# Microstructures and anti-phase boundaries
in long-range lattice systems

Andrea Braides, Edoardo Voglino and Matteo Zanardini
SISSA, via Bonomea 265, Trieste, Italy<EMAIL_ADDRESS>
###### Abstract

We study the effect of long-range interactions in non-convex one-dimensional
lattice systems in the simplified yet meaningful assumption that the relevant
long-range interactions are between $M$-neighbours for some $M\geq 2$ and are
convex. If short-range interactions are non-convex we then have a competition
between short-range oscillations and long-range ordering. In the case of a
double-well nearest-neighbour potential, thanks to a recent result by Braides,
Causin, Solci and Truskinovsky, we are able to show that such a competition
generates $M$-periodic minimizers whose arrangements are driven by an
interfacial energy. Given $M$, the shape of such minimizers is universal, and
independent of the details of the energies, but the number and shapes of such
minimizers increases as $M$ diverges.

Keywords: lattice systems, long-range interactions, non-convex energies,
discrete-to-continuum, interfaces, Gamma-convergence.

MSC Class (2020): 39A12, 34K31, 82B20, 49J45, 46E39.

††Preprint SISSA 08/2024/MATE

##  1 Introduction

In this paper we study the behaviour of boundary-value minimum problems
related to one-dimensional long-range lattice energies, which in the greatest
generality can be stated as the asymptotic behaviour as $n\to+\infty$ of
solutions $u^{n}=\\{u^{n}_{i}\\}_{i}$ of the minimization of functionals of
the form

$E_{n}(u)=\sum_{k=1}^{n}\sum_{i=k}^{n}\psi_{k}(u_{i}-u_{i-k})$ (1)

on $n+1$-tuples $u=\\{u_{i}\\}_{i}$ of parameters with $u_{i}\in\mathbb{R}$,
subjected to boundary conditions $u_{0}=0$ and $u_{n}=n\ell$. In this
generality, the problem is very complex and leads to a variety of different
issues with competing short and long-range oscillations and concentration
effect, except for the trivial case when all $\psi_{k}$ are convex, for which
the minimizer is essentially unique and coincides with the linear function
$u_{i}=i\ell$, except for boundary effects, which are asymptotically
negligible upon some technical assumptions on $\psi_{k}$. Nevertheless, an
averaged description of minimizers is possible in the spirit of
$\Gamma$-convergence. To this end, we regard the energies $E_{n}$ as defined
on functions $u\colon[0,1]\to\mathbb{R}$, with domain the piecewise-affine
functions defined by $u(x)=\frac{1}{n}u_{\lfloor x/n\rfloor}$ for some
$(n+1)$-tuple $\\{u_{i}\\}_{i}$, in such a way that the derivative of $u$ on
$((i-1)/n,i/n)$ is $u_{i}-u_{i-1}$. The $\Gamma$-convergence of such energies
can then be studied with respect to the $L^{1}$-convergence. Upon some growth
hypotheses on $\psi_{k}$ that ensure that limits of sequences $u_{n}$ with
energy of order $n$ belong to some $W^{1,p}(0,1)$, the $\Gamma$-limit of
$\frac{1}{n}E_{n}$ can be written as

$F(u)=\int_{0}^{1}\psi_{\rm hom}(u^{\prime})dt,$ (2)

for a convex function $\psi_{\rm hom}$ resulting from a non-linear
homogenization process (see [6], and [3] for the higher-dimensional case).

The $\Gamma$-convergence above only ensures that the (interpolations of the)
minimizers of boundary-value problems for $E_{n}$ converge to the
corresponding minimizer, or to one of the minimizers, of the analog continuum
boundary-value problem related to $F$, but brings no further information on
their behaviour, which may depend on $\ell$. Note that if $\psi_{\rm hom}$ is
strictly convex at $\ell$ then the unique minimizer is the linear function
$u(x)=\ell x$, while at non-strictly convex points discrete solutions may
converge to a particular choice among minimizers.

A particular class of energies for which an analysis of $\psi_{\rm hom}$
leading to a description of the convergence of discrete minimizers has been
possible is the one studied in [5], where the non-convexity is confined to
nearest-neighbour interactions governed by $\psi=\psi_{1}$, and the long-range
potentials are quadratic; that is, $\psi_{k}(z)=a_{k}z^{2}$, with $a_{k}$ non-
negative, and the energies can be written as

$E_{n}(u)=\sum_{i=1}^{n}\psi(u_{i}-u_{i-1})+\sum_{k=2}^{n}\sum_{i=k}^{n}a_{k}(u_{i}-u_{i-k})^{2}$
(3)

In that case, the properties of minimizers can be linked to properties of the
sequence $a_{k}$. In particular, in [5] the case of double-well $\psi$ is
studied, for which a prototype is

$\psi(z)=\min\\{(z-1)^{2},(z+1)^{2}\\},$ (4)

in which case it is possible to describe the patterns of the minimizers by
tracing whether the value $z_{i}=u_{i}-u_{i-1}$ lies in one “well” (i.e.,
$z_{i}\leq 0$) or the other one (i.e., $z_{i}\geq 0$). As such patterns of
minimizers are concerned, we recall the following interesting characterization
of minimizers of problems (3) in the case when $a_{k}=0$ for all $k\geq 2$
except for one value $k=M$: either

  * (a)

minimizers $u_{i}$ are such that $z_{i}=u_{i}-u_{i-1}$ tend (for $n$ large) to
be $M$-periodic with average $z$, and take only two values, one, for $m$
indices in the period, in one well and the second one, for $M-m$ indices in
the period, in the second well, or

  * (b)

$z_{i}$ defined as above tend (for $n$ large) to be a mixture of two periodic
functions as in (1) with some $z^{\prime}$ and $z^{\prime\prime}$ in the place
of $z$ and for two consecutive values $m$ and $m+1$ between $0$ and $M$.

This characterization extends a formula known before when only $a_{2}\neq 0$
(see [6]), in which case we have the only three possibilities that either the
parameters $z_{i}$ tend to take a constant value in the first or in the second
well, or that we have a $2$-periodic pattern mixing values in both wells.

The appearance of microstructure is a recurring feature of non-convex
variational systems. Such microstructures may be driven by a scale-free
relaxation phenomenon described by convexification or quasiconvexification of
the original energy densities (see e.g. the books by Buttazzo [9] or Dacorogna
[10]), or present more regular patterns at a specific scale due to competing
long-range and short-range effects (as in the seminal paper by S. Müller [13];
see also [1]). Minimizers of $E_{n}$ are similar to the latter, with
oscillations both driven by short- and long-range microscopic interactions,
and by mesoscopic non-convexity.

Simple examples of variational problems exhibiting microscopic oscillations
are lattice systems defined on “spin functions”; i.e., functions taking only a
finite number of values, the traditional choice being $-1$ and $1$. If the
energies are “frustrated”; that is, the system presents interaction potentials
that cannot be all separately minimized at the same time by a single function,
then minimization may produce periodically modulated phases (see [2, Chapter
7]). Often, the determination of the period and shape of such minimizers is a
non-trivial matter as in the case of infinite-range antiferromagnetic systems
studied by Giuliani et al. [12], and has interesting continuum counterparts
(see e.g. [11]).

In the case of double-well problems, the location of the parameter in one or
the other well relaxes the strict constraint that the parameter takes two
values; that is, the constraint $z\in\\{-1,1\\}$ is replaced by a potential
$\psi(z)$ where $\psi$ is a strictly positive continuous function minimized in
$\\{-1,1\\}$. To distinguish between them, we will call the first type of
parameters “hard spins” and the second ones “soft spins”. For the prototypical
double-well potential $\psi$ in (4) it is clear that the two “wells” coincide
with $z$ negative and $z$ positive. If also long-range interactions are taken
into account, minimization for soft spins may produce patters analogous to
those for frustrated hard spins. Furthermore, the “soft” approach allows to
include more easily boundary-value problems as above.

In this paper we carry on a fine analysis of the energies $E_{n}$ in (3) by
examining not only minimizers, but also parameters $u_{i}$ whose energy in (3)
differs from the minimum by a finite quantity bounded as $n$ tends to
$+\infty$. This is done by using a development by $\Gamma$-convergence [4,
Section 1.10] and [7], and is performed for double-well $\psi$ and $a_{k}\neq
0$ only for $k=M$, so that the descriptions (a) and (b) above provide the
value of minima. The meaningful definition of convergence for functions $u$
depends on whether we are in case (a) or (b) above. For simplicity of
illustration consider that the boundary datum $\ell$ is such that case (a)
holds for some $m$. Then, given a sequence $u^{n}_{i}$ with bounded energy,
the sequence is compact in the following sense: there are a finite number of
indices $i^{n}_{j}$, which we may suppose to converge after scaling; that is,
$i^{n}_{j}/n\to x_{j}$, such that in the intervals in the complement of such
indices, each function coincides with a $M$-periodic minimizer
$\overline{u}_{i}$ as in (a), up to an arbitrary small error. Hence, up to
subsequences, each such sequence determines a continuum limit ${\bf u}$ whose
derivative $\mathbf{u}^{\prime}$ takes values in the finite set ${\bf M}_{m}$
of $M$-periodic minimizers as in (1). A similar argument holds in case (b),
for which we can conclude that the continuum limit ${\bf u}$ has derivative
with values in ${\bf M}_{m}\cup{\bf M}_{m+1}$. Once such a piecewise-constant
limit is defined, we will prove that the $\Gamma$-limit has the form

$F(u)=\sum_{t\in S({\bf u}^{\prime})}\Phi({\bf u}^{\prime}(t^{-}),{\bf
u}^{\prime}(t^{+})),$ (5)

where $S({\bf u}^{\prime})$ denotes the discontinuity set of ${\bf
u^{\prime}}$ in $(0,1)$. This shows that minimization may give rise to
microscopic patterns ${\bf M}_{m}$, microscopic incompatibility may give rise
to interfaces between elements of ${\bf M}_{m}$ (anti-phase boundaries) or/and
interfaces between elements of ${\bf M}_{m}$ and ${\bf M}_{m+1}$ (macroscopic
interfaces). In order to avoid boundary effects, the analysis will be carried
out under some periodicity assumptions.

It is interesting to note that, even though the values of the slopes of
microscopic minimizers in ${\bf M}_{m}$ depends on the average slope, or
boundary datum, $\ell$, the set ${\bf M}_{m}$ has a ‘universal’ form, and its
elements are in correspondence with $M$-tuples with $m$ values equal to $1$
and $(M-m)$ values equal to $0$. A final remark is that the presence of
$M$-th-neighbour interaction is often compared to that of a singular
perturbation with a term containing the $M$-th derivative for continuum
double-well problems. However, while in the continuum case the resulting
phase-transition energy is essentially independent of $M$ (see [8, 14]), in
the discrete case our result shows an increasing complexity of minimizers as
$M$ increases.

##  2 Statement of the result

We will fix $M\in\mathbb{N}$ with $M\geq 2$ and functions
$\psi_{1},\psi_{M}\colon\mathbb{R}\to[0,+\infty)$ satisfying the coerciveness
condition

$\lim_{\left|z\right|\rightarrow\infty}\frac{\psi_{1}(z)}{\left|z\right|}=+\infty.$
(6)

We want to study the overall behaviour of functionals with competing nearest-
neighbour and M-th-neighbour interactions driven by the potential $\psi_{1}$
and $\psi_{M}$, respectively, of the form

$\sum_{i}\psi_{1}\left({u_{i+1}-u_{i}}\right)+\sum_{i}\psi_{M}\Big{(}\frac{u_{i+M}-u_{i}}{M}\Big{)},$
(7)

defined on discrete functions indexed on $\mathbb{Z}$. In our assumptions the
potential $\psi_{1}$ will be a double-well energy density which favours
oscillations through non-convexity, while $\psi_{M}$ is a convex potential
favouring long-range ordering.

###  2.1 Analysis at the bulk scaling

We preliminary analyze a scaled version, whose analysis will suggest a
renormalization argument. We use a standard scaling procedure that allows to
use an analytic approach by $\Gamma$-convergence, introducing a reference
interval $[0,1]$ and the small parameter $\varepsilon_{n}=\frac{1}{n}$ with
$n\in\mathbb{N}$. The energies above, when we take into account the
interaction involved on $n+1$ sites, now parameterized by $\varepsilon_{n}i$
with $i\in\\{0,\ldots,n\\}$, take the form

$\sum_{i=0}^{n-1}\psi_{1}\Big{(}\frac{u_{i+1}-u_{i}}{\varepsilon_{n}}\Big{)}+\sum_{i=0}^{n-1}\psi_{M}\Big{(}\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\Big{)}.$
(8)

In this notation $u_{i}=u(\varepsilon_{n}i)$. Note that in the last sum we
also take into account the values of $u_{i}$ for $i\in\\{n,\ldots,n+M-1\\}$.
In the sequel, in order not to have boundary effects, we will define $u_{i}$
for all values of $i$ using some periodic conditions.

After this parameterization, we can identify such discrete functions with the
piecewise-affine interpolation on $[0,1]$ of the sites
$(i\varepsilon_{n},u_{i})$. We define the space of such functions

$\mathcal{A}_{n}(0,1)=\\{u:[0,1]\rightarrow\mathbb{R}\text{ continuous, and
affine on
}(i\varepsilon_{n},(i+1)\varepsilon_{n}),\,i\in\\{0,\dots,n-1\\}\\},$

and the scaled functionals

$E_{n,M}(u)=\sum_{i=0}^{n-1}\varepsilon_{n}\psi_{1}\Big{(}\frac{u_{i+1}-u_{i}}{\varepsilon_{n}}\Big{)}+\sum_{i=0}^{n-1}\varepsilon_{n}\psi_{M}\Big{(}\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\Big{)}.$
(9)

Since

$\sum_{i=0}^{n-1}\varepsilon_{n}\psi_{1}\Big{(}\frac{u_{i+1}-u_{i}}{\varepsilon_{n}}\Big{)}=\int_{0}^{1}\psi_{1}(u^{\prime})\,dt,$

condition (6) ensures that functionals $E_{n,M}$ are equicoercive in
$W^{1,1}(0,1)$; namely, that if $u^{n}$ is bounded in $L^{1}(0,1)$ and
$E_{n,M}(u_{n})\leq C<+\infty$, then, up to subsequences, $u^{n}$ converge
weakly in $W^{1,1}(0,1)$ and strongly in $L^{1}(0,1)$. The $\Gamma$-limit of
$E_{n,M}$ with respect to this convergence is described in the following
result, where we also consider periodic conditions. To that end we fix
$\ell\in\mathbb{R}$ and define

$W^{1,1}_{\\#,\ell}(0,1)=\Big{\\{}u\in W^{1,1}_{\rm
loc}(\mathbb{R}):\;u(t)-\ell t\text{ is }L\text{-periodic}\Big{\\}},$

whose discrete counterpart is

$\mathcal{A}_{n,\ell}^{\\#}(0,1)=\\{u\in
W^{1,1}_{\\#,\ell}(0,1):\,{u|}_{[0,1]}\in\mathcal{A}_{n}(0,1)\\}.$

######  Theorem 2.1.

The functionals $E_{n,M}$ $\Gamma$-converge, with respect to the
$L^{1}$-topology, to the functional defined on $W^{1,1}(0,1)$ by

$\int_{0}^{1}\psi_{0}^{**}(u^{\prime}(t))\;dt,$ (10)

where $\psi_{0}$ is given by:

$\psi_{0}(z)=\psi_{M}(z)+\frac{1}{M}\min\left\\{\sum_{k=1}^{M}\psi_{1}(z_{k}):\;\sum_{k=1}^{M}z_{k}=Mz,\,z_{1},\dots
z_{M}\in\mathbb{R}\right\\}.$ (11)

Furthermore, the convergence is compatible with the addition of periodic
condition; that is, with fixed $\ell\in\mathbb{R}$, the functionals defined by

$E^{\ell}_{n,M}(u)=\left\\{\begin{aligned}
&E_{n.M}(u)&\qquad&u\in\mathcal{A}_{n,\ell}^{\\#}(0,1)\\\
&+\infty&\quad&\text{otherwise}\end{aligned}\right.$

$\Gamma$-converge to

$E^{\ell}_{M}(u)=\left\\{\begin{aligned}
&\int_{0}^{1}\psi_{0}^{**}(u^{\prime}(t))\;dt&\qquad&u\in
W^{1,1}_{\\#,\ell}(0,1)\\\ &+\infty&\quad&\text{otherwise in
$L^{1}(0,1)$.}\end{aligned}\right.$

The proof of this result can be found in [5] (see also [6] for the case
$M=2$).

###  2.2 Microscopic analysis

The main point of the analysis at the bulk scaling is the definition of
$\psi_{0}$ which will allow to renormalize energies (7) by subtracting the
affine term $r_{\ell}$ given by the tangent to $\psi^{**}_{0}$ at $\ell$ and
rewriting the sum as

$\sum_{i=0}^{n-1}\bigg{(}\psi_{M}\Big{(}\frac{u_{i+M}-u_{i}}{M}\Big{)}+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}({u_{i+k+1}-u_{i+k}})-r_{\ell}\Big{(}\frac{u_{i+M}-u_{i}}{M}\Big{)}\bigg{)}.$
(12)

This will be formalized as the computation of a higher-order $\Gamma$-limit
starting from $E_{n,M}$. We will consider periodic boundary conditions. We
also make the simplifying assumption that $n$ is a multiple of $M$, the
general case requiring a more complex notation being stated explicitly in
Section 3.2, taking into account possible mismatch at the boundary due to
incommensurability. In this case, after noting that $\min
E^{\ell}_{M}=\psi_{0}^{**}(\ell)$, and letting $m=m_{\ell}$ denote the tangent
to $\psi^{**}_{0}$ at $\ell$, we can consider the energies

$\displaystyle E^{1}_{n,M}(u)$ $\displaystyle=$ $\displaystyle
E^{1,\ell}_{n,M}(u):=\frac{E^{\ell}_{n,M}(u)-\min
E^{\ell}_{M}}{\varepsilon_{n}}=\frac{E^{\ell}_{n,M}(u)-\psi_{0}^{**}(\ell)}{\varepsilon_{n}}$
(13) $\displaystyle=$
$\displaystyle\sum_{i=0}^{n-1}\bigg{(}\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\left(\frac{u_{i+k+1}-u_{i+k}}{\varepsilon_{n}}\right)-\psi_{0}^{**}(\ell)\bigg{)}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n{-1}}\bigg{(}\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\left(\frac{u_{i+k+1}-u_{i+k}}{\varepsilon_{n}}\right)-\psi_{0}^{**}(\ell)-m\Big{(}\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}-\frac{\ell}{\varepsilon_{n}}\Big{)}\bigg{)}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n{-1}}\bigg{(}\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\left(\frac{u_{i+k+1}-u_{i+k}}{\varepsilon_{n}}\right)-r_{\ell}\Big{(}\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\Big{)}\bigg{)},$

where we have used that $\sum_{i={0}}^{{n-1}}(u_{i+M}-u_{i})=M\ell$ thanks to
the $n$-periodicity of $u_{i}-\ell\varepsilon_{n}i$.

Until now we have made no assumptions on $\psi_{1}$ and $\psi_{M}$. We study a
particular case in which $\psi_{M}$ is a strictly convex function and
$\psi_{1}$ is a double-well potential of the form

$\psi_{1}(z)=\min\\{W_{1}(z),W_{2}(z)\\},$

where $W_{1}$ and $W_{2}$ are two smooth convex functions. Note that this is
not a very restrictive hypothesis since in the determination of the
$\Gamma$-limit only the values of $W_{1}$ and $W_{2}$ close to the bottom of
the wells will be taken into account, so that more general $\psi_{1}$ of
double-well type can be included in this analysis. We also make the assumption
that $W_{1}$ and $W_{2}$ satisfy

$W_{i}(z)\geq cz^{2}-\frac{1}{c},$

for some $c>0$. This structure provides a useful representation of $\psi_{0}$.
Indeed, we can distinguish two sets
$A_{1}=\\{x\in\mathbb{R}\,|\,\psi_{1}(x)=W_{1}(x)\\}$ and
$A_{2}=\\{x\in\mathbb{R}\,|\,\psi_{1}(x)=W_{2}(x)\\}$. Then, we can rewrite
the minimum problem in $\psi_{0}$ as follows:

$\displaystyle\min\Bigg{\\{}\sum_{k=1}^{M}\psi_{1}(z_{k}):\;\sum_{k=1}^{M}z_{k}=Mz\Bigg{\\}}$
(14) $\displaystyle=$ $\displaystyle\min_{0\leq j\leq
M}\,\min\Bigg{\\{}\sum_{k=1}^{j}W_{1}(z_{k})+\sum_{k=j+1}^{M}W_{2}(z_{k}):\;\sum_{k=1}^{M}z_{k}=Mz,\,z_{1},\dots,z_{j}\in
A_{1},\,z_{j+1},\dots,z_{M}\in A_{2}\Bigg{\\}}$ $\displaystyle=$
$\displaystyle\min_{0\leq j\leq
M}\,\min\Big{\\{}jW_{1}(z_{1})+(M-j)W_{2}(z_{2}):\;jz_{1}+(M-j)z_{2}=Mz,\,z_{1}\in
A_{1},\,z_{2}\in A_{2}\Big{\\}},$

where the last equality follows from Jensen inequality.

Figure 1: representation of $\psi_{0}$.

We can define $\overline{\psi}_{j}(z)=\psi_{M}(z)+\frac{1}{M}f_{j}(z)$, where
$f_{j}(z)$ is the value of the inner minimum problem in (14). In this way
$\psi_{0}$ can be represented as follows:

$\psi_{0}(z)=\min_{0\leq j\leq M}\overline{\psi}_{j}(z).$

See Fig. 1 for a typical form of $\psi_{0}$.

A non-trivial result in [5] (see Theorem 4.1 therein) shows that
$\psi_{0}^{**}$ will alternate non-degenerate intervals
$K_{j}=(z^{l}_{j},z^{r}_{j})$ in which
$\psi_{0}^{**}(z)=\psi_{0}(z)=\overline{\psi}_{j}(z)$ and intervals
$J_{j}=[z^{r}_{j-1},z^{l}_{j}]$, for $j\geq 1$, in which
$\psi_{0}^{**}(z)=r_{j}(z)$ is a straight line. In this notation
$z^{l}_{0}=-\infty$ and $z^{r}_{M}=+\infty$.

Referring to (13), this description gives that if $\ell\in K_{j}$ then

$r_{\ell}(z)=\psi_{0}(\ell)+\psi^{\prime}_{0}(\ell)(z-\ell)=\overline{\psi}_{j}(\ell)+\overline{\psi}_{j}^{\prime}(\ell)(z-\ell),$

while if $\ell\in J_{j}$ then $r_{\ell}$ is the affine function whose graph
passes through $(z^{r}_{j-1},\overline{\psi}_{j-1}(z^{r}_{j-1}))$ and
$(z^{l}_{j},\overline{\psi}_{j}(z^{l}_{j}))$.

Note that if we consider

$\mathcal{E}_{n}^{i}(u)=\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\left(\frac{u_{i+k+1}-u_{i+k}}{\varepsilon_{n}}\right)-r_{\ell}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right),$
(15)

then these values are still positive since
$\mathcal{E}^{i}_{n}(u)\geq\psi_{0}^{**}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)-r_{\ell}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)\geq
0$.

Before stating the convergence result we need some definitions.

######  Defn 2.2.

We define $\mathbf{M}^{\alpha}\subset\mathbb{R}^{M}$ as the set of minimizers
for the problem

$\min\left\\{\sum_{i=1}^{M}\psi_{1}(z_{i}):\;\sum_{i=1}^{M}z_{i}=M\alpha\right\\}.$
(P)

We will then set

$\mathbf{M}_{\alpha}=\begin{cases}\mathbf{M}^{\alpha}&\text{ if
}\alpha\in\text{int}(\\{z\in\text{dom}\,\psi_{0}\,|\,\psi_{0}(z)=\psi_{0}^{**}(z)\\})\\\
\bigcup_{z\in
J_{j}\cap\\{\psi_{0}(z)=\psi_{0}^{**}(z)\\}}\mathbf{M}^{z}&\text{ if
}\alpha\in J_{j}.\end{cases}$

######  Remark 2.3.

$\mathbf{M}^{\alpha}$, and consequently also $\mathbf{M}_{\alpha}$, is closed
under permutation, that is, if $z=(z_{1},\dots,z_{M})\in\mathbf{M}^{\alpha}$,
then for any permutation $\sigma$ also $(z_{\sigma(1)},\dots,z_{\sigma(M)})$
belongs to $\mathbf{M}^{\alpha}$.

######  Remark 2.4.

Using the notation for $K_{j}$ and $J_{j}$, Definition 2.2 can be simplified
as

$\mathbf{M}_{\alpha}=\begin{cases}\mathbf{M}^{\alpha}&\text{ if
}\alpha\in\bigcup_{j=0}^{M}K_{j}\\\
\mathbf{M}^{z^{r}_{j-1}}\cup\mathbf{M}^{z^{l}_{j}}&\text{ if }\alpha\in
J_{j},\ j\in\\{1,\ldots,M\\}.\end{cases}$

Since $\psi_{1}(z)=\min\\{W_{1}(z),W_{2}(z)\\}$ then we can study the
cardinality of the sets $\mathbf{M^{\alpha}}$ because, for a fixed
$j\in\\{0,\dots M\\}$, the minimum problem

$\min\Big{\\{}jW_{1}(z_{1})+(M-j)W_{2}(z_{2}):\;jz_{1}+(M-j)z_{2}=M\alpha,\,z_{1}\in
A_{1},\,z_{2}\in A_{2}\Big{\\}}\qquad\qquad(P_{j})$

admits a unique solution by strict convexity. Each minimizer $(z_{1},z_{2})$
of $(P_{j})$ corresponds to $\binom{M}{j}$ minimizers for (P); that is, the
number of possible $M$-tuple such that $j$ entries are $z_{1}$ and $M-j$ are
$z_{2}$. This shows that indeed for each $\alpha\in[z^{l}_{j},z^{r}_{j}]$ the
set ${\bf M}^{\alpha}$ can be identified with a set ${\bf M}_{j}$ depending
only on $j$, and with this notation ${\bf M}_{\alpha}={\bf M}_{j-1}\cup{\bf
M}_{j}$ if $\alpha\in J_{j}$.

In order to study the limit behaviour of $E_{n,M}^{1}$, we need a notion of
convergence of discrete $u^{n}$ to a vectorial function on the continuum.

######  Defn 2.5.

Given $u\in\mathcal{A}_{n,\ell}^{\\#}(0,L)$, we consider the subintervals

$I_{j}=[(j-1)\varepsilon_{n},j\varepsilon_{n}]\qquad\text{for }j\in\mathbb{Z}$

and for $i\in\mathbb{Z}$ we define the set $A_{i}=\bigcup_{k=1}^{M}I_{iM+k}$,
which is the union of $M$ such consecutive intervals. For
$k\in\\{1,\dots,M\\}$, we define $u_{k}$ the $k$-th M-interpolation of $u$ as
the piecewise-affine function obtained extending the slope $z^{Mi+k}$, where

$z^{j}:=\frac{u_{j}-u_{j-1}}{\varepsilon_{n}},$

to the whole $A_{i}$; that is, $u_{k}$ is defined by

$\begin{cases}u_{k,0}=u_{0},\\\ u^{\prime}_{k}(t)=z^{Mi+k}\quad&\text{if }t\in
A_{i}\\\ \end{cases}$

######  Defn 2.6.

We say that a sequence of discrete functions $(u^{n})_{n}$ converges to
$\overline{\mathbf{u}}=(\overline{u}_{1},\dots,\overline{u}_{M})$ in a
functional topology (for instance $L^{1}(0,1)$ or $L^{\infty}(0,1)$) if for
each $k$ the $k$-th interpolation $u_{k}^{n}$ converges to $\overline{u}_{k}$
in that topology.

######  Theorem 2.7 (Equi-coerciveness of the energies).

If a sequence $(u^{n})_{n\in\mathbb{N}}$ satisfies
$\sup_{n}E^{1}_{n,M}(u^{n})<+\infty$, then, up to addition of a constant and
extraction of a subsequence, the sequence converges uniformly to some
$M$-tuple of piecewise-affine functions
$\mathbf{\overline{u}}=(\overline{u}_{1},\dots,\overline{u}_{M})$ with
$\overline{\mathbf{u}}^{\prime}(t)\in\mathbf{M}_{\ell}$ for almost every $t$.
Moreover, there exists a finite set $S\subset(0,1)$ such that $u^{n}$
converges in $W^{1,\infty}_{\rm loc}((0,1)\setminus S)$ to
$\mathbf{\overline{u}}$ and the jump set $S(\overline{\mathbf{u}}^{\prime})$
is contained in $S$.

This compactness theorem justifies the use of the convergence in Definition
2.6 for the $\Gamma$-limit of $E^{1}_{n,M}$. In order to describe it we define
the following transition energy.

######  Defn 2.8.

Let $\ell\in\mathbb{R}$ and $\mathbf{z}=(z_{1},z_{2},\dots
z_{M}),\mathbf{z^{\prime}}=(z^{\prime}_{1},z^{\prime}_{2},\dots
z^{\prime}_{M})\in\mathbf{M}_{\ell}$. The transition energy between
$\mathbf{z}$ and $\mathbf{z^{\prime}}$ is defined by

$\displaystyle\Phi^{(\ell)}(\mathbf{z},\mathbf{z^{\prime}}):=\inf_{N\in\mathbb{N}}\min_{u}\bigg{\\{}$
$\displaystyle\sum_{i\in\mathbb{Z}}\Big{(}\psi_{M}\Big{(}\frac{u_{i+M}-u_{i}}{M}\Big{)}+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}(u_{i+k+1}-u_{i+k})-r_{\ell}\Big{(}\dfrac{u_{i+M}-u_{i}}{M}\Big{)}\Big{)}:$
$\displaystyle\;\;u:\mathbb{Z}\rightarrow\mathbb{R},\,u_{i}=u_{\mathbf{z}}(i)\text{
for }i\leq-N,\,u_{i}=u_{\mathbf{z^{\prime}}}(i)\text{ for }i\geq N\Big{\\}},$
(16)

where $r_{\ell}$ is the tangent line to $\psi_{0}^{**}$ at $\ell$ computed at
the point $x$ and, for any $\mathbf{z}\in\mathbb{R}^{M}$, $u_{\mathbf{z}}$ is
the piecewise-affine function $\mathbb{R}\longrightarrow\mathbb{R}$ defined as
follows:

$\begin{cases}u^{\prime}_{\mathbf{z}}(t)=z_{k}\quad\text{for
}t\in(k-1,k)+M\mathbb{Z}\\\ u_{\mathbf{z}}(0)=0\end{cases}$ (17)

With this definition we can state the $\Gamma$-convergence result as follows

######  Theorem 2.9 (first-order $\Gamma$-limit with periodic boundary
conditions).

Assume that $\psi_{M}$ is a $C^{1}$ strictly convex function, $\psi_{1}$ is of
class $C^{1}$ in a neighbourhood of $\mathbf{M}_{\ell}$ and $\psi_{0}^{**}$ is
differentiable in $\ell$. We define the domain

$\displaystyle
D_{\ell}^{\\#}=\\{\mathbf{u}:\mathbb{R}\rightarrow\mathbb{R}^{M}\,|\,$
$\displaystyle\mathbf{u}\text{ (locally) piecewise
affine},\sum_{k=1}^{M}u_{k}(t)-M\ell t\text{ 1-periodic,}$
$\displaystyle\mathbf{u}^{\prime}\text{ 1-periodic and
}\mathbf{u}^{\prime}(t)\in\mathbf{M}_{\ell}\;\text{ for all
}t\in\mathbb{R}\setminus S(\mathbf{u}^{\prime})\\}.$

Then $E^{1}_{n,M}$ $\Gamma$-converges, with respect to the $L^{\infty}$
topology, to

$E^{1}_{M}(\mathbf{u})=\sum_{t\in
S(\mathbf{u}^{\prime})\cap(0,1]}\Phi^{(\ell)}(\mathbf{u}^{\prime}(t-),\mathbf{u}^{\prime}(t+)),$
(18)

with domain $D_{\ell}^{\\#}$.

##  3 Proof of the results

In this section, for a greater generality, we consider $L>0$ and functions
parameterized on $[0,L]$ instead of $[0,1]$. In this case
$\varepsilon_{n}=\frac{L}{n}$, and we extend all the notation introduced in
the case $L=1$. With fixed $\ell\in\mathbb{R}$, the energies we consider are
directly written in the form

$\displaystyle
E^{1}_{n,M}(u):=\sum_{i=0}^{n{-1}}\bigg{(}\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\left(\frac{u_{i+k+1}-u_{i+k}}{\varepsilon_{n}}\right)-r_{\ell}\Big{(}\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\Big{)}\bigg{)},$
(19)

defined on $\mathcal{A}^{\\#}_{n}(0,L)$. Note that
$E^{1}_{n,M}(u)=\sum_{i={0}}^{{n-1}}\mathcal{E}^{i}_{n}(u)$, with
$\mathcal{E}^{i}_{n}$ be given by (15).

###  3.1 Compactness

We can state the compactness result as in Theorem 2.7 independently of the
boundary condition as follows.

######  Proposition 3.1.

Let $\mathcal{E}^{i}_{n}$ be given by (15). If

$\sup_{n}\sum_{i={0}}^{{n-1}}\mathcal{E}^{i}_{n}(u^{n})=:C<+\infty,$

then $u^{n}$ satisfies the claim of Theorem 2.7 with the interval $[0,L]$ in
the place of the interval $[0,1]$.

###### Proof.

Let $\eta>0$, then we define the set

$I_{n}(\eta):=\\{i\in\\{0,1,\dots
n-1\\}\,:\,\mathcal{E}^{i}_{n}(u^{n})>\eta\\},$

and, as a consequence of the bound on the sum, we get that

$\sup_{n}\\#I_{n}(\eta)\leq C(\eta)\sim C/\eta<+\infty.$

Therefore, if $i\notin I_{n}(\eta)$, by adding and subtracting
$\psi_{0}\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)$ to
$\mathcal{E}^{i}_{n}(u^{n})$, we obtain the two inequalities

$0\leq\psi_{M}\left(\frac{u_{i+M}^{n}-u_{i}^{n}}{M\varepsilon_{n}}\right)+\frac{1}{M}\left(\sum_{k=0}^{M-1}\psi_{1}\left(\frac{u^{n}_{i+k+1}-u^{n}_{i+k}}{\varepsilon_{n}}\right)\right)-\psi_{0}\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)\leq\eta$
$0\leq\psi_{0}\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)-r_{\ell}\left(\frac{u^{n}_{i+M}-u_{i}^{n}}{M\varepsilon_{n}}\right)\leq\eta.$

Note that, if $z$ and $(z_{1},\dots,z_{M})$ satisfy

  1. a)

$\psi_{M}\left(z\right)+\frac{1}{M}\left(\sum_{k=0}^{M-1}\psi_{1}\left(z_{k}\right)\right)-\psi_{0}\left(z\right)\leq\eta,\quad\text{for
}\sum_{k=0}^{M-1}z_{k}=Mz$

  2. b)

$\psi_{0}\left(z\right)-r\left(z\right)\leq\eta$

then $(z_{1},\dots,z_{M})$ is close to a minimizing $M$-tuple for the min in
the definition of $\psi_{0}$, while $\psi_{0}(z)$ is close to $r(z)$, the
tangent line of $\psi_{0}^{**}$ in $\ell$. This means that if
$\psi_{0}(\ell)=\psi_{0}^{**}(\ell)$ then $z$ is close to $\ell$, while if
$\ell$ is in some $J_{j}$ then $z$ is close to either $z^{r}_{j-1}$ or
$z^{l}_{J}$. Hence, for $\eta$ small enough we can find $\varepsilon>0$ so
that if (a) and (b) are satisfied then

$\text{dist}\left((z_{1},\dots,z_{M}),\mathbf{M}_{\ell}\right)\leq\varepsilon<\frac{1}{2}\min\\{\left|\mathbf{z}^{\prime}-\mathbf{z}^{\prime\prime}\right|\,:\,\mathbf{z}^{\prime},\mathbf{z}^{\prime\prime}\in\mathbf{M}_{\ell}\\}.$

This entails that for each $i\notin I_{n}(\eta)$ there exists a unique
$\mathbf{z}^{n}_{i}=(z^{n}_{i,1},\dots,z^{n}_{i,M})\in\mathbf{M}_{\ell}$ such
that

$\left|\left(\frac{u^{n}_{i+1}-u^{n}_{i}}{\varepsilon_{n}},\dots,\frac{u^{n}_{i+M}-u^{n}_{i+M-1}}{\varepsilon_{n}}\right)-\mathbf{z}^{n}_{i}\right|\leq\varepsilon,\implies\;\left|\frac{u^{n}_{i+k}-u^{n}_{i+k-1}}{\varepsilon_{n}}-z^{n}_{i,k}\right|\leq\varepsilon,\quad$
(20)

for all $k\in\\{1,\dots,M\\}$. Since $I_{n}(\eta)$ is a finite set, we can
identify $N_{n}$ pairs of indices $0=j_{0}\leq
i_{1}<j_{1}<i_{2}<j_{2}<\dots<i_{N_{n}}<j_{N_{n}}\leq i_{N_{n}+1}=n$ such that
the slopes of the $M$-interpolations of $u^{n}$ are close to an element of
$[\mathbf{z}_{i_{k}}^{n}]$ in the sense of (20) for all indices between
$i_{k}$ and $j_{k}$, and they are not otherwise.

Since $C\geq E^{1}_{n,M}(u^{n})\geq\eta(N_{n}-1)$, we must have that $N_{n}$
is bounded with respect to $n$. Then, we can assume, up to extracting
subsequences, that $N_{n}$ is constantly equal to $N$. Similarly, since
$\mathbf{M}_{\ell}$ is a finite set, we can assume
$\mathbf{z}_{i_{k}}^{n}=\mathbf{z}_{k}$. With fixed $k\in\\{2,\dots,N\\}$, for
each $n$ we chose an index $\tilde{i}_{k}\in\\{j_{k-1}+1,\dots,i_{k}-1\\}$ and
define the sequence $\\{\tilde{x}^{n}_{k}\\}$ such that
$\tilde{x}^{n}_{k}=\tilde{i}_{k}\varepsilon_{n}$, then, up to subsequences, we
can assume $x^{n}_{k}\rightarrow x_{k}$. However, $\left|i_{k}-j_{k-1}\right|$
must be bounded independently of $n$ and $k$, since

$C\geq E^{1}_{n,M}(u^{n})\geq\sum_{k=1}^{N+1}\eta\left|i_{k}-j_{k-1}\right|.$

Then we have
$\left|\tilde{x}^{n}_{k}-i\varepsilon_{n}\right|\leq\frac{C}{\eta}\varepsilon_{n}$
for any $i\in\\{j_{k-1}+1,\dots,i_{k}-1\\}$. Therefore, the whole
$\\{j_{k-1}+1,\dots,i_{k}-1\\}\varepsilon_{n}$ is converging to $x_{k}$.

Finally, we define the sets $S=\bigcup_{k=1}^{N}\\{x_{k}\\}$ and, for a fixed
small $\delta$, $S_{\delta}=\bigcup_{k=1}^{N}(x_{k}-\delta,x_{k}+\delta)$. By
(20), we have that for $n$ large enough the $s$-th $M$-interpolation satisfies

$\sup_{t\in(0,1)\setminus
S_{\delta}}\left|(u_{s}^{n})^{\prime}(t)-Z_{s}(t)\right|\leq\varepsilon,$ (21)

where $\mathbf{Z}(t)=\left(Z_{1}(t),\dots,Z_{M}(t)\right)$ is a piecewise-
constant function such that $\mathbf{Z}(t)=\mathbf{z}_{k}$ for
$t\in(x_{k-1},x_{k})$.

Note that from the equicoerciveness of $E_{n,M}$ and hence also of
$E^{1}_{n,M}$, we obtain that $u_{n,s}$ are a precompact sequence in
$H^{1}(0,L)$, so that we can suppose that they converge uniformly. But from
(21), the subsequence is such that $(u_{s}^{n})^{\prime}$ is converging to
$Z_{s}$. In conclusion, by applying a diagonal argument, we have proved that,
up to subsequences, $u^{n}$ converges in $W^{1,\infty}_{\rm
loc}((0,L)\setminus S)$ (in the sense of Definition 2.6) to a function
$\mathbf{u}$ such that $\mathbf{u}^{\prime}(t)=Z(t)\in\mathbf{M}_{\ell}$ and
the jump set $S(\mathbf{u}^{\prime})\subseteq S$. ∎

Note that in this proof we have not used the periodicity condition. If it is
taken into account that we have to note that $S(\mathbf{u}^{\prime})$ can also
contain the point $0$.

###  3.2 Computation of the Gamma-limit

We now compute the $\Gamma$-limit subjected to periodic boundary conditions,
without the simplifying assumption that $n$ be a multiple of $M$ used for
presentation purposes in the previous section. We will show that the limit
exists and can be characterized if more in general $n=q$ modulo $M$. The
energy will have the same form as in the case $q=0$, but the characterization
of the domain of the $\Gamma$-limit will depend on $q$, since
$M$-interpolations do not necessarily inherit the periodicity condition of
$u$. Note in particular that the limit as $n\to+\infty$ does not exist.

For $n\in\mathbb{N}$ we let $q\in\\{0,\dots,M-1\\}$ denote its class modulus
$M$ ($n\equiv q$ mod $M$), then given $u$ a piecewise-affine function in
$W^{1,1}_{\\#,\ell}(0,L)$, for any index $Mi+q$ with
$q\in\\{0,1,\dots,M-1\\}$, its $M$-interpolations $u_{k}$ will satisfy:

$\frac{u_{k,Mi+q}-u_{k,Mi+q-1}}{\varepsilon_{n}}=\frac{u_{Mi+k}-u_{Mi+k-1}}{\varepsilon_{n}}=\frac{u_{n+Mi+k}-u_{n+Mi+k-1}}{\varepsilon_{n}}.$

Since $n=Mr+q$, we write

$\frac{u_{k,Mi+k}-u_{k,Mi+k-1}}{\varepsilon_{n}}=\frac{u_{M(i+r)+q+k}-u_{M(i+r)+q+k-1}}{\varepsilon_{n}},$

which is linked to the $q+k$ interpolation (mod $M$) shifted of
$\left\lfloor\frac{n+k}{M}\right\rfloor$. Thus, we have

$\displaystyle u_{k}^{\prime}(t)=u_{k+q}^{\prime}(t+Mr\varepsilon_{n})\quad$
$\displaystyle\text{if }q+k\leq M$ (22) $\displaystyle
u_{k}^{\prime}(t)=u_{k+q-M}^{\prime}(t+M(r+1)\varepsilon_{n})\quad$
$\displaystyle\text{if }q+k>M$

######  Proposition 3.2.

Consider $n(r)=Mr+q$ for a fixed $q\in\\{0,\dots,M-1\\}$. Consider a sequence
$(u^{n(r)})_{r\in\mathbb{N}}$ such that $\sup_{n}E^{1}_{n,M}(u^{n})\leq
C<+\infty$. Then there exists a finite set $S\subset(0,L]$ for which $u^{n}$
(up to subsequences) converges in $W^{1,\infty}_{\rm
loc}(\mathbb{R}\setminus(S+\mathbb{Z}))$ to a $\mathbf{u}=(u_{1},\dots,u_{M})$
as in Theorem 2.7 such that for any $k=1,\dots,M$ we have

$u_{k}^{\prime}(t)=u_{q+k}^{\prime}(t+L)\quad\text{if }q+k\leq M,\qquad
u_{k}^{\prime}(t)=u_{k+q-M}^{\prime}(t+L)\quad\text{if }q+k>M$

for almost every $t\in S(\mathbf{u}^{\prime})$.

###### Proof.

The convergence is given by Proposition 3.1. What remains to prove is the
periodicity property. Suppose at first $q+k\leq M$. Fix $t\notin
S(\mathbf{u}^{\prime})$ and suppose $t+L\notin S(\mathbf{u}^{\prime})$, then
there exists a $\delta>0$ such that
$I_{\delta}(t)=[t-\delta,t+\delta]\subset\mathbb{R}\setminus
S(\mathbf{u}^{\prime})$ and $I_{\delta}(t+L)\subset\mathbb{R}\setminus
S(\mathbf{u}^{\prime})$. For every $k$ fixed, $(u^{n}_{k})^{\prime}$ converges
uniformly on $I_{\delta}(t)$ and $(u^{n}_{k+q})^{\prime}$ on
$I_{\delta}(t+L)$. Since $Mr\varepsilon_{n}=L\left(1-\frac{q}{n}\right)$, we
can rewrite (22) as

$(u_{k}^{n})^{\prime}(t)=(u^{n}_{k+q})^{\prime}\Big{(}t+L-\frac{Lq}{n}\Big{)}$

Taking the limit in $n$, the left-hand side is converging to
$u^{\prime}_{k}(t)$, while for the right-hand side we can notice that, since
$\mathbf{u}^{\prime}$ is piecewise constant,
$u^{\prime}_{k+q}(t+L)=u_{k+q}^{\prime}\Big{(}t+L-\frac{Lq}{n}\Big{)}$ for
every $n$ large enough, so that

$\displaystyle\left|(u^{n}_{k+q})^{\prime}\Big{(}t+L-\frac{Lq}{n}\Big{)}-u^{\prime}_{k+q}(t+L)\right|$
$\displaystyle=$
$\displaystyle\left|(u^{n}_{k+q})^{\prime}\Big{(}t+L-\frac{Lq}{n}\Big{)}-u_{k+q}^{\prime}\Big{(}t+L-\frac{Lq}{n}\Big{)}\right|$
$\displaystyle\leq$ $\displaystyle\sup_{s\in
I_{\delta}(t+L)}\left|(u^{n}_{k+q})^{\prime}(s)-u_{k+q}^{\prime}(s)\right|\xrightarrow[n\rightarrow+\infty]{}0.$

The case $q+k>M$ follows similarly. ∎

######  Remark 3.3.

Note that at the limit we have the boundary conditions
$u_{k}^{\prime}(0^{+})=u_{q+k}^{\prime}(L^{+})$, where $q+k$ is intended
modulus $M$. This holds also if $0\in S(\mathbf{u}^{\prime})$.

######  Proposition 3.4.

Given $(u^{n})_{n\in\mathbb{N}}$ such that
$\sup_{n}E^{1}_{n,M}(u^{n})=:C<+\infty$. Let $0=x_{0}<x_{1}<\dots<x_{N}=L$ and
let $\alpha_{1},\dots,\alpha_{N}\in\mathbb{R}$ be such that
$\mathbf{M}^{\alpha_{j}}\subseteq M_{\ell/L}$ and $u^{n}$ converges to
$\mathbf{u}$ satisfying

$\mathbf{u}^{\prime}(t)=\mathbf{z}^{\alpha_{j}}\in\mathbf{M}^{\alpha_{j}},\quad\text{for
}t\in(x_{j-1},x_{j}).$

Then, $\frac{\ell}{L}=\sum_{j=1}^{N}\alpha_{j}(x_{j}-x_{j-1})$.

###### Proof.

Suppose $L=1$. By the boundary conditions we have

$\displaystyle\ell=(u^{n}_{n}-u^{n}_{M\lfloor n/M\rfloor})+\sum_{i=0}^{\lfloor
n/M\rfloor}u^{n}_{M(i+1)}-u^{n}_{Mi}=(u^{n}_{n}-u^{n}_{M\lfloor
n/M\rfloor})+\sum_{i=0}^{\lfloor
n/M\rfloor}M\varepsilon_{n}\frac{u^{n}_{M(i+1)}-u^{n}_{Mi}}{M\varepsilon_{n}}.$

Now, fixed $\delta>0$, we define
$I_{\delta}^{j}=(x_{j}-\delta,x_{j}+\delta)\cap[0,1]$, for $j=0,\dots,N$, and
we call $S_{\delta}=\cup_{j=0}^{N}I_{\delta}^{j}$. From Proposition 3.1 the
interpolation derivative $(u^{n}_{k})^{\prime}(t)$ converges to
$z_{k}^{\alpha_{j}}$ on $[x_{j-1}+\delta,x_{j}-\delta]$. Then for an index $i$
such that
$[Mi\varepsilon_{n},M(i+1)\varepsilon_{n}]\subset[x_{j-1}+\delta,x_{j}-\delta]$,
we have

$\frac{u^{n}_{M(i+1)}-u^{n}_{Mi}}{M\varepsilon_{n}}=\frac{1}{M}\sum_{m=Mi}^{M(i+1)-1}\frac{u^{n}_{m+1}-u^{n}_{M}}{\varepsilon_{n}}=\frac{1}{M}\sum_{k=1}^{M}\frac{u^{n}_{k,Mi+1}-u^{n}_{k,Mi}}{\varepsilon_{n}}\xrightarrow[n\rightarrow+\infty]{}\frac{1}{M}\sum_{k=1}^{M}z^{\alpha_{j}}_{k}=\alpha_{j}$

Let $k^{+}_{j,n}$ be the smallest index $k$ such that $Mk\varepsilon_{n}\geq
x_{j}+\delta$ and similarly let $k^{-}_{j,n}$ be the largest index $k$ such
that $Mk\varepsilon_{n}\leq x_{j}-\delta$, so that
$\lim_{n}\varepsilon_{n}Mk^{-}_{j,n}=x_{j}-\delta$ and
$\lim_{n}\varepsilon_{n}Mk^{+}_{j,n}=x_{j}+\delta$. We also define
$k^{-}_{0,n}=0$ and $k^{+}_{N,n}=\lfloor\frac{n}{M}\rfloor$. Then,

$\displaystyle\left|\sum_{i=k^{+}_{j-1,n}}^{k^{-}_{j,n}-1}M\varepsilon_{n}\frac{u^{n}_{M(i+1)}-u^{n}_{Mi}}{M\varepsilon_{n}}-\alpha_{j}(x_{j}-x_{j-1}-2\delta)\right|$
$\displaystyle\leq$
$\displaystyle\left|\sum_{i=k^{+}_{j-1,n}}^{k^{-}_{j,n}-1}M\varepsilon_{n}\left(\frac{u^{n}_{M(i+1)}-u^{n}_{Mi}}{M\varepsilon_{n}}-\alpha_{j}\right)\right|$
$\displaystyle\hskip-56.9055pt+\left|\alpha_{j}(x_{j}-\delta-\varepsilon_{n}Mk^{-}_{j,n})\right|+\left|\alpha_{j}(x_{j-1}+\delta-\varepsilon_{n}Mk^{+}_{j-1,n})\right|,$

which tends to $0$ as $n$ tends to infinity. On the other hand, since
$\sup_{n}E^{1}_{n,M}(u^{n})=C$, there exists a constant $C_{1}$ independent
from $n$ such that for every index $i$ we have

$\mathcal{E}_{n}^{i}(u^{n})\leq
C\implies\psi_{1}\left(\frac{u^{n}_{i+1}-u^{n}_{i}}{\varepsilon_{n}}\right)-r_{\ell}\left(\frac{u^{n}_{i+1}-u^{n}_{i}}{\varepsilon_{n}}\right)\leq
C_{1},$

where we used the definition of $\mathcal{E}_{n}^{i}$ in (15). Thus, from the
superlinear growth of $\psi_{1}$, we must have that there exists an $R>0$ such
that $\left|\frac{u^{n}_{i+1}-u^{n}_{i}}{\varepsilon_{n}}\right|\leq R$. Then,
we can conclude

$\displaystyle\left|\ell-\sum_{j=1}^{N}\sum_{i=k^{+}_{j-1,n}}^{k^{-}_{j,n}-1}M\varepsilon_{n}\frac{u^{n}_{M(i+1)}-u^{n}_{Mi}}{M\varepsilon_{n}}\right|$
$\displaystyle=$
$\displaystyle\left|u^{n}_{n}-u^{n}_{Mk^{+}_{N,n}}+\sum_{j=0}^{N}\sum_{i=k^{-}_{j,n}}^{k^{+}_{j,n}-1}M\varepsilon_{n}\frac{u^{n}_{M(i+1)}-u^{n}_{Mi}}{M\varepsilon_{n}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{i=Mk^{+}_{N,n}}^{n-1}\varepsilon_{n}\frac{u^{n}_{i+1}-u^{n}_{i}}{\varepsilon_{n}}+\sum_{j=0}^{N}\sum_{i=Mk^{-}_{j,n}}^{Mk^{+}_{j,n}-1}\varepsilon_{n}\frac{u^{n}_{i+1}-u^{n}_{i}}{\varepsilon_{n}}\right|$
$\displaystyle\leq$ $\displaystyle
R\varepsilon_{n}\bigg{(}M\sum_{j=0}^{N}(k^{+}_{j,n}-k^{-}_{j,n})+n-Mk^{+}_{N,n}\bigg{)}.$

Sending $n$ to $+\infty$ we have

$\left|\ell-\sum_{j=1}^{N}\alpha_{j}(x_{j}-x_{j-1}-2\delta)\right|\leq
2RN\delta$

Finally, by the arbitrariness of $\delta>0$ we must have
$\ell=\sum_{j=1}^{N}\alpha_{j}(x_{j}-x_{j-1})$. ∎

######  Remark 3.5.

Let $\Phi=\Phi^{(\ell)}$ be as in Definition 2.8. We note that the minimum is
well defined because of Weierstrass’ Theorem. Moreover, the infimum in
$N\in\mathbb{N}$ in (2.8) can be replaced with the limit as
$N\rightarrow+\infty$ because of the decreasing monotonicity. The terms inside
the sums are $0$ for $i\geq N$ or $i\leq-N-M$ when
$\mathbf{z},\mathbf{z}^{\prime}\in M_{\ell}$. The function
$\Phi\colon\mathbf{M}_{\ell}\times\mathbf{M}_{\ell}\rightarrow[0,\infty]$ is a
subadditive function. In particular the functional defined by the right-hand
side of (18) is lower-semicontinuous.

######  Remark 3.6.

Given $\xi\in\mathbb{R}$, we can consider

$\displaystyle\Phi(\mathbf{z},\mathbf{z}^{\prime},\xi):=\inf_{N\in\mathbb{N}}\min_{u}\bigg{\\{}$
$\displaystyle\sum_{i\in\mathbb{Z}}\bigg{(}\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}(u_{i+k+1}-u_{i+k})-r_{\ell}\Big{(}\dfrac{u_{i+M}-u_{i}}{M}\Big{)}\bigg{)}:$
$\displaystyle
u:\mathbb{Z}\rightarrow\mathbb{R},\,u=u_{\mathbf{z}}+\xi_{1}\text{ for
}i\leq-N,\,u=u_{\mathbf{z}^{\prime}}+\xi_{2}\text{ for }i\geq
N,\xi=\xi_{2}-\xi_{1}\Big{\\}}$ (23)
$\displaystyle=\inf_{N\in\mathbb{N}}\min_{u}\bigg{\\{}$
$\displaystyle\sum_{i\in\mathbb{Z}}\bigg{(}\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}(u_{i+k+1}-u_{i+k})-r_{\ell}\Big{(}\dfrac{u_{i+M}-u_{i}}{M}\Big{)}\bigg{)}$
$\displaystyle u:\mathbb{Z}\rightarrow\mathbb{R},\,u=u_{\mathbf{z}}\text{ for
}i\leq-N,\,u=u_{\mathbf{z}^{\prime}}+\xi\text{ for }i\geq N\Big{\\}}.$ (24)

The two formulas are equal because the sums which are involved are invariant
under vertical translations of $u$.

Now note that
$\Phi(\mathbf{z},\mathbf{z}^{\prime},\xi)=\Phi(\mathbf{z},\mathbf{z}^{\prime})$
for every $\xi\in\mathbb{R}$, under the assumption that $\psi_{M},\psi_{1}$
are $C^{1}$ functions in a neighbourhood of $\mathbf{M}_{\ell}$. Indeed, if
$(u^{N})_{N}$ is a minimizing sequence in the definition of
$\Phi(\mathbf{z},\mathbf{z}^{\prime},\xi)$, then we can consider the following
new sequence

$\tilde{u}^{2N}=\begin{cases}u_{\mathbf{z}^{\prime}}&\text{ for }i\geq 2N\\\
u_{\mathbf{z}^{\prime}}+2\xi-i\dfrac{\xi}{N}&\text{ for }N\leq i\leq 2N\\\
u^{N}&\text{ for }i\leq N.\end{cases}$

Observe that varying $N\in\mathbb{N}$, all $\tilde{u}^{2N}$’s are competitors
in the definition of $\Phi(\mathbf{z},\mathbf{z}^{\prime})$. Let
$\mathbf{z}\in\mathbf{M}^{\alpha}$ and
$\mathbf{z}^{\prime}\in\mathbf{M}^{\alpha^{\prime}}$ wih
$\alpha,\alpha^{\prime}\in\mathbb{R}$ such that $\mathbf{M}^{\alpha}\subseteq
M_{\ell}$ and $\mathbf{M}^{\alpha^{\prime}}\subseteq M_{\ell}$. We compute the
difference between the involved sums for $\tilde{u}^{2N}$ and the ones for
$u_{N}$ by

$\sum_{i=N}^{2N}\bigg{(}\psi_{M}\Big{(}\alpha^{\prime}-\dfrac{\xi}{N}\Big{)}+\dfrac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\Big{(}z_{k}^{\prime}-\dfrac{\xi}{N}\Big{)}-r_{\ell}\Big{(}\alpha^{\prime}-\dfrac{\xi}{N}\Big{)}\bigg{)}+o\Big{(}\frac{1}{N}\Big{)}.$
(25)

Now observe that the $C^{1}$ function
$f(\eta)=\psi_{M}\left(\alpha^{\prime}-\eta\right)+\dfrac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\left(z_{k}^{\prime}-\eta\right)-r_{\ell}(\alpha^{\prime}-\eta)$
is always non-negative and it is equal to $0$ if $\eta=0$, so that $\eta=0$ is
a minimum point for $f$. Hence by Fermat’s Theorem $f^{\prime}(0)=0$ and so

$\lim_{N\rightarrow+\infty}\sum_{i=N}^{2N}\bigg{(}\psi_{M}\Big{(}\alpha^{\prime}-\dfrac{\xi}{N}\Big{)}+\dfrac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\Big{(}z_{k}^{\prime}-\dfrac{\xi}{N}\Big{)}-r_{\ell}\Big{(}\alpha^{\prime}-\dfrac{\xi}{N}\Big{)}\bigg{)}=0.$

In particular, we proved that
$\Phi(\mathbf{z},\mathbf{z^{\prime}})\leq\Phi(\mathbf{z},\mathbf{z^{\prime}},\xi)$.
Arguing in the same way one can show the reverse inequality.

######  Remark 3.7.

A last useful remark is that $\Phi(\mathbf{z},\mathbf{z}^{\prime})$ is stable
under cyclic permutations of the entries of $\mathbf{z}$ and
$\mathbf{z}^{\prime}$. More precisely, if we define for $q=1,\dots,M-1$ the
cyclic permutation $\sigma_{q}:\mathbf{M}_{\ell}\rightarrow\mathbf{M}_{\ell}$
given by

$\sigma_{q}:\mathbf{z}=(z_{1},\dots,z_{M})\longmapsto\sigma_{q}(\mathbf{z}):=(z_{M+1-q},\dots,z_{M},z_{1},\dots,z_{M-q}),$
(26)

then
$\Phi(\sigma_{q}(\mathbf{z}),\sigma_{q}(\mathbf{z}^{\prime}))=\Phi(\mathbf{z},\mathbf{z}^{\prime})$.
Indeed, if $u$ is a competitor for the minimum problem in
$\Phi(\mathbf{z},\mathbf{z}^{\prime})$, then the translation
$(T_{q}u)_{i}=u_{i-q}$ preserves the value of the objective function in the
definition of $\Phi$ and it satisfies

$T_{q}u=u_{\sigma_{q}(\mathbf{z})}\text{ for }i\leq-N-q,\quad\quad
T_{q}u=u_{\sigma_{q}(\mathbf{z}^{\prime})}\text{ for }i\geq N-q.$

Thus, $T_{q}u$ is a competitor for the minimum problem in
$\Phi(\sigma_{q}(\mathbf{z}),\sigma_{q}(\mathbf{z}^{\prime}))$. Clearly we can
argue in the same way with the inverse translation, proving the claim.

######  Theorem 3.8 (first-order $\Gamma$-limit with periodic boundary
conditions).

Assume that $\psi_{M}$ is a non-degenerate strictly convex $C^{1}$ function,
$\psi_{1}$ is of class $C^{1}$ in a neighbourhood of $\mathbf{M}_{\ell/L}$ and
$\psi_{0}^{**}$ is differentiable in $\frac{\ell}{L}$. Constraining $n\equiv
q\;\text{mod }M$, we define the domain

$\displaystyle
D_{q}^{\\#}=\\{\mathbf{u}:\mathbb{R}\rightarrow\mathbb{R}^{M}\,|\,$
$\displaystyle\mathbf{u}\text{ piecewise
affine},\,\mathbf{u}^{\prime}(t)\in\mathbf{M}_{\ell/L}\;\hbox{ for all
}t\in\mathbb{R}\setminus S(\mathbf{u}^{\prime}),$
$\displaystyle\\#(S(\mathbf{u}^{\prime})\cap(0,L])<+\infty,\,\sum_{k=1}^{M}u_{k}(t)-M\frac{\ell}{L}t\;\text{
L-periodic,}$ $\displaystyle u^{\prime}_{k}(t)=u^{\prime}_{[q+k]_{\text{mod
}M}}(t+L)\text{ for }k=0,\dots,M-1\\}.$

Then $E^{1}_{n,M}$ $\Gamma$-converges, with respect to $L^{\infty}$ topology,
to

$E^{1}_{q}(\mathbf{u})=\sum_{t\in
S(\mathbf{u}^{\prime})\cap(0,1]}\Phi^{(\ell/L)}(\mathbf{u}^{\prime}(t-),\mathbf{u}^{\prime}(t+)),$

with the domain dom$(E_{q}^{1,\\#})=D^{\\#}_{q}$.

###### Proof.

We first consider the case in which $L=1$ and $n\equiv 0$ mod $M$.

Lower bound. Let $u^{n}\rightarrow u$ in $L^{\infty}(0,1)$ be such that
$E^{1,\ell}_{n,M}(u^{n})\leq C<+\infty$ for every $n\in\mathbb{N}$. Then, by
Proposition 3.2, there exists a finite set $S:=\\{x_{1},\dots
x_{N}\\}\subset(0,1]$, with $0<x_{1}<\dots x_{N-1}<x_{N}\leq 1$, and there
exist $\mathbf{z}_{1},\dots,\mathbf{z}_{N}\in M_{\ell}$ such that $u^{n}$ (up
to subsequences) converges in $W^{1,\infty}_{\rm
loc}(\mathbb{R}\setminus(S+\mathbb{Z}))$ to a M-tuple $\mathbf{u}\in
D^{\\#}_{0}$ such that
$\mathbf{u}^{\prime}(t)=\mathbf{z}_{j}\in\mathbf{M}^{\alpha_{j}}$ for
$t\in(x_{j-1}+k,x_{j}+k)$ and for all $k\in\mathbb{Z}$ for
$j\in\\{1,\dots,N\\}$, with $x_{0}=x_{N}-1$. For $j\in\\{1,...,N\\}$, let
$(k_{n}^{j})_{n}$ be a sequence of natural numbers divisible by $M$ such that

$\lim_{n\rightarrow+\infty}k_{n}^{j}\varepsilon_{n}-x_{j}=0.$ (27)

Moreover, let $(h^{j}_{n})_{n}$ be a sequence in $M\mathbb{N}$ such that

$\lim_{n\rightarrow+\infty}\varepsilon_{n}h_{n}^{j}=\frac{x_{j}+x_{j-1}}{2}.$
(28)

Now, we can write

$\displaystyle
E^{1}_{n,M}(u^{n})=\sum_{j=1}^{N}\sum_{i=h_{n}^{j}}^{h_{n}^{j+1}-1}\left(\psi_{M}\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{s=0}^{M-1}\psi_{1}\left(\frac{u^{n}_{i+s+1}-u^{n}_{i+s}}{\varepsilon_{n}}\right)-r\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)\right),$
(29)

where $r=r_{\ell}$, and we have used the notation $h_{n}^{N+1}=h^{1}_{n}+n$
and that by periodicity we can choose the endpoints $h_{n}^{1}$ and
$h_{n}^{1}+n$ without changing the sum. In order to recover $\Phi$, we define

$\tilde{u}^{n}_{i}=\left\\{\begin{aligned}
&u_{\mathbf{z}_{j}}(i)-u_{\mathbf{z}_{j+1}}(h^{j}_{n}-k^{j}_{n})+\frac{u^{n}_{h^{j}_{n}}}{\varepsilon_{n}}\quad&\text{for
}i\leq h^{j}_{n}-k^{j}_{n}\\\
&\frac{u^{n}_{i+k^{j}_{n}}}{\varepsilon_{n}}\quad&\text{for
}h^{j}_{n}-k^{j}_{n}\leq i\leq h^{j+1}_{n}-k^{j}_{n}\\\
&u_{\mathbf{z}_{j+1}}(i)-u_{\mathbf{z}_{j+1}}(h^{j+1}_{n}-k^{j}_{n})+\frac{u^{n}_{h^{j+1}_{n}}}{\varepsilon_{n}}\quad&\text{for
}i\geq h^{j+1}_{n}-k^{j}_{n}.\end{aligned}\right.$ (30)

Since $\mathbf{z}_{j}\in\mathbf{M}^{\alpha_{j}}$, we note that for $i\geq
h^{j+1}_{n}-k^{j}_{n}$, we have

$\psi_{M}\left(\frac{\tilde{u}^{n}_{i+M}-\tilde{u}^{n}_{i}}{M}\right)+\frac{1}{M}\sum_{s=0}^{M-1}\psi_{1}\left(\tilde{u}^{n}_{i+s+1}-\tilde{u}^{n}_{i+s}\right)=\psi_{0}(\alpha_{j+1})=r(\alpha_{j+1})=r\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)$

and similarly for $i\leq h^{j}_{n}-k^{j}_{n}-M$. For $h^{j}_{n}-k^{j}_{n}\leq
i\leq h^{j+1}_{n}-k^{j}_{n}-M$, instead, we have

$\tilde{u}^{n}_{i+1}-\tilde{u}^{n}_{i}=\frac{u^{n}_{i+k^{j}_{n}+1}-u^{n}_{i+k^{j}_{n}}}{\varepsilon_{n}}\quad\text{and}\quad\frac{\tilde{u}^{n}_{i+M}-\tilde{u}^{n}_{i}}{M}=\frac{u^{n}_{i+k^{j}_{n}+M}-u^{n}_{i+k^{j}_{n}}}{M\varepsilon_{n}}.$

Then, defining
$\xi^{n}_{j}=u_{\mathbf{z}_{j+1}}(h^{j+1}_{n}-k^{j}_{n})-u_{\mathbf{z}_{j+1}}(h^{j}_{n}-k^{j}_{n})-\frac{u^{n}_{h^{j+1}_{n}}-u^{n}_{h^{j}_{n}}}{\varepsilon_{n}}$,
we can rewrite

$\displaystyle\sum_{i=h_{n}^{j}}^{h_{n}^{j+1}-1}\left(\psi_{M}\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{s=0}^{M-1}\psi_{1}\left(\frac{u^{n}_{i+s+1}-u^{n}_{i+s}}{\varepsilon_{n}}\right)-r\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)\right)$
$\displaystyle=$ $\displaystyle\
\sum_{i\in\mathbb{Z}}\Bigg{(}\psi_{M}\left(\frac{\tilde{u}^{n}_{i+M}-\tilde{u}^{n}_{i}}{M}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}(\tilde{u}_{i+k+1}^{n}-\tilde{u}^{n}_{i+k})-r\left(\frac{\tilde{u}^{n}_{i+M}-\tilde{u}^{n}_{i}}{M}\right)\Bigg{)}+\omega_{n}$
$\displaystyle\geq$ $\displaystyle\
\Phi\left(\mathbf{z}_{j},\,\mathbf{z}_{j+1},\,\xi_{j}^{n}\right)+\omega_{n}=\Phi\left(\mathbf{z}_{j},\,\mathbf{z}_{j+1}\right)+\omega_{n}.$

The error $\omega_{n}$ comes from the difference in behaviour between
$\tilde{u}^{n}_{i}$ and $u^{n}_{i+k^{j}_{n}}$ when
$h^{j+1}_{n}-k^{j}_{n}-M<i<h^{j+1}_{n}-k^{j}_{n}$ or
$h^{j}_{n}-k^{j}_{n}-M<i<h^{j}_{n}-k^{j}_{n}$; more precisely, $\omega_{n}$ is
the sum, for those $i$, of terms of the kind

$\begin{cases}\psi_{M}\bigg{(}\dfrac{u^{n}_{i+k^{j}_{n}+M}-u^{n}_{i+k^{j}_{n}}}{M\varepsilon_{n}}\bigg{)}-\psi_{M}\left(\dfrac{\tilde{u}^{n}_{i+M}-\tilde{u}^{n}_{i}}{M}\right)&\text{for
}h^{j+1}_{n}-k^{j}_{n}-M<i<h^{j+1}_{n}-k^{j}_{n},\\\
-\psi_{M}\left(\dfrac{\tilde{u}^{n}_{i+M}-\tilde{u}^{n}_{i}}{M}\right)&\text{for
}h^{j}_{n}-k^{j}_{n}-M<i<h^{j}_{n}-k^{j}_{n}.\end{cases}$

and similar terms for $\psi_{1}$. To show that $\lim_{n}\omega_{n}=0$, from
the continuity of $\psi_{M}$, $\psi_{1}$ and $r$, it is sufficient to show
that

$\begin{cases}\lim_{n}\left|\dfrac{u^{n}_{i+k^{j}_{n}+1}-u^{n}_{i+k^{j}_{n}}}{\varepsilon_{n}}-u_{\mathbf{z}_{j+1}}(i+1)-u_{\mathbf{z}_{j+1}}(i)\right|=0,&\text{for
}i\in(h^{j+1}_{n}-k^{j}_{n}-M,h^{j+1}_{n}-k^{j}_{n}+M)\\\
\lim_{n}\left|\dfrac{u^{n}_{i+k^{j}_{n}+1}-u^{n}_{i+k^{j}_{n}}}{\varepsilon_{n}}-u_{\mathbf{z}_{j}}(i+1)-u_{\mathbf{z}{j}}(i)\right|=0,&\text{for
}i\in(h^{j}_{n}-k^{j}_{n}-M,h^{j}_{n}-k^{j}_{n}+M)\end{cases}.$

Let $i_{M}$ denote the residual class of $i$ with respect to $M$. Then, by the
definition of $u_{\mathbf{z}_{j+1}}$ as in (17) we have

$u_{\mathbf{z}_{j+1}}(i+1)-u_{\mathbf{z}_{j+1}}(i)=(\mathbf{z}_{j+1})_{i_{M}+1}.$

On the other hand, since there exists a compact around the midpoint of
$(x_{j},x_{j+1})$ which contains
$\varepsilon_{n}(h^{j+1}_{n}-M,h^{j+1}_{n}+M)$ for all $n$, we can use the
locally uniform convergence of $\mathbf{u}_{n}^{\prime}$ to
$\mathbf{u}^{\prime}$ to gain

$\dfrac{u_{i+1+k_{n}^{j}}^{n}-u^{n}_{i+1+k_{n}^{j}}}{\varepsilon_{n}}=(\mathbf{u}^{n})^{\prime}_{i_{M}+1}(\varepsilon_{n}(i+k_{n}^{j}))\longrightarrow\mathbf{u}^{\prime}_{i_{M}+1}\left(\frac{x_{j+1}+x_{j}}{2}\right)=(\mathbf{z}_{j+1})_{i_{M}+1}\qquad\text{for
}n\rightarrow+\infty.$

The case $i\in(h^{j}_{n}-k^{j}_{n}-M,h^{j}_{n}-k^{j}_{n}+M)$ is analogous.
This proves the claim, and consequently that $\lim_{n}\omega_{n}=0$. In
particular, putting everything together we conclude

$\liminf_{n}E^{1}_{n,M}(u^{n})\geq\sum_{j=1}^{N}\Phi\left(\mathbf{z}_{j},\,\mathbf{z}_{j+1}\right).$

In the case $q\neq 0$, the only thing that changes is that we cannot define
$h^{N+1}_{n}=h^{1}_{n}+n$, because in this way it will not be divisible by
$M$, thus, we take $h^{N+1}_{n}$ the highest index multiple of $M$ below
$h^{1}_{n}+n$. Consequently, in the decomposition (29) there will appear $q$
residual terms of the kind:

$\psi_{M}\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{s=0}^{M-1}\psi_{1}\left(\frac{u^{n}_{i+s+1}-u^{n}_{i+s}}{\varepsilon_{n}}\right)-r\left(\frac{u^{n}_{i+M}-u^{n}_{i}}{M\varepsilon_{n}}\right),$

but since $i\varepsilon_{n}$ is close to $1+\frac{x_{1}+x_{0}}{2}$; that is,
far from any critical point $x_{j}$, by continuity we have that they tend to
$0$.

Upper bound. Let $\mathbf{u}$ be such that $E^{1}_{0}(\mathbf{u})<\infty$ and
suppose , without loss of generality, that $\mathbf{u}(0)=0$, then by
definition there exist $N\in\mathbb{N}$,
$\mathbf{z}_{1},\dots\mathbf{z}_{N}\in\mathbf{M}_{\ell}$ and we can write
$S({\mathbf{u}}^{\prime})\cap(0,1]=\\{x_{1},\dots x_{N}\\}$, with
$0<x_{1}<\dots x_{N-1}<x_{N}\leq 1$,
$\mathbf{u}^{\prime}(t)=\mathbf{z}_{j}\in\mathbf{M}^{\alpha_{j}}$ for
$t\in(x_{j-1}+k,x_{j}+k)$ and for all $k\in\mathbb{Z}$, with $x_{0}=x_{N}-1$.
Up to translations, which do not change the energy, we can always suppose that
$1\notin S(\mathbf{u}^{\prime})$; that is, $x_{N}<1$. Moreover, we have that

$E^{1}_{0}(\mathbf{u})=\sum_{j=1}^{N}\Phi(\mathbf{z}_{j-1},\mathbf{z}_{j}).$

We first consider the case $q=0$. With fixed $\varepsilon>0$, there exists
$\tilde{N}=\tilde{N}(\varepsilon)\in\mathbb{N}$ multiple of $M$ and $N$
discrete functions $v^{j}$ such that

$\displaystyle\Phi(\mathbf{z}_{j-1},\mathbf{z}_{j})$ $\displaystyle\leq$
$\displaystyle\sum_{i\in\mathbb{Z}}\psi_{M}\Big{(}\frac{v^{j}_{i+M}-v^{j}_{i}}{M}\Big{)}+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}(v^{j}_{i+k+1}-v^{j}_{i+k})-r_{\ell}\Big{(}\dfrac{v^{j}_{i+M}-v^{j}_{i}}{M}\Big{)}$
(31) $\displaystyle\leq$
$\displaystyle\Phi(\mathbf{z}_{j-1},\mathbf{z}_{j})+\varepsilon$

and

$v^{j}=\begin{cases}u_{\mathbf{z}_{j-1}}&\text{ for }i\leq-\tilde{N}\\\
u_{\mathbf{z}_{j}}&\text{ for }i\geq\tilde{N}\end{cases}$

for every $j\in\\{1,\ldots N\\}$. Consider now the sequence of functions
$(u^{n})_{n}$ defined as follows

$u^{n}_{i}=\begin{cases}\varepsilon_{n}v^{1}_{i-k^{n}_{N}+n}-\varepsilon_{n}v^{1}_{n-k^{n}_{N}}&\text{
for }0\leq i\leq k^{n}_{1}-\tilde{N}\\\
\varepsilon_{n}v^{2}_{i-k^{n}_{1}}+\varepsilon_{n}D_{2}&\text{ for
}k^{n}_{1}-\tilde{N}\leq i\leq k^{n}_{2}-\tilde{N}\\\ \dots\\\
\varepsilon_{n}v^{N}_{i-k^{N-1}_{n}}+\varepsilon_{n}D_{N}&\text{ for
}k^{n}_{N-1}-\tilde{N}\leq i\leq k^{n}_{N}-\tilde{N}\\\
\varepsilon_{n}v^{1}_{i-k^{N}_{n}}+\varepsilon_{n}D_{1}&\text{ for
}k^{n}_{N}-\tilde{N}\leq i\leq n,\end{cases}$ (32)

where $k^{n}_{j}:=\min\\{k\in\mathbb{N}:k\geq x_{j}n$ and $k$ is multiple of
$M\\}$ and

$\displaystyle
D_{2}=v^{1}_{k_{1}^{n}-k_{N}^{n}+n-\tilde{N}}-v^{2}_{-\tilde{N}}-v^{1}_{n-k^{n}_{N}}$
$\displaystyle
D_{3}=D_{2}+v^{2}_{k^{n}_{2}-k^{n}_{1}-\tilde{N}}-v^{3}_{-\tilde{N}}$
$\displaystyle\dots$ $\displaystyle
D_{N}=D_{N-1}+v^{N-1}_{k^{n}_{N-1}-k^{n}_{N-2}-\tilde{N}}-v^{N}_{-\tilde{N}}$
$\displaystyle
D_{1}=D_{N}+v^{N}_{k^{n}_{N}-k^{n}_{N-1}-\tilde{N}}-v^{1}_{-\tilde{N}}.$

We note that
$u^{n}_{n}-u^{n}_{0}=\sum_{j=1}^{N}\alpha_{j}(k_{j}^{n}-k^{n}_{j-1})\varepsilon_{n}$,
which in general is different from $\ell$; therefore, in order to adjust the
periodicity conditions, we apply a linear correction term
$\tilde{u}^{n}_{i}=u^{n}_{i}+\delta_{n}\frac{i}{n}$, where
$\delta_{n}=\ell-\sum_{j=1}^{N}\alpha_{j}(k_{j}^{n}-k^{n}_{j-1})\varepsilon_{n}$.
Since now the the boundary condition is satisfied, we can extend
$\tilde{u}^{n}$ to $\mathbb{R}$ such that $\tilde{u}^{n}(t)-\ell t$ is
$1$-periodic. From Proposition 3.4 and the fact that $|nx_{j}-k^{n}_{j}|\leq
M$ we have

$\delta_{n}=\sum_{j=1}^{N}\alpha_{j}(x_{j}-x_{j-1}-k^{n}_{j}\varepsilon_{n}+k^{n}_{j-1}\varepsilon_{n})=O(\varepsilon_{n})\implies\exists
C>0:\;\ (k^{n}_{j}-k^{n}_{j-1})\delta_{n}\leq C\ \forall j,n.$ (33)

Since $\mathbf{u}^{n}$ is converging uniformly to $\mathbf{u}$ and
$\delta_{n}x\rightarrow 0$ uniformly in $x\in[0,1]$, we have that
$\tilde{u}^{n}\rightarrow\mathbf{u}$ in $L^{\infty}([0,1])$ in the sense of
Definition 2.6. As for the convergence of the energy, if we let

$\mathcal{E}_{i}^{n}(u)=\psi_{M}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right)+\frac{1}{M}\sum_{k=0}^{M-1}\psi_{1}\left(\frac{u_{i+k+1}-u_{i+k}}{\varepsilon_{n}}\right)-r_{\ell}\left(\frac{u_{i+M}-u_{i}}{M\varepsilon_{n}}\right),$

then, by (31) we have

$E^{1}_{0}(\mathbf{u})+\varepsilon\geq
E^{1}_{n,M}(\tilde{u}^{n})+\sum_{j=1}^{N}\Bigg{(}R_{n}^{j}-\sum_{i=k^{n}_{j-1}+\tilde{N}}^{k^{n}_{j}-\tilde{N}-M}\mathcal{E}^{n}_{i}(\tilde{u}^{n})\Bigg{)},$

where
$R_{n}^{j}=\sum_{i=-\tilde{N}-M+1}^{\tilde{N}-1}\mathcal{E}^{n}_{i}(\varepsilon_{n}v^{j})-\mathcal{E}^{n}_{i+k^{n}_{j-1}}(\tilde{u}^{n})$.
However, this $R_{n}^{j}$ involves only a finite number of indices,
independently from $n$; thus, since
$\frac{\tilde{u}^{n}_{i+M}-\tilde{u}^{n}_{i}}{M\varepsilon_{n}}=\frac{v^{j}_{i+M}-v^{j}_{i}}{M}+\delta_{n}$
and
$\frac{\tilde{u}^{n}_{i+1}-\tilde{u}^{n}_{i}}{\varepsilon_{n}}=v^{j}_{i+1}-v^{j}_{i}+\delta_{n}$,
by continuity of $\psi_{1},\psi_{M}$ and $r_{\ell}$ we have

$\lim_{n}\sum_{j=1}^{N}R^{j}_{n}=0.$

Instead, for $i\in\\{k^{n}_{j-1}-\tilde{N},\dots,k^{n}_{j}-\tilde{N}-M\\}$ we
have $v^{j}_{i-k^{n}_{j-1}}=u_{\mathbf{z}_{j}}(i-k^{n}_{j-1})$ and then

$\mathcal{E}^{n}_{i}(\tilde{u}^{n})=\psi_{M}\left(\alpha_{j}+\delta_{n}\right)+\frac{1}{M}\sum_{k=1}^{M}\psi_{1}\left((\mathbf{z}_{j})_{k}+\delta_{n}\right)-r_{\ell}\left(\alpha_{j}+\delta_{n}\right)=:f_{j}(\delta_{n}).$

Note that we lost the dependence on $i$, so that, by (33), we have

$\sum_{i=k^{n}_{j-1}+\tilde{N}}^{k^{n}_{j}-\tilde{N}-M}\mathcal{E}^{n}_{i}(\tilde{u}^{n})=(k^{n}_{j}-k^{n}_{j-1}-M)f_{j}(\delta_{n})=(k^{n}_{j}-k^{n}_{j-1}-M)\delta_{n}\left(\frac{f_{j}(\delta_{n})}{\delta_{n}}\right)\leq
C\frac{f_{j}(\delta_{n})}{\delta_{n}}.$

By Fermat’s Theorem we can conclude that
$\lim_{n}\frac{f_{j}(\delta_{n})}{\delta_{n}}=f^{\prime}(0)=0$; in conclusion

$E^{1}_{0}(\mathbf{u})+\varepsilon\geq\limsup_{n}E^{1}_{n,M}(\tilde{u}^{n}),$

and the claim is proved by the arbitrariness of $\varepsilon$.

When $q\neq 0$, the proof proceeds similarly, but we have to pay attention to
the boundary mismatch in the cycle of slopes in $\tilde{u}$. This can be fixed
by taking into account that, by Remark 3.7, we can write the energy as

$E^{1}_{q}(\mathbf{u})=\sum_{j=1}^{N}\Phi(\sigma_{q}(\mathbf{z}_{j-1}),\sigma_{q}(\mathbf{z}_{j})),$

where $\sigma_{q}$ is defined as in (26). Then, we can find some $\tilde{N}$
multiple of $M$ and some $v_{j}$ as in (31) with the condition

$v^{j}=\begin{cases}u_{\sigma_{q}(\mathbf{z}_{j-1})}&\text{for
}i\leq-\tilde{N}\\\ u_{\sigma_{q}(\mathbf{z}_{j})}&\text{for
}i\geq\tilde{N}.\end{cases}$

Hence, we define $k^{n}_{j}:=\min\\{k\in\mathbb{N}:k\geq x_{j}n$ and $k\equiv
q$ mod $M\\}$ and we choose $u^{n}$ as in (32). In this way, the
interpolations $\mathbf{u}^{n}$ are still converging uniformly to $\mathbf{u}$
because

$u_{\sigma_{q}(\mathbf{z}_{j})}(i+k_{j}^{n}+1)-u_{\sigma_{q}(\mathbf{z}_{j})}(i+k_{j}^{n})=u_{\sigma_{q}(\mathbf{z}_{j})}(i+q+1)-u_{\sigma_{q}(\mathbf{z}_{j})}(i+q)=u_{\mathbf{z}_{j}}(i+1)-u_{\mathbf{z}_{j}}(i).$

Since $n-k_{N}^{n}$ is divisible by $M$ and
$\frac{u^{n}_{n}-u^{n}_{n-1}}{\varepsilon_{n}}=\sigma_{q}(\mathbf{z_{1}})_{M}$,
we can extend $u^{n}$ by periodicity on the whole $\mathbb{R}$ defining a
recovery sequence. ∎

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