Stability Criteria for Multiphase Partitioning Problems with Volume Constraints

We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign of the principal eigenvalue of the Jacobi operator. We thus derive certain stability criteria, and in particular we recapture the Sternberg-Zumbrun result on the instability of the disconnected phases in the more general setting of several phases.


Introduction
The partitioning of a set into a number of subsets (the "phases") so that the dividing hypersurface (the "interface") has minimal area, is a problem of geometric analysis and calculus of variations. It is of high importance in the physical sciences and engineering because of its relation to surface tension. Examples include a variety of phenomena [1] ranging from the annealing of metals (Mullins [2]) to the segregation of biological species (Ei et al [3]). Two phase systems formed by the mixing of two polymers or a polymer and a salt in water are used for the separation of cells, membranes, viruses, proteins, nucleic acids, and other biomolecules. The partitioning between the two phases is dependent on the surface properties of the materials. An overview of the physical aspects of the subject is offered in [4]. Early studies of the mathematical problem of partitioning include Nitsche's paper [5,6], and Almgren's Memoir [7] (see also White[8].) Paul Fife was one of the top applied mathematicians of his time with significant and lasting contributions to Diffuse Waves, Diffuse Interfaces, Stefan problems and Phase Field Models. His monographs [9,10] "Dynamics of Internal Layer and Diffuse Interfaces" and "Mathematical Aspects of Reactions and Diffusive Systems" and the IMA volume [11] "Dynamical Issues in Combustion Theory" are classics. Fife with his collaborators studied extensively the dynamical problems related to the generation of partition and to their coarsening. For a sample see [12,13,14,15].
Sternberg-Zumbrun [16], treated the static problem and proved that disconnected two phase partitions of convex sets are always unstable. The S-Z formulas with little notation changes are given in the next theorem. For the reader's convenience some (*) The first author was partially supported through the project PDEGE (Partial Differential Equations Motivated by Geometric Evolution), co-financed by the European Union European Social Fund (ESF) and national resources, in the framework of the program Aristeia of the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF). 1 details are given to make the exposition self-contained. Throughout this paper, we take Ω ⊂ R N to be a bounded domain with smooth boundary Σ = ∂Ω. Definition 1. Let M be a n-dimensional C 1 submanifold of R N with boundary and V an open subset of R N such that V ∩ M = ∅. A variation of M is a collection of diffeomorphisms (ξ t ) t∈I , In place of the ξ t we often consider their extensions by identity to all of R N . With each variation we associate the first and second variation fields known [17] as velocity and acceleration fields, ξ t , ξ tt denoting first and second partial derivative in t.
As M ⊂ Ω, ∂M ⊂ Σ, admissible variations of M should respect the rigidity of the boundary of Ω. In this connection S-Z suggested that admissible variations of M be obtained by solving the ODE (1.1) dξ dt = w(ξ), ξ(0) = x for any given first variation vector field w and then setting ξ t (x) = ξ x (t), where ξ x is the solution of (1.1) for the initial condition ξ x (0) = x. The requirement for rigid container walls is satisfied by selecting w so that w(p) ∈ T p Σ for all p ∈ Σ, T p X denoting as usual the tangent space of X. In this paper we consider only normal variations, i.e. those satisfying w(p) ∈ N p (M ) for all p ∈ M . N p is the normal bundle of M at p. By a partitioning of Ω we mean a collection M = (M i ) m i=1 of C 2 hypersurfaces with boundary (which is again C 2 ), which are non-intersecting and their boundaries lie in Σ = ∂Ω. Additionally, by a minimal partitioning M we mean a critical point of the area functional A under the assumed volume constraints, i.e.
for all variations preserving Σ and the volume of the phases. In this equation M t = ξ t (M ), ξ t being a variation, and A(M ) is the area of M .
be a minimal 2-phase partitioning in Ω. Then for any normal variation of M , which preserves Σ and the volume of the phases, i.e.ˆM f = 0, the second variation of area of M is given by where f is the projection of the first variation field w on the unit normal field N of M , |B M | 2 is the norm of the second fundamental form B M associated with M and σ = II Σ (N, N ) is the scalar version of the second fundamental form of Σ.
The orientation of M is selected so that σ 0 on Σ for convex Ω.
Using this formula with r = 2, S-Z proved that for convex Ω, when σ = 0 on Σ, every disconnected two phase partitioning is necessarily unstable. Recall that by definition a minimal partitioning is stable when δ 2 A(M ) > 0 for all variations w = 0 preserving Σ and the volume of the phases.
In the following section an extension of the S-Z formula to m-phase problems is given in Proposition 4. The instability of disconnected multiphase partitions follows as an application of this. In Section 3 we develop the spectral theory of the bilinear form expressing the second variation of area, which is the main tool for proving our stability/instability results. The main statement in this section is Proposition 17 which states that, for normalized variations, the minimum of the second variation of area is given by the principal eigenvalue. The difficulties in obtaining this result are (i) that the the boundary integral´∂ M f 2 cannot be bounded above by´M f 2 , and (ii) that admissible variations need not satisfy the boundary condition (3.5) of the related eigenvalue problem. They were handled by developing an interpolation estimate for the boundary integral in Lemma 15. An extension of Proposition 17 to m-phase partitioning problems is immediate. Proposition 18 gives a characterization of all connected m-phase partitionings by reduction to the two phase case.
The last two sections are devoted to applications. In Section 4 we prove the existence of unstable partitionings in R N when Ω satisfies hypothesis (H) (see Section 4) and that spherical partitionings are stable when they do not make contact with the boundary of Ω. As a byproduct we also prove that spherical partitionings in bounded sets are never absolute minimizers of the area functional under volume constraint. Applications to 2-dimensional problems have been included in Section 5. The main results here are criteria for instability, Proposition 22, and stability, Proposition 25. Proposition 26 shows that sufficiently small partitions are stable. For related work see [18,19,20].

Multiphase Partitioning Problems
A more general functional may be used for more than two phases: The coefficients γ i > 0 have the physical meaning of surface energy density. The summation in (2.1) extends over all interfaces constituting the partition problem. The collection M = (M i ) r i=1 will be considered oriented, its orientation being determined by the orientations of the M i . There are 2 r possible orientations for M . Most of the following formulas depend on the orientation of M . Admissible variations for the functional in (2.1) are those preserving phase volume. They can be directly obtained from general variations by rendering them volume preserving (see [16]). As this is a highly involved process for multiphase systems, we use the method of Lagrange multipliers, which, as it turns out, is more convenient. In this connection we consider the following modified (weighted) functional: In this formula |·| denotes volume, m is the number of phases, P j is the number of distinct sets phase j is split (indexed by k), V j is the volume of phase j and λ j is the Lagrange multiplier corresponding to the volume constraint for the j-th phase. Since Example 3. In the case of a disconnected 3-phase partition the weighted area functional is given by (see Fig. 2.1) The volume constants V j were dropped as they play no part in the variational process.
The following proposition extends Theorem 2 to more than two phases. γ j κ j = 0 (iii) Each M i is normal to Σ, i.e. on each M i ∩ Σ we have N i · N Σ = 0 or N i (p) ∈ T p Σ for all p ∈ ∂M i . N Σ is the normal field of Σ.
(iv) For any admissible variation of M , i.e. one preserving Σ and the volume of the phases, the second variation of area of M is given by Proof. For concreteness we consider the disconnected 3-phase partitioning of N ∂Ω jk being the unit outward normal field of ∂Ω jk , which is easily established, we obtain: Let H i be the mean curvature vector field of M i , ν i the unit tangent field of M i which is normal to ∂M i (also known as "conormal" field) and f i = w · N i the normal component of the variation field on M i (tangential components are irrelevant and are disregarded from the outset). Let (·) ⊤ denote projection on the tangent space of a manifold. Application of the identity [17] the divergence theorem for manifolds [17],´M i div M w ⊤ =´∂ Mi w·ν i , and reordering of terms give:

Standard arguments render
which prove (i) and (ii) and w · ν i = 0, which proves (iii). By variation of the Lagrange multipliers we recover the volume constancy constraints: For the proof of (iv) we start by Simon's general formula for the second variation of area [17] (2.7) In this formula w, a are the first and second variation fields, (·) ⊤ denotes projection on the tangent space of M i , (·) ⊥ denotes projection on the normal space of M i , E 1 , · · · , E N −1 are the basis vector fields in a chart, g rs = E r · E s are the covariant components of the metric tensor of M i and g kl its contravariant components. Summation convention applies throughout this paper. The notation ·, · is alternatively used to denote scalar product in lengthier expressions. Recalling that w is normal In this equation B rk = N, D Er E k are the covariant components of the second fundamental form tensor II(u, v) = N, D u v (u, v are tangent vector fields) and B s r = g sk B rk . By (2.5) as w ⊤ = 0 we obtain Combination of (2.5) with the equality a = D w w, which is obtained by taking the time derivative of (1.1), gives for div Mi a: By Lemma 9 we have for the second variation of the |Ω jk |: Replacing for δ 2 A(M i ), δ 2 |Ω jk | by the above equalities and rearranging give the following expression for δ 2 A ⋆ (M ): By (2.6) the integral on the second row assumes the form On the second and third equalities we have used the identity [17] div R n X = div M X + N · D N X, and Equation (2.8); X is any differentiable field on M . Application of the divergence theorem givesˆM In a similar fashion we reformulate the last two rows in the expression for δ 2 A ⋆ (M ) and this completes the proof of (iv).

Remark 5. The convention for the second fundamental form of
Remark 6. The equation of (ii) of the proposition depends on orientation choice. In the case of a disconnected 2-phase partition we have γ 1 = γ 2 = γ 12 , the interfacial energy density of phases 1 and 2, and (ii) reduces to κ 1 + κ 2 = 0, which, in the 2-dimensional case, implies the interfaces are circular arcs of equal radii. This restricts considerably the number of possible realizations of minimal disconnected multiphase partitions.

Example 7.
Let Ω be an ellipse centered at 0 with major and minor semiaxes a, b. For a > x 0 > 0 the tangents at (x 0 , ±y 0 ) have equations These lines intersect at . Hence the radius of the circular arc, which intersects the ellipse at right angles, is given by This is a monotone decreasing function of x 0 and from this it follows, with simple geometric arguments, that all possible minimal, disconnected, 2-phase partitionings of an ellipse, are pairs of transversal circular arcs symmetric about the x or y-axis.
As an application of Proposition 4 we prove the instability of disconnected 3phase partitions in a convex set. Given any such partition, we choose a variation which is constant on each interface. The volume constraints are satisfied if we By (iv) of Proposition 4 we obtain Generalization to an arbitrary number of phases and phase splittings is immediate: be any stable m-phase partitioning of Ω. Further, assume that Ω is strictly convex, in particular II ∂Ω (N i , N i ) > 0 at all points of ∂M i ∩ Σ, i = 1, · · · , r. Then M is necessarily connected.
We close this section by proving the Lemma that was used in the proof of part (iv) of Proposition 4. Lemma 9. In the setting of Proposition 4, the second variation of volume of any distinct phase Ω j is given by In this equation N ∂Ωj is the unit outward normal field of ∂Ω j and M j denotes collectivelly the interfacial part of ∂Ω j , i.e. M j = ∂Ω j \∂Ω.
where Jξ t is the Jacobian of ξ t . For the second variation of this functional we have dy.
Application of the rule of differentiation of determinants and straight-forward manipulations give We are using greek indices for vector components and coordinates in the surrounding space R N , and latin for submanifolds. Summation convention is applicable to greek indices as well. Formula (2.9) follows from this equality, the identity Gauss's theorem in R N and a = D w w. Since the variation preserves ∂Ω, the integral over ∂Ω j \M j drops.

Spectral Analysis of the 2nd Variation of Area Form
To keep the length of formulas to a minimum and focus on the essence of the argument, we present the details for the two phase partitioning problem and then indicate how the results generalize to more phases.
3.1. Two phase partitioning problem. Let M be the interface of a two phase partitioning problem in Ω, which is assumed minimal, i.e. δA(M ) = 0. For linearized stability we naturally study the minimal eigenvalue of the bilinear form As a matter of convenience, we introduce Lagrange multipliers and the corresponding modified functional 3), is that it satisfies the following inhomogeneous PDE with Neumann boundary condition: in a local coordinate system q 1 , · · · , q N −1 , where g = det(g ij ), g ij is the metric tensor and the comma operator denotes partial derivative in the respective coordinate, i.e. f ,i = ∂f ∂q i = D Ei f . The summation convention on pairs of identical indices is assumed throughout this paper. As M is fixed, g ij is fixed and (3.4) is a linear equation.
Remark 12. The C 2 condition on f can be relaxed by considering the weak form of (3.4), (3.5).
Proof. The first variation of J ⋆ is given by By Green's formula for manifolds we obtain When φ is a C ∞ function with compact support in the interior of M , the second integral on the right side drops and by the fundamental lemma of the calculus of variations we obtain (3.4). When the support of φ intersects the boundary of M , Two are the relevant problems: (a) given a partitioning, to show that it is unstable, i.e. to find a particular admissible variation f such that J(f ) < 0, and (b) to prove that a partitioning M is stable, i.e. for any admissible variation f = 0 we have J(f ) > 0. The proposition next shows that for problems of the first category it suffices to find a negative eigenvalue µ < 0 of problem (3.4).  In particular, if µ < 0, M is unstable.
Remark 14. Proposition 13 implies that no lower bound is necessary for the functional J, in order to conclude that a minimal partitioning is unstable, when a negative eigenvalue is at hand. This is in contrast with problems of category (b).
Proof. Multiplication of (3.4) by f and integration over M gives in view of (3.2) and (3. 3):ˆM Application of Green's formula on the first integral gives By (3.1) and (3.5) we obtain J(f ) = µ. The second assertion follows trivially from this.
In class (b) we need to know in advance that the functional J has a minimum under the conditions (3.2) and (3.3 where c BT is a constant depending on M . Now using this estimate for the boundary integral and the estimates |B M | 2 b 2 0 , σ σ 0 , b 0 and σ 0 being certain constants depending on M and Σ, we obtain from which we can conclude coercivity of J if σ 0 < 1/c 2 BT . In this way we can prove (see [23], Theorem 1.2, p.4) that for sufficiently small principal curvatures of Σ in a neighborhood of ∂M , J has a minimum, which is necessarily a critical point of J ⋆ , hence an eigenfunction of (3.4) with BC (3.5). As a consequence, by (3.6), if J ⋆ has no non-positive eigenvalues, we can draw the conclusion that M is stable.
We can drop the hypothesis of sufficiently small principal curvatures of Σ in a neighborhood of ∂M by replacing (3.7) by an interpolation estimate. The standard notation for Sobolev spaces is used: |u| As an example we prove (3.9) directly for a bounded hypersurface M of R N with boundary.
Example 16. Assume that there is a x 0 ∈ R N such that x 0 · ν(p) > 0 for all p ∈ ∂M . Without loss of generality set x 0 = 0. For u ∈ C ∞ (M ), x being the position vector in R N , by (2.5) By the compactness of ∂M and M there are positive constants c 0 , c 1 , c 2 such that x · ν c 0 , |x| 2 c 1 and |H| c 2 . To compute div M x apply the definition of operator div M (Simon [17]) in a chart with basis vector fields E 1 , · · · , E N −1 : By a density argument we extend to u ∈ H 1 (M ). Estimate

3.2.
Three and more phases. Since disconnected partitionings in strictly convex sets are always unstable according to Proposition 8, we need only consider connected partitionings. The functional J for the (m + 1)-phase partitioning problem reads: The volume constraints are the normalization condition is and the corresponding modified functional is Proposition 10 extends without difficulty to connected multiphase problems, with (3.4) holding on each inteface in the following form: The proof is straight-forward and follows by the fact that the M i appear individually in the form of the second variation of area J and the constraints.
Examples for stable and unstable multiphase partitionings are easily constructed from 2-phase partitionings by means of Proposition 18. Therefore our applications focus on 2-phase partitionings.

Some Applications to Partitioning Problems in R N
As applications of the spectral analysis of the 2nd variation of area, we derive in this section some general conclusions concerning partitionings in R N . 4.1. Stability of N -dimensional spherical partitionings. The stability of partitionings in which one phase has the shape of a sphere, M = S N −1 , has a direct physical meaning. It is the basis for modeling the stability of emulsions, i.e. suspensions of small liquid droplets or deformable solid particles in a surrounding fluid. As ∂M = ∅, all boundary integrals are absent, in particular the boundary condition (3.5). In 3-dimensions one may directly proceed to the solution of (3.4) in spherical polar coordinates using periodic conditions in place of (3.5). The spectral theory of the operator ∆ M is well-known (see for example [24], Ch. V, §8, p. 314; Ch. VII, §5, pp 510-512 and [25] §IV.2). Its eigenvalues are given by λ l = −l(l + 1), l = 0, 1, · · · and the corresponding eigenvectors are the spherical harmonics Y m l , m = 0, ±1, · · · ± l, the first of which are Since the eigenvectors of J satisfy the volume constancy condition (3.2), the first of these is not admissible. Integration of (3.4) in view of (3.2) gives λ = 0. Thus the minimal eigenvalue of J, as obtained by l = 1, is µ 1 = l(l + 1) − (N − 1) = 0, which whould imply neutral stability. A more careful examination of these eigenvectors reveals that they are not true variations, but translations along the axes of the coordinate system. On discarding them and procceding to the next available eigenvalue, l = 2, we have which implies stabilty. For N > 3 (see [25]) λ l = −l(l + N − 2), l = 0, 1, · · · and again µ 1 = 0. Discarding translations again yields stability. In the next section we prove that the same situation pertains also to the 2-dimensional case. For the following proposition, no regularity and convexity conditions are necessary for Ω.

Proposition 19.
Let Ω ⊂ R N be an open set and Ω 1 = B(x 0 , R) a ball such that Ω 1 ⊂ Ω. The two-phase partitioning of Ω defined by M = ∂Ω 1 is stable.
Note that balls are never absolute minimizers of partitioning problems in bounded sets. This is most conveniently seen by moving the ball until it makes contact with the boundary, M ∩ ∂Ω = ∅. Then it is clear that the translated ball is not even minimal, since at the contact point the normal of M is not tangent to ∂Ω. This implies the existence of a variation which decreases the ball's area and this proves that the original ball is not an absolute minimizer.

4.2.
Existence of unstable two phase partitionings. The existence of stable partitionings is guaranteed by the existence of absolute minimizers (see [8] and [23] Th. 1.4, p. 6). Here we give a proof of existence of unstable partitionings in R N under certain conditions on Ω. Let M be a minimal two phase partitioning of Ω. We assume Ω is convex and has the following property: for all sufficiently large σ there is a convex set Ω σ containing M and satisfying the condition (H) Ω σ contacts Σ = ∂Ω along ∂M , i.e. T p ∂Ω σ = T p Σ at all p ∈ ∂M , and and M ∩ ∂Ω = {p 1 , p 2 }. If C 1 , C 2 be the circles of radius 1 σ contained in Ω and contacting Σ = ∂Ω at p 1 , p 2 respectively, then the convex hull of M ∪ C 1 ∪ C 2 = Ω σ satisfies (H).
(ii) Consider a rhombus and a circular arc with its center positioned at one of its vertices. Let Ω be the solid obtained by the revolution of the rhombus about the axis of the rombus that passes through the center of the circular arc. Let M be the surface obtained by the revolution of the arc. By elementary geometric arguments similar to (i) it is easily established that for all sufficiently large σ there is a convex set Ω σ containing M and satisfying condition (H).
Proposition 21. Let M be a minimal partitioning of Ω in R N . Assume that Ω satisfies condition (H). Then there is a convex set Ω ⋆ for which M is an unstable partitioning.
Proof. Assume M is stable or neutrally stable, for otherwise there is nothing to prove. Let Let ǫ > 0 small and f be a variation of M satisfying (3.3), (3.2) and such that J(f ) < µ 1 + ǫ. We can assume that f is not identically vanishing on ∂M , for if and then we have f + f 0 = f 0 = 0 on ∂M and < J(f ).
Since Ω, M were assumed to satisfy (H) there is a σ > 0 and Ω σ =: Ω ⋆ such that and k is a positive number to be fixed later. We have Choosing k > µ 1 + ǫ completes the proof.

Application to 2-Dimensional Partitioning Problems
Two-dimensional partitionings are particularly simple, for in this case where s is the arc length of M and the integrals over ∂M reduce to numbers. The boundary condition (3.5) reduces to (I) f (s) = − λ 2k 2 + C sin(ks) + D cos(ks), The constants λ, C, D in each case are determined by the two conditions of (5.1) and one of (3.2), which form a linear homogeneous system of three equations in these three variables. The condition for existence of solutions of this system is obtained by setting its determinant to 0, which gives a nonlinear equation for k. With a solution for k at hand, we can determine the eigenvalue µ by the last column in the above table of possible solutions for f and the eivenvectors (λ, C, D), each determining an eigenfunction f of problem (3.4). Not all three cases (I)-(III) need to be considered, depending on the problem under study.

5.2.
Case II: µ < −κ 2 , k > 0. By (5.1) s = 0, L and (3.2) we obtain the system As previously we obtain the system By the first of equations (5.4), C = −σ 1 D, and the remaining two equations give a system, the solvability of which is equivalent to the equation As a consequence, when it happens that the length of the interface has one of the following two values the partitioning is unstable. We will prove a more general result in Proposition 22.

5.4.
Stability and instability criteria. The following proposition generalizes the previous result.

Proposition 22.
Let Ω be a bounded, convex, open subset of R 2 and M a minimal two phase partitioning of Ω with length L. Assume there is a neighborhood of the points ∂M in Σ = ∂Ω which is a C 2 curve and the curvatures of Σ at these points are σ 1 , σ 2 . If L satisfies the condition Proof. We only have to prove the inequalities. Letting B = λ 2k 2 , x = Lk, a = σ 1 L and b = σ 2 L, the solvability condition for system (5.3) of case (II) (in the variables B, C, D) is Performing operations we obtain where ρ m = m mod 2 and p 0 = p, q 0 = q, p 1 = q, q 1 = p. Assuming the validity of (5.7) for n we will prove its validity for n + 1. Clearly all f (i) have the same form with f . The coefficient of cosh x in f (n+1) is a sum of numerical multiples of the derivatives p (n+1) , · · · , p (0) = p and q (n+1) , · · · , q (0) = q and it is obtained by taking the derivative of f (n) in (5.7). We have On the second equality we used the change of summation index j = i − 1 and the identity q ρn−j . Application of the binomial theorem and a second redefinition of the sumation index by j = i + 1, prove the assertion.
By (5.7) with p(x) = 4 1 + 1 The expansion of D is Thus D(x) > 0 in a neighborhood ]0, δ[, δ > 0, when ab − 4a − 4b + 12 < 0. Using the definitions of a, b in this inequality we obtain Proof. When σ 1 = σ 2 = 0, by (5.5) it is clear that (5.4) has no solution and it is easily checked that (5.3) has only the trivial solution. System (5.2) reduces to λ = C = 0, sin(kL) D = 0 which has nontrivial solutions only when sin(kL) = 0, i.e. kL = nπ, n ∈ N. By the restriction 0 < k κ < 1 for case (I) it follows that κL > nπ. As ω = κL is the angle of the circular sector defined by M and the tangents to ∂Ω at the extremities of M , by the convexity of Ω we have ω < π. The case k = κ correspomding to µ = 0 gives also no eigenvalues. Thus we have only positive eigenvalues by case (I), which are given by k n = nπ L or µ n = n 2 π 2 L 2 − κ 2 > 0 and this according to Proposition 17 implies stability.
As a next application of equations (5.2)-(5.4) we give a proof of the stability of circles. This is essentially a variational proof of the well-known fact that among all 2-dimensional geometric shapes of equal area, circles have least perimeter. By remarking that when f is an eigenfunction of (3.4) then also f ⋆ defined by f ⋆ (s) = f (L−s) is an eigenfunction for the same eigenvalue, we obtain the equation 4 (σ 1 + σ 2 ) 2 we obtain (σ 1 + σ 2 )L > 12, and this completes the proof.
We conclude by proving that sufficiently small partitions are stable.

Proposition 26.
Let Ω and M be as in Proposition 22. If L = |M | is sufficiently small, then M is stable.
Proof. If L < L 0 , where L 0 = 2 σ1+σ2− √ σ 2 1 +σ 2 2 −σ1σ2 σ1σ2 , case (III) has no solution. As in the proof of Proposition 25 we can prove that case (II) has also no solution, and thus there is no eigenvalue in the range µ −κ 2 . We are looking for eigenvalues in the range −κ 2 < µ 0 or equivalently 0 < k κ. The determinant of the linear system (5.2) in the variables B = − λ 2k 2 , C, D is given by x sin x + σ 1 + σ 2 k x 2 cos x + σ 1 σ 2 k 2 x 3 sin x with x = kL. We expand sin x, cos x into a Taylor series about 0: Now assume k is a root of D(kL) = 0 in ]0, κ] for each L L 0 , i.e.