|u^{n}\|^{2}_{L^{2}(\Omega)}\leq M$ we have that

$\displaystyle|\tilde{u}_{\varepsilon}^{n}(x)-\tilde{u}_{\varepsilon}^{n}(y)|^{2}\geq$
$\displaystyle\Big{|}\nabla\tilde{u}_{\varepsilon}^{n}(x)\cdot\frac{x-y}{|x-y|}\Big{|}^{2}|x-y|^{2}$
$\displaystyle-\|u^{n}\|^{2}_{L^{1}(\Omega)}\|\eta_{\varepsilon}\|^{2}_{C^{2}(\mathbb{R}^{d})}|x-y|^{3}-\|u^{n}\|_{L^{1}(\Omega)}^{2}\|\eta_{\varepsilon}\|^{2}_{C^{2}(\mathbb{R}^{d})}|x-y|^{4}$
$\displaystyle\geq$
$\displaystyle\Big{|}\nabla\tilde{u}_{\varepsilon}^{n}(x)\cdot\frac{x-y}{|x-y|}\Big{|}^{2}|x-y|^{2}-C(\varepsilon,M)(|x-y|^{3}+|x-y|^{4})\,,$

Therefore, for every $x\in\mathbb{R}^{d}$ , setting
$\delta:=\mathrm{dist}(x,\partial B_{R})$ , we get

(2.12)
$\displaystyle(1-s_{n})\int_{B_{R}}\frac{|\tilde{u}_{\varepsilon}^{n}(x)-\tilde{u}_{\varepsilon}^{n}(y)|^{2}}{|x-y|^{d+2s_{n}}}\,\mathrm{d}y\geq(1-s_{n})\int_{B_{\delta}(x)}\frac{|\tilde{u}_{\varepsilon}^{n}(x)-\tilde{u}_{\varepsilon}^{n}(y)|^{2}}{|x-y|^{d+2s_{n}}}\,\mathrm{d}y$
$\displaystyle\geq$
$\displaystyle(1-s_{n})\int_{B_{\delta}(x)}\Big{|}\nabla\tilde{u}_{\varepsilon}^{n}(x)\cdot\frac{x-y}{|x-y|}\Big{|}^{2}|x-y|^{2(1-s_{n})-d}\,\mathrm{d}y$
$\displaystyle-(1-s_{n})C(\varepsilon,M)\int_{B_{\delta}(x)}\frac{|x-y|^{3}+|x-y|^{4}}{|x-y|^{d+2s_{n}}}\,\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{\omega_{d}}{2}\delta^{2(1-s_{n})}|\nabla\tilde{u}_{\varepsilon}^{n}(x)|^{2}-(1-s_{n})C(\varepsilon,M,d)\,,$

where in the last equality we integrated over spherical boundaries from 0 to
$\delta$, using that
$\int_{\mathbb{S}^{d-1}}|\nabla\tilde{u}_{\varepsilon}^{n}(x)\cdot\theta|^{2}\,\mathrm{d}\theta=\omega_{d}|\nabla\tilde{u}_{\varepsilon}^{n}(x)|^{2}$
. By integrating (2.12) over $B_{R}$, we get (2.11).

By Claim 1 and Claim 2, for every $\varepsilon>0$ and for every $R>0$ we have
that

$\liminf_{n\to+\infty}(1-s_{n})F^{s_{n}}(u^{n})\geq\frac{\omega_{d}}{4}\liminf_{n\to+\infty}\int_{B_{R}}|\nabla\tilde{u}_{\varepsilon}^{n}(x)|^{2}(\mathrm{dist}(x,\partial
B_{R}))^{2(1-s_{n})}\,\mathrm{d}x\,,$

whence, using that for every $\varepsilon>0$ the sequence
$\\{\tilde{u}_{\varepsilon}^{n}\\}_{n\in\mathbb{N}}$ is equi-Lipschitz, we get
that, up to a (not relabeled) subsequence,

(2.13)
$\liminf_{n\to+\infty}(1-s_{n})F^{s_{n}}(u^{n})\geq\frac{\omega_{d}}{4}\liminf_{n\to+\infty}\int_{B_{R}}|\nabla\tilde{u}_{\varepsilon}^{n}(x)|^{2}\,\mathrm{d}x\,.$

Notice that
$\tilde{u}^{n}_{\varepsilon}\to\tilde{u}_{\varepsilon}:=\tilde{u}\ast\eta_{\varepsilon}$
in $L^{2}(\mathbb{R}^{d})$ as $n\to+\infty$ . By (2.13) and by (2.1), we have
that in fact
$\tilde{u}^{n}_{\varepsilon}\rightharpoonup\tilde{u}_{\varepsilon}$ in
$H^{1}(B_{R})$ .

In conclusion, by (2.13), we deduce that for every $\varepsilon>0$

$\liminf_{n\to+\infty}(1-s_{n})F^{s_{n}}(u^{n})\geq\frac{\omega_{d}}{4}\int_{B_{R}}|\nabla\tilde{u}_{\varepsilon}(x)|^{2}\,\mathrm{d}x\,,$

whence the claim follows sending first $\varepsilon\to 0$ (using that
$\tilde{u}_{\varepsilon}\to\tilde{u}$ in $H^{1}(\mathbb{R}^{d})$ as
$\varepsilon\to 0$) and then $R\to+\infty$ . ∎

###  2.3. Proof of the $\Gamma$-limsup inequality

The proof of the $\Gamma$-limsup inequality relies on the pointwise
convergence of $(1-s)F^{s}$ to $F^{1}$ (as $s\to 1$) for smooth functions with
compact support and on the density of smooth functions in $H^{1}_{0}(\Omega)$
. As for the pointwise convergence we recall the following result, proved in
[22] in a more general setting.

######  Theorem 2.7.

For every $v\in\mathrm{C}_{\mathrm{c}}^{\infty}(\mathbb{R}^{d})$ it holds

$\lim_{s\rightarrow
1^{-}}(1-s)\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|v(x)-v(y)|^{2}}{|x-y|^{d+2s}}\,\mathrm{d}x\,\mathrm{d}y=\frac{\omega_{d}}{2}\int_{\mathbb{R}^{d}}|\nabla
v(x)|^{2}\,\mathrm{d}x\,.$

With Theorem 2.7 on hand we can prove Theorem 2.1(iii) using standard density
arguments in $\Gamma$-convergence.

###### Proof of Theorem 2.1(iii).

It is enough to prove the claim only for $u\in H^{1}_{0}(\Omega)$ . For every
$u\in H_{0}^{1}(\Omega)$ there exists
$\\{u^{k}\\}_{k\in\mathbb{N}}\subset\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$
such that $u^{k}\rightarrow u$ (as $k\rightarrow+\infty$) in $H^{1}(\Omega)$ .
In view of Theorem 2.7 we have that for every $k\in\mathbb{N}$

$\displaystyle\lim_{n\rightarrow+\infty}(1-s_{n})F^{s_{n}}(u^{k})=$
$\displaystyle\lim_{n\rightarrow+\infty}\frac{1-s_{n}}{2}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|\tilde{u}^{k}(x)-\tilde{u}^{k}(y)|^{2}}{|x-y|^{d+2s_{n}}}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{\omega_{d}}{4}\int_{\mathbb{R}^{d}}|\nabla\tilde{u}^{k}(x)|^{2}\,\mathrm{d}x=\frac{\omega_{d}}{4}\int_{\Omega}|\nabla
u^{k}(x)|^{2}\,\mathrm{d}x\,.$

Therefore by a standard diagonal argument there exists
$\\{k_{n}\\}_{n\in\mathbb{N}}$ such that

$\lim_{n\rightarrow+\infty}u^{k_{n}}=u,\quad\limsup_{n\rightarrow+\infty}(1-s_{n})F^{s_{n}}(u^{k_{n}})\leq\frac{\omega_{d}}{4}\int_{\Omega}|\nabla
u(x)|^{2}\,\mathrm{d}x=F^{1}(u)\,,$

i.e., (2.2). ∎

##  3\. Minimizing movements for $\lambda$-convex functionals defined on a
Hilbert space

In this section we develop the general theory that will allow us to study the
stability of the $s$-fractional heat flow as $s\to 0^{+}$ and $s\to 1^{-}$ .
Throughout this section $\mathscr{H}$ is a generic Hilbert space,
$\langle\cdot,\cdot\rangle_{\mathscr{H}}$ is the inner product of
$\mathscr{H}$ and $|\cdot|_{\mathscr{H}}$ is the norm induced by such a scalar
product. In the abstract setting of this section, we denote by $\dot{v}$ the
time derivative of any function $v$ from a time interval with values in
$\mathscr{H}$.

######  Definition 3.1 ($\lambda$-convexity, $\lambda$-positivity,
$\lambda$-coercivity).

Let $\lambda>0$ . We say that a function
$\mathcal{F}:\mathscr{H}\rightarrow(-\infty,+\infty]$ is $\lambda$-convex if
the function $f(\cdot)+\frac{\lambda}{2}|\cdot|_{\mathscr{H}}^{2}$ 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
$x\in\mathscr{H}$ , and we say that $\mathcal{F}$ is $\lambda$-coercive if the
sublevels of the function
$\mathcal{F}(\cdot)+\frac{\lambda}{2}|\cdot|_{\mathscr{H}}^{2}$ are bounded.

######  Remark 3.2.

We notice that if $\mathcal{F}$ is $\lambda$-positive, then $\mathcal{F}$ is
$\tilde{\lambda}$-coercive for every $\tilde{\lambda}>\lambda$ .

######  Proposition 3.3.

Let $\mathcal{F}:\mathscr{H}\rightarrow(-\infty,+\infty]$ be a proper,
strongly lower semicontinuous function which is $\lambda$-convex and
$\lambda$-positive for some $\lambda>0$ . Then for every
$0<\tau<\frac{1}{2\lambda}$ and for every $y\in\mathscr{H}$ the problem

(3.1)
$\min\biggl{\\{}\mathcal{F}(x)+\frac{1}{2\tau}|x-y|_{\mathscr{H}}^{2}:\;x\in\mathscr{H}\biggr{\\}}$

admits a unique solution.

###### Proof.

We preliminarily notice that, since $\mathcal{F}$ is $\lambda$-convex and
strongly lower semicontinuous, then the function
$\mathcal{F}(\cdot)+\frac{1}{2\tau}|\cdot|^{2}_{\mathscr{H}}$ is strictly
convex and strongly lower semicontinuous and, in turn, weakly lower
semicontinuous. Clearly, this implies that also
$\mathcal{F}(\cdot)+\frac{1}{2\tau}|\cdot-y|^{2}_{\mathscr{H}}$ is weakly
lower semicontinuous. Moreover, by Remark 3.2, we have that $\mathcal{F}$ is
$\frac{1}{2\tau}$-coercive.

Since $\mathcal{F}$ is proper,

$0\leq\inf\biggl{\\{}\mathcal{F}(x)+\frac{1}{2\tau}|x-y|_{\mathscr{H}}^{2}:\;x\in\mathscr{H}\biggr{\\}}\leq
M\,,$

for some $M>0$ . Let $\\{x_{k}\\}_{k\in\mathbb{N}}\subset\mathscr{H}$ be a
sequence such that

(3.2)
$\lim_{k\rightarrow+\infty}\mathcal{F}(x_{k})+\frac{1}{2\tau}|x_{k}-y|_{\mathscr{H}}^{2}=\inf\biggl{\\{}\mathcal{F}(x)+\frac{1}{2\tau}|x-y|_{\mathscr{H}}^{2}:\;x\in\mathscr{H}\biggr{\\}}\,.$

By triangle inequality, for $k$ sufficiently large, we have

$\displaystyle 2M\geq$
$\displaystyle\mathcal{F}(x_{k})+\frac{1}{2\tau}|x_{k}-y|_{\mathscr{H}}^{2}\geq\mathcal{F}(x_{k})+\frac{1}{4\tau}|x_{k}|_{\mathscr{H}}^{2}-\frac{1}{2\tau}|y|_{\mathscr{H}}^{2}$

whence, in view of the $\frac{1}{2\tau}$-coercivity of the function
$\mathcal{F}$ , we deduce that, up to a subsequence,
$x_{k}\overset{\mathscr{H}}{\rightharpoonup}x_{\infty}$ for some
$x_{\infty}\in\mathscr{H}$ . Therefore, by (3.2) and by the weak lower
semicontinuity of the function
$\mathcal{F}(\cdot)+\frac{1}{2\tau}|\cdot-y|^{2}_{\mathscr{H}}$ , we obtain

$\inf\biggl{\\{}\mathcal{F}(x)+\frac{1}{2\tau}|x-y|_{\mathscr{H}}^{2}:\;x\in\mathscr{H}\biggr{\\}}=\lim_{k\rightarrow+\infty}\mathcal{F}(x_{k})+\frac{1}{2\tau}|x_{k}-y|_{\mathscr{H}}^{2}\\\
\geq\mathcal{F}(x_{\infty})+\frac{1}{2\tau}|x_{\infty}-y|_{\mathscr{H}}^{2}\,,$

i.e., that $x_{\infty}$ is a minimizer of the problem in (3.1).

Finally, the uniqueness of the solution is a consequence of the strict
convexity of the functional $\mathcal{F}(\cdot)+\frac{1}{2\tau}|\cdot-y|^{2}$
. ∎

For every function $\mathcal{F}:\mathscr{H}\rightarrow(-\infty,+\infty]$ we
denote by $D(\mathcal{F})$ the set of all $x\in\mathscr{H}$ such that
$\mathcal{F}(x)\in\mathbb{R}$.

######  Definition 3.4 (Fréchet subdifferential).

For $\mathcal{F}:\mathscr{H}\rightarrow(-\infty,+\infty]$ and $x\in
D(\mathcal{F})$, the Fréchet subdifferential of $\mathcal{F}$ at $x$ is
defined as

$\partial\mathcal{F}(x):=\biggl{\\{}v\in\mathscr{H}:\liminf_{y\to
x}\frac{\mathcal{F}(y)-\mathcal{F}(x)-\langle
v,y-x\rangle_{\mathscr{H}}}{|y-x|_{\mathscr{H}}}\geq 0\biggl{\\}}\,.$

######  Remark 3.5.

Whenever $\mathcal{F}$ is a $\lambda$-convex function it holds that

(3.3)
$\partial\mathcal{F}(x)=\biggl{\\{}v\in\mathscr{H}:\,\mathcal{F}(y)-\mathcal{F}(x)-\langle
v,y-x\rangle_{\mathscr{H}}\geq-\frac{\lambda}{2}\,|y-x|_{\mathscr{H}}^{2}\quad\text{for
every }y\in\mathscr{H}\biggr{\\}}\,.$

Indeed, for a convex function $\phi$, $v\in\partial\phi(x)$ if and only if
$\phi(y)-\phi(x)-\langle v,y-x\rangle_{\mathscr{H}}\geq 0$ for every
$y\in\mathscr{H}$, namely the Fréchet subdifferential coincides with the usual
subdifferential of convex analysis. Then, being $\mathcal{F}$ $\lambda$-convex
and since
$\partial\Big{(}\phi+\frac{\lambda}{2}|\cdot|_{\mathscr{H}}^{2}\Big{)}=\partial\phi+\lambda\,\cdot$,
it holds that $v\in\partial\mathcal{F}(x)$ if and only if

$\mathcal{F}(y)+\frac{\lambda}{2}|y|^{2}_{\mathscr{H}}-\mathcal{F}(x)-\frac{\lambda}{2}|x|^{2}_{\mathscr{H}}-\langle
v+\lambda x,y-x\rangle_{\mathscr{H}}\geq 0\quad\text{for every
}y\in\mathscr{H}\,,$

which coincides with the condition in (3.3) since
$|y-x|^{2}_{\mathscr{H}}=|y|^{2}_{\mathscr{H}}-|x|^{2}_{\mathscr{H}}-2\langle
x,y-x\rangle_{\mathscr{H}}$.

Let $\mathcal{F}:\mathscr{H}\rightarrow(-\infty,+\infty]$ be a proper,
strongly lower semicontinuous function which is $\lambda$-positive and
$\lambda$-convex, for some $\lambda>0$ , and let $x_{0}\in D(\mathcal{F})$ .
For every $0<\tau<\frac{1}{2\lambda}$, we denote by
$\\{x_{k}^{\tau}\\}_{k\in\mathbb{N}}$ the discrete-in-time evolution for
$\mathcal{F}$ with initial datum $x_{0}$ , defined by

(3.4) $x_{0}^{\tau}:=x_{0}\,,\qquad
x_{k+1}^{\tau}\in\arg\\!\min\biggl{\\{}\mathcal{F}(x)+\frac{1}{2\tau}|x-x_{k}^{\tau}|^{2}_{\mathscr{H}}\biggr{\\}}\quad\textrm{for
every }k\in\mathbb{N}\cup\\{0\\}\,.$

Since $x_{0}\in D(\mathcal{F})$ , then $x_{k}^{\tau}\in D(\mathcal{F})$ for
every $k\in\mathbb{N}$ . Furthermore, we define the piecewise-affine
interpolation $x^{\tau}:[0,+\infty)\rightarrow\mathscr{H}$ of
$\\{x^{\tau}_{k}\\}_{k\in\mathbb{N}}$ as

(3.5)
$x^{\tau}(t):=x_{k}^{\tau}+\frac{x_{k+1}^{\tau}-x_{k}^{\tau}}{\tau}(t-k\tau),\quad
t\in[k\tau,(k+1)\tau).$

######  Theorem 3.6.

Let $\mathcal{F}:\mathscr{H}\rightarrow(-\infty,+\infty]$ be a proper,
strongly lower semicontinuous function which is $\lambda$-convex and
$\lambda$-positive, for some $\lambda>0$ . Let moreover $x_{0}\in
D(\mathcal{F})$ . Then, there exists a unique solution $x\in
H^{1}([0,+\infty);\mathscr{H})$ to the following Cauchy problem

(3.6) $\begin{dcases}\dot{x}(t)\in-\partial\mathcal{F}(x(t))\quad\text{for
a.e.\ $t\in[0,+\infty)$},\\\ x(0)=x_{0}\,.\end{dcases}$

Moreover, for every $T>0$, $x^{\tau}\rightharpoonup x$ in
$H^{1}([0,T];\mathscr{H})$ , where $x^{\tau}$ is defined in (3.5) for
$0<\tau<\frac{1}{2\lambda}$ . Furthermore,

(3.7) $\displaystyle\|\dot{x}\|^{2}_{L^{2}((0,T);\mathscr{H})}\leq 4^{8\lambda
T+4}(|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2})\textrm{ for every
}T>0\,,$ (3.8) $\displaystyle|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}\leq
8\tau\,12^{8\lambda
t+4}\big{(}|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2}\big{)}\textrm{
for every }t\geq 0\,,\,\tau<\frac{1}{16\lambda}\,.$

###### Proof.

Uniqueness. Let $T>0$ and let $x_{1},x_{2}\in H^{1}([0,T];\mathscr{H})$
satisfy the Cauchy problem (3.6) up to time $T$. We first observe that

(3.9) $\langle
y_{1}-y_{2},v_{1}-v_{2}\rangle_{\mathscr{H}}\leq\lambda\,|y_{1}-y_{2}|_{\mathscr{H}}^{2}\quad\textrm{for
every
}y_{1},y_{2}\in\mathscr{H}\,,\,-v_{i}\in\partial\mathcal{F}(y_{i})\textrm{ for
}i=1,2\,.$

Indeed, by (3.3), we have

$\mathcal{F}(y)-\mathcal{F}(y_{1})+\langle
v_{1},y-y_{1}\rangle_{\mathscr{H}}\geq-\frac{\lambda}{2}\,|y-y_{1}|_{\mathscr{H}}^{2}\,,\quad
y\in\mathscr{H}\,,$

which, for $y=y_{2}$ implies

(3.10) $\mathcal{F}(y_{2})-\mathcal{F}(y_{1})+\langle
v_{1},y_{2}-y_{1}\rangle_{\mathscr{H}}\geq-\frac{\lambda}{2}\,|y_{2}-y_{1}|_{\mathscr{H}}^{2};$

analogously

(3.11) $\mathcal{F}(y_{1})-\mathcal{F}(y_{2})+\langle
v_{2},y_{1}-y_{2}\rangle_{\mathscr{H}}\geq-\frac{\lambda}{2}\,|y_{2}-y_{1}|_{\mathscr{H}}^{2}\,.$

Therefore, (3.9) follows by summing (3.10) and (3.11).

Finally, by formula (3.9) we have

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}|x_{1}(t)-x_{2}(t)|_{\mathscr{H}}^{2}=2\langle\dot{x}_{1}(t)-\dot{x}_{2}(t),x_{1}(t)-x_{2}(t)\rangle_{\mathscr{H}}\leq
2\lambda\,|x_{1}(t)-x_{2}(t)|_{\mathscr{H}}^{2}\quad\text{ for a.e.
$t\in[0,T]$,}\,$

which, by Gronwall’s Lemma, implies

$|x_{1}(t)-x_{2}(t)|_{\mathscr{H}}^{2}\leq|x_{0}-x_{0}|_{\mathscr{H}}e^{2\lambda
t}=0\quad\text{ for a.e. $t\in[0,T]$\,,}$

i.e., $x_{1}(t)=x_{2}(t)$ for a.e. $t\in[0,T]$. We notice that the solution is
in $C^{0,\frac{1}{2}}([0,T];\mathscr{H})$ by the Sobolev embedding of
$H^{1}([0,T];\mathscr{H})$ into $C^{0,\frac{1}{2}}([0,T];\mathscr{H})$, so
that $x_{1}(t)=x_{2}(t)$ when passing to the continuous representatives.
Existence. We first prove that for every $T>0$ the functions $x^{\tau}$
defined in (3.5) converge (as $\tau\to 0$) weakly in
$H^{1}([0,T];\mathscr{H})$ to some function $x\in H^{1}([0,T];\mathscr{H})$
and then we show that the limit $x$ satisfies (3.6) up to time $T$.

By (3.4) we have that

(3.12)
$\mathcal{F}(x_{k+1}^{\tau})+\frac{1}{2\tau}|x_{k+1}^{\tau}-x_{k}^{\tau}|_{\mathscr{H}}^{2}\leq\mathcal{F}(x_{k}^{\tau}),\quad\text{for
every $k\in\mathbb{N}$\,,}$

which together with the $\lambda$-positivity of $\mathcal{F}$ implies that

(3.13)
$\displaystyle\sum_{k=0}^{K}\frac{1}{\tau}\left|x^{\tau}_{k+1}-x^{\tau}_{k}\right|_{\mathscr{H}}^{2}\leq
2\sum_{k=0}^{K}\left(\mathcal{F}(x^{\tau}_{k})-\mathcal{F}(x^{\tau}_{k+1})\right)$
$\displaystyle=$ $\displaystyle
2(\mathcal{F}(x^{\tau}_{0})-\mathcal{F}(x^{\tau}_{K+1}))$ $\displaystyle=$
$\displaystyle
2\Big{(}\mathcal{F}(x^{\tau}_{0})+\frac{\lambda}{2}|x^{\tau}_{K+1}|_{\mathscr{H}}^{2}-\frac{\lambda}{2}|x^{\tau}_{K+1}|_{\mathscr{H}}^{2}-\mathcal{F}(x^{\tau}_{K+1})\Big{)}$
$\displaystyle\leq$ $\displaystyle
2\left(\mathcal{F}(x_{0})+\frac{\lambda}{2}|x^{\tau}_{K+1}|_{\mathscr{H}}^{2}\right)\qquad\textrm{for
every }K\in\mathbb{N}\,.$

Set $\hat{T}=\frac{1}{8\lambda}$ and let $0<\tau\leq\frac{1}{16\lambda}$ . We
set $\hat{K}:=\left\lceil\frac{\hat{T}}{\tau}\right\rceil$ ; by (3.13), we
have

(3.14)
$\begin{split}\int_{0}^{\hat{T}}|\dot{x}^{\tau}(t)|_{\mathscr{H}}^{2}\,\mathrm{d}t\leq&\sum_{k=0}^{\hat{K}}\frac{1}{\tau}\left|x^{\tau}_{k+1}-x^{\tau}_{k}\right|_{\mathscr{H}}^{2}\leq
2\left(\mathcal{F}(x_{0})+\frac{\lambda}{2}|x^{\tau}_{\hat{K}+1}|_{\mathscr{H}}^{2}\right)\,.\end{split}$

Moreover, by triangle and Jensen inequalities and using again (3.13), we get

$\displaystyle\frac{1}{2}|x_{\hat{K}+1}^{\tau}|_{\mathscr{H}}^{2}-|x_{0}|_{\mathscr{H}}^{2}\leq|x_{\hat{K}+1}^{\tau}-x_{0}|^{2}_{\mathscr{H}}$
$\displaystyle\leq$
$\displaystyle\tau(\hat{K}+1)\sum_{k=1}^{\hat{K}+1}\frac{1}{\tau}\left|x^{\tau}_{k}-x^{\tau}_{k-1}\right|_{\mathscr{H}}^{2}=\tau(\hat{K}+1)\sum_{k=0}^{\hat{K}}\frac{1}{\tau}\left|x^{\tau}_{k+1}-x^{\tau}_{k}\right|_{\mathscr{H}}^{2}$
$\displaystyle\leq$ $\displaystyle
2(\hat{T}+2\tau)\Big{(}\mathcal{F}(x_{0})+\frac{\lambda}{2}|x_{\hat{K}+1}^{\tau}|_{\mathscr{H}}^{2}\Big{)}\,,$

which, recalling that $0<2\tau\leq\frac{1}{8\lambda}=\hat{T}$ , implies that

(3.15)
$|x^{\tau}_{\hat{K}+1}|^{2}_{\mathscr{H}}\leq\frac{2}{\lambda}\mathcal{F}(x_{0})+4|x_{0}|^{2}_{\mathscr{H}}\,.$

By (3.14) and (3.15), we have that, for every $\tau$ small enough,

(3.16) $\|\dot{x}^{\tau}\|^{2}_{L^{2}((0,\hat{T});\mathscr{H})}\leq
4(\mathcal{F}(x_{0})+\lambda|x_{0}|_{\mathscr{H}}^{2})\,.$

Iterating the estimates in (3.15) and (3.16), using also that
$\mathcal{F}(x^{\tau}_{k})$ is not increasing with respect to $k$, we deduce
that for every $j\in\mathbb{N}$

$\begin{split}|x^{\tau}(j\hat{T})|_{\mathscr{H}}^{2}&\leq
4^{j}\Big{(}\frac{1}{\lambda}|\mathcal{F}(x_{0})|+|x_{0}|_{\mathscr{H}}^{2}\Big{)},\\\
\|\dot{x}^{\tau}\|^{2}_{L^{2}((0,j\hat{T});\mathscr{H})}&\leq
4^{j+3}(|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2})\,.\end{split}$

In particular, for every $T>0$, we have that

(3.17) $\|\dot{x}^{\tau}\|^{2}_{L^{2}((0,T);\mathscr{H})}\leq 4^{8\lambda
T+4}(|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2})\,.$

Therefore, for every $T>0$, $\|x^{\tau}\|_{H^{1}([0,T];\mathscr{H})}$ is
uniformly bounded and hence, up to a subsequence, $x^{\tau}\rightharpoonup x$
in $H^{1}([0,T];\mathscr{H})$ for some $x\in H^{1}([0,T];\mathscr{H})$ ; this,
in particular, implies the convergence in
$C^{0,\frac{1}{2}}([0,T];\mathscr{H})$ and hence that $x(0)=x_{0}$ . Passing
to the limit in (3.17) we readily get (3.7).

Now we aim at proving that $x$ solves (3.6) up to time $T$, for every $T>0$ ,
that is

(3.18) $\dot{x}(t)\in-\partial\mathcal{F}(x(t))\qquad\textrm{ for almost every
}t\in(0,T)\,.$

To this end, we define the piecewise-constant interpolation
$\tilde{x}^{\tau}:[0,+\infty)\to\mathscr{H}$ of
$\\{x_{k}^{\tau}\\}_{k\in\mathbb{N}}$ as

$\tilde{x}^{\tau}(t):=x_{k+1}^{\tau},\quad t\in[k\tau,(k+1)\tau)\,,$

and we notice that, by minimality, for $\tau$ small enough,

(3.19)
$\dot{x}^{\tau}(t)\in-\partial\mathcal{F}(\tilde{x}^{\tau}(t))\qquad\textrm{for
almost every }t\in[0,+\infty)\,.$

We claim that

(3.20) $\tilde{x}^{\tau}\overset{\mathscr{H}}{\rightarrow}x\quad\textrm{ in
}L^{2}((0,T);\mathscr{H})\,,\quad\textrm{ for every }T>0\,.$

Indeed, by triangle inequality, we have that

(3.21)
$\begin{split}\|\tilde{x}^{\tau}-x\|^{2}_{L^{2}((0,T);\mathscr{H})}&\leq
2\|x^{\tau}-x\|^{2}_{L^{2}((0,T);\mathscr{H})}+2\|\tilde{x}^{\tau}-x^{\tau}\|^{2}_{L^{2}((0,T);\mathscr{H})}\\\
&\leq
2\|x^{\tau}-x\|^{2}_{L^{2}((0,T);\mathscr{H})}+2\tau^{2}\|\dot{x}^{\tau}\|^{2}_{L^{2}((0,T);\mathscr{H})}\,,\end{split}$

where in the last inequality we have used that

$x^{\tau}(t)-\tilde{x}^{\tau}(t)=\frac{x_{k+1}^{\tau}-x_{k}^{\tau}}{\tau}(t-(k+1)\tau)=\dot{x}^{\tau}(t)(t-(k+1)\tau),\quad\textrm{for
every }t\in(k\tau,(k+1)\tau)\,.$

Therefore, by (3.21) and (3.7) we get

$\|\tilde{x}^{\tau}-x\|_{L^{2}((0,T);\mathscr{H})}^{2}\leq
2\|x^{\tau}-x\|^{2}_{L^{2}((0,T);\mathscr{H})}+2\tau^{2}4^{8\lambda
T+4}(|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2})\,,$

which, sending $\tau\to 0$ and recalling that $x^{\tau}\rightharpoonup x$ in
$H^{1}([0,T];\mathscr{H})$, implies (3.20).

With (3.20) on hand, we are in a position to prove (3.18). Let $t_{0}\in(0,T)$
be a Lebesgue point of the function $\dot{x}:[0,T)\to\mathscr{H}$. By (3.19),
we have that

(3.22)
$\mathcal{F}(y)\geq\mathcal{F}(\tilde{x}^{\tau}(t))-\langle\dot{x}^{\tau}(t),y-\tilde{x}^{\tau}(t)\rangle_{\mathscr{H}}-\frac{\lambda}{2}|y-\tilde{x}^{\tau}(t)|_{\mathscr{H}}^{2}\,\qquad\textrm{for
every }y\in\mathscr{H}\,.$

Let $y\in\mathscr{H}$ and $h>0$ ; by integrating (3.22) in the interval
$(t_{0},t_{0}+h)$ and dividing by $h$ , we obtain

$\mathcal{F}(y)\geq\frac{1}{h}\int_{t_{0}}^{t_{0}+h}\mathcal{F}(\tilde{x}^{\tau}(t))\,\mathrm{d}t-\frac{1}{h}\int_{t_{0}}^{t_{0}+h}\langle\dot{x}^{\tau}(t),y-\tilde{x}^{\tau}(t)\rangle_{\mathscr{H}}\,\mathrm{d}t-\frac{1}{h}\int_{t_{0}}^{t_{0}+h}\frac{\lambda}{2}|y-\tilde{x}^{\tau}(t)|_{\mathscr{H}}^{2}\,\mathrm{d}t\,,$

which, sending $\tau\to 0$ , and using the strong lower semicontinuity of
$\mathcal{F}$ , the weak $L^{2}$-convergence of $\dot{x}^{\tau}$ to $\dot{x}$
, and (3.20), yields

$\mathcal{F}(y)\geq\frac{1}{h}\int_{t_{0}}^{t_{0}+h}\mathcal{F}({x}(t))\,\mathrm{d}t-\frac{1}{h}\int_{t_{0}}^{t_{0}+h}\langle\dot{x}(t),y-{x}(t)\rangle_{\mathscr{H}}\,\mathrm{d}t-\frac{1}{h}\int_{t_{0}}^{t_{0}+h}\frac{\lambda}{2}|y-{x}(t)|_{\mathscr{H}}^{2}\,\mathrm{d}t\,.$

Now, since $x\in C^{0,\frac{1}{2}}([0,T];\mathscr{H})$ and since $t_{0}$ is a
Lebesgue point for $\dot{x}$ , sending $h\to 0$ in the formula above, and
using again that $\mathcal{F}$ is strongly lower semicontinuous, by the
arbitrariness of $y$ , we get (3.18).

Finally, we prove that (3.8) holds true. Let
$\eta^{\tau}:[0,+\infty)\to(0,\tau]$ be the function defined by
$\eta^{\tau}(t)=(k+1)\tau-t$ for every $t\in[k\tau,(k+1)\tau)$ . By (3.19),

$\dot{x}^{\tau}(t)\in-\partial\mathcal{F}(x^{\tau}(t+\eta^{\tau}(t)))\qquad\textrm{
for every }t>0\,,$

which, using (3.6) and (3.9), yields

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}$
$\displaystyle=$ $\displaystyle 2\langle
x(t)-x^{\tau}(t),\dot{x}(t)-\dot{x}^{\tau}(t)\rangle_{\mathscr{H}}$
$\displaystyle=$ $\displaystyle 2\langle
x(t)-x^{\tau}(t+\eta^{\tau}(t)),\dot{x}(t)-\dot{x}^{\tau}(t)\rangle_{\mathscr{H}}$
$\displaystyle+2\langle
x^{\tau}(t+\eta^{\tau}(t))-x^{\tau}(t),\dot{x}(t)-\dot{x}^{\tau}(t)\rangle_{\mathscr{H}}$
$\displaystyle\leq$ $\displaystyle
4\lambda|x(t)-x^{\tau}(t+\eta^{\tau}(t))|^{2}_{\mathscr{H}}+2|x^{\tau}(t+\eta^{\tau}(t))-x^{\tau}(t)|_{\mathscr{H}}|\dot{x}(t)-\dot{x}^{\tau}(t)|_{\mathscr{H}}$
$\displaystyle\leq$ $\displaystyle
8\lambda|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}+8\lambda|x^{\tau}(t+\eta^{\tau}(t))-x^{\tau}(t)|^{2}_{\mathscr{H}}$
$\displaystyle+2|x^{\tau}(t+\eta^{\tau}(t))-x^{\tau}(t)|_{\mathscr{H}}(|\dot{x}(t)|_{\mathscr{H}}+|\dot{x}^{\tau}(t)|_{\mathscr{H}})$
$\displaystyle\leq$ $\displaystyle
8\lambda|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}+\tau(8\lambda\tau+3)(|\dot{x}^{\tau}(t)|^{2}_{\mathscr{H}}+|\dot{x}(t)|_{\mathscr{H}}^{2})\,,$
$\displaystyle\leq$ $\displaystyle
8\lambda|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}+4\tau(|\dot{x}^{\tau}(t)|^{2}_{\mathscr{H}}+|\dot{x}(t)|_{\mathscr{H}}^{2})\,,$

where in the last inequality we have used that $\tau\leq\frac{1}{16\lambda}$
(recall also
$|x^{\tau}(t+\eta^{\tau}(t))-x^{\tau}(t)|_{\mathscr{H}}=\eta^{\tau}(t)|\dot{x}^{\tau}(t)|_{\mathscr{H}}\leq\tau|\dot{x}^{\tau}(t)|_{\mathscr{H}}$)
. By integrating the equation above and using (3.7) and (3.17), we get

$\displaystyle|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}\leq$
$\displaystyle\int_{0}^{t}8\lambda|x(s)-x^{\tau}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s+4\tau\int_{0}^{t}(|\dot{x}^{\tau}(s)|^{2}_{\mathscr{H}}+|\dot{x}(s)|_{\mathscr{H}}^{2})\,\mathrm{d}s$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}8\lambda|x(s)-x^{\tau}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s+8\tau
4^{8\lambda t+4}(|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2})\,.$

Setting $\alpha(s):=8\tau 4^{8\lambda
s+4}(|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2})$ and
$\beta(s):=8\lambda$ for every $s>0$ , we have

$\displaystyle|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}\leq$
$\displaystyle\int_{0}^{t}\beta(s)|x(s)-x^{\tau}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s+\alpha(t)\,.$

Noticing that $\alpha$ is non-decreasing, by Gronwall Lemma we get

$\displaystyle|x(t)-x^{\tau}(t)|^{2}_{\mathscr{H}}\leq$
$\displaystyle\alpha(t)e^{\int_{0}^{t}\beta(s)\,\mathrm{d}s}=8\tau 4^{8\lambda
t+4}(|\mathcal{F}(x_{0})|+\lambda|x_{0}|_{\mathscr{H}}^{2})e^{8\lambda t}\,,$

thus providing (3.8). ∎

For every vectorial space $\mathscr{V}$ we denote by $\mathscr{V}^{*}$ the
algebraic dual space of $\mathscr{V}$ and by $\mathscr{V}^{\prime}$ the
topological dual space of $\mathscr{V}$.

######  Proposition 3.7.

Let $\mathcal{F}:\mathscr{H}\to(-\infty,+\infty]$ be a proper lower
semicontinuous function which is $\lambda$-convex, for some $\lambda>0$ and
let $x\in D(\mathcal{F})$ . Let $\hat{\mathscr{H}}$ be a dense subspace of
$\mathscr{H}$ . If there exists $T\in(\hat{\mathscr{H}})^{*}$ such that

(3.23) $\lim_{t\to
0}\frac{\mathcal{F}(x+t\varphi)-\mathcal{F}(x)}{t}=T(\varphi)\qquad\textrm{for
every }\varphi\in\hat{\mathscr{H}}\,,$

then, either $\partial\mathcal{F}(x)=\emptyset$ or
$\partial\mathcal{F}(x)=\\{v\\}$ , where $v$ is the (unique) element in
$\mathscr{H}$ satisfying $T(\varphi)=\langle v,\varphi\rangle_{\mathscr{H}}$
for every $\varphi\in\hat{\mathscr{H}}$ . In particular,
$T\in(\hat{\mathscr{H}})^{\prime}$ and $v$ is its unique continuous extension
to $\mathscr{H}^{\prime}$.

###### Proof.

Since $\hat{\mathscr{H}}$ is dense in $\mathscr{H}$, in order to get the claim
it is enough to prove that for every $v\in\partial\mathcal{F}(x)$

(3.24) $\langle v,\varphi\rangle_{\mathscr{H}}=T(\varphi)\qquad\textrm{for
every }\varphi\in\hat{\mathscr{H}}\,.$

To this purpose, we notice that every $v\in\partial\mathcal{F}(x)$ satisfies

$\mathcal{F}(x+t\varphi)-\mathcal{F}(x)-t\langle
v,\varphi\rangle_{\mathscr{H}}\geq-t^{2}\frac{\lambda}{2}|\varphi|^{2}_{\mathscr{H}}\qquad\textrm{for
every }\varphi\in\hat{\mathscr{H}},\,t\in\mathbb{R}\,,$

which, dividing by $t$ , yields

$\lim_{t\to 0}\frac{\mathcal{F}(x+t\varphi)-\mathcal{F}(x)}{t}=\langle
v,\varphi\rangle_{\mathscr{H}}\qquad\textrm{for every
}\varphi\in\hat{\mathscr{H}}\,;$

therefore, in view of (3.23), we get (3.24). ∎

The following Theorem provides a convergence result for gradient-flows
associated to a $\Gamma$-converging sequence of functions satisfying the
assumptions of Theorem 3.6.

######  Theorem 3.8.

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
$x_{0}^{n}\in D(\mathcal{F}^{n})$ for every $n\in\mathbb{N}$ ,
$S:=\sup_{n\in\mathbb{N}}\mathcal{F}^{n}(x_{0}^{n})<+\infty$ 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 $\mathcal{F}^{n}$ are equicoercive and $\Gamma$-converge to some
proper function $\mathcal{F}^{\infty}$ with respect to the weak
$\mathscr{H}$-convergence (as $n\to+\infty$) . 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.

Then, $x_{0}^{\infty}\in D(\mathcal{F}^{\infty})$ and, for every $T>0$ , the
solutions $x^{n}$ to the Cauchy problem

(3.25)
$\begin{cases}\dot{x}(t)\in-\partial\mathcal{F}^{n}(x(t))\qquad\text{for a.e.\
}t\in(0,T)\,,\\\ x(0)=x_{0}^{n}\end{cases}$

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

(3.26)
$\begin{cases}\dot{x}(t)\in-\partial\mathcal{F}^{\infty}(x(t))\qquad\text{for
a.e.\ }t\in(0,T)\,,\\\ x(0)=x^{\infty}_{0}\,.\end{cases}$

Furthermore, if

(3.27)
$\lim_{n\to+\infty}\mathcal{F}^{n}(x^{n}_{0})=\mathcal{F}^{\infty}(x^{\infty}_{0})\,,$

then, we have that

(3.28) $x^{n}\to x^{\infty}\qquad\textrm{(strongly) in
}H^{1}([0,T];\mathscr{H})\quad\textrm{ for every }T>0\,,$ (3.29)
$x^{n}(t)\overset{\mathscr{H}}{\to}x^{\infty}(t)\quad\textrm{and}\quad\mathcal{F}^{n}(x^{n}(t))\to\mathcal{F}^{\infty}(x^{\infty}(t))\qquad\textrm{for
every }t\geq 0\,.$

###### Proof.

We preliminarily notice that, if either (a) or (b) is satisfied, then the
function $\mathcal{F}^{\infty}$ is strongly lower semicontinuous,
$\lambda$-convex and $\lambda$-positive and $x^{\infty}_{0}\in
D(\mathcal{F}^{\infty})$ . Moreover, by Theorem 3.6, for every
$n\in\mathbb{N}$ there exists a unique solution $x^{n}$ to (3.25).

Let $0<\tau<\frac{1}{2\lambda}$ and let
$\\{x^{\infty,\tau}_{k}\\}_{k\in\mathbb{N}}$ denote the discrete-in-time
evolution in (3.4) for $x_{0}:=x_{0}^{\infty}$ and
$\mathcal{F}:=\mathcal{F}^{\infty}$ . Analogously, for every $n\in\mathbb{N}$
, let $\\{x^{n,\tau}_{k}\\}_{k\in\mathbb{N}}$ denote the discrete-in-time
evolution in (3.4) for $x_{0}:=x_{0}^{n}$ and $\mathcal{F}:=\mathcal{F}^{n}$ .
By Proposition 3.3, $\\{x^{\infty,\tau}_{k}\\}_{k\in\mathbb{N}}$ and
$\\{x^{n,\tau}_{k}\\}_{k\in\mathbb{N}}$ are uniquely determined. Furthermore,
for every $k\in\mathbb{N}$ we set

$\displaystyle\mathcal{I}^{n,\tau}_{k}(\cdot)$ $\displaystyle:=$
$\displaystyle\mathcal{F}^{n}(\cdot)+\frac{1}{2\tau}|\cdot-x^{n,\tau}_{k-1}|_{\mathscr{H}}^{2}\qquad\textrm{for
every }n\in\mathbb{N}\,,$ $\displaystyle\mathcal{I}^{\infty,\tau}_{k}(\cdot)$
$\displaystyle:=$
$\displaystyle\mathcal{F}^{\infty}(\cdot)+\frac{1}{2\tau}|\cdot-x^{\infty,\tau}_{k-1}|_{\mathscr{H}}^{2}\,.$

We first show that, if either (a) or (b) is satisfied, then for every
$k\in\mathbb{N}$

(3.30)
$\mathcal{F}^{n}(x^{n,\tau}_{k})\to\mathcal{F}^{\infty}(x^{\infty,\tau}_{k})\quad\textrm{and}\quad|x^{n,\tau}_{k}-x^{\infty,\tau}_{k}|_{\mathscr{H}}\to
0\qquad\textrm{as }n\to+\infty\,.$

By finite induction, it is enough to show (3.30) for $k=1$ . We distinguish
the two cases in which either (a) or (b) holds true.

Assume first that (a) holds true. By the assumptions on $x^{n}_{0}$ , we have
that

$\mathcal{I}^{n,\tau}_{1}(x^{n,\tau}_{1})\leq\mathcal{F}^{n}(x^{n}_{0})\leq
S\,,$

whence, using that for $\frac{1}{2\tau}>\lambda$ the functions
$\mathcal{I}^{n,\tau}_{1}(\cdot)$ are equicoercive, we deduce that, up to a
subsequence, $x^{n,\tau}_{1}\overset{\mathscr{H}}{\rightharpoonup}y_{1}$ for
some $y_{1}\in\mathscr{H}$ . Moreover, since
$|x_{0}^{n}-x_{0}^{\infty}|_{\mathscr{H}}\to 0$ as $n\to+\infty$ and since the
functions $\mathcal{F}^{n}$ $\Gamma$-converge to the function
$\mathcal{F}^{\infty}$ with respect to the weak $\mathscr{H}$-convergence, we
have that

(3.31)
$\mathcal{I}^{\infty,\tau}_{1}(y_{1})\leq\liminf_{n\to+\infty}\mathcal{F}^{n}(x^{n,\tau}_{1})+\frac{1}{2\tau}\liminf_{n\to+\infty}|x^{n,\tau}_{1}-x^{n}_{0}|_{\mathscr{H}}^{2}\leq\liminf_{n\to+\infty}\mathcal{I}^{n,\tau}_{1}(x^{n,\tau}_{1})\,.$

Furthermore, since the $\Gamma$-limsup inequality is satisfied with respect to
the strong $\mathscr{H}$-convergence, there exists
$\\{\bar{x}^{n,\tau}_{1}\\}_{n\in\mathbb{N}}\subset\mathscr{H}$ such that

(3.32)
$\bar{x}^{n,\tau}_{1}\overset{\mathscr{H}}{\to}x^{\infty,\tau}_{1}\quad\textrm{
and
}\quad\mathcal{F}^{n}(\bar{x}^{n,\tau}_{1})\to\mathcal{F}(x^{\infty,\tau}_{1})\,,$

where $x^{\infty,\tau}_{1}$ is the unique solution to the problem (3.4) with
$\mathcal{F}=\mathcal{F}^{\infty}$ and $k=1$ . Therefore, by (3.31) and
(3.32), we get

$\displaystyle\mathcal{I}^{\infty,\tau}_{1}(y_{1})\leq$
$\displaystyle\liminf_{n\to+\infty}\mathcal{F}^{n}(x^{n,\tau}_{1})+\frac{1}{2\tau}\liminf_{n\to+\infty}|x^{n,\tau}_{1}-x_{0}^{n}|_{\mathscr{H}}^{2}$
$\displaystyle\leq$
$\displaystyle\liminf_{n\to+\infty}\mathcal{I}^{n,\tau}_{1}(x^{n,\tau}_{1})\leq\limsup_{n\to+\infty}\mathcal{I}^{n,\tau}_{1}(x^{n,\tau}_{1})$
$\displaystyle\leq$
$\displaystyle\lim_{n\to+\infty}\mathcal{I}^{n,\tau}_{1}(\bar{x}^{n,\tau}_{1})=\mathcal{I}^{\infty,\tau}_{1}(x^{\infty,\tau}_{1})\,,$

whence, by the minimality of $x^{\infty,\tau}_{1}$ , we deduce that all the
inequalities above are in fact equalities and, in particular, that $y_{1}$ is
a minimizer of $\mathcal{I}^{\infty,\tau}_{1}$; in view of the uniqueness of
the minimizer of $\mathcal{I}^{\infty,\tau}_{1}$, we deduce that
$y_{1}=x^{\infty,\tau}_{1}$ . By Urysohn Lemma, this implies that the whole
sequence $\\{x^{n,\tau}_{1}\\}_{n\in\mathbb{N}}$ weakly converges to
$x^{\infty,\tau}_{1}$ . Moreover, using that

$\mathcal{F}^{\infty}(x^{\infty,\tau}_{1})+\frac{1}{2\tau}|x^{\infty,\tau}_{1}-x^{\infty}_{0}|^{2}_{\mathscr{H}}=\liminf_{n\to+\infty}\mathcal{F}^{n}(x^{n,\tau}_{1})+\frac{1}{2\tau}\liminf_{n\to+\infty}|x^{n,\tau}_{1}-x_{0}^{n}|_{\mathscr{H}}^{2}\,,$

since

$\mathcal{F}^{\infty}(x^{\infty,\tau}_{1})\leq\liminf_{n\to+\infty}\mathcal{F}^{n}(x^{n,\tau}_{1})\quad\textrm{
and
}\quad\frac{1}{2\tau}|x^{\infty,\tau}_{1}-x^{\infty}_{0}|^{2}_{\mathscr{H}}\leq\frac{1}{2\tau}\liminf_{n\to+\infty}|x^{n,\tau}_{1}-x_{0}^{n}|_{\mathscr{H}}^{2}\,,$

we deduce that

$\mathcal{F}^{\infty}(x^{\infty,\tau}_{1})=\lim_{n\to+\infty}\mathcal{F}^{n}(x^{n,\tau}_{1})\quad\textrm{
and
}\quad|x^{\infty,\tau}_{1}-x^{\infty}_{0}|_{\mathscr{H}}=\lim_{n\to+\infty}|x^{n,\tau}_{1}-x^{n}_{0}|_{\mathscr{H}}\,,$

which implies (3.30) (for $k=1$ and then for all $k\in\mathbb{N}$).

Assume now that (b) holds true. As above (recall $\lambda<\frac{1}{2\tau}$) we
have that

$\mathcal{F}^{n}(x^{n,\tau}_{1})+\lambda|x^{n,\tau}_{1}-x^{n}_{0}|^{2}_{\mathscr{H}}\leq\mathcal{I}^{n,\tau}_{1}(x^{n,\tau}_{1})\leq
S\,,$

whence, by the strong compactness property of the functions
$\mathcal{F}^{n}(\cdot)+\frac{\lambda}{2}|\cdot|^{2}_{\mathscr{H}}$ we deduce
that, up to a subsequence, $x^{n,\tau}_{1}\overset{\mathscr{H}}{\to}y_{1}$ for
some $y_{1}\in\mathscr{H}$ . Moreover, since
$|x_{0}^{n}-x_{0}^{\infty}|_{\mathscr{H}}\to 0$ as $n\to+\infty$ , we have
that the functionals $\mathcal{I}^{n,\tau}_{1}$ $\Gamma$-converge with respect
to the strong-$\mathscr{H}$ convergence to the functional
$\mathcal{I}^{\infty,\tau}_{1}$ . By the fundamental theorem of
$\Gamma$-convergence and by the uniqueness of the minimizer of the problem
(3.4) with $\mathcal{F}=\mathcal{F}^{\infty}$ and $k=1$ , we get that
$y_{1}=x^{\infty,\tau}_{1}$ , that the whole sequence
$\\{x^{n,\tau}_{1}\\}_{n\in\mathbb{N}}$ strongly converges to
$x^{\infty,\tau}_{1}$ , and that (3.30) is satisfied for $k=1$ . This
concludes the proof of (3.30) for both the cases (a) and (b).

Now we show that for every $T>0$ , $x^{n}\rightharpoonup x^{\infty}$ in
$H^{1}([0,T];\mathscr{H})$ , where $x^{\infty}$ is the unique solution to
(3.26) . To this end, we first notice that, for $n$ large enough,
$|\mathcal{F}^{n}(x_{0}^{n})|\leq S+2\lambda|x_{0}|^{2}$, so that
$|\mathcal{F}^{n}(x_{0}^{n})|+\lambda|x_{0}^{n}|^{2}\leq
S+4\lambda|x_{0}|^{2}$; therefore, by (3.7)

$\|\dot{x}^{n}\|^{2}_{L^{2}((0,T);\mathscr{H})}\leq 4^{8\lambda
T+4}(S+4\lambda|x_{0}|_{\mathscr{H}}^{2})\,,$

so that, up to a subsequence, $x^{n}\rightharpoonup\bar{x}$ in
$H^{1}([0,T];\mathscr{H})$ , for some $\bar{x}\in H^{1}([0,T];\mathscr{H})$ .
Now we show that $\bar{x}=x^{\infty}$ .

For every $0<\tau<\frac{1}{2\lambda}$ , let $x^{\infty,\tau}$ and $x^{n,\tau}$
($n\in\mathbb{N}$) denote the piecewise affine interpolations defined in
(3.5), of $\\{x^{\infty,\tau}_{k}\\}_{k\in\mathbb{N}}$ and
$\\{x^{n,\tau}_{k}\\}_{k\in\mathbb{N}}$, respectively. By (3.30), we have that

(3.33)
$\lim_{n\to+\infty}|x^{n,\tau}(t)-x^{\infty,\tau}(t)|_{\mathscr{H}}=0\qquad\textrm{for
every }t>0\,,0<\tau<\frac{1}{2\lambda}\,.$

Let $t>0$ . For every $0<\tau<\frac{1}{2\lambda}$ , by triangle inequality and
by (3.8) we have that

(3.34) $\displaystyle|x^{n}(t)-x^{\infty}(t)|_{\mathscr{H}}\leq$
$\displaystyle|x^{n}(t)-x^{n,\tau}(t)|_{\mathscr{H}}+|x^{n,\tau}(t)-x^{\infty,\tau}(t)|_{\mathscr{H}}+|x^{\infty,\tau}(t)-x^{\infty}(t)|_{\mathscr{H}}$
$\displaystyle\leq$ $\displaystyle 16\tau 12^{8\lambda
t+4}\big{(}S+4\lambda|x_{0}|_{\mathscr{H}}^{2}\big{)}+|x^{n,\tau}(t)-x^{\infty,\tau}(t)|_{\mathscr{H}}\,;$

therefore, sending first $n\to+\infty$ and then $\tau\to 0$ in (3.34) and
using (3.33), we get that $x^{n}(t)\overset{\mathscr{H}}{\to}x^{\infty}(t)$ as
$n\to+\infty$ . By the uniqueness of the limit we deduce that
$\bar{x}=x^{\infty}$ and that the whole sequence
$\\{x^{n}\\}_{n\in\mathbb{N}}$ weakly converges in $H^{1}([0,T];\mathscr{H})$
to $x^{\infty}$ . Finally, we prove that (3.28) and (3.29) hold true. By
(3.34), the first part of (3.29) is satisfied. Moreover, by [27, formula
(1.10)] (notice that, as observed in [27], the formula applies also for
$\lambda$-convex energies), we have that, for every $t>0$ ,

(3.35)
$\displaystyle\mathcal{F}^{n}(x^{n}_{0}(t))-\mathcal{F}^{n}(x^{n}(t))=\frac{1}{2}\int_{0}^{t}|\dot{x}^{n}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s\qquad\textrm{for
every }n\in\mathbb{N}\,,$
$\displaystyle\mathcal{F}^{\infty}(x^{\infty}_{0}(t))-\mathcal{F}^{\infty}(x^{\infty}(t))=\frac{1}{2}\int_{0}^{t}|\dot{x}^{\infty}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s\,,$

which, using (3.27), the $\Gamma$-liminf inequality (that holds true in both
the cases (a) and (b)) and the weak $H^{1}$-convergence of $x^{n}$ to
$x^{\infty}$, implies

$\displaystyle\mathcal{F}^{\infty}(x^{\infty}_{0}(t))-\liminf_{n\to+\infty}\mathcal{F}^{n}(x^{n}(t))\leq\mathcal{F}^{\infty}(x^{\infty}_{0}(t))-\mathcal{F}^{\infty}(x^{\infty}(t))=\frac{1}{2}\int_{0}^{t}|\dot{x}^{\infty}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s$
$\displaystyle\leq$
$\displaystyle\liminf_{n\to+\infty}\frac{1}{2}\int_{0}^{t}|\dot{x}^{n}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s\leq\limsup_{n\to+\infty}\frac{1}{2}\int_{0}^{t}|\dot{x}^{n}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s$
$\displaystyle\leq$
$\displaystyle\limsup_{n\to+\infty}\mathcal{F}^{n}(x^{n}_{0}(t))-\liminf_{n\to+\infty}\mathcal{F}^{n}(x^{n}(t))=\mathcal{F}^{\infty}(x^{\infty}_{0}(t))-\liminf_{n\to+\infty}\mathcal{F}^{n}(x^{n}(t))\,.$

Therefore, all the inequalities above are actually equalities; in particular,

(3.36)
$\frac{1}{2}\int_{0}^{t}|\dot{x}^{\infty}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s=\lim_{n\to+\infty}\frac{1}{2}\int_{0}^{t}|\dot{x}^{n}(s)|^{2}_{\mathscr{H}}\,\mathrm{d}s\,,$

which, together (3.27) and (3.35), yields

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

thus obtaining also the second part of (3.29). Finally, by (3.36), we obtain
also (3.28), thus concluding the proof of the theorem. ∎

##  4\. Convergence of the $s$-fractional heat flows

This section is devoted to the proof of the stability of the $s$-fractional
heat flows as $s\to 0^{+}$ and $s\to 1^{-}$. In the first part, we define the
$s$-fractional laplacian for $s\in(0,1)$ and for $s=0$ . The second part
contains the convergence theorems, which are the main results of the paper. In
this section we denote by $v_{t}$ the partial time derivative of a function
$v$ .

###  4.1. The $s$-fractional laplacian for $s\in(0,1)$ and for $s=0$

For every $s\in(0,1)$ and for every
$\psi\in\mathrm{C}_{\mathrm{c}}^{\infty}(\mathbb{R}^{d})$ the $s$-fractional
laplacian of $\psi$ is defined by

$(-\Delta)^{s}\psi(x):=\int_{\mathbb{R}^{d}}\frac{2\psi(x)-\psi(x+z)-\psi(x-z)}{|z|^{d+2s}}\,\mathrm{d}z,\qquad
x\in\mathbb{R}^{d}\,.$

In [17, Lemma 3.2] it is proven that the above integral is finite, that
$(-\Delta)^{s}\psi\in L^{\infty}(\mathbb{R}^{d})$ , and that

(4.1) $(-\Delta)^{s}\psi(x)=2\lim_{r\to 0^{+}}\int_{\mathbb{R}^{d}\setminus
B_{r}(0)}\frac{\psi(x)-\psi(x+z)}{|z|^{d+2s}}\,\mathrm{d}z\,.$

For every $u\in\mathcal{H}^{s}_{0}(\Omega)$ we define the $s$-fractional
laplacian of $u$ by duality as

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

Here and below $\langle\cdot,\cdot\rangle$ denotes the standard scalar product
in $L^{2}$. Clearly, the $s$-fractional laplacian is nothing but the first
variation of the squared Gagliardo $s$-norm, as shown below.

######  Proposition 4.1.

Let $s\in(0,1)$ . For every $u\in\mathcal{H}^{s}_{0}(\Omega)$ and for every
$\varphi\in\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$ we have

(4.3) $\lim_{t\to
0}\frac{F^{s}(u+t\varphi)-F^{s}(u)}{t}=\langle(-\Delta)^{s}u,\varphi\rangle.$

###### Proof.

We have

$\displaystyle\lim_{t\to 0}\frac{F^{s}(u+t\varphi)-F^{s}(u)}{t}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{(\tilde{u}(x)-\tilde{u}(y))(\tilde{\varphi}(x)-\tilde{\varphi}(y))}{|x-y|^{d+2s}}\,\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=$ $\displaystyle\int_{\Omega}u(x)\lim_{r\to
0^{+}}\int_{\mathbb{R}^{d}\setminus
B_{r}(0)}\frac{\tilde{\varphi}(x)-\tilde{\varphi}(x+z)}{|z|^{d+2s}}\,\mathrm{d}z\,\mathrm{d}x$
$\displaystyle+\int_{\Omega}u(y)\lim_{r\to 0^{+}}\int_{\mathbb{R}^{d}\setminus
B_{r}(0)}\frac{\tilde{\varphi}(y)-\tilde{\varphi}(y-z)}{|z|^{d+2s}}\,\mathrm{d}z\,\mathrm{d}y$
$\displaystyle=$ $\displaystyle\langle
u,(-\Delta)^{s}\tilde{\varphi}\rangle=\langle(-\Delta)^{s}u,\varphi\rangle,$

where we have used the change of variable $z=y-x$ , (4.1) and (4.2). ∎

For every $\psi\in\mathrm{C}^{\infty}_{\mathrm{c}}(\mathbb{R}^{d})$ we define
the $0$-fractional laplacian of $\psi$ as

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

We notice that $(-\Delta)^{0}\psi$ is well-defined for every
$\psi\in\mathrm{C}^{\infty}_{\mathrm{c}}(\mathbb{R}^{d})$ since

$\int_{B_{1}}\frac{|2\psi(x)-\psi(x+z)-\psi(x-z)|}{|z|^{d}}\,\mathrm{d}z\leq
2\int_{B_{1}}\frac{|\psi(x+z)-\psi(x)|}{|z|^{d}}\,\mathrm{d}z\\\ \leq
C[\psi]_{0,1}\,.$

and

$\displaystyle\int_{\mathbb{R}^{d}\setminus\overline{B}_{1}}\frac{|\psi(x+z)|}{|z|^{d}}\,\mathrm{d}z\leq\|\psi\|_{L^{1}}\,.$

######  Remark 4.2.

In [11] the notion of logarithmic laplacian $L_{\Delta}$ has been introduced
as follows

$\mathrm{L}_{\Delta}\psi(x):=c_{d,1}(-\Delta)^{0}\psi(x)+c_{d,2}\psi(x)\,,$

where $c_{d,1}$ and $c_{d,2}$ are specific constant depending only on the
dimension $d$ . Such a logarithmic laplacian would correspond to renormalizing
the Gagliardo $s$-seminorm of $\psi$ by removing all but a finite amount of
the blowing up quantity $\frac{\|\psi\|^{2}_{L^{2}}}{s}$ .

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

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

Clearly, the $0$-fractional laplacian is the first variation of the functional
$\hat{F}^{0}$ , as shown in the following result (we recall that the functions
$G^{0}_{1}$ and $J^{0}_{1}$ have been introduced in (1.5) and (1.6),
respectively).

######  Proposition 4.3.

For every $u\in\mathcal{H}^{0}_{0}(\Omega)$ and for every
$\varphi\in\mathrm{C}^{\infty}_{\mathrm{c}}(\Omega)$ we have

(4.5) $\displaystyle\lim_{t\to 0}\frac{G^{0}_{1}(u+t\varphi)-G^{0}_{1}(u)}{t}$
$\displaystyle=$
$\displaystyle\Big{\langle}u,\int_{B_{1}}\frac{2\tilde{\varphi}(x)-\tilde{\varphi}(x+z)-\tilde{\varphi}(x-z)}{|z|^{d}}\,\mathrm{d}z\Big{\rangle}$
(4.6) $\displaystyle\lim_{t\to 0}\frac{J^{0}_{1}(u+t\varphi)-J^{0}_{1}(u)}{t}$
$\displaystyle=$
$\displaystyle\Big{\langle}u,-2\int_{\mathbb{R}^{d}\setminus\overline{B}_{1}}\frac{\tilde{\varphi}(x+z)}{|z|^{d}}\,\mathrm{d}z\Big{\rangle}\,,$

so that

(4.7) $\lim_{t\to
0}\frac{\hat{F}^{0}(u+t\varphi)-\hat{F}^{0}(u)}{t}=\langle(-\Delta)^{0}u,\varphi\rangle\,.$

###### Proof.

Fix $u\in\mathcal{H}^{0}_{0}(\Omega)$ and
$\varphi\in\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$ . Then, using the change
of variable $z=y-x$ , we have

$\displaystyle\lim_{t\to
0}\frac{G^{0}_{1}(u+t\varphi)-G^{0}_{1}(u)}{t}=\iint_{\mathcal{B}_{1}}\frac{(\tilde{u}(x)-\tilde{u}(y))(\tilde{\varphi}(x)-\tilde{\varphi}(y))}{|x-y|^{d}}\,\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega}u(x)\int_{B_{1}}\frac{\tilde{\varphi}(x)-\tilde{\varphi}(x+z)}{|z|^{d}}\,\mathrm{d}z\,\mathrm{d}x+\int_{\Omega}u(y)\int_{B_{1}}\frac{\tilde{\varphi}(y)-\tilde{\varphi}(y-z)}{|z|^{d}}\,\mathrm{d}z\,\mathrm{d}y$
$\displaystyle=$
$\displaystyle\Big{\langle}u,\int_{B_{1}}\frac{2\tilde{\varphi}(x)-\tilde{\varphi}(x+z)-\tilde{\varphi}(x-z)}{|z|^{d}}\,\mathrm{d}z\Big{\rangle}\,,$

i.e., (4.5). Moreover, using again the change of variable $z=y-x$, we obtain

$\displaystyle\lim_{t\to
0}\frac{J^{0}_{1}(u+t\varphi)-J^{0}_{1}(u)}{t}=-2\iint_{\mathbb{R}^{2d}\setminus\overline{\mathcal{B}}_{1}}\frac{\tilde{u}(x)\tilde{\varphi}(y)}{|x-y|^{d}}\,\mathrm{d}y\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega}u(x)\Big{(}-2\int_{\mathbb{R}^{d}\setminus\overline{B}_{1}(x)}\frac{\tilde{\varphi}(y)}{|x-y|^{d}}\,\mathrm{d}y\Big{)}\,\mathrm{d}x\,,$
$\displaystyle=$
$\displaystyle\Big{\langle}u,-2\int_{\mathbb{R}^{d}\setminus\overline{B}_{1}}\frac{\tilde{\varphi}(x+z)}{|z|^{d}}\,\mathrm{d}z\Big{\rangle}\,,$

namely, (4.6). Finally, (4.7) follows from (4.5) and (4.6), using (4.4). ∎

###  4.2. The main results

Here we first state and then prove the convergence results for the parabolic
flows corresponding to the (either scaled or renormalized) $s$-Gagliardo
seminorms as $s\to 0$ and $s\to 1$. These follow by collecting the preparatory
results of the previous sections, and Lemma 4.7 for the first order
convergence as $s\to 0^{+}$.

We start with the convergences as $s\to 0^{+}$.

######  Theorem 4.4.

Let $\\{s_{n}\\}_{n\in\mathbb{N}}\subset(0,1)$ be such that $s_{n}\to 0^{+}$
as $n\to+\infty$ . Let $u^{0}_{0}\in L^{2}(\Omega)$ and let
$\\{u_{0}^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such that
$u^{n}_{0}\in\mathcal{H}^{s_{n}}_{0}(\Omega)$ ,
$S:=\sup_{n\in\mathbb{N}}s_{n}F^{s_{n}}(u^{n}_{0})<+\infty$ and $u^{n}_{0}\to
u^{0}_{0}$ in $L^{2}(\Omega)$ . Then, for every $n\in\mathbb{N}$ there exists
a unique solution $u^{n}\in H^{1}([0,+\infty);L^{2}(\Omega))$ to

(4.8) $\begin{dcases}u_{t}(t)=-s_{n}(-\Delta)^{s_{n}}u(t)\qquad\textrm{for
a.e.\ }t\in[0,+\infty)\\\ u(0)=u^{n}_{0}\,,\end{dcases}$

and such a solution satisfies $(-\Delta)^{s_{n}}u^{n}(t)\in L^{2}(\Omega)$ for
$a.e.\,t\geq 0$ . Moreover, for every $T>0$, $u^{n}\rightharpoonup u^{0}$ in
$H^{1}([0,T];L^{2}(\Omega))$ as $n\to+\infty$ , where $u^{0}\in
H^{1}([0,T];L^{2}(\Omega))$ is the unique solution to

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

Furthermore, if

$\lim_{n\to+\infty}s_{n}F^{s_{n}}(u^{n}_{0})=F^{0}(u^{0}_{0})\,,$

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

$\|u^{n}(t)-u^{0}(t)\|_{L^{2}(\Omega)}\to 0\quad\textrm{and}\quad
s_{n}F^{s_{n}}(u^{n}(t))\to F^{0}(u^{0}(t))\qquad\textrm{for every }t\geq
0\,.$

######  Theorem 4.5.

Let $\\{s_{n}\\}_{n\in\mathbb{N}}\subset(0,1)$ be such that $s_{n}\to 0^{+}$
as $n\to+\infty$ . Let $u^{0}_{0}\in L^{2}(\Omega)$ and let
$\\{u^{n}_{0}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such that
$u^{n}_{0}\in\mathcal{H}^{s_{n}}_{0}(\Omega)$ ,
$S:=\sup_{n\in\mathbb{N}}\hat{F}^{s_{n}}(u^{n}_{0})<+\infty$ and $u^{n}_{0}\to
u^{0}_{0}$ in $L^{2}(\Omega)$ . Then, for every $n\in\mathbb{N}$ there exists
a unique solution $u^{n}\in
H^{1}([0,+\infty);\mathcal{H}^{s_{n}}_{0}(\Omega))$ to

(4.10)
$\begin{dcases}u_{t}(t)=-\Big{[}(-\Delta)^{s_{n}}u(t)-\frac{d\omega_{d}}{s_{n}}u(t)\Big{]}\qquad\textrm{for
a.e.\ }t\in(0,T)\,,\\\ u(0)=u^{n}_{0}\,,\end{dcases}$

and such a solution satisfies $(-\Delta)^{s_{n}}u^{n}(t)\in L^{2}(\Omega)$ for
$a.e.\,t\geq 0$ . Moreover, $u^{0}_{0}\in\mathcal{H}^{0}_{0}(\Omega)$ and, for
every $T>0$, $u^{n}\to u^{0}$ in $H^{1}([0,T];L^{2}(\Omega))$ as $n\to+\infty$
, where $u^{0}\in H^{1}([0,T];\mathcal{H}^{0}_{0}(\Omega))$ is the unique
solution to

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

and such a solution satisfies $(-\Delta)^{0}u^{0}(t)\in L^{2}(\Omega)$ for
$a.e.\,t\geq 0$ . Furthermore, if

$\lim_{n\to+\infty}\hat{F}^{s_{n}}(u^{n}_{0})=\hat{F}^{0}(u^{0}_{0})\,,$

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

$\|u^{n}(t)-u^{0}(t)\|_{L^{2}(\Omega)}\to
0\quad\textrm{and}\quad\hat{F}^{s_{n}}(u^{n}(t))\to\hat{F}^{0}(u^{0}(t))\qquad\textrm{for
every }t\geq 0\,.$

The result below shows the convergence toward the classical heat equation as
$s\to 1^{-}$ of the rescaled in time $s$-fractional heat equations.

######  Theorem 4.6.

Let $\\{s_{n}\\}_{n\in\mathbb{N}}\subset(0,1)$ be such that $s_{n}\to 1^{-}$
as $n\to+\infty$ . Let $u^{\infty}_{0}\in L^{2}(\Omega)$ and let
$\\{u_{0}^{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\Omega)$ be such that
$u^{n}_{0}\in\mathcal{H}^{s_{n}}_{0}(\Omega)$ ,
$S:=\sup_{n\in\mathbb{N}}(1-s_{n})F^{s_{n}}(u^{n}_{0})<+\infty$ and
$u^{n}_{0}\to u^{\infty}_{0}$ in $L^{2}(\Omega)$ . Then, for every
$n\in\mathbb{N}$ there exists a unique solution $u^{n}\in
H^{1}([0,+\infty);L^{2}(\Omega))$ to

(4.12)
$\begin{dcases}u_{t}(t)=-(1-s_{n})(-\Delta)^{s_{n}}u(t)\qquad\textrm{for a.e.\
}t\in[0,+\infty)\\\ u(0)=u^{n}_{0}\,,\end{dcases}$

and such a solution satisfies $(-\Delta)^{s_{n}}u^{n}(t)\in L^{2}(\Omega)$ for
$a.e.\,t\geq 0$ . Moreover, $u^{\infty}_{0}\in H^{1}_{0}(\Omega)$ , and, for
every $T>0$, $u^{n}\rightharpoonup u^{\infty}$ in $H^{1}([0,T];L^{2}(\Omega))$
as $n\to+\infty$ , where $u^{\infty}\in H^{1}([0,T];H^{1}_{0}(\Omega))$ is the
unique solution to

(4.13) $\begin{dcases}u_{t}(t)=\frac{\omega_{d}}{2}\Delta
u(t)\qquad\textrm{for a.e.\ }t\in[0,+\infty)\\\
u(0)=u^{\infty}_{0}\,.\end{dcases}$

Furthermore, if

$\lim_{n\to+\infty}(1-s_{n})F^{s_{n}}(u^{n}_{0})=F^{1}(u^{\infty}_{0})\,,$

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

$\|u^{n}(t)-u^{\infty}(t)\|_{L^{2}(\Omega)}\to
0\quad\textrm{and}\quad(1-s_{n})F^{s_{n}}(u^{n}(t))\to
F^{1}(u^{\infty}(t))\qquad\textrm{for every }t\geq 0\,.$

We first prove Theorem 4.4.

###### Proof of Theorem 4.4.

By the very definition of $F^{s}$ in (1.1), we have that for every
$n\in\mathbb{N}$
$D(s_{n}F^{s_{n}})=\mathcal{H}^{s_{n}}_{0}(\Omega)\neq\emptyset$ and that the
functionals $s_{n}F^{s_{n}}$ are strongly lower semicontinuous,
$\lambda$-positive and $\lambda$-convex for every $\lambda>0$ . Moreover, by
combining Proposition 4.1 with Proposition 3.7 for
$\mathcal{F}=s_{n}F^{s_{n}}$ , $\mathscr{H}=L^{2}(\Omega)$, and
$\hat{\mathscr{H}}=\mathrm{C}^{\infty}_{\mathrm{c}}(\Omega)$, we have that for
every $u\in\mathcal{H}^{s_{n}}_{0}(\Omega)$ , either
$\partial(s_{n}F^{s_{n}})(u)=\emptyset$ or
$\partial(s_{n}F^{s_{n}})(u)=\\{(-\Delta)^{s_{n}}u\\}$ with
$s_{n}(-\Delta)^{s_{n}}u\in L^{2}(\Omega)$ . Therefore, by Theorem 3.6, there
exists a unique solution to the Cauchy problem (4.8), with
$s_{n}(-\Delta)^{s_{n}}u\in L^{2}(\Omega)$ for $a.e.\,t\geq 0$, for every
$n\in\mathbb{N}$. Furthermore, for every $u\in L^{2}(\Omega)$ we have that

(4.14) $\lim_{t\to 0}\frac{F^{0}(u+t\varphi)-F^{0}(u)}{t}=d\omega_{d}\langle
u,\varphi\rangle_{L^{2}(\Omega)}\qquad\textrm{for every }\varphi\in
L^{2}(\Omega)\,,$

whence we deduce that $\partial F^{0}(u)=\\{d\omega_{d}\,u\\}$ . As a
consequence, there exists a unique solution to the problem (4.9). Finally, the
stability claims follow by applying Theorem 3.8 with
$\mathcal{F}^{n}=s_{n}F^{s_{n}}$ and $\mathcal{F}^{\infty}=F^{0}$ , once
noticed that, in view of Theorem 1.2, assumption (a) is satisfied. ∎

In order to prove Theorem 4.5, we provide below a lemma showing uniform
$\lambda$-convexity of the underlying functionals.

######  Lemma 4.7.

For every $\lambda>2|\Omega|$ , the functionals $\hat{F}^{s}$ are
$\lambda$-positive and $\lambda$-convex for every $s\in[0,1)$ .

###### Proof.

As for the $\lambda$-positivity it is enough to notice that, by the very
definition of $\hat{F}^{s}$ in (1.8) and (1.9) and by (1.7), recalling that
$G^{s}_{1}\geq 0$ for every $s\in[0,1)$ , we have that

$\hat{F}^{s}(u)+\frac{\lambda}{2}\|u\|_{L^{2}(\Omega)}^{2}\geq
G_{1}^{s}(u)+\left(\frac{\lambda}{2}-|\Omega|\right)\|u\|_{L^{2}(\Omega)}^{2}\geq
0\,.$

Now we show that the functionals $\hat{F}^{s}$ are $\lambda$-convex for every
$s\in[0,1)$ . We preliminarily notice that the functionals $G^{s}_{1}$ are
convex for every $s\in[0,1)$ . Therefore, it is enough to show that the
functionals $J^{s}_{1}$ are $\lambda$-convex. To this end, for every
$u,v\in\mathcal{H}_{0}^{0}(\Omega)$ we define the function

$f:\mathbb{R}\rightarrow\mathbb{R},\qquad
f(t):=J_{1}^{s}(u+tv)+\frac{\lambda}{2}\|u+tv\|_{L^{2}(\Omega)}^{2}$

and we claim that $\frac{\,\mathrm{d}^{2}}{\,\mathrm{d}t^{2}}f(t)\geq 0$ for
every $t\in\mathbb{R}$ . Indeed, since

$J_{1}^{s}(u+tv)=J_{1}^{s}(u)-2t\iint_{\mathbb{R}^{2d}\setminus\overline{\mathcal{B}}_{1}}\frac{\tilde{u}(x)\tilde{v}(y)}{|x-y|^{d+2s}}\,\mathrm{d}x\,\mathrm{d}y+t^{2}J_{1}^{s}(v)$

and

$\|u+tv\|_{L^{2}(\Omega)}^{2}=\|u\|_{L^{2}(\Omega)}^{2}+2t\int_{\Omega}u(x)v(x)\,\mathrm{d}x+t^{2}\|v\|_{L^{2}(\Omega)}^{2}\,,$

by (1.7) we have

$\frac{\,\mathrm{d}^{2}}{\,\mathrm{d}t^{2}}f(t)=2J_{1}^{s}(v)+\lambda\|v\|_{L^{2}(\Omega)}^{2}\geq(-2|\Omega|+\lambda)\|v\|_{L^{2}(\Omega)}^{2}\geq
0\,,$

which implies the $\lambda$-convexity of the functional $J^{s}_{1}$ and then
the $\lambda$-convexity of $\hat{F}^{s}$ . ∎

###### Proof of Theorem 4.5.

Let $\lambda>2|\Omega|$ be fixed. Then, by the very definition of
$\hat{F}^{s}$ in (1.8) for every $n\in\mathbb{N}$ we have that
$D(\hat{F}^{s_{n}})=\mathcal{H}^{s_{n}}_{0}(\Omega)\neq\emptyset$ and, by
Remark 1.5 and Lemma 4.7, that the functionals $\hat{F}^{s_{n}}$ are strongly
lower semicontinuous, $\lambda$-positive and $\lambda$-convex. Moreover, by
(4.3) and by (4.14), for every $u\in\mathcal{H}^{s_{n}}_{0}(\Omega)$ and for
every $\varphi\in\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$ we have

$\lim_{t\to
0}\frac{\hat{F}^{s_{n}}(u+t\varphi)-\hat{F}^{s_{n}}(u)}{t}=\langle(-\Delta)^{s_{n}}u-\frac{d\omega_{d}}{s_{n}}u,\varphi\rangle_{L^{2}(\Omega)}\,,$

which, by applying Proposition 3.7 with $\mathcal{F}=\hat{F}^{s_{n}}$ ,
$\mathscr{H}=L^{2}(\Omega)$ and
$\hat{\mathscr{H}}=\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$ , implies that
for every $u\in\mathcal{H}^{s_{n}}(\Omega)$ either
$\partial\hat{F}^{s_{n}}(u)=\emptyset$ or
$\partial\hat{F}^{s_{n}}(u)=\\{(-\Delta)^{s_{n}}u-\frac{d\omega_{d}}{s_{n}}u\\}$
with $(-\Delta)^{s_{n}}u-\frac{d\omega_{d}}{s_{n}}u\in L^{2}(\Omega)$ .
Analogously, by Lemma 4.3 and by Proposition 3.7, we have that for every
$u\in\mathcal{H}^{0}_{0}(\Omega)$ either $\partial\hat{F}^{0}(u)=\emptyset$ or
$\partial\hat{F}^{0}(u)=\\{(-\Delta)^{0}u\\}$ with $(-\Delta)^{0}u\in
L^{2}(\Omega)$ .

Therefore, by Theorem 3.6, the solutions to the problems (4.10)
($n\in\mathbb{N}$) and (4.11) are uniquely determined, and the righthand sides
in (4.10) and (4.11) belong to $L^{2}$ for $a.e.\,t$. Finally, the stability
claim follows by applying Theorem 3.8 with $\mathcal{F}^{n}=\hat{F}^{s_{n}}$
and $\mathcal{F}^{\infty}=\hat{F}^{0}$ , once noticed that, in view of Theorem
1.4, assumption (b) is satisfied. ∎

It lasts to prove Theorem 4.6. Also in this case, this follows from the
general results already discussed.

###### Proof of Theorem 4.6..

By the very definition of $F^{s}$ in (1.1), we have that for all
$n\in\mathbb{N}$
$D((1-s_{n})F^{s_{n}})=\mathcal{H}_{0}^{s_{n}}(\Omega)\neq\emptyset$ and that
the functional $(1-s_{n})F^{s_{n}}$ is strongly lower semicontinuous,
$\lambda$-positive and $\lambda$-convex for every $\lambda>0$. Now, by
combining Proposition 4.1 with Proposition 3.7 for
$\mathcal{F}=(1-s_{n})F^{s_{n}}$, $\mathscr{H}=L^{2}(\Omega)$ and
$\hat{\mathscr{H}}=\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$, we have that for
every $u\in\mathcal{H}_{0}^{s_{n}}(\Omega)$, either $(1-s_{n})\partial
F^{s_{n}}(u)=\emptyset$ or $(1-s_{n})\partial
F^{s_{n}}(u)=\\{(1-s_{n})(-\Delta)^{s_{n}}u\\}$ with
$(1-s_{n})(-\Delta)^{s_{n}}u\in L^{2}(\Omega)$. Furthermore, for every $u\in
H_{0}^{1}(\Omega)$ and for all
$\varphi\in\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$ we have that

$\lim_{h\rightarrow
0}\frac{F^{1}(u+h\varphi)-F^{1}(u)}{h}=\frac{\omega_{d}}{2}\langle\nabla
u,\nabla\varphi\rangle_{L^{2}(\Omega)}=:\frac{\omega_{d}}{2}\langle(-\Delta)u,\varphi\rangle_{L^{2}(\Omega)}\,;$

therefore, by applying Proposition 3.7 with $\mathcal{F}=F^{1}$,
$\mathscr{H}=L^{2}(\Omega)$ and
$\hat{\mathscr{H}}=\mathrm{C}_{\mathrm{c}}^{\infty}(\Omega)$ we have that
either $\partial F^{1}(u)=\emptyset$ or $\partial F^{1}(u)=\\{(-\Delta u)\\}$
with $(-\Delta u)\in L^{2}(\Omega)$. Now, by Theorem 3.6, the solutions to the
problems (4.12) ($n\in\mathbb{N}$) and (4.13) are uniquely determined, and the
righthand sides in (4.12) and (4.13) belong to $L^{2}$ for $a.e.\,t$. Finally,
the stability claim follows by applying Theorem 3.8 with
$\mathcal{F}^{n}=(1-s_{n})F^{s_{n}}$ and $\mathcal{F}^{\infty}=F^{1}$, once
noticed that, in view of Theorem 2.1, assumption (b) is satisfied. ∎

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