# Beyond the classical Cauchy-Born rule

Andrea Braides
SISSA, via Bonomea 265, Trieste, Italy

Andrea Causin and Margherita Solci
DADU, Università di Sassari
piazza Duomo 6, 07041 Alghero (SS), Italy

Lev Truskinovsky
PMMH, CNRS - UMR 7636 PSL-ESPCI,
10 Rue Vauquelin, 75005 Paris, France

###### Abstract

Physically motivated variational problems involving non-convex energies are
often formulated in a discrete setting and contain boundary conditions. The
long-range interactions in such problems, combined with constraints imposed by
lattice discreteness, can give rise to the phenomenon of geometric frustration
even in a one-dimensional setting. While non-convexity entails the formation
of microstructures, incompatibility between interactions operating at
different scales can produce nontrivial mixing effects which are exacerbated
in the case of incommensuration between the optimal microstructures and the
scale of the underlying lattice. Unraveling the intricacies of the underlying
interplay between non-convexity, non-locality and discreteness, represents the
main goal of this study. While in general one cannot expect that ground states
in such problems possess global properties, such as periodicity, in some cases
the appropriately defined ‘global’ solutions exist, and are sufficient to
describe the corresponding continuum (homogenized) limits. We interpret those
cases as complying with a Generalized Cauchy-Born (GCB) rule, and present a
new class of problems with geometrical frustration which comply with GCB rule
in one range of (loading) parameters while being strictly outside this class
in a complimentary range. A general approach to problems with such ‘mixed’
behavior is developed.

##  1 Introduction

Variational problems emerging from applications are often both discrete and
non-convex. Important examples include one-dimensional boundary-value problems
with translation-invariant energy densities describing pairwise interactions.
Such problems constitute the main subject of this paper.

The representative energies for this class of problems can be written in the
following generic form

$F(w;k)=\min\biggl{\\{}\sum_{i,j=0}^{k}f_{i-j}(u_{i}-u_{j}):u_{0}=0,u_{k}=w\biggr{\\}},$
(1.1)

where for every $n$ natural number $f_{n}$ is a potentially nonconvex energy
governing interactions between the lattice points at distance $n$, and the
minimum is searched among $k+1$-arrays $(u_{0},\ldots,u_{k})$. We may assume
that $f_{0}=0$. As the parameter $k$ increases and more interactions are taken
into account, a question arises about the behavior of minimal arrays
$(u^{k}_{0},\ldots,u^{k}_{k})$ and of the corresponding minimal energy. One of
the most important issues concerns the existence of a continuum limit of the
type $F_{\rm hom}(u)=\int_{I}f_{\rm hom}(u^{\prime})\,dt,$ with $I$ an
interval in which the nodes $i$ in (1.1) are identified as a discrete subset
(e.g., $I=[0,1]$ where the discrete subset is ${1\over
k}\mathbb{Z}\cap[0,1]$). The single function $f_{\rm hom}$ is expected to
carry, in a condensed way, all the relevant information about the infinite set
of functions $f_{n}$ from (1.1).

To track the asymptotic behavior of the minimum values in (1.1) we can use the
average derivative $z=w/k$ as a parameter, and scale the energy by $k$. Then,
under assumptions on a suitably fast decay of $f_{n}$ with respect to $n$, it
can be shown that the limiting energy density $f_{\rm hom}$ exists and can be
expressed by the formula

$f_{\rm hom}(z)=\lim_{k\to+\infty}{1\over
k}\min\biggl{\\{}\sum_{i,j=0}^{k}f_{i-j}(u_{i}-u_{j}):u_{0}=0,u_{k}=kz\biggr{\\}}.$
(1.2)

Moreover, it can be shown that the function $f_{\rm hom}$ is convex in the
parameter $z$. This result represents a particular case of a more general
variational theory for limits of lattice energies (see e.g. [3]); it can be
also seen as a zero-temperature limit of the analogous result in Statistical
Physics ([81, 79]). However, formula (1.2) is only a formal homogenization
result in a discrete-to-continuum setting which is usually non-constructive.
In this paper we are raising the issue of the actual _computability_ of
$f_{\rm hom}(z)$.

Explicit formulas for $f_{\rm hom}(z)$ in terms of $f_{n}$ are known only in
few cases, most of which are mentioned below. In general, it is known that the
behavior of minimizing arrays $(u^{k}_{0},\ldots,u^{k}_{k})$ at fixed $z$, may
be complex, including equi-distribution (‘crystallization’; see e.g. [66]),
periodic oscillations [24, 49], development of discontinuities (fracture in
lattice models [84, 27]) or defects (internal boundary layers [22]).

A robust approach to the computation of $f_{\rm hom}(z)$ is known under the
name of Cauchy-Born (CB) rule and is applicable under some restrictive
conditions ([41, 15]). It is based on the assumption that the homogenized
energy can be computed using the affine interpolations $u_{j}=zj$ and relying
_exclusively_ on problems with finite $k$. Various sufficient conditions for
the validity of the Cauchy-Born rule have been obtained by a number of authors
mostly in the context of local minimizers [40, 58, 67, 76, 85, 34, 86]. While
those results are usually valid only for subsets of loading parameters, they
are often applicable for dimensions higher than one. They are of considerable
interest, first of all, for the development of numerical methods because the
applicability of the classical CB rule makes such methods extremely efficient,
even if for a limited set of boundary conditions. The difference of our
approach to (1.2) is that we are interested in _global_ minimization (viewed
as a zero temperature limit of a statistically equilibrium response) and
consider the possibility that the conventional CB rule is operative only in a
_subset_ of the loading parameters while in the complementary subset the CB
strategy should be appropriately _generalized_ or even completely ruled out.

The main reason for the failure of the classical Cauchy-Born rule is the
geometrical frustration caused by incompatible optimality demands imposed by
(generically non-convex and long range) potentials $f_{n}$ with positive
integer $n$ and the discreteness of the lattice. More specifically, while non-
convexity entails the formation of microstructures, incompatibility between
interactions operating at different scales can produce nontrivial mixing
effects which are exacerbated in the case of incommensuration between the
optimal microstructures and the scale of the underlying lattice. Unraveling
the intricacies of the underlying interplay between non-convexity, non-
locality and discreteness, represents the main goal of this study.

If the classical Cauchy-Born rule fails, the natural task is to search for a
nontrivial generalization of the Cauchy-Born rule. In this perspective, we
pose the problem of finding the conditions for which the minimal arrays in
(1.1) have ‘global’ features in the sense that solving a ‘local’ problem on a
_finite_ domain opens the way towards describing the limit in (1.2). More
specifically, the question is whether the limiting energy $f_{\rm hom}(z)$ can
be approximately computed by solving a finite set of ‘cell’ problems modeled
on (1.1) and potentially producing _non-affine_ optimal configurations. The
validity of the so-interpreted generalized Cauchy-Born (GCB) rule would then
require that even if the implied ‘local’ problems could be solved only on some
subsets of parameters, the knowledge of the corresponding solutions would
ensure the recovery of the macroscopic (homogenized) energy in the _whole_
range of loading parameters.

Note that in local problems like (1.1) the presence of interactions $f_{n}$
with $n\in\\{1,\ldots,k\\}$ requires $k$ boundary conditions on each side. By
fixing parametrically only the average strain $z$ in (1.2) we effectively
assume that the remaining boundary conditions are natural. This simplifying
assumption may stay on the way of acquiring, for the given ‘local’ problem,
the corresponding ‘global’ features. That is why we will understand the
‘local’ GCB problem as having the right boundary conditions to ensure the
recovery of the macroscopic energy $f_{\rm hom}(z)$. The simplest case is when
the value of $f_{\rm hom}(z)$ can be achieved on arrays such that $i\mapsto
u_{i}-zi$ is periodic with a given period, but in general one should be
allowed to adjust boundary conditions accordingly while keeping in mind that
these changes should not affect the minimizers in an asymptotic sense.

We now illustrate the main difficulties on the way of generalizing the
classical CB rule with some known cases. We start with the simplest example
where the conventional CB rule works trivially. It is the case of convex
nearest-neighbor (NN) interactions; i.e., when $f_{n}=0$ for all $n\geq 2$,
and $f_{1}=f$ is a strictly convex function. In this case, the unique
minimizer of the problem in (1.2) is the affine interpolation $u^{k}_{j}=zj$.
It is independent of $k$ and hence ‘global’: in this case the classical
Cauchy-Born rule is applicable in its simplest form, and $f_{\rm
hom}(z)=f(z)$.

If we make the above example only a little more complex considering also
convex next-to-nearest-neighbour (NNN) interactions; i.e., $f_{n}=0$ for all
$n\geq 3$, with $f_{1}$ and $f_{2}$ convex functions, we loose this exact
characterization of the minimal arrays. However, the discrepancy between
$u^{k}_{j}$ and $zj$ decays fast away from the endpoints $j=0$ and $j=k$ of
the array. A slight adjustment of the boundary-value problems, say by imposing
additional boundary conditions $u_{1}=z$ and $u_{k-1}=z(k-1)$ (which do not
influence the asymptotic value of the minima in (1.2)) reestablishes the
affine interpolations $u^{k}_{j}=zj$ as minimizers, so that $f_{\rm
hom}(z)=2(f_{1}(z)+f_{2}(2z))$. In this case the classical Cauchy-Born rule is
applicable, given that we modify boundary conditions in the ‘cell’ problem.
Note that this analysis extends to any sufficiently fast decaying set of
convex potentials $f_{n}$, giving $f_{\rm
hom}(z)=2\sum_{n=1}^{\infty}f_{n}(nz)$.

Even if we abandon the convex setting, we may still easily describe the
behavior of minimum problems in (1.2) in the case of nearest-neighbor
interaction, with $f_{1}=f$. It can be shown that $f_{\rm hom}$ in (1.2) is
given by the convexification $f^{**}$ of the NN potential [25]. However the
classical Cauchy-Born rule in this case has to be properly generalized.
Suppose, for instance, that the potential $f$ has a double-well form. In this
case the relaxation points towards configurations containing mixtures of the
two energy wells. Since in this setting there are no obstacles to simple
mixing, the relaxation strategy providing $f_{\rm hom}$ is straightforward.
Indeed, for each $z$ there exist $z_{1}$, $z_{2}$, $\theta\in[0,1]$ such that
$f^{**}(z)=\theta f(z_{1})+(1-\theta)f(z_{2})$. Hence, we can construct a
function $u^{z}:\mathbb{Z}\to\mathbb{R}$ with
$u^{z}_{i}-u^{z}_{i-1}\in\\{z_{1},z_{2}\\}$, $u^{z}_{0}=0$ and
$|u^{z}_{i}-iz|\leq C$. Such $u^{z}$ may be chosen periodic, if $\theta$ is
rational, or quasiperiodic (loosely speaking, as the trace on $\mathbb{Z}$ of
a periodic function with an irrational period) otherwise. In both cases we
obtain ‘local’ minimizers with ‘global’ properties which allows one to talk
about the applicability of the GCB rule.

The situation is more complex in the case when non-convexity is combined with
frustrated (incompatible) interactions. To show this effect in the simplest
setting it is sufficient to account for nearest-neighbor and next-to-nearest-
neighbor interactions only and we make the simplest nontrivial choice by
assuming that $f_{1}$ is a ‘double-well’ potential and that $f_{2}$ is a
convex potential. In this case the homogenized potential $f_{\rm hom}$ is also
known explicitly [24, 77]. Its domain can be subdivided in three zones: two
zones of ‘convexity’ where minimizers are trivial (as for convex potentials)
and a zone where (approximate) minimizers in (1.2) are two-periodic functions
with $u^{z}_{i}-u^{z}_{i-1}\in\\{z_{1},z_{2}\\}$ and $z_{1}+z_{2}=z$ (in a
sense, a constrained non-convex case as above). Hence, in these three zones we
have minimizers with a ‘global’ form because the macroscopic energy can be
obtained by solving elementary ‘cell’ problems.

One can say that in the two zones of ‘convexity’ the classical CB rule is
applicable. In the ‘two-periodic’ third zone we see that the homogeneity of
the minimizers is lost but an appropriately augmented GCB rule still holds.
For the remaining values of $z$ no ‘local’ GCB rule is applicable since in
those cases the unique (up to reflections) minimizer is a ‘two-phase’
configuration with affine and two-periodic minimizers coexisting while being
separated by a single ‘interface’ [22]. The frustration (incompatibility)
manifests itself in this case through the impossibility of the penalty-free
accommodation of next-to-nearest interactions across such an internal boundary
layer. As a consequence, as $k$ diverges, such minimizers tend to an affine
interpolation between the ‘convex’ and ‘oscillating’ zones which delivers the
correct value of $f_{\rm hom}(z)$ without being a solution of any finite
‘cell’ problem. Effectively, the ‘representative cell’ in this case has an
infinite size and therefore no GCB-type ‘local’ description of the macroscopic
state is available. A somewhat similar situation is encountered in continuum
homogenization of both random [62] and strongly nonlinear [23, 72] elastic
composites.

In what follows, we interpret the loss of ‘locality’ in homogenization
problems, which was illustrated above on the simplest example, as a failure of
the GCB rule. To shed some light on the mechanism of this phenomenon, we
consider below a class of analytically transparent discrete problems combining
nonconvexity with geometrical frustration.

More specifically, given the complexity of a general asymptotic analysis for
even one-dimensional problems of this type, we limit our attention to a class
of discrete functionals of type (1.1) with $f_{1}(z)={1\over
2}f(z)+m_{1}z^{2}$, where the function $f(z)$ is non-convex, and quadratic
$f_{n}(z)=f_{-n}(z)=m_{n}z^{2}$ for $n\geq 2$. The coefficients $m_{n}$ which
introduce nonlocality and frustration, are assumed to be non negative and
sufficiently integrable. In other words, we suppose that the non-convexity is
‘localized’ in the nearest-neighbor interactions, while all other interactions
are quadratic. The positivity of the infinite sequence ${\bf m}=\\{m_{n}:n\geq
1\\}$ is chosen to ensure that the implied quadratic ‘penalty’ is a measure of
the distance of the configuration $u_{i}$ from the affine configuration
$L_{z}(i)=zi$ and can be then seen as a _non-local version_ of the gradient of
$u-L_{z}$. One can also say that such penalization brings anti-ferromagnetic
interactions; an alternative, ferromagnetic-type quadratic penalty, was
considered, for instance, in [80].

The advantage of this choice of $f_{n}$ is that the ensuing problem can
exhibit both ‘local’ (GCB) and ‘global’ behavior depending on the structure of
the sequence of scalar parameters $m_{n}$. Therefore our goal will be to use
the chosen class of functionals to characterize the difference between CB, GCB
and non-GCB problems in terms of such sequences. We show that in this
naturally limited but still sufficiently rich framework one can _precisely_
specify the factors preventing the GCB-type description of the macroscopic
energy and pointing instead towards the non-GCB nature of the minimizers.
Moreover, the considered example allow us to _abstract_ some general technical
tools which can facilitate the detection and the characterization of the non-
GCB asymptotic behavior in more general minimization problems.

We reiterate that even in the absence of an adequate ‘cell’ problem, the
ensuing value of $f_{\rm hom}(z)$ is fully determined by the homogenization
formula which in our case takes the form $f_{\rm hom}(z)=\widehat{Q}_{\bf
m}f(z)$ where

$\widehat{Q}_{\bf m}f(z)=\lim_{k\to+\infty}{1\over
k}\min\Bigl{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\sum_{i,j=0}^{k}m_{i-j}(u_{i}-u_{j})^{2}:u_{0}=0,u_{k}=kz\Bigr{\\}}\,.$
(1.3)

The nontrivial part of the mapping $\widehat{Q}_{\bf m}f$, accentuating the
nonlinearity of the problem, is carried by the operator $Q_{\bf
m}f(z)=\widehat{Q}_{\bf m}f(z)-2\sum_{n\geq 1}m_{n}n^{2}z^{2}.$ Thus, if $f$
is convex, this mapping, to which we refer as the ${\bf m}$-transform of $f$,
is the identity; actually, the same remains true even if $f$ is
$2m_{1}$-convex, in the sense that the function $z\mapsto f(z)+2m_{1}z^{2}$ is
convex. If, however, the function $f$ is not $2m_{1}$-convex, the ${\bf
m}$-transform of $f$ is nontrivial. Thus, the function $Q_{\bf m}f(z)$ is in
general non-convex and $Q_{\bf m}f(z)>f^{**}(z)$ for some $z$; the non-
convexity of $Q_{\bf m}f(z)$ depends sensitively and ‘nonlocally’ on the
penalizing sequence $\bf m$.

Indeed, recall that $\widehat{Q}_{\bf m}f$ can be viewed as an _operator_
acting on the non-convex function $f$ and producing an ${\bf m}$-dependent
function which effectively represents a constrained relaxation of $f$. In the
same vein, the function $Q_{\bf m}f$ represents a nonlocally _constrained
convexification_ of $f$. Interpreted in such a way, the construction of
$Q_{\bf m}f$ is reminiscent of energy quasiconvexification in continuum
elasticity. The latter deals with minimization of the functionals $\int f({\bf
F})d\bf x,$ where ${\bf F}$ is a matrix field. The role of nonlocal constraint
in such problems is played by the condition ${\rm curl\,\bf{F}}=0$, which is
highly nontrivial in a multidimensional setting [63]. In a one-dimensional
setting this whole construction can be imitated through the introduction of a
penalizing kernel $\bf m$ mimicking the Green’s function of the constraint. As
in the case of continuum elasticity, such a penalization can introduce
incompatibility, which in a discrete setting can lead to geometrical
frustration.

One of the goals of this paper will be to link the degree of the non-convexity
of the function $Q_{\bf m}f$ with the breakdown of the GCB rule. For instance,
in the parametric domain where periodic microstructures are optimal, one can
also expect the convexity of the function $Q_{\bf m}f$. Topologically
different periodic microstructures will exist in finite intervals of $z$ where
they can be ‘stretched’ to secure the commensurability with the lattice. In
such intervals the corresponding minimizers posses ‘global’ properties and the
GCB rule is respected. However, in general, when $z$ is varied continuously,
the optimal microstructure will change discontinuously and the domain of
applicability of the GCB rule can _coexist_ with the domains where it breaks
down. The challenge is to identify the conditions on $\bf m$, when, for
instance, the knowledge of the intervals where GCB rule is applicable, allows
one to re-construct the ${\bf m}$-transform of a given non-convex function $f$
also for $z$ where the GCB rule is non-applicable.

In this paper we are not attempting to solve the problem posed above in its
full generality and instead focus on a physically interesting sub-class of
non-convex functions $f$ allowing one to construct _explicit_ solutions of the
minimization problem for several important classes of penalizing kernels $\bf
m$.

Specifically, we aim at the development of a comprehensive theory for _bi-
convex_ functions $f$. More precisely, we assume that there is a value
$z=z^{*}$ such that the restrictions of $f$ to $(-\infty,z^{*}]$ and
$[z^{*},+\infty)$ are both convex; well-known examples of bi-convex functions
are the quadratic double-well potential ($f(z)=(|z|-1)^{2}$ with $z^{*}=0$),
used for the description of phase transitions, and the truncated quadratic
potential ($f(z)=z^{2}$ if $z\leq 1$ and $f(z)=1$ if $z\geq 1$), which is used
in Fracture Mechanics. In what follows we often refer to the two convex
branches of $f$ as _microscopic phases_.

An important property of the bi-convex functions $f$ is that, independently of
the choice of the kernels $\bf m$, the mapping $\widehat{Q}_{\bf m}f$ is
largely characterized by a phase function $\theta=\theta(z)$ which represents
the asymptotic volume fraction of one of the ‘phases’ in the limiting
minimizer, say the limit of the percentage of indices $i$ for which
$u^{k}_{i}-u^{k}_{i-1}\geq z^{*}$. When $f$ is convex, then $\theta=0$ or
$\theta=1$ and when its is bi-convex, the central question will be to describe
for a given $\bf m$ the form of $\theta(z)$. As we show, the applicability of
GCB can be related to the emergence of the ${\bf m}$-dependent ‘steps’ on the
graph of the function $\theta$ represented by the values $\overline{\theta}$
for which $\\{z:\theta(z)=\overline{\theta}\\}$ is a non-degenerate interval.
In what follows we refer to such intervals as _locking_ states and to the
corresponding GCB-type microstructures as _mesoscopic phases_. This
characterization is justified by the fact that in the locking states the form
of minimizers is _stable_ in the sense that the set of indices $i$ at finite
$k$ such that that $u^{k}_{i}-u^{k}_{i-1}\geq z^{*}$ is independent of $z$, up
to an asymptotically negligible fraction. Therefore, the implied ‘staircase’
structure of the function $\theta$ is not a feature of the discrete problem
only as it survives in the continuum limit. As we show, the locking states
have the desired global properties, and for such states an appropriate finite
‘cell’ problem can be formulated and solved. In other words, in such states
the GCB rule is operative and the computation of the macroscopic energy energy
can be made explicit.

In this paper we have chosen to illustrate all these effects by considering
penalization kernels $\bf m$ amenable to fully _explicit_ study. Our analysis
shows that a rather comprehensive picture can be obtained based on the
analysis of just two archetypal _classes_ of kernels.

The first class of analytically transparent kernels contains ‘concentrated’
(compact, localized, narrow banded, etc.) parametric sequences $\bf m$ defined
by the condition that there exists $M\geq 2$ such that $m_{n}=0$ if $n\geq 2$
and $n\neq M$; here $M$ plays the role of a parameter. We prove that for such
kernels (and independently of $f$, as long as it is non-convex) locking states
do exist and correspond to $\theta_{n}={n\over M}$ with
$n\in\\{0,\ldots,M\\}$. Minimizers in this case, representing mesoscopic
phases, are $M$-periodic. Moreover, we prove that the associated phase
function $\theta$ is piecewise affine, interpolating locally between the
locking states $\theta_{n-1}$ and $\theta_{n}$. Thus, while for $\theta$ that
is not a locking state we do not have GCB-type minimizers (with ‘global’
properties), the whole mapping $\widehat{Q}_{\bf m}f$ can be recovered from
the knowledge of its value at those $z$ corresponding to locking states where
the GCB rule is operative.

The second class of analytically transparent kernels contains exponentially
decaying sequences $\bf m$ which we write in the parametric form
$m_{n}=e^{-\sigma n}$ with $\sigma>0$ playing the role of a parameter
analogous to $M$ in the first class. Here again we can give a complete
description of the relaxed problem, for instance, when $f$ is a truncated
convex potential ($f$ is constant in $[z^{*},+\infty)$). Given this particular
structure of non-convex potentials (describing, for instance, lattice
fracture), locking states are either $\theta=0$ or $\theta\in\\{{1\over
k}:k\in\mathbb{N}\\}$. In the latter case, minimizers are $k$-periodic and
therefore of GCB-type, which means that they posses ‘global’ properties.
Interestingly, we show that in each period such minimizers have a single
difference $u^{k}_{i}-u^{k}_{i-1}$ exceeding the threshold $z^{*}$ (single
‘crack’). Again, we prove that the set of mesoscopic phases is sufficiently
rich to provide the ‘building blocks’ whose simple mixtures allow one to
construct the whole mapping $\widehat{Q}_{\bf m}f$. An important difference
with the case of ‘concentrated’ kernels is that now the optimal ‘simple’
mixtures of ‘global’ (or GCB) states are not unique optimal microstructures.
More precisely, we show that even for _non-locking_ values of $z$ one can
build optimal minimizers which are of GCB-type. For all values of $z$ such
minimizers are _quasiperiodic_ and therefore posses the desired ‘global’
properties, thus broadening the spectrum of possible GCB-type microstructures.

All these explicit results, which also include an analytical study of the
intricate role of the parameters $\sigma$ and $M$, can be obtained because for
these two classes of kernels (concentrated and exponential) one can
reformulate the original non-additive (non-local) minimum problem with
presumably complex mixing properties as an additive (local) problem with no
mixing effects at all. For concentrated kernels this is achieved by rewriting
the non-additive problem as a superposition of additive problems. For
exponential kernels the reduction of complexity is due to the mapping of a
scalar problem with long-range interactions on a vectorial problem with only
nearest-neighbor interactions.

Variational problems with energies like (1.3) have been studied extensively in
the physical literature where they emerged independently in different settings
ranging from conventional magnetic and mechanical systems [8, 59] to discotic
liquid crystals [36, 50, 57]. In such problems the optimal periodicity of a
microstructure representing the ground state (global minimum of the energy)
competes with the periodicity of the lattice, and the geometrical frustration
emerges when the two periodicities are incompatible (for instance,
incommensurate). Since the interactions in actual physical systems are very
complex, the main focus was on the study of simplified discrete models such as
Frenkel-Kontorova model [30] or ANNNI model [82]. A prototypical Ising model
with antiferromagnetic long-range interactions, which is the simplest problem
of this same type was considered in [9]. Two explicit solutions for the class
of problems with exponential kernels studied in the present paper, were found
in [74, 75].

In the mathematical literature discrete and continuous variational models with
antiferromagnetic interactions were considered in [25, 24, 77, 78, 33, 51]. An
important link was established by S. Aubry and J. Mather between variational
problems of type (1.3) and the quasiperiodic trajectories of discrete
dynamical systems. Recent mathematical results extending Aubry-Mather theory
can be found in [12, 52, 43, 48].

In the present paper we reformulate the problems studied previously in the
framework of the theory of dynamical systems, as problems of the calculus of
variations. This change of perspective allows one to apply powerful
homogenization results providing direct access to the corresponding continuum
limits. The goal is to demonstrate how, already in one-dimensional problems,
the the interplay between discreteness and non-convexity compromises the
classical Cauchy-Born rule and precludes the use of conventional ‘cell’
problems for computation of the relaxed energies.

In the context of discrete-to-continuum transitions, the obtained results
bring new understanding of the role of the frustrated non-local interactions
in the determination of homogenized energies. While the case of ferromagnetic
interactions has been extensively studied before, here we show that the
introduction of anti-ferromagnetic interactions brings fundamentally new
effects, most importantly the emergence of mesoscopic phases resulting in the
locking of the minimizers on lattice-commensurate microstructures. While these
effects, which are clearly lattice-induced, appear to be ‘strongly discrete’,
they affect the structure of the continuum energy and, in this sense, do not
disappear in the course of discrete-to-continuum transition.

Instead of the focus on Euler-Lagrange equations, characteristic of the theory
of dynamical systems, our main tools are the direct methods of the calculus of
variations. In particular, we obtained our main results through the use of the
novel bounds resulting either from the judicial choice of periodic test
functions or from cluster minimization. In this sense our results complement
and broaden the findings made in the dynamical systems framework.

One result of this type is the characterization of the continuum limit when
non-local interactions are concentrated on $M$-neighbors. The analysis of this
case highlights the increasing difficulty of dealing with geometrical
frustration and non-commensurability effects as progressively more distant
interactions are incorporated, and suggests the possibility of scale-free
patterns even in the case of finite-range interaction kernels. It complements
the results of Aubry [6], who showed that long-range interactions favor hyper-
uniform solutions. Another result, allowing one to relate the regularity of
the relaxed energies in $\theta$ with the existence of periodic solutions, can
be viewed as an extension of the link between regularity and the rotation
number established by Mather in the framework the dynamical systems approach
[70].

In addition to explicit computations of global minimizers we also posed the
problem of finding the $\Gamma$-equivalent continuum _approximations_ of the
corresponding lattice problems [29]. Here we imply the construction of the
asymptotic continuum theories accounting for the lattice scale. We succeed in
constructing such an approximation in the case of an exponential kernel while
also showing that the conventional formal asymptotic limit, which neglects the
underlying geometric frustration, underestimates the intricacies of the
interplay between non-convexity, non-locality and discreteness and produces
only a lower bound for $Q_{\bf m}f$. This explicit example serves as a
cautionary tale demonstrating in which form the finite scale lattice effects
can survive homogenization and affect the macroscopic variational problem.

##  2 Nonlocal discrete problems and their relaxation

In this paper we study the asymptotic behaviour of particular nonlocal
discrete problems parameterized by the number of nodes involved. This can be
viewed as a discrete-to-continuum homogenization process by introducing a
small parameter $\varepsilon$ and suitable scalings of the energies. However,
with an abuse of terminology, we choose to label this process as the
computation of a relaxed functional.

Following the usual terminology, a functional $\overline{\Phi}$ is the
relaxation of an original functional $\Phi$ if, loosely speaking, infimum
problems involving $\Phi$ have the same value as infimum problems involving
$\overline{\Phi}$, and the latter admit solution (given that the corresponding
problem is coercive), see e.g. [35, 18]. In the context of the Calculus of
Variations, the relaxed functional is usually obtained by a lower-
semicontinuous envelope with respect to some topology, it is stable under
continuous perturbations, and often (but not always) is stable with respect to
closed constraints, such as fixed boundary values or imposed integral
constraints. Moreover, if the original functional depends on some energy
density, often (but not always) the relaxed functional can be characterized by
a new energy density obtained as a transformation (convexification,
quasiconvexification, sub-additive or $BV$-elliptic envelope, etc.) of the
original energy density, so that relaxation of an energy can be viewed as an
operation on an energy density. In our case we deal with a sequence of minimum
problems, so it would be correct to talk about homogenization or
$\Gamma$-convergence rather than relaxation. Nevertheless, we would like to
highlight properties of the homogenized continuum energy in the same spirit of
a lower-semicontinuous envelope, and hence we choose the terminology of
relaxation.

We focus on the relaxation of nonlocal discrete functionals of type (1.3).
They involve a non-convex function $f$ and contain a ‘penalization kernel’
$\mathbf{m}$. The idea is to single out the local (nearest-neighbour)
interaction in the general discrete-to-continuum problem, and consider the
corresponding potential $f$ as the function that needs to be ‘relaxed’. The
nonlocal (beyond nearest-neighbour) interactions are assumed to be linear. The
corresponding quadratic term in the energy brings the simplest penalization
into the relaxation process. We show that even such a simple penalization may
still carry incompatibility and may even lead to geometrical frustration. In
what follows, with a slight abuse of terminology, we will be referring to
(1.3) as a $\mathbf{m}$-dependent relaxation of a non-convex energy density
$f$. Before giving the formal definitions, we make some preliminary comments
distinguishing _penalized_ relaxation from _non-penalized_ relaxation.

###  2.1 Nearest-neighbour interaction and quadratic penalization

As it is well known, the convexification of a function $f$ can be seen as the
result of a discrete-to-continuum relaxation process in a local setting
involving nearest-neighbour interactions only. To be more specific, for any
$k\in\mathbb{N}$ and $z\in\mathbb{R}$ we introduce the set

$\mathcal{A}(k;z)=\\{u\colon[0,k]\cap\mathbb{N}\to\mathbb{R}\ \hbox{ such that
}u(0)=0,u(k)=kz\\}$ (2.1)

of admissible test functions satisfying boundary conditions. Here the
parameter $z$ represents the affine boundary conditions $u(i)=L_{z}(i)$, where
$L_{z}(i)=iz$.

######  Proposition 2.1 (a characterization of the convex envelope).

Let $f\colon\mathbb{R}\to\mathbb{R}$. Then, the convex envelope of $f$ is

$\displaystyle
f^{\ast\ast}(z)=\displaystyle\lim_{k\to+\infty}\frac{1}{k}\inf\bigg{\\{}\sum_{i=1}^{k}f(u(i)-u(i-1)):\
u\in\mathcal{A}(k;z)\bigg{\\}}.$

It is useful in this context to interpret Proposition 2.1 as a consequence of
discrete-to-continuum $\Gamma$-convergence (see e.g. [18, Ch. 4.2]). Indeed,
define for a given bounded interval $I$ and for any $\varepsilon>0$ the set of
indices $\mathcal{I}_{\varepsilon}(I)$ and the set of discrete functions
$\mathcal{A}_{\varepsilon}(I)$ given by

$\mathcal{I}_{\varepsilon}(I)=\\{i\in\mathbb{Z}:\varepsilon i\in
I\\},\quad\mathcal{A}_{\varepsilon}(I)=\\{u\colon\varepsilon\mathcal{I}_{\varepsilon}(I)\to\mathbb{R}\\},$
(2.2)

respectively. Here and in the sequel, $u_{i}$ denotes the value $u(\varepsilon
i)$, and we identify $u\in\mathcal{A}_{\varepsilon}(I)$ with its piecewise-
constant extension in $I$. Having defined

$F^{0}_{\varepsilon}(u;I)=\varepsilon\sum_{i,i-1\in\mathcal{I}_{\varepsilon}(I)}f\Big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\Big{)}$
(2.3)

for $u\in\mathcal{A}_{\varepsilon}(I)$, the $\Gamma$-limit with respect to the
$L^{2}$-convergence of $F^{0}_{\varepsilon}$ is the functional
$F^{0}(u,I)=\int_{I}f^{**}(u^{\prime})\,dt$ for $u\in H^{1}(I)$. Then,
choosing $\varepsilon_{k}=\frac{1}{k}$, by the convergence of minimum problems
we get

$\displaystyle f^{\ast\ast}(z)$ $\displaystyle=$
$\displaystyle\min\\{F^{0}(u;(0,1)):u(0)=0,u(1)=z\\}$ $\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\min\\{F^{0}_{\varepsilon_{k}}(u;(0,1)):u(0)=0,u(1)=z\\},$

which is the desired formula up to a change of variable.

######  Remark 2.2 (additivity).

Note that the problems defining $f^{\ast\ast}$ are additive, in the sense
that, setting

$\mu(k,z)=\inf\bigg{\\{}\sum_{i=1}^{k}\\!f(u_{i}-u_{i-1}):\
u\in\mathcal{A}(k;z)\bigg{\\}},$

we have
$\mu(k,z)=\min\big{\\{}\mu(k_{1},z_{1})+\mu(k_{2},z_{2}):k_{1}+k_{2}=k,\
k_{1}z_{1}+k_{2}z_{2}=kz\big{\\}}$.

We now add to the nearest-neighbour term, described by a non-convex function
$f$, a quadratic long-range term which brings the simplest penalization of
global inhomogeneity while promoting uniformity in the sense of averages.

To this end we introduce a sequence $\mathbf{m}=\\{m_{n}\\}_{n\in\mathbb{N}}$
such that

$m_{n}\geq 0\ \ \hbox{\rm for any }n\ \ \hbox{\rm and }\ \
m_{n}=o(n^{-\beta})_{n\to+\infty}\ \ \hbox{\rm for some }\ \ \beta>3.$ (2.4)

Such penalization has an ‘antiferromagnetic’ character, in that it in fact
favors local oscillations induced by the non-convexity of $f$.

Figure 1: representation of exponential and concentrated kernels.

In the sequel, an important role will be played by the two special families of
kernels: exponential, $m_{n}=e^{-\sigma n}$, and concentrated at some $M$,
$m_{n}=0$ for all $n$ except $n=1$ and $n=M$ with $M\geq 2$; in the latter
example one can similarly account for a parameter $\sigma$ by using the new
definitions, $m_{1}^{\sigma}=(1/\sigma)m_{1}$ and
$m_{M}^{\sigma}=(1/\sigma)m_{M}$, see Fig. 1.

Before formally defining the penalized energy, we need to make some
assumptions on $f$. These assumptions will be used to obtain the existence of
the limit of minimum problems. Note that the hypotheses can be relaxed, but
they are stated as follows in order to avoid unnecessary technicalities. Our
first simplifying assumption is that the (non-convex) potential
$f\colon\mathbb{R}\to[0,+\infty)$ is non-negative and that it satisfy a
quadratic growth hypothesis; namely,

$0\leq f(z)\leq c(z^{2}+1)\ \ \hbox{ for some }c>0.$ (2.5)

In addition to (2.5), we will also assume that the function $f$ satisfies

${1\over c}z^{2}\leq f(z)+m_{1}z^{2}.$ (2.6)

Note that hypothesis (2.6) is automatically satisfied if $m_{1}>0$. We will
point out specifically in which of the cases assumption (2.6) is not
necessary.

######  Definition 2.3 (relaxation with kernel $\bf m$).

For all $z\in\mathbb{R}$ we set

$\widehat{Q}_{\bf
m}f(z)=\lim_{k\to+\infty}\frac{1}{k}\inf\bigg{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}:\
u\in\mathcal{A}(k;z)\bigg{\\}}.$

The function $\widehat{Q}_{\mathbf{m}}f$ is well defined since the limit
exists by known discrete-to-continuum results (see formula (2.9) below). For
this existence the growth condition is essential; however, in some cases we
will use this formula also for some degenerate $f$ for which the limit exists.
Note that, except for the case when only nearest-neighbours are involved, the
minimum problems defining $\widehat{Q}_{\mathbf{m}}f$ are not additive in the
sense of Remark 2.2.

###  2.2 General properties of $\widehat{Q}_{\bf m}f(z)$

In this section, we list some properties of the relaxation with kernel $\bf m$
derived from its variational nature.

######  Remark 2.4 (nearest-neighbour interactions).

By Proposition 2.1, the convex envelope of $f$ can be viewed as
$\widehat{Q}_{\bf 0}f$, where $\bf m=\bf 0$ is the trivial kernel $m_{n}=0$
for any $n\geq 1$; that is,

$\widehat{Q}_{\bf 0}f(z)=f^{\ast\ast}(z).$ (2.7)

More in general, again by Proposition 2.1, we obtain that $\widehat{Q}_{\bf
m}f(z)=(f(z)+2m_{1}z^{2})^{**}$ if $m_{n}=0$ for any $n\geq 2$. Note that in
these cases we have no non-additivity effects.

######  Remark 2.5 ($\widehat{Q}_{\bf m}f$ as a $\Gamma$-limit).

The fact that $\widehat{Q}_{\bf m}f$ is well defined and some of its key
properties follow by the fact that the functional $F$ defined by
$F(u)=\int_{I}\widehat{Q}_{\bf m}f(u^{\prime})\,dt$ for $I$ bounded interval
and $u\in H^{1}(I)$ can be interpreted as the $\Gamma$-limit of a suitable
sequence of discrete functionals $F_{\varepsilon}$. Indeed, consider the
functionals

$F_{\varepsilon}(u;I)=\varepsilon\sum_{i,i-1\in\mathcal{I}_{\varepsilon}(I)}f\Big{(}\frac{u_{i}-u_{i-1}}{\varepsilon}\Big{)}+\varepsilon\sum_{i,j\in\mathcal{I}_{\varepsilon}(I)}m_{|i-j|}\Big{(}\frac{u_{i}-u_{j}}{\varepsilon}\Big{)}^{2}$
(2.8)

defined in $\mathcal{A}_{\varepsilon}(I)$, with $\mathcal{I}_{\varepsilon}(I)$
and $\mathcal{A}_{\varepsilon}(I)$ as in (2.2). Such functionals can be
rewritten as

$F_{\varepsilon}(u;I)=\sum_{h\geq 1}\
\sum_{j,j+h\in\mathcal{I}_{\varepsilon}(I)}\varepsilon\,f^{h}\Big{(}\frac{u_{j+h}-u_{j}}{\varepsilon
h}\Big{)}$

where $f^{1}(z)=f(z)+2m_{1}z^{2}$ and $f^{h}(z)=2z^{2}h^{2}m_{h}$ if $h>1$.
With this notation, functionals $F_{\varepsilon}$ satisfy the hypotheses of
[3, Theorem 6.3]; that is, $f^{1}(z)\geq c_{1}z^{2}$ with $c_{1}>0$, and
$f^{h}(z)\leq c_{h}z^{2}$ with $\sum_{h}c_{h}<+\infty$. The lower bound
follows by the growth hypothesis (2.6), and the upper bound by (2.5) and by
hypothesis (2.4) on $\bf m$. Hence, the $\Gamma$-limit of $F_{\varepsilon}$
with respect to the $L^{2}$-convergence is represented by the functional
$F(u,I)=\int_{I}f_{\rm hom}(u^{\prime})\,dt$, where $f_{\rm hom}$ satisfies
the homogenization formula

$f_{\rm
hom}(z)=\lim_{k\to+\infty}\frac{1}{k}\inf\bigg{\\{}\sum_{h=1}^{k}\sum_{j=0}^{k-h-1}f^{h}\big{(}\frac{u_{j+h}-u_{j}}{h}\big{)}:u\in\mathcal{A}(k;z)\bigg{\\}}.$
(2.9)

Rewriting this formula, we get that the function $f_{\rm hom}$ coincides with
the function $\widehat{Q}_{\bf m}f$ introduced in Definition 2.3, which proves
that it is well-defined as a limit.

######  Remark 2.6.

Note that, while condition (2.6) can be relaxed by requiring that $f$ has a
superlinear growth (not necessarily quadratic), it cannot be dropped
altogether. Indeed, if $f=0$, $m_{2}\neq 0$ and $m_{n}=0$ otherwise, then the
limit in Definition 2.3 does not exist.

The following proposition states the convexity of $\widehat{Q}_{\bf m}f$,
which is ensured by the lower semicontinuity of the $\Gamma$-limit.

######  Proposition 2.7 (convexity of $\widehat{Q}_{\bf m}f$).

Let $\bf m$ be as in (2.4) and let $f\colon\mathbb{R}\to[0,+\infty)$ be a non-
negative function satisfying (2.5) and (2.6). Then the function
$\widehat{Q}_{\bf m}f$ is convex.

In the following remark we highlight that the boundary conditions can be
transformed in conditions on a boundary layer, which are more convenient for
computations.

######  Remark 2.8 (alternative statements of boundary conditions).

The boundary conditions $u_{0}=0$ and $u_{k}=kz$ can be replaced by conditions
on a boundary layer. We state two different equivalent possibilities, that
will both be used in the proofs. In the first one the boundary layer is a
small portion of the whole domain, parameterized by a small $\delta$, which
then we let tend to $0$, as follows

$\displaystyle\hskip-170.71652pt\widehat{Q}_{\bf m}f(z)\\!=\\!\lim_{\delta\to
0}\liminf_{k\to+\infty}\frac{1}{k}\inf\bigg{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\\!\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}\\!:u\in\mathcal{A}_{\delta}(k;z)\bigg{\\}}\hskip-28.45274pt\
$ $\displaystyle=\\!\lim_{\delta\to
0}\limsup_{k\to+\infty}\frac{1}{k}\inf\bigg{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\\!\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}\\!:u\in\mathcal{A}_{\delta}(k;z)\bigg{\\}},$
(2.10)

where

$\mathcal{A}_{\delta}(k;z)=\\{u\in\mathcal{A}(k;z):u_{i}=iz\ \hbox{\rm if }\
i\leq\delta k\ \hbox{\rm and }\ i\geq(1-\delta)k\\}.$

In the second one the double limit is replaced by a $k$-depending boundary
layer at a mesoscopic scale, as follows

$\displaystyle\widehat{Q}_{\bf
m}f(z)=\lim_{k\to+\infty}\frac{1}{k}\inf\bigg{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}:\
u\in\mathcal{A}_{k^{\alpha}}(k;z)\bigg{\\}},$ (2.11)

with $\alpha\in(-1,0)$.

These formulas can be proved by an argument which is customary to variational
treatments of homogenization problems (see e.g. [3]). In proving formulas
(2.8) and (2.11), it is necessary to use the growth hypothesis (2.6). In case
it does not hold, the limits in formulas (2.8) and (2.11) may be different
from the limit in the definition of $\widehat{Q}_{\bf m}f$.

We now give some general estimates on $\widehat{Q}_{\bf m}f$.

######  Remark 2.9 (estimates by decomposition for $\widehat{Q}_{\bf m}f$).

If $\bf m=\bf m^{\prime}+\bf m^{\prime\prime}$; that is,
$m_{n}=m^{\prime}_{n}+m^{\prime\prime}_{n}$ for all $n$, and $f=g+h$, then we
have

$\widehat{Q}_{\bf m}f(z)\geq\widehat{Q}_{\bf m^{\prime}}g(z)+\widehat{Q}_{\bf
m^{\prime\prime}}h(z).$

In Remark 2.4 we have examined the case when ${\bf m}={\bf 0}$. It may be of
interest to consider the case when conversely $f=0$ as in the following lemma.
If $\bf m$ is as in (2.4), then we set

$a_{\mathbf{m}}=2\sum_{n=1}^{+\infty}m_{n}n^{2}.$ (2.12)

######  Lemma 2.10 (minimization of the quadratic part).

Let $m_{1}>0$, so that (2.6) is satisfied with $f=0$. Then we have
$\widehat{Q}_{\bf m}0(z)=a_{\bf m}z^{2}$.

###### Proof.

By using $u_{i}=iz$ as a test function in the definition of $\widehat{Q}_{\bf
m}0(z)$ we get the inequality $\widehat{Q}_{\bf m}0(z)\leq a_{\bf m}z^{2}$,
after noting that

$\lim_{k\to+\infty}\frac{1}{k}\sum_{i,j=0}^{k}m_{|i-j|}(i-j)^{2}=\lim_{k\to+\infty}\frac{2}{k}\sum_{n=1}^{k}(k-n+1)m_{n}n^{2}=2\sum_{n=1}^{+\infty}m_{n}n^{2}=a_{\mathbf{m}}.$

It then suffices to prove that for all fixed $N$ we have

$\widehat{Q}_{\bf m}0(z)\geq 2\sum_{n=1}^{N}m_{n}n^{2}z^{2}.$

With fixed $\alpha\in(-1,0)$, let $u$ be a test function for the problem in
(2.11) with $f=0$ for $k^{1+\alpha}>N$. We then have

${1\over k}\Bigl{(}\sum_{i=1}^{k}2m_{1}(u_{i}-u_{i-1})^{2}\Bigr{)}\geq
2m_{1}z^{2}.$ (2.13)

If $n\in\\{2,\ldots,N\\}$ and $\ell\in\\{0,\ldots,n-1\\}$, let
$i_{\ell}=\lceil\frac{k-\ell}{n}\rceil$. We can rewrite the energy due to
interactions at distance $n$ as

$\displaystyle{1\over
k}2m_{n}\sum_{\ell=0}^{n-1}\sum_{i=1}^{i_{\ell}}(u_{\ell+in}-u_{\ell+(i-1)n})^{2}$
$\displaystyle\geq$ $\displaystyle{1\over
k}2m_{n}\sum_{\ell=0}^{n-1}i_{\ell}\Bigl{(}{1\over
i_{\ell}}\sum_{i=1}^{i_{\ell}}(u_{\ell+in}-u_{\ell+(i-1)n})\Bigr{)}^{2}$
$\displaystyle=$ $\displaystyle{1\over
k}2m_{n}\sum_{\ell=0}^{n-1}i_{\ell}\Bigl{(}\frac{u_{\ell+i_{\ell}n}-u_{\ell}}{i_{\ell}}\Bigr{)}^{2}={1\over
k}2m_{n}\sum_{\ell=0}^{n-1}i_{\ell}n^{2}z^{2}$ $\displaystyle\geq$
$\displaystyle
2m_{n}\frac{n}{k}\Bigl{\lceil}\frac{k-n}{n}\Bigr{\rceil}n^{2}z^{2}=2m_{n}(1+o_{k}(1))n^{2}z^{2},$

where we have used the convexity inequality and the boundary condition
$u_{j}=jz$ close to the boundary. Summing up for $n\in\\{2,\ldots,N\\}$ and
using (2.13), we prove the claim. ∎

In the following proposition we compare $\widehat{Q}_{\bf m}f$ with the convex
envelope of $f$ and with $f$ itself (to be more accurate, taking into account
the case that $f$ is not lower semicontinuous, with the lower-semicontinuous
envelope of $f$).

######  Proposition 2.11 (trivial bounds for $\widehat{Q}_{\bf m}f$).

Let $\bf m$ be as in (2.4) and let $f\colon\mathbb{R}\to[0,+\infty)$ be a non-
negative function satisfying (2.5) and (2.6). The inequalities

$f^{\ast\ast}(z)+a_{\mathbf{m}}z^{2}\leq\widehat{Q}_{\bf
m}f(z)\leq\big{(}f(z)+a_{\mathbf{m}}z^{2}\big{)}^{\ast\ast}\leq\overline{f}(z)+a_{\mathbf{m}}z^{2}$
(2.14)

hold, where $\overline{f}$ denotes the lower-semicontinuous envelope of $f$;
i.e., the largest lower-semicontinuous function not larger than $f$.

###### Proof.

By using $u_{i}=iz$ as a test function in the definition of $\widehat{Q}_{\bf
m}f(z)$ we get the inequality $\widehat{Q}_{\bf m}f(z)\leq f(z)+a_{\bf
m}z^{2}$ as in the first part of the proof of Lemma 2.10. Since
$\widehat{Q}_{\bf m}f$ is continuous by Proposition 2.7, this ensures that
$\widehat{Q}_{\bf m}f(z)\leq\overline{f}(z)+a_{\bf m}z^{2}$. Since
$\widehat{Q}_{\bf m}f(z)$ is convex, we also obtain $\widehat{Q}_{\bf
m}f(z)\leq(f(z)+a_{\mathbf{m}}z^{2})^{\ast\ast}$. The lower bound is obtained
by using Remark 2.9 with the choice $g=f$, $h=0$, $\bf m^{\prime}=\bf 0$ and
$\bf m^{\prime\prime}=\bf m$. This gives

$\widehat{Q}_{\bf m}f(z)\geq\widehat{Q}_{\bf 0}f(z)+\widehat{Q}_{\bf
m}0(z)=f^{\ast\ast}(z)+a_{\bf m}z^{2}$

since $\widehat{Q}_{\bf m}0(z)=a_{\bf m}z^{2}$ by Lemma 2.10, and
$\widehat{Q}_{\bf 0}f(z)=f^{\ast\ast}(z)$. ∎

######  Corollary 2.12.

If $f$ is convex, then $\widehat{Q}_{\bf m}f(z)=f(z)+a_{\bf m}z^{2}$.

######  Remark 2.13 (non-sharpness of lower bounds by decomposition).

If we apply Corollary 2.12 to the estimate in Remark 2.9 with $h$ convex and
${\bf m}^{\prime\prime}\neq{\bf 0}$, then the estimate gives an equality only
if $\widehat{Q}_{\bf m}f(z)=f(z)+a_{\bf m}z^{2}$.

###  2.3 Lower bound: optimization on nearest-neighbour clusters

Rather remarkably, one can explicitly compute $\widehat{Q}_{\bf m}f$ when
there is only one non-zero coefficient $m_{M}$ of $\bf m$ beside nearest
neighbours. The computation is obtained by optimizing on clusters of nearest
neighbours of length $M$. As a consequence one can obtain lower bound for a
general $\bf m$, which are in general not sharp but however useful.

For any given $\lambda\geq 0$, we set

$f_{\lambda}(z)=f(z)+\lambda z^{2}.$ (2.15)

In particular $f_{2m_{1}}(z)=f(z)+2m_{1}z^{2}$ describes the total energy due
to nearest-neighbour interactions. We first rewrite Corollary 2.12 in terms of
the effect of the convexity of this contribution.

######  Proposition 2.14 (convex nearest-neighbour interactions).

Let $f$ be such that $f_{2m_{1}}$ is convex. Then

$\widehat{Q}_{\bf m}f(z)=f(z)+a_{\mathbf{m}}z^{2}.$

More in general, for an arbitrary $f$ this equality holds at all $z$ such that
$f_{2m_{1}}(z)=f_{2m_{1}}^{**}(z)$.

###### Proof.

Applying Remark 2.9 with $g=f$, $h=0$ and $\bf m^{\prime}$ defined as
$m^{\prime}_{1}=m_{1}$ and $m^{\prime}_{n}=0$ if $n\geq 2$, for all $z$ such
that $f_{2m_{1}}(z)=f_{2m_{1}}^{**}(z)$ we have

$\widehat{Q}_{\bf m}f(z)\geq\widehat{Q}_{\bf m^{\prime}}f(z)+\widehat{Q}_{\bf
m^{\prime\prime}}0(z)=f^{**}_{2m_{1}}(z)+a_{\bf
m^{\prime\prime}}z^{2}=f_{2m_{1}}(z)+a_{\bf
m^{\prime\prime}}z^{2}=f(z)+a_{\mathbf{m}}z^{2},$

where we have used Remark 2.4, Lemma 2.10 and the convexity hypothesis. The
converse inequality holds by Proposition 2.11. ∎

Now, we can define nearest-neighbour cluster energies. More precisely, for any
integer $M\geq 2$ we define

$P^{M}\\!f(z)=\frac{1}{M}\min\Big{\\{}\sum_{j=1}^{M}f_{2m_{1}}(z_{j}):\sum_{j=1}^{M}z_{j}=Mz\Big{\\}}+2m_{M}M^{2}z^{2}.$
(2.16)

For completeness of notation, we also set $P^{1}\\!f(z)=f_{2m_{1}}(z)$.

Note that if $M\geq 2$ and $f_{2m_{1}}$ is convex then
$P^{M}\\!f(z)=f(z)+2m_{1}z^{2}+2m_{M}M^{2}z^{2}$.

######  Definition 2.15 (concentrated kernels).

Let $M\geq 1$. We say that a kernel $\bf m$ is concentrated at $M$ if
$m_{n}=0$ if $n\not\in\\{1,M\\}$.

######  Proposition 2.16 (relaxation with concentrated kernel).

If $\bf m$ is concentrated at $M$, then $\widehat{Q}_{\bf
m}f=(P^{M}\\!f)^{\ast\ast}$.

###### Proof.

Remark 2.4 proves the claim for $M=1$. Now, assume $M\geq 2$. We can use
formula (2.11) for the computation of $\widehat{Q}_{\bf m}f(z)$; in
particular, we may suppose that test functions satisfy $u_{i}=zi$ if $i\leq M$
and $i\geq k-M$. Let $u$ be a minimizer; using the notation in the proof of
Lemma 2.10 with $i_{\ell}=\lceil\frac{k-\ell}{M}\rceil$, we can write,

$\displaystyle\sum_{i=1}^{k}\Bigl{(}f(u_{i}-u_{i-1})+2m_{1}(u_{i}-u_{i-1})^{2}\Bigr{)}+2m_{M}\sum_{i=M}^{k}(u_{i}-u_{i-M})^{2}$
$\displaystyle=$
$\displaystyle\sum_{\ell=0}^{M-1}\sum_{i=1}^{Mi_{\ell}}{1\over
M}\Bigl{(}f(u_{i}-u_{i-1})+2m_{1}(u_{i}-u_{i-1})^{2}\Bigr{)}+2m_{M}\sum_{\ell=0}^{M-1}\sum_{i=1}^{i_{\ell}}(u_{\ell+iM}-u_{\ell+(i-1)M})^{2}+C_{z},$

where $C_{z}$ is a constant taking into account extra boundary interactions,
with $|C_{z}|\leq MC(1+z^{2})$ independent of $k$. We then estimate

$\displaystyle\hskip-28.45274pt\sum_{\ell=0}^{M-1}\sum_{i=1}^{Mi_{\ell}}{1\over
M}\Bigl{(}f(u_{i}-u_{i-1})+2m_{1}(u_{i}-u_{i-1})^{2}\Bigr{)}+2m_{M}\sum_{\ell=0}^{M-1}\sum_{i=1}^{i_{\ell}}(u_{\ell+iM}-u_{\ell+(i-1)M})^{2}$
$\displaystyle\geq$
$\displaystyle\sum_{\ell=0}^{M-1}\sum_{i=1}^{i_{\ell}}P^{M}f\Bigl{(}{u_{\ell+iM}-u_{\ell+(i-1)M}\over
M}\Bigr{)}\geq\sum_{\ell=0}^{M-1}\sum_{i=1}^{i_{\ell}}(P^{M}f)^{**}\Bigl{(}{u_{\ell+iM}-u_{\ell+(i-1)M}\over
M}\Bigr{)}$ $\displaystyle\geq$
$\displaystyle\sum_{\ell=0}^{M-1}i_{\ell}(P^{M}f)^{**}\Bigl{(}\frac{u_{\ell+i_{\ell}M}-u_{\ell}}{i_{\ell}M}\Bigr{)}=\sum_{\ell=0}^{M-1}i_{\ell}(P^{M}f)^{**}(z)$
$\displaystyle\geq$ $\displaystyle
M\Bigl{\lceil}\frac{k-M}{M}\Bigr{\rceil}(P^{M}f)^{**}(z)\,.$

Dividing by $k$ and taking the limit as $k\to+\infty$ we obtain the lower
bound.

To prove that the lower bound is sharp it suffices to choose a minimizer
$z_{1},\ldots,z_{M}$ for $P^{M}f(z)$, extend it by $M$-periodicity and define
a test function $u$ on $\\{0,\ldots,k\\}$ with $k=nM$ by setting $u_{0}=0$,
$u_{i}-u_{i-1}=z$ if $i\in\\{1,\ldots,M\\}\cup\\{k-M+1,\ldots,k\\}$, and
$u_{i}-u_{i-1}=z_{i}$ otherwise. Using this test function and letting
$k\to+\infty$, we obtain $\widehat{Q}_{\bf m}f\leq P^{M}f$. Since
$\widehat{Q}_{\bf m}f$ is convex, we finally get $\widehat{Q}_{\bf
m}f\leq(P^{M}f)^{**}$. ∎

######  Remark 2.17 (general concentrated interactions).

In the previous proposition we have considered quadratic interactions between
$M$th neighbours. Actually, it is not necessary to assume quadraticity or even
convexity of these interactions, and the same proof shows that

$\lim_{k\to+\infty}{1\over
k}\min\Bigl{\\{}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\sum_{i=M}^{k}g(u_{i}-u_{i-M}):u_{0}=0,u_{k}=kz\Bigr{\\}}=\psi^{**}(z),$
(2.17)

where $f,g\colon\mathbf{R}\to[0,+\infty)$ are such that $f$ is of quadratic
growth and $g$ satisfies a quadratic bound from above, and $\psi$ is defined
by

$\psi(z)=\frac{1}{M}\min\bigg{\\{}\sum_{j=1}^{M}f(z_{j}):\sum_{j=1}^{M}z_{j}=Mz\bigg{\\}}+g(z).$
(2.18)

######  Remark 2.18 (periodic recovery sequences and multiplicity of
minimizers).

Note that if $P^{M}f(z)=(P^{M}f)^{**}(z)$ and $\\{z_{i}\\}$ is a minimizer for
$P^{M}f(z)$ extended by $M$-periodicity, a function $u$ with $u_{0}=0$,
$u_{i}-u_{i-1}=z_{i}$ gives a recovery sequence for the $\Gamma$-limit of the
functionals (2.8) at $u(x)=zx$. Note that $u_{i}-zi$ is $M$-periodic.

We also observe that if $\\{z_{1},\ldots,z_{M}\\}$ is a minimizer, then any
permutation of its values gives a minimizer.

######  Proposition 2.19 (a lower bound for general $\bf m$).

Let $\bf m$ be any kernel; then for any $M$ the following estimate holds

$\displaystyle\widehat{Q}_{\bf
m}f(z)\geq(P^{M}\\!f)^{\ast\ast}(z)+2\sum_{\begin{subarray}{c}n\geq 2\\\ n\neq
M\end{subarray}}n^{2}m_{n}z^{2},$ (2.19)

and in particular we have $\displaystyle\widehat{Q}_{\bf m}f(z)\geq\sup_{M\geq
1}\biggl{(}(P^{M}\\!f)^{\ast\ast}(z)+2\sum_{\begin{subarray}{c}n\geq 2\\\
n\neq M\end{subarray}}n^{2}m_{n}z^{2}\biggr{)}$.

###### Proof.

Inequality (2.19) is obtained by using Remark 2.9 with ${\bf
m^{\prime}}=(m_{1},0,\dots,0,m_{M},0,\dots)$, Proposition 2.16, and the fact
that $\widehat{Q}_{\bf
m^{\prime\prime}}0(z)=2\sum_{n\not\in\\{1,M\\}}n^{2}m_{n}z^{2}$. If $M=1$, the
estimate is an immediate consequence of Remarks 2.4 and 2.9. ∎

###  2.4 Upper bound: optimization over periodic patterns

In order to give an upper bound for $\widehat{Q}_{\bf m}f$, it is of interest
to consider minimum problems on sets of $N$-periodic functions. We will see
that when the value $\widehat{Q}_{\bf m}f(z)$ is obtained by this periodic
minimization, which can be interpreted as a Cauchy-Born approach, it is
possible to deduce further structural properties of the relaxed functional.

For $N\in\mathbb{N}$ we define

$\widehat{R}^{N}_{\bf m}f(z)=\frac{1}{N}\inf\big{\\{}F^{\\#}(u;[0,N]):\
i\mapsto u_{i}-zi\hbox{ is }N\hbox{-periodic}\big{\\}},$ (2.20)

where

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

Note that each site $i\in\\{1,\dots,N\\}$ interacts with all $j\in\mathbb{Z}$.
Using periodic functions as test functions in the $\Gamma$-limit, we see that
$\widehat{R}^{N}_{\bf m}f(z)\geq\widehat{Q}_{\bf m}f(z)$ for all $N$, so that,
setting

$\widehat{R}_{\bf m}f(z)=\Big{(}\inf_{N}\widehat{R}^{N}_{\bf
m}f(z)\Big{)}^{\ast\ast},$

we obtain a bound for the $\bf m$-relaxation of $f$. More specifically, we can
write

$f(z)+a_{\bf m}z^{2}\geq\widehat{R}^{N}_{\bf m}f(z)\geq\widehat{R}_{\bf
m}f(z)\geq\widehat{Q}_{\bf m}f(z)\geq a_{\bf m}z^{2}$ (2.21)

where $N$ is arbitrary; the first estimate is obtained by taking $u_{i}=iz$.

An application of Remark 2.8 to boundary conditions allows one to show that in
(2.20) we can asymptotically neglect the interaction terms with sites outside
$[0,N]$. Then, we have the following proposition.

######  Proposition 2.20.

For all $z\in\mathbb{R}$ we have $\widehat{R}_{\bf
m}f(z)=\lim\limits_{N\to+\infty}\widehat{R}^{N}_{\bf m}f(z)=\widehat{Q}_{\bf
m}f(z)$.

Accordingly, the $\bf m$-relaxation can be alternatively defined as a limit of
minimum problems constructed on periodic functions.

######  Remark 2.21 (global periodic solutions).

Note that in general the equality in Proposition 2.20 is not attained at
finite $N$. However, in some cases the knowledge of $\widehat{R}^{N}_{\bf m}f$
for some finite $N$ is sufficient for the description of $\widehat{Q}_{\bf
m}f$. A notable case is that of nearest and next-to-nearest neighbor
interactions, for which a general formula for $\widehat{Q}_{\bf m}f$ can be
proven using this approach. In the notation above that formula simply reads
$\widehat{Q}_{\bf m}f=(\widehat{R}^{2}_{\bf m}f)^{**}$ [26, 77]. In
particular, if $f$ is a double-well energy with minimum value $0$ attained for
$z\in\\{-1,1\\}$ then in a neighbourhood of $0$ we have $\widehat{Q}_{\bf
m}f(z)=\widehat{R}^{2}_{\bf m}f(z)$; that is, the minimum for
$\widehat{Q}_{\bf m}f(z)$ is reached on functions with $u_{i}-zi$ 2-periodic,
up to an error due to the boundary conditions and vanishing as $k\to+\infty$.
In this sense, such problems have ‘global’ solutions and are therefore
solvable by the application of the GCB rule.

###  2.5 The $\bf m$-transform of $f$

In view of Proposition 2.11, in order to compare $\widehat{Q}_{\bf m}f$ with
$f$ we can subtract the quadratic term. In this way, the bounds in (2.14) are
rewritten as

$f^{\ast\ast}(z)\leq\widehat{Q}_{\bf
m}f(z)-a_{\mathbf{m}}z^{2}\leq\overline{f}(z).$ (2.22)

This suggests to interpret the function $\widehat{Q}_{\bf
m}f(z)-a_{\mathbf{m}}z^{2}$ as an independent operator acting on $f$. We then
give the following definition.

######  Definition 2.22 ($\mathbf{m}$-transform of $f$).

Let $\bf m$ be as in (2.4) and let $f\colon\mathbb{R}\to[0,+\infty)$ satisfy
(2.5) and (2.6). The ${\mathbf{m}}$-transform of $f$ is the function $Q_{\bf
m}f\colon\mathbb{R}\to[0,+\infty)$ defined as

$Q_{\bf m}f(z)=\widehat{Q}_{\bf m}f(z)-a_{\mathbf{m}}z^{2}.$ (2.23)

Given that, by (2.22),

$f^{\ast\ast}(z)\leq Q_{\bf m}f(z)\leq\overline{f}(z),$ (2.24)

the $\mathbf{m}$-transform of $f$ can be viewed as an $\mathbf{m}$-dependent
interpolation between $f$ and $f^{**}$.

We start the study of the $\mathbf{m}$-transform with the observation that at
$z$ fixed the construction of $Q_{\bf m}f(z)$ can be interpreted in a
variational sense as a minimization problem with a penalization term involving
a distance from the affine function $L_{z}$. This claim is justified by
Remarks 2.23 and 2.24 below.

######  Remark 2.23 (variational definition of $Q_{\bf m}f$).

Note that, when $u_{i}=iz$, then

$a_{\mathbf{m}}z^{2}=\lim_{k\to+\infty}{1\over
k}\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}.$

Hence, we have the equality

$\displaystyle
Q_{\mathbf{m}}f(z)=\displaystyle\\!\lim_{k\to+\infty}\frac{1}{k}\inf\bigg{\\{}\sum_{i=1}^{k}\\!f(u_{i}-u_{i-1})+\\!\sum_{i,j=0}^{k}\\!m_{|i-j|}\big{(}(u_{i}-u_{j})^{2}-(i-j)^{2}z^{2}\big{)}\\!:\\!u\in\mathcal{A}(k;z)\bigg{\\}}.$
(2.25)

######  Remark 2.24 (interpretation of the penalty term as a distance).

If $m_{1}>0$, then the last sum in (2.25) is a measure of the distance from
$u_{i}$ to the affine function $L_{z}(i)=iz$. To show this, we first note that
by Remark 2.8 we can restrict to test functions $u$ such that $u_{i}=iz$ for
$i\leq k^{\alpha+1}$ and $i\geq k-k^{\alpha+1}$ for some $\alpha\in(-1,0)$.

Now, for any $\ell\in\\{1,\dots,k\\}$ we consider the sum of the terms with
$|i-j|=\ell$, obtaining

$\displaystyle\sum_{|i-j|=\ell}\big{(}(u_{i}-u_{j})^{2}-(i-j)^{2}z^{2}\big{)}$
$\displaystyle=\sum_{|i-j|=\ell}\big{(}(u_{i}-iz)-(u_{j}-jz)\big{)}^{2}+2z\sum_{|i-j|=\ell}((u_{i}-iz)-(u_{j}-jz))(i-j)$
$\displaystyle=\sum_{|i-j|=\ell}\big{(}(u_{i}-iz)-(u_{j}-jz)\big{)}^{2}+4z\ell\sum_{i-j=\ell}((u_{i}-iz)-(u_{j}-jz))$
$\displaystyle=\sum_{|i-j|=\ell}\big{(}(u_{i}-iz)-(u_{j}-jz)\big{)}^{2}+4z\ell\sum_{r=0}^{\ell-1}\big{(}(u_{k_{r,\ell}}-k_{r,\ell}z)-(u_{r}-rz)\big{)},$

where $k_{r,\ell}=r+\ell\lfloor\frac{k-r}{\ell}\rfloor$.

If $\ell\leq k^{\alpha+1}$, then $r\leq k^{\alpha}k$ and
$k_{r,\ell}=r+\ell\lfloor\frac{k-r}{\ell}\rfloor\geq
k-\ell\geq(1-k^{\alpha})k$, so that
$u_{k_{r,\ell}}-u_{r}=\ell\lfloor\frac{k-r}{\ell}\rfloor=\ell\lfloor\frac{k}{\ell}\rfloor$,
and the last term in the sum vanishes, so that

$\displaystyle\frac{1}{k}\sum_{|i-j|\leq
k^{\alpha+1}}\big{(}(u_{i}-u_{j})^{2}-(i-j)^{2}z^{2}\big{)}=\frac{1}{k}\sum_{|i-j|\leq
k^{\alpha+1}}\big{(}(u_{i}-iz)-(u_{j}-jz)\big{)}^{2}.$

Now, we fix $\delta>0$. Recalling the decay condition (2.4) on $\bf m$, there
exists $\ell_{\delta}$ such that for $\ell>\ell_{\delta}$ we have
$m_{\ell}<\delta\ell^{-\beta}$. If $k$ is such that
$k^{\alpha+1}>\ell_{\delta}$, then

$\displaystyle\frac{1}{k}\sum_{|i-j|>k^{\alpha+1}}^{k}\\!m_{|i-j|}\big{(}(u_{i}-u_{j})^{2}-(i-j)^{2}z^{2}\big{)}\leq\frac{2}{k}\sum_{\ell>k^{\alpha+1}}^{k}\sum_{i=\ell}^{k}\\!m_{\ell}(u_{i}-u_{i-\ell})^{2}$
$\displaystyle\leq\frac{2}{k}\sum_{\ell>k^{\alpha+1}}\ell^{2}m_{\ell}\sum_{i=1}^{k}\\!(u_{i}-u_{i-1})^{2}\leq\frac{2\delta}{k}\sum_{\ell>k^{\alpha+1}}\ell^{2-\beta}\sum_{i=1}^{k}\\!(u_{i}-u_{i-1})^{2}\,.$

Note that in our computations we limit to $u$ satisfying
$\sum_{i=1}^{k}\\!(u_{i}-u_{i-1})^{2}\leq Ck$ by (2.6), so that this term is
negligible as $k\to+\infty$. Likewise, we obtain

$\displaystyle{2m_{1}\over k}\sum_{i=1}^{k}\\!(u_{i}-u_{i-1}-z)^{2}$
$\displaystyle\leq$ $\displaystyle{1\over
k}\sum_{\ell=1}^{k}\sum_{|i-j|=\ell}m_{|i-j|}\big{(}(u_{i}-u_{j})^{2}-(i-j)^{2}z^{2}\big{)}$
$\displaystyle\leq$ $\displaystyle{2\over
k}\Bigl{(}\sum_{\ell=1}^{\infty}\ell^{2}m_{\ell}\Bigr{)}\sum_{i=1}^{k}(u_{i}-u_{i-1}-z)^{2}\,.$

This double inequality shows that the quadratic part is equivalent to the
square of the $L^{2}$ norm of the derivative of $u-L_{z}$, where $u$ is
identified with the piecewise-affine function on $(0,1)$ with
$u^{\prime}=u_{i}-u_{i-1}$ on $({i-1\over k},{i\over k})$.

Some general algebraic properties deriving from the definition of $Q_{\bf m}f$
are the following.

######  Remark 2.25 (properties of $Q_{\bf m}$).

(i) $Q_{\bf m}(f+g)\geq Q_{s{\bf m}}f+Q_{(1-s){\bf m}}g$ for all $s\in(0,1)$;

(ii) if $g$ is convex $Q_{\bf m}(f+g)\geq(Q_{\mathbf{m}}f)+g$;

(iii) if $g$ is affine then $Q_{\bf m}(f+g)=(Q_{\bf m}f)+g$;

(iv) if $r\geq 0$, then $Q_{\bf m}(rf)(z)=rQ_{{\bf m}/r}f(z);$

(v) if $r\in\mathbb{R}$ and $(f\circ L_{r})(z)=f(rz)$ then $Q_{\bf m}(f\circ
L_{r})(z)=Q_{{\bf m}/r^{2}}f(rz);$

(vi) if $\lambda\in\mathbb{R}$ and we denote $(f\circ
T_{\lambda})(z)=f(z-\lambda)$ then $Q_{\bf m}(f\circ T_{\lambda})(z)=Q_{\bf
m}f(z-\lambda).$

Properties (i)-(v) follow directly from the definition of $Q_{\bf m}f$. We
give some details for the proof of (vi), since for this we have to modify the
boundary condition of the test functions, using (2.11) in Remark 2.8. For any
test function $u$ for $Q_{\bf m}(f\circ T_{\lambda})(z)$ we consider
$u^{\lambda}$ given by $u^{\lambda}_{i}=u_{i}-\lambda i$, which is a test
function for $Q_{\bf m}f(z-\lambda)$ obtaining

$\displaystyle\sum_{i=1}^{k}f(u_{i}-u_{i-1}-\lambda)+\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}-\sum_{i,j=0}^{k}m_{|i-j|}(i-j)^{2}z^{2}$
$\displaystyle\hskip
14.22636pt=\sum_{i=1}^{k}f(u^{\lambda}_{i}-u^{\lambda}_{i-1})+\sum_{i,j=0}^{k}m_{|i-j|}(u^{\lambda}_{i}-u^{\lambda}_{j})^{2}-\sum_{i,j=0}^{k}m_{|i-j|}(i-j)^{2}(z-\lambda)^{2}$
$\displaystyle\hskip
28.45274pt+2\lambda\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j}-z(i-j))(i-j).$

Then, (vi) holds if we show that

$\lim_{k\to+\infty}\frac{1}{k}\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})(i-j)=a_{\bf
m}z.$

Now, we note that in the sum $\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})(i-j)$ we
can regroup the terms with $|i-j|=\ell$ and obtain a telescopic sum whose
ending terms are in the boundary layer. Hence, since for each $\ell$ these
sums are exactly $\ell$, we have

$\displaystyle\lim_{k\to+\infty}\frac{1}{k}\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})(i-j)$
$\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\frac{2}{k}\sum_{\ell=1}^{k}\ell(m_{\ell}(u_{k}-u_{0})\ell)$
$\displaystyle=$
$\displaystyle\lim_{k\to+\infty}\frac{2}{k}kz\sum_{\ell=1}^{k}m_{\ell}\ell^{2}=a_{\bf
m}z,$

concluding the proof of (vi).

######  Definition 2.26 (stability under $\bf m$-transform).

We say that $z$ is a point of $\bf m$-stability for $f$ if $Q_{\bf
m}f(z)=f(z)$. If this equality holds for all $z$, we say that $f$ is ${\bf
m}$-stable.

######  Remark 2.27 (global properties of points of stability).

Let $z$ be a point of $\bf m$-stability for $f$. Then, the value of
$\widehat{Q}_{\bf m}f(z)$ is realized by choosing the affine function
$u\in\mathcal{A}(k;z)$ given by $u_{i}=iz$ in each minimum problem in
Definition 2.3.

We recall that $f_{\lambda}(z)=f(z)+\lambda z^{2}$ as in (2.15).

######  Proposition 2.28 ($\bf m$-stability and convexity).

(i) if $f$ is $\bf m$-stable then $f_{a_{\bf m}}$ is convex;

(ii) if $f_{2m_{1}}$ is convex then $f$ is $\bf m$-stable.

###### Proof.

Claim (i) follows from the definition of $\bf m$-stability since $f_{a_{\bf
m}}=\widehat{Q}_{\bf m}f$. Claim (ii) is given by Proposition 2.14. ∎

######  Remark 2.29 (‘moderately’ non-convex functions are $\bf m$-stable).

The proposition above implies that if $f$ is ‘moderately non-convex’ then it
is also $\bf m$-stable. This is valid in particular if $f$ is twice
differentiable and

$\inf_{z}f^{\prime\prime}(z)>-4m_{1}.$ (2.26)

Figure 2: the function $Q_{\bf m}f$ in Remark 2.30 with $f(z)=(1-z^{2})^{2}$
for different values of $m_{1}<1$.

######  Remark 2.30 (nearest-neighbour interactions).

By Remark 2.4 we get that

  1. (i)

if $m_{n}=0$ for any $n\geq 1$, then $Q_{\bf m}f(z)=\widehat{Q}_{\bf
m}f(z)=f^{\ast\ast}(z);$

  2. (ii)

if $m_{n}=0$ for any $n\geq 2$, then $Q_{\bf
m}f(z)=(f(z)+2m_{1}z^{2})^{**}-2m_{1}z^{2}.$

In the second case, we note that in general if $m_{1}\neq 0$ both inequalities
in (2.24) may be strict for some values of $z$. For example, if
$f(z)=(1-z^{2})^{2}$ and $m_{1}\leq 1$, then

$Q_{\bf m}f(z)=\begin{cases}(1-z^{2})^{2}&\hbox{\rm if }\
z\leq-\sqrt{1-m_{1}}\\\ m_{1}(2-m_{1})-2m_{1}z^{2}&\hbox{\rm if }\
|z|\leq\sqrt{1-m_{1}}\\\ (1-z^{2})^{2}&\hbox{\rm if }\
z\geq\sqrt{1-m_{1}},\end{cases}$

and both inequalities are strict for $|z|<\sqrt{1-m_{1}}$ (see Fig. 2).
Conversely, if $m_{1}\geq 1$ then $Q_{\bf m}f(z)=f(z)$ for any $z$; in
particular in this case $f$ is $\bf m$-stable (but not convex).

######  Remark 2.31 (regularity properties).

From equality (2.23) we deduce that for any $\bf m$ the operator $Q_{\bf m}$
has the same regularity properties of $\widehat{Q}_{\bf m}$; that is, $Q_{\bf
m}f$ has the regularity properties of a convex function. In particular,
$Q_{\bf m}f$ is locally Lipschitz, which is then a necessary condition for $f$
to be $\bf m$-stable. Note that by (2.22) the convexity of $f$ is a sufficient
condition for the stability with respect to any $\bf m$.

######  Proposition 2.32.

Let $Q_{\bf m}^{0}f=f$ and define iteratively $Q_{\bf m}^{n}f=Q_{\bf m}(Q_{\bf
m}^{n-1}f)$. Then the sequence $Q_{\bf m}^{n}f$ is non-increasing and its
limit $Q_{\bf m}^{\infty}f$ is $\bf m$-stable.

###### Proof.

The sequence is non-increasing by (2.22). Moreover $Q_{\bf m}^{n}f\geq f^{**}$
for all $n$. Since the functions $Q_{\bf m}^{n}f$ are equi-Lipschitz
continuous by Remark 2.31, they converge uniformly on compact sets to their
limit $Q_{\bf m}^{\infty}f$ by Ascoli-Arzelà’s Theorem. Since $Q_{\bf m}$ is
continuous with respect to the uniformly convergence on compact sets, we have
$Q_{\bf m}^{\infty}f=\lim_{n}Q_{\bf m}^{n}f=Q_{\bf m}(\lim_{n}Q_{\bf
m}^{n-1}f)=Q_{\bf m}(Q_{\bf m}^{\infty}f)$. ∎

The following proposition states that for non-trivial kernel concentrated at
$M\geq 2$ stable functions are only $f$ such that $f_{2m_{1}}$ is convex,
which is a trivial condition implying stability by Proposition 2.28(ii).
Moreover, iteration of the $\bf m$ transform gives a strictly decreasing
sequence.

######  Proposition 2.33.

Let $\bf m$ be a non-trivial kernel concentrated at $M\geq 2$; that is, with
$m_{M}\neq 0$. In this case:

(i) $f$ is ${\bf m}$-stable if and only if $f_{2m_{1}}$ is convex;

(ii) if $f_{2m_{1}}$ is not convex then for any $n$, there exists $z$ such
that $Q_{\bf m}^{n}f(z)>Q_{\bf m}^{n+1}f(z)$;

(iii) $Q_{\bf m}^{\infty}f(z)=f_{2m_{1}}^{\ast\ast}(z)-2m_{1}z^{2}$.

###### Proof.

(i) By Proposition 2.28 we only have to prove that the convexity of
$f_{2m_{1}}$ is necessary for the ${\bf m}$-stability of $f$. We then suppose
that $f$ is $\bf m$-stable and $f_{2m_{1}}$ is not convex, and show that there
exists $\overline{z}$ such that $f(\overline{z})>Q_{\bf m}f(\overline{z})$,
contradicting the ${\bf m}$-stability of $f$.

From Proposition 2.14 we have that $\widehat{Q}_{\bf m}f(z)=f_{a_{\bf m}}(z)$
for all $z$ such that $f_{2m_{1}}(z)=f_{2m_{1}}^{\ast\ast}(z)$. We consider a
maximal interval where $f_{2m_{1}}>f_{2m_{1}}^{\ast\ast}$. By the growth
conditions on $f$ and its continuity (since we suppose that it is $\bf
m$-stable) this interval is a bounded open interval $(S_{0},S_{M})$, and we
have $\widehat{Q}_{\bf m}f(S_{0})=f_{a_{\bf m}}(S_{0})$ and $\widehat{Q}_{\bf
m}f(S_{M})=f_{a_{\bf m}}(S_{M})$.

Note that, upon setting

$r(z)=f_{2m_{1}}(S_{0})+\frac{f_{2m_{1}}(S_{M})-f_{2m_{1}}(S_{0})}{S_{M}-S_{0}}(z-S_{0}),$

for $z\in[S_{0},S_{M}]$ we have

$\frac{1}{M}\min\Big{\\{}\sum_{j=1}^{M}f_{2m_{1}}(z_{j}):\sum_{j=1}^{M}z_{j}=Mz\Big{\\}}\geq
f_{2m_{1}}^{\ast\ast}(z)=r(z),$

with equality if and only if minimal $z_{j}$ belong to $\\{S_{0},S_{M}\\}$ for
all $j$, which implies that $z\in\\{S_{h}:h\in\\{0,\dots,M\\}\\}$, where

$S_{h}=S_{0}+h\frac{S_{M}-S_{0}}{M}.$

We then have

$P^{M}\\!f(S_{h})=r(S_{h})+2m_{M}M^{2}S_{h}^{2}.$

Since

$P^{M}\\!f(z)\geq(P^{M}\\!f)^{\ast\ast}(z)\geq
f_{2m_{1}}^{**}(z)+2m_{M}M^{2}z^{2}=r(z)+2m_{M}M^{2}z^{2},$

and $Q_{\bf m}f=(P^{M}\\!f)^{\ast\ast}$ by Proposition 2.16, in particular we
have

$\widehat{Q}_{\bf
m}f(S_{h})=(P^{M}\\!f)^{\ast\ast}(S_{h})=P^{M}\\!f(S_{h})=r(S_{h})+2m_{M}M^{2}S_{h}^{2},$

from which we get

$Q_{\bf m}f(S_{h})=r(S_{h})-2m_{1}S_{h}^{2}.$

If $h\in\\{1,\ldots,M-1\\}$ we have

$f(S_{h})+2m_{1}S_{h}^{2}=f_{2m_{1}}(S_{h})>f_{2m_{1}}^{\ast\ast}(S_{h})=r(S_{h})$

which implies

$f(S_{h})>r(S_{h})-2m_{1}S_{h}^{2}=Q_{\bf m}f(S_{h}),$

which contradicts the stability of $f$.

Note that indeed

$\widehat{Q}_{\bf m}f(z)>r(z)+2m_{M}M^{2}z^{2}\hbox{ if }z\in(S_{h},S_{h+1}).$
(2.27)

To check this observe that, since $\widehat{Q}_{\bf
m}f(z)=(P^{M}\\!f)^{\ast\ast}(z)$, there exist $z_{1},z_{2}\in[S_{h},S_{h+1}]$
and $t\in[0,1]$ such that $z=tz_{1}+(1-t)z_{2}$ and

$\displaystyle\widehat{Q}_{\bf m}f(z)$ $\displaystyle=$ $\displaystyle
t\,P^{M}\\!f(z_{1})+(1-t)\,P^{M}\\!f(z_{2})$ $\displaystyle\geq$
$\displaystyle
t\,r(z_{1})+(1-t)r(z_{2})+t\,2m_{M}M^{2}z_{1}^{2}+(1-t)\,2m_{M}M^{2}z_{2}^{2},$

and we get (2.27) unless $z_{1}=z_{2}=z$. The latter case is ruled out, as we
would have $\widehat{Q}_{\bf m}f(z)=t\,P^{M}\\!f(z)$; that is,
$z\in\\{S_{h},S_{h+1}\\}$.

(ii) We fix $h\in\\{0,\ldots,M-1\\}$ and consider any interval
$(S_{h},S_{h+1})$ as defined in the proof of claim (i) above. If
$z\in(S_{h},S_{h+1})$, by (2.27) we have

$\displaystyle Q_{\bf m}f(z)+2m_{1}z^{2}$ $\displaystyle=$
$\displaystyle\widehat{Q}_{\bf m}f(z)-2m_{M}M^{2}z^{2}$ (2.28)
$\displaystyle>$ $\displaystyle
r(z)=f^{**}_{2m_{1}}(z)=(f(z)+2m_{1}z^{2})^{**}$ $\displaystyle\geq$
$\displaystyle(Q_{\bf m}f(z)+2m_{1}z^{2})^{**}\,.$

Hence, each $(S_{h},S_{h+1})$ is an interval of non-convexity of $Q_{\bf
m}f(z)+2m_{1}z^{2}$ and we may repeat the argument of the proof of claim (i)
to show that $Q_{\bf m}^{f}(z)>Q^{2}_{\bf m}f(z)$ in $M-1$ equi-spaced points
in $(S_{h},S_{h+1})$. The argument can be then used iteratively.

(iii) If $f_{2m_{1}}$ is convex the claim is trivial. Suppose otherwise. By
(2.28) we have

$(f(z)+2m_{1}z^{2})^{\ast\ast}=(Q_{\bf m}f(z)+2m_{1}z^{2})^{\ast\ast}=r(z)$

for $z\in[S_{0},S_{M}]$, and, iterating the argument also

$r(z)=(f(z)+2m_{1}z^{2})^{\ast\ast}=(Q^{n}_{\bf
m}f(z)+2m_{1}z^{2})^{\ast\ast},$

for $z\in[S_{0},S_{M}]$ and $n\geq 1$. As in the proof of claim (ii) above we
have

$Q^{n}_{\bf m}f(z)+2m_{1}z^{2}=r(z)\ \ \hbox{\rm if }\
z=S_{0}+\frac{k}{M^{n}}(S_{M}-S_{0})$

for all $k\leq M^{n}$, and then

$Q^{\infty}_{\bf m}f(z)+2m_{1}z^{2}=r(z)\ \ \hbox{\rm if }\
z=S_{0}+\frac{k}{M^{n}}(S_{M}-S_{0})$

for some $n$ and for all $k\leq M^{n}$. By density, the equality then extends
to all $z\in[S_{0},S_{M}]$. Arguing in this way in each interval of non-
convexity of $f_{2m_{1}}$ we conclude. ∎

######  Corollary 2.34.

The same claims of the previous proposition hold if $\bf m$ is such that
$M\geq 2$ exists such that $m_{n}=0$ if $n\not\in\\{1,M\mathbb{N}\\}$.

###### Proof.

The proof follows by noting that

$\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}\leq
2m_{1}\sum_{i=1}^{k}(u_{i}-u_{i-1})^{2}+\tilde{m}_{M}\sum_{{i,j=0,\
|i-j|=M}}^{k}(u_{i}-u_{j})^{2},$

where $\tilde{m}_{M}=\sum_{j=1}^{\infty}j^{2}m_{jM}$, and arguing by
comparison, applying the previous proposition to the kernel $\bf\tilde{m}$
where $\tilde{m}_{1}=m_{1}$ and $\tilde{m}_{n}=0$ if $n\not\in\\{1,M\\}$ ∎

Proposition 2.33 does not hold for ‘incommensurate’ kernels; i.e., such that
there are interactions not multiple of a common $M>1$. In the example below we
treat a paradigmatic case.

######  Example 2.35 (incommensurability and non-trivial $\bf m$-stability).

Let $\bf m$ be such that $m_{n}\neq 0$ if and only if $n\in\\{2,3\\}$.

Let $k\in\mathbb{N}$ and consider the quadratic function

$G(z_{1},\dots,z_{k})=2m_{2}\sum_{i=1}^{k}(z_{i}+z_{i+1})^{2}+2m_{3}\sum_{i=1}^{k}(z_{i}+z_{i+1}+z_{i+2})^{2}$

defined on $k$ periodic sequences $\\{z_{i}\\}_{i\in\mathbb{Z}}$. Noting that

$2m_{2}(z_{i+1}+z_{i+2})^{2}+2m_{3}(z_{i}+z_{i+1}+z_{i+2})^{2}\geq\min\\{m_{2},m_{3}\\}z_{i}^{2},$

we obtain that

$H_{c}(z_{1},\dots,z_{k})=G(z_{1},\dots,z_{k})-2c\sum_{i=1}^{k}z_{i}^{2}\geq
0$

for any $c\in(0,\frac{\min\\{m_{2},m_{3}\\}}{2})$. Hence, since $H_{c}$ is a
symmetric non-negative $2$-homogeneous polynomial of degree $2$, it is convex.
Then

$\frac{1}{k}\min\Big{\\{}H_{c}(z_{1},\dots,z_{k}):\sum_{i=1}^{k}z_{i}=kz\Big{\\}}=8m_{2}z^{2}+18m_{3}z^{2}-2cz^{2}.$
(2.29)

Now, we suppose that $f(z)+2cz^{2}$ is convex for some
$c\in(0,\frac{\min\\{m_{2},m_{3}\\}}{2})$. Then for any $k$

$\displaystyle\frac{1}{k}\min\Big{\\{}\sum_{i=1}^{k}f(z_{i})+G(z_{1},\ldots,z_{k}):\sum_{i=1}^{k}z_{i}=kz\Big{\\}}$
$\displaystyle=\frac{1}{k}\min\Big{\\{}\sum_{i=1}^{k}(f(z_{i})+2cz_{i}^{2})+H_{c}(z_{1},\dots,z_{k}):\sum_{i=1}^{k}z_{i}=kz\Big{\\}}$
$\displaystyle=f(z)+2cz^{2}+8m_{2}z^{2}+18m_{3}z^{2}-2cz^{2}=f(z)+a_{\bf
m}z^{2}.$

Note that by Remark 2.8 in the definition of $Q_{\bf m}f$ we can take
$u_{i}-u_{i-1}=z$ for $i=1,2,3$ and $i=k,k-1,k-2$, and consider the function
$u_{i}-u_{i-1}$ extended by $k$-periodicity. Indeed, the minimum problem in
(2.11) is estimated from below by the periodic problem up to a term
$O(\frac{1}{k})$. Hence, $\widehat{Q}_{\bf m}f(z)\geq f(z)+a_{\bf m}z^{2}$,
and $f$ is $\bf m$-stable, since the other inequality is true by (2.22). Note
that this implies that in general the condition $f_{2m_{1}}$ convex is not
necessary for the $\bf m$-stability of $f$, since in this case it suffices
that $f_{2m_{1}+2c}$ be convex.

######  Definition 2.36 (effective strength of nearest-neighbour interaction).

Let

$G_{k}(z_{1},\dots,z_{k})=\sum_{n=1}^{+\infty}2m_{n}\sum_{i=1}^{k}\Big{(}\sum_{j=i}^{i+n}z_{j}\Big{)}^{2}$

defined on $k$-periodic sequences $\\{z_{j}\\}_{j\in\mathbb{Z}}$. We define
the effective strength of nearest-neighbour interaction $m_{1}^{\rm eff}$ for
$\bf m$ as the supremum of all constant $c$ such that

$G_{k}(z_{1},\dots,z_{k})\geq 2c\sum_{i=1}^{k}z_{i}^{2}$

for all $k\in\mathbb{N}$ and for all $\\{z_{j}\\}_{j\in\mathbb{Z}}$.

######  Remark 2.37 (lower bound with $m_{1}^{\rm eff}$).

Note that $m_{1}^{\rm eff}\geq m_{1}$. The two values coincide if and only if
$\bf m$ satisfies the generalized concentration hypothesis of Corollary 2.34.
Repeating the argument in Example 2.35, we obtain that a sufficient condition
for the $\bf m$ stability of a function $f$ is the convexity of
$f_{2m_{1}^{\rm eff}}$. Moreover, we have the estimate

$Q_{\bf m}f(z)\geq f^{\ast\ast}_{2m_{1}^{\rm eff}}(z)-2m_{1}^{\rm eff}z^{2}.$
(2.30)

This can be achieved again following Example 2.35, estimating $f_{2m_{1}^{\rm
eff}}$ with its convex envelope.

####  2.5.1 Interpolation by parameterized kernels

The penalization kernel $\bf m$ may depend on a scale parameter $\sigma$,
measuring either the range or the scale of incompatibility. Of particular
interest are kernels that tend to $0$ as $\sigma\to+\infty$, while they loose
their summability as $\sigma\to 0$. Kernels $\bf m$ with such a dependence on
a scale parameter $\sigma$ can be used to interpolate between the extreme
bounds in (2.24).

A suitable class of such kernels is constructed as follows. Let
$m\colon[0,+\infty)\to[0,+\infty)$ be a continuous non-increasing function
such that $m$ is strictly positive up to some $\overline{x}>0$, and

$\int_{0}^{+\infty}x^{2}m(x)\,dx<+\infty.$

These conditions are satisfied by $m(x)=e^{-x}$; in this case, by setting
$m_{n}=m^{\sigma}_{n}=m(\sigma n)$, we obtain the exponential kernels
$m_{n}=e^{-\sigma n}$ studied in more detail in Section 5.

The following proposition holds.

######  Proposition 2.38.

Let $m\colon[0,+\infty)\to[0,+\infty)$ be as above, and for all $\sigma>0$
consider the kernel ${\bf m}^{\sigma}=\\{m(\sigma n)\\}_{n}$. Let
$f\colon\mathbb{R}\to[0,+\infty)$ satisfy growth assumptions (2.5) and
$\eqref{crescitasotto}$. Then,

$\lim_{\sigma\to+\infty}Q_{{\bf m}^{\sigma}}f(z)=f^{\ast\ast}(z)\ \ \hbox{\rm
and }\ \ \lim_{\sigma\to 0^{+}}Q_{{\bf m}^{\sigma}}f(z)=\overline{f}(z).$
(2.31)

###### Proof.

Setting $\displaystyle a_{{\mathbf{m}}^{\sigma}}=2\sum_{n=1}^{+\infty}m(\sigma
n)n^{2}$, we obtain that

$a_{{\mathbf{m}}^{\sigma}}\leq
2\int_{0}^{+\infty}m(\sigma(x+1))(x+1)^{2}\,dx\leq\frac{2}{\sigma}\int_{0}^{+\infty}m(y)y^{2}\,dy=\frac{C}{\sigma}\to
0\ \ \hbox{\rm as }\ \sigma\to+\infty.$

Then, the first equality in (2.31) follows directly from Proposition 2.11, as
we have

$f^{\ast\ast}(z)\leq Q_{{\bf
m}_{\sigma}}f(z)\leq\psi_{\sigma}^{\ast\ast}(z)-a_{{\mathbf{m}}^{\sigma}}z^{2}\leq
f(z)$

where $\psi_{\sigma}(z)=f(z)+a_{{\mathbf{m}}^{\sigma}}z^{2}$. Since
$a_{{\mathbf{m}}^{\sigma}}$ decreases to $0$ as $\sigma\to+\infty$, then there
exists a convex function $\psi$ such that

$f^{\ast\ast}(z)\leq\psi(z)=\lim_{\sigma\to+\infty}\psi^{\ast\ast}_{\sigma}(z)=\lim_{\sigma\to+\infty}\Big{(}\psi^{\ast\ast}_{\sigma}(z)-a_{{\mathbf{m}}^{\sigma}}z^{2}\Big{)}\leq
f(z).$

Hence, $\psi(z)=f^{\ast\ast}(z)=\lim\limits_{\sigma\to+\infty}Q_{{\bf
m}^{\sigma}}f(z)$.

Now, we prove the second limit in (2.31). Since (2.14) holds, it is sufficient
to show that $\lim\limits_{\sigma\to 0}Q_{{\bf
m}^{\sigma}}f(z)\geq\overline{f}(z)$. Up to scaling, we can suppose that
$\overline{x}=1$ and $m(1)=1$. Since $m$ is non-increasing, it is sufficient
to prove the desired equality for $m=\chi_{[0,1]}$. The function
$a_{{\mathbf{m}}^{\sigma}}$ is non-increasing with respect to $\sigma$; hence,
for any $z$ there exists the limit of $Q_{{\bf m}^{\sigma}}f(z)$ as $\sigma\to
0$. Let $\sigma_{k}\to 0$ as $k\to+\infty$ and let $u^{k}$ be a minimizer in
$[0,k]$ for the minimum problem in the formula of $Q_{{\bf m}^{\sigma_{k}}}$
in Remark 2.23; that is, $u^{k}$ is an admissible minimizer for
$G^{\sigma_{k}}(u)$ defined by

$G^{\sigma_{k}}(u)=\sum_{i=1}^{k}\\!f(u_{i}-u_{i-1})+\\!\sum_{i,j=0}^{k}\\!m(\sigma_{k}|i-j|)\big{(}(u_{i}-u_{j})^{2}-(i-j)^{2}z^{2}\big{)}.$

Let $N_{k}=\lfloor\frac{1}{\sigma_{k}}\rfloor$. By Remark 2.8, we can assume
that the test functions $u$, defined for $i\in\mathbb{Z}$, satisfy $u_{i}=iz$
for $i\leq N_{k}$ and $i\geq k-N_{k}$. Reasoning as in Remark 2.24, for any
$\ell=1,\dots,N_{k}$ and $r=1,\dots,\ell$ we have

$\sum_{i=1}^{\lfloor
k/l\rfloor}\Big{(}(u^{k}_{i\ell+r}-u^{k}_{(i-1)\ell+r})^{2}-z^{2}\ell^{2}\Big{)}=\sum_{i=1}^{\lfloor
k/l\rfloor}\Big{(}u^{k}_{i\ell+r}-u^{k}_{(i-1)\ell+r}-z\ell\Big{)}^{2}\geq 0.$

We now define a discrete function $w^{k}$ by setting
$w^{k}_{i}=u^{k}_{i}-u^{k}_{i-1}-z$. For any $1\leq n\leq N_{k}$, we can write

$w^{k}_{i}=\sum_{j=i}^{i+n}w^{k}_{j}-\sum_{j=i+1}^{i+n}w^{k}_{j},$

so that, by summing over $n$

$\displaystyle N_{k}\frac{1}{k}\sum_{i=1}^{k}(w^{k}_{i})^{2}$
$\displaystyle\leq$
$\displaystyle\frac{2}{k}\sum_{n=1}^{k}\sum_{i=1}^{k}\big{(}(u^{k}_{i+n}-u^{k}_{i-1}-(n+1)z)^{2}+(u^{k}_{i+n}-u^{k}_{i}-nz)^{2}\big{)}$
$\displaystyle\leq$ $\displaystyle\frac{4}{k}G^{\sigma_{k}}(u^{k}),$

recalling that $m(\sigma_{k}n)=1$ if $\sigma_{k}n\leq 1$ and $0$ otherwise.
Now, let $\tilde{u}^{k}$ denote the piecewise-affine extension to $[0,1]$ of
the discrete function defined by
$\tilde{u}^{k}(\frac{i}{k})=\frac{1}{k}u^{k}_{i}$, so that
$(\tilde{u}^{k})^{\prime}-z=w^{k}_{i}$ in each interval
$(\frac{i-1}{k},\frac{i}{k})$. Since $\frac{4}{k}G^{\sigma_{k}}(u^{k})$ is
equibounded, we obtain that

$\int_{0}^{1}((\tilde{u}^{k})^{\prime}-z)^{2}\,dt=\frac{1}{k}\sum_{i=1}^{k}(w^{k}_{i})^{2}\leq\frac{C}{N_{k}}\to
0\ \ \hbox{\rm as }\ \ k\to+\infty.$

Hence, $\tilde{u}^{k}\to zx$ in $H^{1}(0,1)$. We get

$\displaystyle\lim_{k\to+\infty}Q_{{\bf m}^{\sigma_{k}}}f(z)$
$\displaystyle\geq$
$\displaystyle\liminf_{k\to+\infty}\frac{1}{k}\sum_{i=1}^{k}f(u^{k}_{i}-u^{k}_{i-1})\geq\liminf_{k\to+\infty}\frac{1}{k}\sum_{i=1}^{k}\overline{f}(u^{k}_{i}-u^{k}_{i-1})$
$\displaystyle=$
$\displaystyle\liminf_{k\to+\infty}\frac{1}{k}\sum_{i=1}^{k}\overline{f}(w_{i}^{k}+z)=\liminf_{k\to+\infty}\int_{0}^{1}\overline{f}(\tilde{u}_{k}^{\prime})\,dt\geq\overline{f}(z),$

by the lower-semicontinuity of the functional
$w\mapsto\int_{0}^{1}\overline{f}(w^{\prime})\,dt$ with respect to the strong
$H^{1}$-convergence. ∎

######  Remark 2.39 (‘singular’ kernels depending on $\sigma$).

If $\bf m$ is a kernel concentrated at some $M\geq 2$, with $m_{M}\neq 0$, we
consider a different type of parameter dependence. In this case, we can set
${\bf m}^{\sigma}=\\{m_{n}^{\sigma}\\}_{n}=\\{\phi(\sigma)m_{n}\\}_{n}$, with
$\phi$ decreasing and such that $\lim_{\sigma\to 0^{+}}\phi(\sigma)=+\infty$
and $\lim_{\sigma\to+\infty}\phi(\sigma)=0$; for instance, we may consider

$m_{n}^{\sigma}=\frac{1}{\sigma}m_{n}.$

Since $m_{n}$ is not decreasing, this case cannot be treated directly by
applying the result of the above proposition. However, the same argument as in
the proof of Proposition 2.38 can be used as well giving

$\lim_{\sigma\to+\infty}Q_{{\bf m}^{\sigma}}f(z)=f^{\ast\ast}(z).$ (2.32)

As for the limit as $\sigma\to 0^{+}$, we can follow the proof up to the
definition of $w_{i}^{k}$, obtaining

$\displaystyle\frac{1}{k}\sum_{i=1}^{k}(w^{k}_{i})^{2}$ $\displaystyle\leq$
$\displaystyle\frac{2}{k}\sum_{n=1,M}^{k}\sum_{i=1}^{k}\big{(}(u^{k}_{i+n}-u^{k}_{i-1}-(n+1)z)^{2}+(u^{k}_{i+n}-u^{k}_{i}-nz)^{2}\big{)}$
$\displaystyle\leq$
$\displaystyle\max\Big{\\{}\frac{\sigma_{k}}{m_{1}},\frac{\sigma_{k}}{m_{M}}\Big{\\}}\frac{4}{k}G^{\sigma_{k}}(u^{k}),$

and we can conclude exactly as above, proving that

$\lim_{\sigma\to 0^{+}}Q_{{\bf m}^{\sigma}}f(z)=\overline{f}(z).$ (2.33)

Note that if $m_{1}=0$ equality (2.33) in general does not hold (while (2.33)
is always valid). As an example, we refer to Remark 4.6.

In general, for $\sigma$-dependent kernels equalities (2.31) are achieved only
asymptotically. However, in some cases they are reached for some finite values
of $\sigma>0$. To highlight this fact, we give the following definition.

######  Definition 2.40 (critical transition value of $\sigma$).

Let $f\colon\mathbb{R}\to[0,+\infty)$ be a continuous function satisfying
growth assumptions (2.5) and $\eqref{crescitasotto}$. Let $\\{\bf
m^{\sigma}\\}_{\sigma>0}$ be a family of parameterized kernels. We define the
critical transition value of $\sigma$ by setting

$\sigma_{c}=\sigma_{c}(f)=\sup\\{\sigma>0:Q_{\bf m^{\tau}}f=f\ \hbox{\rm for
all }\tau<\sigma\\}.$

We set $\sigma_{c}=0$ if $Q_{\bf m^{\sigma}}f<f$ for any $\sigma$.

######  Example 2.41 (existence of positive critical transition values).

Let $f(z)=(1-z^{2})^{2}$ and let $\bf m$ be concentrated at some $M\geq 2$, as
in Remark 2.39. We set $m_{1}^{\sigma}=\frac{m_{1}}{\sigma}$ and
$m_{M}^{\sigma}=\frac{m_{M}}{\sigma}$. By Remark 2.29, we have that
$\sigma_{c}=m_{1}$ is the critical transition value. Note that, conversely,
for any $\sigma$ we have that $Q_{{\bf m}^{\sigma}}f(z)>f^{\ast\ast}(z)$ for
some values of $z$, hence the limit in (2.32) is only reached at $+\infty$.

In the sequel, an important role will be played by the two special families of
kernels depending on $\sigma$, exponential and concentrated at some $M$,
introduced above and already illustrated in Fig. 1, for which the computation
of $Q_{\bf m}f$ can be performed analytically. In both cases, we will be able
trace explicitly the role of the scale parameter $\sigma$ characterizing the
range/strength of the penalization kernel.

##  3 Description of minimizers by a phase function

In this section, we focus on the important case of ‘generalized double well’
potentials when the domain of a function $f$ can be subdivided in two sub-
domains of convexity, which in what follows we refer to as $A$ and
$\mathbb{R}\setminus A$. We call such potentials bi-convex and refer to the
two convex branches of $f$ as phases. Given that some microstructures in such
models can be interpreted as a ‘phase mixtures’, it will be convenient to
introduce the ‘volume-fraction parameter’ $\theta$ representing the percentage
of indices $i$ such that $u_{i}-u_{i-1}$ is in the set $A$. The computation of
minima with prescribed volume fraction $\theta$ gives an upper bound for
$\widehat{Q}_{\bf m}f$.

With the introduction of $\theta$, one can proceed in two steps. The first
step involves the computation of the function $\widehat{Q}_{\bf m}f(\theta,z)$
which is obtained by a constrained minimization with prescribed $\theta$. Then
the function $\widehat{Q}_{\bf m}f$ can be obtained by a one-dimensional
optimization of $\widehat{Q}_{\bf m}f(\theta,z)$ over $\theta$, which also
defines the phase function $\theta(z)$ such that $\widehat{Q}_{\bf
m}f(\theta(z),z)=\widehat{Q}_{\bf m}f(z)$. In the problems of interest the
function $\theta(z)$ will have a complex ‘staircase’ structure reflecting the
existence of the locking states at the values of $\theta$ that are stable
under variation of $z$.

######  Remark 3.1 (constrained minimization and the structure of the phase
function).

To understand the role of the constrained minimization producing the function
$\widehat{Q}_{\bf m}f(\theta,z)$ and to reveal the link between the shape of
the phase function $\theta(z)$ and the structure of the relaxed energy
$\widehat{Q}_{\bf m}f$, it will be instructive to consider first the case when
only $M$-neighbour interactions are taken into account. We recall that in this
case there exists $M\geq 2$ such that $m_{M}\neq 0$ and $m_{n}=0$ for any
$n\neq M$.

Proposition 2.16 gives a formula for $\widehat{Q}_{\bf m}f(z)$. If $f$ is bi-
convex, we can subdivide its computation by introducing a dependence on the
fraction $\theta$ of $z_{i}=u_{i}-u_{i-1}$ belonging to the convexity region
$A$. More precisely, for any $n=0,\dots,M$ we can first compute the minimum at
a fixed fraction $\theta_{n}=\frac{n}{M}$ of $z_{i}$ belonging to $A$. Using
the convexity, such minimum problems reduce to the computation of

$\left.\begin{array}[]{ll}P^{M,n}(z)=&\displaystyle\min\Big{\\{}(1-\theta_{n})f(z^{-})+\theta_{n}f(z^{+}):z^{-}\leq
z^{\ast},z^{+}\geq z^{\ast},\\\ &\hskip
56.9055pt\displaystyle(1-\theta_{n})z^{-}+\theta_{n}z^{+}=z\Big{\\}}+2m_{M}(Mz)^{2}.\end{array}\right.$
(3.1)

The optimal bounds are then completely characterized by the functions
$P^{M,n}$, in the sense that

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

We will show that all the $M+1$ values $\theta_{n}$ are locking states in the
sense above. These values of $\theta$ are particularly relevant since the
shape of $\widehat{Q}_{\bf m}f(z)$ will be shown to depend exclusively on
‘phase mixtures’ with ‘volume fraction’ $\theta_{n}$. Another property enjoyed
by $\theta_{n}$ is that the minimum problems corresponding to values of $z$
for which $\theta(z)=\theta_{n}$ admit periodic solutions.

###  3.1 Phase-constrained relaxation and related properties

We now give some precise definitions, and obtain some general bounds valid for
any choice of $f$ and $\bf m$.

Let $z^{\ast}\in\mathbb{R}$ and let $A=[z^{\ast},+\infty)$. For a given
$\theta\in\mathbb{Q}\cap[0,1]$ and $N\in\mathbb{N}$ we consider the set of
test functions $u$ with a percentage $\theta$ of indices $i$ such that
$u_{i}-u_{i-1}\in A$. Since we need a closed condition, the form of the
constraint is given as follows:

$\displaystyle\mathcal{V}(N;\theta)=\\{u\colon[0,N]\cap\mathbb{Z}\to\mathbb{R}:\\#\\{i:u_{i}-u_{i-1}>z^{\ast}\\}\leq\theta
N,$ (3.2) $\displaystyle\hskip
142.26378pt\\#\\{i:u_{i}-u_{i-1}<z^{\ast}\\}\leq(1-\theta)N\\}.$

For any $z\in\mathbb{R}$ we can then define the function

$\displaystyle\widehat{Q}_{\bf
m}f(\theta,z)=\liminf_{\begin{subarray}{c}N\to+\infty\\\ \theta
N\in\mathbb{N}\end{subarray}}\frac{1}{N}\inf\Big{\\{}F_{1}(u;[0,N]):u\in\mathcal{A}(N;z)\cap\mathcal{V}(N;\theta)\Big{\\}},$
(3.3)

where $F_{1}$ is the (non-scaled) functional defined for
$u\colon[0,N]\cap\mathbb{Z}\to\mathbb{R}$ by

$F_{1}(u;[0,N])=\sum_{i=1}^{N}f(u_{i}-u_{i-1})+\sum_{i,j=0}^{N}m_{|i-j|}(u_{i}-u_{j})^{2}$
(3.4)

(see (2.8) with $\varepsilon=1$ and $I=[0,N]$). In the notation
$\widehat{Q}_{\bf m}f(\theta,z)$ we omit the dependence on $z^{\ast}$. Note
that a corresponding definition could be given also for a more general set
$A$.

In order to obtain bounds for $Q_{\bf m}f$, we also define

$Q_{\bf m}f(\theta,z)=\widehat{Q}_{\bf m}f(\theta,z)-a_{\mathbf{m}}z^{2}.$
(3.5)

######  Theorem 3.2 (optimization over the phase fraction).

The following equality holds:

$\inf_{\theta\in\mathbb{Q}\cap[0,1]}\widehat{Q}_{\bf
m}f(\theta,z)=\widehat{Q}_{\bf m}f(z).$

###### Proof.

It is sufficient to prove that $\widehat{Q}_{\bf
m}f(z)\geq\inf_{\theta\in\mathbb{Q}\cap[0,1]}\widehat{Q}_{\bf m}f(\theta,z)$.
To this end, with $\eta>0$ fixed we choose $\delta>0$, $k\in\mathbb{N}$ and
$u$ an admissible test function for the minimum in (2.8) such that

$\frac{1}{k}\Bigl{(}\sum_{i=1}^{k}f(u_{i}-u_{i-1})+\sum_{i,j=0}^{k}m_{|i-j|}(u_{i}-u_{j})^{2}\Bigr{)}\leq\widehat{Q}_{\bf
m}f(z)+\eta.$

Setting

$\theta=\frac{\\#\\{i:\ u_{i}-u_{i-1}\geq z^{\ast}\\}}{k},$

we extend $u$ to $\mathbb{Z}$ so that $u_{i}-zi$ is $k$-periodic. Since
$u\in\mathcal{A}(Nk;z)\cap\mathcal{V}(Nk;\theta)$, we can use it as a test
function for

$\frac{1}{Nk}\inf\Big{\\{}F_{1}(v;[0,Nk]):v\in\mathcal{A}(Nk;z)\cap\mathcal{V}(Nk;\theta)\Big{\\}}$
(3.6)

in the computation of $\widehat{Q}_{\bf m}f(\theta,z)$.

We subdivide the estimate of $F_{1}(v;[0,Nk])$ by grouping interactions in
three (partially overlapping) different subsets taking into account the
location of the interacting sites in the subintervals $[(r-1)k,rk]$ for
$r\in\\{1,\ldots,N\\}$.

i) (interactions within a single subinterval) $i,j\in[(r-1)k,rk]$ for some
$r\in\\{1,\ldots,N\\}$. Summing over all $i,j$ and $r$ gives the contribution

${1\over k}F_{1}(u;[0,k])$ (3.7)

to (3.6).

ii) (interactions between different intervals, but not close to the endpoints)
$i\in I_{r}^{\delta}=[(r-1)k+k\delta,rk-k\delta]\cap\mathbb{Z}$, $j\in
I_{s}=[(s-1)k,sk]\cap\mathbb{Z}$ for some $r,s\in\\{1,\ldots,N\\}$ with $r\neq
s$.

Let $i^{\prime}=i-(r-1)k$ and $j^{\prime}=j-(s-1)k$. We can write

$\displaystyle(u_{i}-u_{j})^{2}$ $\displaystyle=$
$\displaystyle(u_{i^{\prime}}-u_{j^{\prime}}+z(r-s)k)^{2}\leq
2(u_{i^{\prime}}-u_{j^{\prime}})^{2}+2z^{2}(r-s)^{2}k^{2}$ $\displaystyle\leq$
$\displaystyle
2(i^{\prime}-j^{\prime})\sum_{l=j^{\prime}+1}^{i^{\prime}}(u_{l}-u_{l-1})^{2}+2z^{2}(r-s)^{2}k^{2}$

(we can suppose for simplicity that $j^{\prime}<i^{\prime}$). By (2.6) we have
that

$\sum_{l=j^{\prime}+1}^{i^{\prime}}(u_{l}-u_{l-1})^{2}\leq\sum_{l=1}^{k}(u_{l}-u_{l-1})^{2}\leq
c\,F_{1}(u;[0,k])\leq Ck,$

so that $(u_{i}-u_{j})^{2}\leq 2Ck^{2}+2z^{2}(r-s)^{2}k^{2}$.

We may suppose that $k$ is large enough, so that $m_{l}\leq{\eta\over
l^{\beta}}$ if $l\geq k\delta$, where $\beta$ is the decay exponent of $\bf
m$. Note that $|i-j|\geq\big{|}|s-r|+\delta-1\big{|}k\geq\delta k$. Hence,
summing over such $i,j$, $r$ and $s$ we obtain

$\displaystyle{1\over Nk}\sum_{r\neq s}\sum_{i\in I^{\delta}_{r}}\sum_{j\in
I_{s}}m_{|i-j|}(u_{i}-u_{j})^{2}$ $\displaystyle\leq$ $\displaystyle{1\over
N}\sum_{r\neq s}2k(C+z^{2}(r-s)^{2})\sum_{i\in I^{\delta}_{r}}\sum_{j\in
I_{s}}m_{|i-j|}$ (3.8) $\displaystyle\leq$ $\displaystyle 2{1\over
N}\sum_{r\neq
s}(C+z^{2}(r-s)^{2}){\eta\over\big{|}|s-r|+\delta-1\big{|}^{\beta}}k^{3-\beta}$
$\displaystyle\leq$ $\displaystyle
2\sum_{n=1}^{\infty}(C+z^{2}n^{2}){\eta\over|n+\delta-1|^{\beta}}k^{3-\beta}\leq\widetilde{C}\eta\,.$

iii) (interactions between different intervals, close to the endpoints)
$i,j\in J^{\delta}_{r}=[rk-k\delta,rk+k\delta]\cap\mathbb{Z}$ for some
$r\in\\{1,\ldots,N-1\\}$.

For such $i,j$ we have $u_{i}-u_{j}=z(i-j)$. Hence, we have

$\displaystyle{1\over Nk}\sum_{r=1}^{N-1}\sum_{i,j\in
J^{\delta}_{r}}m_{|i-j|}(u_{i}-u_{j})^{2}$ $\displaystyle=$
$\displaystyle{z^{2}\over Nk}\sum_{r=1}^{N-1}\sum_{i,j\in
J^{\delta}_{r}}m_{|i-j|}(i-j)^{2}$ (3.9) $\displaystyle\leq$
$\displaystyle{z^{2}\over k}\sum_{-k\delta\leq l\leq
k\delta}\sum_{n\in\mathbb{Z}}m_{n}n^{2}\leq\widetilde{C}\delta.$

By (3.7)–(3.9) we obtain the estimate

${1\over Nk}F_{1}(u;[0,Nk])\leq{1\over
k}F_{1}(u;[0,k])+\widetilde{C}(\eta+\delta)\leq\widehat{Q}_{\bf
m}f(z)+C(\eta+\delta).$

Taking the liminf as $N\to+\infty$, by the arbitrariness of $\eta$ and
$\delta$ we obtain the claim. ∎

We now study the general properties of $\widehat{Q}_{\bf m}f(\theta,z)$ as a
function of $\theta$. To that end, we write $\theta$ as the quotient of
(coprime) integer numbers $p$ and $q$, so that

$\displaystyle\widehat{Q}_{\bf
m}f(\theta,z)=\liminf_{k\to+\infty}\frac{1}{kq}\inf\Big{\\{}F_{1}(u;[0,kq]):u\in\mathcal{A}(kq;z)\cap\mathcal{V}(kq;\theta)\Big{\\}}.$

We will need to develop some technical ideas related to the possibility of
modifying boundary conditions. We note that the usual cut-off argument as in
Remark 2.8 cannot be directly followed, since forcing the test function to
satisfy the affine condition $u_{i}=iz$ near the boundary may be incompatible
with the constraint. Still, we can modify the argument with a compatible
condition remaining close to the affine function near the boundary.

To make this precise, for any $\delta>0$ we introduce the set

$\widetilde{\mathcal{A}}_{\delta}(N;z)=\\{u\in\mathcal{A}(N;z):\
|u_{i}-u_{i-1}|\leq|z^{\ast}|+2|z|\ \hbox{\rm if }i\leq\delta N\ \hbox{\rm and
}\ i\geq(1-\delta)N\\}$

and state the following result.

######  Lemma 3.3 (compatible boundary conditions).

The following equality holds

$\displaystyle\widehat{Q}_{\bf m}f(\theta,z)=\lim_{\delta\to
0}\liminf_{k\to+\infty}\frac{1}{kq}\inf\Big{\\{}F_{1}(u;[0,kq]):u\in\mathcal{V}(kq;\theta)\cap\widetilde{\mathcal{A}}_{\delta}(kq;z)\Big{\\}}$

for any $\theta=\frac{p}{q}\in\mathbb{Q}\cap[0,1]$ and $z\in\mathbb{R}$.

###### Proof.

Let $z\in\mathbb{R}$; we may suppose without loss of generality $z\leq
z^{\ast}$. Let $u\in\mathcal{V}(kq;\theta)\cap{\mathcal{A}}(kq;z)$ be a test
function. We modify $u$ separately close to the two endpoints $i=0$ and
$i=kq$. Let $u^{z}$ be a function with $u^{z}_{0}=0$, and such that
$u^{z}_{i}-u^{z}_{i-1}=z^{\ast}$ if $u_{i}-u_{i-1}\geq z^{\ast}$ and
$u^{z}_{i}-u^{z}_{i-1}=2z-z^{\ast}$ if $u_{i}-u_{i-1}\leq z^{\ast}$. By a cut-
off argument as in Remark 2.8 we can modify $u$ on $[0,2kq{\delta}]$ in a
function $\widetilde{u}$ in such a way that $\widetilde{u}_{i}=u^{z}$ on
$[0,kq{\delta}]$, and
$\widetilde{u}_{i}-\widetilde{u}_{i-1}\not\in\\{u_{i}-u_{i-1},u^{z}_{i}-u^{z}_{i-1}\\}$
except for at most $kq{\delta}/N$ for a given arbitrary $N$. Since
$u^{z}_{i}-u^{z}_{i-1}=z^{\ast}$ on a strictly positive percentage of points
in $[0,kq{\delta}]$ (hence, we can suppose larger than $kq{\delta}/N$), up to
slightly modifying $\widetilde{u}$ on such points we have that $\widetilde{u}$
satisfies the constraint; i.e., $\widetilde{u}\in\mathcal{V}(kq;\theta)$. The
same argument can be repeated close to $i=kq$. Note that the energy of $u^{z}$
is comparable to that of the affine function $zi$, so that we obtain an
estimate for the energy of $\widetilde{u}$. ∎

This lemma allows to prove the convexity of $\widehat{Q}_{\bf m}f$ in both
variables.

######  Proposition 3.4 (convexity of $\widehat{Q}_{\bf m}f$).

The function

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

is convex; more precisely,

$(1-t)\widehat{Q}_{\bf m}f(\theta_{1},z_{1})+t\,\widehat{Q}_{\bf
m}f(\theta_{2},z_{2})\geq\widehat{Q}_{\bf
m}f((1-t)\theta_{1}+t\theta_{2},(1-t)z_{1}+tz_{2})$

for any $t\in[0,1]\cap\mathbb{Q}$,
$\theta_{h}=\frac{p_{h}}{q_{h}}\in[0,1]\cap\mathbb{Q}$ and
$z_{h}\in\mathbb{R}$.

###### Proof.

For any $k\in\mathbb{N}$ and $\delta>0$ we define

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

With fixed $\delta>0$, we choose sequences $k^{1}_{N},k^{2}_{N}\to+\infty$
(omitting the dependence on $\delta$) such that

$\liminf_{k\to+\infty}\widehat{Q}_{\bf
m}f_{k}^{\delta}(\theta_{h},z_{h})=\lim_{N\to+\infty}\widehat{Q}_{\bf
m}f_{k_{N}^{h}}^{\delta}(\theta_{h},z_{h})$

for $h=1,2$. We set $M_{N}^{h}=k_{N}^{h}q_{h}$. Recalling Lemma 3.3, for any
fixed $\eta>0$ we find $\delta_{\eta}>0$ such that for
$0<\delta<\delta_{\eta}$ small enough there exists a test function
$u^{h}\in\widetilde{\mathcal{A}}_{\delta}(M_{N}^{h};z_{h})$ (again omitting
the dependencies) such that

$\widehat{Q}_{\bf m}f(\theta_{h},z_{h})\geq\lim_{N\to+\infty}\widehat{Q}_{\bf
m}f_{k_{N}^{h}}^{\delta}(\theta_{h},z_{h})-\eta=\lim_{N\to+\infty}\frac{1}{M_{N}^{h}}F_{1}(u^{h};[0,M_{N}^{h}])-\eta.$
(3.10)

Setting $M_{N}=nM_{N}^{1}M_{N}^{2}$, we define a test function $u$ in
$[0,M_{N}]\cap\mathbb{N}$ by means of suitable translations of $u^{1}$ and
$u^{2}$. More precisely, we set $t=\frac{m}{n}$ and

$u_{i}=\left\\{\begin{array}[]{ll}\widehat{u}^{1}_{i}&\hbox{\rm if
}i\in[0,(n-m)M_{N}^{1}M_{N}^{2}]\\\
\widehat{u}^{2}_{i-(n-m)M_{N}^{1}M_{N}^{2}}+\widehat{u}^{1}_{(n-m)M_{N}^{1}M_{N}^{2}}&\hbox{\rm
if }i\in((n-m)M_{N}^{1}M_{N}^{2},M_{N}]\end{array}\right.$

where $\widehat{u}^{h}\colon[0,M_{N}^{h}]\cap\mathbb{N}\to\mathbb{R}$ is given
by

$\widehat{u}^{h}=u^{h}_{i-(j-1)M_{N}^{1}}+(j-1)M_{1}z_{1}\ \ \hbox{\rm if
}i\in[(j-1)M^{h}_{N},jM^{h}_{N}]$

with $j=1,\dots,(n-m)M_{N}^{2}$ if $h=1$ and $j=1,\dots,mM_{N}^{1}$ if $h=2$.
The function $u$ is an admissible test function for
$\widehat{Q}f_{k_{N}^{1}k_{N}^{2}}(\theta,z)$, where

$\theta=(1-t)\theta_{1}+t\theta_{2}=\frac{(n-m)q_{2}p_{1}+mq_{1}p_{2}}{nq_{1}q_{2}}=\frac{p}{q}\
\ \hbox{\rm and }\ z=(1-t)z_{1}+tz_{2}.$

Indeed, $M_{N}=k_{N}^{1}k_{N}^{2}q$, and

$\displaystyle\frac{\\#\\{i:u_{i}-u_{i-1}\geq
z^{\ast}\\}}{M_{N}}=\frac{(n-m)N_{2}k^{1}_{N}p_{1}+mN_{1}k^{2}_{N}p_{2}}{M}=\theta;$

the boundary conditions are satisfied since $u_{M_{N}}=M_{N}z$. We get

$\frac{1}{M_{N}}F_{1}(u;[0,M_{N}])\geq\widehat{Q}_{\bf
m}f_{k_{N}^{1}k_{N}^{2}}(\theta,z).$ (3.11)

Since $u^{h}\in\widetilde{\mathcal{A}}_{\delta}(M_{N}^{h};z)$, recalling that
$m_{|i-j|}=o(|i-j|^{-\beta})$ with $\beta>3$ we obtain

$\displaystyle\frac{1}{M_{N}}F_{1}(u;[0,M_{N}])$ $\displaystyle\leq$
$\displaystyle\frac{(n-m)M_{N}^{2}M_{N}^{1}}{M_{N}}F_{1}(u^{1};[0,M_{N}^{1}])+\frac{mM_{N}^{1}M_{N}^{2}}{M_{N}}F_{1}(u^{2};[0,M_{N}^{2}])$
$\displaystyle+c(\delta)o(1)_{N\to+\infty}+C\delta$ $\displaystyle=$
$\displaystyle\frac{n-m}{n}\widehat{Q}f_{k_{N}^{1}}^{\delta}(\theta_{1},z_{1})+\frac{m}{n}\widehat{Q}f_{k_{N}^{2}}^{\delta}(\theta_{2},z_{2})+c(\delta)o(1)_{N\to+\infty}+C\delta.$

Taking the $\liminf$ as $N\to+\infty$ and recalling (3.10) and (3.11) we get

$\displaystyle\widehat{Q}_{\bf m}f(\theta,z)$ $\displaystyle\leq$
$\displaystyle\liminf_{N\to+\infty}\widehat{Q}_{\bf
m}f_{k_{N}^{1}k_{N}^{2}}(\theta,z)$ $\displaystyle\leq$
$\displaystyle\liminf_{N\to+\infty}\frac{1}{M_{N}}F_{1}(u;[0,M_{N}])$
$\displaystyle\leq$
$\displaystyle\liminf_{N\to+\infty}\Big{(}\frac{n-m}{n}\widehat{Q}_{\bf
m}f_{k_{N}^{1}}^{\delta}(\theta_{1},z_{1})+\frac{m}{n}\widehat{Q}_{\bf
m}f_{k_{N}^{2}}^{\delta}(\theta_{2},z_{2})\Big{)}+C\delta$ $\displaystyle\leq$
$\displaystyle\frac{n-m}{n}\widehat{Q}_{\bf
m}f(\theta_{1},z_{1})+\frac{m}{n}\widehat{Q}_{\bf
m}f(\theta_{2},z_{2}))+\eta+C\delta.$

Since $\eta>0$ is arbitrary and $\delta\in(0,\delta_{\eta})$, this concludes
the proof. ∎

###  3.2 Phase function and locking states

By the convexity of the function $\theta\mapsto\widehat{Q}_{\bf
m}f(\theta,z)$, we can extend it (and consequently also $Q_{\bf
m}f(\theta,z)$) to the irrational values of $\theta\in(0,1)$ by continuity.
This naturally leads to a definition which singles out some critical values
for $\theta$ remaining ‘stably optimal’ for a range of values of the loading
parameter $z$.

######  Definition 3.5 (locking states).

We say that $\theta$ is a locking state for $f$ and ${\bf m}$ if the set
$\\{z:Q_{\bf m}f(\theta,z)=Q_{\bf m}f(z)\\}$ contains an open interval.

The special values of $\theta$, for which the relaxed energy $\widehat{Q}_{\bf
m}f(\theta,z)$ can be obtained by considering periodic minimizers, play a
particular role in the construction of $\widehat{Q}_{\bf m}f$. Usually, the
arrangements of such minimizers remain optimal over an interval of the values
of $z$ and the corresponding $\theta$ are locking states (see Remark 3.23).
The analysis of some model examples from this standpoint will show how the
knowledge of such special values of $\theta$ can allow one to compute the
whole relaxed energy $\widehat{Q}_{\bf m}f(z)$ (for instance for concentrated
kernels).

We can now introduce a ‘phase function’ as follows.

######  Definition 3.6 (phase function).

We define the phase (multi)function $\Theta(z)$ by

$\Theta(z)=\big{\\{}\theta\in[0,1]:\hbox{\rm sc}(Q_{\bf m}f)(\theta,z)=Q_{\bf
m}f(z)\big{\\}},$

where sc$(Q_{\bf m}f)$ denotes the lower semicontinuous envelope of $Q_{\bf
m}f(\theta,z)$ with respect to $\theta$. In order to define a phase function
$\theta(z)$, we select $\theta(z)$ as the minimum of the set $\Theta(z)$.

######  Remark 3.7 (a selection issue).

Note that in order to have $\theta$ well defined we have made a choice of
$\theta(z)$ as a minimum in the case when $\Theta(z)$ is not a singleton. This
is an arbitrary choice and may lead to some difficulty in the interpretation
of this value, for example in cases where the dependence on $\theta\in[0,1]$
is symmetric, or in degenerate cases (see for instance items (b) and (c) with
the corresponding examples in Section 3.3.1).

######  Remark 3.8 (locking states as the ‘steps’ (constancy intervals)
developed by $\theta(z)$).

The definition of the phase function $\theta(z)$ allows one to to interpret
locking states as the values $\overline{\theta}$ for which
$\theta^{-1}(\overline{\theta})$ contains an open interval.

######  Remark 3.9 (possible non-semicontinuity at the extreme points).

Note that $\hbox{\rm sc}(Q_{\bf m}f)(\theta,z)$ differs from $Q_{\bf
m}f(\theta,z)$ only at most for $\theta\in\\{0,1\\}$, by the continuity of
$Q_{\bf m}f(\theta,z)$ in $(0,1)$. If the function $\theta\mapsto Q_{\bf
m}(\theta,z)$ is lower semicontinuous in $0$ and $1$, then the multi-function
$\Theta(z)$ coincides with the set

$\overline{\Theta}(z)=\\{\theta\in[0,1]:Q_{\bf m}f(\theta,z)=Q_{\bf
m}f(z)\\}.$

In general, the set $\overline{\Theta}(z)$ can be empty, in which case, by
Definition 3.6, $\Theta(z)$ is a singleton and $\theta(z)=0$ (or $1$) if there
exists $\theta_{n}\to 0$ (or $1$, respectively) such that $Q_{\bf
m}f(\theta_{n},z)\to Q_{\bf m}f(z)$ (see Example 3.15 below).

######  Proposition 3.10.

If $\widehat{Q}_{\bf m}f$ is affine in an open interval $I$ and
$\Theta(z)=\\{\theta(z)\\}$ for all $z\in I$, then $\theta$ is affine in $I$.

###### Proof.

Let $z_{1},z_{2}\in I$, $\theta_{1}=\theta(z_{1})$, and
$\theta_{2}=\theta(z_{2})$. For $t\in(0,1)$, Proposition 3.2, the convexity of
$\widehat{Q}_{\bf m}f(\theta,z)$ and the hypothesis imply that

$\displaystyle\widehat{Q}_{\bf m}f(tz_{1}+(1-t)z_{2})$ $\displaystyle\leq$
$\displaystyle\widehat{Q}_{\bf
m}f(t\theta_{1}+(1-t)\theta_{2},tz_{1}+(1-t)z_{2})$ $\displaystyle\leq$
$\displaystyle t\widehat{Q}_{\bf m}f(\theta_{1},z_{1})+(1-t)\widehat{Q}_{\bf
m}f(\theta_{2},z_{2})$ $\displaystyle=$ $\displaystyle t\widehat{Q}_{\bf
m}f(z_{1})+(1-t)\widehat{Q}_{\bf m}f(z_{2})$ $\displaystyle=$
$\displaystyle\widehat{Q}_{\bf m}f(tz_{1}+(1-t)z_{2}),$

and the claim follows.∎

######  Remark 3.11 (locking states and periodic microstructures).

The definition of locking state is formally disconnected from the periodicity
of the associated minimizers. However, the two notions are perhaps related.
Indeed, if the value of the minimum energy $\widehat{Q}_{\bf m}f(z)$ is
reached by some periodic minimizer with a given ‘pattern’ or microstructure
(describing the arrangement of $u_{i}-u_{i-1}$ in the two energy wells), then
one can expect the same pattern to be optimal also for small perturbations of
$z$ (with of course, a small variation of the values of $u$). This would then
entail that the corresponding $\theta$ is a locking state, however, the
formalization of this statement remains unproven even if it holds in all our
examples.

###  3.3 Phase-constrained analysis for decoupled interactions

In this section we focus on the two extreme cases when the effects of $f$ and
$\bf m$ can be decoupled; namely, either when $\bf m$ vanishes or when $f$ is
convex. A comparison with these cases will highlight how for general $f$ and
$\bf m$ the interplay between non-convexity and non-locality gives rise to
complex superposition effects. Such effects will be analyzed in the following
sections in two particularly meaningful examples.

####  3.3.1 Convexification as an envelope of phase-constrained problems

We start by considering the case when the kernel $\bf m$ vanishes. We know
that in this case

$Q_{\bf m}f(z)=\widehat{Q}_{\bf m}f(z)=f^{\ast\ast}(z)$

for any $z$. We can still focus on the dependence of the partially relaxed
energy on the volume fraction $\theta$, which is already non trivial.
Moreover, it shows some features that we will later encounter in more complex
examples.

In this section, we will use ${\bf 0}$ instead of ${\bf m}$ in the notation.
Suppose that while $f\colon\mathbb{R}\to[0,+\infty)$ is not convex, there
exists $z^{\ast}\in\mathbb{R}$ such that the restrictions of $f$ to
$(-\infty,z^{\ast}]$ and $[z^{\ast},+\infty)$ are convex. For such $f$, we now
compute both $Q_{\bf 0}f(\theta,z)$ and $\Theta(z)$.

######  Remark 3.12 (growth condition).

The growth condition from below on $f(z)+2m_{1}z^{2}$ assumed in the previous
sections, in this case would imply a growth condition on $f$. Nevertheless,
for the results of this section it is not necessary, and below we also treat
cases where it is not satisfied, showing some non-continuity effects.

Let $f_{0}$ and $f_{1}$ denote the restrictions of $f$ to $(-\infty,z^{\ast}]$
and to $[z^{\ast},+\infty)$, respectively. For $\theta\in(0,1)$, by using the
convexity of $f_{0}$ and $f_{1}$ we get

$\displaystyle Q_{\bf 0}f(\theta,z)=\inf\\{(1-\theta)f_{0}(t)+\theta
f_{1}(s):t\leq z^{\ast},\ s\geq z^{\ast},\ (1-\theta)t+\theta s=z\\}.$

As for the limit cases $\theta=0$ and $\theta=1$, we have

$Q_{\bf 0}f(0,z)=\begin{cases}f_{0}(z)&\hbox{\rm if }z\leq z^{\ast}\\\
+\infty&\hbox{\rm if }z>z^{\ast}\end{cases}\quad\hbox{\rm and }\quad Q_{\bf
0}f(1,z)=\begin{cases}+\infty&\hbox{\rm if }z<z^{\ast}\\\ f_{1}(z)&\hbox{\rm
if }z\geq z^{\ast}.\end{cases}$

We subdivide the subsequent analysis in dependence of the shape of the
function $f^{\ast\ast}(z)$ representing the convex envelope of $f$; more
precisely, on whether the ‘non-convexity set’ $\\{z:f^{\ast\ast}(z)<f(z)\\}$
is a bounded interval, a half line or the whole line. Note that in this set
$f^{\ast\ast}$ is affine.

##### Case (a): the non-convexity set is a bounded interval.

We suppose that there exist $z_{0}\in(-\infty,z^{\ast}]$ and
$z_{1}\in[z^{\ast},+\infty)$ such that

$f^{\ast\ast}(z)=\begin{cases}f(z)&\hbox{\rm if }\
z\in\mathbb{R}\setminus(z_{0},z_{1})\\\ r(z)&\hbox{\rm if }\
z\in[z_{0},z_{1}],\end{cases}$ (3.12)

where $r$ is affine and $r(z)<f(z)$ in $(z_{0},z_{1})$, then $Q_{\bf 0}f(z)$
is obtained as a minimum of $Q_{\bf 0}f(\theta,z)$. In this case, $\Theta(z)$
is a single value $\theta(z)$ for any $z$, and

$\theta(z)=\begin{cases}0&\hbox{\rm if }z\leq z_{0}\\\
\displaystyle\frac{z-z_{0}}{z_{1}-z_{0}}&\hbox{\rm if }z_{0}\leq z\leq
z_{1}\\\ 1&\hbox{\rm if }z\geq z_{1}.\end{cases}$

Note that trivially $Q_{\bf 0}f(z)$ is the convex envelope of the minimum of
the two functions $Q_{\bf 0}f(0,z)$ and $Q_{\bf 0}f(1,z);$ that is, of
$\min\\{Q_{\bf 0}f(\theta,z):\theta\ \hbox{\rm is a locking state}\\}$, since
the only locking states are $0$ and $1$.

Note moreover that, if $\lim\limits_{z\to+\infty}\frac{f(z)}{z}=+\infty$ and
$f^{\prime}_{-}(z^{\ast})$ is finite, then the formula giving $Q_{\bf
0}f(\theta,z)$ can be simplified for $z$ large enough. Indeed, there exists
$z^{+}$ such that for any $\theta\in(0,1)$

$\displaystyle Q_{\bf 0}f(\theta,z)=(1-\theta)f_{0}(z^{\ast})+\theta
f_{1}\Big{(}\frac{z-(1-\theta)z^{\ast}}{\theta}\Big{)}\ \ \hbox{\rm if}\ z\geq
z^{+}.$

Correspondingly, if $\lim\limits_{z\to-\infty}\frac{f(z)}{|z|}=+\infty$ and
$f^{\prime}_{+}(z^{\ast})$ is finite then, for any $\theta\in(0,1)$,

$\displaystyle Q_{\bf 0}f(\theta,z)=(1-\theta)f_{0}\Big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\Big{)}+\theta f_{1}(z^{\ast})\ \ \hbox{\rm if}\ z\leq
z^{-}$

for $|z^{-}|$ large enough.

Figure 3: Graph of $Q_{\bf 0}f^{t}(\theta,z)$ for different values of
$\theta$.

Figure 4: Graph of the phase function $\theta(z)$ for the function $f^{t}$ in
Example 3.13.

######  Example 3.13 (double-well bi-quadratic potential).

For any $t>1$ we define

$f^{t}(z)=\begin{cases}z^{2}&\hbox{\rm if }\ z\leq 1\\\
\displaystyle\Big{(}\frac{z-{t}}{1-{t}}\Big{)}^{2}&\hbox{\rm if }\ z\geq
1.\end{cases}$

If $\theta\in(0,1)$, we get

$Q_{\bf
0}f^{t}(\theta,z)=\begin{cases}\vspace{2mm}\displaystyle\frac{(z-\theta)^{2}}{1-\theta}+\theta&\displaystyle\hbox{\rm
if }\ z\leq\frac{1-\theta{t}}{1-{z_{0}}}\\\
\vspace{2mm}\displaystyle\frac{(z-\theta{t})^{2}}{1-\theta+\theta(1-{t})^{2}}&\displaystyle\hbox{\rm
if }\ \frac{1-\theta{t}}{1-{t}}\leq z\leq 1+\theta{t}({t}-1)\\\
\displaystyle\frac{(z-1+\theta(1-{t}))^{2}}{\theta(1-{t})^{2}}+1-\theta&\displaystyle\hbox{\rm
if }\ z\geq 1+\theta{t}({t}-1)\end{cases}$

(see Fig. 3 and Fig. 4 for the graph of $Q_{\bf 0}f^{t}(\theta,z)$ and
$\theta(z)$, respectively, with different values of $\theta$ and ${t}$ fixed).

Figure 5: Graph of $Q_{\bf 0}f^{t}(\theta,z)$ with $\theta$ fixed and
increasing values of ${t}$.

In Fig. 5 we picture the graph for a fixed $\theta$ and increasing values of
${t}$.

######  Remark 3.14 (fracture as limit of phase transitions).

If $f^{t}$ is defined as in Example 3.13, then for any fixed $\theta\in(0,1)$

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

This limit function is $Q_{\bf 0}f(\theta,z)$ for $f$ the truncated parabola
(see Example 3.15 below with $\tilde{f}(z)=z^{2}$). This asymptotic behaviour
is illustrated in Fig. 5 above.

From a mechanical standpoint, in the limit as $t\to+\infty$ we can recover the
case fracture as limit of phase-transitions problems as the second well gets
moves to the right and its curvature diminishes. For a mechanical
interpretation of this phenomenon, we refer to [84]. In that perspective, also
the energies at fixed $\theta$ are of interest, because it is the case when
something prevents cracks from localization. The resulting constrained
material becomes ‘tension free’.

##### Case (b): the non-convexity set is a half line.

Let $f^{\ast\ast}(z)<f(z)$ on a half-line, and assume that the half-line is
bounded from below, the other case being symmetric.

By the convexity properties of $f_{0}$ and $f_{1}$, up to the subtraction of
the affine function asymptotic to $f_{1}$ at $+\infty$, it is not restrictive
to assume that $\lim\limits_{z\to+\infty}f_{1}(z)\in[\min f_{0},+\infty)$, so
that $f^{\ast\ast}=\min f_{0}$ in $[z_{0}^{\rm min},+\infty)$, where
$z_{0}^{\rm min}$ is the largest minimizer of $f_{0}$ in $(-\infty,z^{\ast}]$.

For any $\theta\in(0,1)$ and $z\geq(1-\theta)z_{0}^{\rm min}+\theta z^{\ast}$,
we can use $z_{0}^{\rm min}$ and $\frac{z-(1-\theta)z_{0}^{\rm min}}{\theta}$
as test values for $Q_{\bf 0}f(\theta,z)$. If $z>z_{0}^{\rm min}$, taking the
limit as $\theta\to 0$ we get

$Q_{\bf 0}f(z)=f_{0}(z_{0}^{\rm min})=\lim_{\theta\to
0}\Big{(}(1-\theta)f_{0}(z_{0}^{\rm min})+\theta
f_{1}\Big{(}\frac{z-(1-\theta)z_{0}^{\rm
min}}{\theta}\Big{)}\Big{)}=\lim_{\theta\to 0}Q_{\bf 0}f(\theta,z).$

Since $Q_{\bf 0}f(\theta,z)=+\infty$ for $z>z^{\ast}\geq z_{0}^{\rm min}$, the
function $\theta\mapsto Q_{\bf 0}f(\theta,z)$ is not lower semicontinuous in
$0$. If we also assume that $\lim\limits_{z\to+\infty}f_{1}(z)>\min f_{0}$,
then

$Q_{\bf 0}f(\theta,z)>Q_{\bf 0}f(z)\ \ \hbox{\rm for any }\ \ \theta\in[0,1]\
\ \hbox{\rm and }\ \ z>z_{0}^{\rm min}$

and $\overline{\Theta}(z)=\emptyset$ (see Remark 3.9). Since for $z\leq
z^{\ast}$ we have $Q_{\bf 0}f(0,z)=Q_{\bf 0}f(z)$, it follows that
$\theta(z)=0$ for any $z\leq z^{\ast}$.

######  Example 3.15 (truncated convex potentials).

Figure 6: (a) $Q_{\bf 0}f(\theta,z)$ for a truncated convex potential and (b)
$\theta\mapsto Q_{\bf 0}f(\theta,z)$ for different values of $z$.

Let $f$ be the truncated convex given by

$f(z)=\begin{cases}\tilde{f}(z)&\hbox{\rm if }z\leq z^{\ast}\\\
\tilde{f}(z^{\ast})&\hbox{\rm if }z\geq z^{\ast},\end{cases}$ (3.13)

where $\tilde{f}$ is a convex function such that the only minimum point of
$\tilde{f}$ is $0$ with $\tilde{f}(0)=0$, and $z^{\ast}>0$. In particular, we
can take $\tilde{f}(z)=z^{2}$, in which case $f$ is called a truncated
quadratic potential. For $\theta\in(0,1)$ we get

$\displaystyle Q_{\bf
0}f(\theta,z)=\begin{cases}\vspace{1mm}\displaystyle\theta
f(z^{\ast})+(1-\theta)f\Big{(}\frac{z-\theta z^{\ast}}{1-\theta}\Big{)}&\hbox{
if }z<\theta z^{\ast}\\\ \displaystyle\theta f(z^{\ast})&\hbox{ if
}z\geq\theta z^{\ast}.\end{cases}$

For all such $f$ the graphs of $Q_{\bf 0}f(\theta,z)$ and of $Q_{\bf 0}f(z)$
have the form as those pictured in Fig. 6(a). In Fig. 6(b) the function
$\theta\mapsto Q_{\bf 0}f(\theta,z)$ is represented for two different values
of $z$, highlighting the lack of lower semicontinuity in $0$ if $z>0$. Note
that for any $\theta\in(0,1)$ we have $Q_{\bf 0}f(\theta,z)>f(z)$ in
$(-\infty,\theta z^{\ast}]$. Moreover, the optimal volume fraction $\theta(z)$
is always equal to zero, even though $Q_{\bf 0}f(\theta,z)=Q_{\bf 0}f(0,z)$
only if $z\leq 0$ (see Remark 3.17 below).

##### Case (c): the non-convexity set is the whole line.

If $f^{\ast\ast}<f$ in the whole $\mathbb{R}$, then in our hypothesis it is
constant, and as in case (b) it is not restrictive to suppose that both
$\lim\limits_{z\to-\infty}f(z)$ and $\lim\limits_{z\to+\infty}f(z)$ are
finite, so that

$Q_{\bf
0}f(z)=\min\Bigl{\\{}\lim\limits_{z\to-\infty}f(z),\lim\limits_{z\to+\infty}f(z)\Bigr{\\}}.$

For $\theta\in(0,1)$,

$Q_{\bf
0}f(\theta,z)=(1-\theta)\lim_{z\to-\infty}f(z)+\theta\lim_{z\to+\infty}f(z).$

The function $\theta\mapsto Q_{\bf 0}f(\theta,z)$ is not lower semicontinuous
in $0$ if $z>z^{\ast}$ and in $1$ if $z<z^{\ast}$.

If $\lim\limits_{z\to-\infty}f(z)=\lim\limits_{z\to+\infty}f(z)$, then $Q_{\bf
0}f(\theta,z)=Q_{\bf 0}f(z)$ for any $\theta\in(0,1)$ and $z\in\mathbb{R}$,
hence $\Theta(z)=[0,1]$ for any $z$ and $\theta(z)=0$ for any $z$.

If $\lim\limits_{z\to+\infty}f(z)<\lim\limits_{z\to-\infty}f(z)$, then for any
$z$ we have $\Theta(z)=\\{1\\}$ and $\theta(z)=1$. Conversely, if
$\lim\limits_{z\to+\infty}f(z)>\lim\limits_{z\to-\infty}f(z)$, then for any
$z$ we have $\Theta(z)=\\{0\\}$ and $\theta(z)=0$. Note that
$\overline{\Theta}(z)=\emptyset$ at least for any $z<z^{\ast}$ in the first
case, and at least for any $z>z^{\ast}$ in the second.

We give some simple examples of case (c), highlighting the difference between
$\Theta$ and $\overline{\Theta}$ due to the lack of semicontinuity at the
endpoints.

######  Example 3.16.

$\bullet$ If $f(z)=\min\\{1,e^{-z}\\}$, then $Q_{\bf 0}f(z)=0$ and $Q_{\bf
0}f(\theta,z)=1-\theta$ for $\theta\in(0,1)$. Since $Q_{\bf 0}f(0,z)$ and
$Q_{\bf 0}f(1,z)$ are strictly positive, then $\overline{\Theta}(z)=\emptyset$
for any $z$. In this case, there is no locking state.

$\bullet$ If $f(z)=\max\\{\min\\{1,2e^{-z}-1\\},0\\}$, then $Q_{\bf 0}f(z)=0$
and $Q_{\bf 0}f(\theta,z)=1-\theta$ for $\theta\in(0,1)$ as in the previous
case. In this case, $Q_{\bf 0}f(1,z)=0$ if $z\geq\log 2$, hence
$\overline{\Theta}(z)=\emptyset$ for any $z<\log 2$ and
$\overline{\Theta}(z)=\\{1\\}$ if $z\geq\log 2$. The only locking state is
$\theta=1$.

$\bullet$ If $f(z)=e^{-|z|}$, then $Q_{\bf 0}f(z)=Q_{\bf 0}f(\theta,z)=0$ for
any $\theta\in(0,1)$ and $z\in\mathbb{R}$. The set
$\overline{\Theta}(z)=(0,1)$ for any $z$, while $\Theta(z)=[0,1]$. The only
locking state is $\theta=0$.

######  Remark 3.17 (locking states in the degenerate cases).

While we still have that trivially $Q_{\bf 0}f(z)$ is the convex envelope of
$\min\\{Q_{\bf 0}f(0,z),Q_{\bf 0}f(1,z)\\}$, in the examples of cases (b) and
(c) nor both values $\theta=0$ and $\theta=1$ are regarded as locking states.
In the last of Examples 3.16 this is due to the arbitrary choice of defining
$\theta(z)$ as an infimum. As a consequence, the notion of locking state is
not relevant in the computation of $Q_{\bf 0}f$, in the sense that we cannot
recover $Q_{\bf 0}f(z)$ from the only knowledge of $Q_{\bf 0}f(\theta,z)$ for
$\theta$ locking states. In Example 3.15, indeed we have the only locking
state $\theta=0$ but $Q_{\bf 0}f(0,z)=+\infty$ for $z>0$.

####  3.3.2 Convex potentials: phase-constrained interpolation

We now consider the second extreme case; that is, when the function $f$ is
convex on all $\mathbb{R}$ and the kernel $\bf m$ is arbitrary. As we noticed
in Proposition 2.11, in this case the function $Q_{\bf m}f$ is trivially equal
to $f$ for any choice of $\bf m$. Nevertheless, the results of the constrained
minimization producing the functions $Q_{\bf m}f(\theta,z)$ are non-trivial
even in this case. They provide further information regarding the general
structure of the dependence of $Q_{\bf m}f(\theta,z)$ on the phase variable
$\theta$. Moreover, such examples can serve as comparison limit cases for non-
convex energies $f$.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a convex function while ${\bf m}$ can
be arbitrary. In this case, we would need the growth hypothesis
$\lim\limits_{z\to\pm\infty}f(z)+2m_{1}z^{2}=+\infty$ only to use some
technical result concerning the variation of the boundary conditions. We fix
an arbitrary $z^{\ast}\in\mathbb{R}$ and define $A=[z^{\ast},+\infty)$.

As for $\theta=0$ and $\theta=1$, by definition we have

$\displaystyle Q_{\bf m}f(0,z)=\begin{cases}f(z)&{\rm if}\ z\leq z^{\ast}\\\
+\infty&{\rm if}\ z>z^{\ast}\end{cases}\quad{\rm and}\quad Q_{\bf
m}f(1,z)=\begin{cases}+\infty&{\rm if}\ z<z^{\ast}\\\ f(z)&{\rm if}\ z\geq
z^{\ast}.\end{cases}$

In particular, $Q_{\bf m}f(z)=Q_{\bf m}f(0,z)$ for $z\leq z^{\ast}$ and
$Q_{\bf m}f(z)=Q_{\bf m}f(1,z)$ for $z>z^{\ast}$. Moreover, the following
proposition holds.

######  Proposition 3.18.

For $\theta\in(0,1)$, we have

$Q_{\bf m}f(\theta,z)=\begin{cases}\vspace{1mm}\displaystyle\theta
f(z^{\ast})+(1-\theta)f\Big{(}\frac{z-\theta z^{\ast}}{1-\theta}\Big{)}+a_{\bf
m}\frac{\theta}{1-\theta}(z-z^{\ast})^{2}&{\rm if}\ z<z^{\ast}\\\
\displaystyle(1-\theta)f(z^{\ast})+\theta
f\Big{(}\frac{z-(1-\theta)z^{\ast}}{\theta}\Big{)}+a_{\bf
m}\frac{1-\theta}{\theta}(z-z^{\ast})^{2}&{\rm if}\ z\geq
z^{\ast}.\end{cases}$ (3.14)

###### Proof.

We fix $z<z^{\ast}$. Let $z_{i}$ be such that $\sum_{i=1}^{kq}z_{i}=kqz$, and

$\overline{z}=\frac{1}{\\#I}\sum_{i\in I}(f(z_{i})+2m_{1}z_{i}^{2}),$

where $I=\\{i:z_{i}\geq z^{*}\\}$ and $\\#I=\theta kq$. Since $f$ is convex,
we get

$\displaystyle\frac{1}{kq}\sum_{i=1}^{kq}(f(z_{i})+2m_{1}z_{i}^{2})=\frac{1}{kq}\Bigl{(}\sum_{i\in
I}(f(z_{i})+2m_{1}z_{i}^{2})+\sum_{i\not\in
I}(f(z_{i})+2m_{1}z_{i}^{2})\Bigr{)}$
$\displaystyle\geq\theta(f(\overline{z})+2m_{1}(\overline{z})^{2})+(1-\theta)\Big{(}f\big{(}\frac{z-\theta\overline{z}}{1-\theta}\big{)}+2m_{1}\big{(}\frac{z-\theta\overline{z}}{1-\theta}\big{)}^{2}\Big{)}$
$\displaystyle\geq\theta(f(z^{\ast})+2m_{1}(z^{\ast})^{2})+(1-\theta)\Big{(}f\big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\big{)}+2m_{1}\big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\big{)}^{2}\Big{)},$ (3.15)

since $\overline{z}\geq z^{*}$ and
$\frac{z-\theta\overline{z}}{1-\theta}\leq\frac{z-\theta z^{\ast}}{1-\theta}$.

Let $M\in\mathbb{N}$ and $n\leq M$ be fixed. We define $n$ partitions of the
interval $[0,kq]$ given by the set of points

$P_{j}=\Big{\\{}hn+j:h=0,\dots,\Big{\lfloor}\frac{kq-j}{n}\Big{\rfloor}-1\Big{\\}},\
j=0,\dots,n-1.$

Let $u$ be an admissible test function for $\widehat{Q}_{\bf m}f(\theta,z)$.
Recalling Lemma 3.3, we can suppose
$u\in\mathcal{V}(kq;\theta)\cap\widetilde{\mathcal{A}}_{\delta}(kq;z)$. With
fixed $n$ and $j$, let $\widetilde{z}$ and $\widetilde{\theta}$ be defined by

$\displaystyle u_{\lfloor\frac{kq-j}{n}\rfloor
n+j}-u_{j}=\Big{\lfloor}\frac{kq-j}{n}\Big{\rfloor}n\widetilde{z}$
$\displaystyle\widetilde{\theta}\Big{\lfloor}\frac{kq-j}{n}\Big{\rfloor}n=\\#\Big{\\{}i\in\Big{[}j,\Big{\lfloor}\frac{kq-j}{n}\Big{\rfloor}n+j\Big{]}\cap\mathbb{Z}:u_{i}-u_{i-1}\geq
z^{\ast}\Big{\\}}.$

Since $u\in\widetilde{\mathcal{A}}_{\delta}(kq;z)$ and $n\leq\delta kq$, we
obtain $(kq-2n)|z-\widetilde{z}|\leq 4n|z|+2n|z|$. Moreover
$(kq-2n)|\theta-\widetilde{\theta}|\leq 2n+2n\theta$, so that (uniformly with
respect to $n$ and $j$)

$\widetilde{z}=z+o(1)_{k\to+\infty},\ \
\widetilde{\theta}=\theta+o(1)_{k\to+\infty}.$ (3.16)

In particular, if $k$ is large enough then $\widetilde{z}<z^{\ast}$. By
substituting to any $z_{i}\geq z^{\ast}$ the value $z^{\ast}$ and to any
$z_{i}<z^{\ast}$ the value
$\frac{\widetilde{z}-\widetilde{\theta}z^{\ast}}{1-\widetilde{\theta}}$, the
convexity of the square gives

$\frac{1}{kq}\sum_{i\in
P_{j}}(z_{i+1}+\dots+z_{i+n})^{2}\geq\frac{1}{kq}\Big{\lfloor}\frac{\theta
kq}{n}\Big{\rfloor}(nz^{\ast})^{2}+\frac{1}{kq}\Big{\lfloor}\frac{(1-\widetilde{\theta})kq}{n}\Big{\rfloor}\Big{(}n\frac{\widetilde{z}-\widetilde{\theta}z^{\ast}}{1-\widetilde{\theta}}\Big{)}^{2}.$

Hence, recalling (3.16)

$\displaystyle\sum_{i,j=0}^{kq}m_{|i-j|}(u_{i}-u_{j})^{2}$ $\displaystyle\geq$
$\displaystyle 2\sum_{n=1}^{M}m_{n}n\Big{(}\Big{\lfloor}\frac{\theta
kq}{n}\Big{\rfloor}(nz^{\ast})^{2}+\Big{\lfloor}\frac{(1-\theta)kq}{n}\Big{\rfloor}(n\frac{z-\theta
z^{\ast}}{1-\theta})^{2}\Big{)}$ $\displaystyle+o(1)_{k\to+\infty}$

which, together with (3.3.2), gives the estimate

$\displaystyle\frac{1}{kq}F_{1}(u;[0,kq])$ $\displaystyle\geq$
$\displaystyle\theta f(z^{\ast})+(1-\theta)f\Big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\Big{)}$
$\displaystyle+2\sum_{n=1}^{M}m_{n}\Big{(}\theta(nz^{\ast})^{2}+(1-\theta)(n\frac{z-\theta
z^{\ast}}{1-\theta})^{2}\Big{)}+o(1)_{k\to+\infty}.$

We obtain that

$\widehat{Q}_{\bf m}f(\theta,z)\geq\theta
f(z^{\ast})+(1-\theta)f\Big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\Big{)}+2\sum_{n=1}^{M}m_{n}\Big{(}\theta(nz^{\ast})^{2}+(1-\theta)\big{(}n\frac{z-\theta
z^{\ast}}{1-\theta}\big{)}^{2}\Big{)}.$

Since $M$ is arbitrary, we conclude that

$Q_{\bf m}f(\theta,z)\geq\theta f(z^{\ast})+(1-\theta)f\Big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\Big{)}+a_{\bf
m}\Big{(}\theta(z^{\ast})^{2}+(1-\theta)\big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\big{)}^{2}\Big{)}-a_{\bf m}z^{2},$

which gives the lower bound for (3.14) in the case $z<z^{\ast}$.

As for the upper estimate, we define a test function $\overline{u}$ by setting

$\overline{u}_{i}=\begin{cases}\vspace{1mm}z^{\ast}i&{\rm if}\ i\leq\theta
kq\\\ \displaystyle z^{\ast}\theta kq+\frac{z-\theta
z^{\ast}}{1-\theta}(i-\theta kq)&{\rm if}\ i>\theta kq;\end{cases}$

since $m_{n}=o(n^{-\beta})_{n\to+\infty}$ with $\beta>3$ we obtain

$\displaystyle\frac{1}{kq}F_{1}(\overline{u};[0,kq])$ $\displaystyle=$
$\displaystyle\theta f(z^{\ast})+(1-\theta)f\Big{(}\frac{z-\theta
z^{\ast}}{1-\theta}\Big{)}$
$\displaystyle+2\sum_{n=1}^{\infty}m_{n}\Big{(}\theta(nz^{\ast})^{2}+(1-\theta)(n\frac{z-\theta
z^{\ast}}{1-\theta})^{2}\Big{)}+o(1)_{k\to+\infty},$

which gives the upper bound for $k\to+\infty$. Similar arguments allow one to
prove (3.14) for $z>z^{\ast}$ or $z=z^{\ast}$. ∎

Figure 7: the phase multifunction $\Theta(z)$ in the convex case.

Note that the phase multifunction $\Theta(z)$ is given by (see Fig. 7)

$\Theta(z)=\begin{cases}\\{0\\}&\hbox{\rm if }z<z^{\ast}\\\ [0,1]&\hbox{\rm if
}z=z^{\ast}\\\ \\{1\\}&\hbox{\rm if }z>z^{\ast}.\end{cases}$ (3.17)

Here $0$ and $1$ are the only locking states.

A particular interesting sub-case in this general class of problems is
represented by semi-degenerate quadratic-affine functions, often used in
theories of plasticity. Assume for instance that for all $\tau\in\mathbb{R}$
the function $\ell^{\tau}\colon\mathbb{R}\to\mathbb{R}$ is defined as

$\ell^{\tau}(z)=\begin{cases}z^{2}&\hbox{\rm if }\ z\leq 1\\\
2\tau(z-1)+1&\hbox{\rm if }\ z>1.\end{cases}$ (3.18)

Using the general expression for $Q_{\bf m}f(\theta,z)$ in (3.14), we can now
obtain an explicit formula for $Q_{\bf m}\ell^{\tau}(\theta,z)$ in the convex
case $\tau\geq 1$, with the natural choice $A=[1,+\infty)$.

######  Example 3.19 (convex-affine potentials).

Figure 8: $Q_{\bf m}\ell^{\tau}(\theta,z)$ for increasing values of $\theta$.

Let $\ell^{\tau}$ be defined as in (3.18). In the convex case $\tau\geq 1$,
for any $\theta\in(0,1)$ we have

$Q_{\bf
m}\ell^{\tau}(\theta,z)=\begin{cases}\vspace{2mm}\displaystyle\frac{1+a_{\bf
m}\theta}{1-\theta}z^{2}-\frac{2(1+a_{\bf
m})\theta}{1-\theta}z+\frac{(1+a_{\bf m})\theta}{1-\theta}&\hbox{\rm if }z\leq
1\\\ \displaystyle\frac{a_{\bf m}(1-\theta)}{\theta}z^{2}-\Big{(}\frac{2a_{\bf
m}(1-\theta)}{\theta}-\tau\Big{)}z+1-2\tau+\frac{a_{\bf
m}(1-\theta)}{\theta}&\hbox{\rm if }z\geq 1\end{cases}$

These constructions are illustrated in Fig. 8.

In Sections 4 and 5 we will also treat the non-convex case of $\ell^{\tau}$;
that is, $\tau<1$, with particular choices of the interaction kernel $\bf m$.
Note that all the general results concerning $\ell^{\tau}$ still hold if we
take a convex $\tilde{f}$ instead of the quadratic term.

###  3.4 Spin representation and optimal microstructures

We observe that for bi-convex problems a more detailed way to describe the
behaviour of extremal functions is by using a two-value function which labels
the position of the strain variable, whether in one or in the other of the two
convex zones of $f$. Such ‘spin function’ can be viewed as a characteristic
function of the microstructure of an extremal. Note that periodic spin