Least-perimeter Partitions of the Disk into Three Regions of Given Areas

LEAST-PERIMETER PARTITIONS OF THE DISK INTO THREE REGIONS OF GIVEN AREAS ANTONIO CA˜ NETE AND MANUEL RITOR´ E Abstract. We prove that the unique least-perimeter way of partitioning the unit 2-dimen- sional disk into three regions of prescribed areas is by means of the standard graph described in Figure 1. Introduction Partitioning problems in the Calculus of Variations have multiple applications in physical sciences. They can model multitude of natural phenomena such as the shape of a cellular tissue, the interface of separation between fluids, and many others, as described in the treatise by D’Arcy Thompson [Th]. In this work we consider the isoperimetric problem of partitioning a planar disk into three regions of given areas with the least possible perimeter, and we prove that the standard configuration in Figure 1, consisting of three circular arcs or segments meeting orthogonally the boundary of the disk, and meeting in threes at 120 degrees in an interior vertex, is the only solution to this problem. Figure 1. The least-perimeter partition of the disk into three given areas In addition to the above conditions, the solution must satisfy a certain balancing condition on the geodesic curvatures of the circles. This condition will be stated precisely in the next section. Existence and regularity of solutions for this problem are guaranteed by the results of F. Morgan [M2], who showed that the minimizer, in the interior of the disk, is composed of smooth curves of constant geodesic curvature meeting in threes at 120 degree angles. Date: June 25, 2003. 2000 Mathematics Subject Classification. 49Q10, 51M25, 52A38, 52A40. Key words and phrases. Isoperimetric partition, stability, stable. Both authors have been supported by MCyT-Feder research project BFM2001-3489. 2 A. CA˜ NETE AND M. RITOR´ E Boundary regularity also follows from [M2] although it is not explicitly stated in his work. Existence and regularity in higher dimension were studied by F. Almgren [Alm]. The least-perimeter way of partitioning a disk Dinto two regions of given areas is by means of an arc of circle or segment that meets orthogonally ∂D. From the existence and regularity results in next section it follows that there is a solution, which is a smooth, possibly nonconnected, embedded curve with constant geodesic curvature that meets ∂D orthogonally. Such a curve must be connected, since otherwise we could rotate one component with respect to the center of the disk until it touches a second one, thus producing a non-allowed singularity. On the other hand, as the curve has constant geodesic curvature, it must be part of a circle or of a line. Figure 2. The least-perimeter partition of the disk into two given areas The isoperimetric problem consisting of enclosing ngiven areas in the disk or in the plane with the least possible perimeter has a strong complexity which is derived not from the geometry of the individual components of the solution (they can be described in terms of circles or lines) but from their large number. The planar double bubble conjecture was proved by J. Foisy et al. [FABH], who showed in 1993 that the standard planar double bubble uniquely minimizes perimeter in R2. Assuming that the studied regions are connected, C. Cox et al. [CHH] proved in 1994 that the standard planar triple bubble uniquely minimizes perimeter in the plane for any three given areas. R. P. Devereaux [D] studied in 1998 the planar triple bubble conjecture under the hypothesis that all the regions have the same pressure. W. Wichiramala finally proved the planar triple bubble conjecture in 2002 in his Ph. D. Thesis [W]. J. Masters [M] proved in 1996 the double bubble conjecture in S2. Interesting preliminary work was carried out by Bleicher [B1], [B2], [B3]. Concerning boundary problems, G. Hruska et al. [HLPS] have obtained some results for planar bubbles in corners. Also results on tori and cones have been obtained in [CHLL], [BL]. In higher dimensions, J. Hass and R. Schlafly [HS] proved the double bubble conjecture in R3for equal volumes. The general conjecture was settled by M. Hutchings et al. [HMRR]. For higher dimensional Euclidean spaces Reichardt et al. [RHLS] have obtained a proof of the double bubble conjecture in R4and partial results in higher dimensional Euclidean spaces. In the three-dimensional torus, M. Carrion et al. [CCWB] have provided numerical evidence for a double bubble conjecture with ten types of solutions. In the three-dimensional sphere and the three-dimensional hyperbolic space, A. Cotton and D. Freeman [CF] have also obtained partial results on the conjecture that the standard double bubble in these spaces uniquely minimizes perimeter. LEAST-PERIMETER PARTITIONS OF THE DISK 3 Planar bubbles are also of great interest to physicists. Interesting articles focusing on physical aspects of the problem are [GJJF] and [CGFM]. The most interesting mathematical open question for these problems is to show that the minimizing configurations must have connected regions, either in R2or in the disk. In addition, in the planar problem, one should also be able to prove that the exterior region is connected, i.e., that there are no empty chambers. We have organized this paper in several sections. In Section 1 we give precise definitions, compute the first and second variations of length for graphs, recall existence and regularity results for the problem of minimizing perimeter while partitioning the disk into given areas, and state some properties which minimizing graphs must satisfy. In Section 2 we obtain a bound on the number of components of the largest pressure region determined by a graph which minimizes perimeter up to second order. We conclude that a minimizing configuration must have one of ten possible types, described in Figure 4. In Section 3 we prove the necessary results to discard the possibilities obtained in Section 2, which allow us to prove our Main Theorem in Section 4. In a final section, we indicate further lines of research and give several conjectures. All the pictures in this paper have been made by using Surface Evolver, a software developed by Ken Brakke (http://www.susqu.edu/facstaff/b/brakke/). 1. Preliminaries 1.1. Notation. Let D⊂R2be the closed unit disk in centered at the origin. An admissible graph C⊂Dconsists of vertices and curves so that at every interior vertex (that is, a vertex in the interior of D) three curves of Cmeet and at every boundary vertex (a vertex in ∂D) just one curve of Cmeets ∂D. We shall also assume that Cinduces a decomposition of the open unit disk into nregions Ri, 1 ⩽i⩽n, possibly nonconnected. An m-component is a connected component of a region with medges. If Riand Rjare adjacent regions, we will denote by Cij ⊂Cthe (not necessarily connected) curve separating them. Let I(i) = {j6=i;Rjtouches Ri}. With this notation ∂Ri∩int(D) = [ j∈I(i) Cij. We shall denote by Nij the normal vector to the curve Cij pointing into the region Ri, and by hij the geodesic curvature of the curve Cij with respect to the normal Nij. Astandard graph consists in three circular arcs or lines segments meeting at an interior vertex at 120 degree angles, reaching orthogonally ∂D, and so that the sum of the geodesic curvatures is zero. Given npositive numbers a1,...,ansuch that Pn i=1 ai=π, the isoperimetric profile is the function I(a1,...,an) defined as the infimum of the lengths of all admissible graphs separating regions in the disk of areas a1,...,an. We will say that an admissible graph Cis minimizing for prescribed areas a1,...,anif I(a1,...,an) is attained by C. 4 A. CA˜ NETE AND M. RITOR´ E 1.2. Variational formulae. Given an admissible graph C⊂D, we will consider smooth one-parameter variations ϕt:C→Dfor tsmall, which satisfy ϕt(∂D)⊂∂D. We will denote by X=dϕt/dt|t=0 the associated infinitesimal vector field, which is smooth on every curve Cij. Note that X(p) is tangent to ∂D for each pin ∂D. Let uij =X·Nij be the normal component of Xon Cij. Given such a variation, it is easy to check that the derivative of the area Aiof Riat t= 0 is given by (1.1) dAi dt t=0 =−X j∈I(i)ZCij uij. For the derivative of length for such a variation we have Proposition 1.1 (First variation of length [HMRR, Lemma 3.1]).Consider an admissible graph C⊂D, and a smooth variation ϕt:C→Dwith associated vector field X. Then the first derivative of the length of ϕt(C)at t= 0 is given by (1.2) dL dt t=0 =−1 2X i∈{1,...,n} j∈I(i) ZCij hijuij +X p∈∂Cij X(p)·νij(p), where νij(p)is the inner conormal to Cij in p. We will say that an admissible graph is stationary if (1.2) vanishes for any area-preserving variation. From Proposition 1.1 it is easy to prove the following Proposition 1.2. Let C⊂Dbe a stationary graph. Then the following conditions are satisfied (i) The geodesic curvature hij is constant on Cij. (ii) The edges of Cmeet in threes at 120-degree angles in interior vertices. (iii) The balancing condition: three edges Cij ,Cjk,Cki meeting in an interior vertex satisfy (1.3) hij +hjk +hki = 0. (iv) The edges of Cmeet ∂D orthogonally at boundary vertices. Condition (ii) implies that, in some interior vertex where the three curves Cij,Cjk,Cki meet, the normals add up to zero, i. e., Nij +Njk +Nki = 0. This implies that the normal components of the vector field Xmust satisfy (1.4) uij +ujk +uki = 0. Given a stationary graph C, and a function u:Si,j Cij →R, with uij =u|Cij , satisfying condition (1.4) on every interior vertex, it is always possible to find a vector field Xon C, so that uij =X·Nij and Xis tangent to ∂D in each boundary vertex. Associated to Xone can also find a one-parameter variation ϕt:C→D, for tsmall enough, so that ϕt(p) = expp(tX(p)) for any pout of an arbitrarily small neighbourhood of ∂D. The argument is as follows: fix an arbitrary neighbourhood Uof ∂D that does not contain interior vertices of C. Modify Xso that it is normal to Cin U. Let νbe the inner normal to ∂D. Extend it to Uso that it is tangent to the edges of C. Also extend Xto a vector field on U∩D by means of the exponential mapping. Let λbe a smooth function equal to 1 near ∂D with LEAST-PERIMETER PARTITIONS OF THE DISK 5 support in U. Consider the vector field Y=X−(X·(λν)) λν and the local one-parameter group ψtgenerated by Y. Since νis tangent to Cand Xis normal to Cin U, we have that Y=Xon C. Moreover, for p∈∂D, the vector Y(p) is tangent to ∂D. Hence the deformation ψt(C∩U) has initial velocity vector field Xand keeps Cinside the disk. The variation ψt(C) has the further property that coincides with expp(tX(p)) in Uout of the support of λ. Now we simply define ϕt(p) equal to expp(tX(p)) out of the support of λ, and equal to ψt(p) in U. The balancing condition (1.3) allows us to define a pressure pion every region Ri, starting from a given region, so that (1.5) hij =pi−pj. These pressures are determined up to an additive constant. The first variation formula of length can be rewritten in terms of pressures in the following way: if Cis a stationary graph, then the first variation of length for an arbitrary variation is given by: (1.6) dL dt = n X i=1 pi dAi dt . Observe that the indetermination of the pressures up to some additive constant does not affect the above formula since Pn i=1 dAi/dt = 0 for any variation of the regions Ri, as Pn i=1 Ai(t) = area(D) along the variation. Let us prove now the second variation formula of length Proposition 1.3 (Second variation of length).Let Cbe a stationary graph and let {ϕt}be a variation with associated vector field Xpreserving areas up to second order. Then the second derivative of length at t= 0 is given by −1 2X i=1,...,n j∈I(i) ZCij (u′′ ij +h2 ijuij)uij +X p∈∂Cij p∈int(D) −qiju2 ij +uij ∂uij ∂νij (p)(1.7) +X p∈∂Cij p∈∂D u2 ij +uij ∂uij ∂νij (p), where qij(p) = (hki +hkj)(p)/√3, and Rkis the third region touching the vertex p. Proof. Differentiating the integral terms in equation (1.2), we get d dtt=0ZCij hijuij=ZCij (u′′ ij +h2 ijuij)uij +hij d dtt=0ZCij uij, but since the variation preserves areas up to second order, it follows that X i∈{1,...,n} j∈I(i) hij d dtt=0ZCij uij= 2 n X i=1 pi d2 Ai dt2t=0 = 0. Differentiating now the second term in equation (1.2), we get d dtt=0 (X·νij) = (DXX·νij) + uij hij (X·νij) + uij ∂uij ∂νij . 6 A. CA˜ NETE AND M. RITOR´ E For p∈int(D), the first term vanishes since νij +νjk +νki = 0, and after some calculations as in [HMRR], the second one can be seen as −qiju2 ij, where qij = (hki +hkj)/√3. For p∈∂D, since the configuration is stationary, the edges meet ∂D orthogonally, so that DXX(p)·νij (p) equals u2 ij times the geodesic curvature of ∂D, and (X·νij )(p) = 0.  The condition that the variation must preserve area up to second order is not really needed as we can show in the next Lemma Lemma 1.4. Let C⊂Dbe a stationary graph. Given smooth functions uij :Cij →Rsuch that (1.1) and (1.4) are satisfied (a variation that preserves area up to first order is given), there is a variation {ϕt}of Cwhich leaves constant the area of the regions enclosed by ϕt(C) and such that the normal components of the initial velocity vector field Xare the functions uij. Proof. Let Xbe a vector field on C, smooth over each curve Cij, such that X·Nij =uij. Let ψt:C→Dbe a one-parameter variation of Cassociated to Xsuch that ψt(p) = expp(tX(p)) out of a small neighbourhood Uof ∂D which does not contain interior vertices of C. We label the regions Riso that Ritouches Ri+1 for i= 1, ...,n−1. Choose positive functions viwith support in the interior of Ci(i+1) and out of U. The variation induced by the vector field viNi(i+1) decreases the area of Ri, increases the area of Ri+1 and leaves constant the area of the remaining regions. Consider the variation equal to (t, s1,...,sn−1)7−→ expptX(p) + n−1 X i=1 siviNi(i+1)(p),in C∩(D−U), and equal to ψt(p) for p∈C∩U. Consider the function (A1,...,An−1) of (t, s1,...,sn−1), given by the areas of the deformation of the regions R1,...,Rn−1. The Jacobian ∂(A1,...,An−1) ∂(s1,...,sn−1) is lower triangular, with non-vanishing entries in the principal diagonal, so that the matrix is regular. The Implicit Function Theorem allows us to find smooth functions s1(t),...,sn−1(t) such that Ai(t, s1(t),...,sn−1(t)) is constant for all i. The initial velocity vector field of such a variation is equal to Xon C∩U, and to X+ Pn−1 i=1 s′ i(0) viNi(i+1) on C∩(D−U). As s′ i(0) = 0 since ψtpreserves areas up to first order, we conclude that Xis the initial velocity vector field.  Remark 1.5.A variation of a stationary graph Cby stationary graphs preserves the angles between edges at interior vertices and the orthogonality condition at boundary vertices. Given a variation preserving the area of all the regions up to first order, we can modify it by Lemma 1.4 so that the areas enclosed are constant along the deformation. From the second variation formula we get that the second derivative of length is given by d2 L dt2=X α dpα dt dAα dt , where αlabels the components of the stationary graph (regions can be disconnected), and dpα/dt is the derivative of the pressure of the component αwith respect to the considered variation. Take into account that the quantity u′′ ij +h2 ijuij, the derivative of the geodesic LEAST-PERIMETER PARTITIONS OF THE DISK 7 curvature hij, only depends on uij , the normal component of the variational vector field X, and that the modification needed in Lemma 1.4 to preserve areas only modifies the acceleration of the variation. The angle-preserving condition depends only on the initial velocity vector field. In general, if the areas are not preserved up to second order, the second derivative of length, for a deformation of a stationary graph by stationary graphs, is given by d2 L dt2=X α dpα dt dAα dt +pα d2 Aα dt2, which can also be obtained by differentiating equation (1.6). Remark 1.6.For a variation such that the angles between the edges are preserved, we have DX(νij +νjk +νki) = 0 (since νij +νjk +νki = 0 for all t), so the boundary term in the second variation formula vanishes. 1.3. Admissible functions and the index form. Let Cbe a stationary graph. We say a function u:Si,j Cij →Ris admissible if the restrictions uij =u|Cij lie in the Sobolev space W1,2(Cij), and verify that at any interior vertex p,uij(p) + ujk(p) + uki(p) = 0. These functions correspond to variations of Cwhich have as normal components of the associated vector field Xthe functions uij . These variations will preserve areas if, for each i, X j∈I(i)ZCij uij = 0. An admissible function uis said to be a Jacobi function if the associated variation preserves the geodesic curvatures of each edge Cij and the angles in each vertex. The fact that the geodesic curvatures are preserved means that the restrictions uij to Cij verify u′′ ij +h2 ijuij = 0. It is clear that the normal component of the Killing vector field generated by the rotations about the origin gives a Jacobi function. From equation (1.7), we define the index form, that is, the induced bilinear form defined on the space of admissible functions, by Q(u, v) = −1 2X i=1,...,n j∈I(i) ZCij (u′′ ij +h2 ijuij)vij (1.8) +X p∈∂Cij p∈int(D) −qijuij +∂uij ∂νij (p)vij(p) + X p∈∂Cij p∈∂D uij +∂uij ∂νij (p)vij(p), where qij are the functions defined in Proposition 1.3. We will say a stationary graph Cis stable if Q(u, u)⩾0 for any admissible function u whose associated variation preserves areas, and unstable if it is not stable. It is clear that a minimizing configuration must be stable. 8 A. CA˜ NETE AND M. RITOR´ E 1.4. Existence and Regularity. From the results of F. Morgan [M2], we obtain the following Theorem 1.7 (Existence and Regularity [M2, Th. 2.3]).Let D⊂R2be a closed disk, and let a1,...,anbe ngiven areas such that Pn i=1 ai= area(D). Then there exists a graph separating Dinto nregions of areas a1,...,an. Moreover such a graph consists of constant geodesic curvature curves meeting in threes in the interior of Dat 120 degree angles, satisfying the balancing condition (1.3) for the geodesic curvatures, and meeting ∂D, one at a time, in an orthogonal way. Proof. From the results in [M2] one gets the existence of a solution and the regularity in the interior of the disk with just triple points as possible singularities. One also gets that there is a finite number of components (and hence of curves) in the minimizing configuration. For the boundary regularity, we only need to prove that at every point of ∂D, at most one curve of the minimizing configuration arrives, at 90 degrees. If one or several curves meet ∂D at pand at least one of them is not orthogonal to ∂D, then the first variation formula implies that the graph is not stationary. Suppose now that several curves meet orthogonally ∂D at p. We order them counter-clockwise and we consider the first one, C, which is the common boundary of components Ωiand Ωj, with Ωia boundary one. Make a small deformation in the interior of Cwhich implies a loss of area δfor Ωi. In order to preserve the areas, it is possible to choose a point qnear pin C, join qto the second curve C′, which is in the boundary of Ωj, and eliminate the part of Cbetween pand q. It can be checked that this new configuration, for δsmall enough, reduces perimeter. Then we get the desired regularity in the boundary of D. 1.5. Some properties of minimizing graphs. We now give and recall some results on minimizing graphs that will be used to prove our main theorem Lemma 1.8. Given npositive numbers a1,...,ansuch that Pn i=1 ai=π, we have (1.9) I(a1,...,an)⩽n. Moreover, equality is never achieved for n⩾4. If equality holds in the case n= 3 then the standard graph consisting of three line segments dividing the disk into three regions of equal areas is minimizing. Proof. We can divide the disk into regions of given areas a1,...,anby using appropriate n radii. This gives (1.9). For n⩾4, this configuration has a prohibited singularity at the origin, so that it cannot be minimizing. If equality holds in (1.9) for n= 3, the configuration must be stationary, so that the three radii meet in 120 degrees, and the configuration is the standard one for equal areas.  Lemma 1.9. A minimizing graph must be connected. Proof. On a nonconnected graph, we can rotate one of the components until it touches another one creating an irregular meeting, so the graph cannot be minimizing.  Remark 1.10.Let C⊂Dbe a minimizing graph, and Ω a connected component of D−C. Lemma 1.9 implies that ∂Ω∩∂D has to be connected. Lemma 1.11 ([FABH, Lemma 2.4]).On a minimizing graph, there are no 2-components. LEAST-PERIMETER PARTITIONS OF THE DISK 9 2. A bound on the number of components of the largest pressure region Lemma 2.1. Let Cbe a stable graph separating Dinto nregions. Then the region of largest pressure has at most n−1nonhexagonal components. Proof. Assume R1is the region of largest pressure and suppose it has at least nnonhexagonal components, Ω1,...,Ωn. For each i, consider the variation given by ui= 1 on ∂Ωi, extended by zero to the whole graph. If Ωiis a boundary component then Q(ui, ui) = −X j∈I(1) ZC1j∩∂Ωi h2 1j+X p∈∂C1j∩∂Ωi p∈int(D) −q1j(p) + X p∈∂C1j∩∂Ωi p∈∂D 1<0, since, for pin C1j∩C1k∩∂Ωi, we have q1j(p) + q1k(p) = hk1+hkj +hj1+hjk √3(p) = hk1+hj1 √3(p)⩽0, as R1has the largest pressure. If Ωiis an interior component then Q(ui, ui) can be computed as above except that the last summand does not appear. So we get Q(ui, ui)⩽0, and equality holds if and only if Ωiis bounded by segments. It is easy to obtain, from Gauss- Bonnet Theorem, that Ωihas to be an hexagon. In the case of three regions, this only occurs if the three pressures are equal. Hence, in our case we can find some nontrivial linear combination uof ui, such that the induced variation preserves areas up to first order and Q(u, u)<0.  Lemma 2.2. Let C⊂Dbe a minimizing graph separating Dinto three regions. Then Cis one of the graphs in Figure 4. Proof. Suppose first that all the pressures are equal. If all the components touch the boundary of Dthen Cis standard. If there is an interior component, then it is hexagonal. It is easy to see that the edges leaving the vertices of the hexagon meet ∂D (otherwise we could find two different parallel rays meeting orthogonally ∂D). This implies that the graph is like in Figure 3. This graph has two regions with three nonhexagonal convex components, and so it is unstable by Lemma 2.1. 2 3 2 13 32 Figure 3. A graph with an hexagonal interior component 16 A. CA˜ NETE AND M. RITOR´ E We now eliminate configurations (4) and (5). These configurations present an interior 4-component of R1, with three incident edges meeting the exterior of the disk. If we extend the fourth edge, it will meet ∂D orthogonally by Proposition 3.3, and we will obtain a configuration of type (3). Hence, the interior 4-component has two orthogonal symmetries meeting at the origin, and we conclude as before the existence of four nodal regions. Consider now configuration (6). Fix an interior 4-component Ω of R1. If we extend the edge leaving the boundary of Ω that does not reach ∂D, it will meet ∂D orthogonally due to the existence of a symmetry of Ω which is in fact a symmetry of the disk. In this way we obtain a configuration of type (3). As above, Ω will have two orthogonal symmetries meeting at 0 and so we can get four nodal regions yielding instability. Consider now configuration (7). Applying Lemma 3.9, the two 4-components will be symmetric about two lines r1, r2passing through the center of the disk (the corresponding edges are not cocircular). Let q1, q2be the intersection points of each line with the interior edges of these components, that will be zeros of the Jacobi field uinduced by the oneparameter group of rotations about the origin. The reflection of q1with respect to r2lies in the boundary of the 3-component of R1and it is not a vertex of the configuration. This point is clearly also a zero of u. Then uhas four nodal regions and the configuration is unstable. By Proposition 3.11, configuration (8) is not minimizing. Configuration (9) is unstable: if the top and bottom edges of each component are cocircular then the configuration is not minimizing by Lemma 3.8. Otherwise we can find an horizontal symmetry of the graph, which is also a symmetry of the disk by Lemma 3.9. Each interior 4- component has a vertical symmetry so that the interior components of R1are identical. Using the function equal to +1 on one of these components, equal to −1 on the other component, and zero otherwise, we have obtained a function satisfying the mean value conditions (1.1) which is negative for the index form. So this configuration is unstable. We could also use the method of Proposition 3.11 to see that this configuration is nonminimizing. So the only remaining possibility is configuration (10), the standard one. Uniqueness for given areas comes from Theorem 3.6.  5. Final remarks In this paper we have obtained that the problem of dividing the disk into three areas has a unique solution in which all regions are connected, as in the problem of partitioning the disk into two areas. It is natural to conjecture that Conjecture 5.1.A minimizing graph separates the disk into connected regions. If we consider the problem for nregions, with n⩾4, by Lemma 2.1 the region of largest pressure will have at most n−1 nonhexagonal connected components and we can obtain by combinatorial arguments a list of all possible minimizing configurations. Of course the number of candidates increases very rapidly when the number of regions increases. We believe that the following conjectures are true Conjecture 5.2.The least perimeter way of dividing the unit disk into four regions of prescribed areas is given by configuration (1) of Figure 6. LEAST-PERIMETER PARTITIONS OF THE DISK 17 Conjecture 5.3.The least perimeter way of dividing the unit disk into five regions of prescribed areas is given by configuration (2) of Figure 6. (1) (2) Figure 6. The conjectural configurations for n= 4 and n= 5 For each case, we believe that there is another possibly stable configuration: for n= 4, the configuration with three boundary regions surrounding an interior one of three edges, and for n= 5, the one consisting in four boundary regions surrounding an interior region of four edges. But estimates we have done using Surface Evolver (Ken Brakke, 1992) for equal areas show that they are nonminimizing. Furthermore, for n= 4, if any of the areas tends to zero, we should obtain in the limit the standard configuration for three areas, which also discards the configuration described above at least for some areas. In the case n= 5 we should have the same behaviour. For n= 6 we give the following conjecture Conjecture 5.4.The least perimeter way of dividing the unit disk into six regions of prescribed areas is given by configuration of Figure 7. Figure 7. The conjectural configuration for n= 6 As before, we belive that the configurations of Figure 8 below are stable, but estimates done with the Surface Evolver considering equal areas show that they are nonminimizing. 18 A. CA˜ NETE AND M. RITOR´ E (1) (2) (3) Figure 8. Some other configurations for n= 6 References [Alm] F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Bull. Amer. Math. Soc. 81 (1975), 151–154. MR 50 #14438 [B1] M. N. Bleicher, Isoperimetric divisions into several cells with natural boundary, Intuitive geometry (Si´ofok, 1985), Colloq. Math. Soc. J´anos Bolyai, vol. 48, North-Holland, Amsterdam, 1987, pp. 63– 84. MR 88j:52023 [B2] ,Isoperimetric networks in the Euclidean plane, Studia Sci. Math. 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[Th] D’Arcy Wentworth Thompson, On Growth and Form: The Complete Revised Edition, Dover Publications, 2002. [W] Wacharin Wichiramala, The Planar Triple Bubble Problem, Ph. D. Thesis, University of Illinois, Urbana-Champaign, 2002. Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Ciencias, Universidad de Granada, E-18071 Granada (Espa˜ na) E-mail address:antoni[email protected] Departamento de Geometr´ ıa y Topolog´ ıa, Facultad de Ciencias, Universidad de Granada, E-18071 Granada (Espa˜ na) E-mail address:[email protected]