With a growing world population and the concentration of citizens in big cities new methods of agriculture are required. Vertical farming attracts more attention in mending these growing problems. To enable a widespread use of low-cost hydroponic systems this study investigates minimal requirements for plants (different herbs and vegetables) in such a hydroponic vertical farming system and the suitability of textiles as sustainable substrates. Therefore, this study aims to investigate plant stress levels, germination rates and water usage in a low-cost hydroponic system with no special lightning in principle comparison with indoor cultivation in soil. The results of the pulse-amplitude-modulation (PAM) measurements as measure of photosynthetic performance indicate that the plants were equally stressed in hydroponic and in soil cultivation. In this respect, the photosynthetic quantum yield in both cultivation systems is on average only slightly lower than the values expected under optimal conditions. It was observed that chive and lovage not only had a significantly higher germination rate in the hydroponic system but also accumulated significantly more fresh as well as dry biomass, while spinach, thyme and marjoram showed higher germination rates in soil cultivation. The water consumption in the setup was considerably higher for the hydroponic system compared to indoor soil cultivation.
Citation: Bennet Brockhagen, Fabian Schoden, Jan Lukas Storck, Timo Grothe, Christian Eßelmann, Robin Böttjer, Anke Rattenholl, Frank Gudermann. Investigating minimal requirements for plants on textile substrates in low-cost hydroponic systems[J]. AIMS Bioengineering, 2021, 8(2): 173-191. doi: 10.3934/bioeng.2021016
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With a growing world population and the concentration of citizens in big cities new methods of agriculture are required. Vertical farming attracts more attention in mending these growing problems. To enable a widespread use of low-cost hydroponic systems this study investigates minimal requirements for plants (different herbs and vegetables) in such a hydroponic vertical farming system and the suitability of textiles as sustainable substrates. Therefore, this study aims to investigate plant stress levels, germination rates and water usage in a low-cost hydroponic system with no special lightning in principle comparison with indoor cultivation in soil. The results of the pulse-amplitude-modulation (PAM) measurements as measure of photosynthetic performance indicate that the plants were equally stressed in hydroponic and in soil cultivation. In this respect, the photosynthetic quantum yield in both cultivation systems is on average only slightly lower than the values expected under optimal conditions. It was observed that chive and lovage not only had a significantly higher germination rate in the hydroponic system but also accumulated significantly more fresh as well as dry biomass, while spinach, thyme and marjoram showed higher germination rates in soil cultivation. The water consumption in the setup was considerably higher for the hydroponic system compared to indoor soil cultivation.
In this paper we compare quenched stochastic two-scale convergence [38] with the notion of stochastic unfolding [30,19], which is equivalent to stochastic two-scale convergence in the mean [6]. In particular, we introduce the concept of stochastic two-scale Young measures to relate quenched stochastic two-scale limits with the mean limit and discuss examples of convex homogenization problems that can be treated with two-scale convergence in the mean, but not conveniently in the quenched setting of two-scale convergence.
Two-scale convergence has been introduced in [32,1,25] for homogenization problems (partial differential equations or variational problems) with periodic coefficients. The essence of two-scale convergence is that the two-scale limit of an oscillatory sequence captures oscillations that emerge along the sequence and that are to leading order periodic on a definite microscale, typically denoted by
In this paper we are interested in stochastic homogenization, i.e. problems with random coefficients with a stationary distribution. The first stochastic homogenization result has been obtained by Papanicolaou and Varadhan in [33] (and independently by Kozlov [23]) for linear, elliptic equations with stationary and ergodic random coefficients on
Eωε(u)=∫QV(τxεω,∇u(x))−f(x)u(x)dx |
where
Ehom(u)=∫QVhom(∇u(x))−f(x)u(x)dx, |
where
● In the mean setting, minimizers
● In the quenched setting, one studies the limiting behavior of a minimizer
Similarly, two variants of stochastic two-scale convergence have been introduced as generalizations of periodic two-scale convergence (for the sake of brevity, we restrict the following review to the Hilbert-space case
● In [6,2] the mean variant has been introduced as follows: We say that a sequence of random fields
limε→0∫Ω×Quε(ω,x)φ(τxεω,x)dP(ω)dx=∫Ω×Qu(ω,x)φ(ω,x)dP(ω)dx, | (1) |
for all admissible test functions
● More recently, Zhikov and Pyatnitskii introduced in [38] a quenched variant: We say that a sequence
limε→0∫Quε(x)φ(τxεω0,x)dx=∫Ω×Qu(ω,x)φ(ω,x)dP(ω)dx, |
for all admissible test functions
Similarly to the periodic case, stochastic two-scale convergence in the mean can be rephrased with help of a transformation operator, see [34,19,30], where the stochastic unfolding operator
Tεu(ω,x)=u(τ−xεω,x), | (2) |
has been introduced. As in the periodic case, it is a linear isometry and it turns out that for a bounded sequence
In the present paper we compare the different notions of stochastic two-scale convergence. Although the mean and quenched notion of two-scale convergence look quite similar, it is non-trivial to relate both. As a main result, we introduce stochastic two-scale Young measures as a tool to compare quenched and mean limits, see Theorem 3.12. The construction invokes a metric characterization of quenched stochastic two-scale convergence, which is a tool of independent interest, see Lemma 3.6. As an application we demonstrate how to lift a mean two-scale homogenization result to a quenched statement, see Section 4.3. Moreover, we present two examples that can only be conveniently treated with the mean notion of two-scale convergence. In the first example, see Section 4.1, the assumption of ergodicity is dropped (as it is natural in the context of periodic representative volume approximation schemes). In the second example we consider a model that invokes a mean field interaction in form of a variance-type regularization of a convex integral functional with degenerate growth, see Section 4.2.
Structure of the paper. In the following section we present the standard setting for stochastic homogenization. In Section 3 we provide the main properties of the stochastic unfolding method, present the most important facts about quenched two-scale convergence and present our results about Young measures. In Section 4 we present examples of stochastic homogenization and applications of the methods developed in this paper.
In the following we briefly recall the standard setting for stochastic homogenization. Throughout the entire paper we assume the following:
Assumption 2.1. Let
(i)(Group property).
(ii)(Measure preservation).
(iii)(Measurability).
We write
Lemma 2.2 (Stationary extension). Let
‖Sφ‖Lp(Ω×Q)=|Q|1p‖φ‖Lp(Ω). |
We say
every shift invariant A∈F (i.e. τxA=A for all x∈Rd) satisfies P(A)∈{0,1}. |
In this case the celebrated Birkhoff's ergodic theorem applies, which we recall in the following form:
Theorem 2.3(Birkhoff's ergodic Theorem [12,Theorem 10.2.II]). Let
limε→0∫QSφ(ω,xε)dx=|Q|⟨φ⟩. | (3) |
Furthermore, if
Stochastic gradient. For
Diφ=limh→0Uheiφ−φh, |
which we refer to as stochastic derivative.
Lppot(Ω):=¯R(D)⊂Lp(Ω)d | (4) |
the closure of the range of
Lpinv(Ω)={φ∈Lp(Ω):Uxφ=φfor all x∈Rd}, |
and denote by
⟨⋅⟩ is ergodic ⇔ Lpinv(Ω)≃R ⇔ Pinvf=⟨f⟩. |
Random fields. We introduce function spaces for functions defined on
Xa⊗Y:={n∑i=1φiηi:φi∈X,ηi∈Y,n∈N} |
in
Xa⊗Y:={n∑i=1φiηi:φi∈X,ηi∈Y,n∈N} |
in
In the following we first discuss two notions of stochastic two-scale convergence and their connection through Young measures. In particular, Section 3.1 is devoted to the introduction of the stochastic unfolding operator and its most important properties. In Section 3.2 we discuss quenched two-scale convergence and its properties. Section 3.3 presents the results about Young measures.
In the following we briefly introduce the stochastic unfolding operator and provide its main properties, for the proofs and detailed studies we refer to [19,30,31,34].
Lemma 3.1([19,Lemma 3.1]). Let
Tε:Lp(Ω×Q)→Lp(Ω×Q) |
such that
∀u∈Lp(Ω)a⊗Lp(Q):(Tεu)(ω,x)=u(τ−xεω,x)a.e.inΩ×Q. |
Moreover, its adjoint is the unique linear isometric isomorphism
Definition 3.2 (Unfolding and two-scale convergence in the mean). The operator
Tεuε→u weakly (strongly) in Lp(Ω×Q). |
In this case we write
Remark 1 (Equivalence to stochastic two-scale convergence in the mean). Stochastic two-scale convergence in the mean was introduced in [6]. In particular, it is said that a sequence of random fields
limε→0⟨∫Quε(ω,x)φ(τxεω,x)dx⟩=⟨∫Qu(ω,x)φ(ω,x)dx⟩, | (5) |
for any
⟨∫Quε(T∗εφ)dx⟩=⟨∫Q(Tεuε)φdx⟩, | (6) |
which proves the equivalence.
We summarize some of the main properties:
Proposition 1 (Main properties). Let
(i)(Compactness, [19,Lemma 3.4].) If
(ii)(Limits of gradients, [19,Proposition 3.7]) Let
uε2⇀uinLp(Ω×Q),∇uε2⇀∇u+χinLp(Ω×Q)d. | (7) |
If, additionally,
(iii)(Recovery sequences, [19,Lemma 4.3]) Let
uε2→u,∇uε2→∇u+χinLp(Ω×Q). |
If additionally
In this section, we recall the concept of quenched stochastic two-scale convergence (see [38,16]). The notion of quenched stochastic two-scale convergence is based on the individual ergodic theorem, see Theorem 2.3. We thus assume throughout this section that
⟨⋅⟩isergodic. |
Moreover, throughout this section we fix exponents
●
●
We denote by
A:={φ(ω,x)=φΩ(ω)φQ(x):φΩ∈DΩ,φQ∈DQ} |
the set of simple tensor products (a countable set), and by
D0:={m∑j=1λjφj:m∈N,λ1,…,λm∈Q,φ1,…,φm∈A}. |
We finally set
D:=spanA=spanD0and¯D:=span(DQ) |
(the span of
Ω0the set of admissible realizations; |
it is a set of full measure determined by the following lemma:
Lemma 3.3. There exists a measurable set
lim supε→0‖(T∗εφ)(ω0,⋅)‖Lr(Q)≤‖φ‖Brandlimε→0∫QT∗ε(φφ′)(ω0,x)dx=⟨∫Q(φφ′)(ω0,x)dx⟩. |
Proof. This is a simple consequence of Theorem 2.3 and the fact that
For the rest of the section
The idea of quenched stochastic two-scale convergence is similar to periodic two-scale convergence: We associate with a bounded sequence
Definition 3.4 (quenched two-scale limit, cf. [38,17]). Let
limε→0∫Quε(x)(T∗εφ)(ω0,x)dx=∫Ω∫Qu(x,ω)φ(ω,x)dxdP(ω). | (8) |
Lemma 3.5 (Compactness). Let
‖u‖Bp≤lim infε→0‖uε‖Lp(Q), | (9) |
and
(For the proof see Section 3.2.1).
For our purpose it is convenient to have a metric characterization of two-scale convergence.
Lemma 3.6 (Metric characterization). (i)Let
d(U,V;Lin(D)):=∑j∈N2−j|U(φj)−V(φj)||U(φj)−V(φj)|+1 |
is complete and separable.
(ii)Let
Jω0ε:Lp(Q)→Lin(D),(Jω0εu)(φ):=∫Qu(x)(T∗εφ)(ω0,x)dx,J0:Bp→Lin(D),(J0u)(φ):=⟨∫Quφ⟩. |
Then for any bounded sequence
(For the proof see Section 3.2.1).
Remark 2. Convergence in the metric space
Lemma 3.7 (Strong convergence implies quenched two-scale convergence). Let
(For the proof see Section 3.2.1).
Definition 3.8 (set of quenched two-scale cluster points). For a bounded sequence
We conclude this section with two elementary results on quenched stochastic two-scale convergence:
Lemma 3.9 (Approximation of two-scale limits). Let
(For the proof see Section 3.2.1).
Similar to the slightly different setting in [17] one can prove the following result:
Lemma 3.10 (Two-scale limits of gradients). Let
uε2⇀ω0uand∇uε2⇀ω0∇u+χasε→0. |
Proof of Lemma 3.5. Set
Uε(φ):=∫Quε(x)(T∗εφ)(ω0,x)dx. |
Note that for all
lim supε→0|uε(φ)|≤lim supε→0‖uε‖Lp(Q)‖T∗εφ(ω0,⋅)‖Lq(Q)≤C0‖φ‖Bq. | (10) |
Since
‖u‖Bp=‖U‖(Bq)∗≤C0=lim infε→0‖uε‖Lp(Q). |
Since
∫Quε(x)φQ(x)dx=uε(1ΩφQ)→U(1ΩφQ)=⟨∫Qu(ω,x)φQ(x)dx⟩=∫Q⟨u(x)⟩φQ(x)dx. |
Since
Proof of Lemma 3.6. We use the following notation in this proof
(i) Argument for completeness: If
Argument for separability: Consider the (injective) map
(ii) Let
Proof of Lemma 3.7. This follows from Hölder's inequality and Lemma 3.3, which imply for all
limsupε→0∫Q|(uε(x)−u(x))T∗εφ(ω0,x)|dx≤limsupε→0(‖uε−u‖Lp(Q)(∫Q|T∗εφ(ω0,x)|qdx)1q)=0. |
Proof of Lemma 3.9. Since
∫Qvδ,εT∗εφ(ω0,x)=∫QT∗ε(vδφ)(ω0,x)→⟨∫Qvδφ⟩. |
By appealing to the metric characterization, we can rephrase the last convergence statement as
d(Jω0εvδ,ε,J0u;Lin(D))≤d(Jω0εvδ,ε,J0vδ;Lin(D))+d(J0vδ,J0u;Lin(D)). |
The second term is bounded by
In this section we establish a relation between quenched two-scale convergence and two-scale convergence in the mean (in the sense of Definition 3.2). The relation is established by Young measures: We show that any bounded sequence
⟨⋅⟩ is ergodic. |
Also, throughout this section we fix exponents
Definition 3.11. We say
ω↦νω(B)is measurable for all B∈B(Bp), |
where
Theorem 3.12. Let
νω0is concentrated onCP(ω0,(uε(ω0,⋅))), |
and
lim infε→0‖uε‖pBp≥∫Ω(∫Bp‖v‖pBpdνω(v))dP(ω). |
Moreover, we have
uε2⇀uwhereu:=∫Ω∫Bpvdνω(v)dP(ω). |
Finally, if there exists
uε(ω)2⇀ωˆu(ω)forP−a.a.ω∈Ω. |
(For the proof see Section 3.3.1).
In the opposite direction we observe that quenched two-scale convergence implies two-scale convergence in the mean in the following sense:
Lemma 3.13. Consider a family
(i)There exists
(ii)There exists a sequence
(iii)There exists a bounded sequence
Then
(For the proof see Section 3.3.1).
To compare homogenization of convex integral functionals w.r.t. stochastic two-scale convergence in the mean and in the quenched sense, we appeal to the following result:
Lemma 3.14. Let
liminfε→0∫Ω∫Qh(τxεω,x,uε(ω,x))dxdP(ω)≥∫Ω∫Bp(∫Ω∫Qh(˜ω,x,v(˜ω,x))dxdP(˜ω))dνω(v)dP(ω). | (11) |
(For the proof see Section 3.3.1).
Remark 3. In [18,Lemma 5.1] it is shown that
uε2⇀ω0u⇒lim infε→0∫Qh(τxεω0,x,uε(x))dx≥∫Ω∫Qh(ω,x,u(ω,x))dxdP(ω). | (12) |
We first recall some notions and results of Balder's theory for Young measures [4]. Throughout this section
Definition 3.15. ● We say a function
● A function
● A sequence
● A Young measure in
Theorem 3.16.([4,Theorem I]). Let
lim infε→0∫Ωh(ω,sε(ω))dP(ω)≥∫Ω∫Mh(ω,ξ)dμω(ξ)dP(ω), | (13) |
provided that the negative part
In order to apply the above theorem we require an appropriate metric space in which two-scale convergent sequences and their limits embed:
Lemma 3.17. (i)>We denote by
d((U1,ε1,r1),(U2,ε2,r2);M):=d(U1,U2;Lin(D))+|ε1−ε2|+|r1−r2| |
is a complete, separable metric space.
(ii)For
U={Jω0εufor someu∈Lp(Q)with‖u‖Lp(Q)≤rin the caseε>0,J0ufor someu∈Bpwith‖u‖Bp≤rin the caseε=0. | (14) |
Then
(iii)Let
{‖u‖Lp(Q)=supφ∈¯D, ‖φ‖Bq≤1|U(φ)|ifε>0,‖u‖Bp=supφ∈D, ‖φ‖Bq≤1|U(φ)|ifε=0. | (15) |
(iv)For
‖(U,ε,r)‖ω0:={(supφ∈¯D, ‖φ‖Bq≤1|U(φ)|p+ε+rp)1pif(U,ε,r)∈Mω0,ε>0,(supφ∈D, ‖φ‖Bq≤1|U(φ)|p+rp)1pif(U,ε,r)∈Mω0,ε=0, |
is lower semicontinuous on
(v)For all
(vi)For all
(vii)Let
Proof.(i)This is a direct consequence of Lemma 3.6 (i) and the fact that the product of complete, separable metric spaces remains complete and separable.
(ii)Let
Case 1:
W.l.o.g. we may assume that
‖u0‖Lp(Q)≤lim infk‖uk‖Lp(Q)≤limkrk=r0. | (16) |
Moreover,
Uk(φΩφQ)=∫Quk(x)φQ(x)φΩ(τxεkω0)dx→∫Qu0(x)φQ(x)φΩ(τxε0ω0)dx=Jω0ε0(φΩφQ) |
and thus (by linearity)
Case 2:
In this case there exist a bounded sequence
‖u0‖Bp≤lim infk‖uεk‖Bp≤r0. | (17) |
This implies that
Case 3:
There exists a bounded sequence
‖u0‖Bp≤lim infk‖uk‖Lp(Q)≤r0. | (18) |
Furthermore, Lemma 3.6 implies that
(iii)We first argue that the representation (14) of
To see (15) let
{‖u‖Lp(Q)=supφ∈¯D, ‖φ‖Bq≤1|∫QuφdxdP|=supφ∈¯D, ‖φ‖Bq≤1|U(φ)|if ε>0,‖u‖Bp=supφ∈D, ‖φ‖Bq≤1|∫Ω∫QuφdxdP|=supφ∈D, ‖φ‖Bq≤1|U(φ)|if ε=0. |
(iv)Let
(v)This follows from the definition and duality argument (15).
(vi)Let
● Case 1:
● Case 2:
● Case 3:
In all of these cases we deduce that
(vii)This is a direct consequence of (ii) – (vi), and Lemma 3.6.
Now we are in position to prove Theorem 3.12
Proof of Theorem 3.12. Let
Step 1. (Identification of
sε(ω):={(Jωεuε(ω,⋅),ε,‖uε(ω,⋅)‖Lp(Q))if ω∈Ω0(0,0,0)else. |
We claim that
h(ω,(U,ε,r)):={‖(U,ε,r)‖pωif ω∈Ω0 and (U,ε,r)∈Mω,+∞else. |
From Lemma 3.17 (iv) and (vi) we deduce that
∫Ωh(ω,sε(ω))dP(ω)=2‖uε‖pBp+ε. |
We conclude that
Step 2. (Compactness and definition of
∫Lin(D)f(ξ)dμ1,ω(ξ)=∫Mf(ξ1)dμω(ξ), |
for all
Step 3. (Lower semicontinuity estimate). Note that
h(ω,(U,ε,r)):={supφ∈¯D,‖φ‖Bq≤1|U(φ)|pif ω∈Ω0 and (U,ε,r)∈Mω,ε>0,supφ∈D,‖φ‖Bq≤1|U(φ)|pif ω∈Ω0 and (U,ε,r)∈Mω,ε=0,+∞else. |
defines a normal integrand, as can be seen as in the proof of Lemma 3.17. Thus Theorem 3.16 implies that
lim infε→0∫Ωh(ω,sε(ω))dP(ω)≥∫Ω∫Mh(ω,ξ)dμω(ξ)dP(ω). |
In view of Lemma 3.17 we have
Step 4. (Identification of the two-scale limit in the mean). Let
h(ω,(U,ε,r)):={U(φ)if ω∈Ω0,(U,ε,r)∈Mω,+∞else. |
defines a normal integrand. Since
limε→0∫Ω∫Quε(ω,x)(T∗εφ)(ω,x)dxdP(ω)=limε→0∫Ωh(ω,sε(ω))dP(ω)=∫Ω∫Bph(ω,v)dνω(v)dP(ω)=∫Ω∫Bp⟨∫Qvφ⟩dνω(v)dP(ω). |
Set
limε→0∫Ω∫Quε(ω,x)(T∗εφ)(ω,x)dxdP(ω)=⟨∫Quφ⟩. |
Since
Step 5. Recovery of quenched two-scale convergence. Suppose that
h(ω,(U,ε,r)):=−d(U,J0˜u(ω);Lin(D)) |
is a normal integrand and
lim supε→0∫Ωd(Jωεuε(ω,⋅),J0˜u(ω);Lin(D))dP(ω)=−lim infε→0∫Ωh(ω,sε(ω))dP(ω)≤−∫Ω∫Bph(ω,J0v)dνω(v)dP(ω)=−∫Ωh(ω,J0˜u(ω))dP(ω)=0. |
Thus, there exists a subsequence (not relabeled) such that
Proof of Lemma 3.14. Step 1. Representation of the functional by a lower semicontinuous integrand on
For all
¯h(ω0,s):={∫Qh(τxεω,x,(πω0s)(x))dxif s=(U,ε,s) with ε>0,∫Ω∫Qh(ω,x,(πω0s)(x))dxdP(ω)if s=(U,ε,s) with ε=0. |
We extend
¯h(ω0,sk)=∫Qh(τxεkω0,uk(ω0,x))dx, |
and
¯h(ω0,s0)=∫Ω∫Qh(ω,x,u0(ω,x))dxdP(ω). |
Since
Step 2. Conclusion. As in Step 1 of the proof of Theorem 3.12 we may associate with the sequence
lim infε→0∫Ω∫Qh(τxεω0,uε(ω0,x))dxdP(ω)=lim infε→0∫Ω¯h(ω,sε(ω))dP(ω)≥∫Ω∫M¯h(ω,ξ)dμω(ξ)dP(ω)=∫Ω∫Bp(∫Ω∫Qh(˜ω,x,v(˜ω,x))dxdP(˜ω))dνω(v)dP(ω). |
Proof of Lemma 3.13. By (ii) and (iii) the sequence
In this section we revisit a standard model example of stochastic homogenization of integral functionals from the viewpoint of stochastic two-scale convergence and unfolding. In particular, we discuss two examples of convex homogenization problems that can be treated with stochastic two-scale convergence in the mean, but not with the quenched variant. In the first example in Section 4.1 the randomness is nonergodic and thus quenched two-scale convergence does not apply. In the second example, in Section 4.2, we consider a variance-regularization to treat a convex minimization problem with degenerate growth conditions. In these two examples we also demonstrate the simplicity of using the stochastic unfolding operator. Furthermore, in Section 4.3 we use the results of Section 3.3 to further reveal the structure of the previously obtained limits in the classical ergodic case with non-degenerate growth with help of Young measures. In particular, we show how to lift mean homogenization results to quenched statements.
In this section we consider a nonergodic stationary medium. Such random ensembles arise naturally, e.g., in the context of periodic representative volume element (RVE) approximations, see [13]. For example, we may consider a family of i.i.d. random variables
ω:Rd→R,ω(x)=∑i∈Zd1i+y+◻(x)¯ω(⌊x⌋), |
where
In this section we consider the following situation. Let
(A1)
(A2)
(A3)There exists
1C|F|p−C≤V(ω,x,F)≤C(|F|p+1) |
for a.a.
We consider the functional
Eε:Lp(Ω)⊗W1,p0(Q)→R,Eε(u)=⟨∫QV(τxεω,x,∇u(ω,x))dx⟩. | (19) |
Under assumptions (A1)-(A3), in the limit
E0:(Lpinv(Ω)⊗W1,p0(Q))×(Lppot(Ω)⊗Lp(Q))→R,E0(u,χ)=⟨∫QV(ω,x,∇u(ω,x)+χ(ω,x))dx⟩. | (20) |
Theorem 4.1 (Two-scale homogenization). Let
(i)(Compactness and liminf inequality.) Let
uε2⇀uinLp(Ω×Q),∇uε2⇀∇u+χinLp(Ω×Q), | (21) |
lim infε→0Eε(uε)≥E0(u,χ). | (22) |
(ii)(Limsup inequality.) Let
uε2→uinLp(Ω×Q),∇uε2→∇u+χinLp(Ω×Q), | (23) |
lim supε→0Eε(uε)≤E0(u,χ). | (24) |
Proof of Theorem 4.1. (i) The Poincaré inequality and (A3) imply that
⟨∫QV(τxεω,x,v(ω,x))⟩=⟨∫QV(ω,x,Tεv(ω,x))⟩for any v∈Lp(Ω×Q), | (25) |
and thus using the convexity of
lim infε→0Eε(uε)=lim infε→0⟨∫QV(ω,x,Tε∇uε)⟩≥E0(u,χ). |
(ii) The existence of a sequence
limε→0Eε(uε)=limε→0⟨∫QV(ω,x,Tε∇uε)⟩=E0(u,χ). |
This concludes the claim, in particular, we even show a stronger result stating convergence of the energy.
Remark 4 (Convergence of minimizers). We consider the setting of Theorem 4.1. Let
Iε:Lp(Ω)⊗W1,p0(Q)→R,Iε(u)=Eε(u)−⟨∫Quεfεdx⟩, |
where
limε→0minIε=limε→0Iε(uε)=I0(u,χ)=minI0, |
where
Remark 5 (Uniqueness). If
In this section we consider homogenization of convex functionals with degenerate growth. More precisely, we consider an integrand
(A3')There exists
⟨λ−1p−1⟩p−1<C | (26) |
and
λ(ω)|F|p−C≤V(ω,x,F)≤C(λ(ω)|F|p+1) |
for a.a.
Moreover, we assume that
Eε:L1(Ω×Q)→R∪{∞},Eε(u)=⟨∫QV(τxεω,x,∇u)dx⟩, |
for
‖u‖λε:=⟨∫Qλ(τxεω)|∇u|pdx⟩1p. |
Recently, in [29,20,21] it shown that
Ehom:L1(Q)→R∪{∞},Ehom(u):=∫QVhom(x,∇u(x))dx, |
for
Vhom(x,F)=infχ∈Lppot(Ω)⟨V(ω,x,F+χ(ω))⟩, | (27) |
for
1C′|F|p−C′≤Vhom(x,F)≤C′(|F|p+1). | (28) |
One of the difficulties in the proof of the homogenization result for
Vhom,δ(x,F)=infχ∈Lppot(Ω)⟨V(ω,x,F+χ(ω))+δ|χ(ω)|p⟩. |
It is simple to show that the infimum on the right-hand side is attained by a unique minimizer. We also consider the corresponding regularized homogenized integral functional
Ehom,δ:L1(Q)→R∪{∞},Ehom,δ(u):=∫QVhom,δ(∇u)dx, |
for
Lemma 4.2. Let
limδ→0Vhom,δ(x,F)=Vhom(x,F). | (29) |
Moreover,
(i)If
lim infδ→0Ehom,δ(uδ)≥Ehom(u). |
(ii)For any
uδ→ustrongly inL1(Q),Ehom,δ(uδ)→Ehom(u). |
Proof. Let
⟨V(ω,x,F+χη)⟩≤Vhom(x,F)+η. |
We have
Vhom,δ(x,F)≤⟨V(ω,x,F+χη)+δ|χη|p⟩≤Vhom(x,F)+η+δ⟨|χη|p⟩. |
Letting first
We further consider a sequence
lim infδ→0Ehom,δ(uδ)≥lim infδ→0Ehom(uδ)≥Ehom(u). |
The first inequality follows by (29) and the second is a consequence of the fact that
If
limδ→0Ehom,δ(u)=Ehom(u). |
This means that (ii) holds.
In the following we introduce a variance regularization of the original functional
Eε,δ(u)=⟨∫QV(τxεω,x,∇u(x))+δ|∇u(x)−⟨∇u(x)⟩|pdx⟩, | (30) |
for
Lemma 4.3. Let
⟨∫Q|∇u|⟩p+δ⟨∫Q|∇u|p⟩≤C(Eε,δ(u)+1). |
Proof. By Jensen's and Hölder's inequalities we have
⟨∫Q|∇u|dx⟩p≤|Q|p−1∫Q⟨|∇u|⟩p≤|Q|p−1⟨λ−1p−1ε⟩p−1⟨∫Qλε|∇u|p⟩, |
where we use the notation
⟨∫Q|∇u|dx⟩p≤C(Q,p)(Eε,δ(u)+1). |
In the end, using the variance-regularization we obtain
2−p⟨∫Q|∇u|p⟩≤⟨∫Q|∇u−⟨∇u⟩|p⟩+∫Q⟨|∇u|⟩p≤Cδ(Eε,δ(u)+1)+C(Eε,δ(u)+1). |
This concludes the proof.
The regularization on the
Lemma 4.4. Let
(i)If
lim infδ→0Eε,δ(uδ)≥Eε(u). |
(ii)For any
uδ→ustronglyinL1(Ω×Q),Eε,δ(uδ)→Eε(u). |
Proof. (i) Let
⟨∫Q∇uδηdx⟩=⟨∫Qλ1pε∇uδλ−1pεηdx⟩→⟨∫Qψλ−1pεηdx⟩as ε→0. |
This means that
lim infδ→0Eε,δ(uδ)≥lim infδ→0Eε(uδ)≥Eε(u). |
(ii) For an arbitrary
uη→ustrongly in L1(Ω)⊗W1,10(Q),⟨∫Qλε|∇uη−∇u|pdx⟩→0. |
Using this and the dominated convergence theorem, we conclude that
limη→0Eε(uη)=Eε(u). |
This in turn yields
lim supη→0lim supδ→0|Eε,δ(uη)−Eε(u)|=0. |
We extract a diagonal sequence
The homogenization of the regularized functional
Theorem 4.5. Let
(i)Let
uε2⇀uinLp(Ω×Q),∇uε2⇀∇u+χinLp(Ω×Q). |
(ii)If
lim infε→0Eε,δ(uε)≥Ehom,δ(u). |
(iii)For any
Tεuε→ustrongly inL1(Ω×Q),Eε,δ(uε)→Ehom,δ(u). |
Proof. (i) The statement follows analogously to the proof of Theorem 4.1 (i).
(ii) Let
lim infε→0Eε,δ(uε)≥Ehom,δ(u). |
(ii) This part is analogous to Theorem 4.1 and Remark 5.
The results of Lemmas (4.2) and (4.4), Theorem (4.5) and [29, 20,21] can be summarized in the following commutative diagram:
Eε,δ(δ→0)→Eε(ε→0)8pt↓↓(ε→0)8ptEhom,δ(δ→0)→Ehom |
The arrows denote Mosco convergence in the corresponding convergence regimes.
In this section we demonstrate how to lift homogenization results w.r.t. two-scale convergence in the mean to quenched statements at the example of Section 4.1. Throughout this section we assume that
Eωε(u):=∫QV(τxεω,x,∇u(x))dx, |
with
Before presenting the main result of this section, we remark that in the ergodic case, the limit functional (20) reduces to a single-scale energy
Ehom:W1,p0(Q)→R,Ehom(u)=∫QVhom(x,∇u(x))dx, |
where the homogenized integrand
Vhom(x,F)=infχ∈Lppot(Ω)⟨V(ω,x,F+χ(ω))⟩. | (31) |
In particular, we may obtain an analogous statement to Theorem 4.1 where we replace
Theorem 4.6. Let
uε(ω,⋅)⇀uweaklyinW1,p(Q),uε(ω,⋅)2⇀ωu,∇uε(ω,⋅)2⇀ω∇u+χ,andminEωε=Eωε(uε(ω,⋅))→E0(u,χ)=minE0. |
Remark 6 (Identification of quenched two-scale cluster points). If we combine Theorem 4.6 with the identification of the support of the Young measure in Theorem 3.12 we conclude the following: There exists a subsequence such that
In the proof of Theorem 4.6 we combine homogenization in the mean in form of Theorem 4.1, the connection to quenched two-scale limits via Young measures in form of Theorem 3.12, and a recent result described in Remark 3 by Nesenenko and the first author.
Proof of Theorem 4.6. Step 1. (Identification of the support of
Since
limε→0minEε=limε→0Eε(uε)=E0(u,χ)=minE0. | (32) |
In particular, the sequence
limε→0Eε(uε)≥∫Ω∫B(∫Ω∫QV(˜ω,x,ξ2)dxdP(˜ω))νω(dξ)dP(ω). |
In view of (32) and the fact that
minE0≥∫Ω∫B0E0(ξ1,ξ2−∇ξ1)νω(dξ)dP(ω)≥minE0∫Ω∫B0νω(dξ)dP(ω). |
Since
Step 2. (The strictly convex case).
The uniqueness of
lim infε→0Eωε(uε(ω,⋅))≥E0(u,χ)=minE0. |
On the other hand, since
limε→0minEωε=limε→0Eωε(uε(ω,⋅))=E0(u,χ)=minE0. |
The authors thank Alexander Mielke for fruitful discussions and valuable comments. MH has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 "Scaling Cascades in Complex Systems", Project C05 "Effective models for materials and interfaces with multiple scales". SN and MV acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 405009441.
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