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Review

An NAD+ dependent/sensitive transcription system: Toward a novel anti-cancer therapy

  • Received: 12 November 2019 Accepted: 27 February 2020 Published: 02 March 2020
  • Cancer is widely known as a “genetic disease” because almost all cancers involve genomic mutations. However, cancer could also be referred to as a metabolic disease because it mainly depends on glycolysis to produce adenosine triphosphate (ATP) and is frequently accompanied by dysfunctions in the mitochondria, which are the primary organelle required in all eukaryotic cells. Importantly, almost all (99%) mitochondrial proteins are encoded by host nuclear genes. Not only cancer but also aging-related diseases, including neurodegenerative diseases, are associated with a decline in mitochondrial functions. Importantly, the nicotinamide adenine dinucleotide (NAD+) level decreases with aging. This molecule not only plays important roles in metabolism but also in the DNA repair system. In this article, we review 5′-upstream regions of mitochondrial function-associated genes to discuss the possibility of whether NAD+ is involved in transcriptional regulations. If the expression of these genes declines with aging, this may cause mitochondrial dysfunction. In this regard, cancer could be referred to as a “transcriptional disease”, and if so, novel therapeutics for cancer will need to be developed.

    Citation: Fumiaki Uchiumi, Akira Sato, Masashi Asai, Sei-ichi Tanuma. An NAD+ dependent/sensitive transcription system: Toward a novel anti-cancer therapy[J]. AIMS Molecular Science, 2020, 7(1): 12-28. doi: 10.3934/molsci.2020002

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  • Cancer is widely known as a “genetic disease” because almost all cancers involve genomic mutations. However, cancer could also be referred to as a metabolic disease because it mainly depends on glycolysis to produce adenosine triphosphate (ATP) and is frequently accompanied by dysfunctions in the mitochondria, which are the primary organelle required in all eukaryotic cells. Importantly, almost all (99%) mitochondrial proteins are encoded by host nuclear genes. Not only cancer but also aging-related diseases, including neurodegenerative diseases, are associated with a decline in mitochondrial functions. Importantly, the nicotinamide adenine dinucleotide (NAD+) level decreases with aging. This molecule not only plays important roles in metabolism but also in the DNA repair system. In this article, we review 5′-upstream regions of mitochondrial function-associated genes to discuss the possibility of whether NAD+ is involved in transcriptional regulations. If the expression of these genes declines with aging, this may cause mitochondrial dysfunction. In this regard, cancer could be referred to as a “transcriptional disease”, and if so, novel therapeutics for cancer will need to be developed.


    We consider a mixture of mesoscale size particles within an ambient fluid that contains locally aligned microscopic scale rod-like molecules, that is a nematic liquid crystals. This type of mixture, which is usually referred to as a nematic colloid material, has emerged in the recent years as the material of choice for testing a number of exciting hypothesis in the design of new materials. An overview of the field, and its applications, from the physical point of view is available in the reviews [17,27].

    The mathematical studies of such systems are still relatively few and focus on two extreme situations:

    ● the effect produced by one colloidal particle, particularly related to the so-called 'defect patterns' that is the strong distortions produced at the interface between the particle and the ambient nematic fluid;

    ● the collective effects produced by the presence of many particle, fairly uniformly distributed, with a focus on the homogenised material.

    In the first direction one should note that defects appear because of the anchoring conditions at the boundary of the particles, which generate topological obstructions [1,2,8,9,10,11,28]. Indeed, there have been a number of works, identifying several physically relevant regimes [1] and the influence of external fields [2].

    Our work focuses on the second direction, namely on long-scale effects produced by the effects of a large number of particles, namely on the homogenisation regime. There have been a couple of works in this direction, on which our builds, namely [5,6,7]. The main novelty of our approach, compared to those in [5,6,7], is that we allow for a much larger class of surface energy densities. We do not assume that the surface energy density is bounded from below and we do consider surface energy densities of quartic growth, which is the maximal growth compatible with the Sobolev embeddings. Surface energy densities of quartic growth have been proposed in the physical literature [3,15,24,26].

    We focus on a regime in which the total volume of the particles is much smaller than that of the ambient nematic environment, known as the dilute regime. Our aim is to provide a mathematical understanding of statements from the physical literature e.g., [19,25] showing that in such a regime the colloidal nematics behave like a homogenised, standard nematic material, but with different (better) properties than those of the original nematic material.

    In our previous work [12], we provided a first approach to these issues and we showed that using periodically distributed identical particles, one can design a suitable surface energy to obtain an apriori designed potential, that models the main physical properties of the material (in particular the nematic-isotropic transition temperature).

    The purpose of these notes is two-fold: on the one hand, to extend the main results of [12] to polydisperse and inhomogenoeus nematic colloids; on the other hand, to explore a regime of parameters that differs from the one considered in [12].

    Realistically, a set of colloidal inclusions will hardly be identical: The particles will differ in their size, shape, or charge. In order to account for polydispersity, we will consider several populations of colloidal inclusions, which may differ in their shape and properties. Moreover, we will not require the centres of mass of the inclusions to be homogeneously distributed in space. In mathematical terms, let P1, P2, , PJ be subsets of R3 (the reference shapes of the inclusions), and let ΩR3 be a bounded, smooth domain (the container). We define

    Pjε:=Njεi=1(xi,jε+εαRi,jεPj)forj{1,,J}, (1.1)

    where α is a positive number, the xi,jε's are points in Ω and the Ri,jε are rotation matrices that satisfy suitable assumptions (see Section 2.1). As in [7,12], we work in the dilute regime, namely we assume that α>1 so that the total volume occupied by the inclusions, |Pε|ε3α3, tends to zero as ε0. However, we also assume that α<3/2 so that the total surface area of the inclusions, σ(Pε)ε2α3, diverges as ε0. We define Pε:=jPjε and Ωε:=ΩPε (the space that is effectively occupied by the nematic liquid crystal). In accordance with the Landau-de Gennes theory, the nematic liquid crystal is described by a tensorial order parameter, that is, a symmetric, trace-free (3×3)-matrix field Q. We consider the free energy functional

    Fε[Q]:=Ωε(fe(Q)+fb(Q))dx+ε32αJj=1Pjεfjs(Q,ν)dσ. (1.2)

    Here, fe, fb are suitable elastic and bulk energy densities (in the Landau-de Gennes theory, fe is typically a positive definite, quadratic form in Q and fb is a quartic polynomial in Q; see Section 2.1), fjs is a surface anchoring energy densities (which may vary for different species of inclusions), and ν denotes the exterior unit normal to Ω. We prove a convergence result for local minimisers of Fε to local minimsers of the effective free energy functional:

    F0[Q]:=Ω(fe(Q)+fb(Q)+fhom(Q,x))dx.

    The "homogenised potential" fhom, which keeps memory of the surface integral, is explicitly computable in terms of the fjs's, the distribution of the centres of mass xi,jε and the rotations Ri,jε. As an application of this result, we show that polydisperse inclusions may be used to mimic the effects of an applied electric field. More precisely, for a pre-assigned parameter WR and a pre-assigned symmetric matrix P, we may tune the shape Pjε of the inclusions and the surface energy densities, so to have in the limit

    fhom(Q)=Wtr(QP).

    When P has the form P=EE for some ER3, this expression may be interpreted as an electrostatic energy density induced by the "effective field" E, up to terms that do not depend on Q.

    Moving beyond the issue of polydispersity we consider another physically restrictive assumption we made in [12], namely concerning the anchoring strength. In (1.2), the scaling of parameters is chosen so to have a factor of ε32α in front of the surface integral, which compensates exactly the growth of the surface area, σ(Pε)ε2α3. However, other choices of the scaling are possible, corresponding to different choices of the anchoring strength at the boundary of the inclusions. One can easily check that having a weaker anchoring, say of the type ε2α3+δ with δ>0 will lead to a vanishing of the homogenized term, so the main interest is to understand what happens for stronger anchoring.To illustrate this possibility, we study the asymptotic behaviour, as ε0, of minimisers of

    Fε,γ[Q]:=Ωε(fe(Q)+fb(Q))dx+ε32αγJj=1Pjεfjs(Q,ν)dσ,

    where γ is a positive parameter. This scaling corresponds to a much stronger surface anchoring and we expect the behaviour of minimisers to be dominated by the surface energy, as ε0. Indeed, we will show that for γ small enough the functionals Fε,γ Γ-converge to the constrained functional

    ˜F(Q):={Ω(fe(Q)+fb(Q))dxiffhom(Q(x),x)=0fora.e.xΩ+otherwise,

    as ε0.

    This result leaves a number of interesting of open problems, the most immediate ones being what is the optimal range of γ for which this holds and, directly related to this, if one gets a different limit for large values of γ.

    The paper is organized as follows: in the following, in Section 2.1 we consider the polydisperse setting and the general homogenisation result. The main results of this section, namely Theorem 2.1 and Proposition 2.2 are presented after the introduction of the mathematical setting, in Subsection 2.1. The proof of the results is provided in Subsection 2.3 after a number of preliminary results, need in the proof, provided in Subsection 2.2.

    In Section 3 we provide an application of the results in Section 2.1, namely showing that, in a polydispersive regime, one can obtain a linear term in the homogenised potential (Proposition 3.2).

    Finally, in Section 4 we study the case when the scaling of the anchoring strength is ε32αγ with γ suitably small, and provide in Theorem 4.3 the Γ-converegence result mentioned above. Its proof is done at the end of the section after a number of preliminary results.

    The Landau-de Gennes Q-tensor. In the Landau-de Gennes theory, the local configuration of a nematic liquid crystal is represented by a symmetric, symmetric, trace-free, real (3×3)-matrix, known as the Q-tensor, which describes the anisotropic optical properties of the medium [13,21]. We denote by S0 the set of matrix as above. For Q, PS0, we denote QP:=tr(QP). This defines a scalar product on S0, and the corresponding norm will be denoted by |Q|:=(tr(Q2))1/2=(i,jQij)1/2.

    The domain. let P1, P2, , PJ be subsets of R3 (the reference shapes of the inclusions), and let ΩR3 be a bounded, smooth domain (the container). We define Pjε as in (1.1), where α, xi,jε, Ri,jε satisfy the following assumptions:

    H1. 1<α<3/2.

    H2. There exists a constant λΩ>0 such that

    dist(xi,jε,Ω)+12inf(h,k)(i,j)|xh,kεxi,jε|λΩε

    for any ε>0, any j{1,,J} and any i{1,,Njε}.

    H3. For any j{1,,J}, there exists a non-negative function ξjL(Ω), such that

    μjε:=ε3Njεi=1δxi,jεξjdxasmeasuresinR3,asε0.

    H4. For any j{1,,J}, there exists a Lipschitz-continuous map Rj:¯ΩSO(3) such that Ri,jε=Rj(xi,jε) for any ε>0 and any i{1,,Njε}.

    H5. For any j{1,,J}, PjR3 is a compact, convex set whose interior contains the origin.

    The assumption (H2) is a separation condition on the inclusions. As a consequence of (H2), the number of the inclusions, for each population j, is Njεε3. Therefore, the total volume of the inclusions in each population is bounded by Njεε3ε3α30, because of (H1). Thus, we are in the diluted regime, as in [7,12]. We define

    Pε:=Jj=1Pjε,Ωε:=ΩPε.

    The assumption H5 guarantees that Ωε is a Lipschitz domain.

    The free energy functional. For QH1(Ωε,S0), we consider the free energy functional

    Fε[Q]:=Ωε(fe(Q)+fb(Q))dx+ε32αJj=1Pjεfjs(Q,ν)dσ. (2.1)

    The surface anchoring energy densities depend on j, as colloids that belong to different populations may have different surface properties. For the rest, our assumptions for the elastic energy density fe, bulk energy density fb, and surface energy densities fjs are the same as in [12]. We say that a function f:S0R3R is strongly convex if there exists θ>0 such that the function S0R3Df(D)θ|D|2 is convex.

    H6. fe:S0R3[0,+) is differentiable and strongly convex. Moreover, there exists a constant λe>0 such that

    λ1e|D|2fe(D)λe|D|2,|(fe)(D)|λe(|D|+1)

    for any DS0R3.

    H7. fb:S0R is continuous, non-negative and there exists λb>0 such that 0fb(Q)λb(|Q|6+1) for any QS0.

    H8. For any j{1,,J}, the function fjs:S0×S2R is locally Lipschitz-continuous. Moreover, there exists a constant λs>0 such that

    |fjs(Q1,ν1)fjs(Q2,ν2)|λs(|Q1|3+|Q2|3+1)(|Q1Q2|+|ν1ν2|)

    for any j{1,,J} and any (Q1,ν1), (Q2,ν2) in S0×S2.

    A physically relevant example of elastic energy density fe that satisfies (H6) is given by

    fLdGe(Q):=L1kQijkQij+L2jQijkQik+L3jQikkQij (2.2)

    (Einstein's summation convention is assumed), so long as the coefficients L1, L2, L3 satisfy

    L1>0,L1<L3<2L1,35L1110L3<L2 (2.3)

    (see e.g., [13,20]). The assumption (H7) is satisfied by the quartic Landau-de Gennes bulk potential, given by

    fLdGb(Q):=atr(Q2)btr(Q3)+c(tr(Q2))2+κ(a,b,c)

    where aR, b>0, c>0 are coefficients depending on the material and the temperature and κ(a,b,c)R is a constant, chosen in such a way that inffLdGb=0. An example of surface energy density that satisfies (H8) is the Rapini-Papoular type energy density:

    fs(Q,ν):=Wtr(QQν)2withQν:=ννId3

    and W a (typically positive) parameter. However, (H8) allows for much more general surface energy densities, which may not be positive and may have up to quartic growth in Q (for examples, see e.g., [3,15,24,26] and the references therein). In addition to (H8), physically relevant surface energy densities must satisfy symmetry properties (frame-indifference, invariance with respect to the sign of ν) but these will play no rôle in our analysis.

    The homogenised potential. For any j{1,,J}, let us define fjhom:S0ׯΩR as

    fjhom(Q,x):=Pjfjs(Q,Rj(x)νPj)dσfor(Q,x)S0ׯΩ, (2.4)

    where νPj denotes the inward-pointing unit normal to Pj, and Rj:¯ΩSO(3) is the map given by (H4). Finally, let

    fhom(Q,x):=Jj=1ξj(x)fjhom(Q,x)for(Q,x)S0ׯΩ, (2.5)

    where ξjL(Ω) is the function given by (H3). Our candidate homogenised functional is defined for any QH1(Ω,S0) as

    F0[Q]:=Ω(fe(Q)+fb(Q)+fhom(Q,x))dx. (2.6)

    The convergence result. The assumptions (H1)–(H8) are not enough to guarantee that global minimisers of Fε exist and actually, it may happen that Fε is unbounded from below [12, Lemma 3.6]. Instead, our main result focus on the asymptotic behaviour of local minimisers. Given gH1/2(Ω,S0), we let H1g(Ωε,S0) — respectively, H1g(Ω,S0) — be the set of maps QH1(Ωε,S0) — respectively, QH1(Ω,S0) — that satisfy Q=g on Ω, in the sense of traces. For each QH1g(Ωε,S0), we define the map EεQH1g(Ω,S0) by EεQ:=Q on Ωε and EεQ:=Qi,jε on Pi,jε, where Qi,jε is the unique solution of Laplace's problem

    {ΔQi,jε=0inPi,jεQi,jε=QonPiε. (2.7)

    Theorem 2.1. Suppose that the assumptions (H1)–(H8) are satisfied. Suppose, moreover, that Q0H1g(Ω,S0) is an isolated H1-local minimiser for F0 — that is, there exists δ0>0 such that

    F0[Q0]<F0[Q]

    for any QH1g(Ω,S0) such that QQ0 and QQ0H1(Ω)δ0. Then, for any ε small enough, there exists an H1-local minimiser Qε for Fε such that EεQεQ0 strongly in H1(Ω) as ε0.

    The proof of Theorem 2.1 follows a variational approach, and is based on the following fact:

    Proposition 2.2. Let QεH1g(Ωε,S0) be such that EεQεQ weakly in H1(Ω) as ε0. Then, there holds

    Ω(fe(Q)+fb(Q))dxlim infε0Ωε(fe(Qε)+fb(Qε))dx (2.8)
    Ωfhom(Q,x)dx=limε0ε32αJj=1Pjεfjs(Qε,ν)dσ. (2.9)

    Proposition 2.2 can be reformulated as a Γ-convergence result. Indeed, from Proposition 2.2 we immediately have F0Γlim infε0Fε (with respect to a suitable topology, induced by the operator Eε). A trivial recovery sequence suffices to obtain the opposite Γ-lim sup inequality, thanks to (2.9) and the fact that in the dilute limit, |Pε|0. Theorem 2.1 follows from Proposition 2.2 by general properties of the Γ-convergence.

    Throughout the paper, we will write AB as a short-hand for ACB, where C is a positive constant, depending only on the domain, the boundary datum and the free energy functional (2.1), but not on ε.

    The main technical tool is the following trace inequality, which is adapted from [7,Lemma 4.1].

    Lemma 2.3 ([12,Lemma 3.1]). Let PR3 be a compact, convex set whose interior contains the origin. Then, there exists a constant C=C(P)>0 such that, for any a>0, b2a and any uH1(bPaP), there holds

    (aP)|u|4dσCbPaP(|u|2+|u|6)dx+Ca2b3bPaP|u|4dx.

    Given an inclusion Pi,jε=xi,jε+εαRi,jεPj, we consider ˆPi,jε:=xi,jε+μεRi,jεPj, where μ>0 is a small (but fixed) parameter. By taking μ small enough, we can make sure that the ˆPi,jε's are pairwise disjoint. Then, by applying Lemma 2.3 component-wise on ˆPi,jεPi,jε and summing the corresponding inequalities over i and j, we deduce

    Lemma 2.4. For any QH1(Ωε,S0), there holds

    ε32αPε|Q|4dσε32αΩε(|Q|2+|Q|6)dx+Ωε|Q|4dx.

    Another tool is the harmonic extension operator, Eε:H1(Ωε,S0)H1(Ω,S0), defined by (2.7).

    Lemma 2.5. The operator Eε:H1(Ωε,S0)H1(Ω,S0) satisfies the following properties.

    (i). There exists a constant C>0 such that (EεQ)L2(Ω)CQL2(Ωε) for any ε>0 and any QH1(Ωε,S0).

    (ii). If the maps QεH1(Ω,S) converge H1(Ω)-strongly to Q as ε0, then Eε(Qε|Ωε)Q strongly in H1(Ω) as ε0, too.

    Proof. For any i, j, consider the inclusion Pi,jε=xi,jε+εαRi,jεPj and let Ri,jε:=xi,jε+2εαRi,jεPj. Let Rε:=i,jRi,jε. The properties of Laplace equation, combined with a scaling argument (see, e.g., [12,Lemma 3.4]), imply that

    (EεQ)L2(Pε)QL2(RεPε). (2.10)

    Statement (ⅰ) then follows immediately. To prove Statement (ⅱ), take a sequence QεH1(Ω,S0) that converges strongly to Q as ε0. Then,

    Qε(Eε(Qε|Ωε))L2(Ω)QεL2(Pε)+(Eε(Qε|Ωε))L2(Pε)(2.10)QεL2(Rε)QL2(Rε)+QQεL2(Ω).

    Both terms in the right-hand side converge to 0 as ε0, because |Rε|ε3α30, and Statement (ii) follows.

    The proof of Theorem 2.1 is largely similar to that of [12, Theorem 1.1]. We reproduce here only some steps of the proof, either because there require a modification or because they will be useful in Section 4.

    Remarks on the lower semi-continuity of Fε. Even before we address the asymptotic analysis as ε0, we should make sure that, for fixed ε>0, the functional Fε is sequentially lower semi-continuous with respect to the weak topology on H1(Ωε,S0). If the surface energy density fs is bounded from below, then the surface integral is lower semi-continuous by Fatou lemma. If fs has subcritical growth, that is |fs(Q)||Q|p+1 for some p<4, then the lower semi-continuity of the surface integral follows from the compact Sobolev embedding H1/2(Ωε,S0)Lp(Ωε,S0) and Lebesgue's dominated convergence theorem. However, our assumption (H8) allows for surface energy densities that have quartic growth and are unbounded from below, e.g. fs(Q):=|Q|4. In this case, the surface integral alone may not be sequentially weakly lower-semi continuous [12, Lemma 3.10]. However, lower semi-continuity may be restored at least on bounded subsets of H1(Ωε,S0), when ε is small:

    Proposition 2.6 ([12,Proposition 3.9]). Suppose that the assumptions (H1)–(H8) are satisfied. For any M>0 there exists ε0(M)>0 such that following statement holds: if 0<εε0(M), QkQ weakly in H1g(Ωε,S0) and if QkL2(Ωε)M for any k, then

    Fε[Q]lim infk+Fε[Qk].

    The proof carries over from [12], almost word by word, using Lemma 2.4. Essentially, the loss of lower semi-continuity that may arise from the surface integral can be quantified, with the help of Lemma 2.4 and the bound on Qk. However, since the surface integral is multiplied by a small factor ε32α, this loss of lower semi-continuity is compensated by the strong convexity of the elastic term fe, for ε sufficiently small.

    Pointwise convergence of the surface energy terms. For ease of notation, let us define

    Jε[Q]:=ε32αJj=1Pjεfjs(Q,ν)dσ (2.11)
    J0[Q]:=Ωfhom(Q,x)dxforQH1g(Ω,S0) (2.12)

    We state some properties of the functions fjhom:S0ׯΩR, fhom:S0ׯΩR, defined by (2.5), (2.4) respectively.

    Lemma 2.7. For any j{1,,J}, the function fjhom is locally Lispchitz-continuous, and there holds

    |fjhom(Q,x)||Q|4+1,|fjhom(Q,x)||Q|3+1 (2.13)

    for any (Q,x)S0ׯΩ. Moroever, the function fhom satisfies

    |fhom(Q1,x)fhom(Q2,x)|(|Q1|3+|Q2|3+1)|Q1Q2| (2.14)

    for any Q1, Q2S0 and any x¯Ω.

    Proof. Using the definition (2.4) of fjhom, and the assumption (H8), we obtain

    |fjhom(Q1,x1)fjhom(Q2,x2)|Pj|fjs(Q1,Rj(x1)νPj)fjs(Q2,Rj(x2)νPj)|dσPj(|Q1|3+|Q2|3+1)(|Q1Q2|+|(Rj(x1)Rj(x2))νPj|)dσ

    Since Rj is Lipschitz-continuous by (H4), we deduce

    |fjhom(Q1,x1)fjhom(Q2,x2)|(|Q1|3+|Q2|3+1)(|Q1Q2|+|x1x2|) (2.15)

    and (2.14) follows. We multiply the previous inequality by ξj, where ξj is given by (H3), take x1=x2=x, and sum over j. Since ξjL(Ω) by Assumption (H3), we obtain

    |fhom(Q1,x)fhom(Q2,x)|(2.5)Jj=1ξjL(Ω)|fjhom(Q1,x)fjhom(Q2,x)|(|Q1|3+|Q2|3+1)|Q1Q2|.

    Let us introduce the auxiliary quantity

    ˜Jε[Q]:=ε32αJj=1Njεi=1Pi,jεfjs(Q(xi,jε),ν)dσ. (2.16)

    Lemma 2.8. For any bounded, Lipschitz map Q:¯ΩS0, there holds

    ˜Jε[Q]=Jj=1R3fjhom(Q(x),x)dμjε(x) (2.17)

    (where the measures μjε are defined by (H3)), and

    |Jε[Q]˜Jε[Q]|εα(Q3L(Ω)+1)QL(Ω). (2.18)

    Proof. For any i and j, consider the single inclusion Pi,jε:=xi,jε+εαRi,jεPjε. Since ν(x)=Ri,jενPj(εα(Ri,jε)T(xxi,jε)) for any xPi,jε, by a change of variable we obtain

    ˜Jε[Q]=ε3Jj=1Njεi=1Pjfjs(Q(xi,jε),Ri,jενP)dσ(H4)=ε3Jj=1Njεi=1Pjfjs(Q(xi,jε),Rj(xi,jε)νP)dσ(2.5)=ε3Jj=1Njεi=1fjhom(Q(xi,jε),xi,jε).

    Now, (2.17) follows from the definition of μjε, (H3). On the other hand, by decomposing the integral on Pj as a sum of integrals over the boundary of each inclusion, we obtain

    |Jε[Q]˜Jε[Q]|ε32αJj=1Njεi=1Pi,jε|fjs(Q(x),ν)fjs(Q(xi,jε),ν)|dσ(x)(H8)ε32αNjεi=1Jj=1Pi,jε(|Q(x)|3+|Q(xi,jε)|3+1)|Q(x)Q(xi,jε)|dσ(x)

    Since Q is assumed to be Lipschitz continuous and the diameter of Pi,jε is εα, we have |Q(x)Q(xi,jε)|εαQL(Ω). This implies

    |Jε[Q]˜Jε[Q]|ε3ασ(Pε)(Q3L(R3)+1)QL(Ω).

    Finally, we note that σ(Pε)ε2α3, because there are O(ε3) inclusions (as a consequence of (H2)) and the surface area of each inclusion is O(ε2α). Thus, the lemma follows.

    Since μjεξjdx by Assumption (H3), as an immediate consequence of Lemma 2.8 and (2.5), (2.12) we obtain

    Proposition 2.9. For any bounded, Lipschitz map Q:¯ΩS0, there holds Jε[Q]J0[Q] as ε0.

    Once Proposition 2.9 is proved, the rest of the proof of Proposition 2.2 and Theorem 2.1 follows exactly as in [12].

    Proposition 3.1. There exist (possibly disconnected) shapes PkR3, k{1,2,3,4,5,6} such that taking as surface energy the Rapini-Papoular surface energy fs(Q,ν)=tr(QQν)2 with Qν=νν13Id where ν is the exterior unit-normal, we have:

    fPkhom(Q)=(23+tr(Q2))σ(Pk)2tr(QMk) (3.1)

    where

    Mk=(π3+π2)Idπ2ekek,k{1,2,3}
    M4=(π3+π2)Idπ2e3e3+23(e1e2+e2e1),
    M5=(π3+π2)Idπ2e2e2+23(e1e3+e3e1),
    M6=(π3+π2)Idπ2e1e1+23(e2e3+e3e2),

    with Mk,k{1,2,3,4,5,6} a basis in the linear space of 3×3 symmetric matrices.

    Proof. By formula (2.4) we have:

    fPkhom(Q)=Pk(tr(Q2)2tr(QQν)+tr(Qν)2)dσ (3.2)

    hence we readily get (3.1) with Mk=Pkν(x)ν(x)dσ(x).

    Let us take for 1i,j3 with ij the 'potato wedges' domains

    Ω+ij:={x=(x1,x2,x3)R3:|x|1, xi0, xj0} (3.3)

    as candidates for 'parts of' our shapes Pk's (see Figure 1).

    Figure 1.  The 'potato wedge' domain Ω+23.

    We calculate the term

    Ω+12ννdσ=Ω+12{x1=0}ννdσ:=I1+Ω+12{x2=0}ννdσ:=I2+Ω+12{x1x2>0}ννdσ:=I3

    Then:

    I1=e1e1{x22+x231,x1=0}dσ=π2e1e1 (3.4)

    where e1:=(1,0,0). Similarly we get I2=π2e2e2 with e2:=(0,1,0). Finally:

    (I3)ij=Ω+12{x1x2>0}xixjdσ(x),i,j{1,2,3}. (3.5)

    Because x1x3, respectively x2x3 are odd functions in the variable x3 along the domain Ω+12{x1x2>0} we have that (I3)13=(I3)31=(I3)23=(I3)32=0. Furthermore:

    (I3)12=(I3)21=Ω+12{x1x2>0}x1x2dσx=π0(π20sin3θcosφsinφdφ)dθ=π0sin3θdθπ20cosφsinφdφ=4312=23 (3.6)

    Similarly we get: (I3)11=(I3)22=(I3)33=π3. Summarizing the last calculations, we get:

    Ω+12ννdσ=(π3+π223023π3+π2000π3) (3.7)

    Analogous calculations provide

    Ω+13ννdσ=(π3+π20230π30230π3+π2) (3.8)
    Ω+23ννdσ=(π3000π3+π223023π3+π2) (3.9)

    Similarly we define, for 1i<j3,

    Ωij:={x=(x1,x2,x3)R3:|x|1, xi0, xj0} (3.10)

    and we have:

    Ω12ννdσ=(π3+π223023π3+π2000π3) (3.11)

    respectively

    Ω13ννdσ=(π3+π20230π30230π3+π2) (3.12)
    Ω23ννdσ=(π3000π3+π223023π3+π2) (3.13)

    We take then:

    P1:=Ω+23(Ω23(0,1,0)),P2:=Ω+13(Ω13(1,0,0)),P3:=Ω+12(Ω12(1,0,0))

    and, respectively

    P4:=Ω+12,P5:=Ω+13,P6:=Ω+23.

    Proposition 3.2. Let P be a 3×3 symmetric matrix, not necessarily traceless, and WR. There exists a family of JPN shapes Pj and corresponding surface energy strengths ij,j{1,,JP} such that, taking for each shape the Rapini-Papoular surface energy with corresponding intensity ij, i.e.,

    fjs(Q,ν)=Wijtr(QQν)2 (3.14)

    with Qν:=νν13Id and ν is the exterior unit-normal, the corresponding homogenised potential is:

    fPhom(Q)=WαP(13+12tr(Q2))+Wtr(QP) (3.15)

    with αPR explicitly computable in terms of the shapes volumes and the surface energy strengths.

    Proof. We take Mk, k{1,,6}, as provided in Proposition 3.1, to be a linear basis in the spaces of 3×3 symmetric matrices. Then there exists ak:=PMk, k{1,,6} such that

    P=6k=1akMk

    Let P1,,PJP be the connected components of P1,,Pk. Each Pj is a compact, convex set of the form (3.3) or (3.10) (see Figure 1). For j{1,,JP}, we define the corresponding intensities as ij=12ak where k=k(j) is such that PjPk. Then, noting that the homogenised potentials corresponding to each species will add together to provide the homogenised porential corresponding to all the species, we get:

    fPhom(Q)=127k=1akfPkhom(Q)=W(13+12tr(Q2))6k=1ikσ(Pk)+Wtr(QP) (3.16)

    hence we obtain the claimed (3.15) with αP:=6k=1ikσ(Pk).

    Remark 3.3. We can, without loss of generality, drop the constant term in a bulk potential, since adding a constant to an energy functional does not change the minimiser. In particular in fPhom we can ignore the term W3αP.

    We wish now to choose the surface energy densities fjs of Rapini-Papoular type and the shapes of the colloidal particles, in such a way that given the symmetric 3×3 matrix P and WR, local minimisers of the Landau-de Gennes functional

    Fε[Q]=Ωε(fLdGe(Q)+atr(Q2)btr(Q3)+c(tr(Q2))2)dx+ε32αJj=1Pjεfjs(Q,ν)dσ (3.17)

    (with fLdGe given by (2.2)) converge to local minimisers of the homogenised functional

    F0[Q]=Ω(fLdGe(Q)+atr(Q2)btr(Q3)+c(tr(Q2))2+Wtr(PQ))dx. (3.18)

    We will assume that 1<α<3/2 and the centres of the inclusions, xi,jε, satisfy (H2) and that they are uniformly distributed, i.e., they satify (H3) with ξj=1. We also assume that all inclusions of the same family are parallel to each other, that is, we take Ri,jε=Id for any i, j, ε (in particular, (H4) is satisfied with Rj=Id).

    Remark 3.4. One could also choose colloidal particles and corresponding surface energies that modify the b and c coefficients, but for this it would not suffice to use Rapini-Papoular type of surface energies (see for instance Section 2.2 in [12]).

    Corollary 3.5. Let (a,b,c)R3 with c>0. Let aR, W>0, and let P be a symmetric, 3×3 matrix. Suppose that the inequalities (2.3) are satisfied. Then, there exists a family of shapes Pj and a corresponding surface energy fjs for each of them, such that for any isolated local minimiser Q0 of the functional F0 defined by (3.18), and for ε>0 small enough, there exists a local minimiser Qε of the functional Fε, defined by (3.17), such that EεQεQ0 strongly in H1(Ω,S0).

    Proof. This statement is a particular case of our main result, Theorem 2.1. If (2.3) holds and c>0, c>0, then the conditions (H6)–(H7) are satisfied.

    We take JP species Pj,j{1,,JP} and surface energies given by (3.14), as in Proposition 3.1. Each Pj is a compact, convex set of the form (3.3) or (3.10), so (H5) is satisfied (up to translations) and (H8) is satisfied too. The homogenised potential corresponding to these is:

    fPhom(Q)=WαP(13+12tr(Q2))+Wtr(QP) (3.19)

    where αP:=6k=1PMk2σ(Pk) (and the Mk,k{1,,6} are those from Proposition 3.1). We further take one more species, of spherical colloids PJP+1:=B1, and define the surface energy density

    fJP+1s(Q,ν):=14π(a+WαP2a)(νQ2ν). (3.20)

    This produces (see also for instance Remark 2.9 in [12]) a homogenised potential

    fsphhom(Q):=(a+WαP2a)tr(Q2)

    Then the homogenised potential for all the JP+1 species is

    fhom(Q)=fsphhom(Q)+fPhom(Q)=(aa)tr(Q2)+btr(Q3)+c(tr(Q2))2+Wtr(PQ)W3αP

    and, since we can, without loss of generality, see Remark 3.3 drop the constant term W3αP, the corollary follows from Theorem 2.1.

    The purpose of this section is to study the asymptotic behaviour, as ε0, of minimisers of a functional with a different choice of the scaling for the surface anchoring strength. We consider the free energy functional:

    Fε,γ[Q]:=Ωε(fe(Q)+fb(Q))dx+ε32αγJj=1Pjεfjs(Q,ν)dσ. (4.1)

    (where ν(x) denotes as usually the exterior normal at the point x on the boundary), with α(1,32) and

    K1. 0<γ<1/4.

    Due to the extra factor εγ in front of the surface integral, we cannot apply Proposition 2.6 to obtain the lower semi-continuity of Fε,γ for fixed ε. Therefore, in contrast with the previous sections, we assume boundedness from below on the surface term.

    K2. fs0.

    Remark 4.1. Under the assumption (K2), the sequential weak lower semi-continuity of Fε (for fixed ε) follows from the compact embedding H1/2(Ωε)L2(Ωε) and Fatou's lemma. Therefore, a routine application of the direct method of the Calculus of Variations shows that minimisers of exist, for any .

    As a consequence of (K2) and of (H3), the function is non-negative, too. In fact, we will also assume that

    K3. for any .

    Recall that, for any , the measures are supposed to converge weakly to a non-negative function . We need to prescribe a rate of convergence. We express the rate of convergence in terms of the -norm (that is, the dual Lipschitz norm, also known as flat norm in some contexts):

    (4.2)

    K4. There exists a constant such that for any .

    Remark 4.2. The assumption (K4) is satisfied if the inclusions are periodically distributed. Consider, for simplicity, the case , and suppose that the centres of the inclusions, , are exactly the points such that . Let . Then, for any , we have

    Moreover, , because . Therefore, (K4) holds.

    Finally, we assume some regularity on the boundary datum .

    K5. is bounded and Lipschitz.

    -convergence to a constrained problem. We can now define the candidate limit functional. Let

    (4.3)

    and ,

    Theorem 4.3. Suppose that the assumptions (H1)–(H8), (K1)–(K5) are satisfied. Then, the following statements hold.

    i. Given a family of maps such that , there exists a non-relabelled sequence and such that weakly in ,

    ii. For any , there exists a sequence of maps such that strongly in and

    Remark 4.4. The theorem is only meaningful when is non-empty, and it may happen that is empty even if for any .

    Before we give the proof of Theorem 4.3, we state some auxiliary results. We first recall some properties of the convolution, which will be useful in constructing the recovery sequence.

    Lemma 4.5. For any and , there exists a smooth map that satisfies the following properties:

    (4.4)
    (4.5)
    (4.6)

    Moreover, if are Borel subsets of such that , then

    (4.7)

    Proof. Let us take a non-negative, even function , supported in the unit ball , such that . Let . By a change of variable, we see that

    (4.8)

    Let be defined as the convolution . Then, by Young's inequality, we have

    The other inequality in (4.4) is obtained in a similar way. The condition (4.6) is a well-known property of convolutions.

    Let us prove (4.5). Let be the Fourier transform* of . Then, is smooth and rapidly decaying (that is, it belongs to the Schwartz space ) and, in particular, it is Lipschitz continuous. Moreover, . By the properties of the Fourier transform, we have . By applying Plancherel theorem, we obtain

    *We adopt the convention .

    It only remains to prove (4.7). Let be the indicator function of (i.e., on and elsewhere). Observe that on , because is supported on the ball of radius and, by assumption, . Then, Young inequality implies

    Lemma 4.6. Let a bounded, smooth domain, and let be a bounded, Lipschitz map. For any and , there exists a bounded, Lipschitz map that satisfies the following properties:

    (4.9)
    (4.10)
    (4.11)
    (4.12)
    (4.13)

    Proof. Since is bounded and smooth, we can extend to a bounded, Lipschitz map , still denoted for simplicity, in such a way that , . Let . Then, , and we extend to a new map by setting on . By applying Lemma 4.5 to , we construct a family of smooth maps that satisfies (4.4)–(4.7). We define

    The map is bounded, Lipschitz, and on . We claim that satisfies (4.10)–(4.13) with ; the lemma will follow by taking . First, we note that (4.10) is a consequence of the extension of to the whole and (4.4) together with the Gagliardo-Nirenberg-Soboleve inequality. Then we check (4.11). Clearly . Using the chain rule, and keeping in mind that the function is -Lipschitz, we see that

    (4.14)

    and (4.11) follows, with the help of (4.4).

    We pass to the proof of (4.12). Let . Since is a compact, smooth manifold, for sufficiently small the set is diffeomorphic to the product . We identify and denote the variable in as . We apply Poincaré inequality to the map on each slice :

    By integrating with respect to , we obtain

    (4.15)

    and hence,

    Finally, let us prove (4.13). Combining (4.7) and (4.15), we deduce

    (4.16)

    Therefore, we have

    and both terms in the right-hand side converge to zero as , due to (4.6).

    Lemma 4.7. For any , there exists a sequence in that converges -strongly to and satisfies

    (the functionals , are defined in (2.11), (2.12) respectively). The constant implied in front of the right-hand side depends on the -norms of and , as well as , , , with .

    Proof. Let us fix a small . Let be a positive parameter, to be chosen later, and let be the Lipschitz map given by Lemma 4.6. We have

    (4.17)

    where is defined by (2.16). We will estimate separately all the terms in the right-hand side.

    First, let us estimate the difference . This can be achieved with the help of Lemma 2.8:

    (4.18)

    (here and througout the rest of the proof, the constant implied in front of the right-hand side may depend on the -norms of and ).

    As for the second term, , we write in the form (2.17) and we re-write using (2.5), (2.12):

    (4.19)

    To estimate the terms at the right-hand side, we apply Lemma 2.7 and Lemma 4.6:

    and

    Injecting these inequalities into (4.19), and using that by Assumption (4.2), we obtain

    (4.20)

    Finally, the term . We apply Lemma 2.7 and the Hölder inequality:

    The sequence is bounded in , thanks to Sobolev embedding and to Lemma 4.6. Therefore,

    (4.21)

    Combining (4.17), (4.18), (4.20) and (4.21), we deduce

    Keeping into account that , we see that the optimal choice of is , and the lemma follows.

    Lemma 4.8. Let be a family of maps, such that strongly in , as . Then,

    Proof. By Sobolev embedding, we have strongly in . Then, up to extraction of a non-relabelled subsequence, we find functions , such that

    (4.22)

    Let be the the indicator function of (i.e., on and elsewhere). Thanks to (H6), (H7) and to (4.22), we have

    Moreover, since , converges to strongly in and we may extract a further subsequence so to have a.e. Then, the lemma follows from Lebesgue's dominated converge theorem.

    Proof of Theorem 4.3. Let , for , be a family of maps such that . We first extract a (non-relabelled) subsequence , so that is achieved as a limit; this allows us to pass freely to subsequences, in what follows. Thanks to (H6), (H7), (K2), we have . By Lemma 2.5, there is a (non-relabelled) subsequence and such that weakly in . By Proposition 2.2, there holds

    and

    so belongs to the class defined by (4.3). Thus, Statement (i) is proved.

    We now prove Statement (ⅱ). Let be fixed. We can suppose without loss of generality that , otherwise the statement is trivial. Due to Lemma 4.7, there is a sequence such that strongly in and

    (4.23)

    Let . By Lemma 2.5, strongly in . Using Lemma 4.8 and (4.23), and recalling that , we conclude that

    so the proof is complete.

    The work of A. Z. is supported by the Basque Government through the BERC 2018-2021 program, by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym "DESFLU".

    The authors declare that there is no conflict of interest regarding the publication of this article.


    Acknowledgments



    We are grateful to Takuma Komori, Hikaru Kondo, Yoriko Kirinoki, Naoya Osamura, Kanako Shimono, and Maiko Tanaka for outstanding technical assistance and discussion.

    Conflict of interest



    The authors declare no conflict of interest.

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