functions determine a corresponding rational volume fraction $\theta$.

To illustrate the geometry of microstructures we restate periodic minimum
problems for bi-convex functions in terms of a spin representation. This will
allow us to rewrite non-convex minimum problems as minima of a family of
convex problems, and to obtain a better control of the geometry of minimizers.
We will use this formulation in some explicit examples in the next sections,
to characterize optimal periodic geometries.

We begin by formally introducing the spin variable
$\underline{s}\in\\{-1,1\\}^{N}$ parameterizing the location of the argument
of a bi-convex function $f$. The corresponding volume fraction is then

$\theta={1\over 2N}\sum_{j=1}^{N}(1-s_{j}).$

Let $f_{-1},f_{1}\colon\mathbb{R}\to\mathbb{R}$ be such that

$f(z)=\min\\{f_{-1}(z),f_{1}(z)\\}=\begin{cases}f_{-1}(z)&\hbox{\rm if }\
z\in\mathbb{R}\setminus A\\\ f_{1}(z)&\hbox{\rm if }\ z\in A.\end{cases}$
(3.19)

The slight difference in the notation with respect to previous sections, where
the two functions were denoted by $f_{0}$ and $f_{1}$, is due to the focus on
individual components of the spin vector taking the values $-1$ and $1$. While
the definitions and properties will hold without any further assumptions, in
the applications we will consider the ‘natural’ case when $A$ is a half-line
and the functions $f_{-1},f_{1}$ are convex.

Omitting the dependence on $A$, for $N\in\mathbb{N}$ we define
$\widehat{R}^{N}_{\bf m}f\colon\\{-1,1\\}^{N}\times\mathbb{R}\to\mathbb{R}$ by
setting

$\widehat{R}^{N}_{\bf
m}f(\underline{s},z)=\frac{1}{N}\inf\big{\\{}F^{\\#}(u,\underline{s};[0,N]):\
i\mapsto u_{i}-zi\hbox{ is }N\hbox{-periodic}\big{\\}},$ (3.20)

where

$F^{\\#}(u,\underline{s};[0,N])=\sum_{i=1}^{N}f_{s_{i}}(u_{i}-u_{i-1})+\sum_{i=1}^{N}\sum_{j\in\mathbb{Z}}m_{|i-j|}(u_{i}-u_{j})^{2}.$

Note that $\widehat{R}^{N}_{\bf m}f$ depends on the choice of $f_{1}$ and
$f_{-1}$ and not only on their minimum $f$.

######  Remark 3.20 (regularity with respect to $z$).

If $f_{1}$ and $f_{-1}$ are of class $C^{1}(\mathbb{R})$ then the function
$z\mapsto\widehat{R}^{N}_{\bf m}f(\underline{s},z)$ is of class
$C^{1}(\mathbb{R})$ for any fixed $\underline{s}\in\\{-1,1\\}^{N}$. This is a
direct consequence of the Euler-Lagrange equations characterizing the minumum
points of $F^{\\#}$.

Now we add the phase constraint, minimizing over all $\underline{s}$
corresponding to a given volume fraction, which eventually will give an
alternative chatacterization of $\widehat{Q}_{\bf m}f(\theta,z)$. More
precisely, fixed $\theta=\frac{p}{q}\in\mathbb{Q}\cap[0,1]$, for any $N\in
q\mathbb{Z}$ we define the function

$\Phi^{N}_{\bf m}f(\theta,z)=\min\\{\widehat{R}^{N}_{\bf
m}f(\underline{s},z):\ \underline{s}\in\mathcal{S}_{N}(\theta)\\},$ (3.21)

where $\mathcal{S}_{N}(\theta)$ is the set of admissible spin vectors

$\mathcal{S}_{N}(\theta)=\\{\underline{s}\in\\{-1,1\\}^{N}:\
\\#\\{i:s_{i}=1\\}=\theta N\\}$

and again we omit the dependence on $A$. Moreover, we define

$\Phi_{\bf m}f(\theta,z)=\liminf_{N\to+\infty}\Phi^{N}_{\bf m}f(\theta,z).$

The following proposition states that the analysis of $\widehat{Q}_{\bf
m}f(\theta,z)$ can be reduced to the periodic spin formulation giving
$\Phi_{\bf m}f(\theta,z)$.

######  Proposition 3.21 (periodic spin characterization of $\widehat{Q}_{\bf
m}f(\theta,z)$).

The following equality holds:

$\Phi_{\bf m}f(\theta,z)=\widehat{Q}_{\bf m}f(\theta,z)$

In particular the function $(\theta,z)\mapsto\Phi_{\bf m}f(\theta,z)$ is
convex.

###### Proof.

The inequality $\Phi_{\bf m}f(\theta,z)\geq\widehat{Q}_{\bf m}f(\theta,z)$
directly follows by definition. Conversely, given a minimum point $u$ for

$\displaystyle\widehat{Q}^{\delta,Nq}_{\bf
m}f(\theta,z)=\frac{1}{Nq}\inf\Big{\\{}F_{1}(u;[0,Nq]):u\in\widetilde{\mathcal{A}}_{\delta}(Nq;z)\cap\mathcal{V}(Nq;\theta)\Big{\\}},$

we can extend it to $\mathbb{Z}$ so that $u_{i}-zi$ is $Nq$-periodic. Using
this extended test function in the definition of $\Phi^{Nq}_{\bf
m}f(\theta,z)$, with the same computations as in the proof of Lemma 3.3 we
obtain

$\Phi^{Nq}_{\bf m}f(\theta,z)\leq\widehat{Q}^{\delta,Nq}_{\bf
m}f(\theta,z)+o(1)$

as $N\to+\infty$ and $\delta\to 0$. ∎

We are interested in those $\theta$ for which the constrained relaxation
$\widehat{Q}_{\bf m}f(\theta,z)$ is characterized by periodic minimization;
that is, for which there is an interval of $z$ such that the corresponding
optimal spin function $\underline{s}$ is periodic. Such $\underline{s}$ will
be locally $z$-independent, and this will allow to derive regularity
properties for $\widehat{Q}_{\bf m}f(\theta,z)$. For those special values of
$\theta$, we think of such functions $\widehat{Q}_{\bf m}f(\theta,\cdot)$ as
describing energy meta-wells. For brevity of notation, we directly say that
the corresponding value of $\theta$ is an energy well. As we are going to show
below, this concept is closely related to that of a locking state.

######  Definition 3.22 (energy meta-wells).

Let $f$ be as in (3.19) and let $\Phi^{N}_{\bf m}$ be as in (3.21). The value
$\theta\in[0,1]\cap\mathbb{Q}$ is an energy well of $f$ at $z$ (related to the
sequence ${\bf m}$) if there exists $N$ such that $N\theta\in\mathbb{Z}$ and

$\Phi^{N}_{\bf m}f(\theta,z)=\Phi_{\bf m}f(\theta,z).$ (3.22)

We say that $\theta$ is an energy well of $f$ in an open interval $I$ if there
exists $N$ such that (3.22) holds for all $z\in I$; if such $I$ exists, we say
that $\theta$ is a non-degenerate energy well of $f$. If $I=\mathbb{R}$, we
simply say that $\theta$ is an energy well of $f$.

Note that the definition a priori depends on $f_{1}$ and $f_{-1}$. However,
the condition that $f=\min\\{f_{1},f_{-1}\\}$ implies that in the minimization
procedure we may assume $f_{1}=+\infty$ outside $A$ and $f_{-1}=+\infty$
inside $A$, which shows that the definition indeed only depends on $f$.

######  Remark 3.23 (energy meta-wells and periodic solutions).

By Proposition 3.21 we also have that if $\theta\in[0,1]\cap\mathbb{Q}$ is an
energy well of $f$ at $z$ then

$\Phi^{N}_{\bf m}f(\theta,z)=\widehat{Q}_{\bf m}f(\theta,z).$

This implies the existence of periodic minimizers; that is, of test function
$u_{i}$ minimizing $\widehat{Q}_{\bf m}f(\theta,z)$ with $u_{i}-zi$
$N$-periodic.

######  Remark 3.24.

If $\theta$ is an energy well of $f$ at $z$, then there exists $N$ such that

$\Phi^{kN}_{\bf m}f(\theta,z)=\Phi^{N}_{\bf m}f(\theta,z)=\Phi_{\bf
m}f(\theta,z)$

for any $k\geq 1$.

We now examine the regularity of $\Phi_{\bf m}f$ at fixed $\theta$.

######  Proposition 3.25 (differentiability with respect to $z$).

If $\theta$ is an energy well of $f$ in an open interval $I$, then the
function $z\mapsto\Phi_{\bf m}f(\theta,z)$ is differentiable at any $z\in I$.

###### Proof.

Given $\theta$ an energy well in $I$ and the corresponding $N$ as in
Definition 3.22, note that $z\mapsto\Phi_{\bf m}f(\theta,z)$ is the minimum of
a finite number of $C^{1}$ functions, corresponding to
$\underline{s}\in\mathcal{S}_{N}(\theta)$. Since $z\mapsto\Phi_{\bf
m}f(\theta,z)$ is convex the derivatives of these functions must agree at the
intersections. ∎

A central question in the description of $\widehat{Q}_{\bf m}f$ is the
reduction to a set $X$ of $\theta$ such that the claim of Theorem 3.2 holds
taking the infimum only on $X$ and such that the computation of
$\widehat{Q}_{\bf m}f(\theta,z)$ can be carried on for $\theta\in X$. This is
the case for concentrated kernel. We will see in the examples that $\theta$ in
these $X$ are often energy wells. The following proposition shows that if such
an energy well is ‘essential’ then it is a locking state.

######  Proposition 3.26 (energy wells and locking states).

Let $X\subset[0,1]\cap\mathbb{Q}$ be such that

$\Big{(}\inf_{\theta\in X}\\{\widehat{Q}_{\bf
m}f(\theta,z)\\}\Big{)}^{\ast\ast}=\widehat{Q}_{\bf m}f(z)$ (3.23)

for all $z$, and let $\theta^{\ast}\in X$ be an energy well that is essential
in (3.23); that is, such that

$\Big{(}\inf_{\theta\in X\setminus\\{\theta^{\ast}\\}}\\{\widehat{Q}_{\bf
m}f(\theta,z)\\}\Big{)}^{\ast\ast}>\widehat{Q}_{\bf m}f(z)$ (3.24)

for some $z$. Then, $\theta^{\ast}$ is a locking state.

###### Proof.

We recall that $\widehat{Q}_{\bf m}f(\theta,z)=\Phi_{\bf m}f(\theta,z)$ by
Proposition 3.21. Since $\theta^{\ast}$ is an energy well, by Proposition
3.25, the function $z\mapsto\Phi_{\bf m}f(\theta^{\ast},z)$ is differentiable.
By the essentiality condition (3.24), that function cannot be tangent to
$\widehat{Q}_{\bf m}f$ in an isolated point, nor can be transversal to it.
Hence, it must coincide with $\widehat{Q}_{\bf m}f$ in an interval.∎

As for regularity properties of $Q_{\bf m}f$ with respect to $\theta$, we note
that in general locking states are points where the characterization of the
energy changes. This suggests that we may have a jump in the derivative at
these points.

######  Conjecture 3.27 (Non differentiability at the energy wells).

If $X\subset[0,1]\cap\mathbb{Q}$ is such that (3.23) holds for all $z$ and
$\theta^{\ast}\in X$ is an energy well satisfying (3.24), hen the function
$\theta\mapsto Q_{\bf m}f(\theta,z)$ is not differentiable in $z$ at
$\theta^{\ast}$.

This conjecture is reminiscent of regularity properties in dynamical systems,
where the global structure of minimizers can be used in the proofs, as in the
work of J. Mather [70]. Anyway, we will prove that it holds in the case
studies (see Remark 4.5 for the $M$-th neighbour case, and Remark 5.26 for the
truncated convex potential and exponential kernel).

######  Remark 3.28 (Generalized Cauchy-Born (GCB) states).

The spin representation of a microstructure allows one to effectively
parametrize periodic minimizers. Such a representation can be expected to
exist for locking states which can be viewed as examples of ‘global’
solutions. We can also interpret such states as respecting the generalized
Cauchy-Born (GCB) rule. To make the notion of the GCB rule more general we may
refer to the possibility of computing the macroscopic energy by solving an
appropriate boundary value problem on a finite representative ‘cell’. The
question arises in which cases any minimizer can be viewed as a GCB state in
the above sense or as a simple mixture (a convex combination) of such states.
We will see in the next sections that for broad classes of physically
interesting non-convex energies $f$ and the penalization kernels $\bf m$ only
GCB states are relevant.

##  4 Relaxation with concentrated-kernel penalization

In this section, we analyze the relaxation of a general bi-convex function $f$
with a concentrated kernel $\bf m$. We recall that in this case there exists
$M\neq 2$ such that $m_{n}=0$ for all $n\geq 2$ except for $n=M$ and that such
penalization leads to a non-additive problem (see Definition 2.15). We show
that the optimal microstructures in this case are restricted to periodic
states, corresponding to a fraction $\theta_{n}={n\over M}$ for
$n\in\\{0,\dots,M\\}$, and compatible mixtures of such periodic states
corresponding to neighbouring values of the phase fractions $\theta_{n}$ and
$\theta_{n+1}$, in other words, to first and second order laminates.

Following the notation of Section 3, let $z^{\ast}\in\mathbb{R}$,
$A=[z^{\ast},+\infty)$, and let $f\colon\mathbb{R}\to\mathbb{R}$ be such that
the restrictions of $f$ to $(-\infty,z^{\ast}]$ and $[z^{\ast},+\infty)$ are
convex. In this section, we again use the notation

$f_{2m_{1}}(z)=f(z)+2m_{1}z^{2}$ (4.1)

for the overall nearest-neighbour interactions.

We assume that growth hypothesis (2.6) holds, so that
$f_{2m_{1}}(z)\to+\infty$ as $z\to\pm\infty$. Note that the analysis can also
cover the degenerate case when this condition is not satisfied. As a model, in
Remark 4.6 we will consider the case of a truncated quadratic potential $f$
with $m_{1}=0$, highlighting the effect of degeneracy.

###  4.1 Formulas for the relaxation

In the case of a bi-convex $f$, formula of Proposition 2.16 describing
$\widehat{Q}_{\bf m}f$ can be further specified as follows

$\widehat{Q}_{\bf m}f(z)=\big{(}\min_{n}P^{M,n}(z)\big{)}^{\ast\ast},$ (4.2)

where for any $n\in\\{0,\dots,M\\}$ we let $\theta_{n}=\frac{n}{M}$ and
introduce

$\left.\begin{array}[]{ll}P^{M,n}(z)=&\displaystyle\min\Big{\\{}(1-\theta_{n})f_{2m_{1}}(z^{-})+\theta_{n}f_{2m_{1}}(z^{+}):z^{-}\leq
z^{\ast},z^{+}\geq z^{\ast},\\\ &\hskip
56.9055pt\displaystyle(1-\theta_{n})z^{-}+\theta_{n}z^{+}=z\Big{\\}}+2m_{M}(Mz)^{2}.\end{array}\right.$
(4.3)

Now we prove that for any rational $\theta$ the constrained function
$\widehat{Q}_{\bf m}f(\theta,z)$, defined in (3.3), can be also characterized
in terms of the functions $P^{M,n}$, which themselves correspond to particular
values of $\theta$, in the sense that $P^{M,n}(z)=\widehat{Q}_{\bf
m}f(\theta_{n},z)$.

######  Theorem 4.1 (shape of $\widehat{Q}_{\bf m}f$ and of the phase function
$\theta$).

There exists an ordered family of disjoint intervals $(s_{n}^{-},s_{n}^{+})$,
where $s_{0}^{-}=-\infty$ and $s_{M}^{+}=+\infty$, such that

(i) $\widehat{Q}_{\bf m}f(z)=P^{M,n}(z)$ in $(s_{n}^{-},s_{n}^{+})$ and it is
affine in each of the remaining intervals; that is, between $s_{n}^{+}$ and
$s_{n+1}^{-}$ for each $n$;

(ii) $\theta(z)=\theta_{n}$ in $(s_{n}^{-},s_{n}^{+})$ and it is affine in
each of the remaining intervals.

(iii) the set of the locking states of $f$ is $\\{\theta_{n}\\}$ and

$\widehat{Q}_{\bf m}f(z)=\big{(}\min\\{\widehat{Q}_{\bf
m}f(\theta,z):\theta\hbox{ is a locking state}\\}\big{)}^{\ast\ast}\,.$

###### Proof.

The proof of (i) and (ii) will follow from Lemma 4.4 below, while (iii) is
obtained by (4.2). ∎

######  Remark 4.2.

Note that if $\theta(z)=\theta_{n}$ then the value of $\widehat{Q}_{\bf
m}f(z)$ is attained on periodic minimizers. The phase function $\theta$ can be
explicitly written as

$\theta(z)=\begin{cases}\vspace{2mm}0&\hbox{\rm if }\ z\leq s_{0}^{+}\\\
\displaystyle\theta_{n}+\frac{1}{M}\frac{z-s_{n}^{+}}{s_{n+1}^{-}-s_{n}^{+}}&\hbox{\rm
if }\ s_{n}^{+}\leq z\leq s_{n+1}^{-}\\\ \displaystyle\theta_{n}&\hbox{\rm if
}\ s_{n}^{-}\leq z\leq s_{n}^{+}\\\ 1&\hbox{\rm if }\ s_{M}^{-}\leq
z.\end{cases}$ (4.4)

Moreover, if we write the convex envelope of the minimum of $P^{M,n}$ and
$P^{M,n+1}$ as

$\min\big{\\{}P^{M,n},P^{M,n+1}\big{\\}}^{\ast\ast}(z)=\begin{cases}P^{M,n}(z)&\hbox{\rm
if }\ z\leq s_{n}^{+}\\\ r^{M,n}(z)&\hbox{\rm if }\ s_{n}^{+}\leq z\leq
s_{n+1}^{-}\\\ P^{M,n+1}(z)&\hbox{\rm if }\ s_{n+1}^{-}\leq z,\end{cases}$
(4.5)

where $r^{M,n}$ is the interpolating affine function

$\displaystyle r^{M,n}(z)$ $\displaystyle=$ $\displaystyle
P^{M,n}(s_{n}^{+})+\frac{P^{M,n+1}(s_{n+1}^{-})-P^{M,n}(s_{n}^{+})}{s_{n+1}^{-}-s_{n}^{+}}(z-s_{n}^{+}),$

then $\widehat{Q}_{\bf m}f(z)=r^{M,n}(z)$ if $z\in[s_{n}^{+},s_{n+1}^{-}]$.
Note that this characterization of $\widehat{Q}_{\bf m}f$ holds under
assumption (2.6), while it may fail if this condition is dropped, as we show
in Remark 4.6 below.

The main technical point of this section is Lemma 4.4 giving an explicit
formula for the constrained minimizations involving only pairs of successive
locking states. The proof of this fact relies on the following algebraic
lemma.

######  Lemma 4.3 (an algebraic lemma).

Let $n\in[0,M-1]\cap\mathbb{N}$. If
$\theta\in\big{[}\frac{n}{M},\frac{n+1}{M}\big{]}$, then there exist
coefficients $\alpha^{n}_{k}$, $k=0,\dots,M$, such that
$\alpha^{n}_{k}\in[0,I_{k}]$ for any $k$ and

$\begin{cases}\displaystyle\sum_{k=0}^{M}\alpha^{n}_{k}=\frac{M\theta-n}{M}Nq\\\
\displaystyle\sum_{k=0}^{M}k\alpha^{n}_{k}=(n+1)\frac{M\theta-n}{M}Nq.\end{cases}$
(4.6)

###### Proof.

The linear system (4.6) has infinitely many solutions depending on $M-1$
parameters. We have to show that there exists one solution in
$\Pi_{k=0}^{M}[0,I_{k}]$. To this end, it is sufficient to show that the
hyperplane given by the equation

$H_{\lambda}(\alpha_{0}^{n},\dots,\alpha_{M}^{n})=\sum_{k=0}^{M}(\lambda+k)\alpha^{n}_{k}-(\lambda+n+1)\frac{M\theta-n}{M}Nq=0$

intersects $\Pi_{k=0}^{M}[0,I_{k}]$ for any $\lambda\in\mathbb{R}$, which
happens if for any $\lambda\in\mathbb{R}$ there exist two points
$v,w\in\mathbb{R}^{M+1}$ such that $H_{\lambda}(v)\ H_{\lambda}(w)\leq 0.$
Since $n\leq M\theta\leq n+1$ and

$H_{\lambda}(0,\dots,0)=-(\lambda+n+1)\frac{M\theta-n}{M}Nq,\quad
H_{\lambda}(I_{0}^{n},\dots,I_{M}^{n})=(\lambda+n)\frac{(n+1-M\theta)}{M}Nq,$

we get $H_{\lambda}(0,\dots,0)\ H_{\lambda}(I_{0}^{n},\dots,I_{M}^{n})\leq 0$
if $\lambda\leq-(n+1)\ \hbox{ or }\lambda\geq-n$.

For the remaining cases, we note that by (4.8)

$(n+1)\sum_{k=0}^{M}(M-k)I_{k}-(M-(n+1))\sum_{k=0}^{M}kI_{k}=\big{(}(n+1)-M\theta\big{)}Nq.$

Since $(n+1)(M-k)-(M-(n+1))k=M(n+1-k)\leq 0$ if $k\geq n+1$, we get

$\sum_{k=0}^{n}(n+1-k)I_{k}\geq\frac{(n+1)-M\theta}{M}Nq.$

If we choose $v=(v_{0},\dots,v_{k})$ with $v_{k}=0$ if $k\leq n$ and
$v_{k}=I_{k}$ if $k>n$ we obtain

$\displaystyle H_{-(n+1)}(v)$ $\displaystyle=$
$\displaystyle-(n+1)\Big{(}\sum_{k=0}^{M}I_{k}-\sum_{k=0}^{n}I_{k}\Big{)}+\sum_{k=0}^{M}kI_{k}-\sum_{k=0}^{n}kI_{k}$
$\displaystyle=$
$\displaystyle\frac{M\theta-(n+1)}{M}Nq+\sum_{k=0}^{n}(n+1-k)I_{k}\geq 0.$

Noting that

$H_{-n}(v)=\frac{M\theta-(n+1)}{M}Nq+\sum_{k=0}^{n}(n-k)I_{k}-\frac{M\theta-(n+1)}{M}Nq\geq
0,$

it follows that $H_{\lambda}(v)\geq 0$ for any $\lambda\in(-(n+1),-n)$. Since
$H_{\lambda}(0,\dots,0)\leq 0$, this concludes the proof of Lemma 4.3. ∎

Now, we state the interpolation lemma.

######  Lemma 4.4 (interpolation between locking states).

Let $\theta\in[\theta_{n},\theta_{n+1}]\cap\mathbb{Q}$, with $n$ integer such
that $0\leq n<M$, and $\theta_{n}=\frac{n}{M}$ as above. Then the following
formula holds:

$\displaystyle\widehat{Q}_{\bf
m}f(\theta,z)=\displaystyle\min\Big{\\{}M(\theta_{n+1}-\theta)P^{M,n}(w_{n})+M(\theta-\theta_{n})P^{M,n+1}(w_{n+1}):$
$\displaystyle\hskip
85.35826ptM(\theta_{n+1}-\theta)w_{n}+M(\theta-\theta_{n})w_{n+1}=z\Big{\\}}.$
(4.7)

We mention that in view of growth condition (2.6) the minimum in (4.4) is
achieved.

###### Proof.

Up to scaling, we suppose $m_{M}=1$ for notational convenience. Since Lemma
3.3 holds, for $u\in\mathcal{A}(Nq;z)$, if $F_{1}$ is the non-scaled
functional given by (3.4), we can estimate $F_{1}(u;[0,Nq])$ as

$\displaystyle F_{1}(u;[0,Nq])$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{Nq}f(z_{i})+2m_{1}\sum_{i=1}^{Nq}(z_{i})^{2}+\sum_{i,j=0,\
|i-j|=M}^{Nq}(u_{i}-u_{j})^{2}$ $\displaystyle\geq$
$\displaystyle\sum_{j=1}^{Nq}f_{2m_{1}}(z_{j})+\frac{M}{Nq}\sum_{i\in
M\mathbb{Z}\cap[M,Nq]}\Big{(}\sum_{j=i-M+1}^{i}z_{j}\Big{)}^{2}+o(1)_{N\to+\infty},$

where and $z_{i}=u_{i}-u_{i-1}$.

It is not restrictive to assume $Nq\in M\mathbb{N}$. For any $i\geq M$ we
define

$\displaystyle J^{+}(i)$ $\displaystyle=$
$\displaystyle\\{j\in\\{i-M+1,\dots,i-1,i\\}:z_{j}\geq z^{\ast}\\}$
$\displaystyle J^{-}(i)$ $\displaystyle=$
$\displaystyle\\{j\in\\{i-M+1,\dots,i-1,i\\}:z_{j}<z^{\ast}\\}.$

Moreover, for any $k=0,\dots,M$ we set

$\mathcal{I}_{k}=\\{i\in M\mathbb{Z}\cap[M,Nq]:\\#J^{+}(i)=k\\},$

and we denote the cardinality of $\mathcal{I}_{k}$ by $I_{k}$. Note that

$M\sum_{k=0}^{M}I_{k}=Nq,\ \ \sum_{k=0}^{M}(M-k)I_{k}=(1-\theta)Nq\ \ \hbox{
and }\ \ \sum_{k=0}^{M}kI_{k}=\theta Nq.$ (4.8)

Let $\psi_{-1}$ and $\psi_{1}$ denote the restrictions of $f_{2m_{1}}$ to
$(-\infty,z^{\ast})$ and $[z^{\ast},+\infty)$ respectively. Then, by
separating the contributions in each $\mathcal{I}_{k}$, thanks to the
convexity of $\psi_{-1}$, $\psi_{1}$ and of the square we have

$\displaystyle\sum_{j=1}^{Nq}f_{2m_{1}}(z_{j})+\frac{M}{Nq}\sum_{i\in
M\mathbb{Z}\cap[M,Nq]}\Big{(}\sum_{j=i-M+1}^{i}z_{j}\Big{)}^{2}$
$\displaystyle=\sum_{k=0}^{M}\sum_{i\in\mathcal{I}_{k}}\Big{(}\sum_{j\in
J^{-}(i)}\psi_{-1}(z_{j})+\sum_{j\in
J^{+}(i)}\psi_{1}(z_{j})\Big{)}+M\sum_{k=0}^{M}\sum_{i\in\mathcal{I}_{k}}\Big{(}\sum_{j\in
J^{-}(i)}z_{j}+\sum_{j\in J^{+}(i)}z_{j}\Big{)}^{2}\Big{)}$
$\displaystyle\geq\sum_{k=0}^{M}I_{k}\Big{(}(M-k)(\psi_{-1}(w_{k}^{-})+k\psi_{1}(w_{k}^{+})+M\big{(}(M-k)w_{k}^{-}+kw_{k}^{+}\big{)}^{2}\Big{)}$
(4.9)

where $w_{M}^{-}=w_{0}^{+}=0$ and

$\displaystyle
w_{k}^{-}=\frac{1}{(M-k)I_{k}}\sum_{i\in\mathcal{I}_{k}}\sum_{j\in
J^{-}(i)}z_{j},\ \ \ \
w_{k}^{+}=\frac{1}{kI_{k}}\sum_{i\in\mathcal{I}_{k}}\sum_{j\in J^{+}(i)}z_{j}$

otherwise.

We now may conclude the proof of the lower bound by applying Lemma 4.3 to
(4.1), regrouping the terms therein so as to compare that expression with
$P^{M,n}$. Noting that

$\sum_{k=0}^{M}(M-k)\alpha_{k}^{n}=\frac{(M\theta-n)(M-(n+1))}{M}Nq,$

we get by convexity that

$\displaystyle\sum_{k=0}^{M}\alpha_{k}^{n}\Big{(}(M-k)(\psi_{-1}(w_{k}^{-})+k\psi_{1}(w_{k}^{+})+M\big{(}(M-k)w_{k}^{-}+kw_{k}^{+}\big{)}^{2}\Big{)}$
$\displaystyle\geq$
$\displaystyle\Big{(}\sum_{k=0}^{M}(M-k)\alpha_{k}^{n}\Big{)}\psi_{-1}(z_{n+1}^{-})+\Big{(}\sum_{k=0}^{M}k\alpha_{k}^{n}\Big{)}\psi_{1}(z_{n+1}^{+})$
$\displaystyle+M\Big{(}\sum_{k=0}^{M}\alpha_{k}^{n}\Big{)}\Big{(}\frac{\big{(}\sum_{k=0}^{M}(M-k)\alpha_{k}^{n}\big{)}z_{n+1}^{-}+\big{(}\sum_{k=0}^{M}k\alpha_{k}^{n}\big{)}z_{n+1}^{+}}{\sum_{k=0}^{M}\alpha_{k}^{n}}\Big{)}^{2}$
$\displaystyle\geq$
$\displaystyle\frac{(M\theta-n)}{M}Nq\Big{(}(M-(n+1))\psi_{-1}(z_{n+1}^{-})+(n+1)\psi_{1}(z_{n+1}^{+})$
$\displaystyle+M\big{(}(M-(n+1))z_{n+1}^{-}+(n+1)z_{n+1}^{+}\big{)}^{2}\Big{)},$

where

$z_{n+1}^{-}=\frac{\sum_{k=0}^{M}(M-k)\alpha_{k}^{n}w_{k}^{-}}{\sum_{k=0}^{M}(M-k)\alpha_{k}^{n}},\quad
z_{n+1}^{+}=\frac{\sum_{k=0}^{M}k\alpha_{k}^{n}w_{k}^{+}}{\sum_{k=0}^{M}k\alpha_{k}^{n}}.$

Hence,

$\displaystyle\sum_{k=0}^{M}\alpha_{k}^{n}\Big{(}(M-k)(\psi_{-1}(w_{k}^{-})+k\psi_{1}(w_{k}^{+})+M\big{(}(M-k)w_{k}^{-}+kw_{k}^{+}\big{)}^{2}\Big{)}$
$\displaystyle\geq$
$\displaystyle(M\theta-n)NqP^{M,n+1}\Big{(}\Big{(}1-\frac{n+1}{M}\Big{)}z_{n+1}^{-}+\frac{n+1}{M}z_{n+1}^{+}\Big{)}.$

Correspondingly we obtain

$\displaystyle\sum_{k=0}^{M}(I_{k}-\alpha_{k}^{n})\Big{(}(M-k)(\psi_{-1}(w_{k}^{-})+k\psi_{1}(w_{k}^{+})+M\big{(}(M-k)w_{k}^{-}+kw_{k}^{+}\big{)}^{2}\Big{)}$
$\displaystyle\geq$
$\displaystyle(n+1-M\theta)NqP^{M,n}\Big{(}\Big{(}1-\frac{n}{M}\Big{)}z_{n}^{-}+\frac{n}{M}z_{n}^{+}\Big{)},$

where

$z_{n}^{-}=\frac{\sum_{k=0}^{M}(M-k)(I_{k}-\alpha_{k}^{n})w_{k}^{-}}{\sum_{k=0}^{M}(M-k)(I_{k}-\alpha_{k}^{n})},\quad
z_{n}^{+}=\frac{\sum_{k=0}^{M}k(I_{k}-\alpha_{k}^{n})w_{k}^{+}}{\sum_{k=0}^{M}k(I_{k}-\alpha_{k}^{n})}.$

Noting that

$\displaystyle(n+1-M\theta)\Big{(}(M-n)z_{n}^{-}+nz_{n}^{+}\Big{)}$
$\displaystyle\hskip
56.9055pt+(M\theta-n)\Big{(}(M-(n+1))z_{n+1}^{-}+(n+1)z_{n+1}^{+}\Big{)}=Mz,$

for $\theta\in\big{[}\frac{n}{M},\frac{n+1}{M}\big{]}$ we then have, up to a
negligible term,

$\displaystyle F_{1}(u;[0,Nq])$ $\displaystyle\geq$
$\displaystyle\min\Big{\\{}(n+1-M\theta)P^{M,n}(w_{n})+(M\theta-n)P^{M,n+1}(w_{n+1}):$
$\displaystyle\hskip
28.45274pt(n+1-M\theta)w_{n}+(M\theta-n)w_{n+1}=z\Big{\\}}$

which concludes the proof of the lower bound in (4.4).

Figure 9: construction of the upper bound for $M=3$ and $n=1$.

As for the upper bound, let
$\theta=\frac{p}{q}\in[\frac{n}{M},\frac{n+1}{M}]$, $z\in\mathbb{R}$ be fixed
and $(w_{n},w_{n+1})$ be a minimizer of (4.4). For all $k\geq 1$ we define a
test function $u\colon[0,kMq]\cap\mathbb{Z}\to\mathbb{R}$ constructed as
follows. Let $w_{n}^{\pm}$ be a minimizer of the problem defining
$P^{M,n}(w_{n})$ in (4.3), and let $w_{n+1}^{\pm}$ be a minimizer of the
corresponding problem defining $P^{M,n+1}(w_{n+1})$. We set $u_{0}=0$, and

$\displaystyle u_{i}-u_{i-1}$ $\displaystyle=$
$\displaystyle\begin{cases}w_{n}^{+}&\hbox{ if }i\in\\{1,\dots,n\\}\hbox{ mod
}M\\\ w_{n}^{-}&\hbox{ if }i\in\\{n+1,\dots,M\\}\hbox{ mod }M\end{cases}\ \ \
\ \ \hbox{for }\ i\leq kMq(\theta_{n+1}-\theta)$ $\displaystyle u_{i}-u_{i-1}$
$\displaystyle=$ $\displaystyle\begin{cases}w_{n+1}^{+}&\hbox{ if
}i\in\\{1,\dots,n+1\\}\hbox{ mod }M\\\ w_{n+1}^{-}&\hbox{ if
}i\in\\{n+1,\dots,M\\}\hbox{ mod }M\end{cases}\ \ \,\hbox{for
}i>kMq(\theta_{n+1}-\theta)$

(see Fig. 9). Note that $u(kMq)=kMqz$ and $u\in\mathcal{V}(kMq;\theta)$, so
that $u$ is an admissible test function for the computation of
$\widehat{Q}_{\bf m}f(\theta,z)$, and the upper bound follows. ∎

######  Remark 4.5 (Non-differentiability at locking states).

From formula (4.4) we deduce that for all $z$ the function $\theta\mapsto
Q_{\bf m}f(\theta,z)$ is differentiable at any
$\theta\not\in\\{\theta_{1},\dots,\theta_{M-1}\\}$, whereas instead

$\frac{\partial(Q_{\bf
m}f)}{\partial\theta}(\theta_{n}^{+},z)\neq\frac{\partial(Q_{\bf
m}f)}{\partial\theta}(\theta_{n}^{-},z)$

except possibly for some critical values of $z$. Indeed, in the computation of
the left-hand side derivative of $Q_{\bf m}f$ at $\theta=\theta_{n}$ we use
$P^{M,n-1}$ while for the right-hand side we use $P^{M,n+1}$, whose values are
generically different at the minimum points of (4.4).

###  4.2 Computation of $Q_{\bf m}f$ for prototypical non-convex energies

We now apply Theorem 4.1 to some prototypical $f$; namely, truncated quadratic
potential and double-well potential.

####  4.2.1 Truncated quadratic potential

We consider a special case of the truncated convex potentials introduced in
Example LABEL:trconintro with $\tilde{f}(z)=z^{2}$ and $z^{\ast}=1$; that is,
let $f\colon\mathbb{R}\to\mathbb{R}$ be defined by

$f(z)=\begin{cases}z^{2}&\hbox{ if }z\leq 1\\\ 1&\hbox{ if }z>1,\end{cases}$
(4.10)

and let $A=[1,+\infty)$. Note the growth assumption (2.6) implies that
$m_{1}>0$.

In this case, we have

$Q_{\bf m}f(z)=\begin{cases}\displaystyle z^{2}&\hbox{\rm if }\ z\leq
s^{+}_{0}\\\ \displaystyle r^{M,n}(z)-2(m_{1}+m_{M}M^{2})z^{2}&\hbox{\rm if }\
s_{n}^{+}\leq z\leq s_{n+1}^{-}\\\
\displaystyle\frac{2m_{1}(1-\theta_{n})}{2m_{1}+\theta_{n}}z^{2}+\theta_{n}&\hbox{\rm
if }\ s_{n}^{-}\leq z\leq s_{n}^{+}\\\ \displaystyle 1&\hbox{\rm if }\
s_{M}^{-}\leq z,\end{cases}$ (4.11)

where the points $s_{n}^{+}$ and $s_{n}^{-}$ in Theorem 4.1 are

$s_{n}^{\pm}=s_{n}^{\pm}(m_{1},m_{M})=\frac{2m_{1}+\theta_{n}}{\sqrt{2m_{1}(2m_{1}+1)}}\
\sqrt{\frac{m_{1}(2m_{1}+1)+m_{M}M^{2}(2m_{1}+\theta_{n})\pm
m_{M}M}{m_{1}(2m_{1}+1)+m_{M}M^{2}(2m_{1}+\theta_{n})}}$ (4.12)

and $r^{M,n}$ is the affine interpolating function in Remark 4.2. The formula
for $Q_{\bf m}f$ is obtained by explicitly computing the functions
$P^{M,n}(z)$ (see Appendix B).

Figure 10: $Q_{\bf m}f(z)$ and $\theta(z)$ in the cases $M=2$ (a) and $M=3$
(b).

In Figure 10 (a)–(b), we show the structure of the functions $Q_{\bf m}f(z)$
and $\theta(z)$ in the cases $M=2$ and $M=3$, respectively. Note that in the
first case $\theta_{1}=\frac{1}{2}$ corresponds to periodic minimizers of
period $2$ and in the second case $\theta_{1}=\frac{1}{3}$ and
$\theta_{2}=\frac{2}{3}$ correspond to the two possible periodic minimizers of
period $3$. In the affine regions, we have mixtures of two periodic solutions,
corresponding to neighbouring locking states.

######  Remark 4.6 (Degenerate case with $m_{1}$=0).

The computation of $Q_{\bf m}f$ for the truncated quadratic potential $f$ can
be performed also in the degenerate case where the growth hypothesis (2.6)
does not hold; that is, supposing $m_{1}=0$. Note that in this case there is
no coercivity on the nearest-neighbour interactions.

Figure 11: $Q_{\bf m}f$ and $\theta$ in a degenerate case.

The construction in Theorem 4.1 becomes degenerate, and we obtain the formula

$Q_{\bf m}f(z)=\begin{cases}\displaystyle z^{2}&\displaystyle\hbox{\rm if }\
z\leq z_{M}^{-}\\\ \displaystyle
2\sqrt{2m_{M}M(1+2m_{M}M^{2})}z-2m_{M}M-2m_{M}M^{2}z^{2}&\displaystyle\hbox{\rm
if }\ z_{M}^{-}\leq z\leq z_{M}^{+}\\\
\displaystyle\frac{1}{M}&\displaystyle\hbox{\rm if }\ z_{M}^{+}\leq
z,\end{cases}$ (4.13)

where

$z_{M}^{-}=\sqrt{\frac{2m_{M}M}{1+2m_{M}M^{2}}}\ \ \ \hbox{\rm and }\ \
z_{M}^{+}=\sqrt{\frac{1+2m_{M}M^{2}}{2m_{M}M^{3}}}.$

The corresponding phase function is then given by $\theta(z)=0$ if $z\leq
z_{M}^{-}$, $\theta(z)=\frac{1}{M}$ if $z\geq z_{M}^{+}$ and affine otherwise,
so that the locking states are $\theta=0$ and $\theta=\frac{1}{M}$. Hence,
$Q_{\bf m}f(z)$ is obtained as the convex envelope of the minimum of
$P^{M,1}(z)$ and $P^{M,0}(z)$ only.

As for the description of $\theta$ as in (4.4), note that

$\lim_{m_{1}\to 0}s_{0}^{+}(m_{1},m_{M})=z_{M}^{-},\ \ \ \lim_{m_{1}\to
0}s_{1}^{-}(m_{1},m_{M})=z_{M}^{+},$

while we have that as $m_{1}\to 0$ then $s_{n}^{+}(m_{1},m_{M})\to+\infty$ for
any $n\geq 1$ and $s_{n}^{-}(m_{1},m_{M})\to+\infty$ for any $n\geq 2$. This
corresponds to the fact that the sets of $z$ where $\theta(z)>1/M$ tend to
$+\infty$ as $m_{1}\to 0$. In Fig. 11 we picture $Q_{\bf m}f$ and $\theta$.

######  Remark 4.7 (Asymptotic analysis as $M\to+\infty$).

In this remark we highlight the dependence of $\theta=\theta^{M}$ and $Q_{\bf
m}f=Q^{M}f$ on $M$. We show that the limit of the functions $\theta^{M}$ as
$M\to+\infty$ is the phase function of $f$ when the only not vanishing
coefficient is $m_{1}$, and correspondingly for $Q^{M}f(z)$.

Figure 12: Graph of $Q^{M}f(z)$ (for $M=6$) and of the limit function.

Indeed, the following estimates hold

$\displaystyle
s_{n}^{+}\leq\frac{n+2m_{1}M+\frac{1}{2}}{\sqrt{2m_{1}(1+2m_{1})}M}=:\widetilde{s}_{n}^{+},\
\ \
s_{n}^{-}\geq\frac{n+2m_{1}M-1}{\sqrt{2m_{1}(1+2m_{1})}M}=:\widetilde{s}_{n}^{-},$

so that we can define two piecewise-constant functions by setting

$\overline{\theta}^{M}(z)=\begin{cases}0&\hbox{\rm if }\
z\leq\widetilde{s}_{0}^{+}\\\ \theta^{M}(s_{n-1}^{+})&\hbox{\rm if }\
z\in(\widetilde{s}_{n-1}^{+},\widetilde{s}_{n}^{+}]\\\ 1&\hbox{\rm if }\
\widetilde{s}_{M}^{+}<z\end{cases}\ \ \ \hbox{\rm and }\ \ \
\underline{\theta}^{M}(z)=\begin{cases}0&\hbox{\rm if }\
z\leq\widetilde{s}_{0}^{-}\\\ \theta^{M}(s_{n}^{-})&\hbox{\rm if }\
z\in(\widetilde{s}_{n-1}^{-},\widetilde{s}_{n}^{-}]\\\ 1&\hbox{\rm if }\
\widetilde{s}_{M}^{-}<z,\end{cases}$

obtaining that
$\underline{\theta}^{M}(z)\leq\theta^{M}(z)\leq\overline{\theta}^{M}(z)$. The
claim follows noting that

$\lim_{M\to+\infty}\overline{\theta}^{M}(z)=\begin{cases}0&\hbox{\rm if }\
z\leq\sqrt{\frac{2m_{1}}{1+2m_{1}}}\\\
2m_{1}\Big{(}z\sqrt{\frac{1+2m_{1}}{2m_{1}}}-1\Big{)}&\hbox{\rm if }\
\sqrt{\frac{2m_{1}}{1+2m_{1}}}\leq z\leq\sqrt{\frac{1+2m_{1}}{2m_{1}}}\\\
1&\hbox{\rm if }\ \sqrt{\frac{1+2m_{1}}{2m_{1}}}\leq z,\end{cases}$

and the same for $\underline{\theta}^{M}(z)$. Correspondingly

$\lim_{M\to+\infty}Q^{M}f(z)=\begin{cases}z^{2}&\hbox{\rm if }\
z\leq\sqrt{\frac{2m_{1}}{1+2m_{1}}}\\\
-2m_{1}\Big{(}z^{2}-2z\sqrt{\frac{1+2m_{1}}{2m_{1}}}+1\Big{)}&\hbox{\rm if }\
\sqrt{\frac{2m_{1}}{1+2m_{1}}}\leq z\leq\sqrt{\frac{1+2m_{1}}{2m_{1}}}\\\
1&\hbox{\rm if }\ \sqrt{\frac{1+2m_{1}}{2m_{1}}}\leq z\end{cases}$

(see Figure 12). In particular, we note that

$\lim_{M\to+\infty}Q^{M}f(z)=(f_{2m_{1}})^{\ast\ast}(z)-2m_{1}z^{2}=Q_{\bf
m^{\prime}}f(z),$

where ${\bf m^{\prime}}=\\{m_{1},0,\dots\\}$.

Figure 13: example of convex-affine non-convex potentials.

######  Example 4.8 (convex-affine potentials as perturbations of truncated
potentials).

We consider the functions $\ell^{\tau}$ introduced in (3.18) in the non-convex
case $0\leq\tau<1$, as pictured in Figure 13, with nearest and next-to-nearest
neighbour interactions; that is, with $M=2$. To simplify the computations, we
fix $m_{1}=\frac{1}{2}$ and $m_{2}=\frac{1}{4}$. The computation of $Q_{\bf
m}\ell^{\tau}(z)$ involves the values $Q_{\bf m}\ell^{\tau}(\theta,z)$ in the
three locking states $\theta_{0}=0,$ $\theta_{1}=\frac{1}{2}$ and
$\theta_{2}=1$; more precisely, it is sufficient to consider $Q_{\bf
m}\ell^{\tau}(0,z)=\ell^{\tau}(z)$ for $z\leq 1$, $Q_{\bf
m}\ell^{\tau}(1,z)=\ell^{\tau}(z)$ for $z\geq 1$ and

$Q_{\bf
m}\ell^{\tau}\Big{(}\frac{1}{2},z\Big{)}=\frac{1}{3}z^{2}+\frac{4\tau}{3}z+\frac{3-6\tau-\tau^{2}}{6}$

for $\frac{3}{4}\leq z\leq\frac{3}{2}$. Hence

$Q_{\bf m}\ell^{\tau}(z)=\begin{cases}Q_{\bf m}\ell^{\tau}(0,z)&\hbox{\rm if
}z\leq s^{\tau,+}_{0}\\\ r^{\tau}_{1}(z)-3z^{2}&\hbox{\rm if
}s^{\tau,+}_{0}\leq z\leq s^{\tau,-}_{1}\\\ Q_{\bf
m}\ell^{\tau}(\frac{1}{2},z)&\hbox{\rm if }s^{\tau,-}_{1}\leq z\leq
s^{\tau,+}_{1}\\\ r^{\tau}_{2}(z)-3z^{2}&\hbox{\rm if }s^{\tau,+}_{1}\leq
z\leq s^{\tau,-}_{2}\\\ Q_{\bf m}\ell^{\tau}(1,z)&\hbox{\rm if }z\geq
s^{\tau,-}_{2}\end{cases}$

where $r^{\tau}_{1}(z)$ is the common tangent (in $s^{\tau,+}_{0}$ and
$s^{\tau,-}_{1}$) to the parabolas $\widehat{Q}_{\bf m}\ell^{\tau}(0,z)$ and
$\widehat{Q}_{\bf m}\ell^{\tau}(\frac{1}{2},z)$, and correspondingly
$r^{\tau}_{2}(z)$ is the common tangent (in $s^{\tau,+}_{1}$ and
$s^{\tau,-}_{2}$) to the parabolas $\widehat{Q}_{\bf
m}\ell^{\tau}(\frac{1}{2},z)$ and $\widehat{Q}_{\bf m}\ell^{\tau}(1,z)$.

Figure 14: $Q_{\bf m}\ell^{\tau}$ and corresponding phase functions for
increasing values of $\tau\in(0,1)$.

In Fig. 14 we represent $Q_{\bf m}\ell^{\tau}$ for two different values of
$\tau$, also showing the three energies $\widehat{Q}_{\bf
m}\ell^{\tau}(\theta,z)$ when $\theta\in\\{0,\frac{1}{2},1\\}$, and the
corresponding phase function $\theta$. The value of $\tau$ in (b) is larger
than that in (a). Note in particular that if $\tau\to 1$ then
$s_{2}^{\tau,-}-s_{0}^{\tau,+}\to 0$; that is, the locking state
$\theta=\frac{1}{2}$ progressively disappears, and we recover the convex case
(see Example 3.19), while for $\tau=0$ we recover the case of the truncated
quadratic potential with $M=2$.

####  4.2.2 Double-well bi-quadratic potential

Let $f\colon\mathbb{R}\to\mathbb{R}$ be defined by $f(z)=(1-|z|)^{2}$, and let
$A=[0,+\infty)$. By explicitly computing the functions $P^{M,n}$ (see Appendix
B), we obtain for $Q_{\bf m}f(z)$ the formula

$Q_{\bf m}f(z)=\begin{cases}(1+z)^{2}&\hbox{\rm if }\ z\leq s_{0}^{+}\\\
r^{M,n}(z)-2(m_{1}+m_{M}M^{2})z^{2}&\hbox{\rm if }\ s_{n}^{+}\leq z\leq
s_{n+1}^{-}\\\ \displaystyle
z^{2}+2(1-2\theta_{n})z+1-\frac{4\theta_{n}(1-\theta_{n})}{1+2m_{1}}&\hbox{\rm
if }\ s_{n}^{-}\leq z\leq s_{n}^{+}\\\ (1-z)^{2}&\hbox{\rm if }\ s_{M}^{-}\leq
z,\end{cases}$

where

$s_{n}^{\pm}=s_{n}^{\pm}(m_{1},m_{M})=\frac{2\theta_{n}-1}{1+2m_{1}}\pm\frac{2m_{M}M}{(1+2m_{1})(1+2m_{1}+2m_{M}M^{2})}$

and $r^{M,n}$ is the interpolating affine function given in Remark 4.2.

Figure 15: The function $z\mapsto Q^{M}f(z)$ for different values of $M$ and
the limit function.

######  Remark 4.9 (Asymptotic analysis as $M\to+\infty$).

As in Remark 4.7, we highlight the dependence on $M$ by writing
$\theta(z)=\theta^{M}(z)$ and $Q_{\bf m}f(z)=Q^{M}f(z)$. We show that also in
this case the limit of $\theta^{M}(z)$ as $M\to+\infty$ is the phase function
of $f$ when the only not vanishing coefficient is $m_{1}$, and correspondingly
for $Q^{M}f(z)$. Indeed, since the distribution of $s_{n}^{+}$ and $s_{n}^{-}$
is uniform, we can directly deduce that

$\lim_{M\to+\infty}\theta^{M}(z)=\begin{cases}0&\hbox{\rm if }\
z\leq-\frac{1}{1+2m_{1}}\\\ \frac{(1+2m_{1})z+1}{2}&\hbox{\rm if }\
|z|\leq\frac{1}{1+2m_{1}}\\\ 1&\hbox{\rm if }\
z\geq\frac{1}{1+2m_{1}}.\end{cases}$

Correspondingly

$\lim_{M\to+\infty}Q^{M}f(z)=\begin{cases}(1+z)^{2}&\hbox{\rm if }\
z\leq-\frac{1}{1+2m_{1}}\\\ -2m_{1}z^{2}+\frac{2m_{1}}{1+2m_{1}}&\hbox{\rm if
}\ |z|\leq\frac{1}{1+2m_{1}}\\\ (1-z)^{2}&\hbox{\rm if }\
z\geq\frac{1}{1+2m_{1}}\end{cases}$

(see Figure 15). Again, we note that
$\lim\limits_{M\to+\infty}Q^{M}f(z)=Q_{\bf m^{\prime}}f(z)$, where ${\bf
m^{\prime}}=\\{m_{1},0,\dots\\}$.

####  4.2.3 Analysis of $Q_{\bf m}f(\theta,z)$

Figure 16: analysis of $\theta\mapsto Q_{\bf m}f(\theta,z)$ for different
values of $z$ in the truncated quadratic case.

Examining (4.4), which gives the values of $Q_{\bf m}f(\theta,z)$ as
interpolations between neighbouring locking states, we note that $Q_{\bf m}f$
is given by different formulas in different regions of the plane ($\theta,z$).
We briefly examine some feature of this dependence in the simplest meaningful
case $M=2$ (see also Fig. 10(a) and Fig. 15 for a comparison).

In Figures 16 (truncated quadratic potential) and 17 (double-well potential),
we highlight zones with qualitatively different behaviour, distinguished by
colouring. In the same pictures, the graphs of $\theta\mapsto Q_{\bf
m}f(\theta,z)$ are shown for some values of $z$ in the regions of
qualitatively different behaviour. Note that for any fixed $z$ the function
$\theta\mapsto Q_{\bf m}f(\theta,z)$ is differentiable everywhere (including
the points where there is a change of the analytical expression), except for
the point corresponding to the locking state $\theta_{1}=\frac{1}{2}$, where
the left and right derivative are not equal.

Figure 17: The function $\theta\mapsto Q_{\bf m}f(\theta,z)$ for different
values of $z$ (double-well potential).

For the reader’s convenience, in the case of double-well potential we include
an explicit formula which is particularly simple thanks to the symmetry of
$Q_{\bf m}f(\theta,z)$ with respect to $(\frac{1}{2},0)$. We fix
$m_{1}=\frac{1}{2},m_{2}=\frac{1}{4}$, obtaining

$Q_{\bf m}f(\theta,z)=\begin{cases}\frac{3z^{2}}{1-\theta}+2z+1&\hbox{\rm if
}\ z\leq\theta-1\\\ \frac{2z^{2}}{1-\theta}+\theta&\hbox{\rm if }\
\theta-1<z\leq\frac{\theta-1}{2}\\\
z^{2}-2(2\theta-1)z+\theta^{2}-\frac{\theta}{2}+\frac{1}{2}&\hbox{\rm if }\
\frac{\theta-1}{2}<z\leq\frac{2\theta+1}{4}\\\
\frac{12z^{2}}{2\theta+1}-2z+1&\hbox{\rm if }\
\frac{2\theta+1}{4}<z.\end{cases}$

####  4.2.4 Dependence on the scale parameter $\sigma$

As in Remark 2.39, we introduce a dependence of the concentrated kernel $\bf
m$ on the parameter $\sigma$ by setting $m_{1}^{\sigma}=\frac{m_{1}}{\sigma}$
and $m_{M}^{\sigma}=\frac{m_{M}}{\sigma}$, for which we have

$\lim_{\sigma\to 0^{+}}Q_{\bf m^{\sigma}}f(z)=\overline{f}(z)\ \ \hbox{\rm and
}\ \ \lim_{\sigma\to+\infty}Q_{\bf m^{\sigma}}f(z)=f^{\ast\ast}(z)$ (4.14)

for any $f$.

Figure 18: Representation of constancy sets of $\theta$ in the
$z$-${1\over\sigma}$ plane.

In the case the truncated quadratic function $f$ defined by (4.10) and
analyzed in Section 4.2.1, the first limit can be also checked directly
noticing that $s_{n}^{+}(m_{1}^{\sigma},m_{M}^{\sigma})\to 1$ as
${1\over\sigma}\to+\infty$ for any $n$, where $s_{n}^{+}(\cdot,\cdot)$ is
defined in (4.12). Note that if $\theta\in(0,1)$ then $Q_{\bf
m^{\sigma}}f(\theta,z)\to+\infty$ as ${1\over\sigma}\to 0^{+}$. Moreover, for
any $\theta\in(0,1)$ and for any $z$,

$\lim_{\sigma\to+\infty}Q_{{\bf m}^{\sigma}}f(\theta,z)=Q_{{\bf
0}}f(\theta,z)=\begin{cases}\theta+\frac{(z-\theta)^{2}}{1-\theta}&\hbox{\rm
if }\ z\leq\theta\\\ \theta&\hbox{\rm if }\ z\geq\theta.\end{cases}$

In Fig. 18 we picture in the $z$-${1\over\sigma}$ plane the zones where
$\theta(z)=\theta_{n}$ for some $n\in\\{0,\ldots,M\\}$ and those where
$\theta(z)$ is affine for fixed $\sigma$ (in grey) for $M=4$.

As for the double-well potential, if the coefficient $m_{1}$ does not vanish,
then we re-obtain the first limit in (4.14) by noting that

$\lim_{\sigma\to
0^{+}}s_{n}^{+}(m_{1}^{\sigma},m_{M}^{\sigma})=\lim_{\sigma\to
0^{+}}s_{n}^{-}(m_{1}^{\sigma},m_{M}^{\sigma})=0,$ (4.15)

where $s_{n}^{+}$ and $s_{n}^{-}$ are defined in (B.4).

Figure 19: Representation of constancy sets of $\theta$ in the
$z$-${1\over\sigma}$ plane

In Fig. 19 we picture in the $z$-${1\over\sigma}$ plane the zones where
$\theta(z)=\theta_{n}$ for some $n\in\\{0,\ldots,M\\}$ and those where
$\theta(z)$ is affine for fixed $\sigma$ (in grey) for $M=4$.

######  Remark 4.10.

Figure 20: the limit of $Q_{{\bf m}^{\sigma}}f$ for $\sigma\to 0$ in the case
$m_{1}=0$.

If $m_{1}=0$, Remark 2.39 does not apply. Taking the limit for $\sigma\to
0^{+}$, in this case we obtain

$\lim_{\sigma\to 0^{+}}s_{n}^{+}(m_{1}^{\sigma},m_{M}^{\sigma})=z_{n},\ \ \
\hbox{\rm and }\ \ \ \lim_{\sigma\to
0^{+}}s_{n}^{-}(m_{1}^{\sigma},m_{M}^{\sigma})=z_{n-1},$ (4.16)

where we set

$z_{n}=\frac{2n+1-M}{M}.$

The limit function is then given by

$\lim_{\sigma\to 0^{+}}Q_{{\bf
m}^{\sigma}}f(z)=\begin{cases}(1+z)^{2}&\displaystyle\hbox{\rm if }\ z\leq
z_{0}\\\
\displaystyle\big{(}z+(1-2\theta_{n})\big{)}^{2}&\displaystyle\hbox{\rm if }\
z_{n-1}\leq z\leq z_{n}\\\ (1-z)^{2}&\displaystyle\hbox{\rm if }\ z_{M}\leq
z,\end{cases}$

or, equivalently,

$\lim_{\sigma\to 0^{+}}Q_{{\bf m}^{\sigma}}f(z)=\min_{0\leq n\leq
M}\big{\\{}\big{(}z+(1-2\theta_{n})\big{)}^{2}\big{\\}}=\min_{0\leq n\leq
M}\\{Q_{\bf m}f(\theta_{n},z)\\}.$

Figure 21: Representation of $\theta$ in the $z$-${1\over\sigma}$ plane for
$M=4$ (case $m_{1}=0$)

Note that in this case the limit differs from $f$ but coincides with the
minimum among $P^{M,n}(z)-2m_{M}M^{2}z^{2}$ (see Fig. 20), whose
convexification still equals $f^{**}$.

In Fig. 21 we picture in the $z$-${1\over\sigma}$-plane the zones where
$\theta(z)=\theta_{n}$ for some $n\in\\{0,\ldots,M\\}$ and those where
$\theta(z)$ is affine for fixed $\sigma$ (in grey).

##  5 Relaxation with exponential-kernel penalization

The case of concentrated kernels studied in the previous section allowed us to
highlight some properties of $Q_{\bf m}f$, in particular we were able to
characterize the locking states using explicit formulas. Now, we analyze the
effect of the superposition of spatially distributed long-range interactions,
which bring additional complexity to the structure of $Q_{\bf m}f$.

In Section 5.1 we sketch a method for obtaining bounds for a general kernel
$\bf m$ via higher-dimensional embeddings. This method is optimal in the case
when the non-local term $\sum_{i,j}m_{|i-j|}(u_{i}-u_{j})^{2}$ depending on
the given kernel $\bf m$ can be obtained by integrating out the variable $v$
from the simplest additive energy depending on two variables $u$ and $v$; that
is, $a\sum_{i}(v_{i}-v_{i-1})^{2}+b\sum_{i}(u_{i}-v_{i})^{2}$. To have this,
we note that the kernel $\bf m$ must be exponential. Hence, the study of
general exponential kernels will constitute the main goal of this section. The
idea of rewriting the problems defining $\widehat{Q}_{\bf m}f$ as additive
problems in terms of an auxiliary variable has been already used implicitly in
the case of concentrated kernels. Indeed, in that case we introduced coarse-
grained energies depending only on $M$-neighbour interactions $u_{i+M}-u_{i}$
through the functions $P^{M,n}$.

###  5.1 Higher-dimensional embeddings for general $\bf m$

In this section we discuss the possibility of simplifying the quadratic
penalty term in Definition 2.3 for an arbitrary kernel $\bf m$ by introducing
auxiliary variables. This will be later applied to the exponential kernel
defined in (5.12). The idea is to view the long-range interactions
parameterized by an arbitrary $\bf m$ as a projection of short-range
interactions operating in a higher-dimensional space. In other words, we now
suppose that the kernels $\bf m$ can be viewed as the Green’s functions of
some higher-dimensional local problems. Note however that the locality of the
corresponding higher-dimensional problem can be expected only for kernels $\bf
m$ with sufficiently fast rate of decay. To highlight the ideas, we discuss in
detail only the simplest class of projections, where the dimension of the
extended configurational space is doubled. As a result, the nonlocal _scalar_
problem is transformed into a local _vector_ problem.

For each fixed $k\in\mathbb{N}$, we define a quadratic form depending on two
variables as follows. Let $A$ be a $(k+1)\times(k+1)$ matrix and let
$s\in\mathbb{R}$ be a scalar parameter. We set

$H^{k}[A,s](u,v)=2s\langle Av,v\rangle+2s\langle u-v,u-v\rangle,$ (5.1)

where $u,v\colon\\{0,\dots,k\\}\to\mathbb{Z}$ and $\langle\cdot,\cdot\rangle$
denotes the scalar product in $\mathbb{R}^{k+1}$.

The following result restates the definition of $\widehat{Q}_{\bf m}f$ as a
minimum problem involving a quadratic form of type (5.1).

######  Theorem 5.1 (higher-dimensional equivalent formulation).

Let ${\bf m}$ satisfy (2.4) and be such that the function $n\mapsto m_{n}$ is
not increasing for $n$ large enough. Then, there exist a
$(k+1)\times(k+1)$-dimensional matrix $A^{k}_{\bf m}$ and a scalar $s_{\bf m}$
such that, setting $H^{k}_{\bf m}=H^{k}[A^{k}_{\bf m},s_{\bf m}]$ in (5.1),
the following equality holds

$\widehat{Q}_{\bf
m}f(z)=\lim_{k\to+\infty}\frac{1}{k}\min\Big{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+H^{k}_{\bf
m}(u,v):u,v\in\mathcal{A}(k,z)\Big{\\}}$ (5.2)

for all $f\colon\mathbb{R}\to[0,+\infty)$ satisfying growth conditions (2.5)
and (2.6).

The proof of Theorem 5.1 is based on Lemma 5.2 which implies that
asymptotically the quadratic part of the energies in the definition of
$\widehat{Q}_{\bf m}f$ can be viewed as projections of functions of the form
(5.1).

To shorten the notation, we introduce the quadratic function

$J_{\bf m}^{k}(u)=\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2},$ (5.3)

defined on $u\colon\\{0,\dots,k\\}\to\mathbb{Z}$.

To quantify the relation between $J_{\bf m}^{k}$ and the corresponding
$H^{k}_{\bf m}$, we introduce a notion of $L^{2}$ norm for
$u\colon\\{0,\dots,k\\}\to\mathbb{Z}$ by setting

$\|u\|_{k}^{2}=\frac{1}{k}\sum_{i=1}^{k}(u_{i})^{2},$

which coincides with the $L^{2}$ norm of the piecewise-constant function
$\tilde{u}\colon(0,1)\to\mathbb{R}$ defined by $\tilde{u}(t)=u_{i}$ in
$(\frac{i-1}{k}\frac{i}{k}]$.

######  Lemma 5.2 (projection of the quadratic part of the energies).

Let ${\bf m}$ satisfy (2.4) and be such that the function $n\mapsto m_{n}$ is
not increasing for $n$ large enough. Let $J_{\bf m}^{k}$ be as in (5.3). Then,
there exist a $(k+1)\times(k+1)$-dimensional matrix $A^{k}_{\bf m}$ and a
scalar $s_{\bf m}$ such that

$\min\\{H^{k}_{\bf m}(u,v):v\colon\\{0,\dots,k\\}\to\mathbb{R}\\}=J_{\bf
m}^{k}(u)+\|u\|_{k}^{2}\ o\Big{(}\frac{1}{k}\Big{)}$ (5.4)

for all $u\colon\\{0,\dots,k\\}\to\mathbb{R}$, where $H_{\bf m}^{k}$ is
defined in Theorem 5.1.

###### Proof.

We introduce the $(k+1)\times(k+1)$ matrix $M^{k}_{\bf m}=(m_{ij})$ given by
$m_{ij}=m_{|i-j|}$, $i,j=0,\dots,k$. Note that the functional $J^{k}_{\bf m}$
is independent of the choice of $m_{0}$, so that we can choose the value of
$m_{0}$ arbitrarily. We assume that this value is such that the matrix
$M^{k}_{\bf m}$ is invertible.

As a first step, we write the functional $J_{\bf m}^{k}$, up to an
infinitesimal term, as the sum of a suitable quadratic form depending on the
whole series of $m_{n}$ and a residual boundary term. By Lemma A.1 (see
Appendix A), up to a change of variables with $L=1$ and $\varepsilon=1/k$, we
can suppose that $u$ is constant in $[0,k^{\alpha}]$ and in $[k-k^{\alpha},k]$
with a fixed $\alpha\in(\frac{3}{\beta},1)$, where $\beta$ is the decay
parameter of $\bf m$ given by (2.4). Up to translations, we can assume
$u_{0}=0$ and hence $u_{i}=0$ for $i\leq k^{\alpha}$. Setting

$s_{\bf m}=m_{0}+2\sum_{n=1}^{+\infty}m_{n}\ \ \ \hbox{\rm and }\ \ \ s_{\bf
m}^{i}=\sum_{j=0}^{k}m_{ij},$

we get

$s_{\bf m}^{i}-s_{\bf
m}=-\sum_{n=i+1}^{+\infty}m_{n}-\sum_{n=k-i+1}^{+\infty}m_{n},$

so that, using the decay condition $m_{n}=o(n^{-\beta})$, we obtain

$\displaystyle J_{\bf m}^{k}(u)$ $\displaystyle=$ $\displaystyle 2s_{\bf
m}\langle u-\frac{1}{s_{\bf m}}M_{\bf m}^{k}u,u\rangle+2\sum_{i=0}^{k}(s_{\bf
m}^{i}-s_{\bf m})(u_{i})^{2}$ $\displaystyle=$ $\displaystyle 2s_{\bf
m}\langle u-\frac{1}{s_{\bf m}}M_{\bf m}^{k}u,u\rangle-2t_{\bf
m}(u_{k})^{2}+\sum_{i=0}^{k}(u_{i})^{2}\ o(k^{1-\alpha\beta})$
$\displaystyle=$ $\displaystyle 2s_{\bf m}\langle u-\frac{1}{s_{\bf m}}M_{\bf
m}^{k}u,u\rangle-2t_{\bf m}(u_{k})^{2}+\|u\|_{k}^{2}\ o(k^{2-\alpha\beta}),$

where $t_{\bf m}=\sum_{n=0}^{+\infty}n\,m_{n}$. Note that $2-\alpha\beta<-1$
since $\alpha>\frac{3}{\beta}$.

The matrix $A^{k}_{\bf m}$ will be obtained by modifying the matrix $s_{\bf
m}\big{(}M_{\bf m}^{k}\big{)}^{-1}-I$, which gives a minimum for $H^{k}_{\bf
m}$ in $v=\frac{1}{s_{\bf m}}M_{\bf m}^{k}u$, so as to take into account the
boundary contribution. This is done by changing the values $(A^{k}_{\bf
m})_{11}$ and $(A^{k}_{\bf m})_{kk}$ in such a way that they compensate the
boundary terms. We set

$A^{k}_{\bf m}=s_{\bf m}\begin{pmatrix}c_{\bf m}&0&\dots&\dots&0\\\
0&1&0&\dots&\dots\\\ \dots&\dots&\dots&\dots&\dots\\\ \dots&\dots&0&1&0\\\
\dots&\dots&0&0&c_{\bf m}\end{pmatrix}\big{(}M_{\bf m}^{k}\big{)}^{-1}-I,\ \
\hbox{\rm with }\ \ c_{\bf m}=\frac{s_{\bf m}+m_{0}}{2t_{\bf m}+s_{\bf
m}+m_{0}}.$ (5.5)

We can write

$A^{k}_{\bf m}=s_{\bf m}\big{(}M_{\bf m}^{k}\big{)}^{-1}-I-\frac{2t_{\bf
m}s_{\bf m}}{2t_{\bf m}+s_{\bf m}+m_{0}}(e_{0}\otimes e_{0}+e_{k}\otimes
e_{k})\big{(}M_{\bf m}^{k}\big{)}^{-1},$

and we prove that the minimum of $H^{k}_{\bf m}(u,v)$ coincides, up to an
infinitesimal term, with $J_{\bf m}^{k}(u)$. This minimum is attained for
$v^{k,\rm min}$ given by

$v^{k,\rm min}=(A^{k}_{\bf m}+I)^{-1}u=\frac{1}{s_{\bf m}}M_{\bf
m}^{k}u+\frac{2t_{\bf m}}{s_{\bf m}(s_{\bf
m}+m_{0})}u_{k}\begin{pmatrix}m_{k}\\\ m_{k-1}\\\ \dots\\\
m_{0}\end{pmatrix}.$ (5.6)

Then, recalling the decay assumption on $m_{n}$, we get

$\displaystyle H^{k}_{\bf m}(u,v^{k,\rm min})$ $\displaystyle=$ $\displaystyle
2s_{\bf m}\langle u-v^{k,\rm min},u\rangle$ (5.7) $\displaystyle=$
$\displaystyle 2s_{\bf m}\langle u-\frac{1}{s_{\bf m}}M_{\bf
m}^{k}u,u\rangle-2(u_{k})^{2}t_{\bf m}+|u_{k}|\sqrt{k}\|u\|_{k}\
o(k^{-\alpha\beta})$ $\displaystyle=$ $\displaystyle 2s_{\bf m}\langle
u-\frac{1}{s_{\bf m}}M_{\bf m}^{k}u,u\rangle-2(u_{k})^{2}t_{\bf
m}+\|u\|_{k}^{2}\ o(k^{\frac{1}{2}-\alpha\beta}),$

concluding the proof of (5.4) since $\alpha>\frac{3}{\beta}$. ∎

######  Remark 5.3.

Let $u^{k}$ be constant on $[0,k^{\alpha}]$ and $[k-k^{\alpha},k]$. Then the
corresponding $v^{k,\min}$ given by (5.6) satisfies
$|v^{k,\min}_{0}-u^{k}_{0}|+|v^{k,\min}_{k}-u^{k}_{k}|=o(k^{{1\over
2}-\alpha\beta})\|u^{k}\|_{k}$. Hence it can be modified so as to obtain
$\widehat{v}^{k}$ equal to $u^{k}$ in $0$ and $k$ and
$|v^{k,\min}_{i}-v^{k}_{i}|=o(k^{{1\over 2}-\alpha\beta})\|u^{k}\|_{k}$ for
all $i$. By (5.7) we can estimate

$H^{k}_{\bf m}(u,\widehat{v}^{k})\leq H^{k}_{\bf m}(u,v^{k,\rm
min})+\|u^{k}\|^{2}_{k}\ o(k^{1-\alpha\beta}).$

If $\|u^{k}\|_{k}$ are equibounded, then the last term is $o(\frac{1}{k})$
since $\alpha>{3\over\beta}$. Note that we may also construct
$\widehat{v}^{k}$ so that $\widehat{v}^{k}_{i}=u^{k}_{0}$ for $i\leq
k^{\alpha^{\prime}}$ and $\widehat{v}^{k}_{i}=u^{k}_{k}$ if $i\geq
k-k^{\alpha^{\prime}}$ with $\alpha^{\prime}<\alpha$.

###### Proof of Theorem 5.1.

We write

$\widehat{Q}_{\bf
m}f(z)=\lim_{k\to+\infty}\frac{1}{k}\min\Big{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+J_{\bf
m}^{k}(u):u\in\mathcal{A}(k,z)\Big{\\}}\,.$ (5.8)

Let $u^{k}$ denote a minimizer of the problem above, and note that
$\|u^{k}\|_{k}$ are equibounded in view of the growth condition on
$f(z)+m_{1}z^{2}$. Note that thanks to Lemma A.1 we may suppose that the
function $u^{k}$ is constant on $[0,k^{\alpha}]$ and $[k-k^{\alpha},k]$. Then,
applying Lemma 5.2 and Remark 5.3, we obtain the desired result. ∎

In general, the advantage of the rewriting in Theorem 5.1 is not clear.
However, thanks to the two-variable formulation, we can obtain some general
lower bound in suitable hypotheses. In the next section, we will see that for
exponential kernels functionals $H^{k}_{\bf m}$ can be rewritten as nearest-
neighbour energies, which will allow to make these bounds sharp.

######  Remark 5.4 (lower bounds with additive vector energies).

Suppose that there exists $C>0$ such that for all $v\in\mathcal{A}(k;z)$

$\langle A^{k}_{\bf m}v,v\rangle\geq
C\sum_{i=1}^{k}(v_{i}-v_{i-1})^{2}+\|v\|_{k}^{2}\ o(1)_{k\to+\infty}.$ (5.9)

Then, by (5.2), we can bound $\widehat{Q}_{\bf m}f(z)$ from below with limits
of scaled minimum problems for energies of the form

$\sum_{i=1}^{k}f(u_{i}-u_{i-1})+2s_{\bf
m}C\sum_{i=1}^{k}(v_{i}-v_{i-1})^{2}+2s_{\bf
m}\sum_{i=1}^{k}(u_{i}-v_{i})^{2}.$

We will see in the next section that this holds with some particular choices
of the kernel $\bf m$; namely, the exponential kernels.

In view of Remark 5.4, we now focus on bounds for problems involving energies
of the form

$E(u,v;[0,k])=\sum_{i=1}^{k}f(u_{i}-u_{i-1})+a\sum_{i=1}^{k}(v_{i}-v_{i-1})^{2}+b\sum_{i=1}^{k}(u_{i}-v_{i})^{2}$

with $a,b>0$.

We suppose that there exist $z^{*}$ and $\eta$ such that $f$ is convex for
$z\leq z^{*}$ and $f(z)\geq\eta$ for $z>z^{*}$. For any $N\geq 1$ we define

$\displaystyle g_{N}(z)={1\over
N}\Bigl{(}\min\Bigl{\\{}\sum_{i=2}^{N}f(u_{i}-u_{i-1})+a\sum_{i=1}^{N}(v_{i}-v_{i-1})^{2}+b\sum_{i=1}^{N}(u_{i}-v_{i})^{2}$
$\displaystyle\hskip 85.35826ptv_{0}=0,v_{N}=Nz,u_{i}-u_{i-1}\leq z^{*}\hbox{
for }i\geq 2\Bigr{\\}}+\eta\Bigr{)},$ (5.10)

where we limit the interactions $v_{i}-v_{j}$ only to nearest neighbours, and
we allow $u_{i}-u_{i-1}>z^{*}$ only for $i=1$. Note that if $N=1$ then
$g_{1}(z)=az^{2}+\eta$.

We also set $g_{\infty}(z)=f(z)+az^{2}$ with domain $z\leq z^{*}$, which
corresponds to minimal states with $u_{i}-u_{i-1}\leq z^{*}$ for all $i$.

######  Proposition 5.5 (lower bound with nearest-neighbour energies).

We have

$\lim_{k\to+\infty}{1\over
k}\min\Bigr{\\{}E(u,v;[0,k]):u_{k}-u_{0}=v_{k}-v_{0}=kz\Bigl{\\}}\geq\Bigl{(}\inf_{N}g_{N}(z)\Bigr{)}^{**}.$
(5.11)

###### Proof.

The proof is obtained giving a lower bound for the minima

${1\over
k}\min\Bigl{\\{}\sum_{i=1}^{k}f_{\eta}(u_{i}-u_{i-1})+a\sum_{i=1}^{k}(v_{i}-v_{i-1})^{2}+b\sum_{i=1}^{k}(u_{i}-v_{i})^{2}:u_{k}-u_{0}=v_{k}-v_{0}=kz\Bigr{\\}},$

where

$f_{\eta}(z)=\begin{cases}f(z)&\hbox{ if }z\leq z^{*}\\\ \eta&\hbox{ if
}z>z^{*}.\end{cases}$

Consider a minimizer $u$ for such problem. If $u_{i}-u_{i-1}\leq z^{*}$ for
all $i$ then by the convexity of $f$ this minimum equals the value $g_{0}(z)$.
If otherwise $u_{i}-u_{i-1}>z^{*}$ for some $i$, note that we can always
suppose that this holds for $i=1$, by splitting the discrete interval
$\\{0,\ldots,k\\}$ into subsets $\\{i_{k_{j-1}}\ldots,i_{k_{j}}\\}$,
$j=1,\dots,r$, in which $u_{i}-u_{i-1}>z^{*}$ only for $i=i_{k_{j-1}}+1$, we
obtain a lower estimate with

$\sum_{j=1}^{r}{N_{j}\over k}g_{N_{j}}(z_{j})$

where $N_{j}=k_{j}-k_{j-1}$ and $z_{j}={u_{k_{j}}-u_{k_{j-1}}\over N_{j}}$, so
that we have the convex combination

$\sum_{j=1}^{r}{N_{j}\over k}z_{j}=z\,.$

From this estimate (5.11) follows. ∎

We will prove general properties of the functions $g_{N}$ in Section 5.3,
which will allow to describe the structure of their convex envelope and their
optimality in computing $\widehat{Q}_{\bf m}f$.

###  5.2 Reduction to a local problem for the exponential kernel

We now introduce some notation for the exponential kernels. We define

$\mathbf{m}=\mathbf{m}^{\sigma}=\\{m^{\sigma}_{n}\\}=\\{e^{-\sigma n}\\},$
(5.12)

where $\sigma>0$ is a given constant. Highlighting the dependence on the
parameter $\sigma$, we set

$\left.\begin{array}[]{ll}\displaystyle\widehat{Q}_{\sigma}f(z)=\\!\lim_{k\to+\infty}\frac{1}{k}\inf\Big{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\sum_{i,j=0}^{k}e^{-|i-j|\sigma}(u_{i}-u_{j})^{2}:u\in\mathcal{A}(k;z)\Big{\\}},\end{array}\right.$
(5.13)

and introduce the corresponding $\bf m^{\sigma}$-transform of $f$

$Q_{\sigma}f(z)=\widehat{Q}_{\sigma}f(z)-a_{{\bf
m}^{\sigma}}z^{2}=\widehat{Q}_{\sigma}f(z)-\frac{2e^{-\sigma}(1+e^{-\sigma})}{(1-e^{-\sigma})^{3}}z^{2}.$
(5.14)

Let $F_{\varepsilon}^{\sigma}$ denote the non-local functionals of the type
defined in (LABEL:) with exponential kernel $m_{n}=e^{-\sigma n}$; that is,

$F^{\sigma}_{\varepsilon}(u;I)=\varepsilon\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}f\Big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\Big{)}+\varepsilon\sum_{i,j\in\mathcal{I}_{\varepsilon}(I)}e^{-\sigma|i-j|}\Big{(}\frac{u_{i}-u_{j}}{\varepsilon}\Big{)}^{2},$
(5.15)

where $\mathcal{I}_{\varepsilon}(I)=\\{i\in\mathbb{Z}:\varepsilon i\in I\\}$,
$\mathcal{I}^{\ast}_{\varepsilon}(I)=\\{i\in\mathbb{Z}:\varepsilon
i,\varepsilon(i-1)\in I\\}$ and the function $u$ belongs to
$\mathcal{A}_{\varepsilon}(I)=\\{u\colon\varepsilon\mathcal{I}_{\varepsilon}(I)\to\mathbb{R}\\}$
as defined in (2.2). Following the general approach formulated in Section 5.1,
given $a,b>0$ we define the local two-variable energies

$E_{\varepsilon}(u,v;I)=\varepsilon\\!\\!\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}\\!\\!f\Big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\Big{)}+\frac{a}{\varepsilon}\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}\\!\\!(v_{i}-v_{i-1})^{2}+\frac{b}{\varepsilon}\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}\\!\\!(u_{i}-v_{i})^{2}$
(5.16)

for $u,v\in\mathcal{A}_{\varepsilon}(I)$. We will prove an asymptotic
equivalence result between $F_{\varepsilon}^{\sigma}$ and $E_{\varepsilon}$;
more precisely, that the $\Gamma$-limits of the two sequences are the same for
a suitable choice of $a=a_{\sigma}$ and $b=b_{\sigma}$. The $\Gamma$-limit of
$E_{\varepsilon}$ is computed with respect to the convergence
$u^{\varepsilon},v^{\varepsilon}\to u$ defined as the convergence in
$L^{2}(I)$ of the piecewise-constant extensions of $u^{\varepsilon}$ and
$v^{\varepsilon}$ to the function $u\in H^{1}(I)$. The result is obtained, in
the spirit of Section 5.1, by explicitly integrating out the variable $v$.

######  Theorem 5.6 (asymptotic equivalence).

Let

$a_{\sigma}=a_{{\bf
m}^{\sigma}}=\frac{2(1+e^{-\sigma})e^{-\sigma}}{(1-e^{-\sigma})^{3}},\quad\quad
b_{\sigma}=\frac{2(1+e^{-\sigma})}{(1-e^{-\sigma})},$ (5.17)

and set $E^{\sigma}_{\varepsilon}=E_{\varepsilon}$ as defined in (5.16) with
$a=a_{\sigma}$ and $b=b_{\sigma}$. Then the sequence
$E^{\sigma}_{\varepsilon}$ $\Gamma$-converges to the same $\Gamma$-limit as
the sequence $F^{\sigma}_{\varepsilon}$.

######  Remark 5.7 (asymptotic behaviour controlled by $\sigma$).

We can interpret the extremal regimes of strong and weak additivity in terms
of the parameters of the two-parameter energies (5.16). Let
$a_{\sigma},b_{\sigma}$ be given by (5.17). As $\sigma\to 0$ we have both
$a_{\sigma}\to+\infty$ and $b_{\sigma}\to+\infty$, with an increasing strength
of the effect of the term involving the distance of $u$ from the affine
function $zi$. Conversely, when $\sigma\to+\infty$ we have $a_{\sigma}\to 0$,
and the role of this distance term gradually diminishes.

######  Remark 5.8 (equivalence with arbitrary coefficients).

The equivalence result in Theorem 5.6 can be extended to arbitrary pairs
$a,b>0$ up to considering the non-local functionals with kernel $m_{n}=\varrho
e^{-\sigma n}$; that is, the functionals given by

$F^{\varrho,\sigma}_{\varepsilon}(u;I)=\varepsilon\sum_{i,i-1\in\mathcal{I}_{\varepsilon}(I)}f\Big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\Big{)}+\varepsilon\,\varrho\sum_{i,j\in\mathcal{I}_{\varepsilon}(I)}e^{-\sigma|i-j|}\Big{(}\frac{u_{i}-u_{j}}{\varepsilon}\Big{)}^{2},$

with the choices

$\sigma=\sigma_{a,b}=2\sinh^{-1}\Big{(}\frac{1}{2}\sqrt{\frac{b}{a}}\Big{)}\quad\hbox{
and }\quad\varrho=\varrho_{a,b}=\frac{b^{2}}{4a\sinh(\sigma_{a,b})}.$ (5.18)

Indeed, with this definition we get

$\displaystyle\frac{a}{\varrho_{a,b}}=\frac{2(1+e^{-\sigma_{a,b}})e^{-\sigma_{a,b}}}{(1-e^{-\sigma_{a,b}})^{3}}=a_{\sigma}\
\ \hbox{ and }\ \
\frac{b}{\varrho_{a,b}}=\frac{2(1+e^{-\sigma_{a,b}})}{1-e^{-\sigma_{a,b}}}=b_{\sigma},$

so that we can apply Theorem 5.6 obtaining the equivalence between
$\frac{1}{\varrho}F_{\varepsilon}^{\varrho,\sigma}$ and
$\frac{1}{\varrho}E^{\sigma}_{\varepsilon}$. The corresponding (trivial)
generalization of $Q_{\sigma}f$ in (5.14) can be obtained by defining

$\widehat{Q}_{\sigma,\varrho}f(z)=\\!\lim_{k\to+\infty}\frac{1}{k}\inf\Big{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\varrho\sum_{i,j=0}^{k}e^{-|i-j|\sigma}(u_{i}-u_{j})^{2}:u\in\mathcal{A}(k;z)\Big{\\}},$
(5.19)

and setting
$Q_{\sigma,\varrho}f(z)=\widehat{Q}_{\sigma,\varrho}f(z)-a_{\sigma}\varrho
z^{2}$, with $a_{\sigma}$ as in (5.17).

The proof of Theorem 5.6 is based on the following lemma, which allows to
integrate out the variable $v$ by applying the general result of Lemma 5.2 to
the case of exponential kernels.

######  Lemma 5.9.

Let $L>0$ and $k_{\varepsilon}=\lfloor\frac{L}{\varepsilon}\rfloor$. We fix
$\alpha\in(0,1)$ and set
$n_{\varepsilon}=\lfloor(k_{\varepsilon})^{\alpha}\rfloor$. Let
$F^{\sigma}_{\varepsilon}$ be given by (5.15) and $E^{\sigma}_{\varepsilon}$
be given by (5.16) with $a_{\sigma},b_{\sigma}$ as in (5.17) and $I=[0,L]$.
Then, if
$u^{\varepsilon}\in\mathcal{A}_{\varepsilon}=\mathcal{A}_{\varepsilon}([0,L])$
satisfies $u^{\varepsilon}_{i}=u^{\varepsilon}_{0}$ for $i\leq
n_{\varepsilon}$, $u^{\varepsilon}_{i}=u^{\varepsilon}_{k_{\varepsilon}}$ for
$i\geq k_{\varepsilon}-n_{\varepsilon}$, we have

$\min\\{E^{\sigma}_{\varepsilon}(u^{\varepsilon},v;[0,L])\\!:v\in\mathcal{A}_{\varepsilon}^{\\#}(u^{\varepsilon})\\}=F^{\sigma}_{\varepsilon}(u^{\varepsilon};[0,L])+\|u^{\varepsilon}\|^{2}_{L^{2}}\
o(1)_{\varepsilon\to 0}$ (5.20)

where
$\mathcal{A}_{\varepsilon}^{\\#}(u^{\varepsilon})=\\{v\in\mathcal{A}_{\varepsilon}:v_{0}=v_{1}=u^{\varepsilon}_{0},\
v_{k_{\varepsilon}}=v_{k_{\varepsilon}-1}=u^{\varepsilon}_{k_{\varepsilon}}\\}$.

###### Proof.

For $u,v\in\mathcal{A}_{\varepsilon}$, we set

$\displaystyle\hskip-34.14322ptH_{\varepsilon}(u,v)=\frac{a_{\sigma}}{\varepsilon}\sum_{i=1}^{k_{\varepsilon}}(v_{i}-v_{i-1})^{2}+\frac{b_{\sigma}}{\varepsilon}\sum_{i=1}^{k_{\varepsilon}}(u_{i}-v_{i})^{2}=E_{\varepsilon}^{\sigma}(u,v;[0,L])-\varepsilon\sum_{i=1}^{k_{\varepsilon}}f(u_{i}-u_{i-1})$
$\displaystyle\hskip-34.14322ptJ_{\varepsilon}(u)=\frac{1}{\varepsilon}\sum_{i,j=0}^{k_{\varepsilon}}e^{-\sigma|i-j|}(u_{i}-u_{j})^{2}=F_{\varepsilon}^{\sigma}(u,v;[0,L])-\varepsilon\sum_{i=1}^{k_{\varepsilon}}f(u_{i}-u_{i-1}).$

Up to translations, we can assume $u^{\varepsilon}_{0}=0$ (and hence
$u^{\varepsilon}_{i}=0$ for $i\leq L\varepsilon^{-\alpha}$). We introduce the
$(k_{\varepsilon}+1)\times(k_{\varepsilon}+1)$ matrix
$M^{\varepsilon}_{\sigma}=(m_{ij})$ given by
$m_{ij}=m^{\sigma}_{|i-j|}=e^{-\sigma|i-j|}$, $i,j=0,\dots,k_{\varepsilon}$.
Note that $m_{n}^{\sigma}=e^{-\sigma n}$ satisfies
$m_{n}^{\sigma}=o(n^{-\beta})$ for any $\beta$ and in particular for
$\beta>\frac{3}{\alpha}$. In order to apply Lemma 5.2, we compute
$s_{\sigma}=s_{\bf m^{\sigma}}$ and the matrix
$A_{\sigma}^{\varepsilon}=A^{k_{\varepsilon}}_{\bf m}$ given by formula (5.5),
obtaining

$s_{\sigma}=m^{\sigma}_{0}+2\sum_{n=1}^{+\infty}m^{\sigma}_{n}=\frac{1+e^{-\sigma}}{1-e^{-\sigma}}\
\ \ \hbox{\rm and }\ \ \
A_{\sigma}^{\varepsilon}=D^{\varepsilon}_{\sigma}(M^{\varepsilon}_{\sigma})^{-1}-I,$
(5.21)

where $D^{\varepsilon}_{\sigma}$ is the
$(k_{\varepsilon}+1)\times(k_{\varepsilon}+1)$ diagonal matrix with diagonal
$\\{1+e^{-\sigma},s_{\sigma},\dots,s_{\sigma},1+e^{-\sigma}\\}$. Moreover, in
this case we can compute the inverse of the matrix $M^{\varepsilon}_{\sigma}$,
which is the tridiagonal $(k_{\varepsilon}+1)\times(k_{\varepsilon}+1)$ matrix
given by

$(M^{\varepsilon}_{\sigma})^{-1}=\frac{1}{1-e^{-2\sigma}}\begin{pmatrix}1&-e^{-\sigma}&0&\dots&&0\\\
-e^{-\sigma}&1+e^{-2\sigma}&-e^{-\sigma}&0&\dots&0\\\
0&-e^{-\sigma}&1+e^{-2\sigma}&-e^{-\sigma}&0&\dots\\\
\dots&\dots&\dots&\dots&\dots&\dots\\\
0&\dots&\dots&0&-e^{-\sigma}&1\end{pmatrix}.$ (5.22)

Now, to each $u\in\mathcal{A}_{\varepsilon}$ we associate the corresponding
function defined on $\\{0,\dots,k^{\varepsilon}\\}$ by $i\mapsto u(\varepsilon
i)$; with a slight abuse of notation, we still denote this function by $u$.
Setting

$H^{k_{\varepsilon}}_{\sigma}(u,v)=\frac{2s_{\sigma}}{\varepsilon}\langle
A^{\varepsilon}_{\sigma}v,v\rangle+\frac{2s_{\sigma}}{\varepsilon}\langle
u-v,u-v\rangle$

for $u,v\colon\\{0,\dots,k^{\varepsilon}\\}\to\mathbb{R}$, we can then apply
Lemma 5.2 with $k=k^{\varepsilon}$, obtaining

$\displaystyle\min\\{H^{k_{\varepsilon}}_{\sigma}(u^{\varepsilon},v):\
v\colon\\{0,\dots,k_{\varepsilon}\\}\to\mathbb{R}\\}=J_{\varepsilon}(u^{\varepsilon})+\|u^{\varepsilon}\|_{L^{2}}^{2}\
o(1)_{\varepsilon\to 0}.$ (5.23)

We conclude by proving that, up to an infinitesimal term, the minimum of
$H_{\sigma}^{k_{\varepsilon}}(\tilde{u}^{\varepsilon},\cdot)$ on
$\mathcal{A}_{\varepsilon}$ coincides with the minimum of
$H_{\varepsilon}(u^{\varepsilon},\cdot)$ on $\mathcal{A}^{\\#}_{\varepsilon}$.
Indeed, given $u,v\in\mathcal{A}_{\varepsilon}$ we can write

$\displaystyle H_{\sigma}^{k_{\varepsilon}}(u,v)$ $\displaystyle=$
$\displaystyle-\frac{s_{\sigma}}{\varepsilon}\sum_{i,j=0}^{k_{\varepsilon}}(A_{\sigma}^{\varepsilon})_{ij}(v_{i}-v_{j})^{2}+\frac{2s_{\sigma}}{\varepsilon}\sum_{i=0}^{k_{\varepsilon}}\Big{(}\sum_{j=0}^{k_{\varepsilon}}(A_{\sigma}^{\varepsilon})_{ij}\Big{)}v_{i}^{2}+\frac{2s_{\sigma}}{\varepsilon}\sum_{i=0}^{k_{\varepsilon}}(u_{i}-v_{i})^{2}$
(5.24) $\displaystyle=$
$\displaystyle\frac{2(1+e^{-\sigma})e^{-\sigma}}{\varepsilon(1-e^{-\sigma})^{3}}\sum_{i=1}^{k_{\varepsilon}}(v_{i}-v_{i-1})^{2}+\frac{2(1+e^{-\sigma})}{\varepsilon(1-e^{-\sigma})}\sum_{i=0}^{k_{\varepsilon}}(u_{i}-v_{i})^{2}$
$\displaystyle=$ $\displaystyle H_{\varepsilon}(u,v),$

since $\sum_{j=0}^{k_{\varepsilon}}(A_{\sigma}^{\varepsilon})_{ij}=0$ for any
$i$ by (5.21) and (5.22). This formula in particular implies

$\langle
A^{\varepsilon}_{\sigma}v,v\rangle=\frac{e^{-\sigma}}{(1-e^{-\sigma})^{2}}\sum_{i=1}^{k_{\varepsilon}}(v_{i}-v_{i-1})^{2};$

that is, estimate (5.9) with $C=\frac{e^{-\sigma}}{(1-e^{-\sigma})^{2}}$,
which in this case is an equality.

Finally, recalling Remark 5.3 we obtain

$\min\\{H^{k_{\varepsilon}}_{\sigma}(u^{\varepsilon},v):v\colon\\{0,\dots,k_{\varepsilon}\\}\to\mathbb{R}\\}=\min\\{H_{\varepsilon}(u^{\varepsilon},v):v\in\mathcal{A}_{\varepsilon}^{\\#}(u^{\varepsilon})\\}+\|u^{\varepsilon}\|^{2}_{L^{2}}\
o(1)_{\varepsilon\to 0}$

and the claim follows by (5.23). ∎

###### Proof of Theorem 5.6.

Upper estimate. Let $F^{\sigma}(u;[0,L])$ be the $\Gamma$-limit of the
sequence $F_{\varepsilon}^{\sigma}$. Let $u\in L^{2}(0,L)$ be such that
$F^{\sigma}(u;[0,L])<+\infty$ and let
$u^{\varepsilon}\in\mathcal{A}_{\varepsilon}$ be a recovery sequence for the
$\Gamma$-limit $F^{\sigma}(u;[0,L])$. Let $\hat{u}^{\varepsilon}$ be the
sequence given by Lemma A.1 and $v^{\varepsilon,{\rm min}}$ be obtained by
minimization of the minimum problem in (5.20) with
$u^{\varepsilon}=\hat{u}^{\varepsilon}$. Recalling Lemma 5.9, we get

$\displaystyle\limsup_{\varepsilon\to
0}E^{\sigma}_{\varepsilon}(\hat{u}^{\varepsilon},v^{\varepsilon,{\rm
min}};[0,L])$ $\displaystyle\leq$ $\displaystyle\limsup_{\varepsilon\to
0}F^{\sigma}_{\varepsilon}(\hat{u}^{\varepsilon},[0,L])$ $\displaystyle\leq$
$\displaystyle\limsup_{\varepsilon\to
0}F^{\sigma}_{\varepsilon}(u^{\varepsilon};[0,L]).$

This gives the upper estimate for the $\Gamma$-limit of
$E^{\sigma}_{\varepsilon}$.

Lower estimate. Let $u\in H^{1}(0,L)$ and let
$u^{\varepsilon},v^{\varepsilon}$ converge to $u$ in $L^{2}(0,L)$ and be such
that $\sup E^{\sigma}_{\varepsilon}(u^{\varepsilon},v^{\varepsilon};[0,L])\leq
S<+\infty.$ Let $\hat{u}^{\varepsilon},\hat{v}^{\varepsilon}$ be the sequences
given by Lemma A.1(B). Hence

$\liminf_{\varepsilon\to
0}E^{\sigma}_{\varepsilon}(\hat{u}^{\varepsilon},\hat{v}^{\varepsilon};(0,L))\leq\liminf_{\varepsilon\to
0}E^{\sigma}_{\varepsilon}(u^{\varepsilon},v^{\varepsilon};(0,L)).$ (5.25)

Applying Lemma 5.9 we obtain

$\displaystyle\liminf_{\varepsilon\to
0}E^{\sigma}_{\varepsilon}(u^{\varepsilon},v^{\varepsilon};[0,L])$
$\displaystyle\geq$ $\displaystyle\liminf_{\varepsilon\to
0}E^{\sigma}_{\varepsilon}(\hat{u}^{\varepsilon},v^{\varepsilon,{\rm
min}}(\hat{u}^{\varepsilon});[0,L])$ $\displaystyle\geq$
$\displaystyle\liminf_{\varepsilon\to
0}F^{\sigma}_{\varepsilon}(\hat{u}^{\varepsilon};[0,L]).$

This concludes the proof. ∎

By the results in Section 5.1 we can use the equivalence above to give a
useful characterization of $\widehat{Q}_{\sigma}f$.

######  Remark 5.10 (representation of $\widehat{Q}_{\sigma}f$ in terms of
local functionals).

Formula (5.2) in Theorem 5.1 and equality (5.24) prove the following formula
for the function $\widehat{Q}_{\sigma}f$ defined in (5.13):

$\widehat{Q}_{\sigma}f(z)=\lim_{N\to+\infty}\frac{1}{N}\min\\{E_{1}^{\sigma}(u,v;[0,N]):u_{0}=v_{0}=0,u_{N}=v_{N}=Nz\\},$
(5.26)

where $E_{1}^{\sigma}$ is defined by (5.16) with $\varepsilon=1$ and
$a=a_{\sigma},b=b_{\sigma}$ satisfying (5.17).

######  Remark 5.11 (representation of the constrained relaxation in terms of
local functionals).

Formula (5.26) can be extended to constrained problems; namely, we have

$\widehat{Q}_{\sigma}f\Big{(}\frac{p}{q},z\Big{)}=\liminf_{k\to+\infty}\frac{1}{kq}\min\Big{\\{}E_{1}^{\sigma}(u,v;[0,kq]):u,v\in\mathcal{A}(kq;z),u\in\mathcal{V}\Big{(}kq;\frac{p}{q}\Big{)}\Big{\\}},$
(5.27)

where, accordingly with the notation above, $\widehat{Q}_{\sigma}f(\theta,z)$
denotes the constrained relaxation $\widehat{Q}_{{\bf
m}_{\sigma}}f(\theta,z)$, and $\mathcal{V}(kq;\frac{p}{q})$ is the set of
admissible constrained functions defined in (3.2). Indeed, we note that
Theorem 5.1 also holds for constrained relaxation, since we can apply Lemma
5.2 to $u$ satisfying a volume constraint (see Lemma 3.3).

######  Remark 5.12 (non-exponential kernels).

For a general kernel $\bf m$ the matrix $M^{k}=(m_{ij})_{i,j=0}^{k}$ is a
symmetric Toeplitz matrix. Under decay conditions on $m_{n}$ we can apply the
arguments in Section 5.1. However, since $(M^{k})^{-1}$ now is not of the form
(5.22) (for some insight on the problem of the inversion of a general
symmetric Toeplitz matrix we refer, e.g., to [16]), the resulting functional
$H^{k}_{\varepsilon}$ does not depend on nearest neighbours only and the
argument showing the optimality of the bounds can not be completed as above.
However, for particular classes of kernels $\bf m$ the resulting functionals
$H^{k}_{\varepsilon}$ may be still amenable to analysis, even if they involve
next-to-nearest-neighbour interactions and beyond. The analytical transparency
of such functionals will then allow one to extract useful information on the
form of the corresponding $\widehat{Q}_{\bf m}f$.

###  5.3 Truncated convex potential

In this section we show some properties of $\widehat{Q}_{\sigma}f$ and of the
corresponding phase function $\theta$ if $f$ is a general truncated convex
function; that is,

$f(z)=\begin{cases}\tilde{f}(z)&\hbox{\rm if }z\leq z^{\ast}\\\
\tilde{f}(z^{\ast})&\hbox{\rm if }z>z^{\ast},\end{cases}$ (5.28)

where $z^{\ast}>0$ and $\tilde{f}\colon\mathbb{R}\to[0,+\infty)$ is strictly
convex and such that $\tilde{f}(0)=0$. Note that we can suppose that
$\tilde{f}$ satisfies the growth condition

$\tilde{f}(z)\geq c_{1}z^{2}-c_{2}$

in $[0,+\infty)$ for some $c_{1},c_{2}>0$. Using the notation of Section 3, we
set $A=[z^{\ast},+\infty)$.

######  Remark 5.13 (more general $f$).

Note that the condition $\tilde{f}(0)=0$ can be substituted by the hypothesis
that $\tilde{f}$ has a minimum point $z_{\rm min}<z^{*}$, since affine changes
of variables are compatible with the definition of $\widehat{Q}_{\bf m}f$ by
Remark 2.25.

####  5.3.1 Characterization of $\widehat{Q}_{\sigma}f$ in terms of periodic
arrangements

Given the local form of the problem (5.26) formulated in terms of the two-
variable functional $E_{1}^{\sigma}(u,v;[0,N])$, the relaxed energy
$\widehat{Q}_{\sigma}f$ can be obtained by optimizing the location of ‘broken
bonds’; that is, of indices $i$ such that $u_{i}-u_{i-1}\in A$, similarly to
what done in the case of concentrated kernels. The fact that these bonds can
be always considered as either isolated or organized in a ‘broken island’
makes the structure of oscillations (microstructure) compatible with the
lattice. This makes the problem analytically tractable.

Note first that on the complement of the broken bonds the energy coincides
with its ‘convex part’, defined as follows. Given $a,b>0$, for a bounded
interval $I$ and $u,v\in\mathcal{A}_{\varepsilon}(I)$ we introduce the
functional $\tilde{E}_{\varepsilon}$ given by

$\tilde{E}_{\varepsilon}(u,v;I)=\varepsilon\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}\tilde{f}\big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\big{)}+\frac{a}{\varepsilon}\\!\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}\\!(v_{i}-v_{i-1})^{2}+\frac{b}{\varepsilon}\\!\sum_{i\in\mathcal{I}_{\varepsilon}(I)}\\!(u_{i}-v_{i})^{2},$
(5.29)

where we recall that
$\mathcal{I}_{\varepsilon}^{\ast}=\\{i\in\mathbb{Z}:\varepsilon
i,\varepsilon(i-1)\in I\\}$. Note that, since these energies will be used to
compute minimum problems with Dirichlet boundary conditions, we consider the
last term of the sum in the whole
$\mathcal{I}_{\varepsilon}(I)=\\{i\in\mathbb{Z}:\varepsilon i\in I\\}$.

In view of Section 5.1, for all $N\geq 2$ we can write the functions $g_{N}$
introduced in (5.1) with $\eta$ replaced by $\tilde{f}(z^{\ast})$ as

$g_{N}^{a,b}(z)=g_{N}(z)=\frac{1}{N}\big{(}\tilde{f}(z^{\ast})+\min\big{\\{}av_{1}^{2}+\tilde{E}_{1}(u,v;[1,N]):v_{N}=Nz\big{\\}}\big{)}.$
(5.30)

They represent the minimal energy of an array of $N$ bonds, of which the first
one is broken, with given average gradient. By uniformity of notation, we also
set

$g_{1}(z)=\tilde{f}(z^{\ast})+az^{2}\quad\hbox{\rm and}\quad
g_{\infty}(z)=\tilde{f}(z)+az^{2}.$ (5.31)

If $a=a_{\sigma}$ and $b=b_{\sigma}$ are given by (5.17), then we set

$g_{N}^{\sigma}(z)=g_{N}(z)\ \ \ \hbox{\rm and }\ \ \
\tilde{E}_{\varepsilon}^{\sigma}(u,v;I)=\tilde{E}_{\varepsilon}(u,v;I).$

Note that, by using $u_{i}=v_{i}=zi$ as test function in the definition of
$g^{\sigma}_{N}(z)$, we get

$\lim_{N\to+\infty}g^{\sigma}_{N}(z)\leq\tilde{f}(z)+a_{\sigma}z^{2}.$

In the following proposition, based on the analysis of the distribution of
broken bonds in minimizers, we show that $\widehat{Q}_{\sigma}f(z)$,
considered as the infimum of the corresponding constrained functions, can be
described by only using the values $\theta=\frac{1}{N}$, which will be proved
to be the locking states. The full description of this structure will be given
in Proposition 5.23, after a delicate analysis of the general properties of
$g_{N}$.

######  Proposition 5.14 (characterization of $\widehat{Q}_{\sigma}f$ in terms
of periodic arrangements).

Fixed $\sigma>0$, let $a=a_{\sigma}$ and $b=b_{\sigma}$ be given by (5.17). If
$f$ is a truncated convex potential as in (5.28), then

$\widehat{Q}_{\sigma}f(z)=\Big{(}\inf_{N\in\mathbb{N}}\\{g^{\sigma}_{N}\\}\Big{)}^{\ast\ast}(z)\,.$
(5.32)

######  Remark 5.15.

Note that, recalling Remark 5.8, Proposition 5.14 holds for any $a,b>0$ with
$g^{a,b}_{N}$ in place of $g^{\sigma}_{N}$ and $a$ in place of $a_{\sigma}$,
up to substituting $\widehat{Q}_{\sigma}f$ with
$\widehat{Q}_{\sigma_{a,b},\varrho_{a,b}}f$ as defined in (5.19), with
$\sigma_{a,b}$ and $\varrho_{a,b}$ given by (5.18).

###### Proof of Proposition 5.14.

The lower bound is a consequence of Proposition 5.5. To conclude the proof we
show that
$\widehat{Q}_{\sigma}f(z)\leq(\inf_{n\in\mathbb{N}}\\{g^{\sigma}_{n}\\})^{\ast\ast}(z)$.
Since $\widehat{Q}_{\sigma}f$ is convex, it is sufficient to prove that
$\widehat{Q}_{\sigma}f(z)\leq\inf_{n\in\mathbb{N}}\\{g^{\sigma}_{n}(z)\\}$.

We fix $\delta>0$. For $z\in\mathbb{R}$ there exists
$\overline{n}\in\mathbb{N}$ such that
$g^{\sigma}_{\overline{n}}(z)\leq\inf_{n\in\mathbb{N}}\\{g^{\sigma}_{n}(z)\\}+\delta$.
If $\overline{n}=1$, then we can take as test functions $u,v$ given by
$u_{i}=v_{i}=iz$. For any $N\geq 1$ we get

$\frac{1}{N}E^{\sigma}_{1}(u,v;[0,N])\leq\frac{1}{N}\tilde{E}^{\sigma}_{1}(u,v;[0,N])=\tilde{f}(z)+a_{\sigma}z^{2}=g^{\sigma}_{1}(z)$

and the result follows by taking the limit for $N\to+\infty$. Otherwise, let
$\overline{u},\overline{v}\in\mathcal{A}_{1}([1,\overline{n}])$ be such that
$\overline{v}_{\overline{n}}=\overline{n}z$ and

$\tilde{f}(z^{\ast})+a_{\sigma}\overline{v}_{1}^{2}+\tilde{E}^{\sigma}_{1}(\overline{u},\overline{v};[1,\overline{n}])=\overline{n}\,g^{\sigma}_{\overline{n}}(z).$

We extend $\overline{u}$ and $\overline{v}$ in $0$ by setting
$\overline{u}_{0}=\overline{n}z-\overline{u}_{\overline{n}}$ and
$\overline{v}_{0}=0$. It follows that

$\displaystyle E^{\sigma}_{1}(\overline{u},\overline{v};[0,\overline{n}])$
$\displaystyle\leq$ $\displaystyle
E^{\sigma}_{1}(\overline{u},\overline{v};[1,\overline{n}])+b_{\sigma}(\overline{u}_{1}-\overline{v}_{1})^{2}+a_{\sigma}\overline{v}_{1}^{2}+\tilde{f}(z^{\ast})$
$\displaystyle\leq$
$\displaystyle\tilde{E}^{\sigma}_{1}(\overline{u},\overline{v};[1,\overline{n}])+a_{\sigma}\overline{\lambda}^{2}+\tilde{f}(z^{\ast})$
$\displaystyle=$ $\displaystyle\overline{n}\,g^{\sigma}_{\overline{n}}(z).$

For any $N\geq 1$ we choose $u^{N}$ and $v^{N}$ as test functions in
$[0,\overline{n}N]$ defined by setting $u^{N}_{i}$ equal to
$(j-1)\overline{n}z+\overline{u}_{i-(j-1)\overline{n}}$ in each
$[(j-1)\overline{n},j\overline{n})$, $j\in\\{1,\dots,N-1\\}$ and in
$[(N-1)\overline{n},N\overline{n}]$ and correspondingly $v^{N}_{i}$. We get

$\displaystyle\frac{1}{\overline{n}N}E^{\sigma}_{1}(u^{N},v^{N};[0,\overline{n}N])=\frac{1}{\overline{n}N}NE^{\sigma}_{1}(\overline{u},\overline{v};[0,\overline{n}])\leq
g^{\sigma}_{\overline{n}}(z)\leq\inf_{n\in\mathbb{N}}\\{g^{\sigma}_{n}(z)\\}+\delta.$

Letting $N\to+\infty$ the claim follows by the representation formula for
$\widehat{Q}_{\sigma}f$ given in (5.26). ∎

Figure 22: Shape of a minimizer of (5.33) in a ‘broken island’.

######  Remark 5.16 (simplification of the minimal configurations).

Given $u\in\mathcal{A}_{1}([0,N])$, we say that $i\in\\{1,\dots,N\\}$ belongs
to $\mathcal{B}(u)$ (the set of broken indices of $u$) if
$u_{i}-u_{i-1}>z^{\ast}$.

For future reference we show that the solutions of

$\min\big{\\{}E^{\sigma}_{1}(u,v;[0,N])\\!:v_{0}\\!=\\!0,v_{N}\\!=\\!Nz,\\#\mathcal{B}(u)=n\big{\\}}$
(5.33)

can be regrouped and rearranged. Let $(u,v)$ solve (5.33). Note that in the
union of the non-isolated ‘broken intervals’ we can assume that $u$ and $v$
are affine and equal. More precisely, the convexity of the square and a
translation argument allow to prove that there exists $z_{0}$ such that if
$\underline{i}+k+1\in\mathcal{B}(u)$ for $k\in\\{0,\dots,\underline{k}\\}$,
with $\underline{k}\geq 1$ then

$\displaystyle v(\underline{i}+k)=v(\underline{i})+z_{0}k\ \hbox{ for }\
k=0,\dots,...,\underline{k}+1$ $\displaystyle
v(\underline{i}+k)=u(\underline{i}+k)\ \hbox{ for }\ k=1,\dots,\underline{k}$

(see Figure 22). As a second step, we show that if $(u,v)$ solves (5.33) we
can assume that there is at most one ‘broken zone’ for $u$ with length greater
than $1$. To this end, we extend $u$ and $v$ by periodicity by setting
$u(N+j)=u(j)+Nz$ and $u(-j)=u(N-j)-Nz$ for $j=1,\dots,N$, and correspondingly
for $v$.

Figure 23: Construction of $(u,v)$ with isolated broken bonds.

Now we show that the minimum is attained at $(u,v)$ such that if
$i\in\mathcal{B}(u)$, then $i-1\not\in\mathcal{B}(u)$ and
$i+1\not\in\mathcal{B}(u)$, or $j\in\mathcal{B}(u)$ for all
$j\in\\{i+1,\dots,N\\}$. To show this, we suppose that $i_{0},i_{0}+1$,
$i_{0}+k$ and $i_{0}+k+1$ belong to $\mathcal{B}(u)$ for some $i_{0}\geq 1$,
$k\geq 2$ and $i_{0}+k\leq N$, while $i_{0}+j+1\not\in\mathcal{B}(u)$ for
$j\in\\{1,\dots,k-2\\}$.

We modify $u$ and $v$ by setting for $j=0,\dots,k-1$

$\tilde{u}(i_{0}+j)=u(i_{0}+j+1)-z_{0}\ \hbox{ and }\
\tilde{v}(i_{0}+j)=v(i_{0}+j+1)-z_{0}$

(see Figure 23). With this definition

$E^{\sigma}_{1}(\tilde{u},\tilde{v};[0,N])\leq E^{\sigma}_{1}(u,v;[0,N]).$

Figure 24: Distribution of broken bonds.

Thanks to the periodic extension of $u$ and $v$, this proves that in minimum
problem (5.33) we can assume that there exist
$n_{0},n_{1},...,n_{r}\in\mathbb{N}$ with $n_{l}>1$ for any
$l\in\\{1,\dots,r\\}$, $r+n_{0}=n(N,z)$ and $n_{0}=N-\sum_{l=1}^{r}n_{l}$,
such that

$\displaystyle i\in\mathcal{B}(u)\ \hbox{ for all }\ i\in\\{1,\dots,n_{0}\\}\
\hbox{ and }\sum_{l=1}^{j}n_{l}+1\in\mathcal{B}(u)\hbox{ for all }\
j\in\\{1,\dots,r\\}$ (5.34)

(see Figure 24).

This reduces the problem of the computation of the minimum value (5.33) to the
solution of the minimum problem on each (translated) island $[0,n_{j}],$
$j\in\\{1,\dots,r\\}$,

$\min\Big{\\{}E^{\sigma}_{1}(u,v;[0,n_{j}]):v_{0}\\!=\\!0,v_{n_{j}}\\!=\\!z_{j}n_{j},\
\mathcal{B}(u)=\\{1\\}\Big{\\}}$

and in the broken island $[0,n_{0}]$, where

$\min\Big{\\{}E^{\sigma}_{1}(u,v;[0,n_{0}]):v_{0}\\!=\\!0,v_{n_{0}}\\!=\\!z_{0}n_{0},\
\\#\mathcal{B}(u)=n_{0}\Big{\\}}=n_{0}g_{1}(z_{0}),$ (5.35)

with suitable boundary conditions $z_{j}$ satisfying
$\sum_{j=0}^{r}n_{j}z_{j}=Nz$.

Since
$E^{\sigma}_{1}(u,v;[0,n-1])=\tilde{E}^{\sigma}_{1}(u,v;[0,n-1])-b_{\sigma}(u_{0}-v_{0})^{2}$
if $\\#\mathcal{B}(u)=0$, for $n>1$ and $z\in\mathbb{R}$ we have

$\displaystyle\min\Big{\\{}E^{\sigma}_{1}(u,v;[0,n]):v_{0}\\!=\\!0,v_{n}\\!=\\!nz,\
\mathcal{B}(u)=\\{1\\}\Big{\\}}$
$\displaystyle\geq\min_{w\in\mathbb{R}}\Big{\\{}\min\Big{\\{}\tilde{E}^{\sigma}_{1}(u,v;[1,n]):v_{1}=w,v_{n}=nz\Big{\\}}+a_{\sigma}w^{2}+\tilde{f}(z^{\ast})\Big{\\}}$
$\displaystyle=ng^{\sigma}_{n}(z).$

####  5.3.2 General properties of the periodic bounds $g_{N}$

In order to relate the constrained relaxation
$\widehat{Q}_{\sigma}f(\theta,z)$ to $g^{\sigma}_{N}(z)$ and to characterize
the locking states of $f$, we analyze the properties of $g^{\sigma}_{N}(z)$ in
dependence on both $N$ and $z$. Note that in the following results we may
consider general values of $a,b>0$ and not limit to $a_{\sigma},b_{\sigma}$,
so that the results of this section hold for a general $g_{N}$ as defined in
(5.1).

######  Proposition 5.17 (convexity of $g_{N}$).

The functions $g_{N}$ are uniformly strictly convex. More precisely, we have

${1\over 2}g_{N}(z)+{1\over 2}g_{N}(z^{\prime})\geq
g_{N}\Bigl{(}{z+z^{\prime}\over 2}\Bigr{)}+a\Bigl{(}{z-z^{\prime}\over
2}\Bigr{)}^{2}$ (5.36)

for all $z,z^{\prime}\in\mathbb{R}$ and $N\in\mathbb{N}$.

###### Proof.

If $u,v$ and $u^{\prime},v^{\prime}$ are minimizers for $g_{N}(z)$ and
$g_{N}(z^{\prime})$ we can use the functions ${1\over 2}(u+u^{\prime}),{1\over
2}(v+v^{\prime})$ as test functions for $g_{N}({1\over 2}(z+z^{\prime}))$.
Using the convexity of $\tilde{f}$ and the quadraticity of the other terms;
more precisely, that for all $i$ we have (after setting $v_{0}=0$)

$a(v_{i}-v_{i-1})^{2}+a(v^{\prime}_{i}-v^{\prime}_{i-1})^{2}={a\over
2}((v_{i}+v^{\prime}_{i})-(v_{i-1}+v^{\prime}_{i-1}))^{2}+{a\over
2}((v_{i}-v_{i-1})-(v^{\prime}_{i}-v^{\prime}_{i-1}))^{2},$

we get

$\displaystyle{1\over 2}g_{N}(z)+{1\over 2}g_{N}(z^{\prime})$
$\displaystyle\geq$ $\displaystyle g_{N}\Bigl{(}{z+z^{\prime}\over
2}\Bigr{)}+{1\over N}{a\over
4}\sum_{i=1}^{N}((v_{i}-v_{i-1})-(v^{\prime}_{i}-v^{\prime}_{i-1}))^{2}$
$\displaystyle\geq$ $\displaystyle g_{N}\Bigl{(}{z+z^{\prime}\over
2}\Bigr{)}+a\Bigl{(}{1\over 2}{1\over
N}\sum_{i=1}^{N}((v_{i}-v_{i-1})-(v^{\prime}_{i}-v^{\prime}_{i-1}))\Bigr{)}^{2}$
$\displaystyle=$ $\displaystyle g_{N}\Bigl{(}{z+z^{\prime}\over
2}\Bigr{)}+a\Bigl{(}{z-z^{\prime}\over 2}\Bigr{)}^{2}\,,$

as desired. ∎

######  Remark 5.18.

From the previous proposition we deduce that $g_{N}^{\prime\prime}(z)\geq 2a$
at all $z$ where $g_{N}$ is twice differentiable. In particular, we obtain
that $g_{N}(z)\geq{\tilde{f}(z^{\ast})\over N}+az^{2}$ for all $N\geq 1$.

######  Remark 5.19 (symmetry of solutions).

The solutions $u,v$ of the minimum problem

$\min\bigg{\\{}\sum_{i=2}^{N}\tilde{f}({u_{i}-u_{i-1}})+a\sum_{i=2}^{N}(v_{i}-v_{i-1})^{2}+b\sum_{i=1}^{N}(u_{i}-v_{i})^{2}:v_{1}=v^{1},v_{N}=v^{N}\bigg{\\}}$
(5.37)

are symmetric with respect to the centre of the interval, in the sense that

$v_{j+1}-v_{j}=v_{N-j+1}-v_{N-j},\qquad u_{j+1}-u_{j}=u_{N-j+1}-u_{N-j}$
(5.38)

for $1\leq j\leq N-1$. Furthermore, if $N=2M+1$ is odd then

$v_{M+1}=u_{M+1}={v^{N}+v^{1}\over 2}$ (5.39)

while, if $N=2M$ is even then

${v_{M+1}+v_{M}\over 2}={u_{M+1}+u_{M}\over 2}={v^{N}+v^{1}\over 2}.$ (5.40)

Indeed, first note that we may state the boundary condition equivalently as
$v_{N}-v_{1}=V:=v^{N}-v^{1}$. Then, condition (5.38) is a direct consequence
of the strict convexity of the energy and is obtained using

$\overline{v}_{i}={v_{i}-v_{N+1-i}\over
2},\qquad\overline{u}_{i}={u_{i}-u_{N+1-i}\over 2}$ (5.41)

as test functions. To check, e.g., (5.39), note that from (5.38)

$\displaystyle v_{M+1}$ $\displaystyle=$ $\displaystyle
v_{1}+\sum_{j=1}^{M}(v_{j+1}-v_{j})=v_{1}+\sum_{j=1}^{M}(v_{N-j+1}-v_{N-j})$
$\displaystyle=$ $\displaystyle
v_{1}+\sum_{k=M+1}^{N-1}(v_{k+1}-v_{k})=v_{N}-v_{M+1}+v_{1},$

from which the first equality in (5.39) follows. To check the second one, note
that from (5.38) we obtain
$v_{i}+v_{2M+2-i}-2v_{M+1}=u_{i}+u_{2M+2-i}-2u_{M+1}=0$ for all $i$, from
which

${1\over N}\sum_{i=1}^{N}u_{i}=u_{M+1},\qquad{1\over
N}\sum_{i=1}^{N}v_{i}=v_{M+1}.$ (5.42)

Now, considering in place of $u_{i}$ the function

$\overline{u}_{i}=u_{i}+{v^{1}+v^{N}\over 2}-u_{M+1},$

as test functions, the only change in the problem in (5.37) is in the last
sum, for which, using (5.42) and the already proved equality in (5.39) for
$v$, we have

$\displaystyle\sum_{i=1}^{N}(\overline{u}_{i}-v_{i})^{2}=\sum_{i=1}^{N}(u_{i}-v_{i})^{2}-N\Bigl{(}{v^{1}+v^{N}\over
2}-u_{M+1}\Bigr{)}^{2},$

which contradicts the minimality of $u,v$ if the second equality in (5.39)
does not hold. The proof of (5.40) follows the same line with minor
modifications.

######  Proposition 5.20 (convexity properties with respect to $N$ with given
parity).

For all $N_{1},N_{2}\geq 1$ such that $N_{1}+N_{2}$ is even and $N_{1}\neq
N_{2}$, for all $z_{1},z_{2}\in\mathbb{R}\setminus\\{0\\}$ we have

${N_{1}\over N_{1}+N_{2}}g_{N_{1}}(z_{1})+{N_{2}\over
N_{1}+N_{2}}g_{N_{2}}(z_{2})\ >\ g_{N}(z),$ (5.43)

where $N=\frac{N_{1}+N_{2}}{2}\ \ \hbox{\rm and }\
z=\frac{N_{1}z_{1}+N_{2}z_{2}}{N_{1}+N_{2}}$. In particular, we have the
convexity property in $N$

${N_{1}\over N_{1}+N_{2}}g_{N_{1}}(z)+{N_{2}\over N_{1}+N_{2}}g_{N_{2}}(z)\ >\
g_{N}(z),\hbox{ where }N={N_{1}+N_{2}\over 2}\hbox{ and }z\neq 0.$ (5.44)

###### Proof.

We consider the case of $N_{1}$ and $N_{2}$ odd, the case of $N_{1}$ and
$N_{2}$ even following the same line with minor modifications.

Figure 25: construction of the test function $\overline{v}$.

Let $u^{1},v^{1}$ be minimizers for $g_{N_{1}}(z_{1})$ and let $u^{2},v^{2}$
be minimizers for $g_{N_{2}}(z_{2})$. We define $\overline{u},\overline{v}$ by
setting

$\overline{v}_{i}=\begin{cases}v^{1}_{i}+{v^{2}_{1}-v^{1}_{1}\over 2}&\hbox{
if }1\leq i\leq{N_{1}+1\over 2}\\\
v^{2}_{i+\frac{N_{2}-N_{1}}{2}}+\frac{1}{2}(N_{1}z_{1}+N_{2}z_{2})&\hbox{ if
}i\geq{N_{1}+1\over 2},\end{cases}$
$\overline{u}_{i}=\begin{cases}u^{1}_{i}+{v^{2}_{1}-v^{1}_{1}\over 2}&\hbox{
if }1\leq i\leq{N_{1}+1\over 2}\\\
u^{2}_{i+\frac{N_{2}-N_{1}}{2}}+\frac{1}{2}(N_{1}z_{1}+N_{2}z_{2})&\hbox{ if
}i\geq{N_{1}+1\over 2}\end{cases}$

(see Fig. 25). Thanks to Remark 5.19 this is a good definition,
$\overline{v}_{(N_{1}+1)/2}=\overline{u}_{(N_{1}+1)/2}$, and we have
$\overline{v}_{N}=\frac{1}{2}(N_{1}z_{1}+N_{2}z_{2})$, so that these are test
functions for $g_{N}(z)$. Again, by the symmetry properties of $v^{1}$ and
$v^{2}$ in Remark 5.19 we obtain (5.43). Note the strict inequality, which is
proved by noting that $\overline{v}_{i},\overline{u}_{i}$ do not satisfy the
properties of minimizers in Remark 5.19. ∎

From Proposition 5.20 we deduce a general convexity property which holds also
if $N_{1}$ and $N_{2}$ have different parity. Note that this implies that
fractures will be equidistributed up to oscillations of a unit, due to
incommensurability phenomena.

######  Corollary 5.21 (convexity properties with respect to arbitrary $N$).

Let $k,N\geq 2$ be integers, and $w_{k},w_{0}\in\mathbb{R}\setminus\\{0\\}$.
Then

$(N+k)g_{N+k}(w_{k})+Ng_{N}(w_{0})>(N+k-1)g_{N+k-1}(w_{k-1})+(N+1)g_{N+1}(w_{1})$
(5.45)

for some $w_{k-1},w_{1}$ such that
$(N+1)w_{1}+(N+k-1)w_{k-1}=Nw_{0}+(N+k)w_{k}$. Moreover $w_{k-1},w_{1}$ belong
to the interval with endpoints $w_{0}$ and $w_{k}$.

###### Proof.

Let $(w_{1},\dots,w_{k-2})$ be the solution of the linear system given by the
equations

$(N+h)w_{h}+(N+h-2)w_{h-2}=2(N+h-1)w_{h-1}$

for $h=2,\dots,k$. We can repeat the application of (5.43) to each pair
$N_{1}=N+h$, $N_{2}=N+h-2$ with $h=2,\dots,k$, by fixing at each step
$z_{1}=w_{h},$ $z_{2}=w_{h-2}$, obtaining

$\displaystyle(N+k)g_{N+k}(w_{k})-(N+k-1)g_{N+k-1}(w_{k-1})$
$\displaystyle\hskip 85.35826pt>\
(N+k-1)g_{N+k-1}(w_{k-1})-(N+k-2)g_{N+k-2}(w_{k-2})$ $\displaystyle\hskip
85.35826pt>\ (N+1)g_{N+1}(w_{1})-Ng_{N}(w_{0}).$

The last part of the claim follow by induction. ∎

Now we can show an ordering property of the functions $g_{N}$ which allows to
describe the structure of $\widehat{Q}_{\sigma}f$ in terms of the locking
states.

######  Remark 5.22.

If we define the auxiliary functions $\widetilde{g}_{N}(z)=g_{N}(z)-{\eta\over
N}$, then we have $\widetilde{g}_{N}(z)<\widetilde{g}_{N+1}(z)$ for all $N\geq
1$ and $z>0$. This is proved by induction using Proposition 5.20 with
$N_{1}=N-1$, $N_{2}=N+1$ and $z_{1}=z_{2}=z$, after noting that for $N=1$ the
inequality $\widetilde{g}_{1}(z)<\widetilde{g}_{2}(z)$ is implied by
Proposition 5.17 since $\widetilde{g}_{1}(z)=az^{2}$.

####  5.3.3 Characterization of locking states

The convexity properties of $g_{N}(z)$ allow to characterize the locking
states of the function $f$ and to give a description of $Q_{\sigma}f(z)$.

######  Theorem 5.23 (locking states of $Q_{\sigma}f$).

Let $f$ be as in (5.28) and let $m_{n}^{\sigma}=e^{-\sigma n}$. Then the set
of locking states of $Q_{\sigma}f$ is given by

$\Big{\\{}\frac{1}{N}:N\in\mathbb{N},N\geq 1\Big{\\}}\cup\\{0\\}.$

###### Proof.

Step $1$. We prove by induction the monotonicity of the sequence $g_{N}(z)$
for $z$ large enough. By Proposition 5.20 we obtain that if $g_{N}(z)\geq
g_{N-1}(z)$ then

${N-1\over 2N}g_{N}(z)+{N+1\over 2N}g_{N+1}(z)\ \geq\ {N-1\over
2N}g_{N-1}(z)+{N+1\over 2N}g_{N+1}(z)\ >\ g_{N}(z);$

hence,

${N+1\over 2N}g_{N+1}(z)\ >\ \Big{(}1-{N-1\over 2N}\Big{)}g_{N}(z)\ =\
{N+1\over 2N}g_{N}(z)\,.$

Hence, iterating this argument, we get that the sequence $k\mapsto g_{k}(z)$
is not decreasing for $k\geq N-1$ and strictly increasing for $k\geq N$.

Step $2$. Now we show that for $z$ large enough then $g_{2}(z)\geq g_{1}(z)$.
By the growth hypothesis $\tilde{f}(z)\geq c_{1}z^{2}-c_{2}$ we get

$\displaystyle g_{2}(z)$ $\displaystyle\geq$
$\displaystyle\frac{\tilde{f}(z^{\ast})}{2}-\frac{c_{2}}{2}+\frac{1}{2}\min\Big{\\{}c_{1}(u_{2}-u_{1})^{2}+a(2z-v_{1})^{2}+a(v_{1})^{2}$
$\displaystyle\hskip
28.45274pt+b(u_{2}-2z)^{2}+b(u_{1}-v_{1})^{2}:u_{1},u_{2},v_{1}\in\mathbb{R}\Big{\\}}.$

By computing the minimum, we obtain

$\displaystyle g_{2}(z)$ $\displaystyle\geq$
$\displaystyle\frac{\tilde{f}(z^{\ast})}{2}-\frac{c_{2}}{2}+\frac{1}{2}\Big{(}\frac{a(2c_{1}+b)+bc_{1}}{2c_{1}+b}(2z-v_{1})^{2}+a(v_{1})^{2}\Big{)}$

with

$v_{1}=\frac{a(4c_{1}+2b)+2bc_{1}}{a(4c_{1}+2b)+bc_{1}}z.$

Hence for $z$ large enough

$\displaystyle g_{2}(z)$ $\displaystyle\geq$
$\displaystyle\frac{\tilde{f}(z^{\ast})}{2}-\frac{c_{2}}{2}+a\Big{(}1+\frac{bc_{1}(a(4c_{1}+2b)+bc_{1})}{(a(4c_{1}+2b)+bc_{1})^{2}}\Big{)}z^{2}$
$\displaystyle>$ $\displaystyle\tilde{f}(z^{\ast})+az^{2}\ =\ g_{1}(z).$

From this property and Remark 5.18 we deduce that there exists a unique
$z_{1}$ such that $g_{2}(z_{1})=g_{1}(z_{1})$, and hence
$g_{1}(z)=\min_{N}g_{N}(z)$ in $[z_{1},+\infty)$ by Step 1.

Step $3$. By Step 1 $g_{N}(z_{1})>g_{2}(z_{1})=g_{1}(z_{1})$ for all $N\geq
3$. Let $[z_{2},z_{1}]$ be the maximal interval containing $z_{1}$ where
$g_{2}(z)=\min_{N\geq 1}g_{N}(z)=\min_{N\geq 2}g_{N}(z)$. Since in particular
$g_{3}>g_{2}$ in the interval $(z_{2},z_{1}]$ by Remark 5.22, we have
$g_{N}>g_{4}>g_{3}$ for all $N>4$ in the closed interval $[z_{2},z_{1}]$
always by Step 1. This implies that $g_{3}(z_{2})=g_{2}(z_{2})$. Moreover,
note that $g_{4}(z_{2})>g_{2}(z_{2})$, since otherwise we would have
$g_{3}(z_{2})<g_{2}(z_{2})$ by (5.44) with $z=z_{2}$, $N_{1}=2$ and $N_{2}=4$.

Step $4$. We define
$z_{3}=\max\\{z:g_{4}(z)\leq\min\\{g_{3}(z),g_{2}(z),g_{1}(z)\\}$. This is
well defined since $g_{4}(0)<\min\\{g_{3}(0),g_{2}(0),g_{1}(0)\\}$ and we have
$z_{3}<z_{2}$. Note that in $(z_{4},z_{3})$ we have
$\min\\{g_{N}(z):z\in\mathbb{N}\\}\in\\{g_{2}(z),g_{3}(z)\\}$. We then define
iteratively $z_{n}=\max\\{z:g_{n+1}(z)\leq\min\\{g_{k}(z):k\leq n\\}$. Again,
this is a good definition and $z_{n}<z_{n-1}$. In $(z_{n},z_{n-1})$ we have
that $\min\\{g_{N}(z):N\in\mathbb{N}\\}\in\\{g_{n}(z),g_{n-1}(z)\\}$. Indeed,
by Corollary 5.21 if $g_{k}(z)=g_{\ell}(z)$ at some $z$ then $|k-\ell|\leq 1$.
Since $\min\\{g_{N}(z_{n-1}):N\in\mathbb{N}\\}=g_{n}(z_{n-1})$ and we cannot
have $g_{n}(z)=g_{n+1}(z)$ if $z\in(z_{n},z_{n-1})$, the claim follows.

Step $5$. Inequality (5.44) shows that the graph of $g_{N}$ lies below the
graph of the convex envelope of the minimum between $g_{N-1}$ and $g_{N+1}$ in
an open interval. By Proposition 5.14 this proves that $1\over N$ is a locking
state. ∎

Figure 26: pictorial description of Theorem 5.23 for a single choice of
$\sigma$ (shape of $\widehat{Q}_{\sigma}f$ and $\theta$, not to scale)

In order to highlight the dependence on $\sigma$, for any $\sigma>0$ and for
any $N\geq 1$, in the sequel $z_{N}(\sigma)$ will denote the corresponding
value $z_{N}$ given by Theorem 5.23. Moreover, for any $\sigma$ we set
$z_{0}(\sigma)=+\infty$.

Figure 27: relative behaviour of $g^{\sigma}_{N}$ and the final resulting
$\theta$.

######  Remark 5.24 (shape of $Q_{\sigma}f(z)$ and $\theta(z)$).

The graph of the function $Q_{\sigma}f(z)$ possesses infinitely many concave
parabolic arcs, corresponding to the intervals where
$\widehat{Q}_{\sigma}f(z)$ is affine, which accumulates in
$\overline{z}_{\ast}(\sigma)=\inf_{N}z_{N}(\sigma)>0$. Correspondingly, the
phase function $\theta(z)$ is affine, interpolating between consecutive values
$1/N$ (see Fig. 27).

Summarizing, the behaviour of the penalized energy $Q_{\sigma}f(z)$ in terms
of the macroscopic gradient $z$ has the following features:

$\bullet$ (‘unfractured zone’) for $z\leq\overline{z}_{\ast}(\sigma)$ optimal
sequences take into account only the convex part of $f$; i.e., there are no
broken bonds;

$\bullet$ (‘completely microfractured zone’) there exists
$\overline{z}^{\ast}(\sigma)=s_{1}^{-}(\sigma)>z_{1}(\sigma)$ such that for
$z\geq\overline{z}^{\ast}(\sigma)$ (that is, in
$I_{1}(\sigma)=[\overline{z}^{\ast}(\sigma),+\infty)$) the part of the energy
involving the function $f$ is identically $\tilde{f}(z^{\ast})$; i.e., we have
broken bonds for all values of the index $i$;

$\bullet$ (increasingly segmented behavior of the relaxed energy) for values
of the macroscopic gradient between $\overline{z}_{\ast}(\sigma)$ and
$\overline{z}^{\ast}(\sigma)$ the energy $\widehat{Q}_{\sigma}f$ behaves as a
superposition of infinitely many ‘damaged materials’ indexed by the parameter
$N$ representing the microscopic optimal spacing of broken bonds. For the
values $z$ where $\widehat{Q}_{\sigma}f(z)$ is affine, optimal sequences mix
the damaged materials parameterized by $N$ and $N-1$. The point
$\overline{z}_{\ast}(\sigma)$ is an accumulation point for the different
behaviors as $N\to+\infty$.

######  Remark 5.25 (limit behaviours of the damaged zones).

By Proposition 2.38, highlighting the dependence on the parameter $\sigma$, we
deduce that

(i) $\displaystyle\lim_{\sigma\to
0}\overline{z}_{\ast}(\sigma)=\lim_{\sigma\to
0}\overline{z}^{\ast}(\sigma)=z^{\ast}$, corresponding to the extreme non-
additivity case,

(ii) $\displaystyle\lim_{\sigma\to+\infty}\overline{z}_{\ast}(\sigma)=0$ and
$\displaystyle\lim_{\sigma\to+\infty}\overline{z}^{\ast}(\sigma)=+\infty$,
corresponding to full additivity.

######  Remark 5.26 (Generic non differentiability).

Note the generic non differentiability of $\widehat{Q}_{\sigma}f(\theta,z)$
with respect to $\theta$ at the locking states. This is due to the different
definitions of this function in left and right neighbourhoods of each locking
state $\frac{1}{N}$. Indeed, the definition of
$\widehat{Q}_{\sigma}f(\theta,z)$ uses $g^{\sigma}_{N}(z),g^{\sigma}_{N+1}(z)$
in a left neighbourhood and $g^{\sigma}_{N-1}(z),g^{\sigma}_{N}(z)$ in a right
neighbourhood of $\theta=\frac{1}{N}$, respectively, in analogy with the case
of concentrated kernels, as seen in Section 4 (see Remark 4.5).

###  5.4 Properties of optimal microstructures

In the previous section we have shown that $\theta$ of the form $1\over N$
with $N\in\mathbb{N}$ are locking states. We now show that such values
correspond to energy wells, and characterize all
$\widehat{Q}_{\sigma}f(\theta,\cdot)$.

####  5.4.1 Microstructures as interpolations of energy meta-wells

The following proposition reinterprets $g^{\sigma}_{N}$ as the energy of
periodic minimizers for $\theta_{N}=1/N$.

######  Proposition 5.27 ($g^{\sigma}_{N}$ as an energy meta-well).

Let $g_{N}^{\sigma}$ be defined as in (5.30) with $a=a_{\sigma}$ and
$b=b_{\sigma}$ satisfying (5.17). The following equality holds for any
$N\in\mathbb{N}$ and $z\in\mathbb{R}$:

$\Phi^{N}_{\bf m}f\Big{(}\frac{1}{N},z\Big{)}=g^{\sigma}_{N}(z)$

where $\Phi^{N}_{\bf m}f$ is defined in (3.21) with $A=[z^{\ast},+\infty)$,
$f_{-1}=\tilde{f}$, $f_{1}=\tilde{f}(z^{\ast})$ if $z\in A$ and $+\infty$
otherwise, and $m_{n}=e^{-\sigma n}$.

###### Proof.

We first observe that $\Phi^{N}_{\bf m}f(\frac{1}{N},z)=\widehat{R}^{N}_{\bf
m}f(\underline{s}_{N},z)$ where $\underline{s}_{N}=(1,-1,-1,\ldots,-1)$ and
$\widehat{R}^{N}_{\bf m}f$ is defined in (3.20) with

$F^{\\#}(u,\underline{s};[0,N])=\sum_{i=1}^{N}f_{s_{i}}(u_{i}-u_{i-1})+\sum_{i=1}^{N}\sum_{j\in\mathbb{Z}}e^{-\sigma|i-j|}(u_{i}-u_{j})^{2}.$

By extending $\underline{s}_{N}$ by $N$-periodicity, we have

$\displaystyle\widehat{R}^{N}_{\bf m}f(\underline{s}_{N},z)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\min\\{F^{\\#}(u,\underline{s}_{N};[0,N]):u_{i}-zi\
N\hbox{-periodic}\\}$ $\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\frac{1}{kN}\min\\{F^{\\#}(u,\underline{s}_{N};[0,kN]):u_{i}-zi\
N\hbox{-periodic}\\}$ $\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\frac{1}{kN}\min\Big{\\{}\tilde{f}(z^{\ast})k+\sum_{r=1}^{k}\sum_{l=2}^{N}\tilde{f}(u_{N(r-1)+l}-u_{N(r-1)+l-1})$
$\displaystyle+a_{\sigma}\sum_{i=1}^{kN}(v_{i}-v_{i-1})^{2}+b_{\sigma}\sum_{i=1}^{kN}(u_{i}-v_{i})^{2}:u_{i}-zi,v_{i}-zi\
N\hbox{-periodic}\Big{\\}},$

the last equality being a consequence of (5.4), the equivalence result of
Lemma 5.9 and the characterization of the minima given by (5.6), which ensures
that also the minimizing $v$ can be chosen periodic. Hence by the periodicity
we get

$\displaystyle\Phi^{N}_{\bf
m}f\Big{(}\frac{1}{N},z\Big{)}=\frac{1}{N}\min\Big{\\{}\tilde{f}(z^{\ast})+\sum_{i=2}^{N}\tilde{f}(u_{i}-u_{i-1})+a_{\sigma}\sum_{i=1}^{N}(v_{i}-v_{i-1})^{2}+b_{\sigma}\sum_{i=1}^{N}(u_{i}-v_{i})^{2}:$
$\displaystyle\hskip 199.16928ptu_{i}-zi,v_{i}-zi\
N\hbox{-periodic}\Big{\\}}.$

Finally, noting that we can remove the periodicity condition on $u$ and that
we can rewrite the condition on $v$ as a boundary condition, we get the claim.
∎

Let
$I_{N}=I_{N}(\sigma)=\\{z\in\mathbb{R}:\widehat{Q}_{\sigma}f(z)=g^{\sigma}_{N}(z)\\}$.
Note that Remark 5.11 implies that $\widehat{Q}_{\sigma}f(\theta,\cdot)$ can
be described in terms of the convex combination of the functions
$g^{\sigma}_{N}(z)$. In particular, by the convexity of $g^{\sigma}_{N}(z)$
with respect to $N$, we have

$\widehat{Q}_{\sigma}f\big{(}\frac{1}{N},z\big{)}=g^{\sigma}_{N}(z)$ (5.46)

in the whole $\mathbb{R}$.

We are now in the position to characterize
$\widehat{Q}_{\sigma}f(\theta,\cdot)$ as an interpolation between consecutive
energy meta-wells (corresponding to the locking states), as in Lemma 4.4 for
the concentrated kernels.

######  Proposition 5.28 (interpolation between energy wells).

Given $\sigma>0$, suppose that $a=a_{\sigma}$ and $b=b_{\sigma}$ are as in
(5.17). Then, for any $\theta\in\mathbb{Q}\cap(0,1)$ and for any
$z\in\mathbb{R}$ the following equality holds:

$\widehat{Q}_{\sigma}f(\theta,z)=\min\Big{\\{}t(\theta)g^{\sigma}_{N_{\theta}}(z^{\prime})+(1-t(\theta))g^{\sigma}_{N_{\theta}+1}(z^{\prime\prime}):t(\theta)z^{\prime}+(1-t(\theta))z^{\prime\prime}=z\Big{\\}},$
(5.47)

where

$N_{\theta}=\Big{\lfloor}\frac{1}{\theta}\Big{\rfloor}\quad\hbox{\rm and}\quad
t(\theta)=N_{\theta}\Big{(}\theta(N_{\theta}+1)-1\Big{)}.$ (5.48)

###### Proof.

We divide the proof in two steps.

Step $1$: $\theta=\frac{1}{N}$. In this case, the claim becomes (5.46) for all
$z\in\mathbb{R}$. We note that for each $N$ the formula is proved for $z\in
I_{N}$. Moreover, for arbitrary $z$ it can be further simplified as follows.
Let $k\in\mathbb{N}$ be fixed and let $(u,v)$ be a minimizer in (5.27) with
$p=1$ and $q=N$. Since $\widehat{Q}_{\sigma}f(\frac{1}{N},z)$ can be expressed
as in (5.27), it is sufficient to show that for all $k$

$\frac{1}{kN}E^{\sigma}_{1}(u,v;[0,kN])\geq g^{\sigma}_{N}(z).$

It is not restrictive to suppose that $u_{1}-u_{0}\geq z^{\ast}$. By grouping
the interactions, we estimate

$E^{\sigma}_{1}(u,v;[0,kN])\geq\sum_{j=1}^{k}N_{j}g^{\sigma}_{N_{j}}(z_{j})$

where $\sum_{j=1}^{k}N_{j}=kN$ and $\sum_{j=1}^{k}N_{j}z_{j}=kNz$. By
Proposition 5.20, we infer that all even $N_{j}$ are equal to some $N_{\rm
e}$, and the corresponding $z_{j}$ coincide with some $z_{\rm e}$, and the
same holds for odd $N_{j}$ with $N_{\rm o}$ and corresponding $z_{j}$ with
$z_{\rm o}$, so that there exist integers $k_{\rm e}$ and $k_{\rm o}$ such
that

$E^{\sigma}_{1}(u,v;[0,kN])\geq k_{\rm e}N_{\rm e}g^{\sigma}_{N_{\rm
e}}(z_{\rm e})+k_{\rm o}N_{\rm o}g^{\sigma}_{N_{\rm o}}(z_{\rm o})$

where

$k_{\rm e}N_{\rm e}+k_{\rm o}N_{\rm o}=kN\quad\hbox{\rm and }\quad k_{\rm
e}N_{\rm e}z_{\rm e}+k_{\rm o}N_{\rm o}z_{\rm o}=kNz.$

Since $u\in\mathcal{V}(kN,\frac{1}{N})$, we also have $k_{\rm e}+k_{\rm o}=k$.
By (5.45) we deduce that $|N_{\rm e}-N_{\rm o}|=1$, and this is only possible
if either $k_{\rm e}$ or $k_{\rm o}$ vanishes, from which we conclude.

Step $2$: general case. We fix $\theta=\frac{p}{q}$ with $p$ and $q$ coprime
integers satisfying $1<p<q$. Let $k\in\mathbb{N}$ be fixed and let $(u,v)$ be
a minimizer in (5.27). By grouping the interactions as in the case
$\theta=\frac{1}{N}$, thanks to (5.45) we obtain that there exists
$N\in\mathbb{N}$ such that

$k_{1}+k_{2}=kp,\quad k_{1}N+k_{2}(N+1)=kq$ (5.49)

for some $k_{1},k_{2}\in\mathbb{N}$, and

$E^{\sigma}_{1}(u,v;[0,kq])\geq
k_{1}Ng^{\sigma}_{N}(z^{\prime})+k_{2}(N+1)g^{\sigma}_{N+1}(z^{\prime\prime})$

where $z^{\prime},z^{\prime\prime}$ satisfy
$k_{1}Nz^{\prime}+k_{2}(N+1)z^{\prime\prime}=kqz$. Since (5.49) implies
$\frac{q}{p}\geq N>\frac{q}{p}-1$, we deduce that $N=N_{\theta}$ is the unique
integer solution of the equation (with $k_{1}=k(p(N_{\theta}+1)-q)>0$ and
$k_{2}=k(q-pN_{\theta})>0$). Hence

$E^{\sigma}_{1}(u,v;[0,kq])\geq
k_{1}N_{\theta}g^{\sigma}_{N_{\theta}}(z^{\prime})+k_{2}(N_{\theta}+1)g^{\sigma}_{N_{\theta}+1}(z^{\prime\prime}).$
(5.50)

Noting that

$\frac{k_{1}N_{\theta}}{kq}=t(\theta)\quad\hbox{\rm and
}\quad\frac{k_{2}(N_{\theta}+1)}{kq}=1-t(\theta),$

since $\widehat{Q}f_{\sigma}(\theta,z)$ can be expressed as in (5.27) we
obtain, by using (5.50),

$\displaystyle\widehat{Q}_{\sigma}f(\theta,z)\geq\min\Big{\\{}t(\theta)g^{\sigma}_{N_{\theta}}(z^{\prime})+(1-t(\theta))g^{\sigma}_{N_{\theta}+1}(z^{\prime\prime}):t(\theta)z^{\prime}+(1-t(\theta))z^{\prime}=z\Big{\\}}.$

The opposite inequality follows by the equality
$g^{\sigma}_{N}(z)=\widehat{Q}_{\sigma}f(\frac{1}{N},z)$ proved in the case
$\theta=\frac{1}{N}$ and by the convexity of
$\widehat{Q}_{\sigma}f(\theta,z)$. Indeed, noting that

$\frac{t(\theta)}{N_{\theta}}+\frac{1-t(\theta)}{N_{\theta}+1}=\theta,$

for all pairs $(z^{\prime},z^{\prime\prime})$ such that
$t(\theta)z^{\prime}+(1-t(\theta))z^{\prime\prime}=z$, we have

$\displaystyle
t(\theta)g^{\sigma}_{N_{\theta}}(z^{\prime})+(1-t(\theta))g^{\sigma}_{N_{\theta}+1}(z^{\prime\prime})$
$\displaystyle=$ $\displaystyle
t(\theta)\widehat{Q}_{\sigma}f\Big{(}\frac{1}{N_{\theta}},z^{\prime}\Big{)}+(1-t(\theta))\widehat{Q}_{\sigma}f\Big{(}\frac{1}{N_{\theta}+1},z^{\prime\prime}\Big{)}$
$\displaystyle\geq$
$\displaystyle\widehat{Q}_{\sigma}f\Big{(}\frac{t(\theta)}{N_{\theta}}+\frac{1-t(\theta)}{N_{\theta}+1},t(\theta)z^{\prime}+(1-t(\theta))z^{\prime\prime}\Big{)}$
$\displaystyle\geq$ $\displaystyle\widehat{Q}_{\sigma}f(\theta,z)$

as desired. ∎

####  5.4.2 A canonical optimal microstructure uniform at all scales

The description of $\widehat{Q}_{\sigma}f$ that we have obtained in terms of
$g^{\sigma}_{N}$ highlights a number of equivalent minimizers. However, in
this class we can define a set of canonical ground states. These states are
characterized by the corresponding distribution of spins, or, equivalently,
the distribution of broken bonds. Similar sets have independently appeared in
the study of related dynamical systems [7, 70].

In order to describe this optimal distribution of broken bonds, for a given
$\theta\in[0,1]$ we define the set of integers

$A(\theta)=\\{k\in\mathbb{Z}:\lfloor
k\theta\rfloor\neq\lfloor(k+1)\theta\rfloor\\}.$

A characteristic property of the set $A(\theta)$ is its ‘uniformity at all
scales’; that is, the property that for each $M\in\mathbb{N}$ each interval of
length $M$ contains either $\lfloor M\theta\rfloor$ or $\lfloor
M\theta\rfloor+1$ elements of $A(\theta)$. The set $A(\theta)$ can be
described as the most uniformly distributed among sets with such property (up
to translations). Note, for instance, that if
$\frac{1}{N+1}<\theta<\frac{1}{N}$ then the difference between two consecutive
elements of $A(\theta)$ is either $N$ or $N+1$. The set $A(\theta)$ is
periodic if and only if $\theta$ is rational; otherwise it follows a pattern
reminiscent of quasiperiodic functions (see e.g. [14, 65]).

The following proposition states that in the computation of
$\widehat{Q}_{\sigma}f(z)$ we can consider the corresponding minimum problems
only on functions $u$ whose broken sites coincide with $A(\theta(z))$.

######  Proposition 5.29 (optimality of $A(\theta)$).

Let $f$ be as in (5.28). Then, for any $\sigma>0$ and $z\in\mathbb{R}$, the
following equality holds:

$\widehat{Q}_{\sigma}f(z)=\liminf_{\begin{subarray}{c}k\to+\infty\\\ k\in
A(\theta(z))\end{subarray}}\frac{1}{k}\min\\{E^{\sigma}_{1}(u,v;[0,k]):v_{0}=0,v_{k}=zk,u_{i}-u_{i-1}\geq
z^{\ast}\Leftrightarrow i\in A(\theta(z))\\}.$

###### Proof.

For each $N$, we can suppose that the set where
$\widehat{Q}_{\sigma}f(z)=g^{\sigma}_{N}(z)$ is an interval
$I_{N}=[s_{N}^{-},s_{N}^{+}]$. Let $z\in(s_{N+1}^{+},s_{N}^{-})$. Then,
writing

$z=ts_{N}^{-}+(1-t)s_{N+1}^{+},$ (5.51)

we have that

$\widehat{Q}_{\sigma}f(z)=r^{\sigma}_{N+1}(z)=tg^{\sigma}_{N}(s_{N}^{-})+(1-t)g^{\sigma}_{N+1}(s_{N+1}^{+}).$
(5.52)

Recalling the definition of the phase function $\theta(z)$ (see Definition
3.6) and the fact that $\theta(z)$ is affine in each open interval where
$\widehat{Q}_{\sigma}f$ is affine, as stated in Proposition 3.10, we deduce

$\widehat{Q}_{\sigma}f(z)=\widehat{Q}_{\sigma}f(\theta(z),z)\ \ \hbox{and }\ \
\theta(z)=t\frac{1}{N}+(1-t)\frac{1}{N+1},$

where the link between $z,t$ and $N$ is given by (5.51). Hence, using the
local representation given by (5.27), for all $k\in A(\theta(z))$ we can split
the minimum

$\min\\{E^{\sigma}_{1}(u,v;[0,k]):v_{0}=0,v_{k}=zk,u_{i}-u_{i-1}\geq
z^{\ast}\Leftrightarrow i\in A(\theta(z))\\}$

into the sum of the minima

$M_{j}=\min\\{E_{1}(u,v;[i_{j-1},i_{j}]):v_{i_{j-1}}=0,v_{i_{j}}=(i_{j}-i_{j-1})z_{j},u_{i}-u_{i-1}\geq
z^{\ast}\Leftrightarrow i=i_{j}\\},$

where $A(\theta(z))\cap[0,k]=\\{i_{0},i_{1},\dots,i_{n_{k}}\\}$ with
$0=i_{0}<\dots<i_{n_{k}}=k$, and $z_{j}$ are such that
$\sum_{j=1}^{n_{k}}(i_{j}-i_{j-1})z_{j}=kz$.

Furthermore, noting that
$M_{j}=(i_{j}-i_{j-1})g^{\sigma}_{i_{j}-i_{j-1}}(z_{j})$, we obtain by
convexity

$\displaystyle\frac{1}{k}\min\\{E_{1}(u,v;[0,k]):v_{0}=0,v_{k}=zk,u_{i}-u_{i-1}\geq
z^{\ast}\Leftrightarrow i\in A(\theta(z))\\}$ $\displaystyle\hskip
28.45274pt\geq\frac{1}{k}\sum_{j=1}^{n_{k}}(i_{j}-i_{j-1})g^{\sigma}_{i_{j}-i_{j-1}}(z_{j})\geq\frac{1}{k}\sum_{j=1}^{n_{k}}(i_{j}-i_{j-1})\widehat{Q}_{\sigma}f(z_{j})\geq\widehat{Q}_{\sigma}f(z).$
(5.53)

Conversely, fixed $k\in A(\theta(z))$, let $\mathcal{I}_{N}=\\{j\leq
n_{k}:i_{j}-i_{j-1}=N\\}$ and $\mathcal{I}_{N+1}=\\{j\leq
n_{k}:i_{j}-i_{j-1}=N+1\\}$ and $z_{k}^{\pm}$ be such that

$N\\#\mathcal{I}_{N}z_{k}^{-}+(N+1)\\#\mathcal{I}_{N+1}z_{k}^{+}=kz$

and $z_{k}^{-}\to s_{N}^{-},z_{k}^{+}\to s_{N+1}^{+}$ as $k\to+\infty$. Then,
using the minimizers of $g^{\sigma}_{N}(z_{k}^{-})$ and of
$g^{\sigma}_{N+1}(z_{k}^{+})$ to test the minimum problem in (5.4.2), we get
the upper bound

$N\\#\mathcal{I}_{N}g^{\sigma}_{N}(z_{k}^{-})+(N+1)\\#\mathcal{I}_{N+1}g^{\sigma}_{N+1}(z_{k}^{+}).$

Taking the limit as $k\to+\infty$, by (5.52) we obtain the claim. ∎

######  Remark 5.30 (optimality of $A(\theta)$ for the constrained
relaxation).

The same proof shows that for any $\theta$

$\displaystyle\widehat{Q}_{\sigma}f(\theta,z)=\liminf_{\begin{subarray}{c}k\to+\infty\\\
k\in
A(\theta)\end{subarray}}\frac{1}{k}\min\\{E^{\sigma}_{1}(u,v;[0,k]):v_{0}=0,v_{k}=zk,u_{i}-u_{i-1}\geq
z^{\ast}\Leftrightarrow i\in A(\theta)\\}.$

Figure 28: representation of two periodic minimizers

For the sake of illustration, in Fig. 28 we represent two periodic minimizers
(the black dots representing broken bonds) for $\theta=2/5$. In the first case
we have a 15-periodic minimizers, the second array is the ‘canonical’ one,
alternating broken bonds at distance two and three.

######  Remark 5.31 (the $M$-th neighbour case).

In the case of $M$-th only interactions, we focus first on
$\theta=\theta_{k}={k\over M}$, with $k\in\\{0,\ldots,M\\}$ , that is, on
locking states, or, equivalently, on energy wells. The construction in
Proposition 4.4 shows that all periodic spin configurations with period a
submultiple of $M$ compatible with $\theta_{k}$, correspond to optimal
laminates. Indeed, the only requirement on minimizers is that for all
intervals of length $M$ we have an equal number of spins of either type (which
is trivially true). Note in particular that we may choose minimizers with
$u_{i}-u_{i-1}>z^{*}$ exactly for $i\in A(\theta)$ since this set is
$M$-periodic. Now if $\theta$ is not of the form $k/M$, we do not have
periodic optimal minimizers. This is in contrast to the exponential case,
where we do have periodic minimizers for all $\theta\in\mathbb{Q}$.

Figure 29: representation of two periodic minimizers

In Fig. 29 we represent two 5-periodic minimizers (the black dots representing
the elongations larger than $z^{*}$) for $M=5$ and $\theta=2/5$. The second
array is the ‘canonical’ one, alternating broken bonds at distance two and
three.

We note that in some of our examples illustrating periodic minimizers with
‘global’ properties, the canonical periodic microstructures, epitomizing a
generalized Cauchy-Born (GCB) states, are unique. This is true, for instance,
in the case of the exponential kernel ${\bf m}$. Instead, for concentrated
kernels we may have more than one minimal (GCB-type) microstructure. Note also
that in the case of exponential kernels, outside the special regimes where the
minimizers are periodic, we can mix GCB states and, since different GCB states
do not interact, the mixing process is bringing arbitrariness. In particular,
GCB states could be mixed canonically, even though in the examples of interest
in this paper this does not bring any advantages. However, this is not the
general case and when different GCB states interact, their mixtures can become
suboptimal, as in the case of concentrated kernels. We argue that in such
‘strongly non-additive’ cases the non-periodic GCB states with the properties
of our canonical microstructures can become the preferred ones if interaction
happens at all scales (which is not the case for concentrated kernels).

###  5.5 Explicit constructions

In this section we explicitly compute $Q_{\sigma}f$ in a meaningful case,
using the general results of the previous section. This also allows us to
treat some classes of energies more general than truncated potentials.

####  5.5.1 The Novak-Truskinovsky model

Let $f$ be the truncated quadratic potential defined as in (5.28) with
$\tilde{f}(z)=z^{2}$; that is,

$f(z)=\begin{cases}z^{2}&\hbox{\rm if }\ z\leq\sqrt{\eta}\\\ \eta&\hbox{\rm if
}\ z\geq\sqrt{\eta}\\\ \end{cases}$ (5.54)

with $\eta>0$ fixed. By using the computations in [74] and the results of this
section, we obtain an explicit formula for $g^{\sigma}_{N}(z)$, and hence
$Q_{\sigma}f(z)$.

######  Remark 5.32 (explicit computation of minima).

Let $\tilde{E}_{1}$ be defined as in (5.29) with $\tilde{f}(z)=z^{2}$ and
$a,b>0$. Then, by the computations in [74, Sec. 3] we get

$\min\\{\tilde{E}_{1}(u,v;[0,N]):v_{0}=0,v_{N}=N\\}=\frac{N^{2}a(a+1)}{Na+\tanh((N+1)\zeta)\coth(\zeta)-1}$

where

$\zeta=2\sinh^{-1}\\!\Big{(}\frac{1}{2}\sqrt{\frac{b(a+1)}{a}}\Big{)}.$ (5.55)

By using (5.30), we obtain

$g_{N}(z)=c_{N}z^{2}+\frac{\eta}{N},$ (5.56)

where

$c_{N}=\frac{Na(a+1)}{Na+\tanh(N\zeta)\coth(\zeta)}$ (5.57)

and $\zeta$ as in (5.55).

Since we are interested in the analysis of $Q_{\sigma}f$, if $a=a_{\sigma}$
and $b=b_{\sigma}$ satisfy (5.17) we write $g^{\sigma}_{N},c^{\sigma}_{N}$ and
$\zeta_{\sigma}$ in place of $g_{N},c_{N}$ and $\zeta$, respectively. The
interval where $\widehat{Q}_{\sigma}f(z)=g^{\sigma}_{N}(z)$ is given by
$I_{N}(\sigma)=[s_{N}^{-},s_{N}^{+}]$, where

$\left.\begin{array}[]{ll}\displaystyle
s_{N}^{+}=s_{N}^{+}(\sigma)=\displaystyle\sqrt{\frac{\eta}{N(N-1)(c^{\sigma}_{N}-c^{\sigma}_{N-1})}}\sqrt{\frac{c^{\sigma}_{N-1}}{c^{\sigma}_{N}}}\
\ \hbox{\rm if }\ N\geq 2;\ s_{1}^{+}=s_{1}^{+}(\sigma)=+\infty\\\
\displaystyle s_{N}^{-}=\displaystyle
s_{N}^{-}(\sigma)=\displaystyle\sqrt{\frac{\eta}{(N+1)N(c^{\sigma}_{N+1}-c^{\sigma}_{N})}}\sqrt{\frac{c^{\sigma}_{N+1}}{c^{\sigma}_{N}}}\
\ \hbox{\rm if }\ N\geq 1.\end{array}\right.$ (5.58)

Hence,

$\overline{z}_{\ast}(\sigma)=\lim_{N\to+\infty}s_{N}^{\pm}=\sqrt{\frac{a_{\sigma}\eta}{(a_{\sigma}+1)\coth(\zeta_{\sigma})}}\
\ \hbox{\rm and }\
\overline{z}^{\ast}(\sigma)=s_{1}^{-}=\sqrt{\frac{\eta(2a_{\sigma}+b_{\sigma}(a_{\sigma}+1))}{a_{\sigma}b_{\sigma}}}.$
(5.59)

Note that $\overline{z}_{\ast}(\sigma)>\sqrt{a_{\sigma}\eta}$. Concluding, we
have

$Q_{\sigma}f(z)=\begin{cases}z^{2}&\hbox{if
}z\leq\overline{z}_{\ast}(\sigma)\\\
g^{\sigma}_{N}(z)-a_{\sigma}z^{2}&\hbox{if }s_{N}^{-}\leq z\leq s_{N}^{+}\
\hbox{ for some $N\geq 2$}\\\ r^{\sigma}_{N+1}(z)-a_{\sigma}z^{2}&\hbox{if
}s_{N+1}^{+}\leq z\leq s_{N}^{-}\ \hbox{ for some $N$}\\\ \eta&\hbox{if
}z\geq\overline{z}^{\ast}(\sigma),\end{cases}$ (5.60)

where $r^{\sigma}_{N+1}(z)$ is the common tangent to $g^{\sigma}_{N+1}(z)$ and
$g^{\sigma}_{N}(z)$.

Figure 30: representation of $\theta$ in the $z$-${1\over\sigma}$ plane and a
cross section at fixed $\sigma$.

The phase function $\theta$ corresponding to this example is pictured in Fig.
30, where the grey zones between pair of curves denote the pairs in the
$z$-${1\over\sigma}$ plane in which $\theta$ is affine for fixed $\sigma$
between consecutive value of the form ${1\over N}$.

####  5.5.2 Interpolation between varying degrees of non convexity

In this setting it is also of interest to consider a broader class of non
convex convex-affine functions $f$ which includes the convex-constant
functions as particular cases. More specifically, consider the functions
$\ell_{f}^{\tau}$ defined by

$\ell_{f}^{\tau}(z)=\begin{cases}f(z)&\hbox{\rm if }\ z\leq z^{*}\\\
f(z^{*})+\tau f^{\prime}(z^{*})(z-z^{*})&\hbox{\rm if }\ z>z^{*}\end{cases}$
(5.61)

with $0<\tau<1$. In this way we construct an interpolation between the
constrained relaxation of the truncated-convex potential and of the convex
potential which is obtained if beyond $z^{*}$ we smoothly extend $f$ in an
affine way. Accordingly, in (5.61) we have the truncated-convex potential as
above at $\tau=0$, while at $\tau=1$ the function $\ell_{f}^{1}$ is convex.

We can write $\ell_{f}^{\tau}(z)=\Phi^{\tau}(z)+\Gamma^{\tau}(z)$, where

$\Gamma^{\tau}(z)=f(z^{*})+\tau f^{\prime}(z^{*})(z-z^{*})$

and

$\Phi^{\tau}(z)=\begin{cases}f(z)-\tau f^{\prime}(z^{*})(z-z^{*})&\hbox{\rm if
}\ z\leq z^{*}\\\ f(z^{*})&\hbox{\rm if }\ z>z^{*}.\end{cases}$

The function $\Phi^{\tau}$ is a truncated convex potential to which we can
apply the results above, while, by Remark 2.25(iii) we have

$Q_{\sigma}\ell_{f}^{\tau}=Q_{\sigma}(\Phi^{\tau}+\Gamma^{\tau})=Q_{\sigma}(\Phi^{\tau})+\Gamma^{\tau}.$

We can carry on this computation for the quadratic-affine functions
$\ell^{\tau}$ defined in (3.18); that is, $\ell_{f}^{\tau}$ with $f(z)=z^{2}$
and $z^{\ast}=\sqrt{\eta}$. Note that we can equivalently rewrite
$\ell^{\tau}(z)=\widetilde{\Phi}^{\tau}(z)+\widetilde{\Gamma}^{\tau}(z)$,
where $\widetilde{\Gamma}^{\tau}(z)=2\tau z-\tau^{2}$ and

$\widetilde{\Phi}^{\tau}(z)=\begin{cases}(z-\tau)^{2}&\hbox{\rm if }\ z\leq
1\\\ (1-\tau)^{2}&\hbox{\rm if }\ z>1,\end{cases}$

which can be seen as a translation by $\tau$ of the function $\Psi^{\tau}$
given by

$\Psi^{\tau}(z)=\begin{cases}z^{2}&\hbox{\rm if }\ z\leq 1-\tau\\\
(1-\tau)^{2}&\hbox{\rm if }\ z>1-\tau.\end{cases}$

The latter is exactly of the form considered in Example 5.5.1 with
$\eta=\eta^{\tau}=(1-\tau)^{2}$. Its constrained relaxation is then described
in (5.60), and we eventually have

$Q_{\sigma}\ell^{\tau}(z)=(Q_{\sigma}\Psi^{\tau})(z-\tau)+2\tau z-\tau^{2}.$

Note that by (5.59) the endpoints of the interval where the corresponding
$\theta(z)$ is not $0$ or $1$ are

$\overline{z}_{\ast,\tau}(\sigma)=\tau+(1-\tau)\sqrt{\frac{a_{\sigma}}{(a_{\sigma}+1)\coth(\zeta_{\sigma})}}\
\ \hbox{\rm and }\
\overline{z}^{\ast,\tau}(\sigma)=\tau+(1-\tau)\sqrt{\frac{2a_{\sigma}+b_{\sigma}(a_{\sigma}+1)}{a_{\sigma}b_{\sigma}}},$

with $a_{\sigma},b_{\sigma},\zeta_{\sigma}$ as in Example 5.5.1. Note that
$\overline{z}_{\ast,\tau}(\sigma)<1<\overline{z}^{\ast,\tau}(\sigma)$, and
$\lim\limits_{\tau\to
1^{-}}\overline{z}_{\ast,\tau}(\sigma)=\lim\limits_{\tau\to
1^{-}}\overline{z}^{\ast,\tau}(\sigma)=1$.

##  6 Asymptotically equivalent continuum models

The goal of the relaxation of the discrete problems discussed in this paper
was to obtain a homogenized continuum model. We have seen that generically the
presence of nonlocal interactions prevents even the simplest non-convex 1D
problem from being fully characterized by a bulk continuum energy. It follows
from our analysis that the exceptions, when the ‘local’ description also has
‘global’ features and the generalized Cauchy-Born rule is applicable, are
extremely rare. Then the question arises regarding the very nature of the
continuum model which could be considered as asymptotically equivalent to a
discrete model carrying both non-convexity and incompatibility induced by
nonlocal interactions. In this section we present an explicit example showing
that the answer to this question may be nontrivial. While our analysis here
will not be exhaustive, it points towards a new class of hybrid discrete-
continuum variational problems which may be of a considerable interest per se.

In the interest of analytical transparency we focus on the specific
homogenization problem for energies $E_{\varepsilon}$ with the truncated
quadratic potential $f$ given by (5.54); that is, the NT model analyzed in
Example 5.5.1. Our goal will be to find a continuum analog of this problem
allowing one to approximate both the minimal energy and the optimal
microstructure. More specifically we search for the continuum problem which
will be asymptotically $\Gamma$-equivalent to $E_{\varepsilon}$ in the sense
of [29]. In other words, the challenge is to construct a quasi-continuum
problem still carrying some elements of the ‘lost’ discreteness of the
original problem.

To show that the task of constructing such a problem is nontrivial we first
present a naive approach to ‘continualization’ in this setting which has been
proposed phenomenologically and studied extensively in applications [10]. We
show the shortcomings of such an approach and then correct it to match the
exact solution of the discrete problem presented in Section 5.5.

###  6.1 Naive construction

We recall that the original problem is defined on a bounded interval $I$ and
involves two functions $u,v\in\mathcal{A}_{\varepsilon}(I)$. We can write the
corresponding energy function in the form of a sum

$\displaystyle
E_{\varepsilon}(u,v;I)=E_{\varepsilon}^{*}(u;I)+E_{\varepsilon}^{**}(u,v;I)$
(6.1)

where

$\displaystyle
E_{\varepsilon}^{*}(u;I)=\varepsilon\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}f\big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\big{)}$
(6.2)

with $\mathcal{I}_{\varepsilon}^{\ast}=\\{i\in\mathbb{Z}:\varepsilon
i,\varepsilon(i-1)\in I\\}$ and

$\displaystyle
E_{\varepsilon}^{**}(u,v;I)=\frac{a}{\varepsilon}\\!\sum_{i\in\mathcal{I}^{\ast}_{\varepsilon}(I)}\\!(v_{i}-v_{i-1})^{2}+\frac{b}{\varepsilon}\\!\sum_{i\in\mathcal{I}_{\varepsilon}(I)}\\!(u_{i}-v_{i})^{2},$
(6.3)

Assuming now that $I$ is a bounded interval and $\varepsilon>0$, we can
construct for each of the entries in the sum (6.1), viewed _independently_ ,
the asymptotically $\Gamma$-equivalent functionals, defined, respectively, for
$u\in SBV(I)$ and $v\in H^{1}(I)$. This equivalence can be interpreted as a
uniform (with respect to boundary data) approximation up to order
$\varepsilon$ of problems with fixed boundary data for $E_{\varepsilon}^{*}$
and $E_{\varepsilon}^{**}$ by the corresponding problems for some functionals
$G_{\varepsilon}^{*}$ and $G_{\varepsilon}^{**}$, respectively.

A natural choice for such _independently_ equivalent functionals (see [29] for
details) is

$G^{*}_{\varepsilon}(u;I)=\int_{I}\gamma(u^{\prime})^{2}\,dt+\eta\varepsilon\\#S(u),$
(6.4)

and

$G^{**}_{\varepsilon}(u,v;I)=\int_{I}\Big{(}\alpha(v^{\prime})^{2}+\beta\big{(}\frac{u-v}{\varepsilon}\big{)}^{2}\Big{)}\,dt$
(6.5)

for suitable $\alpha,\beta,\gamma,\eta>0$. We recall that here $u$ is a
piecewise-Sobolev function with jump set denoted by $S(u)$. Given (6.4) and
(6.5) it seems natural to assume that the functional

$G_{\varepsilon}(u,v;I)=\int_{I}\Big{(}\gamma(u^{\prime})^{2}+\alpha(v^{\prime})^{2}+\beta\big{(}\frac{u-v}{\varepsilon}\big{)}^{2}\Big{)}\,dt+\eta\varepsilon\\#S(u)$
(6.6)

represents the desired (quasi) continuum analog of the original problem.

We recall the convergence result proved in [21].

######  Remark 6.1 (asymptotic behaviour of the energies $G_{\varepsilon}$).

The $\Gamma$-limit of $G_{\varepsilon}$ with respect to the convergence
$u_{\varepsilon},v_{\varepsilon}\to v$ in $L^{2}(I)$ is given by

$G_{\rm hom}(v)=\int_{I}g_{\rm hom}(v^{\prime})\,dt.$

The integrand $g_{\rm hom}$ is characterized as

$g_{\rm hom}(z)=\inf_{S>0}\Big{\\{}\lambda_{S}z^{2}+\frac{\eta}{S}\Big{\\}}$
(6.7)

where

$\lambda_{S}=\frac{(\alpha+\gamma)\frac{\omega S}{2}}{\frac{\omega
S}{2}+\frac{\gamma}{\alpha}\tanh(\frac{\omega S}{2})}\quad\hbox{ and
}\quad\omega^{2}=\frac{(\alpha+\gamma)\beta}{\alpha\gamma}.$ (6.8)

The function $g_{\rm hom}(z)$ is strictly convex, and the following properties
hold:

(i) $\displaystyle g_{\rm hom}(z)=(\alpha+\gamma)z^{2}$ in $[0,z_{c}]$, where
$z_{c}=\sqrt{\frac{2\eta\omega\alpha}{4\gamma(\alpha+\gamma)}}$;

(ii) $\displaystyle g_{\rm hom}(z)\sim\alpha z^{2}+Cz^{2/3}$ as $z\to+\infty$,
where $C>0$ depends only on $\alpha,\beta,\gamma,\eta$.

###  6.2 Lattice induced interdependence of $E_{\varepsilon}^{*}(u;I)$ and
$E_{\varepsilon}^{**}(u,v;I)$

Now we show that using the above approach, we obtain the discontinuous
function $u$ which provides only formal approximations for the ‘jump sets’ of
the original discrete problems.

Figure 31: comparison between the graph of the function $g_{\rm hom}$ (below)
and that of $Q_{\bf m}f$ after subtraction of the quadratic part

######  Remark 6.2 (non-equivalent scaling behavior).

Note first that the critical value $\overline{z}_{\ast}$ in the NT discrete
model, defined in (5.59), is different from the corresponding critical value
in the continuum problem discussed above. Indeed, if we choose $\gamma=1$ as
in the discrete case, in order for the discrete and continuous energies to be
equivalent up to $\overline{z}_{\ast}$ we need to ‘correct’ the continuum
fracture energy by substituting $\eta$ with an effective fracture toughness
$\frac{\eta}{\cosh\zeta}$ with $\zeta$ given by (5.55). However, such a
correction will not extend the equality of the energy functions beyond the
threshold. In particular, note the different scaling behavior of the two
models as $z$ diverges, see Fig. 31.

It is clear that the proposed _lattice-independent_ approximation of
$E_{\varepsilon}^{*}(u;I)$ and $E_{\varepsilon}^{**}(u,v;I)$ fails because in
general separate uniform approximations of minima for two functionals does not
provide a uniform approximation for the minimum of the sum. More specifically,
in our case functionals $G^{*}_{\varepsilon}$ favor the onset of (at most) one
jump point of $u$, while functionals $G^{**}_{\varepsilon}$, not involving
jump sets, allow for an unbounded number of jumps. While in the
correspondingly tailored regimes we can have good _separate_ approximations,
the sum of the two energies in $E_{\varepsilon}$ optimizes the number and
location of jumps accounting for the lattice induced interaction between
$E_{\varepsilon}^{*}(u;I)$ and $E_{\varepsilon}^{**}(u,v;I)$ and therefore in
a different way than $G_{\varepsilon}$ which does not account for such lattice
induced interaction.

Note that while in the discrete case we have interaction constrained by the
lattice discreteness, in the naive continuum problem such interaction is
lattice-unconstrained, which allows in principle for a richer class of
microstructures. That is why we can obtain in this way at most a lower bound.

###  6.3 A lattice-compatible construction

As we have seen above, the limit of the energies defined in (6.4) when
$\varepsilon\to 0$ has different properties from those of its discrete
counterpart and the failure of this approach is related to the discrete-to-
continuum transition-induced loss of the constraint on the location of the
jumps.

To construct the asymptotically equivalent [29] continuum theory the approach
should be more subtle because the corresponding relaxation procedure should
involve a delicate interplay between continuum limit and discrete energy
minimization, which are tightly coupled.

Indeed, as we have seen above decoupling discrete-to-continuum transition from
the relaxation of a non-convex energy gives rise to a quantitatively and
qualitatively incorrect asymptotic behavior. Apparently the discrete-to-
continuum limit and the incompatibility-constrained non-convex minimization do
not commute and by performing the former independently of the latter we at
best underestimate the relaxed energy. In other words, by neglecting the
discrete constraint we may be able to construct lower bounds (using the naive
approximation). We do not systematically analyze this issue here.

To get an insight on how to fix the problem, it is instructive to compare
(6.7) with formulas (5.56) and (5.57). Note, in particular, that in the latter
the parameter $N$ is discrete while in the former the parameter $S$ is
continuous. This highlights that the discreteness, fundamental in the
construction of the ${\bf m}$-relaxation in the original problem, is
underestimated in the computation of $g_{\rm hom}$. In other words, the
internal physical scale and the lattice scale tend to zero simultaneously but
the value of their ratio is not remembered in the limit.

With this remark in mind, we now look for a modification of the ‘naive’
continuum energies which corrects the non-equivalent behavior, while
maintaining the relevant features associated with the discreteness in the
original functional $E_{\varepsilon}$. Since the energies defined in (6.4)
cannot be equivalent to $E_{\varepsilon}$ mainly because of the discrete
location of the jump points, it is natural to add the constraint that the jump
set $S(u)$ be contained in $\varepsilon\mathbb{Z}$.