# Non-occurrence of gap for one-dimensional non autonomous functionals

Carlo Mariconda &

###### Abstract

Let $F(y):=\displaystyle\int_{t}^{T}L(s,y(s),y^{\prime}(s))\,ds$ be a positive
functional, unnecessarily autonomous, defined on the space
$W^{1,p}([t,T];{\mathbb{R}}^{n})$ ($p\geq 1$) of Sobolev functions, possibly
with prescribed one or two end point conditions. It is important, especially
for the applications, to be able to approximate the infimum of $F$ with the
values of $F$ along a sequence of Lipschitz functions satisfying the same
boundary condition(s). Sometimes this is not possible, i.e., the so called
Lavrentiev phenomenon occurs. This is the case of the innocent like Manià’s
Lagrangian $L(s,y,y^{\prime})=(y^{3}-s)^{2}(y^{\prime})^{6}$ and boundary data
$y(0)=0,y(1)=1$; nevertheless in this situation the gap does not occur with
just the end point condition $y(1)=1$. The paper focuses about the different
set of conditions needed to avoid the gap for problems with just one or with
both end point conditions. Under minimal assumptions on the, possibly extended
valued, Lagrangian we ensure the non-occurrence of the Lavrentiev phenomenon
with just one end point condition. We introduce an additional hypothesis,
satisfied when the Lagrangian is bounded on bounded sets, in order to ensure
the validity of both end point conditions $y_{h}(t)=y(t),y_{h}(T)=y(T)$; the
result gives some new light even in the autonomous case.

†† Carlo Mariconda (corresponding author), ORCID $0000-0002-8215-9394$),
Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-
Civita”, Via Trieste 63, 35121 Padova, Italy; e-mail:
<EMAIL_ADDRESS>_Mathematics Subject Classification (2010):_ 49,
49J45, 49N60

Keywords. Regularity, Lipschitz, minimizing sequence, approximation

##  1 Introduction

#### The state of the art

We consider here a one-dimensional, vectorial functional of the calculus of
variations

$F(y)=\int_{t}^{T}L(s,y(s),y^{\prime}(s))\,ds$

defined on the space of Sobolev functions $W^{1,p}(I,{\mathbb{R}}^{n})$ on
$I:=[t,T]$ with values in ${\mathbb{R}}^{n}$, for some $p\geq 1$. In the paper
the Lagrangian $L(s,y,u)$ is Borel and is assumed to have values in
$[0,+\infty[\cup\\{+\infty\\}$. Following the terminology of control theory we
will refer to $s$ as to the time variable, to $y$ as to the state variable and
to $u$ as to the velocity variable. The space of absolutely continuous
functions provides the correct framework in order to find a minimizer of $F$:
Tonelli’s Theorem ensures its existence for any given boundary data when
$L(s,\cdot,\cdot)$ is lower semicontinuous, $L$ has a superlinear growth and
$L(s,y,\cdot)$ is convex. In order to approximate the value of the infimum of
$F$ (e.g., by means of numerical methods), one may be tempted to approximate
in the space of absolutely continuous functions, a given minimizer $y\in
W^{1,p}(I;{\mathbb{R}}^{n})$ with a sequence of Lipschitz functions
$(y_{h})_{h}$ sharing the same boundary data of $y$, in such a way that
$\displaystyle\lim_{h}F(y_{h})=F(y)$. It turns out however, that this may not
be possible: we say that in this case that the Lavrentiev phenomenon occurs.
This is the case, for instance, for the innocent-looking Manià’s [18] problem

$\min F(y):=\int_{0}^{1}(y^{3}-s)^{2}(y^{\prime})^{6}\,ds:\,y\in
W^{1,1}(I),\,y(0)=0,\,y(1)=1$

described in Example 2.3: clearly the minimum is obtained for $y(s):=s^{1/3}$,
however it can be shown that there is $\varepsilon>0$ such that
$F(z)>\varepsilon$ whenever $z:[0,1]\to{\mathbb{R}}$ is Lipschitz and
$z(0)=y(0)=0,z(1)=y(1)=1$. As it is pointed out in [5], it is interesting to
note that things change drastically if one allows the approximating sequence
to have a different initial datum: indeed the sequence

$y_{h}(s):=\begin{cases}{1}/{h^{1/3}}&\text{ if }s\in[0,1/h],\\\
t^{1/3}&\text{ otherwise},\end{cases}$

converges to $y$ and $F(y_{h})\to F(y)$. Another celebrated example, due to
Ball and Mizel, is provided with a polynomial Lagrangian in $(s,y,u)$ that
satisfies Tonelli’s existence conditions.

When a minimizer exists, a way to exclude the Lavrentiev phenomenon is to
provide conditions that ensure the Lipschitz continuity of the minimizer
itself. In the autonomous case it turns out, starting from the work [13] of
Clarke and Vinter, up to its refinement by Dal Maso and Frankowska [14], that
the hypotheses of Tonelli’s existence theorem (even without convexity) provide
Lipschitzianity of the minimizers. Actually weaker growth conditions than
superlinearity ensure both existence and Lipschitzianity: we refer to the
pioneer work [12] of F. Clarke and to the subsequent papers of Cellina and his
school (see [9, 7, 20]). In the non autonomous case, there are functionals
satisfying Tonelli’s condition whose minimizers are not Lipschitz or worse,
that exhibit the Lavrentiev phenomenon. Condition (S), an additional local
Lipschitz continuity assumption on the first “time” variable of $\Lambda$
(thus always satisfied in the autonomous case), formulated in § 2.5, is enough
to ensure Lipschitzianity: this fact was established by Clarke in [12] and
generalized by Bettiol and Mariconda in [3]. Property (S) is known to be a
sufficient condition for the validity of the Du Bois-Reymond equation; we
refer to [10] for the smooth case, to [12] for the nonsmooth convex case under
weak growth assumptions, to [2, 3] by Bettiol and the author in the general
case.

Many functionals arising from applications do not fulfill, however, known
existence criteria and the previous regularity approach may thus not be
pursued. In this case, looking for the non-occurrence of the Lavrentiev
phenomenon is even more challenging and interesting. This is easy if one
requires some “natural growth assumptions” from above and continuous
Lagrangians; our interest is therefore devoted to Lagrangians that possibly
violate these assumption.
Here again, the autonomous case stands on its own: Alberti and Serra Cassano
in [1, Theorem 2.4] state that, if $L(s,y,u)=\Lambda(y,u)$ is just Borel and
satisfies

$\forall K>0\,\,\,\exists\nu_{0}>0\qquad\Lambda\text{ is bounded on
}B_{K}\times B_{\nu_{0}}$ (${\rm B}_{\Lambda}$)

then, given $y\in W^{1,p}(I,{\mathbb{R}}^{n})$ such that
$\Lambda(y,y^{\prime})\in L^{1}(I;{\mathbb{R}}^{n})$ there is a sequence
$(y_{h})_{h}$ of Lipschitz functions converging to $y$ in
$W^{1,p}(I;{\mathbb{R}}^{n})$ and such that $I(y_{h})\to I(y)$ and
$y_{h}(t)=y(t)$ for all $h$: we say in this case that there is no Lavrentiev
gap at $y$ (here and below $B_{r}$ denotes the closed ball of center the
origin and radius $r$ in ${\mathbb{R}}^{n}$) for the initial prescribed datum.
The violation of Assumption (BΛ) may lead to the occurrence of the Lavrentiev
phenomenon, as shown in Example 3.5. Though [1, Remark 2.8] conjectures that
the result does still hold for the variational problem with both end point
conditions, the proof of [1, Theorem 2.4] actually holds for just one end
point condition: Example 3.4 (Alberti, personal communication), shows that
this is not just a technical issue. The importance of the boundary datum, and
the difficulty of preserving it, was noticed exploring the recent literature
also in the multidimensional case by Bousquet, Treu and Mariconda in [4, 21,
22].
In the non autonomous case the examples (see Manià’s) show that some
additional conditions have to be added: To the author’s knowledge, starting
from Lavrentiev himself in [15], most of the criteria for the avoidance of the
Lavrentiev phenomenon require that $\Lambda$ is locally Lipschitz or Hölder
continuous in the state variable (see [16, 23, 24]). As was pointed out by
Carlson in [6], many of them can actually be obtained as a consequence of a
property introduced by L. Cesari and T.S. Angell in [11]. There are however
few exceptions: the non-occurrence of the Lavrentiev phenomenon was
established for two end point conditions without the above regularity
hypotheses on the state variable by:

  * •

Cellina, Ferriero and Marchini in [8] for a class of real valued Lagrangians
of the form $L(s,y,u)=\Lambda(y,u)\Psi(s,y)$ assuming the continuity of
$\Lambda(y,u)$, $\Psi(s,y)>m>0$ (thus excluding Manià’s Lagrangian) and the
convexity of $u\mapsto\Lambda(y,u)$. Notice that the Lagrangian in Manià’s
example is the product of $\Lambda(y,u)=u^{6}$ with $\Psi(s,y)=(y-s^{3})^{2}$,
but $\Psi$ takes the value 0 (along the minimizer).

  * •

Mariconda in [19] for a general class of Lagrangians $\Lambda(s,y,u)$,
assuming the local Lipschitz Condition (S) on the time variable $s$, radial
convexity in the velocity variable and a linear growth from below in the
velocity variable. It is important to notice that Condition (S) for $\Lambda$
is not an option: the lack of its validity may lead to the Lavrentiev
phenomenon, as in Ball – Mizel example. In the extended valued case it is
assumed, moreover, that $\Lambda$ tends with some uniformity to $+\infty$ as
the distance to the boundary of the effective domain
$\operatorname{Dom}(\Lambda)$ (i.e., the set where $\Lambda$ is finite) tends
to 0. As in [1] the lack of regularity of the Lagrangian is compensated by
some local boundedness condition, including (${\rm B}_{\Lambda}$).

#### The main results

These were the main motivations for this paper:

  1. 1.

Is it true, as suggested by [1, Remark 2.8], that the Lavrentiev phenomenon
for the two end point conditions problem does not occur in the case of a real
valued autonomous Lagrangian that is bounded on bounded sets?

  2. 2.

Find a set of assumptions, possibly smaller than those presented in [19] or
[8], that guarantee, even in the autonomous case, the non-occurrence of the
phenomenon for the two end points conditions problem.

  3. 3.

Find sufficient conditions for the non-occurrence of the Lavrentiev phenomenon
with just one end point condition for non autonomous Lagrangians.

The main results give actually some light on the problems, and apply to the
wider class of Lagrangians of the form

$L(s,y,u)=\Lambda(s,y,u)\Psi(s,y),\quad\Lambda,\Psi\geq 0,$ (1.1)

with different sets of hypotheses for $\Lambda$ and for $\Psi$. All of the
results in the paper assume Condition (S) (just) on $s\mapsto\Lambda(s,y,u)$,
and the continuity of $y\mapsto\Psi(s,y)$. Notice that this class il strictly
larger than the one of functions of the form $L(s,y,u)=\Lambda(s,y,u)$
satisfying (S). For instance $L(s,y,u):=\Lambda(s,y,u)\Psi(s,y)$ with

$\forall s\in[0,1],\forall
y,u\in{\mathbb{R}}\qquad\Lambda(s,y,u):=u^{2},\,\Psi(s,y)=\sqrt{s}$

does not satisfy (S), but $\Lambda$ does. In Section 3 extend the result of
Alberti – Serra Cassano [1, Theorem 2.4] to the wider class of Lagrangians
(1.1), for the variational problem with (just) one end point constraint

$\min\\{F(y):\,y\in
W^{1,p}(I;{\mathbb{R}}^{n}),\,y(t)=X\in{\mathbb{R}}^{n}\\}.$
($\mathcal{P}_{X}$)

Corollary 3.6 shows the non-occurrence of the Lavrentiev phenomenon once
$\Lambda$, other than Condition (BΛ) (obviously adapted to the non autonomous
case), satisfies Condition (S). Regarding $\Psi$, it is enough that, for all
$K>0$,

  * (${\rm B}_{\Psi}$)

$\Psi$ is bounded on $I\times B_{K}$;

  * (${\rm C}_{\Psi}$)

$\Psi(\cdot,z)$ is continuous for every $z\in B_{K}$.

  * (${\rm P}_{\Psi}$)

$\inf\Psi>0$.

The above gives an answer to Question 3. Concerning Question 2, we show that,
give $y\in W^{1,p}(I;{\mathbb{R}}^{n})$ such that $F(y)<+\infty$, there is no
Lavrentiev gap for the two end point conditions problem once one assumes,
moreover, that:

  * (${\rm U}_{y,\Lambda}$)

There is an open subset $U_{y}$ of $y(I)$ such that, for all $r>0$, $\Lambda$
is bounded on $I\times U_{y}\times B_{r}$.

Hypothesis (${\rm U}_{y,\Lambda}$) was conjectured by Alberti in a personal
communication for the autonomous case. As shown in Example 3.4, its lack may
lead to a gap. Notice that Hypothesis (${\rm U}_{y,\Lambda}$) subsumes the
fact that $\Lambda$ is real valued on an infinite strip. Thus, in order to be
satisfied for any admissible trajectory $y$, we require in Corollary 3.6 that
$\Lambda$ is bounded on bounded sets (and thus a fortiori, that $\Lambda$ is
not extended valued). This gives, in any case, a reassuring positive answer to
Question 1, confirming the statement in [1, Remark 2.8].

When $\Psi\equiv 1$, the conclusions of Theorem 3.1 and Corollary 3.6 do not
overlap with those obtained in [19, Corollary 6.7] for the same kind of
Lagrangians. Some extra hypotheses assumed there, e.g., the radial convexity
assumption on $0<r\mapsto\Lambda(s,z,rv)$, reveals to be more suitable for
extended valued Lagrangians and allow to deal with state constraints, out or
reach with the methods of the present paper.

#### A word on the proof

The proof of Theorem 3.1 follows the path of that of [1, Theorem 2.4]. Given
$y\in W^{1,p}(I;{\mathbb{R}}^{n})$ such that $F(y)<+\infty$, we build a
sequence $(y_{h})_{h}$ of Lipschitz functions, satisfying the required
boundary conditions/state constraints, and:

  1. 1.

$\displaystyle\lim_{h\to+\infty}F(y_{h})=F(y)$;

  2. 2.

$y_{h}\to y$ in $W^{1,p}(I;{\mathbb{R}}^{n})$.

We begin by considering a standard approximating sequence $(z_{h})_{h}$ of
Lipschitz functions such that $z_{h}^{\prime}=y^{\prime}$ everywhere except at
most an open subset $A_{h}$ whose measure tends to 0 as $m\to+\infty$. We then
reparametrize each $z_{h}$, i.e., we set $y_{h}:=z_{h}\circ\psi_{h}$ for a
suitable Lipschitz injective function $\psi_{h}:I\to I$. A delicate point
here, is that in general (BΛ) and the assumptions for $\Psi$ alone do not
ensure that $\psi_{h}(I)=I$. This is the point where, for the two end point
conditions problem, Hypothesis (Uy,Λ) plays a role, allowing to build a
suitable change of variables $\varphi_{h}$, satisfying $\varphi_{h}(t)=t$ and
$\varphi_{h}=T$ in such a way that $y_{h}:=z_{h}\circ\psi_{h}$ fulfills the
required properties.

##  2 Notation and Basic assumptions

###  2.1 Basic Assumptions

Let $p\geq 1$. The functional $F$ (sometimes referred as to the “energy”) is
defined by

$\forall y\in W^{1,p}(I,{\mathbb{R}}^{n})\qquad
F(y):=\int_{I}L(s,y(s),y^{\prime}(s))\,ds,$

where $L(s,y,v)$ is of the form $L(s,y,v)=\Lambda(s,y,v)\Psi(s,y)$.

######  Basic Assumptions.

We assume the following conditions.

  * •

$I=[t,T]$ is a closed, bounded interval of ${\mathbb{R}}$;

  * •

$\Lambda:I\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\to[0,+\infty[\cup\\{+\infty\\},\,(s,y,u)\mapsto\Lambda(s,y,u)$
($n\geq 1$) is Borel measurable;

  * •

$\Psi:I\times{\mathbb{R}}^{n}\to[0,+\infty]$ is Borel.

  * •

The effective domain of $\Lambda$, given by

$\operatorname{Dom}(\Lambda):=\\{(s,y,u)\in
I\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}:\,\Lambda(s,y,u)<+\infty\\}$

is of the form $\operatorname{Dom}(\Lambda)=I\times D_{\Lambda}$, with
$D_{\Lambda}\subseteq{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$.

###  2.2 Notation

We introduce the main recurring notation:

  * •

The Euclidean norm of $x\in{\mathbb{R}}^{n}$ is denoted by $|x|$;

  * •

The Lebesgue measure of a subset $A$ of $I=[t,T]$ is $|A|$ (no confusion may
occur with the Euclidean norm);

  * •

If $y:I\to{\mathbb{R}}^{n}$ we denote by $y(I)$ its image, by $\|y\|_{\infty}$
its sup-norm and by $\|y\|_{p}$ its norm in $L^{p}(I;{\mathbb{R}}^{n})$;

  * •

The complement of a set $A$ in ${\mathbb{R}}^{n}$ is denoted by $A^{c}$;

  * •

The characteristic function of a set $A$ is $\chi_{A}$.

  * •

If $x\in{\mathbb{R}}$, we denote by $x^{+}$ its positive part, by $x^{-}$ its
negative part;

  * •

$\operatorname{Lip}(I;{\mathbb{R}}^{n})=\\{y:I\to{\mathbb{R}}^{n},\,y\text{
Lipschitz}\\}$; if $n=1$ we simply write $\operatorname{Lip}(I)$;

  * •

For $p\geq 1$,
$W^{1,p}(I;{\mathbb{R}}^{n})=\\{y:I\to{\mathbb{R}}^{n}:\,y,y^{\prime}\in
L^{p}(I;{\mathbb{R}}^{n})\\}$; if $n=1$ we simply write $W^{1,p}(I)$.

###  2.3 Two variational problems

We shall consider different variational problems associated to the functional
$F$, with different end-point conditions and/or state constraints. Let
$X,Y\in{\mathbb{R}}^{n}$. We define

  * •

$\Gamma_{X}:=\\{y\in W^{1,p}(I,{\mathbb{R}}^{n}):\,y(t)=X\\}$, and the
corresponding variational problem

$\text{Minimize }\\{F(y):\,y\in\Gamma_{X}\\},$ ($\mathcal{P}_{X}$)

whenever $\inf(\mathcal{P}_{X})<+\infty$.

  * •

$\Gamma_{X,Y}:=\\{y\in W^{1,p}(I,{\mathbb{R}}^{n}):\,y(t)=X,\,y(T)=Y\\}$, and
the corresponding variational problem

$\text{Minimize }\\{F(y):\,y\in\Gamma_{X,Y}\\},$ ($\mathcal{P}_{X,Y}$)

whenever $\inf(\mathcal{P}_{X,Y})<+\infty$.

There is no privilege in considering the initial condition $y(t)=X$ instead of
the final one $y(T)=Y$ for the one end point conditions problem: any result
obtained here can be reformulated for a final end point condition variational
problem, with the same set of assumptions.

###  2.4 Lavrentiev gap at a function and Lavrentiev phenomenon

In this paper we consider different boundary data for the same integral
functional.

######  Definition 2.1 (Lavrentiev gap at $y\in W^{1,p}(I;{\mathbb{R}}^{n})$).

Let $y\in W^{1,p}(I;{\mathbb{R}}^{n})$ be such that $F(y)<+\infty$ and let
$\Gamma\in\\{\Gamma_{X},\Gamma_{X,Y}\\}$.
We say that the Lavrentiev gap does not occur at $y$ for the variational
problem corresponding to $\Gamma$ if there exists a sequence
$\left(y_{h}\right)_{h\in{\mathbb{N}}}$ of functions in
$\operatorname{Lip}(I,{\mathbb{R}}^{n})$ satisfying:

  1. 1.

$\forall h\in\mathbb{N}\quad y_{h}\in\Gamma$;

  2. 2.

$\displaystyle\limsup_{h\to+\infty}F(y_{h})=F(y)$;

  3. 3.

$y_{h}\to y$ in $W^{1,p}(I;{\mathbb{R}}^{n})$.

We say that the Lavrentiev phenomenon does not occur for the variational
problem corresponding to $\Gamma$ if

$\inf_{\begin{subarray}{c}y\in W^{1,p}(I;{\mathbb{R}}^{n})\\\
y\in\Gamma\end{subarray}}F(y)=\inf_{\begin{subarray}{c}y\in\operatorname{Lip}(I;{\mathbb{R}}^{n})\\\
y\in\Gamma\end{subarray}}F(y).$ (2.1)

######  Remark 2.2 (Gap and phenomenon).

AA

  * •

Let $y\in W^{1,p}(I;{\mathbb{R}}^{n})$. The non-occurrence of the phenomenon
at $y$ ensures that, given $\varepsilon>0$, there is a Lipschitz function
$\overline{y}$ satisfying the same boundary data and/or constraints such that
$F(\overline{y})\leq F(y)+\varepsilon$. If $L(s,y,\cdot)$ is convex for all
$(s,y)\in I\times{\mathbb{R}}^{n}$, and $L(s,\cdot,\cdot)$ is lower
semicontinuous, the non-occurrence of the Lavrentiev gap at $y$ implies the
convergence of $(y_{h})_{h}$ to $y$ in energy, i.e.,

$\lim_{h\to+\infty}F(y_{h})=F(y):$

Indeed in that case $F$ is weakly lower semicontinuous.

  * •

Of course, the non-occurrence of the Lavrentiev gap along a minimizing
sequence implies the non-occurrence of the Lavrentiev phenomenon for the same
variational problem.

The following celebrated example motivates the need to distinguish problems
with just one end point condition from problems with both end points
conditions.

######  Example 2.3 (Manià’s example [18]).

Consider the problem of minimizing

$F(y)=\int_{0}^{1}(y^{3}-s)^{2}(y^{\prime})^{6}\,ds:\,y\in
W^{1,1}(I),\,y(0)=0,\,y(1)=1.$ ($\mathcal{P}_{0,1}$)

Then $y(s):=s^{1/3}$ is a minimizer and $F(y)=0$. Not only $y$ is not
Lipschitz; it turns out (see [5, §4.3]) that the Lavrentiev phenomenon occurs,
i.e.,

$0=\min
F=F(y)<\inf\\{F(y):\,y\in\operatorname{Lip}([0,1]),\,y(0)=0,y(1)=1\\}.$

However, as it is noticed in [5], the situation changes drastically if one
allows to vary the initial boundary condition along the sequence
$(y_{h})_{h}$. Indeed it turns out that the sequence $(y_{h})_{h}$, where each
$y_{h}$ is obtained by truncating $y$ at $1/h$, $h\in\mathbb{N}_{\geq 1}$, as
follows:

$y_{h}(s):=\begin{cases}{1}/{h^{1/3}}&\text{ if }s\in[0,1/h],\\\
s^{1/3}&\text{ otherwise},\end{cases}$

Figure 1: The function $y_{h}$.

is a sequence of Lipschitz functions satisfying

$y_{h}(1)=y(1)=1,\quad F(y_{h})\to F(y),\quad y_{h}\to y\text{ in
}W^{1,1}([0,1]).$

Therefore, no Lavrentiev phenomenon occurs for the variational problem

$\min F(y)=\int_{0}^{1}(y^{3}-s)^{2}(y^{\prime})^{6}\,ds:\,y\in
W^{1,1}(I),\,\,y(1)=1.$ (2.2)

###  2.5 Condition (S)

We consider the following local Lipschitz condition (S) on the first variable
of $\Lambda$.

######  Condition (S).

For every $K\geq 0$ of ${\mathbb{R}}^{n}$ there are $\kappa,\beta\geq
0,\gamma\in L^{1}[t,T]$, $\varepsilon_{*}>0$ satisfying, for a.e. $s\in I$

$|\Lambda(s_{2},z,v)-\Lambda(s_{1},z,v)|\leq\big{(}\kappa\Lambda(s,z,v)+\beta|v|^{p}+\gamma(s)\big{)}\,|s_{2}-s_{1}|$
(2.3)

whenever $s_{1},s_{2}\in[s-\varepsilon_{*},s+\varepsilon_{*}]\cap I$, $z\in
B_{K}$, $v\in{\mathbb{R}}^{n}$, $(s,z,v)\in\operatorname{Dom}(\Lambda)$.

######  Remark 2.4.

Condition (S) is fulfilled if $\Lambda=\Lambda(y,u)$ is autonomous. In the
smooth setting, Condition (S) ensures the validity of the Erdmann - Du Bois-
Reymond (EDBR) condition. In this more general framework it plays a key role
in Lipschitz regularity under slow growth conditions [12, 3, 19] and ensures
the validity of the (EDBR) for real valued Lagrangians [2, 3].

###  2.6 A useful option: linear growth from below for $\Lambda$

The following additional linear growth from below on $\Lambda$ is not assumed
in the main results; however its validity allows to weaken some of the
hypotheses of Corollary 3.6 below.

  * •

There are $\alpha>0$ and $d\geq 0$ satisfying, for a.e. $s\in[0,T]$ and every
$z\in{\mathbb{R}}^{n},v\in{\mathbb{R}}^{n}$,

$\Lambda(s,z,{v})\geq\alpha|{v}|-d.$ (GΛ)

######  Lemma 2.5.

Let $y\in W^{1,p}(I;{\mathbb{R}}^{n})$ be such that $F(y)<+\infty$. Assume
that $\Lambda$ fulfills (GΛ) and that the infimum of $\Psi$ along the graph of
$y$ is strictly positive, i.e.,

  * (${\rm P}_{y,\Psi}$)

There is $m_{y,\Psi}>0$ such that $\Psi(s,z)\geq m_{y,\Psi}$ for all $s\in
I,z\in y(I)$.

Then

$\|y\|_{1}\leq\dfrac{F(y)+\displaystyle m_{y,\Psi}d(T-t)}{\displaystyle
m_{y,\Psi}\alpha}.$

###### Proof.

From (${\rm P}_{\Psi}$) and (GΛ) we obtain

$\displaystyle F(y)$
$\displaystyle=\int_{t}^{T}\Lambda(s,y(s),y^{\prime}(s))\Psi(s,y(s))\,ds$
$\displaystyle\geq m_{y,\Psi}\,\int_{t}^{T}\Lambda(s,y(s),y^{\prime}(s))\,ds$
$\displaystyle\geq m_{y,\Psi}\alpha\int_{t}^{T}|y^{\prime}(s)|\,ds-
m_{y,\Psi}d(T-t),$

proving that

$\int_{t}^{T}|y^{\prime}(s)|\,ds\leq\dfrac{F(y)+\displaystyle
m_{y,\Psi}d(T-t)}{\displaystyle m_{y,\Psi}\alpha}.$

∎

##  3 Non-occurrence of the Lavrentiev gap/phenomenon for ($\mathcal{P}_{X}$)
and for ($\mathcal{P}_{X,Y}$)

Let $X\in{\mathbb{R}}^{n}$. We consider here problems ($\mathcal{P}_{X}$)
($\mathcal{P}_{X,Y}$).

###  3.1 Nonoccurrence of the Lavrentiev gap

Theorem 3.1 below extends [1, Theorem 2.4] to nonautonomous Lagrangians:
indeed if $\Lambda(s,y,u)=\Lambda(y,u)$ then Condition (S) is fulfilled.

######  Theorem 3.1 (Non-occurrence of the Lavrentiev gap for
($\mathcal{P}_{X}$) and for ($\mathcal{P}_{X,Y}$) at $y\in
W^{1,p}(I;{\mathbb{R}}^{n})$).

Assume that $\Lambda$ satisfies Condition (S) and let $y\in
W^{1,p}([t,T],{\mathbb{R}}^{n})$ be such that

$\Lambda(s,y(s),y^{\prime}(s))\Psi(s,y(s))\in L^{1}([t,T]).$

Moreover, suppose that there is a neighbourhood $\mathcal{O}_{y}$ of $y(I)$ in
${\mathbb{R}}^{n}$ such that:

  * (${\rm B}_{y,\Psi}$)

$\Psi$ is bounded on $I\times\mathcal{O}_{y}$;

  * (${\rm C}_{y,\Psi}$)

$\Psi(\cdot,z)$ is continuous for all $z\in y(I)$;

  * (${\rm B}_{y,\Lambda}$)

There is $\nu_{0}>0$ such that $\Lambda$ is bounded on
$I\times\mathcal{O}_{y}\times B_{\nu_{0}}$.

Moreover, assume that $\Lambda(s,y,y^{\prime})\in L^{1}(I)$. Then:

  1. 1.

There is no Lavrentiev gap for ($\mathcal{P}_{X}$) at $y$.

  2. 2.

Assuming, moreover, that:

     * (${\rm U}_{y,\Lambda}$)

There is an open subset $U_{y}$ of $y(I)$ such that, for all $r>0$, $\Lambda$
is bounded on $I\times U_{y}\times B_{r}$,

then there is no Lavrentiev gap for ($\mathcal{P}_{X,Y}$) at $y$.

######  Remark 3.2.

Notice that in Theorem 3.1, the integrability of $\Lambda(s,y,y^{\prime})$ is
satisfied if, for instance, $\Psi$ satisfies Condition (Py,Ψ) formulated in
Lemma 2.5. Indeed, if $\Psi\geq m_{y,\Psi}$ on $I\times y(I)$, then

$\int_{t}^{T}\Lambda(s,y(s),y^{\prime}(s))\,ds\leq\dfrac{1}{m_{y,\Psi}}F(y)<+\infty.$

Manià example 2.3 shows that the integrability of $\Lambda(s,y,y^{\prime})$ is
not a technical matter for the problem with two and, as shown below, even one
end point condition.

######  Example 3.3.

Consider Manià’s Example 2.3. The Lagrangian may be written as

$L(s,y,u)=(y^{3}-s)^{2}u^{6}=\Lambda(u)\Psi(s,y),\quad\Psi(s,y)=(y^{3}-s)^{2},\,\Lambda(u)=u^{6}.$

Consider the minimizer $y(s):=s^{1/3}$. All of the assumptions of Theorem 3.1
are satisfied, except the integrability of $\Lambda(y^{\prime})$; notice also
that $\Psi$ vanishes along $(s,y(s))$ (see Remark 3.2). As a matter of fact,
the Lavrentiev gap occurs at $y$, for the variational problem with (just)
initial prescribed datum

$\min F(z):=\int_{0}^{1}(z^{3}-s)^{2}(z^{\prime})^{6}\,ds,\quad z(0)=0.$

Indeed, if $y_{h}$ is a sequence of Lipschitz functions that converges to $y$
in $W^{1,1}$ then $y_{h}(1)$ is definitely in any given neighborhood of
$y(1)=1$, say in $[3/4,3/2]$: the arguments of [5, §4.3] show that there is
$\eta>0$ such that $F(y_{h})\geq\eta$ for $h$ big enough.

###  3.2 Examples

The next examples concern the autonomous case, i.e., $\Lambda=\Lambda(z,v)$
and $\Psi\equiv 1$. Example 3.4 below shows that Hypothesis (${\rm
U}_{y,\Lambda}$) is essential for the validity of Claim 2 in Theorem 3.1, when
$\Lambda$ is extended valued. It is a slight modification of an example by G.
Alberti (personal communication).

######  Example 3.4 (Occurrence of the Lavrentiev phenomenon in an autonomous,
convex and l.s.c. problem with both endpoint constraints).

Let $y\in W^{1,1}([0,1];{\mathbb{R}})$ be such that

  * •

$y$ is of class $C^{1}$ in $[0,1[$, $y(0)=0,y(1)=1$;

  * •

$y^{\prime}>0$ on $[0,1[$,

  * •

$y^{\prime}(1):=\displaystyle\lim_{s\to 1^{-}}y^{\prime}(s)=+\infty$.

Such a function exists, e.g., $y(s):=1-\sqrt{1-s},s\in[0,1]$. For every
$z\in[0,1[$ set $q(z):=y^{\prime}(y^{-1}(z))$. Let, for
$(s,y,v)\in[0,1]\times{\mathbb{R}}\times{\mathbb{R}}$,

$\Lambda(s,z,v):=\begin{cases}0&\text{ if }z\in[0,1[\text{ and }v\leq
q(z)\text{ or }z\notin[0,1[,\\\ +\infty&\text{
otherwise},\end{cases}\qquad\Psi\equiv 1,$

Figure 2: The domain of $\Lambda(s,\cdot,\cdot)$ in Example 3.4

and set $F(z):=\displaystyle\int_{0}^{1}\Lambda(s,z(s),z^{\prime}(s))\,ds$ for
every $z\in W^{1,1}([0,1];{\mathbb{R}})$. Clearly $F(y)=\min F=0$. Notice
that:

  * •

$\Lambda$ is autonomous;

  * •

$\Lambda$ is lower semicontinuous on ${\mathbb{R}}^{2}$ and
$\Lambda(s,z,\cdot)$ is convex for all $z\in{\mathbb{R}}$;

  * •

$\Lambda$ is bounded on
${\mathbb{R}}\times{\mathbb{R}}\times]-y^{\prime}(0),y^{\prime}(0)[$;

  * •

$\Lambda$ takes the value $+\infty$ on every set of the form
$[0,1]\times]a,b[\times{\mathbb{R}}$.

Thus $\Lambda,\Psi$ satisfy all the assumptions of Claim 1 in Theorem 3.1. In
particular, there is no Lavrentiev gap at $y$ (and thus phenomenon) for the
problem with just one end point condition: either $y(0)=0$ or $y(1)=1$.
However, condition (Uy,Λ) is not fulfilled.
Claim. $F(z)=+\infty$ for every Lipschitz $z:[0,1]\to{\mathbb{R}}$ satisfying
$z(0)=0,z(1)=1$. Indeed assume the contrary: let $z$ be such a function and
suppose $F(z)<+\infty$. Let $0\leq t_{1}<t_{2}\leq 1$ be such that
$z(t_{1})=0,z(t_{2})=1$ and $z([t_{1},t_{2}])=[0,1]$. Since $F(z)<+\infty$
then

$z^{\prime}(s)\leq q(z(s))\text{ a.e. on }[t_{1},t_{2}].$ (3.1)

Notice that, since $\displaystyle\lim_{s\to 1}q(z(s))=+\infty$ and
$z^{\prime}$ is bounded, then necessarily (3.1) is strict on a non negligible
set. It follows that

$\int_{t_{1}}^{t_{2}}\dfrac{z^{\prime}(s)}{q(z(s))}\,ds<\int_{t_{1}}^{t_{2}}\,ds=t_{2}-t_{1}\leq
1.$

However the change of variable $\zeta=z(s)$ (which is justified, for instance,
by the chain rule [17, Theorem 1.74]), gives

$\displaystyle\int_{t_{1}}^{t_{2}}\dfrac{z^{\prime}(s)}{q(z(s))}\,ds$
$\displaystyle=\int_{0}^{1}\dfrac{1}{q(\zeta)}\,d\zeta$
$\displaystyle=\int_{0}^{1}\dfrac{1}{y^{\prime}(y^{-1}(\zeta))}\,d\zeta$
$\displaystyle=\small{(\tau=y^{-1}(\zeta))}\int_{0}^{1}\dfrac{y^{\prime}(\tau)}{y^{\prime}(\tau)}\,d\tau=1,$

a contradiction, proving the claim.

The violation of Condition (${\rm B}_{y,\Lambda}$) in Theorem 3.1 may cause
the occurrence of the phenomenon, even for one point constraint problems.

######  Example 3.5 (Occurrence of the phenomenon in autonomous scalar
problems with one endpoint constraint).

Let $y$ and $q$ be as in Example 3.4, assume moreover that $y\in C^{2}$ and
$y^{\prime\prime}>0$ on $[0,1[$. Define, for
$(s,z,v)\in[0,1]\times{\mathbb{R}}\times{\mathbb{R}}$,

$\Lambda(s,z,v):=\begin{cases}0&\text{ if }z\in I:=[0,1]\text{ and }v\geq
q(z),\\\ +\infty&\text{ otherwise},\end{cases}$

Figure 3: The domain of $\Lambda$ in Example 3.5

and set $F(z):=\displaystyle\int_{0}^{1}\Lambda(s,z(s),z^{\prime}(s))\,ds$ for
every $z\in W^{1,1}([0,1];{\mathbb{R}})$. Notice that $\Lambda$ is autonomous,
lower semicontinuous and $\Lambda(s,z,\cdot)$ is convex for every
$z\in{\mathbb{R}}$. The Lagrangian here is of the form
$\Lambda(s,z,z^{\prime})\Psi(s,z)$ with $\Psi\equiv 1$. Clearly $\Psi$
satisfies the required assumptions of Theorem 3.1 and $\inf\Psi>0$, whereas
$\Lambda$ takes the on every rectangle $I\times y(I)\times B_{\nu_{0}}$ and
thus violates (${\rm B}_{y,\Lambda}$) in Theorem 3.1.
Claim. The Lavrentiev phenomenon occurs for $F$ with the end point condition
$z(1)=1$.
Clearly $\Lambda(s,y,y^{\prime})=0$ a.e. in $[0,1]$, so that $F(y)=\min F=0$.
However, $F(z)=+\infty$ for every Lipschitz function $z$ satisfying $z(1)=1$.
Indeed, let $z(\cdot)$ be a Lipschitz function on $[0,1]$ satisfying $z(1)=1$.
Let $M\geq 0$ be such that $z^{\prime}\leq M$ a.e. on $[0,1]$ and
$t_{*}\in[0,1[$ be such that $y^{\prime}(t_{*})>M$. Since $y(t_{*})<1=z(1)$
there is, by continuity, $\varepsilon>0$ such that

$\forall s\in[1-\varepsilon,1]\qquad y(t_{*})<z(s).$ (3.2)

Then $\Lambda(s,z(s),z^{\prime}(s))=+\infty$ on $[1-\varepsilon,1]$. Indeed,
if $s\in[1-\varepsilon,1]$, the monotonicity of $y^{\prime}$ implies

$z^{\prime}(s)\leq M<y^{\prime}(t_{*})<y^{\prime}(y^{-1}(z(s))=q(z(s)),$

whence $\Lambda(z(s),z^{\prime}(s))=+\infty$. It follows that $F(z)=+\infty$.

###  3.3 Nonoccurrence of the Lavrentiev phenomenon

We immediately obtain a sufficient condition for the non-occurrence of the
Lavrentiev gap. The hypotheses of Corollary 3.6 ensure that the conditions of
Theorem 3.1 are satisfied for every $y\in W^{1,p}(I;{\mathbb{R}}^{n})$ with
$F(y)<+\infty$.

######  Corollary 3.6 (Non-occurrence of the Lavrentiev phenomenon for
($\mathcal{P}_{X}$)).

Assume the validity of the Basic Assumptions and, moreover, that, for all
$K>0$:

  * (${\rm B}_{\Psi}$)

$\Psi$ is bounded on $I\times B_{K}$;

  * (${\rm C}_{\Psi}$)

$\Psi(\cdot,z)$ is continuous for every $z\in B_{K}$;

  * (${\rm P}_{\Psi}$)

There is $m_{K,\Psi}>0$ such that $\Psi(s,z)\geq m_{K,\Psi}$ for all $s\in
I,z\in B_{K}$;

  * (${\rm B}_{\Lambda}$)

There is $\nu_{0}>0$ such that $\Lambda$ is bounded on $I\times B_{K}\times
B_{\nu_{0}}$.

Then:

  1. 1.

The Lavrentiev phenomenon does not occur for ($\mathcal{P}_{X}$).

  2. 2.

If $\Lambda$ is real valued and,

     * (UΛ)

For all $r>0$, $\Lambda$ is bounded on $I\times B_{K}\times B_{r}$,

then the Lavrentiev phenomenon does not occur for ($\mathcal{P}_{X,Y}$).

  3. 3.

If, in addition to the assumptions, $\Lambda$ fulfills (GΛ) and
$\inf\Psi=m_{\Psi}>0$, the conclusions of Claims 1, 2 hold whenever Hypotheses
(${\rm B}_{\Psi}$), (${\rm C}_{\Psi}$), (${\rm B}_{\Lambda}$) are satisfied
for just one value of $K>K_{0}$, where

$K_{0}:=|X|+\dfrac{\inf{\rm(}\mathcal{P}_{X}{\rm)}+\displaystyle
m_{\Psi}d(T-t)}{\displaystyle m_{\Psi}\alpha}.$ (3.3)

###### Proof.

1\. Let $(\overline{y}_{h})_{h}$ be a minimizing sequence for (PX) such that

$\forall h\in\mathbb{N}\qquad
F(\overline{y}_{h})\leq\inf{\rm(}\mathcal{P}_{X}{\rm)}+\dfrac{1}{h+1}.$

Fix $h\in\mathbb{N}$ and choose $K=K_{h}>0$ in such a way that $B_{K_{h}}$
contains a neighborhood of $\overline{y}_{h}(I)$; the validity of hypotheses
(${\rm B}_{\Lambda}$), (${\rm B}_{\Psi}$), (${\rm C}_{\Lambda}$) implies that
of (${\rm B}_{\overline{y}_{h},\Lambda}$), (${\rm
B}_{\overline{y}_{h},\Psi}$), (${\rm C}_{\overline{y}_{h},\Lambda}$) with
$\mathcal{O}_{h}:=B_{K_{h}}$, and that of (P${}_{\overline{y}_{h},\Psi}$). By
applying Theorem 3.1 we obtain the existence of
$y_{h}\in\operatorname{Lip}(I;{\mathbb{R}}^{n})$ satisfying the boundary
condition $y_{h}(t)=X$ and

$F(y_{h})\leq
F(\overline{y}_{h})+\dfrac{1}{h+1}\leq\inf{\rm(}\mathcal{P}_{X}{\rm)}+\dfrac{2}{h+1}.$

The claim follows.
2\. Consider a minimizing sequence $(\overline{y}_{h})$ for
($\mathcal{P}_{X,Y}$) such that

$\forall h\in\mathbb{N}\qquad
F(\overline{y}_{h})\leq\inf{\rm(}\mathcal{P}_{X,Y}{\rm)}+\dfrac{1}{h+1}.$

Notice that, for each $h\in\mathbb{N}$, the validity of Hypothesis (UΛ)
implies that of (U${}_{\overline{y}_{h},\Lambda}$). We proceed as in Claim 1.
3\. If $({\rm P}_{\Psi}$) and (GΛ) hold, from Lemma 2.5 we may assume in the
proof of Claims 1 and 2 that

$\displaystyle\|\overline{y}_{h}\|_{\infty}$
$\displaystyle\leq|X|+\|\overline{y}_{h}\|_{1}$
$\displaystyle\leq|X|+\dfrac{F(\overline{y}_{h})+\displaystyle
m_{\Psi}d(T-t)}{\displaystyle m_{\Psi}\alpha}$
$\displaystyle\leq|X|+\dfrac{\inf{\rm(}\mathcal{P}_{X}{\rm)}+\frac{1}{h+1}+\displaystyle
m_{\Psi}d(T-t)}{\displaystyle m_{\Psi}\alpha}<K,$

once $h\geq h_{0}$, with $\dfrac{1}{(h_{0}+1)m_{\Psi}\alpha}<K-K_{0}$. We can
thus take, for all $h\geq h_{0}$, $K_{h}:\equiv K$ in remaining part of the
proof. The conclusion follows. ∎

######  Remark 3.7.

Here are some comments concerning the assumptions of Theorem 3.1 and Corollary
3.6.

  * •

Hypothesis (${\rm B}_{\Lambda}$) is required in [1, Theorem 2.4] for
autonomous Lagrangians of the form $L(s,y,u)=\Lambda(y,u)$.

  * •

The validity of (${\rm B}_{\Psi}$) (resp. of (${\rm C}_{\Psi}$)) for every
$K>0$ is of course equivalent to the fact that $\Psi$ is bounded on bounded
sets (resp. that $\Psi(\cdot,z)$ is continuous for every
$z\in{\mathbb{R}}^{n}$). Claim 3 of Corollary 3.6 explains the choice of the
apparently naive formulation of the assumptions.

  * •

The validity of (${\rm B}_{\Lambda}$) for every $K>0$ implies that $\Lambda$
is bounded in $I\times{\mathbb{R}}^{n}\times\\{0\\}$; as shown in Corollary
3.6 this inconvenient is encompassed if$\Lambda$ satisfies (GΛ).

  * •

Hypothesis (Uy,Λ) implies that the effective domain of $\Lambda$ contains the
unbounded strip $I\times U_{y}\times{\mathbb{R}}^{n}$; Hypothesis (UΛ) forces
$\Lambda$ to be real valued or, if (GΛ) holds, at least real valued on
$I\times B_{K_{0}}\times{\mathbb{R}}^{n}$ where $K_{0}$ is given by (3.3).

In the real valued, continuous case, many of the assumptions of Corollary 3.6
are satisfied whenever $\Lambda$ si bounded on bounded sets.

######  Corollary 3.8 (Non-occurrence of the Lavrentiev phenomenon for
($\mathcal{P}_{X}$) – real valued case).

Assume the validity of the Basic Assumptions and, moreover, that $\Lambda$ is
real valued, bounded on bounded sets, and that $\Psi$ is continuous and
strictly positive. Then the Lavrentiev phenomenon does not occur for
($\mathcal{P}_{X,Y}$).

##  4 Proof of Theorem 3.1

Lemma 4.1 is a slight modification of [1, Lemma 2.6].

######  Lemma 4.1.

Let $(g_{h})_{h}$ be a sequence of functions in $L^{1}(I,[0,+\infty[)$
converging to $g\in L^{1}(I)$ a.e. in $I$ and $(E_{h})_{h}$ be a sequence of
measurable subsets of $I$ such that $|I\setminus E_{h}|\to 0$. Then

$\int_{E_{h}}g_{h}\,ds\to\int_{I}g\,ds.$

###### Proof.

Possibly passing to a subsequence, we may assume that the characteristic
functions of $E_{h}$ converge to 1 a.e. in $I$. Fatou’s lemma then yields

$\displaystyle\liminf_{h}\int_{E_{h}}g_{h}\,ds\geq\int_{I}g\,ds.$

Moreover, since $g_{h}\geq 0$ and $E_{h}\subset I$ for every $h$, we have that
$\displaystyle\int_{E_{h}}g_{h}\,ds\leq\int_{I}g_{h}\,ds$ and Fatou’s lemma
gives

$\limsup_{h}\int_{E_{h}}g_{h}\,ds\leq\limsup_{h}\int_{I}g_{h}\,ds\leq\int_{I}g\,ds,$

which concludes the proof. ∎

###### Proof of Theorem 3.1.

Proof of Claim 1.
The first steps of the proof follow the path of the proof of [1, Theorem 2.4],
which are recalled and adapted to the more general Lagrangian considered here.

  * i)

It follows from Assumptions (${\rm B}_{y,\Lambda}$) and (${\rm B}_{y,\Psi}$)
that there are $M,{\nu_{0}}>0$ and a neighbourhood $\mathcal{O}_{y}$ of $y(I)$
such that:

$\Lambda(s,z,u)\leq M,\quad\Psi(s,z)\leq M\text{ whenever
}z\in\mathcal{O}_{y},|u|\leq{\nu_{0}}.$ (4.1)

  * ii)

For every $h\in\mathbb{N}$ there are a Lipschitz function
$z_{h}:I\to{\mathbb{R}}^{n}$ and an open subset $A_{h}$ of $I$ such that (see
Figure 4):

    * $\bullet$

$z_{h}(t)=y(t)$, $z_{h}(T)=y(T)$,

    * $\bullet$

$z_{h}=y,\,z_{h}^{\prime}=y^{\prime}$ in $I\setminus A_{h}$,

    * $\bullet$

$z_{h}$ is affine in each connected component of $A_{h}$,

    * $\bullet$

$|A_{h}|\leq\dfrac{T-t}{2(h+1)}$.

Figure 4: The functions $y_{h}$ and $z_{h}$.

  * iii)

Writing each $A_{h}$ as a countable union of open intervals
$I_{h,k}:=]a_{h,k},b_{h,k}[$, $k\in{J_{h}}\subseteq{\mathbb{N}}$, set

$\alpha_{h,k}:=|I_{h,k}|=b_{h,k}-a_{h,k},\quad\beta_{h,k}:=z_{h}(b_{h,k})-z_{h}(a_{h,k})=y(b_{h,k})-y(a_{h,k}).$

Then

$\sum_{k\in{J_{h}}}|\beta_{h,k}|\leq\sum_{k\in\Lambda_{k}}\int_{I_{h,k}}|y^{\prime}|\,ds=\int_{A_{h}}|y^{\prime}|\,ds\to
0\quad h\to\infty.$

  * iv)

For every $h\in\mathbb{N}$ define $\varphi_{h}\in\ W^{1,1}(I)$ by

$\varphi_{h}(t)=t,\quad\varphi_{h}^{\prime}=\begin{cases}1&\text{ in
}I\setminus A_{h},\\\ \dfrac{|z_{h}^{\prime}|}{\nu_{0}}\vee 1&\text{ in
}A_{h}.\end{cases}$

  * v)

It turns out that $\varphi_{h}:[t,T]$ is injective, $I\subseteq\varphi_{h}(I)$
, $(\varphi_{h}(s))_{h}$ converges uniformly to $Id(s):=s$ on $I$ and
$|\varphi_{h}(I)|\to|I|$.

  * vi)

We have

$\displaystyle|\varphi_{h}(A_{h})|$
$\displaystyle=\int_{\varphi_{h}(A_{h})}1\,ds=\int_{A_{h}}|\varphi_{h}^{\prime}(\tau)|\,d\tau$
$\displaystyle\leq\dfrac{1}{\nu_{0}}\int_{A_{h}}|z_{h}^{\prime}(s)|\,ds+|A_{h}|$
$\displaystyle\leq\dfrac{1}{\nu_{0}}\sum_{k\in
J_{k}}\int_{I_{h,k}}|z_{h}^{\prime}(s)|\,ds+|A_{h}|$
$\displaystyle\leq\dfrac{1}{\nu_{0}}\sum_{k\in J_{k}}|\beta_{h,k}|+|A_{h}|\to
0.$

  * vii)

Let $\psi_{h}:[t,T]\to[t,t_{h}]$ ($t_{h}\leq T$) be the inverse of
$\varphi_{h}$, restricted to $I$. Define $y_{h}:=z_{h}(\psi_{h})$. Then
$y_{h}$ is Lipschitz, $y_{h}(t)=y(t)$ and

$y_{h}^{\prime}=z_{h}^{\prime}(\psi_{h})\psi_{h}^{\prime}=\begin{cases}y^{\prime}(\psi_{h})&\text{
in }I\setminus\varphi_{h}(A_{h}),\\\
\left({\nu_{0}}\dfrac{z_{h}^{\prime}(\psi_{h})}{|z_{h}^{\prime}(\psi_{h})|}\right)\wedge
z_{h}^{\prime}(\psi_{h})&\text{ in }I\cap\varphi_{h}(A_{h}).\end{cases}$

Moreover, $(y_{h})_{h}$ converges to $y$ in $W^{1,p}(I,{\mathbb{R}}^{n})$. We
may thus assume that $y_{h}(I)\subset\mathcal{O}_{y}$ for every $h$.

  * viii)

It remains to show that $(F(y_{h}))_{h}$ converges to $F(y)$ as $h\to+\infty$;
this is where the proof differs from the one of the autonomous case. We write

$F(y_{h})=\underbrace{\int_{I\setminus\varphi_{h}(A_{h})}\Lambda(s,y_{h},y_{h}^{\prime})\Psi(s,y_{h})\,ds}_{P_{1,h}}+\underbrace{\int_{I\cap\varphi_{h}(A_{h})}\Lambda(s,y_{h},y_{h}^{\prime})\Psi(s,y_{h})\,ds}_{P_{2,h}}.$

    * a)

Study of the convergence of $(P_{1,h})_{h}$. From the definition of $y_{h}$,
recalling that $z_{h}=y$ and $\varphi_{h}^{\prime}=1$ out of $A_{h}$, we get

$P_{1,h}=\int_{I\setminus\varphi_{h}(A_{h})}\Lambda(s,y(\psi_{h}),y^{\prime}(\psi_{h}))\Psi(s,y(\psi_{h}))\,ds.$

The change of variable $\tau=\psi_{h}(s)$ and the fact that
$\psi_{h}^{\prime}=1$ on $I\setminus\varphi_{h}(A_{h})$ give

$P_{1,h}=\int_{\psi_{h}(I)\setminus
A_{h}}\Lambda(\varphi_{h}(\tau),y(\tau),y^{\prime}(\tau))\Psi(\varphi_{h}(\tau),y(\tau))\,d\tau.$

Almost everywhere in $I$ we have (omitting the variable $\tau$ in
$y(\tau),y^{\prime}(\tau)$)

$\Lambda(\varphi_{h},y,y^{\prime})=\big{[}\Lambda(\varphi_{h},y,y^{\prime})-\Lambda(\tau,y,y^{\prime})\big{]}+\Lambda(\tau,y,y^{\prime}).$
(4.2)

Choose $h$ big enough in such a way that
$\|\varphi_{h}-Id\|_{\infty}<\varepsilon_{*}$. Condition (S) (with
$K:=\|y\|_{\infty}$) implies that, a.e. in $I$,

$|\Lambda(\varphi_{h},y,y^{\prime})-\Lambda(\tau,y,y^{\prime})|\leq\big{(}\kappa\Lambda(\tau,y,y^{\prime})+\beta|y^{\prime}|^{p}+\gamma(\tau)\big{)}\,\|\varphi_{h}-Id\|_{\infty}.$

Therefore, since $\Lambda(\tau,y,y^{\prime})\in L^{1}(I)$, from (4.1), for all
$h\in\mathbb{N}$ big enough,

$\int_{I\setminus\varphi_{h}(A_{h})}\\!\\!\\!\\!\\!|\Lambda(\varphi_{h},y,y^{\prime})-\Lambda(\tau,y,y^{\prime})|\Psi(\varphi_{h},y)\,d\tau\leq
C_{y}\|\varphi_{h}-Id\|_{\infty},$ (4.3)

for a suitable constant $C_{y}$, possibly depending on $y$. It follows from
(4.2), (4.3) and (4.1) that

$P_{1,h}=\int_{\psi_{h}(I)\setminus
A_{h}}\Lambda(\tau,y,y^{\prime})\Psi(\varphi_{h},y)\,d\tau+\varepsilon_{h},\quad\varepsilon_{h}\to
0.$ (4.4)

Now, since $\Lambda(\tau,y,y^{\prime})\in L^{1}(I)$, we deduce from (4.1) that
the functions

$g_{h}:=\Lambda(\tau,y,y^{\prime})\Psi(\varphi_{h},y)\chi_{\psi_{h}(I)\setminus
A_{h}}$

are in $L^{1}(I)$ and, in view of (${\rm C}_{y,\Psi}$), converge a.e. in $I$
to $\Lambda(\tau,y,y^{\prime})\Psi(\tau,y)$; it follows from Lemma 4.1 that

$\lim_{h\to\infty}P_{1,h}=\int_{I}\Lambda(\tau,y(\tau),y^{\prime}(\tau))\Psi(\tau,y(\tau))\,d\tau.$

    * b)

Study of the convergence of $(P_{2,h})_{h}$. We know that for all $h$,
$y_{h}(I)\subset\mathcal{O}_{y}$ and $|y^{\prime}_{h}|\leq{\nu_{0}}$ a.e. in
$\varphi_{h}(A_{h})\cap I$. It follows from (4.1) that

$\text{For a.e.
}\tau\in\varphi_{h}(A_{h})\quad\Lambda(\tau,y_{h}(\tau),y_{h}^{\prime}(\tau))\Psi(\tau,y_{h}(\tau))\leq
M^{2}.$

Therefore Step vi) implies that

$P_{2,h}\leq M^{2}|\varphi_{h}(A_{h})|\to 0.$

Proof of Claim 2.
We proceed as above, with some additional care on the definition of
$\varphi_{h}$, since we need now that
$\displaystyle\int_{t}^{T}\varphi^{\prime}_{h}(s)\,ds=T-t$. Referring to the
proof of Claim 1:

  * •

We proceed as in Steps i), ii), iii).

  * iv′) We modify the above Step iv) as follows.

    * $\bullet$

Since $y^{-1}(U_{y})$ is open in $I$ and non empty, we may choose
$\overline{h}$ in such a way that

$|y^{-1}(U_{y})|>10|A_{h}|\qquad h\geq\overline{h}.$ (4.5)

Since $y$ coincides with $z_{\overline{h}}$ out of $A_{\overline{h}}$ there is
$\ell>0$ satisfying $\|y^{\prime}\|_{\infty}\leq\ell$ on $I\setminus
A_{\overline{h}}$.

    * $\bullet$

Notice that

$\int_{A_{h}}\left[\left(\dfrac{|z_{h}^{\prime}(s)|}{\nu_{0}}\vee
1\right)-1\right]\,ds\to 0.$

Indeed, it follows from (vi)) that

$\int_{A_{h}}\left[\left(\dfrac{|z_{h}^{\prime}(s)|}{\nu_{0}}\vee
1\right)-1\right]\,ds\leq|\varphi_{h}(A_{h})|\to 0.$

It follows from (4.5) that, for $h$ big enough, there is

$\Sigma_{h}\subset y^{-1}(U_{y})\setminus(A_{h}\cup A_{\overline{h}})$

of sufficiently small measure, more precisely satisfying

$\dfrac{|\Sigma_{h}|}{2}=\int_{A_{h}}\left[\left(\dfrac{|z_{h}^{\prime}(s)|}{\nu_{0}}\vee
1\right)-1\right]\,ds.$

    * $\bullet$

For each $h$ big enough, define $\varphi_{h}$ as follows:

$\varphi_{h}(t)=t,\quad\varphi_{h}^{\prime}=\begin{cases}1&\text{ in
}I\setminus(A_{h}\cup\Sigma_{h}),\\\ \dfrac{|z_{h}^{\prime}|}{\nu_{0}}\vee
1&\text{ in }A_{h},\\\ 1/2&\text{ in }\Sigma_{h}.\end{cases}$

  * v′) From iv′) we now have

$\displaystyle\int_{t}^{T}\varphi_{h}^{\prime}(s)\,ds$
$\displaystyle=|I\setminus(A_{h}\cup\Sigma_{h})|+\int_{A_{h}}\left(\dfrac{|z_{h}^{\prime}(s)|}{\nu_{0}}\vee
1\right)\,ds+\dfrac{|\Sigma_{h}|}{2}$
$\displaystyle=|I|-|A_{h}|-|\Sigma_{h}|+\left(\dfrac{|\Sigma_{h}|}{2}+|A_{h}|\right)+\dfrac{|\Sigma_{h}|}{2}=|I|.$

Thus $\varphi_{h}:I\to I$ is bijective.

  * vii′)

Let $\psi_{h}$ be the inverse of $\varphi_{h}$. Define
$y_{h}:=z_{h}(\psi_{h})$. Then $y_{h}$ is Lipschitz, $y_{h}(t)=y(t)$ and

$y_{h}^{\prime}=z_{h}^{\prime}(\psi_{h})\psi_{h}^{\prime}=\begin{cases}y^{\prime}(\psi_{h})&\text{
in }I\setminus\varphi_{h}(A_{h}\cup\Sigma_{h}),\\\
\left({\nu_{0}}\dfrac{z_{h}^{\prime}(\psi_{h})}{|z_{h}^{\prime}(\psi_{h})|}\right)\wedge
z_{h}^{\prime}(\psi_{h})&\text{ in }\varphi_{h}(A_{h}),\\\
2z_{h}^{\prime}(\psi_{h})=2y^{\prime}(\psi_{h})&\text{ in
}\varphi_{h}(\Sigma_{h}).\end{cases}$

As in Claim 1, it turns out that $(y_{h})_{h}$ converges to $y$ in
$W^{1,p}(I,{\mathbb{R}}^{n})$. Indeed, it follows from the proof of Claim 1
that

$\int_{I\setminus\varphi_{h}(\Sigma_{h})}|y_{h}^{\prime}-y^{\prime}|^{p}\,ds\to
0.$

It remains to prove that
$\|y_{h}^{\prime}-y^{\prime}\|_{L^{p}(\varphi_{h}(\Sigma_{h}))}\to 0$. Since
$|y^{\prime}|\leq\ell$ on $\Sigma_{h}$ we have

$\displaystyle\int_{\varphi(\Sigma_{h})}|y_{h}^{\prime}-y^{\prime}|^{p}\,ds$
$\displaystyle\leq
2^{p}\int_{\varphi(\Sigma_{h})}|y_{h}^{\prime}|^{p}\,ds+2^{p}\int_{\varphi(\Sigma_{h})}|y^{\prime}(s)|^{p}\,ds$
$\displaystyle\leq
2^{p}\int_{\varphi(\Sigma_{h})}2^{p}|y^{\prime}(\psi_{h})|^{p}\,ds+2^{p}\int_{\varphi(\Sigma_{h})}|y^{\prime}(s)|^{p}\,ds$
$\displaystyle\leq
2^{2p}\ell^{p}|\varphi(\Sigma_{h})|+2^{p}\int_{\varphi(\Sigma_{h})}|y^{\prime}(s)|^{p}\,ds\to
0,$

since $\varphi_{h}$ is absolutely continuous and $|\varphi_{h}(\Sigma_{h})|\to
0$, proving the claim.
We may thus assume, as in the proof of Claim 1, that
$y_{h}(I)\subset\mathcal{O}_{y}$ for every $h$ big enough.

  * viii′)

It remains to show that $(F(y_{h}))_{h}$ converges to $F(y)$ as $h\to+\infty$.
We write

$F(y_{h})=\underbrace{\int_{I\setminus\varphi_{h}(A_{h}\cup\Sigma_{h})}*\,ds}_{P_{1,h}}+\underbrace{\int_{\varphi_{h}(A_{h})}*\,ds}_{P_{2,h}}+\underbrace{\int_{\varphi_{h}(\Sigma_{h})}*\,ds}_{P_{3,h}},$

where here $*$ stands for $\Lambda(s,y_{h},y_{h}^{\prime})\Psi(s,y_{h})$. As
in the proof of Claim 1 we get

$P_{1,h}\to F(y),\quad P_{2,h}\to 0.$

It remains to prove that $P_{3,h}\to 0$. Since
$\varphi_{h}^{\prime}\equiv\dfrac{1}{2}$ on $\Sigma_{h}$, the change of
variables $s=\varphi_{\nu}(\tau)$ gives

$\displaystyle P_{3,h}$
$\displaystyle=\int_{\varphi_{h}(\Sigma_{h})}\Lambda(s,y_{h},y_{h}^{\prime})\Psi(s,y_{h})\,ds$
$\displaystyle=\int_{\varphi_{h}(\Sigma_{h})}\Lambda\left(s,y(\psi_{h}),y^{\prime}(\psi_{h})\psi_{h}^{\prime}\right)\Psi(s,y(\psi_{h}))\,ds$
$\displaystyle=\dfrac{1}{2}\int_{\Sigma_{h}}\Lambda\left(\varphi_{h},y,2{y^{\prime}}\right)\Psi(\varphi_{h},y)\,d\tau$

Now, for a.e. $\tau$ on $\Sigma_{h}$, $y(\tau)\in U_{y}$ and
$2|y^{\prime}(\tau)|\leq 2\ell$. Hypotheses (Uy,Λ) and (By,Ψ) imply that
$\Lambda(\varphi_{h},y,2y^{\prime})\Psi(\varphi_{h},y)$ is bounded on
$\Sigma_{h}$: the conclusion follows.

∎

######  Remark 4.2.

Notice that in the proof of Claim 1 of Theorem 3.1, every element of the
sequence $(z_{h})_{h}$ satisfies the boundary conditions
$z_{h}(t)=y(t),z_{h}(T)=y(T)$. However, since $\varphi_{h}^{\prime}\geq 1$ on
$[t,T]$, it may happen that $\varphi_{h}(T)>T$, and this yields
$\psi_{h}(T)<T$; as a consequence $y_{h}(T)=z_{h}(\psi_{h}(T))$ may differ
from $z_{h}(T)=y(T)$.
In presence of a final end point condition of the form $y(T)=Y$, instead of a
initial one, the proof goes as above, replacing the definition of the change
of variables $\varphi_{h}$ by

$\varphi_{h}(T)=T,\quad\varphi_{h}^{\prime}=\begin{cases}1&\text{ in
}I\setminus A_{h},\\\ \dfrac{|z_{h}^{\prime}|}{\nu_{0}}\vee 1&\text{ in
}A_{h}.\end{cases}$

## Acknowledgments

I warmly thank Giovanni Alberti for the mail exchange we had during the
preparation of the paper and for providing the construction of Example 3.4 and
for sharing the conjecture of Condition (Uy,Λ) in Theorem 3.1. I am grateful
to Giulia Treu for her comments on the manuscript and warm encouragement. This
research is partially supported by the Padua University grant SID 2018
“Controllability, stabilizability and infimum gaps for control systems”, prot.
BIRD 187147 and has been accomplished within the UMI Group TAA “Approximation
Theory and Applications”.

## References

  * [1] G. Alberti and F. Serra Cassano.  Non-occurrence of gap for one-dimensional autonomous functionals.  In Calculus of variations, homogenization and continuum mechanics (Marseille, 1993), volume 18 of Ser. Adv. Math. Appl. Sci., pages 1–17. World Sci. Publ., River Edge, NJ, 1994.
  * [2] P. Bettiol and C. Mariconda.  A new variational inequality in the calculus of variations and Lipschitz regularity of minimizers.  J. Differential Equations, 268(5):2332–2367, 2020.
  * [3] P. Bettiol and C. Mariconda.  A Du Bois-Reymond convex inclusion for non-autonomous problems of the Calculus of Variations and regularity of minimizers.  Appl. Math. Optim., 83:2083–2107, 2021.
  * [4] P. Bousquet, C. Mariconda, and G. Treu.  On the Lavrentiev phenomenon for multiple integral scalar variational problems.  J. Funct. Anal., 266:5921–5954, 2014.
  * [5] G. Buttazzo, M. Giaquinta, and S. Hildebrandt.  One-dimensional variational problems, volume 15 of Oxford Lecture Series in Mathematics and its Applications.  The Clarendon Press, Oxford University Press, New York, 1998.  An introduction.
  * [6] D. A. Carlson.  Property (D) and the Lavrentiev phenomenon.  Appl. Anal., 95(6):1214–1227, 2016.
  * [7] A. Cellina.  The classical problem of the calculus of variations in the autonomous case: relaxation and Lipschitzianity of solutions.  Trans. Amer. Math. Soc., 356:415–426 (electronic), 2004.
  * [8] A. Cellina, A. Ferriero, and E. M. Marchini.  Reparametrizations and approximate values of integrals of the calculus of variations.  J. Differential Equations, 193(2):374–384, 2003.
  * [9] A. Cellina, G. Treu, and S. Zagatti.  On the minimum problem for a class of non-coercive functionals.  J. Differential Equations, 127(1):225–262, 1996.
  * [10] L. Cesari.  Optimization—theory and applications, volume 17 of Applications of Mathematics (New York).  Springer-Verlag, New York, 1983.  Problems with ordinary differential equations.
  * [11] L. Cesari and T. S. Angell.  On the Lavrentiev phenomenon.  Calcolo, 22(1):17–29, 1985.
  * [12] F. H. Clarke.  An indirect method in the calculus of variations.  Trans. Amer. Math. Soc., 336:655–673, 1993.
  * [13] F. H. Clarke and R. B. Vinter.  Regularity properties of solutions to the basic problem in the calculus of variations.  Trans. Amer. Math. Soc., 289:73–98, 1985.
  * [14] G. Dal Maso and H. Frankowska.  Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton-Jacobi equations.  Appl. Math. Optim., 48:39–66, 2003.
  * [15] M. Lavrentiev.  Sur quelques problèmes du calcul des variations.  Ann. Mat. Pura App., 4:107–124, 1926.
  * [16] Philip D. Loewen.  On the Lavrentiev phenomenon.  Canad. Math. Bull., 30(1):102–108, 1987.
  * [17] Jan Malý and William P. Ziemer.  Fine regularity of solutions of elliptic partial differential equations, volume 51 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1997.
  * [18] B Manià.  Sopra un esempio di lavrentieff.  Boll. Un. Matem. Ital., 13:147–153, 1934.
  * [19] C. Mariconda.  Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems.  Trans. Amer. Math. Soc., (to appear), 2021.
  * [20] C. Mariconda and G. Treu.  Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth.  Calc. Var. Partial Differential Equations, 29:99–117, 2007.
  * [21] C. Mariconda and G. Treu.  Non-occurrence of a gap between bounded and Sobolev functions for a class of nonconvex Lagrangians.  J. Convex Anal., 27(4):1247–1259, 2020.
  * [22] C. Mariconda and G. Treu.  Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians.  Open Math., 18(1):1–9, 2020.
  * [23] G. Treu and S. Zagatti.  On the Lavrentiev phenomenon and the validity of Euler-Lagrange equations for a class of integral functionals.  J. Math. Anal. Appl., 184(1):56–74, 1994.
  * [24] A. J. Zaslavski.  Nonoccurrence of the Lavrentiev phenomenon for nonconvex variational problems.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 22(5):579–596, 2005.