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Research article

Quality evaluation of digital voice assistants for diabetes management

  • Received: 04 September 2022 Revised: 02 March 2023 Accepted: 14 March 2023 Published: 30 March 2023
  • Background 

    Digital voice assistants (DVAs) are increasingly used to search for health information. However, the quality of information provided by DVAs is not consistent across health conditions. From our knowledge, there have been no studies that evaluated the quality of DVAs in response to diabetes-related queries. The objective of this study was to evaluate the quality of DVAs in relation to queries on diabetes management.

    Materials and methods 

    Seventy-four questions were posed to smartphone (Apple Siri, Google Assistant, Samsung Bixby) and non-smartphone DVAs (Amazon Alexa, Sulli the Diabetes Guru, Google Nest Mini, Microsoft Cortana), and their responses were compared to that of Internet Google Search. Questions were categorized under diagnosis, screening, management, treatment and complications of diabetes, and the impacts of COVID-19 on diabetes. The DVAs were evaluated on their technical ability, user-friendliness, reliability, comprehensiveness and accuracy of their responses. Data was analyzed using the Kruskal-Wallis and Wilcoxon rank-sum tests. Intraclass correlation coefficient was used to report inter-rater reliability.

    Results 

    Google Assistant (n = 69/74, 93.2%), Siri and Nest Mini (n = 64/74, 86.5% each) had the highest proportions of successful and relevant responses, in contrast to Cortana (n = 23/74, 31.1%) and Sulli (n = 10/74, 13.5%), which had the lowest successful and relevant responses. Median total scores of the smartphone DVAs (Bixby 75.3%, Google Assistant 73.3%, Siri 72.0%) were comparable to that of Google Search (70.0%, p = 0.034), while median total scores of non-smartphone DVAs (Nest Mini 56.9%, Alexa 52.9%, Cortana 52.5% and Sulli the Diabetes Guru 48.6%) were significantly lower (p < 0.001). Non-smartphone DVAs had much lower median comprehensiveness (16.7% versus 100.0%, p < 0.001) and reliability scores (30.8% versus 61.5%, p < 0.001) compared to Google Search.

    Conclusions 

    Google Assistant, Siri and Bixby were the best-performing DVAs for answering diabetes-related queries. However, the lack of successful and relevant responses by Bixby may frustrate users, especially if they have COVID-19 related queries. All DVAs scored highly for user-friendliness, but can be improved in terms of accuracy, comprehensiveness and reliability. DVA designers are encouraged to consider features related to accuracy, comprehensiveness, reliability and user-friendliness when developing their products, so as to enhance the quality of DVAs for medical purposes, such as diabetes management.

    Citation: Joy Qi En Chia, Li Lian Wong, Kevin Yi-Lwern Yap. Quality evaluation of digital voice assistants for diabetes management[J]. AIMS Medical Science, 2023, 10(1): 80-106. doi: 10.3934/medsci.2023008

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  • Background 

    Digital voice assistants (DVAs) are increasingly used to search for health information. However, the quality of information provided by DVAs is not consistent across health conditions. From our knowledge, there have been no studies that evaluated the quality of DVAs in response to diabetes-related queries. The objective of this study was to evaluate the quality of DVAs in relation to queries on diabetes management.

    Materials and methods 

    Seventy-four questions were posed to smartphone (Apple Siri, Google Assistant, Samsung Bixby) and non-smartphone DVAs (Amazon Alexa, Sulli the Diabetes Guru, Google Nest Mini, Microsoft Cortana), and their responses were compared to that of Internet Google Search. Questions were categorized under diagnosis, screening, management, treatment and complications of diabetes, and the impacts of COVID-19 on diabetes. The DVAs were evaluated on their technical ability, user-friendliness, reliability, comprehensiveness and accuracy of their responses. Data was analyzed using the Kruskal-Wallis and Wilcoxon rank-sum tests. Intraclass correlation coefficient was used to report inter-rater reliability.

    Results 

    Google Assistant (n = 69/74, 93.2%), Siri and Nest Mini (n = 64/74, 86.5% each) had the highest proportions of successful and relevant responses, in contrast to Cortana (n = 23/74, 31.1%) and Sulli (n = 10/74, 13.5%), which had the lowest successful and relevant responses. Median total scores of the smartphone DVAs (Bixby 75.3%, Google Assistant 73.3%, Siri 72.0%) were comparable to that of Google Search (70.0%, p = 0.034), while median total scores of non-smartphone DVAs (Nest Mini 56.9%, Alexa 52.9%, Cortana 52.5% and Sulli the Diabetes Guru 48.6%) were significantly lower (p < 0.001). Non-smartphone DVAs had much lower median comprehensiveness (16.7% versus 100.0%, p < 0.001) and reliability scores (30.8% versus 61.5%, p < 0.001) compared to Google Search.

    Conclusions 

    Google Assistant, Siri and Bixby were the best-performing DVAs for answering diabetes-related queries. However, the lack of successful and relevant responses by Bixby may frustrate users, especially if they have COVID-19 related queries. All DVAs scored highly for user-friendliness, but can be improved in terms of accuracy, comprehensiveness and reliability. DVA designers are encouraged to consider features related to accuracy, comprehensiveness, reliability and user-friendliness when developing their products, so as to enhance the quality of DVAs for medical purposes, such as diabetes management.



    In the present paper we study the effective conductivity of an n-dimensional periodic two-phase composite, with n{2,3} which from now on we assume fixed. The composite is obtained by introducing into a homogeneous matrix a periodic set of inclusions of a large class of sufficiently smooth shapes. Both the matrix and the set of inclusions are filled with two different homogeneous and isotropic heat conductor materials of conductivity λ and λ+, respectively, with

    (λ+,λ)[0,+[2[0,+[2{(0,0)}.

    We note that the limit case of zero conductivity corresponds to a thermal insulator. On the other hand, if the conductivity tends to +, the material is a perfect conductor. The inclusions' shape is determined by the image of a fixed domain through a diffeomorphism ϕ, and the periodicity cell is a 'box' of edges of lengths q11,,qnn. As it is known, it is possible to define the composite's effective conductivity matrix λeff by means of the solution of a transmission problem for the Laplace equation (see Definition 1.1, cf. Mityushev, Obnosov, Pesetskaya, and Rogosin [43,§5]). The effective conductivity can be thought as the conductivity of a homogeneous material whose global behavior as a conductor is 'equivalent' to the composite. Our aim is to study the dependence of λeff upon the 'triple' ((q11,,qnn),ϕ,(λ+,λ)), i.e., upon the perturbation of the periodicity structure of the composite, of the inclusions' shape, and of the conductivity parameters of each material. A perturbation analysis of the properties of composite materials has been carried out by several authors with different techniques. For example, in Ammari, Kang, and Touibi [5] the authors have exploited a potential theoretic approach in order to investigate the asymptotic behavior of the effective properties of a periodic dilute composite. Then Ammari, Kang, and Kim [3] and Ammari, Kang, and Lim [4] have studied anisotropic composite materials and elastic composites, respectively. The method of Functional Equations has been used to study the dependence on the radius of the inclusions for a wide class of 2D composites. For ideal composites, we mention here, for example, the works of Mityushev, Obnosov, Pesetskaya, and Rogosin [43], Gryshchuk and Rogosin [26], Kapanadze, Mishuris, and Pesetskaya [28]. Contributions to composites with different contact conditions are, for example, Drygaś and Mityushev [20] and Castro, Kapanadze, and Pesetskaya [9,10] (non-ideal composites), Castro and Pesetskaya [11] (composites with inextensible-membrane-type interface). The effect of shapes on the properties of composites has been studied under several different points of view by many authors. For example, Berlyand, Golovaty, Movchan, and Phillips [7] have analyzed the transport properties of fluid/solid and solid/solid composites and have investigated how the curvature of the inclusions affects such properties. Berlyand and Mityushev [8] have studied the dependence of the effective conductivity of two-phase composites upon the polydispersivity parameter. Gorb and Berlyand [25] considered the asymptotic behavior of the effective properties of composites with close inclusions of optimal shape. For 2D composites, we also mention the recent work by Mityushev, Nawalaniec, Nosov, and Pesetskaya [42], where the authors have applied the generalized alternating method of Schwarz in order to study the effective conductivity of two-phase random composites with non-overlapping inclusions whose boundaries are arbitrary Lyapunov's curves. In Lee and Lee [37], the authors have studied how the effective elasticity of dilute periodic elastic composites is affected by its periodic structure. Finally, Pukhtaievych [48] has explicitly computed the effective conductivity of a periodic dilute composite with perfect contact as a power series in the size of the inclusions. In the present paper we perform a regularity analysis of the behavior of the effective conductivity upon joint perturbation of periodicity structure of the composite, of the inclusions' shape, and of the conductivity parameters. Moreover, in contrast with previous contributions we do not confine our attention just to specific shapes.

    We now introduce the geometry of the problem. If q11,,qnn]0,+[, we will use the following notation:

    q=(q11000000qnn), (1)

    and

    Qnj=1]0,qjj[Rn. (2)

    The set Q plays the role of the periodicity cell and the diagonal matrix q plays the role of the periodicity matrix. Clearly |Q|nnj=1qjj is the measure of the fundamental cell Q and qZn{qz:zZn} is the set of vertices of a periodic subdivision of Rn corresponding to the fundamental cell Q. We denote by q1 the inverse matrix of q. We denote by Dn(R) the space of n×n diagonal matrices with real entries and by D+n(R) the set of elements of Dn(R) with diagonal entries in ]0,+[. Moreover, we find it convenient to set

    ˜Q]0,1[n,˜qIn(10000001).

    Then we take

    α]0,1[ and a bounded open connected subset Ω of Rn of class C1,α such that Rn¯Ω is connected. (3)

    The symbol '¯' denotes the closure of a set. For the definition of sets and functions of the Schauder class Ck,α (kN) we refer, e.g., to Gilbarg and Trudinger [23]. Then we consider a class of diffeomorphisms A˜QΩ from Ω into their images contained in the unitary cell ˜Q (see (14)). If ϕA˜QΩ, the Jordan-Leray separation theorem ensures that Rnϕ(Ω) has exactly two open connected components (see, e.g, Deimling [18,Thm. 5.2,p. 26]), and we denote by I[ϕ] the bounded open connected component of Rnϕ(Ω). Since ϕ(Ω)˜Q, a simple topological argument shows that ˜Q¯I[ϕ] is also connected. If qD+n(R), we consider the following two periodic domains (see Figure 1):

    Sq[qI[ϕ]]zZn(qz+qI[ϕ]),Sq[qI[ϕ]]Rn¯Sq[qI[ϕ]].
    Figure 1.  The sets Sq[qI[ϕ]], Sq[qI[ϕ]], and qϕ(Ω) in case n=2.

    The set Sq[qI[ϕ]] represents the homogeneous matrix made of a material with conductivity λ where the periodic set of inclusions ¯Sq[qI[ϕ]] with conductivity λ+ is inserted. Globally, the union of the matrix and the inclusions represents the two-phase composite material we consider.

    With the aim of introducing the definition of the effective conductivity, we first have to introduce a boundary value problem for the Laplace equation. If qD+n(R), ϕC1,α(Ω,Rn)A˜QΩ, and (λ+,λ)[0,+[2, for each j{1,,n} we consider the following transmission problem for a pair of functions (u+j,uj)C1,αloc(¯Sq[qI[ϕ]])×C1,αloc(¯Sq[qI[ϕ]]):

    {Δu+j=0in Sq[qI[ϕ]],Δuj=0in Sq[qI[ϕ]],u+j(x+qeh)=u+j(x)+δhjqjjx¯Sq[qI[ϕ]],h{1,,n},uj(x+qeh)=uj(x)+δhjqjjx¯Sq[qI[ϕ]],h{1,,n},λ+νqI[ϕ]u+jλνqI[ϕ]uj=0on qI[ϕ],u+juj=0on qI[ϕ],qI[ϕ]u+jdσ=0, (4)

    where νqI[ϕ] is the outward unit normal to qI[ϕ] and {e1,,en} is the canonical basis of Rn. We observe that for C1,α functions, the Laplace equations of problem (4) have to be considered in the sense of distributions. Then, by elliptic regularity theory (see for instance Friedman [21,Thm. 1.2,p. 205]), if the Laplace equation is satisfied by a function in the sense of distributions, we know that such a function is of class C in the interior, and that accordingly the Laplace equation is satisfied in the classical sense. As we will see, problem (4) admits a unique solution (u+j,uj) in C1,αloc(¯Sq[qI[ϕ]])×C1,αloc(¯Sq[qI[ϕ]]), which we denote by (u+j[q,ϕ,(λ+,λ)],uj[q,ϕ,(λ+,λ)]). Such family of solutions is needed in order to define the effective conductivity as follows (cf., e.g., Mityushev, Obnosov, Pesetskaya, and Rogosin [43,§5]).

    Definition 1.1. Let qD+n(R), ϕC1,α(Ω,Rn)A˜QΩ, and (λ+,λ)[0,+[2. Then the effective conductivity

    λeff[q,ϕ,(λ+,λ)](λeffij[q,ϕ,(λ+,λ)])i,j=1,,n

    is the n×n matrix with (i,j)-entry defined by

    λeffij[q,ϕ,(λ+,λ)]1|Q|n{λ+qI[ϕ]xiu+j[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]xiuj[q,ϕ,(λ+,λ)](x)dx}i,j{1,,n}.

    Remark 1.2. Under the assumptions of Definition 1.1, by applying the divergence theorem, one can verify that

    λeffij[q,ϕ,(λ+,λ)]=1|Q|n{λ+qI[ϕ]Du+i[q,ϕ,(λ+,λ)](x)Du+j[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]Dui[q,ϕ,(λ+,λ)](x)Duj[q,ϕ,(λ+,λ)](x)dx}i,j{1,,n}.

    Indeed, if we set

    ˜u+k[q,ϕ,(λ+,λ)](x)=u+k[q,ϕ,(λ+,λ)](x)xkx¯Sq[qI[ϕ]]˜uk[q,ϕ,(λ+,λ)](x)=uk[q,ϕ,(λ+,λ)](x)xkx¯Sq[qI[ϕ]]k{1,,n},

    then

    1|Q|n{λ+qI[ϕ]Du+i[q,ϕ,(λ+,λ)](x)Du+j[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]Dui[q,ϕ,(λ+,λ)](x)Duj[q,ϕ,(λ+,λ)](x)dx}=1|Q|n{λ+qI[ϕ]Du+j[q,ϕ,(λ+,λ)](x)D(xi+˜u+i[q,ϕ,(λ+,λ)](x))dx+λQ¯qI[ϕ]Duj[q,ϕ,(λ+,λ)](x)D(xi+˜ui[q,ϕ,(λ+,λ)](x))dx}=1|Q|n{λ+qI[ϕ]Du+j[q,ϕ,(λ+,λ)](x)Dxidx+λ+qI[ϕ]Du+j[q,ϕ,(λ+,λ)](x)D˜u+i[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]Duj[q,ϕ,(λ+,λ)](x)Dxidx+λQ¯qI[ϕ]Duj[q,ϕ,(λ+,λ)](x)D˜ui[q,ϕ,(λ+,λ)](x)dx}=1|Q|n{λ+qI[ϕ]xiu+j[q,ϕ,(λ+,λ)](x)dx+λ+qI[ϕ]Du+j[q,ϕ,(λ+,λ)](x)D˜u+i[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]xiuj[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]Duj[q,ϕ,(λ+,λ)](x)D˜ui[q,ϕ,(λ+,λ)](x)dx}.

    Therefore, in order to conclude that the two definitions are equivalent, we need to show that

    λ+qI[ϕ]Du+j[q,ϕ,(λ+,λ)](x)D˜u+i[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]Duj[q,ϕ,(λ+,λ)](x)D˜ui[q,ϕ,(λ+,λ)](x)dx=0. (5)

    By an application of the divergence theorem for C1 functions (cf. Ziemer [50,Thm. 5.8.2,Rmk. 5.8.3]), we have

    qI[ϕ]Du+j[q,ϕ,(λ+,λ)](x)D˜u+i[q,ϕ,(λ+,λ)](x)dx=qI[ϕ](νqI[ϕ]u+j[q,ϕ,(λ+,λ)](x))˜u+i[q,ϕ,(λ+,λ)](x)dσx (6)

    and

    Q¯qI[ϕ]Duj[q,ϕ,(λ+,λ)](x)D˜ui[q,ϕ,(λ+,λ)](x)dx=Q(νQuj[q,ϕ,(λ+,λ)](x))˜ui[q,ϕ,(λ+,λ)](x)dσxqI[ϕ](νqI[ϕ]uj[q,ϕ,(λ+,λ)](x))˜ui[q,ϕ,(λ+,λ)](x)dσx. (7)

    By the periodicity of ˜ui[q,ϕ,(λ+,λ)] and of ˜uj[q,ϕ,(λ+,λ)], we have

    Q(νQuj[q,ϕ,(λ+,λ)](x))˜ui[q,ϕ,(λ+,λ)](x)dσx=Q(νQxj)˜ui[q,ϕ,(λ+,λ)](x)dσx+Q(νQ˜uj[q,ϕ,(λ+,λ)](x))˜ui[q,ϕ,(λ+,λ)](x)dσx=Q(νQ(x)ej)˜ui[q,ϕ,(λ+,λ)](x)dσx+Q(νQ(x)D˜uj[q,ϕ,(λ+,λ)](x))˜ui[q,ϕ,(λ+,λ)](x)dσx=0, (8)

    since contributions on opposite sides of Q cancel each other. Thus by (6)–(8) we obtain

    λ+qI[ϕ]Du+j[q,ϕ,(λ+,λ)](x)D˜u+i[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]Duj[q,ϕ,(λ+,λ)](x)D˜ui[q,ϕ,(λ+,λ)](x)dx=λ+qI[ϕ](νqI[ϕ]u+j[q,ϕ,(λ+,λ)](x))˜u+i[q,ϕ,(λ+,λ)](x)dσxλqI[ϕ](νqI[ϕ]uj[q,ϕ,(λ+,λ)](x))˜ui[q,ϕ,(λ+,λ)](x)dσx. (9)

    Since the validity of (4) implies that

    ˜u+i[q,ϕ,(λ+,λ)](x)=˜ui[q,ϕ,(λ+,λ)](x)xqI[ϕ]

    and that

    λ+νqI[ϕ]u+j[q,ϕ,(λ+,λ)](x)λνqI[ϕ]uj[q,ϕ,(λ+,λ)](x)=0xqI[ϕ],

    we then deduce by (9) that (5) holds true.

    As a consequence, the effective conductivity matrix of Definition 1.1 coincides with the one analyzed by Ammari, Kang, and Touibi [5,p. 121] for a periodic two-phase composite and which can be deduced by classical homogenization theory (see, e.g., Allaire [1], Bensoussan, Lions, and Papanicolaou [6], Jikov, Kozlov, and Oleĭnik [27], Milton [41]). We emphasize that the justification of the expression of the effective conductivity via homogenization theory holds for 'small' values of the periodicity parameters. For further remarks on the definition of effective conductivity we refer to Gluzman, Mityushev, and Nawalaniec [24,§2.2].

    The main goal of our paper is to give an answer to the following question:

    What can be said on the regularity of the map(q,ϕ,(λ+,λ))λeff[q,ϕ,(λ+,λ)]? (10)

    We answer to the above question by proving that for all i,j{1,,n} there exist ε]0,1[ and a real analytic map

    Λij:D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×]1ε,1+ε[R

    such that

    λeffij[q,ϕ,(λ+,λ)]=δijλ+(λ++λ)Λij[q,ϕ,λ+λλ++λ] (11)

    for all (q,ϕ,(λ+,λ))D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×[0,+[2 (see formula (25) of Theorem 5.1 below). The approach we use was introduced by Lanza de Cristoforis in [31] and then exploited to analyze a large variety of singular and regular perturbation problems (cf., e.g., Lanza de Cristoforis [32], Dalla Riva and Lanza de Cristoforis [15], Dalla Riva [14]).

    In particular, in the present paper we follow the strategy of [39] where we have studied the behavior of the longitudinal flow along a periodic array of cylinders upon perturbations of the shape of the cross section of the cylinders and the periodicity structure, when a Newtonian fluid is flowing at low Reynolds numbers around the cylinders. More precisely, we transform the problem into a set of integral equations defined on a fixed domain and depending on the set of variables (q,ϕ,(λ+,λ)). We study the dependence of the solution of the integral equation upon (q,ϕ,(λ+,λ)) and then we deduce the result on the behavior of λeffij[q,ϕ,(λ+,λ)].

    Formula (11) implies that the effective conductivity λeff[q,ϕ,(λ+,λ)] can be expressed in terms of the conductivity λ of the matrix, of the conductivity λ+ of the inclusions, and of a real analytic function of the periodicity parameters q11,,qnn, of the shape ϕ and of the contrast parameter λ+λλ++λ. In particular, by the real analyticity of Λij, the expression Λij[q,ϕ,λ+λλ++λ] can be written as a convergent power series of q,ϕ,λ+λλ++λ in a suitable Banach space. Also, formula (11) immediately implies that the map

    (q,ϕ,(λ+,λ))λeffij[q,ϕ,(λ+,λ)] (12)

    from D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×[0,+[2 to R admits a real analytic extension around every point (q,ϕ,(λ+,λ))D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×[0,+[2. Such a result implies that if δ0>0 and we have a family of triples {(qδ,ϕδ,(λ+δ,λδ))}δ]δ0,δ0[ in D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×[0,+[2 such that the map δ(qδ,ϕδ,(λ+δ,λδ)) is real analytic, then we can deduce the possibility to expand λeffij[qδ,ϕδ,(λ+δ,λδ)] as a power series in δ, i.e.,

    λeffij[qδ,ϕδ,(λ+δ,λδ)]=k=0ckδk (13)

    for δ close to zero. Moreover, the coefficients (ck)kN in (13) can be constructively determined by computing the differentials of λeffij.

    Furthermore, such a high regularity result can be seen as a theoretical justification which guarantees that differential calculus may be used in order to characterize critical periodicity-shape-conductivity triples (q,ϕ,(λ+,λ)) as a first step to find optimal configurations, under specific constraints. Indeed, if one is interested in finding a triple (q,ϕ,(λ+,λ)) that maximizes (or minimizes) the effective conductivity λeffij[q,ϕ,(λ+,λ)] under given constraints on (q,ϕ,(λ+,λ)), then, since the map in (12) is analytic, one can try to apply differential calculus to characterize critical configurations. In the present paper, by critical we mean critical points of the effective conductivity functional, that we think as a map defined on a suitable function space. In this sense, finding a critical point could be a first step to search for (local) maximum or minimum points under suitable restrictions. If, for example, we have a family {(qδ,ϕδ,(λ+δ,λδ))}δ]δ0,δ0[ in D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×[0,+[2, for some δ0>0, which depends smoothly on δ, then we can apply differential calculus to the map δλeffij[qδ,ϕδ,(λ+δ,λδ)] to find, for example, local minimizers or maximizers. Similar considerations can be done if, more in general, (qδ,ϕδ,(λ+δ,λδ)) depends smoothly on a parameter δ which belongs to a differentiable manifold. Moreover, we note that, in contrast with other approaches that can be applied only to particular shapes as circles or ellipses, our method permits to consider inclusions of a large class of sufficiently smooth shapes.

    As already mentioned, our method is based on integral equations, that are derived by potential theory. However, integral equations could also be deduced by the generalized alternating method of Schwarz (cf. Gluzman, Mityushev, and Nawalaniec [24] and Drygaś, Gluzman, Mityushev, and Nawalaniec [19]), which also allows to produce expansions in the concentration.

    Incidentally, we observe that the are several contributions concerning optimization of effective parameters from many different points of view. For example, one can look for optimal lattices without confining to rectangular distributions. In this direction, Kozlov [29] and Mityushev and Rylko [44] have discussed extremal properties of hexagonal lattices of disks. On the other hand, even if, in wide generality, the optimal composite does not exist (cf. Cherkaev [13]), one can discuss the dependence on the shape under some specific restrictions. For example, one could build composites with prescribed effective conductivity as described in Lurie and Cherkaev [38] (see also Gibiansky and Cherkaev [22]). In Rylko [49], the author has studied the influence of perturbations of the shape of the circular inclusion on the macroscopic conductivity properties of 2D dilute composites. Inverse problems concerning the determination of the shape of equally strong holes in elastic structures were considered by Cherepanov [12]. For an experimental work concerning the analysis of particle reinforced composites we mention Kurtyka and Rylko [30]. Also, we mention that one could apply the topological derivative method as in Novotny and Sokołowski [46] for the optimal design of microstructures.

    Let α, Ω be as in (3). We denote by νΩ the outward unit normal to Ω and by dσ the area element on Ω. We retain the standard notation for the Lebesgue space L1(Ω) of Lebesgue integrable functions. We denote by |Ω|n1 the (n1)-dimensional measure of Ω. To shorten our notation, we denote by Ωfdσ the integral mean 1|Ω|n1Ωfdσ for all fL1(Ω). Also, if X is a vector subspace of L1(Ω) then we set X0{fX:Ωfdσ=0}. Moreover, we note that throughout the paper 'analytic' always means 'real analytic'. For the definition and properties of analytic operators, we refer to Deimling [18,§15].

    Let q,Q be as in (1) and (2). If ΩQ is a subset of Rn such that ¯ΩQQ, we define the following two periodic domains

    Sq[ΩQ]zZn(qz+ΩQ),Sq[ΩQ]Rn¯Sq[ΩQ].

    If u is a real valued function defined on Sq[ΩQ] or Sq[ΩQ], we say that u is q-periodic provided that u(x+qz)=u(x) for all zZn and for all x in the domain of definition of u. If kN, we set

    Ckb(¯Sq[ΩQ]){uCk(¯Sq[ΩQ]):Dγu is bounded γNn s. t. |γ|k},

    and we endow Ckb(¯Sq[ΩQ]) with its usual norm

    uCkb(¯Sq[ΩQ])|γ|ksupx¯Sq[ΩQ]|Dγu(x)|uCkb(¯Sq[ΩQ]),

    where |γ|ni=1γi denotes the length of the multi-index γ(γ1,,γn)Nn. Moreover, if we further take β]0,1], then we set

    Ck,βb(¯Sq[ΩQ]){uCk,β(¯Sq[ΩQ]):Dγu is bounded γNn s. t. |γ|k},

    and we endow Ck,βb(¯Sq[ΩQ]) with its usual norm

    uCk,βb(¯Sq[ΩQ])|γ|ksupx¯Sq[ΩQ]|Dγu(x)|+|γ|=k|Dγu:¯Sq[ΩQ]|βuCk,βb(¯Sq[ΩQ]),

    where |Dγu:¯Sq[ΩQ]|β denotes the β-Hölder constant of Dγu. Similarly, we set

    Ckq(¯Sq[ΩQ]){uCkb(¯Sq[ΩQ]):u is q-periodic},

    which we regard as a Banach subspace of Ckb(¯Sq[ΩQ]), and

    Ck,βq(¯Sq[ΩQ]){uCk,βb(¯Sq[ΩQ]):u is q-periodic},

    which we regard as a Banach subspace of Ck,βb(¯Sq[ΩQ]). The spaces Ckb(¯Sq[ΩQ]), Ck,βb(¯Sq[ΩQ]), Ckq(¯Sq[ΩQ]), and Ck,βq(¯Sq[ΩQ]) can be defined similarly.

    Our method is based on a periodic version of classical potential theory. In order to construct periodic layer potentials, we replace the fundamental solution of the Laplace operator by a q-periodic tempered distribution Sq,n such that

    ΔSq,n=zZnδqz1|Q|n,

    where δqz denotes the Dirac measure with mass in qz (see e.g., [34,p. 84]). The distribution Sq,n is determined up to an additive constant, and we can take

    Sq,n(x)=zZn{0}1|Q|n4π2|q1z|2e2πi(q1z)x

    in the sense of distributions in Rn (see e.g., Ammari and Kang [2,p. 53], [34,§3]). Moreover, Sq,n is real analytic in RnqZn and locally integrable in Rn (see e.g., [34,§3]).

    We now introduce periodic layer potentials. Let ΩQ be a bounded open subset of Rn of class C1,α for some α]0,1[ such that ¯ΩQQ. We set

    vq[ΩQ,μ](x)ΩQSq,n(xy)μ(y)dσyxRn,wq,[ΩQ,μ](x)ΩQνΩQ(x)DSq,n(xy)μ(y)dσyxΩQ,

    for all μC0(ΩQ). Here above, DSq,n(ξ) denotes the gradient of Sq,n computed at the point ξRnqZn. The function vq[ΩQ,μ] is called the q-periodic single layer potential, and wq,[ΩQ,μ] is a function related to the normal derivative of the single layer potential. As is well known, if μC0(ΩQ), then vq[ΩQ,μ] is continuous in Rn and q-periodic, and we set

    v+q[ΩQ,μ]vq[ΩQ,μ]|¯Sq[ΩQ]  vq[ΩQ,μ]vq[ΩQ,μ]|¯Sq[ΩQ].

    We collect in the following theorem some properties of v±q[ΩQ,] and wq,[ΩQ,]. For a proof of statements (ⅰ)–(ⅲ) we refer to [34,Thm. 3.7] and to [16,Lem. 4.2]. For a proof of statements (ⅳ) we refer to [16,Lem. 4.2 (ⅰ),(ⅲ)].

    Theorem 2.1. Let q,Q be as in (1) and (2). Let α]0,1[. Let ΩQ be a bounded open subset of Rn of class C1,α such that ¯ΩQQ. Then the following statements hold.

    (i) The map from C0,α(ΩQ) to C1,αq(¯Sq[ΩQ]) which takes μ to v+q[ΩQ,μ] is linear and continuous. The map from C0,α(ΩQ) to C1,αq(¯Sq[ΩQ]) which takes μ to vq[ΩQ,μ] is linear and continuous.

    (ii) Let μC0,α(ΩQ). Then

    νΩQv±q[ΩQ,μ]=12μ+wq,[ΩQ,μ]on ΩQ.

    Moreover,

    ΩQwq,[ΩQ,μ]dσ=(12|ΩQ|n|Q|n)ΩQμdσ.

    (iii) Let μC0,α(ΩQ)0. Then

    Δvq[ΩQ,μ]=0in RnSq[ΩQ].

    (iv) The operator wq,[ΩQ,] is compact in C0,α(ΩQ) and in C0,α(ΩQ)0.

    In order to consider shape perturbations of the inclusions of the composite, we introduce a class of diffeomorphisms. Let Ω be as in (3) and let Ω be a bounded open connected subset of Rn of class C1,α. We denote by AΩ and by A¯Ω the sets of functions of class C1(Ω,Rn) and of class C1(¯Ω,Rn) which are injective and whose differential is injective at all points of Ω and of ¯Ω, respectively. One can verify that AΩ and A¯Ω are open in C1(Ω,Rn) and C1(¯Ω,Rn), respectively (see, e.g., Lanza de Cristoforis and Rossi [36,Lem. 2.2,p. 197] and [35,Lem. 2.5,p. 143]). Then we find it convenient to set

    A˜QΩ{ϕAΩ:ϕ(Ω)˜Q},A˜Q¯Ω{ΦA¯Ω:Φ(¯Ω)˜Q}. (14)

    If ϕAΩ, the Jordan-Leray separation theorem ensures that Rnϕ(Ω) has exactly two open connected components, and we denote by I[ϕ] the bounded open connected component of Rnϕ(Ω) (see, e.g, Deimling [18,Thm. 5.2,p. 26]).

    We conclude this section of preliminaries with some results on problem (4). By means of the following proposition, whose proof is of immediate verification, we can transform problem (4) into a q-periodic transmission problem for the Laplace equation.

    Proposition 2.2. Let q,Q be as in (1) and (2) and α, Ω be as in (3). Let (λ+,λ)[0,+[2. Let ϕC1,α(Ω,Rn)A˜QΩ. Let j{1,,n}. A pair

    (u+j,uj)C1,αloc(¯Sq[qI[ϕ]])×C1,αloc(¯Sq[qI[ϕ]])

    solves problem (4) if and only if the pair

    (˜u+j,˜uj)C1,αq(¯Sq[qI[ϕ]])×C1,αq(¯Sq[qI[ϕ]])

    delivered by

    ˜u+j(x)=u+j(x)xjx¯Sq[qI[ϕ]],˜uj(x)=uj(x)xjx¯Sq[qI[ϕ]],

    solves

    {Δ˜u+j=0in Sq[qI[ϕ]],Δ˜uj=0in Sq[qI[ϕ]],˜u+j(x+qeh)=˜u+j(x)x¯Sq[qI[ϕ]],h{1,,n},˜uj(x+qeh)=˜uj(x)x¯Sq[qI[ϕ]],h{1,,n},λ+νqI[ϕ]˜u+jλνqI[ϕ]˜uj=(λλ+)(νqI[ϕ])jon qI[ϕ],˜u+j˜uj=0on qI[ϕ],qI[ϕ]˜u+jdσ=qI[ϕ]yjdσy. (15)

    Next, we show that problems (4) and (15) admit at most one solution.

    Proposition 2.3. Let q,Q be as in (1) and (2) and α, Ω be as in (3). Let (λ+,λ)[0,+[2. Let ϕC1,α(Ω,Rn)A˜QΩ. Let j{1,,n}. Then the following statements hold.

    (i) Problem (4) has at most one solution in C1,αloc(¯Sq[qI[ϕ]])×C1,αloc(¯Sq[qI[ϕ]]).

    (ii) Problem (15) has at most one solution in C1,αq(¯Sq[qI[ϕ]])×C1,αq(¯Sq[qI[ϕ]]).

    Proof. By the equivalence of problems (4) and (15) of Proposition 2.2, it suffices to prove statement (ⅱ), which we now consider. By the linearity of the problem, it clearly suffices to show that if (˜u+j,˜uj)C1,αq(¯Sq[qI[ϕ]])×C1,αq(¯Sq[qI[ϕ]]) solves

    {Δ˜u+j=0in Sq[qI[ϕ]],Δ˜uj=0in Sq[qI[ϕ]],˜u+j(x+qeh)=˜u+j(x)x¯Sq[qI[ϕ]],h{1,,n},˜uj(x+qeh)=˜uj(x)x¯Sq[qI[ϕ]],h{1,,n},λ+νqI[ϕ]˜u+jλνqI[ϕ]˜uj=0on qI[ϕ],˜u+j˜uj=0on qI[ϕ],qI[ϕ]˜u+jdσ=0, (16)

    then (˜u+j,˜uj)=(0,0). The case (λ+,λ)]0,+[2 was proved in Pukhtaievych [47,Prop. 1]. Accordingly it suffices to consider the limiting cases λ+=0,λ0 and λ=0,λ+0.

    Let λ+=0,λ0. The fifth equation in (16) implies that

    νqI[ϕ]˜uj=0on qI[ϕ].

    Accordingly, the divergence theorem implies that

    0Q¯qI[ϕ]|D˜uj(y)|2dy=Q˜uj(y)νQ˜uj(y)dσyqI[ϕ]˜uj(y)νqI[ϕ]˜uj(y)dσy=0.

    Indeed, by the q-periodicity of ˜uj one has that

    Q˜uj(y)νQ˜uj(y)dσy=0.

    Then, there exists cR such that ˜uj=c in ¯Sq[qI[ϕ]]. The sixth equation in (16) and the maximum principle for harmonic functions imply that also ˜u+j=c in ¯Sq[qI[ϕ]]. Finally, the seventh equation in (16) implies that c=0, i.e., ˜u+j=0 in ¯Sq[qI[ϕ]] and ˜uj=0 in ¯Sq[qI[ϕ]].

    Next we consider the case λ=0,λ+0. The fifth equation in (16) implies that

    νqI[ϕ]˜u+j=0on qI[ϕ].

    By the uniqueness of the solution of the interior Neumann problem up to constants, there exists cR such that ˜u+j=c in ¯Sq[qI[ϕ]]. The sixth equation in (16) and the periodic analog of the maximum principle for harmonic functions imply that also ˜uj=c in ¯Sq[qI[ϕ]] (see e.g., [45,Prop. A.1]). Finally, the seventh equation in (16) implies that c=0, i.e., ˜u+j=0 in ¯Sq[qI[ϕ]] and ˜uj=0 in ¯Sq[qI[ϕ]].

    In this section, we convert problem (4) into an equivalent integral equation. As done in [39] for the longitudinal flow along a periodic array of cylinders, we do so by representing the solution in terms of single layer potentials, whose densities solve certain integral equations. Therefore, we first start with the following proposition regarding the invertibility of an integral operator that will appear in such integral formulation of problem (4).

    Proposition 3.1. Let q,Q be as in (1) and (2) and α, Ω be as in (3). Let ϕC1,α(Ω,Rn)A˜QΩ. Let γ[1,1]. Let Kγ be the operator defined by

    Kγ[μ]=12μγwq,[qI[ϕ],μ]on qI[ϕ],μC0,α(qI[ϕ]).

    Then the following statements hold.

    (i) Kγ is a linear homeomorphism from C0,α(qI[ϕ])0 to itself.

    (ii) Kγ is a linear homeomorphism from C0,α(qI[ϕ]) to itself.

    Proof. We first consider statement (ⅰ). If γ]1,1[, then statement (ⅰ) follows from Pukhtaievych [47,Prop. 2] by noting that there exists a pair (γ+,γ)]0,+[2 such that

    γ=γ+γγ++γ.

    Accordingly, we have to consider only the limit cases γ{1,1}. We start with the case γ=1. Since by Theorem 2.1 (ⅳ) the operator wq,[qI[ϕ],] is compact in C0,α(qI[ϕ])0, the Fredholm alternative theorem implies that it suffices to show that K1[] is injective. Accordingly let μC0,α(qI[ϕ])0 be such that

    K1[μ]=12μwq,[qI[ϕ],μ]=0on qI[ϕ].

    The jump formula for the normal derivative of the single layer potential of Theorem 2.1 (ⅱ) implies that νqI[ϕ]v+q[qI[ϕ],μ]=0 on qI[ϕ]. Then, the properties of the single layer potential and the proof of Theorem 2.3 imply that there exists cR such that v+q[qI[ϕ],μ]=c in ¯Sq[qI[ϕ]] and vq[qI[ϕ],μ]=c in ¯Sq[qI[ϕ]]. Finally by the jump formula for the normal derivative of the single layer potential of Theorem 2.1 (ⅱ) we have that

    μ=νqI[ϕ]vq[qI[ϕ],μ]νqI[ϕ]v+q[qI[ϕ],μ]=0on qI[ϕ].

    Next, we consider the case γ=1. As before, it suffices to show that K1[] is injective. Accordingly let μC0,α(qI[ϕ])0 be such that

    K1[μ]=12μ+wq,[qI[ϕ],μ]=0on qI[ϕ].

    The jump formula for the normal derivative of the single layer potential of Theorem 2.1 (ⅱ) implies that νqI[ϕ]vq[qI[ϕ],μ]=0 on qI[ϕ]. Then, the properties of the single layer potential and the proof of Theorem 2.3 imply that there exists cR such that v+q[qI[ϕ],μ]=c in ¯Sq[qI[ϕ]] and vq[qI[ϕ],μ]=c in ¯Sq[qI[ϕ]]. Finally by the jump formula for the normal derivative of the single layer potential of Theorem 2.1 (ⅱ) we have that

    μ=νqI[ϕ]vq[qI[ϕ],μ]νqI[ϕ]v+q[qI[ϕ],μ]=0on qI[ϕ].

    Next, we consider statement (ⅱ). The Fredholm alternative theorem and the compactness of wq,[qI[ϕ],] in C0,α(qI[ϕ]) imply that it suffices to show that Kγ is injective in C0,α(qI[ϕ]). To this aim, we show that if μC0,α(qI[ϕ]) is such that

    Kγ[μ]=12μγwq,[qI[ϕ],μ]=0, (17)

    then μC0,α(qI[ϕ])0, i.e., qI[ϕ]μdσ=0. Then statement (ⅰ) would imply that μ=0. So let μC0,α(qI[ϕ]) be such that (17) holds. Theorem 2.1 (ⅱ) implies

    0=qI[ϕ]Kγ[μ]dσ={12γ(12|qI[ϕ]||Q|)}qI[ϕ]μdσ.

    A straightforward computation shows that 12γ(12|qI[ϕ]||Q|)=0 if and only if |qI[ϕ]||Q|12 and γ=112|qI[ϕ]||Q|. Since |qI[ϕ]||Q|]0,1[ one can easily realize that 112|qI[ϕ]||Q|[1,1]. Thus 12γ(12|qI[ϕ]||Q|)0 for all γ[1,1] and then qI[ϕ]μdσ=0.

    We are now ready to show that problem (4) can be reformulated in terms of an integral equation which admits a unique solution.

    Theorem 3.2. Let q,Q be as in (1) and (2) and α, Ω be as in (3). Let (λ+,λ)[0,+[2. Let ϕC1,α(Ω,Rn)A˜QΩ. Let j{1,,n}. Then problem (4) has a unique solution

    (u+j[q,ϕ,(λ+,λ)],uj[q,ϕ,(λ+,λ)])C1,αloc(¯Sq[qI[ϕ]])×C1,αloc(¯Sq[qI[ϕ]]).

    Moreover

    u+j[q,ϕ,(λ+,λ)](x)=v+q[qI[ϕ],μj](x)qI[ϕ]v+q[qI[ϕ],μj](y)dσyqI[ϕ]yjdσy+xjx¯Sq[qI[ϕ]],uj[q,ϕ,(λ+,λ)](x)=vq[qI[ϕ],μj](x)qI[ϕ]vq[qI[ϕ],μj](y)dσyqI[ϕ]yjdσy+xjx¯Sq[qI[ϕ]], (18)

    where μj is the unique solution in C0,α(qI[ϕ])0 of the integral equation

    12μjλ+λλ++λwq,[qI[ϕ],μj]=λ+λλ++λ(νqI[ϕ])jon qI[ϕ]. (19)

    Proof. We first note that, by Proposition 2.3 (ⅱ), problem (4) has at most one solution in C1,αloc(¯Sq[qI[ϕ]])×C1,αloc(¯Sq[qI[ϕ]]). As a consequence, we only need to prove that the pair of functions defined by (18) solves problem (4). Since

    (νqI[ϕ])jC0,α(qI[ϕ])0,

    Proposition 3.1 (ⅰ) implies that there exists a unique solution μjC0,α(qI[ϕ])0 of the integral equation (19). A straightforward computation and the continuity of the single layer potential imply that

    λ+(12μj+wq,[qI[ϕ],μj])λ(12μj+wq,[qI[ϕ],μj])=(λλ+)(νqI[ϕ])jon qI[ϕ],v+q[qI[ϕ],μj]qI[ϕ]v+q[qI[ϕ],μj]dσvq[qI[ϕ],μj]+qI[ϕ]vq[qI[ϕ],μj]dσ=0on qI[ϕ].

    Accordingly, the properties of the single layer potential (see Theorem 2.1) together with Proposition 2.2 imply that the pair of functions defined by (18) solves problem (4).

    The previous theorem provides an integral equation formulation of problem (4) and a representation formula for its solution. We conclude this section by writing the effective conductivity in a form which makes use of the density μj solving equation (19). To do that, we exploit the representation formula given by the previous theorem. Let the assumptions of Theorem 3.2 hold and let u+j[q,ϕ,(λ+,λ)], uj[q,ϕ,(λ+,λ)] and μj be as in Theorem 3.2. Then by the divergence theorem we have

    qI[ϕ]xiu+j[q,ϕ,(λ+,λ)](x)dx=qI[ϕ]u+j[q,ϕ,(λ+,λ)](y)(νqI[ϕ](y))idσy=qI[ϕ](v+q[qI[ϕ],μj](y)qI[ϕ]v+q[qI[ϕ],μj](z)dσzqI[ϕ]zjdσz+yj)(νqI[ϕ](y))idσy=qI[ϕ]v+q[qI[ϕ],μj](y)(νqI[ϕ](y))idσyqI[ϕ](νqI[ϕ](y))idσyqI[ϕ]v+q[qI[ϕ],μj](z)dσzqI[ϕ](νqI[ϕ](y))idσyqI[ϕ]zjdσz+δij|qI[ϕ]|n.

    Similarly, we have

    Q¯qI[ϕ]xiuj[q,ϕ,(λ+,λ)](x)dx=Quj[q,ϕ,(λ+,λ)](y)(νQ(y))idσyqI[ϕ]uj[q,ϕ,(λ+,λ)](y)(νqI[ϕ](y))idσy=δij|Q|nqI[ϕ]vq[qI[ϕ],μj](y)(νqI[ϕ](y))idσy+qI[ϕ](νqI[ϕ](y))idσyqI[ϕ]vq[qI[ϕ],μj](z)dσz+qI[ϕ](νqI[ϕ](y))idσyqI[ϕ]zjdσzδij|qI[ϕ]|n.

    Indeed

    Q(vq[qI[ϕ],μj](y)qI[ϕ]vq[qI[ϕ],μj](z)dσzqI[ϕ]zjdσz+yj)(νQ(y))idσy=Qyj(νQ(y))idσy=δij|Q|n.

    Moreover, by the divergence theorem, we have

    qI[ϕ](νqI[ϕ](y))idσy=0i{1,,n}.

    Accordingly, by the continuity of the single layer potential, we have that

    λeffij[q,ϕ,(λ+,λ)]=1|Q|n{λ+qI[ϕ]xiu+j[q,ϕ,(λ+,λ)](x)dx+λQ¯qI[ϕ]xiuj[q,ϕ,(λ+,λ)](x)dx}=1|Q|n{δijλ|Q|n+(λ+λ)(qI[ϕ]vq[qI[ϕ],μj](y)(νqI[ϕ](y))idσyqI[ϕ](νqI[ϕ](y))idσyqI[ϕ]vq[qI[ϕ],μj](z)dσzqI[ϕ](νqI[ϕ](y))idσyqI[ϕ]zjdσz+δij|qI[ϕ]|n)}=δijλ+(λ++λ){1|Q|n(λ+λ)(λ++λ)(qI[ϕ]vq[qI[ϕ],μj](y)(νqI[ϕ](y))idσy+δij|qI[ϕ]|n)}. (20)

    Thanks to Theorem 3.2, the study of problem (4) can be reduced to the study of the boundary integral equation (19). Therefore, our first step in order to study the dependence of the solution of problem (4) upon the triple (q,ϕ,(λ+,λ)) is to analyze the dependence of the solution μj of equation (19). Since in equation (19) the conductivity parameters λ+ and λ enter only in the quotient λ+λλ++λ, we will study the dependence of μj upon the periodicity q, the shape ϕ, and a parameter γ that will play the role of the contrast parameter λ+λλ++λ.

    Before starting with this plan, we note that equation (19) is defined on the (q,ϕ)-dependent domain qI[ϕ]. Thus, in order to bypass this problem, we first need to provide a reformulation on a fixed domain. More precisely, we have the following lemma.

    Lemma 4.1. Let q,Q be as in (1) and (2) and α, Ω be as in (3). Let (λ+,λ)[0,+[2. Let ϕC1,α(Ω,Rn)A˜QΩ. Let j{1,,n}. Then the function θjC0,α(Ω) solves the equation

    12θj(t)λ+λλ++λqϕ(Ω)DSq,n(qϕ(t)s)νqI[ϕ](qϕ(t))(θjϕ(1))(q1s)dσs=λ+λλ++λ(νqI[ϕ](qϕ(t)))jtΩ, (21)

    if and only if the function μjC0,α(qI[ϕ])0, with μj delivered by

    μj(x)=(θjϕ(1))(q1x)xqI[ϕ] (22)

    solves equation (19). Moreover, equation (21) has a unique solution in C0,α(Ω).

    Proof. The equivalence of equation (21) in the unknown θj and equation (19) in the unknown μj, with μj delivered by (22), is a straightforward consequence of a change of variables. Then, the existence and uniqueness of a solution of equation (21) in C0,α(Ω) follows from Theorem 3.2 and from the equivalence of equations (19) and (21).

    Inspired by Lemma 4.1, for all j{1,,n} we introduce the map

    Mj:D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×]2,2[×C0,α(Ω)C0,α(Ω)

    by setting

    Mj[q,ϕ,γ,θ](t)12θ(t)γqϕ(Ω)DSq,n(qϕ(t)s)νqI[ϕ](qϕ(t))(θϕ(1))(q1s)dσsγ(νqI[ϕ](qϕ(t)))jtΩ, (23)

    for all (q,ϕ,γ,θ)D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×]2,2[×C0,α(Ω). As one can readily verify, under the assumptions of Lemma 4.1, equation (21) can be rewritten as

    Mj[q,ϕ,λ+λλ++λ,θ]=0 on Ω. (24)

    Our aim is to recover the regularity of the solution θ of the above equation upon the periodicity q, the shape ϕ, and the contrast parameter λ+λλ++λ. Our strategy is based on the implicit function theorem for real analytic maps in Banach spaces applied to equation (24). As a first step, we need to prove that Mj is real analytic. In order to do that, we need the following result of [40], where we show that some integral operators associated with the single layer potential and its normal derivative depend analytically upon the periodicity q and the shape ϕ.

    Lemma 4.2. Let α, Ω be as in (3). Then the following statements hold.

    (i) The map from D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×C0,α(Ω) to C1,α(Ω) which takes a triple (q,ϕ,θ) to the function V[q,ϕ,θ] defined by

    V[q,ϕ,θ](t)qϕ(Ω)Sq,n(qϕ(t)s)(θϕ(1))(q1s)dσstΩ,

    is real analytic.

    (ii) The map from D+n(R)×(C1,α(Ω,Rn)A˜QΩ)×C0,α(Ω) to which takes a triple to the function defined by

    is real analytic.

    Next, we state the following technical lemma about the real analyticity upon the diffeomorphism of some maps related to the change of variables in integrals and to the outer normal field (for a proof we refer to Lanza de Cristoforis and Rossi [35,p. 166] and to Lanza de Cristoforis [33,Prop. 1]).

    Lemma 4.3. Let , be as in (3). Then the following statements hold.

    (i) For each , there exists a unique such that and

    Moreover, the map from to is real analytic.

    (ii) The map from to which takes to is real analytic.

    We are now ready to prove that the solutions of (24) depend real analytically upon the triple 'periodicity-shape-contrast'. We do so by means of the following.

    Proposition 4.4. Let , be as in (3). Let . Then the following statements hold.

    (i) For each , there exists a unique in such that

    and we denote such a function by .

    (ii) There exist and a real analytic map from

    to such that

    Proof. The proof of statement (ⅰ) is a straightforward modification of the proof of Lemma 4.1. Indeed, it suffices to replace by in the proof of Lemma 4.1.

    Next we turn to consider statement (ⅱ). As a first step we have to study the regularity of the map . The analyticity in the domain of definition of of the second term in the right hand side of (23) follows from Lemma 4.2 (ⅱ). Moreover, the analyticity in the domain of definition of of the third term in the right hand side of (23) follows from Lemma 4.3 (ⅱ) and from the analyticity of the map from to which takes to . Accordingly, is real analytic from to . By standard calculus in normed spaces, for all , the partial differential of at with respect to the variable is delivered by

    for all . By Lemma 3.1 (ⅱ) and by changing the variable as in the proof of Lemma 4.1, we deduce that is a linear homeomorphism from onto . Accordingly, we can apply the implicit function theorem for real analytic maps in Banach spaces (see, e.g., Deimling [18,Thm. 15.3]), and we deduce the existence of and of as in the statement.

    In this section we prove our main result that answers to question (10) on the behavior of the effective conductivity upon the triple 'periodicity-shape-conductivity'. To this aim, we exploit the representation formula in (20) of the effective conductivity and the analyticity result of Proposition 4.4.

    Theorem 5.1. Let , be as in (3). Let . Let be as in Proposition 4.4. Then there exists a real analytic map from the space to such that

    (25)

    for all .

    Proof. Let and be as in Proposition 4.4. Then, we set to be the map from the space to defined by

    for all . By formula (20) for the effective conductivity, by Proposition 4.4, by Lemma 4.1 and by Theorem 3.2, the only thing that remains in order to complete the proof is to show that the map is real analytic. Lemma 4.3 implies that

    for all . Since

    clearly depends analytically on . Moreover, by the divergence theorem and by Lemma 4.3

    Then, by taking into account that the pointwise product in Schauder spaces is bilinear and continuous, and that the integral in Schauder spaces is linear and continuous, Lemma 4.3 implies that the map from to which takes to is real analytic. Now, by Proposition 4.4, by Lemma 4.2 (ⅰ), by Lemma 4.3, together again with the above considerations, we can conclude that the map is real analytic from to and accordingly the statement holds.

    In the present paper we considered the effective conductivity of a two or three dimensional periodic two-phase composite material. The composite is obtained by introducing into a homogeneous matrix a periodic set of inclusions of a large class of sufficiently smooth shapes. We proved a regularity result for the effective conductivity of such a composite upon perturbations of the periodicity structure, of the shape of the inclusions, and of the conductivities of each material. Namely, we showed the real analytic dependence of the effective conductivity as a functional acting between suitable Banach spaces.

    The consequences of our result are twofold. First, this high regularity result represents a theoretical justification to guarante that differential calculus may be used in order to characterize critical periodicity-shape-conductivity triples as a first step to find optimal configurations, under specific constraints for the triple (cf. Section 1). Second, if and we have a family in such that the map is real analytic, then we can deduce the possibility to expand as a power series in , i.e.,

    for close to zero. Our result implies the possibility of an expansion of this type; then one can apply the method developed in [17] to determine the coefficients , by computing the differentials of .

    Both the authors are members of the 'Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni' (GNAMPA) of the 'Istituto Nazionale di Alta Matematica' (INdAM) and acknowledge the support of the Project BIRD191739/19 'Sensitivity analysis of partial differential equations in the mathematical theory of electromagnetism' of the University of Padova. P.M. acknowledges the support of the grant 'Challenges in Asymptotic and Shape Analysis - CASA' of the Ca' Foscari University of Venice. The authors wish to thank the anonymous referees for many valuable comments that have improved the presentation of the paper.


    Acknowledgments



    The authors would like to thank Ms Wing Lam Chung and Ms Amanda Shue Qi Chia for conducting the pilot test; and Ms Alyssa Elyn Pua and Mr Shandon Shi Kai Ng for performing the quality evaluations. The authors would also like to thank Mr Jovan Kris Kamadjaja, Mr Francis Tee Heng Chia, Mr Shandon Shi Kai Ng and Mdm Betty Gan for lending their smart devices for this study (Google Nest Mini, Samsung Galaxy Note 9, Echo Dot and iPhone 6S respectively).

    Conflict of interest



    No competing financial interests exist. All authors have no conflicts of interest with any of the evaluated apps or their companies in this study. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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