Homogenization of high-contrast composites under differential constraints
-
Elisa Davoli
,
Martin Kružík
und
Valerio Pagliari
Abstract
We derive, by means of variational techniques, a limiting description
for a class of integral functionals under linear differential constraints.
The functionals are designed to encode the energy of a high-contrast composite, that is,
a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix
whose cavities are occupied by a filling with very different physical properties.
Our main result provides a Γ-convergence analysis as the periodicity tends to zero, and
shows that the variational limit of the functionals at stake is the sum of two contributions,
one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions.
As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to the standard topologies in
1 Introduction
The goal of this contribution is to derive a unified limiting description for a class of integral functionals that are inspired by the physics of high-contrast composites and that are evaluated on fields subject to certain differential constraints.
High-contrast materials are characterized by the property that their microscopic physical features may change abruptly from point to point. In the case of a binary periodic medium of this kind, we have two relevant microscales: the periodicity of the microstructure and the high-contrast parameter, which encodes the strong (high-contrast) difference in the properties of the two components. Typically, the scale at which periodicity is observed is much smaller than the size of the specimen. It is hence natural, from a mathematical viewpoint, to study the asymptotics of the quantities that describe the system in the limit of vanishing period. This procedure is called homogenization. It allows to make predictions on the effective character of heterogeneous media when no experiments are available, and eases numerical simulations by reducing the degrees of freedom of the problem. Therefore, mathematical homogenization has been a lively field of investigation for many decades. We refer, e.g., to the monographs [15, 33] for a thorough introduction to the subject.
In this paper we tackle the case in which
both the periodicity and the high-contrast are quantified by a small number
The novelty of our contribution consists
in tackling homogenization of high-contrast materials in the setting of
For
To depict the fine texture of the material in mathematical terms,
we consider the periodicity cell
and
which correspond respectively to the inclusions and to the surrounding matrix (see Figure 2).
Hereafter, we will systematically adopt the adjectives “soft” and “stiff”
to refer respectively to
The subdivision of the unit cube Q
into the “soft” region
The reference set Ω and its microscopic structure.
The collection of the “soft” inclusions
The main goal of this contribution is to characterize the asymptotic behavior as
evaluated on fields
Even though
In the formula above,
Our first main result is presented in Theorem 6, and, loosely speaking, states the following:
Theorem.
The functional
Key techniques to establish the theorem are p-equiintegrability arguments, as well as the notions of two-scale convergence [2] and of periodic unfolding [32, 31, 78, 79].
The proof relies essentially on the possibility of decoupling the behavior of the material in the “soft” and “stiff” contributions.
The Γ-convergence of the “stiff” portion is a corollary of homogenization results in the setting of
The decoupling of the system into regions having different material features hinges on careful compactness and splitting arguments. The latter are in turn based on two essential properties of the differential constraint
The analysis of the differential constraints satisfying the aforementioned Assumptions 1 and 2 is the focus of Section 7. A special case in which the first requirement holds is that of operators
Theorem.
Let Ω be a “nice” set and let
We refer to Theorems 10 and 11 for the precise formulation of the result,
as well as to Section 7 for a broader discussion on the topic.
We point out that assuming the existence of potentials imposes some topological constraints on the set Ω.
An example is easily provided by the case
Before proceeding with the mathematical details of our analysis, we conclude the section with some bibliographical notes.
The very first mathematical analysis of high-contrast materials was developed in the seminal work [8] by G. L. Auriault, who employed formal asymptotic expansions. In the early 2000, V. V. Zhikov [80] introduced a novel approach by extending classical two-scale techniques [2] to the setting of PDEs with rapidly oscillating coefficients. The study of high-contrast problems has been ever since at the center of an extensive scientific effort, with applications ranging from elastodynamics [75] to Maxwell’s equations [13, 27]. A full characterization of the settings of linear elasticity and of conducting materials was provided in [22, 21], while the nonlinear elastic setting was considered in [25, 26] (see also [17]). The effects of microstructure on free-discontinuity functionals have been studied when high-contrast behavior appears in bulk terms only [10, 11], when it is just in surface terms [72, 73], and when both contributions are affected [18]. Mumford–Shah energies with high-contrast surface contributions have been recently characterized in [66]. The analysis of variational models for layered high-contrast materials was undertaken in [29, 30, 39].
As for the notion of
1.1 Structure of the paper
The plan of the paper is as follows.
In Section 2,
we describe the setting of the problem and the related assumptions, and we formulate the precise statements of our main results, i.e., Theorems 6, 10, and 11. To lay the ground for the proofs, we collect auxiliary assertions and properties concerning measurable selection criteria,
2 Setting and main results
This section is devoted to the set-up of our analysis and to the presentation of the main achievements of this contribution. We will first fix the notation and the hypotheses used throughout the paper. The major results will be summarized in Section 2.3.
2.1 Energy functionals
Let
Similarly, we let
denote the characteristic function
We assume that
Each
The function
There exist
for a fixed
There exists
where p is the same as in (H3).
There exists
Remark 1 (Degeneracy of the “soft” component).
We draw again attention to the fact that,
in spite of the standard coercivity and growth of
and the ratio
Owing to the growth conditions prescribed by hypothesis (H3),
the natural environment for our problem is the space
We endow the previous space with the norm of
2.2 The differential constraint
Let
where
Our fundamental requirement on
has rank equal to r.
We observe that for every open set
where
and for
where
When such a relation holds,
we say that u is
In addition to the constant rank hypothesis,
we need to consider further restrictions on the class of operators
Assumption 1 (Null-average).
For all
Assumption 2 (
A
-free extension).
There exist an ε-independent constant
if
We recall that
whenever
Remark 2 (On Assumption 2).
In problems involving perforated media the use of extension techniques is fairly common (see, e.g., [1] or [15, Chapter 19]), and Assumption 2 is akin to others in the literature. For comparison, we mention [46, Definition 1.4]. The main differences consist in: (1) the fact that the extension operator here is assumed to exist on a periodically perforated domain, with constants independent of the periodicity parameter ε, (2) the fact that the existence of an extended p-equiintegrable sequence here is only required when starting from a sequence that also had p-equiintegrability properties. We refer to [46] for a general discussion on the connections between the existence of extension operators from an open set to a bigger surrounding domain, and the theory of DiPerna–Majda measures.
A notion of
As far as our analysis is concerned,
items (1)–(3) in Assumption 2 are among the main ingredients
in the proof of Proposition 1, and
without them we would not be able
to establish high-contrast compactness for sequences with equibounded energy
(see Definition 5).
On the contrary, the preservation of p-equiintegrability is not required at this stage,
whereas it is used in the proof of our main Γ-convergence result
(see the proofs of Propositions 9 and 4).
Nonetheless, point (4) in Assumption 2 could be dispensed with
in some simplified versions of our problem,
e.g., when
For the sake of brevity, it is convenient to give a name to the class of operators addressed in our study:
Definition 3.
Let
It is well known that
the family of constant rank operators is quite large [48];
as for the subclass of the admissible ones,
we prove that it is nonempty in Section 7
where, in particular, we analyze the cases
of the curl, of the
2.3 Main results
The first main result of this paper, Theorem 6 below, is the characterization
of the asymptotic behavior as
Definition 4.
Let X be a set endowed with a notion of convergence.
We say that the family
for any sequence
there exists a sequence
We have already remarked that
the natural domain of the functionals (2.8) is contained in
Definition 5.
We say that a family
Some comments are in order.
The reason why
the definition above is made up of two requirements
is that
there is no standard convergence which is able to capture alone
the behaviors of both components.
On one hand, the convergence of extensions is related to the coercivity of the “stiff” part,
and it is a somehow customary requirement in homogenization
(see Section 2.4);
on the other, the convergence
If we select
As a last remark on Definition 5, note that
it is obvious that
the weak convergence in
It will be convenient to have a special notation
for spaces of functions satisfying the differential constraint encoded by
and
Note that limit points of high-contrast convergent sequences are automatically in
where
and
(2.7)
We are now ready to state our main convergence result.
Theorem 6.
Let
Let us assume that
If a family
For any
For any
By standard Γ-convergence techniques, we find the following corollary of Theorem 6:
Corollary 7.
Under the same assumptions and notation of Theorem 6,
if
then there exists a subsequence of
Our proof of Theorem 6 relies on four main intermediate results:
a compactness analysis combined with the characterization of high-contrast limits of
the identification of an optimal lower bound for the “soft” part of the energy, cf. Proposition 8,
the identification of the limiting description for the “stiff” part, cf. Proposition 9.
The proof of Theorem 6 is exposed in Section 4.
Let us now present shortly the results that we obtain for the “soft” and “stiff” parts of the energy. The splitting procedure in Section 4 yields the functionals
where the densities
Precisely, we establish the following.
Proposition 8 (Γ-limit of the “soft” component).
Let
for every p - equiintegrable sequence
there exists a sequence
We point out that the p-equiintegrability condition in the first part of the previous statement does not affect the generality of the result, as Proposition 4 will prove. For more details on this point, the reader is referred to the splitting argument in Section 4 below.
For what concerns the functionals
Proposition 9 (Γ-limit of the “stiff” component).
Let
The last section is focused on the study of admissible differential operators.
Differently from the other parts of the paper, the analysis of Section 7 encompasses all linear, k-th order, homogeneous differential operators with constant coefficients and constant rank.
It will be useful to consider those operators
Theorem 10 (Existence of potentials for
A
-free maps).
Let
An immediate consequence of this result is that
Assumption 1 holds
whenever the unit cube Q is an
Our third and last main result shows that for operators admitting a potential
satisfying a suitable Korn-type inequality it is possible to define an extension operator preserving
Theorem 11.
Let
where
for all
for any bounded, connected and open set
(2.13)
Then there exist a constant
It is fundamental to observe that the maps under consideration in the theorem above are already a priori in the kernel of the operator
2.4 Comparison with other works
Before getting to the heart of the matter, we take the chance to compare our main convergence result, Theorem 6, with similar ones in the literature.
When
This notion of convergence is inspired by the one considered by X. Pellet, L. Scardia and C. I. Zeppieri in [66] to deal with high-contrast Mumford–Shah energies. Unlike that contribution, aside from the role of the differential constraint
We also stress that, in the absence of growth conditions from below on the “soft” part of the energy, the weak
For a general differential constraint
The example below shows that
Example 12.
Let
we cannot conclude that for all
as the following counterexample proves.
For
for every
In this way, requirements (H1) and (H3)–(H5) are met, and
because
In order to disprove (2.14), we pick
where
For such particular choices, we see that the left-hand side of (2.14) equals 0, while for the right-hand side we find
3 Preliminaries
We gather in this section some preliminary definitions and results to be used later in the paper.
3.1 Measurable selection arguments
We recall here a measurable selection criterion that we will invoke in the proof of Corollary 3. For a thorough discussion on the topic we refer to the book by C. Castaing and M. Valadier [24].
Proposition 1 ([24, Theorem III.6]).
Let S be a multifunction
defined on the measurable space
3.2
𝒜
-free fields and two-scale limits
In this subsection we include some useful tools for the variational analysis of problems under differential constraints.
We recall two results about
(asymptotically)
Lemma 2 (
A
-free decomposition, [48, Lemma 2.15]).
Let
Lemma 3 (
A
-free periodic extension, [45, Lemma 2.8]).
Let
To perform our analysis, we need to track how the differential constraint behaves along sequences that converge in the sense of Definition 5 above. In this respect, the notion of two-scale convergence [65, 2] will be crucial:
Definition 4.
We say that a sequence
In this case we write
We say that
For the sake of completeness, we record in the following lines the properties of two-scale convergence to be exploited in this work; we refer to [2, 78] for further reading.
Lemma 5.
Let
If
If
If
If
is p - equiintegrable and
It is well known that
two-scale convergence in
Lemma 6.
For
where
Moreover, if
We refer to [32, 31, 78, 79] for further properties of the unfolding operator.
A characterization of
weak two-scale limits of
Proposition 7 ([45, Theorem 1.2]).
A function
As a last technical tool, we show that
the unfolding of an
Note that
Lemma 8.
Let
Moreover, if also
Proof.
Let
where
Since
Assume further that
Let now
for almost every
3.3 Fourier analysis
The study of the class of admissible operators in Section 7 is grounded on the theory of Fourier multipliers. For a comprehensive treatment of the matter, we refer to the monographs [76, 51]; here we limit ourselves to a short recollection of useful properties.
We let
be its Fourier transform.
We also denote by
can be extended to a bounded operator from
In the scalar case, the criterion reads:
Proposition 9 ([63, Theorem 2 in Appendix]).
Let
then m is an
A helpful consequence of the previous result is:
Corollary 10.
Let
Multipliers stand as basics examples of pseudo-differential operators.
Their theory, in turn, can be used to characterize Sobolev spaces as follows.
Given
We then say that a distribution u belongs to
and we endow the space with the norm
and that the norms
4 Proof of Theorem 6
This section is devoted to the proof of Theorem 6,
which stands as our principal result
about the asymptotics of the energy functionals
The section is organized as follows. We first address the compactness result in Proposition 1, second we show how to reduce our problem to the sub-problems regarding the “soft” and the “stiff” components, then we prove the Γ-convergence statement in Theorem 6.
4.1 Compactness
The next proposition shows that sequences which are equibounded in energy have converging subsequences with respect to the high-contrast of convergence introduced in Definition 5. Note that, for the result to hold, neither boundary condition nor analogues of loading terms need to be taken into account.
Since according to position (2.8)
only
Proposition 1 (High-contrast limits of
A
-free sequences).
Let
There exist a (non-relabeled) subsequence and two maps
(4.1)The following differential constraints are fulfilled:
(4.2)(4.3)(4.4)If
(4.5)(4.6)(4.7)If additionally
and
Proof.
If
(1) By Lemma 5, since bounded sequences in
For what concerns (4.1),
we observe that,
thanks to the relation between weak two-scale convergence and weak
Further, it holds that
whence, by the first two convergences in (4.9), we find
In particular, also the first equality in (4.1) is satisfied.
By linearity,
the previous considerations also entail that
(2) Since
where the latter equality is due to the choice of the support of ψ. Therefore, the third convergence in (4.9) yields
which in turn implies (4.4).
(3) For every
(4.12)
Since Assumption 2 grants that
i.e., (4.6) is proved.
Finally, to obtain (4.7),
we combine the fact that
(4) By (4.7) and Assumption 1, we deduce that
Then, by (4.5),
where we used (4.5) and (4.1).
This also shows that
As already anticipated in Remark 2, we note that the preservation of p-equiintegrability granted by Assumption 2 is not needed to prove item (3) of the previous proposition. We will instead resort to it to establish Proposition 4 below.
4.2 Splitting
In the light of the of previous subsection, we know that those sequences which are equibounded in energy are precompact with respect to the high-contrast convergence, and that their cluster points fulfill suitable differential constraints. Following the approach of M. Cherdantsev and K. D. Cherednichenko [25], the next step is to show that the asymptotic behavior of the energy along such sequences coincides with the sum of those of the energies of two decoupled systems, one sitting on the “soft” inclusions, the other on the “stiff” matrix.
To favor intuition,
consider a family
Setting
where
(notice that
Lemma 2 (Splitting).
Let
and
Proof.
As a consequence of the fact that
Only the estimates involving the semilimits are now left to prove.
It suffices to show that
Hölder’s inequality yields
We conclude this section
by showing that,
thanks to the
Lemma 3.
Let
Proof.
Despite the dependence of the energy density
We first note that
(4.15) is left unchanged
if we replace
By employing the definition of
Since
where the second equality is a consequence of (4.16),
the second to last inequality is due to (H3),
and the last one follows from
The improved variant of Lemma 2 for the liminf inequality in (4.14) reads as follows.
Proposition 4 (Refined splitting for the liminf).
Suppose
Proof.
The equiboundedness in energy entails that
By Lemma 2,
there exist a (not relabeled) subsequence of
Note that the last property is not mentioned explicitly in the statement of Lemma 2, but it is a byproduct of the construction in its proof (see [48, Lemma 2.15]).
For every
Thanks to (2) above,
the convergence to 0 following from (3) above and (4.19).
In view of the assumptions on
To complete the proof of the corollary, it suffices now show that for
where
Then, in view of (2), of Assumption 2, and of (3), we deduce (4.20). ∎
4.3 Γ-convergence
In this short subsection we tackle the proofs of Theorem 6 and Corollary 7.
Proof of Theorem 6.
Statement (1), concerned with the high-contrast compactness of families with equibounded energy, follows from Proposition 1.
Let now
Finally, we turn to (3).
For
Then, setting
At this stage, the convergence of infima and of minimizers easily follows.
5 Asymptotics for the soft component
This section is devoted to the proof of Proposition 8.
We address the asymptotic analysis of the energy stored in the “soft” component of the system, that is,
where the supremum is meant to run over all open sets that are compactly contained in Ω,
and, for any open
Remark 1 (On the constant
α
0
).
By the definition of
Since
and, more generally, that
when
We separate the discussion of the Γ-liminf and of the Γ-limsup inequalities.
5.1 Γ-liminf inequality
The proof of the Γ-liminf inequality relies on the compactness and splitting results of Section 4, on unfolding techniques, as well as on the following intermediate statement.
Lemma 2.
Let
and for all open
Then it holds
Proof.
The proof is divided into two steps.
Step 1: A simplified setting. We start by establishing an inequality for the case in which the densities on the right-hand side do not depend on ε (cf. [25, Lemma 20]), that is, we first prove that
According to the definition of
In view of (5.3) and (5.4),
Thanks to the p-equiintegrability of
Step 2: The general case.
We adapt the argument in the proof of [37, Theorem 5.14],
relying on the almost everywhere convergence of
Fix
Besides, since
Let now
In view of (H5), for any such
By (5.8),
for every
for every
In view of (5.5) we deduce
where the first inequality is due to (5.6), and the second one to (5.10).
The arbitrariness of
We are now ready to tackle the Γ-liminf inequality for the “soft” contribution; the argument is comparable to the one in [25, Lemma 21].
Proposition 3.
Let
Proof.
If the lower limit of
where
We exhibit a formula for
whence
Also, since the unfolding procedure preserves p-equiintegrability,
where
where
By the definition of
Being z an integer-valued vector,
it holds
because for all
From this, owing to (5.11) and (5.12), we can invoke Lemma 2 to infer
In view of the p-equiintegrability of
5.2 Γ-limsup inequality
We now turn to the Γ-limsup inequality for the energy functional associated to the “soft” portion of the material. The optimality of the lower bound identified in Proposition 3 hinges upon the next proposition.
Proposition 4.
Let
it holds that
We postpone the proof of Proposition 4 to the end of the section, and we discuss next how the Γ-limsup inequality is deduced from it.
Corollary 5.
Let
the upper limit inequality
is satisfied.
Proof.
We first remark that if
Fix now
Proposition 4 yields a sequence
the inequality
is satisfied.
Lemma 5 and Assumption 1 entail that
The conclusion hence follows then by the arbitrariness of δ and by standard properties of Γ-convergence (see, e.g., [14, Section 1.2]). ∎
We now address the proof of Proposition 4.
Proof of Proposition 4.
We construct a family
so that
Step 1: Construction of
Recalling (1.1), we define
and, for
In this way,
defines a measurable function which vanishes on
We firstly check that
From Jensen’s inequality we then deduce
As for the convergence of the sequence
where
that is,
which, combined with (5.13), ensures that
In view of the definition of strong two-scale convergence, cf. Definition 4,
we conclude that
Finally, we show that
If
Step 2: Limsup inequality when w is regular.
We have so far shown that,
if
where
We preliminarily notice that, by construction,
Since
where the last identity is due to the fact that
In the light of the previous equalities, we have
Keeping in mind Lemma 5,
we observe that
Step 3: Recovering the general statement.
Let
On the whole, the results above form a proof of the Γ-convergence of the energies
6 Asymptotics for the stiff component
In this section, we characterize the limiting behavior of the family
Theorem 1 ([45, Theorem 1.1]).
Let
Then, for all
Γ
-
converges with respect to the weak
where, for all
As a corollary of the theorem above,
we derive the asymptotics of
Proof of Proposition 9.
For
and we observe that
with
where a and λ are the constants in (H3), and
We observe that Theorem 1 applies to
in the weak
Notice that the term
for all
Let now
By Lemma 3, we obtain the desired upper limit inequality:
7 Admissible differential constraints
In what follows we deepen our analysis of constant rank operators.
We investigate the existence of Sobolev potentials for
In contrast to the previous sections,
here we take into account also differential operators of order greater than 1:
we treat the broader class
of linear, k-th order, homogeneous differential operators with constant coefficients.
To fix the notation, given
Recall that if
We keep in place the constant rank condition,
which currently reads:
there exists
is the symbol of
7.1 Existence of potentials
This subsection is devoted to the proof of Theorem 10.
Loosely speaking, our goal is to prove that,
given an operator
Definition 1.
Let
Analogous definitions concerning extensions of sequences that are (asymptotically)
Remark 2 (On the topology of
A
-extension domains).
The requirements of Definition 1 may act as implicit restrictions on the topology of the set Ω, according to the specific choice of
we find that
For the proof of Theorem 10
we rely heavily on some recent contributions
by B. Raiţă and coauthors,
who established existence of potentials for
Corollary 3.
Assume that
Proof.
For the desired equality to hold, we just need to prove that
Let us fix
We first observe that
for every
is either empty or a closed (and hence complete) subspace of
being measurable, then for every open set
is measurable as well.
It follows that the multifunction
Summing up,
we recovered a measurable function w
from
where the latter inequality follows from the periodicity of w in its second variable. ∎
We devote the remainder of the section
to the proof of Theorem 10
and to the pertaining tools.
We first highlight that
B. Raiţă [69] has recently pointed out that
the kernel of the symbol of any constant rank operator
coincides with the image of the symbol of a suitable operator
Proposition 4 ([69, Theorem 1 and Lemma 2]).
Let
where
Moreover,
suppose that
Observe that if the operator
A second relevant finding related to constant rank operators was obtained
by A. Guerra and B. Raiţă [52],
who showed that this class is exactly the one
in which a sort of Korn inequality holds.
To state their result,
we need to introduce the projection
where
associates to each
Proposition 5 (Korn-type inequality for constant rank operators [52]).
Let
For future use, we remark that
the conclusion remains valid when
To the purpose of constructing Sobolev potentials for
Lemma 6 ([64, Lemma 3.7]).
Let
In the proof of the lemma the constant rank assumption is fundamental. Indeed, it ensures that the map
Remark 7 (Self-adjointness of the projection operator).
As a consequence of Parseval’s formula and
of the self-adjointness of
We next clarify why
Lemma 8 (Properties of the generalized inverse [52, 23]).
Given
where
Let
If
We can now characterize the properties of the projection
Lemma 9.
Let
For all
If
(7.4)In particular, when
Proof.
When
because the image of
and statement (1) holds.
We now turn to point (2). We argue for
where in the latter inequality we exploited statement (1) in Lemma 8.
Since
For every
whence, by the
From the definition of
Eventually, we are able to establish the existence of potentials in a nonsmooth setting and prove the first main result of this section. We will use the following variant of Poincaré–Wirtinger’s inequality, which can be derived as a corollary of Rellich–Kondrachov’s theorem.
Lemma 10.
Let
where
Proof of Theorem 10.
Let us fix
As a first step,
we approximate
By construction,
the functions
Next, we proceed by applying Proposition 5:
since
where
Since the functions
where
Remark 11.
Let
The first inequality easily follows from the identity
for a suitable
7.2
𝒜
-free extensions
The Γ-convergence analysis that we developed in Sections 4–6 is grounded on the splitting argument contained in Lemma 2, which in turn requires Assumption 2. The latter is designed to tackle the periodically perforated structure featured by our problem. As a preliminary step towards Theorem 11, we tackle the simpler scenario in which the parameter ε is neglected. We establish the following:
Theorem 12 (Existence of
A
-free extensions).
Let
for all
there exist a projection operator on the subspace of
(7.8)
Then there exist a map
and a constant
Here, the projection
Remark 13 (On Korn-type inequalities).
After the first version of this manuscript was completed,
A. Arroyo-Rabasa proved in the preprint [6] that
inequalities of the form (2.13)
(that is, (7.8) for every open bounded D)
hold whenever
We will comment on the relationships
between the previous theorem and the theory developed above
at the end of this section, see Remark 16.
For the moment being,
let us just highlight that
the hypothesis concerning existence of potentials for
Lemma 14 (cf. [1, Lemma 2.6]).
Let
and a constant
Proof.
The proof follows the same lines of [1, Lemma 2.6].
Note that, in order to recover item (3) when
We are now in a position to prove the first main result of this section.
Proof of Theorem 12.
Let us fix
If
By construction, then,
To conclude, we need to show that
c being a constant depending on
for some
Arguing as in the proof of Theorem 12, we ground the study about extensions from perforated domains on the corresponding result for Sobolev functions. When the perforations are detached from the boundary, the following holds:
Proposition 15 (cf. [25, Lemma 8] and references therein).
Let
if
The proposition may be proved
by adapting the strategy in the seminal work by E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale [1].
Their result addresses only the case
Thanks to the previous result,
we obtain that
Assumption 2 is fulfilled
by an operator
Proof of Theorem 11.
Let
where
By the definition of
whence, by the definition on
For what concerns the contribution on
we have
The second inequality is a consequence of the construction of
which holds by assumption. We obtain
On the whole, the last estimate and (7.10) yield
Eventually, we turn to (4).
For any
Since
where the last inequality is a consequence of the p-equiintegrability of
In conclusion, we infer the p-equiintegrability of
whence, owing to point (4) in Proposition 15,
if
In light of our analysis, existence of potentials and of extension operators turn out to be almost equivalent; a key role in this respect is played by (7.8). We elaborate on this point in the next remark.
Remark 16 (Relations between existence of potentials and of extension operators).
Here, we compare the main results of the current section.
We let the operator
Conversely,
if
All in all, we see that,
given a constant rank operator
We conclude with a parade of examples.
Example 17 (Curl).
For the choice
with
The symbol of
and
It is easy to check that Assumption 1 holds. Besides, when Ω is simply connected, in view of Proposition 15 all the conditions in Assumption 2 are fulfilled too.
Example 18 (Operators associated with higher-order gradients).
Let Ω be simply connected. Then, for any
Example 19 (The
curl
curl
operator).
Let
where
(the indices are computed modulo 3) and
The symbol of this operator is
and
When
Example 20 (Divergence).
Let us choose
where, for
thus
Proposition 21 (Extensions and potentials for divergence-free vector fields [57]).
Let
Ω is a
For every
where
It is interesting to notice that the authors of [57] derive item (2) from (1), that is, they prove an implication of the form (7.11).
It is proved in [6, Section 4.1] that
Funding source: Austrian Science Fund
Award Identifier / Grant number: F65
Award Identifier / Grant number: V 662
Award Identifier / Grant number: Y1292
Award Identifier / Grant number: I 4052/19-29646L
Funding source: Grantová Agentura České Republiky
Award Identifier / Grant number: I 4052/19-29646L
Funding source: Ministerstvo Školství, Mládeže a Tělovýchovy
Award Identifier / Grant number: 8J19AT013
Funding statement: We acknowledge support from the Austrian Science Fund (FWF) projects F65, V 662, Y1292, from the FWF-GAČR project I 4052/19-29646L, and from the OeAD-WTZ project CZ04/2019 (MŠMT ČR 8J19AT013). Data sharing is not applicable to this article, as no datasets were generated or analysed during the current study.
Acknowledgements
We are thankful to the anonymous referee for their comments, that helped us improve the quality of the manuscript. The authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Program.
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Artikel in diesem Heft
- Frontmatter
- An inequality for the normal derivative of the Lane–Emden ground state
- Homogenization of high-contrast composites under differential constraints
- Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
- The first Grushin eigenvalue on cartesian product domains
- Lipschitz bounds for integral functionals with (p,q)-growth conditions
- Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
- Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets
- Functional inequalities and applications to doubly nonlinear diffusion equations
- Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
- Stability of the ball under volume preserving fractional mean curvature flow
Artikel in diesem Heft
- Frontmatter
- An inequality for the normal derivative of the Lane–Emden ground state
- Homogenization of high-contrast composites under differential constraints
- Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds
- The first Grushin eigenvalue on cartesian product domains
- Lipschitz bounds for integral functionals with (p,q)-growth conditions
- Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains
- Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets
- Functional inequalities and applications to doubly nonlinear diffusion equations
- Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case
- Stability of the ball under volume preserving fractional mean curvature flow
