Absence of Lavrentievs gap for anisotropic functionals
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arXiv:2210.15217v1 [math.AP] 27 Oct 2022
Absence of Lavrentiev’s gap for anisotropic functionals
Micha l Borowskia, Iwona Chlebickaa,∗, B la˙zej Miasojedowa
aInstitute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Abstract
We establish the absence of the Lavrentiev gap between Sobolev and smooth maps for a non-autonomous
variational problem of a general structure, where the integrand is assumed to be controlled by a function
which is convex and anisotropic with respect to the last variable. This fact results from new results on good
approximation properties of the natural underlying unconventional function space. Scalar and vector-valued
problems are studied.
Keywords: Density of smooth functions, Lavrentiev’s phenomenon, Musielak–Orlicz–Sobolev spaces
1. Introduction
We study the well-posedness of minimization of the following variational functional
F[u] := ZΩ
F(x, u, ∇u)dx , (1)
over an open and bounded set Ω ⊂Rn,n≥1, where F: Ω ×R×Rn→Ris merely continuous with respect
to the second and the third variable. We suppose that there exist constants 0 < ν, β < 1< L such that
νM (x, βξ)≤F(x, z, ξ)≤L(M(x, ξ) + 1),for all x∈Ω, z ∈R, ξ ∈Rn,(2)
for an N-function M: Ω ×Rn→[0,∞) that is continuous with respect to both variables. This function
is called anisotropic because it depends on ξnot necessarily via |ξ|. We focus on the issue whether the
minimizers can be approximated by regular functions in the topology that is natural for the problem. As
an answer, we find sharp conditions on Mensuring the absence of the Lavrentiev gap between Sobolev and
smooth maps that precisely capture the anisotropic version of double-phase and multi-phase functionals. This
fact results from fine approximation properties of the natural underlying unconventional function space.
The natural function space to study minimizers to problems like (1) is a Sobolev type space equipped with
a Luxemburg norm defined by the means of a functional ξ7→ RΩM(x, ξ)dx applied to a distributional gradient
of a function u∈W1,1(Ω) with ∇uin a relevant Musielak–Orlicz spaces. Since the growth of Mdoes not need
to be doubling, it should be taken into account that the Musielak–Orlicz spaces EMand LM(being norm
and modular closure of L∞, respectively) may differ. Consequently, their Sobolev-type versions V1
0EM(Ω)
and V1
0LM(Ω) do not need to coincide. If M, M∗∈∆2, then LM=EMand V1
0LM=V1
0EM=V1,M
0.
∗Corresponding author
Email addresses: m.borowski@mimuw.edu.pl (Micha l Borowski), i.chlebicka@mimuw.edu.pl (Iwona Chlebicka),
b.miasojedow@mimuw.edu.pl (B la˙zej Miasojedow)
1MSC2020: 46E30 (46E40,46A80)
2M.B. and I.C. are supported by NCN grant 2019/34/E/ST1/00120.
Preprint submitted to Elsevier October 28, 2022
The situation when the infimum of a variational problem over a family of regular functions (e.g. smooth
or Lipschitz) is strictly greater than the infimum taken over all functions satisfying the same boundary
conditions is called Lavrentiev’s phenomenon after his seminal paper [43]. To give a flavour of a great deal
of classical results and new developments in this area, let us refer to [3, 4, 9, 10, 13, 14, 15, 28, 31, 32, 46, 53]
and references therein. In our setting, having u0∈V1EM(Ω), if Mis not sufficiently regular, it might happen
that
inf
u0+V1
0EM(Ω) F[u]<inf
u0+C∞
c(Ω) F[u].(3)
Since [53, 55] by Zhikov, it is known that log-H¨older continuity of the variable exponent of M(x, ξ) = |ξ|p(x)
ensures that the above scenario is excluded. In other kinds of isotropic generalized Orlicz growth problems,
the role of this continuity condition is typically played by an assumption (A1’) from [37, 38], embracing condi-
tions from [7, 25, 28, 31] needed for study on M(x, ξ) = |ξ|p+a(x)|ξ|qwith possibly vanishing weight a. Such
conditions are inevitable due to the examples of functions that cannot be approximated [3, 32, 33, 45, 54].
Smooth approximation properties of an inhomogeneous and anisotropic space of Musielak–Orlicz–Sobolev-
type were proven under some far from sharp conditions [17, 19, 35] and improved recently to a truly local and
anisotropic one in [9]. Nonetheless, in the view of [3, 12, 13], there was still some room for progress. Apart from
the intrinsic mathematical interest of Lavrentiev’s phenomenon, we shall indicate that the Musielak–Orlicz–
Sobolev spaces are used as a framework in modelling of modern materials involving electrorheological and
non-Newtonian fluids, thermo-visco-elastic ones, as well as in image restoration processing, see [16, 36, 41, 52].
Henceforth, we provide a precise tool into their analysis. Section 6 describes how our result directly extend
the existence results of [17, 35] and other contributions.
One of the most important feature of the space that we want to include in our analysis is anisotropy.
A weak N-function M: Ω ×Rn→[0,∞) is called isotropic if M(x, ξ) = m(x, |ξ|) with a weak N-function
m: Ω ×[0,∞)→[0,∞) and anisotropic if its dependence on ξis allowed to be more complicated. One can
consider the functions admitting a decomposition called orthotropic (studied e.g. in [11, 29]):
Mx, (ξ1,...,ξn)=
n
X
i=1
Mi(x, |ξi|) with weak N-functions Mi: [0,∞)→[0,∞).
If an anisotropic function is not comparable to any function admitting a decomposition that after an affine and
invertible change of variables has the above form, we call it essentially fully anisotropic. A relevant example
can be found in [22]. Fully anisotropic spaces are considered since [40, 42, 50] and [8, 23, 49]. They serve as
a setting for nonlinear partial differential equations or calculus of variations, see e.g. [2, 5, 24, 51]. Recently,
we can observe more and more attention paid to problems that are both inhomogeneous and anisotropic at
the same time, e.g. [12, 17, 18, 19, 21, 35, 36, 39, 44, 52] and sharp conditions for the density of smooth
functions are strikingly missing in the theory.
Our goal is to detect the optimal conditions that need to be imposed on Mto exclude the case of strict
inequality in (3). The key accomplishments of the present paper yield the absence of Lavrentiev’s phenomenon
(Theorem 1) and the modular density of smooth functions (Theorem 2). Both of the mentioned results holds
for weak N-functions satisfying some balance condition reflecting a log-H¨older continuity of the variable
exponent of a sharp range of powers in the double-phase case. In order to present the conditions, given a
weak N-function M: Ω ×Rn→[0,∞) and a ball B, we define
M−
B(ξ) = ess infy∈BM(y, ξ) and M+
B(ξ) = ess supy∈BM(y, ξ).(4)
2
The considered conditions read as follows.
Isotropic condition (Biso
γ).Let us assume that M: Ω ×[0,∞)→[0,∞) is a weak N-function. Suppose
there exist constants c♦, C♦≥1 such that for every ball Bwith radius r≤1 and for all ξ∈Rn
satisfying |ξ| ≤ c♦rγ−1there holds M+
B(|ξ|)≤M−
B(C♦|ξ|) + 1.
Orthotropic condition (Bort
γ).Let us assume that
M(x, ξ) =
n
X
i=1
Mi(x, |ξi|),
where Mi: Ω ×[0,∞)→[0,∞) are weak N-functions and ξ= (ξ1,...,ξn). Suppose there exist
constants c♦, C♦≥1 such that for every ball Bwith radius r≤1, all i, and for all ξi∈Rsatisfying
|ξi| ≤ c♦rγ−1there holds (Mi)+
B(|ξi|)≤(Mi)−
B(C♦|ξi|) + 1.
The above conditions (Biso
γ) and (Bort
γ) covering the most typical settings are special cases of a fully anisotropic
condition (Bgen
γ), see Section 4. They force that the growth of a weak N-function with respect to the second
variable is not perturbed much in a small spacial region. The first of our main accomplishments yields precise
result on the absence of Lavrentiev’s phenomenon, i.e.,
inf
u0+V1
0LM(Ω) F[u] = inf
u0+C∞
c(Ω) F[u],(5)
whereas the second one – the approximation result in anisotropic Musielak–Orlicz–Sobolev spaces, i.e.,
V1
0LM(Ω) = C∞
c(Ω)mod .
Here C∞
c(Ω)mod stands for the closure in the sequential modular topology of LMof the gradients. Theorem 1
is supplied with its extended versions in Section 4, which are more complicated in the exposition, but cover
more general functionals. We embrace and extend the results on the absence of Lavrentiev’s phenomenon
from [1, 13, 32] to precisely capture local nature of the problem, as well as anisotropy and general growth
of Fwith respect to the last variable. Moreover, in the regions of low growth of Mwe improve the known
anisotropic approximation results of [9, 17, 35]. When the growth is isotropic, we do it up to the known
borderline cases, see [4, 7, 33, 53]. In turn, we supply the existence theory that holds in the absence of
Lavrentiev’s phenomenon [12, 17, 18, 19, 21, 35] with precise information when the methods therein apply.
Let us show two simple examples directly illustrating how our main results extend the state of the art.
Main models. A consequence of our main result, that is Theorem 1, is that for a double-phase functional
Hiso[u] := ZΩ
b(x, u) (|∇u|p+a(x)|∇u|q)dx (6)
where b: Ω ×R→Rsatisfies 0 < ν < b(·,·)< L and is merely continuous with respect to the last variable,
1≤p≤q,a: Ω →[0,∞) such that a∈C0,α(Ω), and for
Hiso(x, ξ) = |ξ|p+a(x)|ξ|q,and V1,Hiso
0(Ω) := {u∈W1,1
0(Ω) : |∇u| ∈ LHiso (Ω)},
3
it holds
inf
u0+V1,Hiso
0(Ω)
Hiso[u] = inf
u0+C∞
c(Ω) Hiso[u],(7)
whenever
q≤p+α . (8)
Consequently, even in the classical isotropic case, Theorem 1 extends the known results. In particular, it
generalizes the very recent contribution [13] relaxing growth by allowing for functionals involving integrands
dependent on three variables (precisely, Flike (1) under the assumption (2) with Msubstituted by Hiso). In
the appearance of more phases, we give more precise bound than [13]. Furthermore, having a priori knowledge
on the regularity of a minimizer Theorem 1 enables to improve the range (8). Namely, for u, u0∈C0,γ (Ω),
γ∈[0,1], we get that (7) holds whenever
q≤p+α
1−γ.
Since we need only continuity of band only with respect to the second variable, we relax the assumptions of
[7, Theorem 4] yielding the absence of Lavrentiev’s gap for Hiso under extra assumptions on the decay of the
modulus of continuity of bwith respect to both variables. The range from (8) cannot be improved for p < n
due to [3]. Our main model, that was not covered by the literature, is the following anisotropic functional
G[u] := ZΩ
G(x, u, ∇u)dx , (9)
where G: Ω ×R×Rn→Ris continuous with respect to all its variables and
νH (x, ξ)≤G(x, z, ξ)≤LH (x, ξ),for all x∈Ω, z ∈R, ξ ∈Rn
and some constants ν, L > 0, where
H(x, ξ) =
n
X
i=1
|ξi|pi+
n
X
i=1
ai(x)|ξi|qi,1≤pi≤qi,ai: Ω →[0,∞), such that ai∈C0,αi(Ω), i = 1,...,n.
(10)
As a consequence of our main result, it holds
inf
u0+V1,H
0(Ω)
G[u] = inf
u0+C∞
c(Ω) G[u] (11)
where V1,H
0(Ω) := {u∈W1,1
0(Ω) : |∇u| ∈ LH(Ω)},
whenever
qi≤pi+αifor i= 1,...,n. (12)
Furthermore, in this case we can also trade a priori known regularity of a minimizer with this range. For
u, u0∈C0,γ(Ω), γ∈[0,1], by Theorem 1 we get that (11) holds whenever
qi≤pi+αi
1−γfor i= 1,...,n.
4
So far, the best known result for the absence of Lavrentiev’s phenomenon in this kind of anisotropic double-
phase spaces was due to [9] and covered the ranges for the exponents qi
pi≤1 + αi
n,i= 1,...,n. Note that
when pi< n, this range of admissible piand qiis smaller than (12). This means that in the current study,
we admit a worse modulus of continuity of Hand still provide (11). See also Remark 4.2.
In Corollary 1.2, we give more examples where the modulus of continuity is essentially improved com-
pared to [9], including anisotropic variable exponent double-phase functionals and functionals Orlicz phases.
Remark 4.1 illustrates our results in the setting of the general growth and full anisotropy. To our best knowl-
edge, there is no anisotropic counterexample available so far in the literature.
Let us pass to presenting our main accomplishments. Function space LMis defined in Section 2. Our
main results concern
V1
0LM(Ω) = f∈W1,1
0(Ω) : ∇f∈LM(Ω; Rn).
Absence of Lavrentiev’s phenomenon. Let us formulate our general result that under a balance condition
Lavrentiev’s phenomenon for Fdoes not occur between Sobolev and smooth maps. It is followed by a long
list of examples being of separate attention in the field.
Theorem 1 (Absence of Lavrentiev’s phenomenon).Let Ωbe a bounded Lipschitz domain in Rnand func-
tional Fbe given by (1) with F: Ω ×R×Rn→Rsatisfying (2) for a weak N-function Mthat is continuous
with respect to both variables and such that M∈∆2. Assume further that Fis measurable with respect to the
first variable and continuous with respect to the second and the third variable. Then we observe the absence
of Lavrentiev’s phenomenon in the following cases.
(i) If γ= 0,u0∈V1LM(Ω), and Msatisfies condition (Biso
γ)or (Bort
γ), then
inf
u0+V1
0LM(Ω) F[u] = inf
u0+C∞
c(Ω) F[u].(13)
(ii) If γ∈(0,1),u0∈V1LM(Ω) ∩C0,γ(Ω), and Msatisfies condition (Biso
γ)or (Bort
γ), then
inf
u0+V1
0LM(Ω)∩C0,γ (Ω) F[u] = inf
u0+C∞
c(Ω) F[u].(14)
(iii) If u0∈V1LM(Ω) ∩C0,1(Ω), then
inf
u0+V1
0LM(Ω)∩C0,1(Ω) F[u] = inf
u0+C∞
c(Ω) F[u].(15)
Remark 1.1. Conditions (Biso
γ)and (Bort
γ)are always satisfied when M(x, ξ) = M(ξ)including anisotropic
functions. In turn, functionals driven by such Mnever face Lavrentiev’s phenomenon.
By a direct application of Theorem 1 in the case of particular choices of Mwe get the following corollary.
Corollary 1.2. Let Ωbe a bounded Lipschitz domain in Rn,γ∈[0,1), and functional Fbe given by (1)
with F: Ω ×R×Rn→Rsatisfying (2) for a weak N-function M. Assume further that Fis measurable with
respect to the first variable and continuous with respect to the second and the third variable. Then we have
the following isotropic consequences of Theorem 1.
5
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arXiv:2210.15217v1[math.AP]27Oct2022AbsenceofLavrentiev’sgapforanisotropicfunctionalsMichalBorowskia,IwonaChlebickaa,∗,Bla˙zejMiasojedowaaInstituteofAppliedMathematicsandMechanics,UniversityofWarsaw,ul.Banacha2,02-097Warsaw,PolandAbstractWeestablishtheabsenceoftheLavrentievgapbetweenSobolevandsmooth...
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