## Abstract

We consider a variant of the isoperimetric problem with a non-local term representing elastic energy. More precisely, our aim is to analyse the optimal energy of an inclusion of a fixed volume the energy of which is determined by surface and elastic energies. This problem has been studied extensively in the physical/metallurgical literature; however, the analysis has mainly been either (i) numerical, or (ii) restricted to a specific set of inclusion shapes, e.g. ellipsoids. In this article, we prove a lower bound for the energy, with no *a priori* hypothesis on the shape (or even number) of the inclusions.

## 1. Introduction

We consider a variant of the isoperimetric problem with a non-local term representing elastic energy. More precisely, our aim is to analyse the optimal energy of an inclusion of a fixed volume the energy of which is the sum of surface and elastic energies. We note that this problem has been studied extensively in the physical/metallurgical literature. However, in this literature, the analysis has mainly been either (i) numerical, or (ii) restricted to a specific set of inclusion shapes, e.g. ellipsoids. Such studies give upper bounds on the minimum energy. In this article, we prove a corresponding lower bound, with no *a priori* hypothesis on the shape (or even number) of the inclusions.

Elastic inclusions can be observed when a material undergoes a phase transformation between two preferred strains, which may, for example, be elicited by a change of temperature. In this case, the phase transformation is initiated by the creation and growth of a small nucleus representing the new material state. The saddle point between the two uniform phases is represented by the critical nucleus the energy of which describes the energy barrier between the two uniform phases.

In the classical theory of nucleation, the size and shape of the critical nucleus is determined by a competition between bulk energy and interfacial energy. While the bulk energy favours the emergence of the new phase, the interfacial energy provides an energy barrier for the creation and growth of the nucleus. In this situation where all the terms contributing to the energy are local, the optimal shape of the inclusion does not depend on its size. The minimizers are well-known: in the simplest situation when the interfacial energy is isotropic, the shape of the inclusion is a sphere. More generally, when the interfacial energy is anisotropic, the minimizers take the form of the well-known Wulff shape (e.g. Wulff 1901; Taylor 1975; Fonseca & Müller 1991).

The elastic energy introduces a length scale into the problem; in particular, the shape of the optimal inclusion depends on its volume. The energy of inclusions in the presence of elastic energy has mainly been studied numerically or assuming an ellipsoidal inclusion shape. Numerical simulations for the morphology and evolution of the nucleus have been given, e.g. by Voorhees *et al.* (1992) and Zhang *et al.* (2007, 2008). In the physical literature, the shape (and the growth) of the inclusion has been optimized within an ansatz, e.g. Khachaturyan (1982), Mura (1982), Brener *et al.* (1999) and Wang & Khachaturyan (1994). While such calculations give much insight, they only apply for certain restricted classes of configurations.

Early work on mathematical analysis of energy-driven pattern formation includes the analysis by Kohn & Müller (1992, 1994) of a toy model in elasticity theory. Related problems have attracted increasing attention in the past years with the analysis of various models (Choksi & Kohn 1998; Choksi *et al.* 1999; Alberti *et al.* 2009; Capella & Otto 2009, submitted). While most of the previous analysis does not address the dependence of the energy on the volume fraction of the different phases, recently the case of extreme volume fraction has gained more attention: in Choksi *et al.* (2008), the intermediate state of a type-I superconductor is studied by ansatz-free analysis for the case of extreme volume fraction. Another recent result addresses the energy scaling in a bulk ferromagnet in the presence of an external field of critical strength (Knüpfer & Muratov 2010). In all of the above models, the energy is characterized by a non-local term and a local, regularizing term of higher order. One special feature of the elastic energy is the fact that the non-local term is anisotropic. Insightful analysis has been developed on determining possible configurations that are free of elastic energy (e.g. Dolzmann & Müller 1995; Müller & Šverák 1996).

Our main result, theorem 2.1, is an ansatz-free lower bound for the energy when the volume of the inclusion phase is fixed. Let us give an overview of the arguments that are used in the proof. There are two main ingredients: the first ingredient is a covering argument which reduces the task to a local problem by identifying a local length scale where elastic and interfacial energies are in balance. The second main ingredient is a lower bound for the elastic energy for the local problem. Here, the analysis takes advantage of the discreteness of the phase function *χ* which describes the shape of the minority phase. In a related context, discreteness of *χ* has been used to prove rigidity results or to give lower bounds on the energy (Dolzmann & Müller 1995; Capella & Otto 2009). While most of the mathematical arguments in this article are based on the sharp-interface description of the elastic energy, we also present the two examples of diffuse-interface models and show how our results extend to these models.

### (a) Structure of the article

In §2, we introduce the model and state our results for both the sharp and diffuse-interface models. In §3, we give the proof of our result for the sharp-interface model. In §4, we discuss the diffuse-interface models. Basic results about geometrically linear elasticity are collected in the appendix A.

### (b) Notation

The following notation will be used throughout the article: by a universal constant, we mean a constant that only depends on the dimension *d*. The symbols and indicate that an estimate holds up to a universal constant. For example, *A*∼*B* says that there are universal constants *c*,*C*>0 such that *cA*≤*B*≤*CA*. The symbols ≪ and ≫ indicate that an estimate requires a small universal constant. If we, for example, say that for *ϵ*≪1, then this means that *A*≤*CB* holds for all *ϵ*≤*ϵ*_{0} where *ϵ*_{0}>0 is a small universal constant.

The mean value of a function *f* on the set *E* is denoted by 〈*f*〉_{E}. For *u*∈*BV* (*E*), the total variation of *u* is sometimes denoted by ∥*Du*∥_{E}. The Fourier transform of *u* is defined by ; in particular, Parseval’s identity holds with constant 1. The set of *d*×*d* matrices is denoted by , the set of symmetric matrices by . The set of ‘compatible strains’ is1.1where the tensor product is defined componentwise by (*u*⊗*v*)_{ij}=*u*_{i}*v*_{j}. Finally, for , the contraction is defined by , and the corresponding matrix norm is given by .

## 2. Model and statement of results

In the framework of geometrically non-linear elasticity, the energy associated with the deformation of an elastic body with domain of reference , *d*≥2, is given by2.1where the non-convex energy density describes the preferred states of the material (e.g. Ball & James 1987; Bhattacharya 2003). Each preferred crystal configuration of the material corresponds to a well of the elastic energy density. We want to analyse the case where the elastic energy has two preferred phases. Furthermore, we use the geometrically linear approximation of the non-linear theory (2.1) studied, e.g. by Khachaturyan (1967) and Roitburd (1969); (see also Kohn 1991; Bhattacharya 1993). In this theory, the deformation is described by the displacement *u*(*x*)=*y*(*x*)−*x*, while the elastic energy is a function of the linear strain *e*(*u*)=1/2(∇*u*+∇^{t}*u*). Also, it is well accepted that on small scales the energy should be complemented by a higher-order term. Therefore, we include a sharp-interface energy penalizing the interfaces between the two phases. Finally, we include a term that captures the energetic favourability of the new phase.

The above considerations motivate considering the energy2.2where . In this model, the two preferred phases are represented by 0 and *F*. The elastic energy ∥*e*(*u*)−*χF*∥^{2} uses a trivial Hooke’s Law; this represents no loss of generality since we seek a lower bound and our focus lies on its scaling law not the prefactor. The characteristic function describes the region occupied by the minority phase; see also, Bhattacharya (2003, ch. 12). It allows us to define the volume of the inclusion2.3The set of admissible functions for the prescribed volume *μ* is given by

### Theorem 2.1 (Scaling of energy)

*Let* *, and define* *, where* *is the set of compatible strains defined in the compatible strain (1.1). In any dimension d≥2 we have the upper bound*2.4*and the lower bounds*2.5*and*2.6*In particular (averaging the two lower bounds), we have*2.7

Equation (2.7) gives the scaling of the minimal energy up to a universal constant. The first term on the right, *δμ*, is related to the incompatibility of the two phases; it vanishes if they are elastically compatible. The second term is independent of the compatibility between the two phases but depends on the magnitude of the ‘eigenstrain’ *F*. For small inclusions, interfacial energy is dominant and the optimal scaling is achieved by a sphere. On the other hand, for larger inclusions, the shape of the minimizer is determined by competition between elastic energy and interfacial energy. In this case, the optimal scaling is achieved by an inclusion with a shape of a thin disc, the large surfaces of which lie in the twin planes between the two phases. Equation (2.7) shows that this well-known picture is energetically optimal.

When with all other parameters held fixed, stronger upper bound (2.4) and first stronger lower bound (2.5) show thatThus, for *η*>0 we have2.8In view of upper bound (2.4), it is natural to guess that the correction defined by equation (2.8) is of order . However, our results fall short of proving this, since the left-hand side of lower bound (2.6) is *e*(*μ*)+*γμ* rather than *e*(*μ*)+(*γ*−*δ*)*μ*. (For a brief discussion why, see the end of §3.)

The energy barrier between the two uniform states in model (2.2) can be calculated by a minimax principle: first, identify the global minimum *e*(*μ*) of all configurations with fixed volume *μ*; then identify the maximum over all *μ*>0. The corresponding minimizing configuration with this volume is the critical nucleus for the phase transformation. Its energy describes the energy barrier between the two uniform phases, in the absence of defects or boundaries.

As a consequence of theorem 2.1, we identify three regimes that characterize the energy scaling and the size of the critical nucleus for model (2.2).

### Theorem 2.2 (Critical nucleus)

*Let e(μ) and δ be as in theorem 2.1. Then*

*— For γ≪δ, we have* *and* *.*

*— For γ≫δ and γ≪∥F∥*^{2}*, we have* *and μ***∼η*^{d}*γ*^{1−2d}*∥F∥*^{2d−2} *for all μ***∈ argmax*_{μ>0}*e(μ).*

*— For γ≫δ and γ≫∥F∥*^{2}*, we have* *and μ***∼η*^{d}*γ*^{−d} *for all μ***∈argmax*_{μ>0}*e(μ).*

Theorem 2.2 is a direct consequence of equation (2.7). In the first of its three regimes the energy barrier is infinite, so there is no critical nucleus of finite size. This excludes a phase transformation in an infinite sample. Note that the first regime can only occur for an incompatible inclusion. In the second and third regimes the size of the critical nucleus is finite. Depending on the relative size of *γ* and ∥*F*∥, the scaling and the size of the critical nucleus is however quite different. In particular, the second regime corresponds to a penny-shaped nucleus while the third regime corresponds to a (approximately) spherical nucleus. The cases in theorem 2.2 do not cover all possible values of the parameters (for example, they do not cover the case *γ*=*δ*). This is because we do not have a lower bound directly analogous to the stronger upper bound (2.4).

We note that the physical relevance of theorem 2.2 relies on the following physical assumptions about the phase transformation: (i) The volume *μ* is supposed to be a continuous function in time. (ii) For fixed time *t* and volume *μ*(*t*), the configuration achieves the optimal shape. (iii) Finite-size effects (such as nucleation at a boundary or corner) are being ignored. (iv) The critical nucleus has a reasonably sharp interface.

Theorems 2.1 and 2.2 address the sharp-interface energy (2.2), but the same ideas can also be used in a diffuse-interface setting. We shall explain this in §4 where we define two diffuse-interface analogies of model (2.2), and . Their minimum-energy scaling laws are the same as that of .

### Theorem 2.3 (Diffuse-interface energies)

*For γ=0, we have*

We remark that our estimates only capture the scaling but not the leading order constant of the minimal energy. Consequently, the results do not give information about the precise shape of the minimizers. While we expect that when *μ* is large the minimizer resembles a thin disc, this does not follow from our analysis. However, our analysis shows that a thin disc (with an appropriate ratio of height and diameter) is optimal in terms of the scaling of the energy. We expect that a more precise estimate of the shape of the optimal inclusion would require the use of the Euler–Lagrange equation for model (2.2).

## 3. Proof of theorem 2.1

In §3*a*, we give two lower bounds on the elastic energy. The proof of theorem 2.1 is then given in §3*b*.

### (a) Lower bounds for the elastic energy

In this section, we give two lower bounds on the elastic energy in propositions 3.1 and 3.5. The main result is the following lower bound for the elastic energy:

### Proposition 3.1

*Let* *d*≥2 *and let* *χ*∈*BV* (*B*_{R},{0,1}). *There exist constants* *c*_{d} *and* *α*_{d} *such that if*3.1*then we have*3.2

Although this estimate does not assume compatibility of *F*, it is most relevant in this case: the case of incompatible *F* leads to a higher elastic energy and is treated in proposition 3.5. The two conditions in equation (3.1) state that the volume fraction of the minority phase should be relatively small in *B*_{R} and that the interfacial energy should be small compared with ∂*B*_{R}. Both conditions in equation (3.1) are necessary: if we did not assume ∥*χ*∥_{L1(BR)}≪1, then the configuration with uniform gradient *F* would not cost any elastic energy and hence would yield a counter example. The second condition in equation (3.1) excludes stripe-like patterns where the stripes are aligned with the twin planes between the strains *F* and 0. Such laminar structures also would not yield any contribution of elastic energy (if *F* is compatible).

We split the proof of proposition 3.1 into two parts. We first prove the case *d*=2 in lemma 3.2, before turning to the case of general *d*≥2 in lemma 3.4. The proof for *d*=2 relies strongly on the fact that the interfacial energy is discrete on the boundary of two-dimensional sets. Since compatibility is a two-dimensional issue (see also lemma A.2), it is not surprising that the case *d*=2 is special.

### Lemma 3.2

*Proposition 3.1 holds for* *d*=2 *and* .

### Proof.

By the rescaling *x*↦*x*/*R* and by a rotation, it is enough to consider *R*=1 and *F*=diag(*λ*_{1},*λ*_{2}). Furthermore, without loss generality, we assume *λ*_{1}≥|*λ*_{2}|. We argue by contradiction and assume that equation (3.2) does not hold; i.e. for some fixed universal but arbitrarily small constant *c*_{2}, we have3.3where we have set *μ*:=∥*χ*∥_{L1(Bα)}. (Here and below we write *α* in place of *α*_{2} for simplicity of notation.) In the following, we will not keep track about the precise form of the constants. Instead, we use the notation ≪ if an estimate holds for a universal but small constant. We thus write equation (3.3) as3.4

*Step 1: notation and choice of Q^{(i)}.* It is more convenient to work with rectangles instead of balls. We cut out three rectangles ,

*i*=1,2,3, where with , see figure 1. Also let . We choose the sets such that , in particular . Furthermore, we may assume that the side lengths of

*Q*

^{(i)}are of order 1, i.e. , |

*I*

_{2}|∼1 (i.e. up to a universal constant). Note that this is possible for ; the argument is not optimized in

*α*. By Fubini’s theorem and by adjusting the sets slightly, we may also assume that, on the boundaries of the sets

*Q*

^{(i)}, there is no concentration of energy and no concentration of the minority phase, i.e. the following integrals are well defined and we have (using equations (3.1) and (3.4))3.5 3.6and 3.7

*Step 2: only majority phase on ∂ Q^{(i)}.* By discreteness of the surface energy on lines and discreteness of

*χ*, equations (3.6) and (3.7) can be improved. In fact,3.8and3.9Indeed, since the sets ∂

*Q*

^{(i)}are one-dimensional, the measure in equation (3.6) is discrete; hence equation (3.6) strengthens to equation (3.8). Furthermore, since

*χ*∈{0,1} and in view of equation (3.8), equation (3.7) strengthens to equation (3.9).

*Step 3: smallness of u_{1} and u_{2} on ∂Q^{(i)}.* In this step, we show that (after normalization)

*u*

_{1}is small on all horizontal boundaries and

*u*

_{2}is small on all vertical boundaries of the sets

*Q*

^{(i)}, i.e.3.10where

*i*=0,1,2,3 and

*j*=0,1. Since the energy only depends on

*e*(

*u*), we may transform

*u*by transformations that only affect the anti-symmetric part of

*Du*. In particular, by the change of variablesandwe may assume that the average of

*u*

_{1}vanishes on both horizontal boundaries, i.e.3.11Similarly, by the change of variables , we may also assume that the average of

*u*

_{2}vanishes on the left vertical boundary, i.e. . By equation (3.9), we now obtain -control of

*u*

_{1}on both horizontal boundaries, i.e.3.12In particular, our supremum control of

*u*

_{1}on the horizontal boundaries yields3.13We next use the cross-diagonal part of the energy to get similar control for

*u*

_{2}on all vertical boundaries of

*Q*

^{(i)}. For this, we note that by Jensen’s inequality, we have3.14Estimates (3.13) and (3.14) together yield |〈∂

_{1}

*u*

_{2}〉

_{Q(i)}|≪

*λ*

_{1}

*μ*for

*i*=1,2,3. Together with our normalization, this shows that

*u*

_{2}is small in average on all vertical boundaries of

*Q*

^{(i)}, i.e.3.15By equations (3.5) and (3.9), we then get control on

*u*

_{2}on all vertical boundaries of

*Q*

^{(i)}, i.e.3.16for

*i*=0,1,2,3. This concludes the proof of equation (3.10).

*Step 4: smallness of u_{1},u_{2} in Q^{(i)}.* In this step, we show that

*u*

_{1}is small in average in all

*Q*

^{(i)}, i.e.3.17We show the argument for

*i*=1. In order to prove equation (3.17), consider an arbitrary horizontal line

*Γ*in

*Q*

^{(i)}and let be the larger of the two rectangles confined by

*Γ*and ∂

*Q*, in particular . We note that on both vertical boundaries of , we have equation (3.16) and hence . Arguing as in the previous step of the proof, it follows that . Since on one horizontal boundary of we have , this yields, 〈

*u*

_{1}〉

_{Γ}≪

*λ*

_{1}

*μ*. This concludes the proof of equation (3.17).

*Step 5: contradiction.* In this step, we use the diagonal part of the elastic energy to derive a contradiction. Indeed, we shall show that3.18contradicting equation (3.17). To see contradiction (3.18), we choose such that . We then haveAveraging over , then yields3.19where the averages are taken in the variables and where we have used in the last estimate. The last term on the right-hand side can be estimated by application of Young’s inequality,which in view of equations (3.4) and (3.19) implies contradiction (3.18). ■

The following result will be needed in the proof for higher space dimensions. Note that this result does not only apply for characteristic functions, but instead works for any taking values in [0,1]:

### Lemma 3.3

*Suppose that the sets* , *i*=1,2,3, *are given as in the proof of proposition 3.2. Furthermore suppose that equation (3.10) holds. Then we have for any* ,3.20

### Proof.

By assumption, the conclusion of step 3 in the proof of lemma 3.2 is satisfied. We can then argue as in steps 4 and 5 to get equation (3.20). ■

We now turn to the proof of proposition 3.1 for general *d*≥2. In this case, the surface energy on the boundaries ∂*Q*^{(i)}, in general, is no longer discrete and hence the argument leading to equations (3.8) and (3.9) cannot be used. The idea is to recover an estimate related to equation (3.9) by averaging out (*d*−1) dimensions and by using an induction argument.

### Lemma 3.4

*Proposition 3.1 holds for all* *d*≥2.

### Proof.

By the rescaling *x*↦*x*/*R* and a rotation, we may assume that *R*=1 and that *F*=diag(*λ*_{1}, … ,*λ*_{d}) where, without loss of generality, *λ*_{1}≥⋯≥*λ*_{d} and *λ*_{1}≥|*λ*_{d}|. Proceeding by induction, we show that proposition 3.1 holds in *d* dimensions if it holds both in (*d*−1) dimensions and two dimensions. The case *d*=2 has been shown in lemma 3.2. By induction hypothesis, we hence assume that the proposition holds for all *d*′ with 2≤*d*′≤*d*−1. We will argue by contradiction and hence (setting ), we assume3.21

*Step 1: notation and choice of Q^{(i)}.* The geometry is a natural generalization of the one used in the proof of lemma 3.2: we choose sets , where and where and with for

*i*=1,2,3 and

*j*=1, … ,

*d*. Also let . We choose the sets such that , in particular . Furthermore, we may assume that the side lengths of

*Q*

^{(i)}are of order 1, i.e. , |

*I*

_{j}|∼1. We also define

*Π*as the extension through

*B*

_{1}of all boundaries of the sets

*Q*

^{(i)}with normal

*e*

_{1}or

*e*

_{2}, i.e.By Fubini’s theorem and by adjusting the sets slightly, we may assume that there is no concentration of energy and no concentration of the minority phase on

*Π*, i.e.3.22 3.23and 3.24

*Step 2: predominantly majority phase on part of boundary.* For *d*=2, we have used that the restriction of the measure ∥∇*χ*∥ on one-dimensional sets is discrete to strengthen equations (3.6) and (3.7) to equations (3.8) and (3.9). This is not possible for *d*≥3. Instead, we claim that we have only a small fraction of the minority phase on all surfaces of ∂*Q*^{(i)} with normal *e*_{1} and *e*_{2}, in the sense of3.25which can be seen as a stronger version of equation (3.24). We present the argument for one estimate in equation (3.25) (the other arguments being analogous), i.e. we show3.26In order to prove equation (3.26), consider where is the projection of the set on the last (*d*−1) components. Note that if *α*_{d} is sufficiently small, then for some , *ρ*∼1, we havewhere is the (*d*−1)-dimensional ball with centre and radius *γ*. Furthermore by equations (3.23) and (3.24), the restriction of *χ* on the set satisfies equation (3.1). Finally, the restriction of *F* to directions in has norm ≥*λ*_{2}. By the induction hypothesis, it hence follows thatconcluding the proof of equation (3.26).

*Step 3: reduction to two dimensions.* In this step, we reduce the argument to the two-dimensional case by averaging out *H*-dependence. For this, we define and bywhere *i*=1,2. Correspondingly, we define the two-dimensional strain matrix . We note that by Jensen’s inequality we have3.27We claim that (after normalization) is small on all horizontal boundaries and is small on all vertical boundaries of the sets , i.e.3.28where *i*=0,1,2,3 and *j*=0,1. Assuming for a moment that equation (3.28) holds, we can apply lemma 3.3 to getwhich contradicts equation (3.21). This concludes the proof of the lemma if equation (3.28) holds.

*Step 4: smallness of on .* It remains to prove equation (3.28), which is done in this step. By a change of coordinates (as in step 3 of the proof of lemma 3.2), we may assume3.29It follows that for *j*=0,1, we have3.30where we have used Jensen’s inequality in the third estimate. From equation (3.30), we get3.31As in equation (3.14), estimate (3.31) yields control of *u*_{2}, i.e. we get for *i*=1,2,3. By equation (3.29), it then follows successively that the average of is small on all vertical boundaries of *Q*^{(i)}, i.e.3.32Now, as before, using control of elastic energy (3.22), we obtainwhere we have used equations (3.32), (3.21) and (3.25) in the last estimate. This concludes the proof of equation (3.28) and hence of lemma 3.4. ■

We next address a lower bound for incompatible strains:

### Proposition 3.5 (Lower bound for incompatible strains)

*We have*

### Proof.

By a rotation we may assume that *F*=diag(*λ*_{1}, … ,*λ*_{d}) and, without loss of generality, *λ*_{1}≥⋯≥*λ*_{d} and *λ*_{1}≥|*λ*_{d}|. Furthermore, we may assume that *u* is a minimizer for fixed *χ*. Then, in view of lemma A.1, the elastic energy can be expressed in Fourier variables aswhere *n*=*ξ*/|*ξ*| and where *Φ*(*n*)=∥*F*∥^{2}−2∥*Fn*∥^{2}+〈*n*,*Fn*〉^{2}. By Parseval’s identity and since *χ* is a characteristic function, it follows thatThe proof is concluded by the lower bound on *Φ* in lemma A.2. ■

### (b) Proof of theorem 2.1

In this section, we give the proof of theorem 2.1. We first note that by rescaling in length, one of the parameters *μ*,*η* and ∥*F*∥ can be scaled to 1. We choose to remove *η*, i.e. we rescale , , and . Furthermore, since the energy dependence on *γ* is trivial, we may assume *γ*=0 in this section. Skipping the hats in the sequel, the non-dimensionalized version of the energy is then given bywhile theorem 2.1 is equivalent to the following.

### Theorem 3.6

*For any d≥2, there is a constant C (depending only on d) such that*3.33*Also, there are constants c*_{1} *and c*_{2} *(depending only on d) such that*3.34*and*3.35

We shall discuss the upper and lower bounds separately.

### Proof of theorem 3.6 (Upper bound).

This result is well understood in the physical literature (e.g. Khachaturyan 1982). We give a simple proof which does not require the use of Fourier transform.

*Part 1: the case of large inclusions.* We first give the construction for the case of larger inclusions, i.e when *μ*≫∥*F*∥^{−2d}. In this case, the shape is determined by a balance of elastic energy and surface energy. The idea is to choose the inclusion to have approximately the shape of a thin disc *Q*_{T,R} with diameter *R* and thickness *T* where *T*≪*R*. The disc is oriented such that the two large surfaces are aligned with one of the twin planes between *F* and 0. Since the desired result is a scaling law, it will be sufficient to specify how *T* and *R* scale with the parameters. In particular, we do not need to optimize the precise shape of the disc. The construction is as follows: let *P* be the projection of *F* on the set of compatible strains. In particular, *P* has the representation *P*=1/2(*v*⊗*n*+*n*⊗*v*) for some with ∥*n*∥=1 and furthermore , see lemma A.2. The construction is symmetric with respect to the cylindrical coordinates *z*:=〈*x*,*n*〉, *r*:=|*x*−〈*x*,*n*〉*n*|.

Consider two points *x*^{(1)}=−*x*^{(2)} on the axis *r*=0 with distance *d* and consider the intersection , where *ρ*>0. Now for any 0<*T*≪*R*, we can adjust *d* and *ρ* such that the intersection is a lens with thickness of order *T* and diameter of order *R*. We next define *χ* byAssuming that the consistency condition *T*≪*R* holds, the interfacial energy for this configuration is estimated by . It remains to choose the deformation *u*. For this, we define by *u*_{0}(0)=0 and ∇*u*_{0}=*v*⊗*n* in *Q*_{T,R}. Outside *Q*_{T,R}, we let *u*_{0} be constant on all lines which are normal to the surface ∂*Q*_{T,R}. In the remaining area, we set *u*_{0}=0, see figure 2. In particular,3.36Let 0≤*ζ*≤1 be a smooth cut-off function such that *ζ*=1 in *B*_{R} and *ζ*=0 outside *B*_{2R}. We choose *ζ* such that furthermore . We then define by *u*:=*ζu*_{0}. Correspondingly, the elastic energy is estimated as follows:whereWe choose *T* such that the volume constraint (2.3) is satisfied, i.e. |*Q*_{T,R}|=*μ* and in particular, *R*^{d−1}*T*∼*μ*. It follows that:where we have chosen the optimal radius *R*=∥*F*∥^{2/(2d−1)}*μ*^{2/(2d−1)}. One can check that *R* and *T*, defined above, satisfy the consistency condition *T*≪*R* if *μ*≫∥*F*∥^{−2d} is satisfied. This concludes the proof of the upper bound for large inclusions.

*Part 2: the case of small inclusions.* It remains to consider the case . In this case the surface energy dominates. Accordingly, we choose the inclusion to have the shape of a ball with volume *μ* and radius of size *R*∼*μ*^{1/d}, i.e. *χ*:=*χ*_{BR}. In particular, the interfacial contribution of the energy is estimated by3.37Now, let *ζ* be a smooth cut-off function with 0≤*ζ*≤1 and such that *ζ*=1 in *B*_{R} and *ζ*=0 outside *B*_{2R}. We may, furthermore, assume that . We define by *u*_{0}(0)=0 and ∇*u*_{0}=*P*. In particular,Now, choosing *u*:=*ζu*_{0}, the elastic energy can be estimated as follows:3.38where we have used *μ*≤∥*F*∥^{−2d} in the third inequality. Estimates (3.37) and (3.38) together yield the upper bound for small inclusions. ■

### Proof of theorem 3.6 (Lower bound).

The first lower bound (3.34) is easy. Proposition 3.5 gives a lower bound for the elastic term, and the isoperimetric inequality gives a lower bound for the surface energy term. Combining the two results gives equation (3.34). The only remaining task is to prove3.39In order to show equation (3.39), we combine an application of proposition 3.1 with a decomposition argument. Let *M*:=*supp* *χ*. Without loss of generality we assume that all *x*∈*M* are points of density 1 of *M*. For all *x*∈*M*, let3.40where *c*_{0} is a sufficiently small universal constant to be fixed later. We note that both sides of the inequality in equation (3.40) are continuous functions in *r*. Furthermore, the inequality in equation (3.40) is not satisfied for *r*=0 (since *x* has density 1 in *M*), but always satisfied in the limit (since ). This ensures the existence of *R*(*x*) satisfying equation (3.40). Furthermore, we observe that *R*=*R*(*x*) satisfies one of the following conditions: Either3.41or3.42Let us explain how definition (3.40) is motivated by the two constructions in the proof of the upper bound: in fact, for each of these constructions, take the smallest ball that covers supp *χ* and consider the density of the minority phase in this ball. Then, up to the constant *c*_{0}, in case (3.41) the density of the minority phase in *B*_{R}(*x*) corresponds to the density of the upper bound construction for the case of small inclusions. Similarly, up to the constant *c*_{0}, the density of the minority phase in case (3.42) corresponds to the density of the upper bound construction for the case of large inclusions.

Trivially, *M* is covered by and by cases (3.41) and (3.42), the radii *R*(*x*) are uniformly bounded in terms of ∥*F*∥ and *μ*. Hence, by Vitali’s covering lemma, there is an at most countable subset of points such that the balls *B*_{Ri/5}(*x*_{i}) are disjoint while *M* is still covered by the balls *B*_{Ri}(*x*_{i}). Let *R*_{i}:=*R*(*x*_{i}), *R*_{i}′:=*R*_{i}/5 and *R*_{i}′′:=*α*_{d}*R*_{i}/5 (where *α*_{d} is the constant from proposition 3.1) and let *B*_{i}:=*B*_{Ri}(*x*_{i}), *B*_{i}′:=*B*_{Ri′}(*x*_{i}) and *B*_{i}′′:=*B*_{Ri′′}(*x*_{i}). In particular, for *i*≠*j* and . It follows that3.43where *E*_{|A}, , is the fraction of the energy localized on the set *A*, i.e.We claim that the following two estimates hold:3.44and3.45The desired lower bound is then a consequence of estimates (3.44) and (3.45). Indeed, we havewhere in the last estimate, we have used that whenever *c*_{i}≥0 and 0≤*β*<1. It remains to prove estimates (3.44) and (3.45). In order to show the estimate (3.44), we note that by the minimality of *R*, the following estimates hold:

If and , then3.46Similarly, if and , thenFinally, if , then3.47Estimates (3.46) and (3.47) together yield estimate (3.44).

In order to prove estimate (3.45), we differentiate between three cases: in the first case, we assume that case (3.41) holds. Since the density of the minority phase is much smaller than 1 in *B*_{i}′′ and hence also in *B*_{i}′, we get by the isoperimetric inequality that3.48Note that in the above formula, the surface energy on ∂*B*_{i}′ is not counted. This version of the isoperimetric inequality applies since we are in a low volume fraction case. It follows thatIn the second case, we assume that case (3.42) holds and furthermore3.49In view of , we immediately obtainIt remains to consider the case when case (3.42) holds and furthermore . In this case, choosing *c*_{0} small enough, the assumptions of proposition 3.1 are satisfied (on *B*_{i}′). An application of this proposition then yieldsThe above estimates yield estimate (3.45) which concludes the proof of equation (3.39) and hence of the theorem. ■

As we noted in §1, it is natural to conjecture that the upper bound (2.4) is within a constant of being optimal. In the context of theorem 3.6 this amounts to the conjecture thatOur methods seem incapable of giving such a conclusion, since the (real-space) argument used to prove equation (3.34) is insensitive to the value of , while the (Fourier and Sobolev-estimate-based) argument used to prove equation (3.35) treats the elastic and perimeter terms separately. To do better, it would seem necessary to find a Fourier-based argument that treats the elastic and perimeter terms together.

## 4. Two diffuse-interface models

Diffuse-interface models are widely used in the literature on elastic phase transformations; recent examples include the work by Poduri & Chen (1996) and Zhang *et al.* (2007, 2008). We present two diffuse-interface variants of our model (2.2) and show that in the absence of bulk energy, i.e. when *γ*=0, the scaling of the minimal energy for the diffuse-interface models is the same as that of the sharp-interface model.

In the first model, the energy is formulated in terms of the strain *e*(*u*). We set4.1with double-well potential given by where *F* is (as usual) a symmetric matrix. The second model we want to discuss is given by4.2where the standard double-well potential *W*_{1}(*t*):=*t*^{2}(1−*t*)^{2} penalizes the deviation of the order parameter from a characteristic function. Model (4.1) is a strain-gradient version of model (2.2). Models like (4.2) are often preferred for numerical work since the minimization over *u* (given ) can be efficiently computed using FFT (Zhang *et al.* 2008).

In model (4.2) the ‘phase’ is determined by a scalar-valued order parameter . In connection with model (4.1) it is convenient to define an analogous scalar-valued order parameter by4.3Note that by the triangle inequality, and4.4As in the sharp-interface setting, we want to characterize the energy of inclusions with fixed volume of the minority phase. For this, we choose4.5for both models (4.1) and (4.2); accordingly, the admissible functions for and areandAnother possibility would be to use the *L*^{1}-norm of in definition (4.5). However, this is not a good notion for our purpose since then minimizers for fixed volume would tend to spread out to infinity (never approaching a phase transformation).

Neglecting bulk energy (i.e. setting *γ*=0), theorem 2.1 can be generalized to the above diffuse-interface models as stated in theorem 2.3. The proof proceeds as follows.

### Proof of theorem 2.3.

By the same rescaling as in §3*b*, we may assume *η*=1. The proof of the upper bound for the two diffuse-interface energies is easy and follows by replacing the sharp interfaces in the constructions in the proof of theorem 3.6 by diffuse-interfaces with thickness of order *η*/∥*F*∥^{2}. Therefore, the only non-trivial task is to prove lower bounds for and .

We focus first on . Given we begin by constructing an appropriate sharp-interface function *χ*. For this, we observe thatIt follows by the co-area formula and Fubini’s theorem that there is *c**∈(1/4,1/2) such that4.6where is defined by *χ*(*x*)=0, if and *χ*(*x*)=1 if . In particular, since and in view of equation (4.5) we have . It remains to give a lower bound for the elastic energy. We have4.7where one easily calculates that4.8By equations (4.6), (4.7) and (4.8), it follows that , whence . Since and since the scaling law (2.7) for the minimal energy is monotone in *μ* when *γ*=0, this means thatOur final task is to bound from below by . Given , we define by equation (4.3), and we note that by definition . It is clear from equation (4.4) thatIt is also easy to see that(the proof uses the fact that is a Lipschitz-continuous function of *e* with Lipschitz constant of order ∥*F*∥^{−1}, and the properties that and ). Thus which concludes the proof. ■

## Acknowledgements

The authors thank the anonymous referees for thoughtful comments which substantially improved the paper. R. Kohn furthermore gratefully acknowledges support from NSF grant DMS-0807347.

## Appendix A

The Fourier representation of the elastic field has been extensively studied, e.g. by Khachaturyan (1982). For the convenience of the reader, we give the derivation following Capella & Otto (2009).

### Lemma A.1

*For any* , *we have (setting* *n*:=*ξ*/|*ξ*|)A1*where*s *Φ*(*n*)=∥*F*∥^{2}−2|*Fn*|^{2}+〈*n*,*Fn*〉^{2}.

### Proof.

Fourier transformation of the elastic energy yieldsA2Taking the first variation in equation (A2), we get that for all , where we used . Since *F* is symmetric, it follows thatMultiplying this equation with *ξ*, one gets with *n*:=*ξ*/|*ξ*|. Inserting this formula into equation (A2), a straightforward calculation then yields equation (A1). ■

The following lemma characterizes the set of compatible strains :

### Lemma A.2

*Suppose that* *has eigenvalues* *λ*_{1}≥⋯≥*λ*_{d} *(counted by multiplicity) with* *λ*_{1}≥|*λ*_{d}|. *Then*A3*where* *Φ*(*n*)=∥*F*∥^{2}−2|*Fn*|^{2}+〈*n*,*Fn*〉^{2}. *In particular*,

### Proof of lemma A.2.

To show the first equality in equation (A3), we write *P*=(*u*⊗*v*+*v*⊗*u*)/2. The equality then follows by using the Euler–Lagrange equation as in the proof of lemma A.1 (with replaced by *u* and *ξ* replaced by *v*). To show the second equality in equation (A3) it suffices to prove thatA4since we just have shown that the extreme right and left terms in equation (A4) are equal. Let *e*_{1} and *e*_{d} be the eigenvectors for *λ*_{1} and *λ*_{d} and setThe choice *P*=(*u*⊗*v*+*v*⊗*u*)/2 then leads to , which yields the left inequality in equation (A4). We turn to the right inequality: note that, substituting , a straightforward calculation yields:where *Φ*_{j}(*n*_{j})≥0 and *Φ*_{ij}(*n*_{i},*n*_{j})≥0. If *λ*_{j}≥0, then . This already yields equation (A4) if *λ*_{d}≥0. Otherwise, there is 2≤*N*≤*d* such that *λ*_{N−1}≥0 and *λ*_{N}<0. Let *α*_{j}:=−*λ*_{j}, hence 0<*α*_{N}≤⋯≤*α*_{d}, and . A short calculation shows that, to conclude the proof of equation (A4), it is enough to showA5whereA6The result now follows by straightforward minimization which is sketched below: Clearly, in order to prove equation (A5), we may assume that *α*_{i}≠*α*_{j} for *i*≠*j* (the general case follows by continuity of equation (A6) in the coefficients *α*_{j}). Derivating equation (A6) in *x*_{j} yields ∂_{j}*φ*=*x*_{N}+⋯+*x*_{d}−*α*_{j} for *j*=*N*, … ,*d* so that at every point, the partial derivative can only vanish in a single direction (since *α*_{i}≠*α*_{j} for *i*≠*j*). It follows that the minimum is not achieved in the interior, but instead at a point where all *x*_{j}=0 except at most one *x*_{i}≠0. A short computation then shows that the minimum of equation (A5) is given by the value and it is achieved by *x*_{N}=⋯=*x*_{d−1}=0, *x*_{d}=*α*_{d}. This yields equation (A5) and thus concludes the proof for the second inequality in equation (A4). ■

- Received June 18, 2010.
- Accepted August 2, 2010.

- © 2010 The Royal Society