## Abstract

A necessary optimality condition for the problem of the minimum-resistance shape for a rigid three-dimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint on the volume of the inclusion is obtained through Betti's reciprocal work theorem. It generalizes Pironneau's optimality condition for the minimum-drag shape for a rigid body immersed into a uniform Stokes flow and is specialized for axisymmetric inclusions in axisymmetric and transversal translations. In both cases of translation, the three-dimensional displacement field is represented in terms of generalized analytic functions, and the three-dimensional elastostatics problem is reduced to boundary-integral equations (BIEs) via the generalized Cauchy integral formula. Minimum-resistance shapes are found in the semi-analytical form of functional series from an iterative procedure coupling the optimality condition and the BIEs. They are compared with the minimum-resistance spheroids and with the minimum-resistance spindle-shaped and lens-shaped bodies. Remarkably, in the axisymmetric translation, the minimum-resistance shapes transition from spindle-like shapes to almost prolate spheroidal shapes as the Poisson ratio changes from 1/2 to 0, whereas in the transversal translation, they are close to oblate spheroidal shapes for any Poisson ratio.

## 1. Introduction

The problem of a rigid/elastic inclusion translated (displaced) in an unbounded isotropic elastic medium has long been a popular research subject owing to its application in metallurgy and mechanics of composite materials [1–8]. Under the assumption of absent external volume forces, negligible inertial and thermal effects, and small deformations, the displacement field **u** of the medium is governed by the Navier equations
1.1where *ν*∈[0,1/2] is the Poisson ratio. The function *ϑ*=−((2−2*ν*)/(1−2*ν*)) div **u** is to be referred to as displacement divergence. If *ν*=1/2, then div **u**=0, and *ϑ* is defined as the limit of −((2−2*ν*)/(1−2*ν*)) div **u** with *ν*→1/2 and div **u**→0. In this case, the elastic medium is called incompressible, and the Navier equations (1.1) become mathematically identical to the Stokes equations for the velocity **u** and pressure *p*=*μϑ* of creeping flows of a viscous incompressible fluid (the so-called Stokes flows), where *μ* is shear viscosity of the fluid.

Further, the inclusion is considered to be a three-dimensional rigid body of fixed (finite) volume, and the direction and magnitude of its translation in the elastic medium is determined by vector **v** (small enough for the linear theory of elasticity to be applicable). In this case, the displacement field **u** is equal to **v** on the surface of the inclusion, *S*, and vanishes at infinity, i.e.
1.2The question of interest is what shape of the inclusion minimizes resistance of the medium. It is closely related to the problem of the ‘maximum-penetration’ cross section for a rigid cylindrical inclusion embedded in the elastic medium [8] and to the one of the minimum-drag shape for a rigid body immersed into a uniform flow of a viscous incompressible fluid [9–12]. Owing to the mathematical equivalence of the Stokes equations and the Navier equations for *ν*=1/2, the minimum-resistance shape of the inclusion in an incompressible elastic medium is determined by Pironneau's optimality condition [9] and coincides with the minimum-drag shape of the body in a uniform Stokes flow [13]. The condition prescribes the absolute value of the normal derivative of the displacement vector **u** to be constant on the surface of the inclusion, which implies that the optimal shape has conic vertices with the angle of 2*π*/3 [9,13]. In this case, the resistance–force ratio of the optimal shape with respect to an equal-volume sphere is 0.95425 [13]. There are similar optimality conditions for the minimum-drag shapes in cases of a viscous incompressible flow with an arbitrary Reynolds number *Re* [11] and of an electrically conducting fluid under the presence of a magnetic field [14]. As *Re* increases, the minimum-drag shape elongates, but retains conic vertices with the same angle of 2*π*/3 for any *Re*, and the drag ratio with respect to an equal-volume sphere for the same *Re* decreases. For the elastic medium, does the minimum-resistance shape elongate or shorten as *ν* decreases, and does it have conic vertices for any *ν*? How does the resistance–force ratio change with *ν*? In addition, if the inclusion is assumed to be axisymmetric and translates perpendicular to its axis of revolution (transversal translation), what shape minimizes the resistance and how does it depend on *ν*?

A quick insight into some of these questions can be gained by analysing the ratio of the resistance force of a prolate spheroid in axisymmetric translation to the resistance force of an equal-volume sphere for the same *ν*, which is determined by
where *d*>1 is the axes ratio of the prolate spheroid; see [15], (3.9b). For each *ν*, *f*_{∥} is minimized with respect to *d*, and optimal *f*_{∥} and *d* correspond to the minimum-resistance prolate spheroid (figure 1). In addition, the ratio of the resistance force of an oblate spheroid in transversal translation to the resistance force of an equal-volume sphere for the same *ν* is determined by
where *d*<1 is the axes ratio of the oblate spheroid; see [15], (6.3b). Similarly, for each *ν*, *f*_{⊥} is minimized with respect to *d*, and optimal *f*_{⊥} and *d* correspond to the minimum-resistance oblate spheroid (figure 2). As *ν* increases, the optimal resistance–force ratios *f*_{∥} and *f*_{⊥} both decrease, and the optimal spheroids elongate (flatten) in the axisymmetric (transversal) translation. For *ν*=1/2, the minimum-resistance shape improves the resistance–force ratio *f*_{∥}=0.95551 of the optimal spheroid by only 0.13%. How does this margin change with a decrease of *ν*?

The posed problem of minimum-resistance shape can be solved in the framework of partial differential equation (PDE) constrained optimization, which offers a general computational scheme through the approach of adjoint equations and finite-element methods (FEMs); see [16] and references therein. However, this scheme has several disadvantages: (i) it truncates and discretizes the unbounded region occupied by the medium, (ii) it represents the body's shape as a set of mesh nodes, and (iii) it may take several hundreds of iterations [17].^{1} An alternative semi-analytical scheme is to reduce the elastostatics problem (1.1)–(1.2) to boundary-integral equations (BIEs) [20–22],^{2} to approximate the surface of the shape by a finite functional series, and to use a necessary optimality condition for the minimum-resistance shape to find series coefficients. Compared with PDE-constrained optimization coupled with an FEM, it deals only with the surface of the inclusion in an analytical form and, as a result, obtains optimal shapes significantly faster and more accurately. It was demonstrated in finding minimum-drag shapes for rigid bodies in Stokes and Oseen flows [14,23,24]. Some shape optimization schemes take advantage of BIEs, but use an FEM method to compute gradient descent directions [25,26].

The suggested semi-analytical scheme has a simple implementation with the approach of generalized analytic functions, which play a role in three-dimensional continuous mechanics similar to that of ordinary analytic functions in the two-dimensional case [27,28].^{3} In a cylindrical coordinate system (*r*,*φ*,*z*), let the three-dimensional displacement field **u** be expanded into a Fourier series with respect to the angular coordinate *φ*. It will be shown that the Fourier harmonics of **u** admit simple representations in terms of *n*th-order *r*-analytic functions , where and *U*=*U*(*r*,*z*) and *V* =*V* (*r*,*z*) are real and imaginary parts, respectively, related by the system^{4}
1.3In contrast to the Kolosov formulae [27,28] (also called Kolosov–Muskhelishvili formulae) and their three-dimensional analogues [35,36], those representations contain no derivatives of the involved generalized analytic functions and, as a result, have an advantage in obtaining closed-form solutions and in deriving BIEs through the generalized Cauchy integral formula [37]. As shown in [14,23,24], similar derivative-free representations provide a simple coordination of all the parts of the optimization scheme: for axisymmetric Stokes and Oseen flows, the drag of the body, the optimality conditions for the minimum-drag shape and the corresponding BIEs are all formulated in terms of the same generalized analytic function.

The goal of this work is twofold: (i) to obtain the necessary optimality condition for the minimum-resistance shape for a fixed-volume rigid inclusion translated in an unbounded isotropic elastic medium and to analyse the optimal shape's dependence on the Poisson ratio *ν* and (ii) to demonstrate the approach of generalized analytic functions in finding the optimal shape, which includes reformulating the optimality condition in terms of generalized analytic functions and obtaining the BIEs based on the generalized Cauchy integral formula.

This work is organized into four sections and three appendices. Section 2 derives the necessary optimality condition for the minimum-resistance shape through Betti's reciprocal work theorem and specializes that condition for axisymmetric inclusions. Then with the necessary optimality condition and with the BIEs based on the generalized Cauchy integral formula, §§3 and 4 find the minimum-resistance axisymmetric shapes for the axisymmetric and transversal translations in the form of functional series and compare them with the minimum-resistance prolate and oblate spheroids, respectively, and with the minimum-resistance spindle and lens for *ν*=0, 1/3 and 1/2. Remarkably, in the axisymmetric translation, as *ν* changes from 1/2 to 0, the minimum-resistance shapes transition from spindle-like shapes to almost prolate spheroidal shapes, and compared with the minimum-resistance prolate spheroids, the reduction in the resistance–force ratio drops from 0.13% to 0.006%, whereas in the transversal translation, the minimum-resistance shapes are almost oblate spheroidal for all *ν*, and compared with the minimum-resistance oblate spheroids, the reduction in the resistance–force ratio is 0.01% for *ν*=1/2 and 0.0001% for *ν*=0. Appendix A proves an auxiliary relationship, appendix B presents the Cauchy integral formula for *n*th-order *r*-analytic functions, and appendix C derives a general representation for an asymmetric displacement field **u** in terms of *n*th-order *r*-analytic functions.

## 2. Optimality condition for minimum-resistance shape

Suppose a rigid three-dimensional inclusion of fixed volume is translated in an unbounded isotropic elastic medium, and suppose the displacement field **u** of the medium satisfies the Navier equations (1.1) with the boundary conditions (1.2). The question is what shape minimizes the resistance force exerted on the inclusion.

### Theorem 2.1 (necessary optimality condition)

*A fixed-volume rigid inclusion has the minimum-resistance shape if*
2.1*where ∂***u***/∂n is the normal derivative, S is the surface of the inclusion, σ is the stress tensor satisfying* div *σ*=0,^{5} *and* *is the stress vector acting on a surface element with the outward normal* . *In addition, if the shape is assumed to be axisymmetric with the axis of revolution that coincides with the z-axis in a cylindrical coordinate system (r,φ,z), then the necessary optimality condition (*2.1*) takes the form
*2.2

### Proof.

The magnitude of the resultant force exerted on the inclusion, which is translated along the vector **v**, is determined by , where *μ* is the shear modulus and
2.3Let be the radius vector representing the surface of the minimum-resistance shape, *S*, and let be the radius vector corresponding to the surface of a shape variation, *S*_{ϵ}, where *ϵ* is a positive small number and is any continuous bounded scalar function satisfying the condition , which follows from the fact that the inclusion has fixed volume (see [11,14]).

Let **u** be the displacement field for the elastic medium with the inclusion having the minimum-resistance shape *S*, and let be a variation of **u**, where satisfies the Navier equations (1.1) and vanishes at infinity. The variation **u**_{ϵ} results in the stress tensor with corresponding to and satisfying .

On *S*_{ϵ}, **u**_{ϵ} should satisfy **u**_{ϵ}=**v**, which implies , so that and, consequently, on *S*, or
2.4

The variation of the functional (2.3) involves variation of both the stress tensor *σ* and the shape *S*:

The functionals *I*_{ϵ} and *J*_{ϵ} are evaluated separately:
where the second equality follows from Betti's reciprocal work theorem [27], and the third one follows from and the boundary condition (2.4). Then
where div *σ*≡0, and is the region bounded by the surfaces *S* and *S*_{ϵ}. Consequently,

The necessary optimality condition requires , so that . However, the volume constraint implies , which holds for arbitrary *f* only if is constant on *S*. If *f* is axisymmetric and has the axis of revolution that coincides with the *z*-axis, then holds only if is a constant function of *r* and *z* on *S*. □

### Remark 2.2

Betti's reciprocal work theorem, used in evaluating the functional *I*_{ϵ}, can be viewed as a particular case of the method of adjoint equations (see [11,14,23,24]).

The stress vector acting on a surface element of the inclusion with the outward unit normal vector is given by

Appendix A shows that if the inclusion is axisymmetric, and **u** is constant on the surface *S* of the inclusion, then
2.5which implies that, in the case of translation of the axisymmetric rigid inclusion, takes the form
2.6

Consequently, with (2.5), (2.6) and *x*=(3−4*ν*)/(4−4*ν*), the optimality condition (2.1) simplifies to
2.7whereas the optimality condition (2.2) takes the form
2.8

## 3. Minimum-resistance shape in axisymmetric translation

Let (*r*,*φ*,*z*) be a cylindrical coordinate system with the basis (**e**_{r},**e**_{φ},**k**), and let the displacement field **u** of an isotropic elastic medium be independent of the angular coordinate *φ* and have zero component **u**_{φ}: **u**=**u**_{r}(*r*,*z*) **e**_{r}+**u**_{z}(*r*,*z*)**k**, i.e. **u** is axially symmetric. In this case, the Navier equations (1.1) take the form
3.1and *ϑ*=−(2−2*ν*)/(1−2*ν*) div **u** implies that
3.2

### Proposition 3.1 (representation of axisymmetric displacement field)

*Let G*_{1}=*G*_{1}(*r*,*z*) *and G*_{2}=*G*_{2}(*r*,*z*) *be zero-order r-analytic functions, defined by* (1.3) *with n*=0. *Then, the components u _{r} and u_{z} of an arbitrary axisymmetric displacement field, where the z-axis is the axis of revolution, can be represented in the form*
3.3

*and the divergence ϑ and the vorticity function ω*≡

*e*

_{φ}⋅curl

**u**

*are determined by*3.4

### Proof.

In this case, **u**_{r} and **u**_{z} can be represented as in the proof of proposition 7 in [33] by
3.5where *a*, *b*, *c* and *d* are real-valued constants, and *U*_{j}=*U*_{j}(*r*,*z*) and *V* _{j}=*V* _{j}(*r*,*z*) are real and imaginary parts of a zero-order *r*-analytic function , respectively, so that Δ_{0}*U*_{j}=0 and Δ_{1}*V* _{j}=0 for *j*=1,2.

Then, (3.2) with (3.5) yields 3.6

It follows from Δ_{0}*U*_{1}=0 and Δ_{1}*V* _{1}=0 that Δ_{1}(*rU*_{1})=2(∂/∂*r*)*U*_{1}, Δ_{1}(*zV* _{1})=2(∂/∂*z*)*V* _{1}, Δ_{0}(*zU*_{1})=2(∂/∂*z*)*U*_{1} and Δ_{0}(*rV* _{1})=2(∂/∂*r*+1/*r*)*V* _{1}, and, consequently, (3.1) with (3.5) and with *U*_{1}=*ϑ* implies
which are dependent equations by virtue of (3.6). Thus, 2*a*+*b*=1−2*x* and *a*+*b*=1−*x*, so that *a*=−*x*, *b*=1, *c*=1, *d*=*x*, and the representations (3.3) and (3.4) follow. □

### Remark 3.2 (uniqueness of *G*_{1} and *G*_{2})

*G*

*G*

The zero-order *r*-analytic functions *G*_{1} and *G*_{2} in (3.3) are determined uniquely except for an interior problem for *ν*=1/2, which admits a homogeneous solution *G*_{1}=*c* and , where *c* is a real-valued constant (see proposition 9(ii) in [33]). Indeed, for *ν*∈[0,1/2), the Navier equations (1.1) with zero displacement boundary conditions have only zero solution; see [27], §3.5.2. Consequently, if there is another pair *H*_{1} and *H*_{2} that represent **u** through (3.3), then (3.3) with the pair *G*_{1}−*H*_{1} and *G*_{2}−*H*_{2} in place of *G*_{1} and *G*_{2}, respectively, should result in zero displacement field **u**. However, the representation (3.4) and **u**≡0 imply that *G*_{1}−*H*_{1}≡0, and then (3.3) along with *G*_{1}≡*H*_{1} yields *G*_{2}−*H*_{2}≡0.

The representations (3.3) and (3.4) extend formulae (36) and (37) in [33] for the velocity and pressure of axisymmetric Stokes flows, whose mathematical model is identical to the Navier equations (1.1) with *ν*=1/2.

Suppose an axisymmetric rigid inclusion is translated along its axis of revolution (the *z*-axis) in an unbounded isotropic elastic medium, so that the displacement field **u** satisfies the Navier equations (1.1) and the boundary conditions (1.2) with **v**=*v*_{z} **k**, where *v*_{z} is a constant. This problem is axially symmetric, and **u**_{r}, **u**_{z}, *ϑ* and *ω* can be represented by (3.3) and (3.4). Let ℓ be the positively oriented piece-wise smooth cross section of the surface of the inclusion, *S*, in the *rz*-half plane (*r*≥0), which is either a closed curve or an open curve with the endpoints lying on the *z*-axis.

### Proposition 3.3 (resistance force in axisymmetric translation)

*The resistance force, exerted on the inclusion in the axisymmetric translation, is given by two equivalent formulae*:
3.7a*and*
3.7b*where* *G*_{1} *is the zero-order r*-*analytic function in* (3.3) *and* (3.4), *and μ is the shear modulus*.

### Proof.

The representations (2.6) and (3.4) imply that the resultant force exerted on the inclusion is determined by
which is equivalent to (3.7a), where d*s* is the curve length element of ℓ, and (∂*r*/∂*n*) d*s*=d*z* and −(∂*z*/∂*n*) d*s*=d*r* for positively oriented ℓ. The formula (3.7b) follows from (3.7a) and proposition 6 in [33]. □

Remarkably, (3.7a) resembles the Kutta–Joukowski airlift formula with an ordinary analytic function.

With the representation (3.3) and with the Cauchy integral formula (B1) for zero-order *r*-analytic functions (see appendix B), the axisymmetric elastostatics problem of the inclusion translation can be reduced to a BIE for the function *G*_{1} as follows. Let *D*^{−} be the open region occupied by the elastic medium, and let be its cross section in the *rz*-half plane (*r*≥0). The boundary of (cross section of *S*) is the positively oriented piece-wise smooth curve ℓ. The representation (3.3) and **u**|_{S}=*v*_{z}**k** yield on ℓ, whereas (3.3), (3.4) and imply that *G*_{1} and *G*_{2} vanish at infinity ( as for *j*=1,2). Suppose that *G*_{1} on ℓ is known, then on ℓ, and with the Cauchy integral formula (B1) for *n*=0, (3.3) and (3.4) furnish representations for **u**_{r}, **u**_{z}, *ϑ* and *ω* in :
where , is the generalized Cauchy operator (B1) for zero-order *r*-analytic functions and the operator is determined by
with and
and
in which and are complete elliptic integrals of the first and second kinds, respectively,^{6} and .

The operator has no Cauchy-type singularity on ℓ, and, consequently, is continuous as *ζ* approaches ℓ from within . This fact and on ℓ yield the BIE for *G*_{1}:
3.8

For *ν*=1/2, (3.8) coincides with the BIE (54) in [33] for the axisymmetric Stokes flow about a rigid body of revolution and has a homogeneous solution, which is an arbitrary real-valued constant (see theorem 10 in [33]). For *ν*∈[0,1/2), (3.8) has no non-zero homogeneous solution—this can be proved similarly to either theorem 10 in [33] or theorem 3.2 in [37].

The kernel *K*_{2} has a logarithmic singularity, and the BIE (3.8) is solved as follows. Let ℓ be parametrized by *ζ*=*ζ*(*t*), *t*∈[−1,1], then on ℓ, *G*_{1} is approximated by a finite functional series
with basis functions and ,^{7} and real-valued coefficients *a*_{1},…,*a*_{m}, *b*_{1},…,*b*_{m} are found by minimizing the total quadratic error of (3.8) on [−1,1]:
which is reduced to a system of linear equations for *a*_{1},…,*a*_{m}, *b*_{1},…,*b*_{m}. For a spheroid with semi-axes *a* and *b*, , *ϕ*_{1k}=*T*_{2k−1}(*t*) and *ϕ*_{2k}=*T*_{2(k−1)}(*t*), where *T*_{k}(*t*) is the Chebyshev polynomial of the first kind, and the solution of (3.8) is in good agreement with the analytical solution *G*_{1}(*t*)=*F*_{z}/(8*μaζ*′(*t*)), *t*∈[−1,1], found in the prolate spheroidal coordinates, where *F*_{z} is the resistance force (see also [15]). With minimum-resistance shape in mind, the representation of *G*_{1} on ℓ is further specialized for fore-and-aft symmetric inclusions with conic vertices.

When ℓ is symmetric with respect to the *r*-axis (fore-and-aft symmetry), it can be parametrized by *ζ*=*ζ*(*t*), *t*∈[0,1], for *z*≥0 and determined by , *t*∈[0,1], for *z*≤0. In this case, , so that on ℓ, *G*_{1} can be parametrized by *G*_{1}=*G*_{1}(*r*(*t*),*z*(*t*))≡*G*_{1}(*t*), *t*∈[0,1], for *z*≥0 and determined by its symmetry condition for *z*≤0. In addition, let the endpoints of ℓ lie on the *z*-axis, and let *η* be the angle between ℓ and the *z*-axis at *ζ*(1). The inclusion has a conic vertex when *η*<*π*/2. As in proposition 2 in [23] (see also proposition 4.1 in [24]), it can be shown that if the equation
3.9where and is the associated Legendre function of the first kind of rank *n* (for *n*=0, the superscript is omitted), has zero *α* in (0,1), then as *t*→1−, and, thus, for fore-and-aft symmetric inclusions with conic vertices, *G*_{1}(*t*) is represented by
3.10for *z*≥0, where all *a*_{k} and *b*_{k} are real-valued, and *a*_{0}=*b*_{0}=0 when either *ν*∈[0,1/2) and *η*≥1/2 or *ν*=1/2 and *η*>*π*/3. For *ν*=1/2 and *η*=*π*/3, in (3.10) is replaced by .

### Remark 3.4

The BIE (3.8) was derived under the assumption of *G*_{1} and *G*_{2} to be Hölder continuous on ℓ, whereas the approximation (3.10) has an integrable singularity at *t*=1. However, as in the case of ordinary analytic functions (see §8.3 in [38], e.g. formulae (8.4), (8.6′) and the one preceding (8.7)), it can be shown that *G*_{1} and *G*_{2} with integrable singularities on ℓ can still be represented in by the generalized Cauchy integral formula (B1). Thus, with (3.10), the BIE (3.8), whose kernels have only a logarithmic singularity, holds for all *ζ*∈ℓ except possibly for *ζ*=*ζ*(1) and .

The solution form (3.10) is tested for a rigid spindle-shaped inclusion, which is an axisymmetric body obtained by revolving a circular segment around the *z*-axis. In the *rz*-half plane (*r*≥0), the cross section of the spindle surface, ℓ, is parametrized by , *t*∈[−1,1], where *η*∈[0,*π*] is half of the angle of the spindle vertex (*η*=*π*/2 corresponds to a sphere), and the multiplier makes the spindle volume to be 4*π*/3. Solutions of (3.8) with (3.10) agree with those in [39,40] obtained in bispherical coordinates through the Fourier integral transform. For *ν*=0, 1/3 and 1/2, the spindle-shaped inclusions that minimize the resistance force *F*_{z} as a function of *η* correspond to *η*=1.22396, 1.03919 and 0.87027, respectively: figure 3*b* shows cross sections of those inclusions in the *rz*-plane, and table 1 presents corresponding resistance coefficients *F*_{z}/(6*πμv*_{z}).^{8}

The next step is to find the minimum-resistance shape for an axisymmetric rigid inclusion of volume 4*π*/3. In this case, the shape optimality conditions (2.7) and (2.8) are equivalent and reduce to
3.11which with the representation (3.4) is conveniently reformulated in terms of the zero-order *r*-analytic function *G*_{1} as
3.12For *ν*=1/2 (which corresponds to *x*=1/2), (3.11) simplifies to the necessary optimality condition for the minimum-drag shape of a rigid body immersed in a uniform Stokes flow [9]:
3.13

For the minimum-resistance (fore-and-aft symmetric) shape, ℓ is approximated in the *rz*-half plane for *z*≥0 by
3.14where *a*_{0},…,*a*_{N}, *b*_{0},…,*b*_{N} are real-valued coefficients and *γ*=*γ*(*a*_{0},…,*a*_{N},*b*_{0},…,*b*_{N}) is the multiplier introduced to satisfy the volume constraint identically^{9} (see [14,41]). For *z*≤0, ℓ is determined by , *t*∈[0,1]. Then, on ℓ, *G*_{1} is represented by (3.10) and is found from (3.8). With the shape optimality condition (3.12) reformulated in the form
3.15where , optimal ℓ can be found iteratively as in [9]. Namely, let the initial shape be that of the minimum-resistance spheroid, i.e. *a*_{0}=*d*^{−1/3}, *b*_{0}=*d*^{2/3}, *a*_{j}=*b*_{j}=0, *j*=1,…,*N* in (3.14), where *d*>1 is the spheroid's axes ratio shown in figure 1*b*. If *ζ*_{k} and *h*_{k} represent ℓ and *h*, respectively, at step *k*, then *ζ*_{k+1} is determined by
3.16where the step size *ϵ*_{k} and coefficients *a*_{0},…,*a*_{N}, *b*_{0},…,*b*_{N}, *γ* in (3.14) for *ζ*_{k+1} are found by algorithm 1 in [41], in which should be replaced by *h*. The accuracy of the shape is controlled through the error in satisfying the shape optimality condition (3.15).

Figure 3*c* shows cross sections of the minimum-resistance shape in the *rz*-plane for *ν*=0, 1/3 and 1/2, and table 2 presents corresponding coefficients *a*_{0},…,*a*_{N}, *b*_{0},…,*b*_{N}, *γ* in (3.14), the constant *C* and the error in satisfying (3.15). Table 1 compares the resistance coefficients *F*_{z}/(6*πμv*_{z}) for a unit sphere, minimum-resistance prolate spheroid, minimum-resistance spindle and minimum-resistance shape for *ν*=0, 1/3 and 1/2. All shapes have volume 4*π*/3. The minimum-resistance shape for *ν*=1/2 coincides with the one for a rigid body immersed into a uniform Stokes flow, which was obtained in [13] based on Pironneau's optimality condition (3.13) and for which *F*_{z}/(6*πμv*_{z}) was reported to be 0.95425 [13]. This shape has conic vertices with the angle of 2*π*/3 (see [9,11]).

### Remark 3.5

For *ν*∈[0,1/2), (3.10) with (3.9) shows that if the inclusion has a conic vertex with the angle *η* between ℓ and the *z*-axis to be less than *π*/2, then on ℓ in the vicinity of the vertex, *G*_{1} is unbounded. However, for *ν*∈[0,1/2), the shape optimality condition (3.12) implies that *G*_{1} is bounded on ℓ, so that, except for *ν*=1/2, the minimum-resistance shape has no conic vertices, and *a*_{0}=*b*_{0}=0 in (3.10).

## 4. Minimum-resistance shape in transversal translation

Suppose an axisymmetric rigid inclusion is translated perpendicular to its axis of revolution in an unbounded isotropic elastic medium (the transversal translation problem). Let (*r*,*φ*,*z*) be a cylindrical coordinate system with the basis (**e**_{r},**e**_{φ},**k**), and let the inclusion's axis of revolution coincide with the *z*-axis. In this case, the displacement field **u** of the elastic medium satisfies the Navier equations (1.1) and the boundary conditions (1.2) with **v**=*v*_{x} ** i**, where

*v*

_{x}is a constant and

**is the basis vector of the**

*i**x*-axis in the Cartesian coordinate system (

*x*,

*y*,

*z*) related to the cylindrical one in the ordinary way.

In the cylindrical coordinates, the first condition in (1.2), i.e. , takes the form
4.1so that the Fourier expansion of **u** and *ϑ* with respect to *φ* contains only the first-order harmonics
4.2

Let ℓ be the cross section of the surface of the inclusion, *S*, in the *rz*-half plane (*r*≥0), which is either a closed curve or an open curve with the endpoints lying on the *z*-axis. ℓ is assumed to be piece-wise smooth and positively oriented. Then, (4.1) and (4.2) imply that
4.3

Proposition C.1 in appendix C obtains representations for the Fourier harmonics of a general asymmetric displacement field in terms of *n*th-order *r*-analytic functions. So, the components *u*_{1r}, *u*_{1φ}, *u*_{1z} and *ϑ*_{1} can be represented by (C3) for *n*=1 with zero-order *r*-analytic functions *G*_{1}=*G*_{1}(*r*,*z*) and *G*_{2}=*G*_{2}(*r*,*z*) and with a first-order *r*-analytic function *G*_{3}=*G*_{3}(*r*,*z*):
4.4a
4.4b
and
4.4cwhere and *x*=(3−4*ν*)/(4−4*ν*). The representations (4.4a)–(4.4c) extend formulae (30a) and (30b) in [34] for the velocity and pressure in transversal Stokes flows.

### Proposition 4.1 (resistance force in transversal translation)

*Let u_{1r}, u_{1φ}, u_{1z} and ϑ_{1} be represented by* (4.4

*a*)–(4.4

*c*).

*Then, the resistance force, exerted on the inclusion in the transversal translation, is given by two equivalent formulae*4.5a

*and*4.5b

*where μ is the shear modulus*.

### Proof.

Let *U*_{j} and *V* _{j} be the real and imaginary parts of *G*_{j}, *j*=1,2,3, in (4.4a) and (4.4b), then the representations (4.4a) and (4.4b) and the boundary conditions (4.3) yield
4.6and
4.7where ∂/∂*n* and ∂/∂*s* are the derivatives in the directions of the outward normal and of the unit tangent vector , respectively, on ℓ and such that ∂*r*/∂*s*=−∂*z*/∂*n* and ∂*z*/∂*s*=∂*r*/∂*n*.

The relationships (2.6), (4.6), (4.7) and (4.4c) imply that the resultant force exerted on the inclusion is determined by
which is equivalent to (4.5a), where d*s* is the curve length element, and (∂*r*/∂*s*) d*s*=d*r* and (∂*z*/∂*s*) d*s*=d*z* for positively oriented ℓ. The formula (4.5b) follows from (4.5a) and proposition 6 in [33]. □

For an arbitrary axisymmetric inclusion, the transversal translation problem can be reduced to two BIEs as follows. Let open region be the cross section of the medium in the *rz*-half plane (*r*≥0). The boundary of is ℓ (cross section of the inclusion's surface). With (4.3), the representations (4.4a) and (4.4b) yield a boundary-value problem for the functions *G*_{1}, *G*_{2} and *G*_{3}:
4.8aand
4.8bwhere *V* _{1}=Im *G*_{1}, *V* _{3}=Im *G*_{3} and (4.8b) is the sum of (4.4b) and the imaginary part of (4.4a). Equation (4.8a) yields on ℓ, and, with (4.4a), can be represented in through the generalized Cauchy integral formula (B1) in terms of the boundary values of *G*_{1} and *G*_{3}:
4.9where and the operator is determined by
with , and being the generalized Cauchy operator (B1) for *n*=0 and *n*=1, respectively, and
in which . Observe that has no Cauchy-type singularity on ℓ, and consequently, it is continuous as *ζ* approaches ℓ from within . This fact and on ℓ yield the BIE for *G*_{1} and *G*_{3}:
4.10

The second BIE can be obtained as follows. A complex-valued function on ℓ is the boundary value of a generalized analytic function in if and only if it satisfies the Sokhotski–Plemelj formula (see remark B.3 in appendix B), which for the function *G*_{3}, vanishing at infinity, takes the form , *ζ*∈ℓ. With the identity , *ζ*∈ℓ, this Sokhotski–Plemelj formula can be rewritten as
4.11where , being combined under a single integral, has no Cauchy-type singularity. In addition, with (4.8b), *V* _{3} can be excluded from (4.10) and (4.11):
4.12Similar to theorem 6 in [34], it can proved that the system of (4.10) and (4.11) with *V* _{3} excluded by (4.12) is complete for determining the boundary values of *G*_{1} and *U*_{3}=Re *G*_{3} and that it has no non-zero homogeneous solution. For *ν*=1/2, the system (4.10)–(4.12) reduces to (37)–(40) in [34] that corresponds to the transversal translation of a rigid body of revolution in a viscous incompressible fluid under the zero Reynolds number assumption.

The system (4.10)–(4.12) is solved similar to (3.8). Namely, ℓ is parametrized by *ζ*=*ζ*(*t*), *t*∈[*t*_{1},*t*_{2}], and, on ℓ, the functions *G*_{1} and *U*_{3} are approximated by finite functional series and , *t*∈[*t*_{1},*t*_{2}], where *ϕ*_{1k}, *ϕ*_{2k}, *ϕ*_{3k}, *k*=1,…,*m*, are real-valued basis functions, and real-valued coefficients *a*_{k}, *b*_{k}, *c*_{k}, *k*=1,…,*m*, are found by minimizing the total quadratic error in satisfying the system (4.10)–(4.12) (see §3). For a spheroid with semi-axes *a* and *b*, , *t*∈[−1,1], and the solution of the system (4.10)–(4.12) agrees with the analytical solution *G*_{1}(*t*)=*F*_{x}/(8*μaζ*′(*t*)) and *G*_{3}(*t*)=(*x*−1)/2 Re[*ζ*(*t*)]*G*_{1}(*t*), *t*∈[−1,1], found in the oblate spheroidal coordinates. For a spindle-shaped inclusion with *η*≥*π*/2 (see §3), it agrees with the one in [40,42], obtained in bispherical coordinates through the Fourier integral transform.^{10} Because, in the transversal translation, the minimum-resistance axisymmetric shape may resemble a lens, this time, the representations of *G*_{1} and *U*_{3} on ℓ are specialized for axisymmetric inclusions that are fore-and-aft symmetric and have a sharp equatorial edge.

Let positively oriented fore-and-aft symmetric ℓ be parametrized by *ζ*=*ζ*(*t*), *t*∈[0,1], for *z*≥0, and be determined by , *t*∈[0,1], for *z*≤0. In this case, and *U*_{3}(*r*,−*z*)=−*U*_{3}(*r*,*z*), so that, on ℓ, *G*_{1} and *U*_{3} can be parametrized by *G*_{1}=*G*_{1}(*r*(*t*),*z*(*t*))≡*G*_{1}(*t*) and *U*_{3}=*U*_{3}(*r*(*t*),*z*(*t*))≡*U*_{3}(*t*), *t*∈[0,1], for *z*≥0 and can be determined by their symmetry conditions for *z*≤0. Let *η* be the angle between ℓ and the *r*-axis at *ζ*(0) (equatorial point). The inclusion has a sharp equatorial edge when *η*<*π*/2. As in [40], it can be shown that if the equation
4.13has zero *α* in (0,1), then and as *t*→0+, and, consequently, for axisymmetric inclusions with fore-and-aft symmetry and sharp equatorial edge, *G*_{1}(*t*) and *U*_{3}(*t*) are represented by
4.14for *z*≥0, where *a*_{0},…,*a*_{m}, *b*_{0},…,*b*_{m}, *c*_{0},…,*c*_{m} are real-valued, and *a*_{0}=*b*_{0}=*c*_{0}=0 when either *ν*∈[0,1/2) and *η*≥*π*/2 or *ν*=1/2 and *η*>*η**=0.894888.^{11} For *ν*=1/2 and *η*=*η**, the terms and *c*_{0}*t*^{α−1} in (4.14) are replaced by and , respectively.

As the BIE (3.8), the system of (4.10) and (4.11) was derived via the generalized Cauchy integral formula (B1), which assumes *G*_{1}, *G*_{2} and *G*_{3} to be Hölder continuous on ℓ, whereas both of the series in (4.14) have integrable singularities at *t*=0. However, it can be shown that, with (4.14), the equations (4.10) and (4.11), whose kernels have only a logarithmic singularity, hold for all *ζ*∈ℓ except possibly for *ζ*=*ζ*(0) (see remark 3.4.)

The solution form (4.14) is tested for a rigid biconvex lens^{12} aligned with the *z*-axis and centred at the origin of the cylindrical coordinate system. In the *rz*-half plane, the cross section of the lens surface, ℓ, is parametrized by , *t*∈[0,1], for *z*≥0, and is determined by , *t*∈[0,1], for *z*≤0, where *η*∈[0,*π*] is half of the angle of the lens's equatorial edge (*η*=*π*/2 corresponds to a sphere), and where the multiplier makes the lens volume to be 4*π*/3. For *ν*=1/2 and *η*≥*π*/2, solutions of the system (4.10)–(4.12) in the form (4.14) agree with those in [43], obtained in toroidal coordinates through the Mehler–Fock integral transform. For *ν*=0, 1/3 and 1/2, the biconvex lens-shaped inclusions that minimize the resistance force *F*_{x} as a function of *η* correspond to *η*=1.43834, 1.35304 and 1.21972, respectively: figure 4*a* shows cross sections of those inclusions in the *rz*-plane for *ν*=0 and *ν*=1/2, and table 3 presents corresponding values of the resistance coefficient *F*_{x}/(6*πμv*_{x}).

For an axisymmetric rigid inclusion that has volume 4*π*/3 and minimizes the resistance force in the transversal translation, the shape is found as in the case of axisymmetric translation. In the *rz*-half plane, ℓ is represented by (3.14) for *z*≥0 and is determined by , *t*∈[0,1], for *z*≤0. With the boundary conditions (4.3), the representation (4.2) implies that
where is the unit tangential vector for the surface *S*, and, consequently, the shape optimality condition (2.8) reduces to
4.15where *C* is an unknown constant. On ℓ, *G*_{1} and *U*_{3} are approximated by (4.14) and are found from the system (4.10)–(4.12) by square error minimization (see §3). With (4.4a) and (4.4b) and with the relationships (4.6) and (4.7), the shape optimality condition (4.15) takes the form
4.16where *U*_{1}=Re *G*_{1}, *V* _{1}=Im *G*_{1}, *U*_{3}=Re *G*_{3} and *r*′, *z*′, *U*′_{1} and *U*′_{3} are the derivatives of *t*. Starting from the shape of a minimum-resistance oblate spheroid (figure 2*b*), ℓ is updated iteratively by (3.16), where the step size *ϵ*_{k} and coefficients *a*_{0},…,*a*_{N}, *b*_{0},…,*b*_{N}, *γ* in (3.14) for *ζ*_{k+1} are found by algorithm 1 in [41], in which is replaced by with and with *h*(*t*) determined by (4.16). As in the axisymmetric translation, the accuracy of the shape is controlled through the error in satisfying the shape optimality condition (4.16).

Figure 4*b* shows cross sections of the minimum-resistance shape in the *rz*-plane for *ν*=0 and 1/2, and table 4 presents corresponding coefficients *a*_{0},…,*a*_{N}, *b*_{0},…,*b*_{N}, *γ* in (3.14), the constant , and the error in satisfying (4.16). Table 3 compares resistance coefficients *F*_{x}/(6*πμv*_{x}) for a unit sphere, minimum-resistance oblate spheroid, minimum-resistance biconvex lens and minimum-resistance shape for *ν*=0, 1/3 and 1/2. All shapes have volume 4*π*/3. Remarkably, in the axisymmetric translation, the minimum-resistance shapes transition from almost prolate spheroidal shapes to spindle-like shapes as *ν* increases from 0 to 1/2, whereas in the transversal translation, the minimum-resistance shapes are close to oblate spheroidal shapes for all *ν*.

## Acknowledgements

We are grateful to the referees for their comments and suggestions, which helped to improve the quality of the paper.

## Appendix A. Proof of the relationship (2.5)

Let ** a** be an arbitrary constant vector, then
A1For an axisymmetric surface

*S*, whose axis of revolution coincides with the

*z*-axis in the cylindrical coordinate system (

*r*,

*φ*,

*z*), the normal and tangential unit vectors are determined by and , respectively. Because the displacement vector

**u**is constant on

*S*, the vector

**b**=[

**a**×

**u**]=

*b*

_{x}

**i**+

*b*

_{y}

**j**+

*b*

_{z}

**k**is constant on

*S*as well. In the cylindrical coordinates,

**b**=

*b*

_{r}

**e**

_{r}+

*b*

_{φ}

**e**

_{φ}+

*b*

_{z}

**k**, where and , and the relationships ∂

*r*/∂

*s*=−∂

*z*/∂

*n*, ∂

*r*/∂

*n*=∂

*z*/∂

*s*yield Consequently, (A1) simplifies to , and since

**is arbitrary, (2.5) follows.**

*a*## Appendix B. Generalized Cauchy integral formula

The Cauchy integral formula for *n*th-order *r*-analytic functions, defined by (1.3), was obtained in [33], theorem 2 and is a particular case of the one for *H*-analytic functions [37], theorem 2.7.

### Theorem B.1 (Cauchy integral formula for nth-order r-analytic functions)

*Let* *be a bounded open region in the rz-half plane (r≥0) in the cylindrical coordinates (r,φ,z), and let* *have a piecewise smooth positively oriented boundary ℓ, which is either closed or an open curve with the endpoints lying on the z-axis (if* *contains a segment of the z-axis). If G is an nth-order r-analytic function in* *and is Hölder continuous on ℓ,*^{13} *then the Cauchy integral formula for G is given by*
B1*where* *is the generalized Cauchy operator,* *and Ω*_{±}*(ζ,τ) are real-valued functions determined by
*B2*in which Γ(⋅) is the gamma function,* *is the hypergeometric function, and* .

### Remark B.2

If is a complement to in the *rz*-half plane, and *G* is an *n*th-order *r*-analytic function in and is Hölder continuous on ℓ, then the Cauchy integral formula for *G* takes the form in , where the Cauchy operator is defined in (B1).

### Remark B.3 (Sokhotski–Plemelj formulae)

Let be a bounded open region in the *rz*-half plane with a piece-wise smooth positively oriented boundary ℓ, and let be a complement to in the *rz*-half plane. If *G*^{+} and *G*^{−} are *n*th-order *r*-analytic functions in and , respectively, and are Hölder continuous on ℓ, and if, in addition, *G*^{−} vanishes at infinity, then in , and when *ζ* approaches ℓ from within , *G*^{±} satisfies the Sokhotski–Plemelj formula on ℓ except points where ℓ is non-smooth. A given function *f*^{±} on ℓ is a boundary value of an *n*th-order *r*-analytic function *G*^{±} in if and only if it satisfies on ℓ.

For zero-order *r*-analytic functions (*n*=0), (B2) simplifies to
B3whereas for first-order *r*-analytic functions (*n*=1), (B2) takes the form^{14}
B4where and are complete elliptic integrals of the first and second kinds, respectively.^{15}

## Appendix C. Representation of asymmetric displacement field

Suppose an axisymmetric elastic body (bounded or unbounded), whose axis of revolution coincides with the *z*-axis in the cylindrical coordinate system (*r*,*φ*,*z*) with basis (**e**_{r},**e**_{φ},**k**), is imposed arbitrary displacement boundary conditions. In this case, the displacement field **u** and divergence *ϑ* are expanded into Fourier series with respect to the angular coordinate *φ*:
C1and
C2where the choice of either upper or lower functions in brackets depends on whether **u** and *ϑ* are even or odd functions with respect to *φ*. The zero-order Fourier harmonic corresponds to the axisymmetric case: *u*_{0φ}≡0, and can be represented by (3.3) with two zero-order *r*-analytic functions *G*_{1} and *G*_{2}. For *n*≥1, the components *u*_{nr}, *u*_{nφ}, *u*_{nz} and *ϑ*_{n} admit representations in terms of two (*n*−1)th-order *r*-analytic functions and one *n*th-order *r*-analytic function.

### Proposition C.1 (representation of asymmetric displacement field)

*Let G*_{n1}=*G*_{n1}(*r*,*z*) *and G*_{n2}=*G*_{n2}(*r*,*z*) *be* (*n*−1)th-*order r*-*analytic functions, and let G*_{n3}=*G*_{n3}(*r*,*z*) *be an n*th-*order r*-*analytic function, defined by* (1.3). *For n*≥1, *the components u*_{nr}, *u*_{nφ}, *u*_{nz} *and ϑ*_{n} *in the Fourier expansions* (C1) *and* (C2) *can be represented by*
C3*where* *and x*=(3−4*ν*)/(4−4*ν*). *For ν*=1/2, (C3) *simplifies to the representations* (16) *and* (17) *in* [34], *which correspond to Stokes flows*.

### Proof.

The proof is similar to that of proposition 2 in [34]. Let , then in terms of and *u*_{nz}, the Navier equations (1.1) are reformulated as
C4where Δ_{n}=(∂^{2}/∂*r*^{2})+(1/*r*)(∂/∂*r*)+∂^{2}/∂*z*^{2}−*n*^{2}/*r*^{2}, and *ϑ*=−((2−2*ν*)/(1−2*ν*))div **u** implies that
C5

As in the proof of proposition 2 in [34], the components and *u*_{nz} for *n*≥1 can be represented by
C6where *a*_{n}, *b*_{n}, *c*_{n}, *d*_{n} and *e*_{n} are real-valued constants, *U*_{nj}(*r*,*z*) and *V* _{nj}(*r*,*z*) are the real and imaginary parts of an (*n*−1)th-order *r*-analytic function for *j*=1,2, and *U*_{n3}(*r*,*z*) and *V* _{n3}(*r*,*z*) are the real and imaginary parts of an *n*th-order *r*-analytic function , so that Δ_{n−1}*U*_{nj}=0 and Δ_{n}*V* _{nj}=0 for *j*=1,2, and Δ_{n}*U*_{n3}=0 and Δ_{n+1}*V* _{n3}=0.

Then, (C5) with (C6) yields
C7It follows from Δ_{n−1}*U*_{n1}=0 and Δ_{n}*V* _{n1}=0 that Δ_{n±1}(*r* *V* _{n1})=2(∂/∂*r*∓*n*/*r*)*V* _{n1}, Δ_{n−1}(*zU*_{n1})=2 ∂*U*_{n1}/∂*z*, Δ_{n}(*rU*_{n1})=2(∂/∂*r*−(*n*−1)/*r*)*U*_{n1} and Δ_{n}(*zV* _{n1})=2 ∂*V* _{n1}/∂*z*, and, consequently, (C4) with (C6) and *U*_{n1}=*ϑ*_{n} implies that
which are dependent in view of (C7). Thus, *a*_{n}=(1+*x*)/(2*n*−1), *b*_{n}=−2((1−*x*)*n*−1)/(2*n*−1), *c*_{n}=((1−*x*)*n*−1)/(2*n*−1), *d*_{n}=((1−*x*)*n*+*x*)/(2*n*−1), and *e*_{n}=1−*x*, and (C3) follows from (C6). □

### Remark C.2 (uniqueness of *G*_{n1}, *G*_{n2} and *G*_{n3})

The functions *G*_{n1}, *G*_{n2} and *G*_{n3} in (C3) are determined up to *cr*^{n−1}, and −*c*(((1−*x*)*n*+1)/(2*n*−1))*r*^{n}, respectively, where *c* is a real-valued constant, and they are determined uniquely provided that at least one of them vanishes at infinity. Indeed, the Navier equations (1.1) with homogeneous boundary conditions have only zero solution: **u**≡0 and *ϑ*≡0 (see [27], §3.5.2). Then, the third equation in (C3) implies *G*_{n1}=*cr*^{n−1}, and together with the sum of the second equation and the imaginary part of the first equation in (C3), it yields Im *G*_{n3}≡0, or, equivalently, *G*_{n3}=*br*^{n}, where *b* is a real-valued constant. Now, it follows from the first equation in (C3) that , which is an (*n*−1)th-order *r*-analytic function if *b*=−(((1−*x*)*n*+1)/(2*n*))*c*. If at least one of *G*_{n1}, *G*_{n2} and *G*_{n3} vanishes at infinity, then *c*=0, and all *G*_{n1}, *G*_{n2} and *G*_{n3} are zero functions.

## Footnotes

↵1 However, gradient flow methods, posed in Hilbert spaces, can have a good convergence for shape optimization problems [18,19].

↵2 The use of BIEs restricts this scheme to problems with either linear or linearized governing equations.

↵3 The theory of analytic functions is also pivotal in two-dimensional shape optimization problems in fluid and structural mechanics [8,29–32].

↵4 The system (1.3) implies that Δ

_{n}*U*=0 and Δ_{n+1}*V*=0, where Δ_{n}=∂^{2}/∂*r*^{2}+(1/*r*)∂/∂*r*+∂^{2}/∂*z*^{2}−*n*^{2}/*r*^{2}is the*n*th-order harmonic operator. Because of this,*n*th-order*r*-analytic functions were called*n*-harmonically analytic in [33,34].↵5 div

*σ*is a vector with components*σ*_{ij,j}.↵6 The function has a logarithmic singularity as

*κ*→1−: .↵7 For

*ν*=1/2, a real-valued constant is a homogeneous solution of (3.8), and, thus, none of*ϕ*_{1k}(*t*) should be constant.↵8 For

*ν*=1/2, an analytical solution for a rigid spindle-shaped body immersed into a uniform Stokes flow yields*η*=0.870465 (see [23,39]); however, the value of the resistance coefficient in table 1 remains the same.↵9 If , then .

↵10 For

*η*<*π*/2,*G*_{1}can also be represented by (3.10), but, as shown in [40],*α*∈(0,1) is found from an equation more complex than (3.9).↵11 The value of

*η** solves , which is the derivative of (4.13) with respect to*α*at*α*=1 for*ν*=1/2.↵12 A rigid biconvex lens is a body of revolution, which is either a union or an intersection of two intersecting equal-sized spheres.

↵13 A function is Hölder continuous on ℓ if |

*f*(*r*_{2},*z*_{2})−*f*(*r*_{1},*z*_{1})|≤*c*|*ζ*_{2}−*ζ*_{1}|^{α}for any and , some*α*∈(0,1] and non-negative constant*c*.↵14

*Ω*_{+}and*Ω*_{−}in (B4) are given by (2.22) in [37], in which the multiplier 1/*κ*^{2}was accidentally missing.↵15 The function has a logarithmic singularity as

*κ*→1−: .

- Received April 2, 2013.
- Accepted September 5, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.