Least-perimeter partitions of the disk

Proceedings of XIII Fall Workshop on Geometry and Physics Murcia, September 20–22, 2004 Publ. de la RSME, Vol. 9 (2005), 141–148 Least-perimeter partitions of the disk Antonio Ca˜ nete [email protected] Departamento de Geometr´ıa y Topolog´ıa Universidad de Granada E-18071 Granada (Espa˜na) Abstract. In this work we study the isoperimetric problem of partitioning a planar disk into nregions of prescribed areas using the least-possible perimeter. We obtain the regularity conditions that must be satisfied by the solutions, and solve completely the problem in the cases of two and three regions. Keywords: Isoperimetric partition, stability. 2000 Mathematics Subject Classification: 49Q10, 51M25. 1. Introduction In the last years, the study of isoperimetric problems has been of great interest not only for its own nature but also for its relation with multitude of physical phenomena. Among them, the ones relating to isoperimetric partitions can model several natural situations. In this work we treat the following partitioning problem: consider a planar disk D,andnpositive numbers a1,...,a nwhose sum is the total area of the disk. Then we want to find the way of dividing the disk into nregions Ri, each one of area ai, with the least possible perimeter. This question explains properly many situations in Nature: the first steps in the process of division of a cell, the shape of the interfaces separating different fluids in a round dish, and other ones (see [4, Ch. VII and VIII]). For this problem, the main trouble is that regions may have several components, since we do not assume them to be connected. Therefore there are many ways of dividing the disk into nregions of the given areas. 142 Least-perimeter partitions of the disk From [3] we obtain the existence of a solution, for any number of regions we consider: Theorem 1. (Existence Theorem) Given a planar disk Dand npositive numbers a1,...,a nsuch that n i=1 ai=area (D), there exists a least-perimeter way of dividing the disk into nregions of areas a1,...,a n, consisting of smooth curves meeting in threes in the interior of the disk, and meeting ∂D only one curve at each time. 2. Regularity Conditions Let D⊂R2be a closed disk, centered at the origin. Along this work, a graph Cwill consist of a finite number of vertices and edges in Dsuch that at every interior vertex (a vertex in the interior of the disk) three edges meet, and at every vertex in ∂D, only one edge arrives. Note that, in view of Theorem 1, this is a natural definition. We shall assume that a graph Cdecomposes the open disk into nregions Ri, possibly nonconnected as commented before, and we shall denote by Cij ⊂ Cthe curve separating two adjacent regions Riand Rj(this curve may not be connected), by Nij the normal vector to Cij pointing into Ri,andbyhij the geodesic curvature of Cij with respect to Nij (as we are working in the plane, the geodesic curvature coincides with the usual curvature of a curve). We will call minimizing graph to the least-perimeter graph dividing the disk into nregions of the given areas. Let ϕt:C→Dbe a smooth variation of a graph C, for t small, such that ϕt(C∩∂D)⊂∂D. Denoting the associated vector field by X=dϕt/dt|t=0, with normal components uij =X·Nij on Cij, then the derivative of the area Aienclosed by a region Riat t=0isequalto dAi dt t=0 =− j∈I(i)Cij uij, where I(i)={j=i;Rjtouches Ri}. Proposition 2. (First variation of length) Given a graph Cand a smooth variation ϕt:C→D, the first derivative of the length functional of ϕt(C)at t=0is equal to dL dt t=0 =−1 2 i∈{1,...,n} j∈I(i) 1Cij hijuij + p∈∂Cij X(p)·νij(p)2,(1) where νij(p)is the inner conormal to Cij in p. Antonio Ca˜ nete 143 A graph will be said stationary if dL dt |t=0 = 0, for any variation preserving the areas. Observe that stationary graphs are critical points for the length functional when the areas Aiare fixed. Since we want to minimize such a functional, it follows that a minimizing graph must be stationary. From Proposition 2 we obtain the regularity conditions that stationary graphs (and then minimizing graphs) must verify: Theorem 3. (Regularity Conditions) Given a stationary graph C, the following regularity conditions must be satisfied: i) The curvature hij is constant on Cij, and accordingly the edges will be circular arcs or line segments. ii) The edges of Cmeet in threes at 120-degree angles in interior vertices. iii) Three edges Cij,Cjk,Cki meeting in an interior vertex satisfy hij +hjk +hki =0. iv) The edges of Cmeet ∂D orthogonally. Proof. By considering appropriate area-preserving variations in (1), the conditions above easily follow. 3. Minimizing graph for two regions The problem for two regions can now be solved from Theorem 3. In this case no triple interior vertices can appear in the minimizing graph (any interior vertex would be surrounded by three different regions), and so any edge will be a circular arc or a line segment meeting ∂D orthogonally. Let us assume that the minimizing graph has more than one edge. Then, by rotating one of them about the origin until touching another one, we will obtain a new minimizing graph (since the perimeter and the areas enclosed are preserved) with a non-allowed vertex in ∂D, which is contradictory. Hence we have the following Theorem 4. Let C⊂Dbe a minimizing graph for two given areas. Then C consists of a circular arc or a line segment meeting orthogonally ∂D. 144 Least-perimeter partitions of the disk Figure 1: The least-perimeter partition of the disk into two given areas 4. Minimizing graph for three regions Now we treat the problem for three regions. Straightforward calculations give the next proposition. Proposition 5. (Second variation of length) For a stationary graph Cand a variation {ϕt}that preserves the areas, the second derivative of the length functional at t=0is given by −1 2 i=1,...,n j∈I(i)1Cij (u ij +h2 ijuij)uij + p∈∂Cij p∈int(D)−qiju2 ij +uij ∂uij ∂νij (p)(2) + p∈∂Cij p∈∂D u2 ij +uij ∂uij ∂νij (p)2, where qij(p)=(hki +hkj)(p)/√3,andRkis the third region touching the vertex p. Let us introduce an important concept in this work: for a region Ri,it is possible to define its pressure pias a real number such that, for any edge Cij ⊂C, hij =pi−pj. Then, for area-preserving variations by stationary graphs (that is, at each instant of the deformation we obtain stationary graphs, satisfying the conditions of Theorem 3), the second variation of length (2), expressed in terms of the pressures, turns d2 L dt2= α dpα dt dAα dt ,(3) where αlabels the components of the graph (recall that regions may have various components). Antonio Ca˜ nete 145 We will say that a stationary graph Cis stable if d2L dt2t=0 ⩾0, for any area-preserving variation. This means that stable graphs are second order local minima for the length when the areas Aiare preserved. Then it is clear that a minimizing graph must be stable. Ahexagonal component of a region is a component bounded by six edges. Fix R1as the region of largest pressure. From Proposition 5 we get the following result: Proposition 6. Consider a stable graph dividing the disk into n regions. Then, R1has at most n−1nonhexagonal components. In the case n= 3, this result allows us to discuss the possible configurations for a minimizing graph, once we have checked, using Gauss-Bonnet Theorem, that hexagonal components cannot occur: Proposition 7. Let C⊂Dbe a minimizing graph dividing Dinto three regions. Then Cis one of the graphs in Figure 2. Configuration 2(10) will be called standard graph. Now we will show how to discard the non-standard possibilities, to conclude that the minimizing graph for three regions is the standard graph. Let us check that configurations 2(1) and 2(2) are unstable, and hence not minimizing. The motive is that both of them have a region with two triangles (components of three edges) touching ∂D. Proposition 8. Given a stationary graph with a triangle Ωtouching ∂D, there exists a variation by stationary graphs such that (i) increases the area of Ω, (ii) decreases the pressure of Ω, keeping the other pressures unchanged, and (iii) leaves invariant the edges of the graph not placed in ∂Ω. Remark. For this variation, we have that dp dt dA dt <0inΩ. Proposition 9. Any stationary graph with a region with two triangles touching the boundary of the disk is unstable. Sketch of the proof. First, consider in each triangle Ωithe variation of Proposition 8. By combining both variations, we can construct another variation by stationary graphs too, preserving the areas. Finally, by using Equation (3) to compute the second variation of length we have d2 L dt2t=0 = α dpα dt dAα dt =dp1 dt dA1 dt +dp2 dt dA2 dt <0, 146 Least-perimeter partitions of the disk β αα 3 (1) α, β ∈{1,2} 3 2 11 2 2 (2) α ββ 3 3 (3) α, β ∈{1,2} 1 1 2 2 3 3 (4) 1 1 α α α ββ (5) α, β ∈{2,3} 23 23 32 1 1 (6) β 1 αα 1 (7) α, β ∈{2,3} 3 3 211 2 (8) 1 2 3 2 3 21 (9) 3 2 1 (10) Figure 2: The ten possible configurations for a minimizing graph Antonio Ca˜ nete 147 since the only pressures that change along the deformation are the ones of the triangles Ω1and Ω2. Therefore, Proposition 9 yields instability of configurations 2(1) and 2(2). A function u:C=3i,j Cij →Ris said a Jacobi function if the restrictions to Cij satisfy u ij +h2 ij uij =0. The following result will show that configurations 2(3) to 2(7) of Figure 2 are unstable: Proposition 10. ([2]) Let Cbe a stationary graph separating the disk into three regions. If there exists a Jacobi function defined on Cwith at least four nodal domains, then Cis unstable. In this case, the normal components of the rotations vector field about the origin constitute a suitable Jacobi function (recall that a nodal domain is a domain in Cwhere that function does not vanish). Then, because of some symmetries we find in these configurations, we can apply Proposition 10 and obtain the instability of all of them. Finally let us see that configuration 2(8) cannot be a minimizing graph. It is possible to construct a new configuration with the same perimeter and enclosing the same areas by this geometric transformation: z 2 ’ Ω1 Ω2 Ω1 ’T v v’ Ω1 ’Ω2Ω2 ’ 1 Ω−T T Ω Figure 3: Geometric transformation creating two non-allowed vertices As the new configuration has two non-allowed vertices, it cannot be minimizing, and so, neither configuration 2(8). The same argument can be applied for configuration 2(9). Then, the nine non-standard possibilities have been discarded, so we obtain the following theorem: Theorem 11. ([1]) Let C⊂Dbe a minimizing graph for three given areas. Then Cis a standard graph. Remark. For two and three regions, the solutions are unique up to rigid motions of the disk. 148 Least-perimeter partitions of the disk Figure 4: The least-perimeter partition of the disk into three given areas Acknowledgments This work has been partially supported by MCyT-Feder research project MTM2004-01387. References [1] A. Ca˜ nete and M. Ritor´ e. Least-perimeter partitions of the disk into three regions of given areas, Indiana Univ. Math. J. 53 (2004), 883-904. [2] M. Hutchings, F. Morgan, M. Ritor´ e and A. Ros. Proof of the double bubble conjecture, Ann. of Math. 155 (2002), 459–489. [3] F. Morgan. Soap bubbles in R2andinsurfaces,Pacific J. Math. 165 (1994), 347–361. [4] D. Wentworth Thompson.On Growth and Form: The Complete Revised Edition, Dover Publications, 2002.