EXISTENCE OF LEAST-ENERGY CONFIGURATIONS OF IMMISCIBLE FLUIDS Brian White

October 7, 1994 revised April 3, 1995 Abstract. This paper shows how to model equilibrium congurations of immiscible uids using Fleming's at chains with coecients in a group. Existence is proved and regularity is discussed briey.

1. Introduction

Consider a container lled with N incompressible immiscible uids. It is convenient to regard the material that the container is made of as uid 0. Associated to each pair of uids i and j there is a surface energy density eij which is positive if i 6= j . Of course eij  eji and eii  0. If we ignore gravity (which we will do until x10), the total energy of a con guration of the uids is: X 0

i
eij  Aij

where Aij is the area of the interface between uid i and uid j . The uids will tend to arrange themselves so as to minimize the energy of the con guration that they form. That is, the uids try to solve the following mathematical problem. Given a bounded open set   R3 with piecewise smooth boundary and given positive numbers V1 V2 : : :  VN such that X Vi = volume()

nd a partition of  into disjoint sets Ui (1  i  N ) that minimizes X 0

i
eij  area(@Ui \ @Uj )

(where U0 = R3 n ) subject to the constraints volume(Ui ) = Vi

(1  i  N )

1991 Mathematics Subject Classication. Primary 49Q20 secondary 53A10. Key words and phrases. Immiscible uids, minimal surfaces, soap bubbles, at chains. The author was partially funded by NSF grant DMS-9207704. Typeset by AMS-TEX 1

Here of course  is region inside the container, and Ui is the region occupied by the uid

i.

Existence and almost everywhere regularity of a minimizing con guration was proved by Almgren Alm1,VI.2] in 1976 assuming a rather strong hypothesis on the coecients eij . This paper presents a simpler existence proof. The hypothesis on the coecients eij is weaker and is physically reasonable. Indeed, the hypothesis is necessary in order for a minimum to exist. The approach used here to model immiscible uids is also useful when analyzing regularity. The regularity theorems are discussed brie y in x11 and will proved in another paper. 2. The Hypothesis on the Coefficients

The hypothesis on the coecients we will use is the following triangle inequality:

eik  eij + ejk

(*)

for j 6= 0.

In practice, this poses no real restriction for the following reason. Suppose for example we are dealing with three uids such that e13 > e12 + e23 , and consider a con guration  = U1  U2  U3 Note that if U1 and U3 have an interface, then we can reduce the energy of the con guration by introducing a very thin layer of uid 2 between uids 1 and 3. In particular, the energy is reduced from X

eij  Aij

to (arbitrarily close to)

X

eij  Aij where e13 = e12 + e23 and eij = eij for the other ij 's. In general, given coecients eij > 0 (\actual energy densities") with eij  eji , we de ne new coecients eij (\eective energy densities") as follows. Consider a complete graph with N + 1 vertices. Let the length of the edge from vertex i to vertex j be eij . Then we let eij be the length of the shortest path from vertex i to vertex j that does not pass

through vertex 0. (Here 0 is excluded because one is not allowed to introduce a layer of the container between two uids.) Then of course the eij 's do satisfy the triangle inequality (*). Furthermore, the in mum of X P

eij  Aij

equals the in mum of eij  Aij . This is because (i) the e -energy of any con guration is less than or equal to its e-energy, and (ii) given any con guration with e -energy E , we can modify it (by introducing suitable thin layers as above) to get a con guration with e-energy as close as we like to E . Abstractly, when the eij 's do not satisfy the triangle inequality, the e-energy functional is not lower semicontinuous. The e -energy is the largest lower semicontinuous functional that is less than or equal to the e-energy. 2

3. Flat Chains with Coefficients in a Group

The proof uses at chains with coecients in a group G as introduced by Fleming F] in 1966, so we begin with a brief explanation of what they are. Let G be an abelian group with a norm j  j, i.e., with a function jj : G ! R such that (1) jgj 0 with equality if and only if g = 0, and (2) jg + hj  jgj + jhj The norm makes G into a metric space by setting dist(g h) = jg ; hj. We will assume that the metric space is complete and separable. (For this paper it suces to consider nite groups.) Fix an ambient space Rn (of course n = 3 is the most physical case) and a compact convex subset K  RnP . For each integer k 0, consider the abelian group of all formal

nite sums of the form i giPi , where each gi 2 G and each Pi is a k-dimensional oriented compact convex polyhedron in K . Next form the quotient group obtained by identifying gP with ;gP~ whenever P and P~ coincide but have opposite orientations. Also, identify gP and gP1 + gP2 whenever P can be subdivided into P1 and P2 . The resulting abelian group Pk (K  G) is called the group of polyhedral k-chains in K with coecients in G. De ne a boundary homomorphism @ : Pk ! Pk;1 in the obvious way. P Note that any polyhedral k-chain T can be written as a linear combination i giPi ] of non-overlapping polyhedra (i.e., polyhedra with disjoint interiors). The weighted area or mass of the chain is then de ned to be X M(T ) = jgij area(Pi ) The boundary operator is not continuous with respect to the mass norm, so one introduces a weaker norm, the at norm: ;  F (T ) = inf M( T ; @Q ) + M( Q ) Q

where the in mum is over polyhedral (k +1)-chains Q. (In Fleming's paper F (T ) is written W (T ).) The at norm makes Pk (K  G) into a metric space. The completion of this metric space is denoted Fk (K  G), and its elements are called at k-chains in K with coecients in G. By uniform continuity, functionals such as the at norm and operations such as addition and boundary extend in a unique way from polyhedral chains to at chains. The mass norm on Pk (K G) extends in a natural way to a lower-semicontuous functional (also called mass) on Fk (K G). Suppose that every closed bounded subset of G is compact. A fundamental compactness theorem F,7.5] for at chains asserts that, given any sequence Ti 2 Fk (K  G) with M(Ti ) and M(@Ti ) uniformly bounded, there is an F -convergent subsequence. (This compactness theorem is a trivial consequence of the deformation theorem for polyhedral chains F,7.3]. The fact that, for certain groups, at chains of nite mass are recti able is much deeperF,10.1].) 3

4. Flat n-Chains in Rn Note that the polyhedral n-chains in K  Rn can be identi ed with the set of piecewise

constant functions

g : Rn ! G

that vanish outside of K . Here two functions that dier only on a set of measure 0 are regarded as the same. \Piecewise constant" means locally constant except along a nite collection of hyperplanes. The identi cation is as follows. Any T 2 F k (K  G) can be written as X T = giPi ] where the Pi 's are non-overlapping and inherit their orientations from Rn. We can then associate to T the function

g : Rn ! G g(x) = gi if x is in the interior of Pi g(x) = 0 if x is not in the interior of any Pi We can then also denote T by K ]xg or Rn]xg. Note that the mass norm of T is equal to the L1 norm of g(). Also, since there are no nonzero (n + 1)-chains in Rn, we see from the de nition of F that Z F (T ) = M(T ) = jg(x)j dx Consequently the F -completion of the polyhedral chains (i.e., the space of at n-chains) is isomorphic to the L1 -completion of the piecewise constant functions. That is, we have an isomorphism g ! K ]xg from L1(K  G) to Fn(K  G), and M(K ]xg) = F (K ]xg) =

Z

jg j

5. The Existence Proof

First suppose that the triangle inequality (*)

eik  eij + ejk

holds for all i, j , and k (including j = 0). In the next section the general case (in which (*) is not assumed for j = 0) will be reduced to this one. We let G be the free Z2 -module with N generators f1  f2 : : :  fN (one for each uid). We wish to de ne a norm on this group such that

jfi ; fj j = eij 4

and

jfij = ei0

Of course the triangle inequality (*) must hold if such a norm exists. In fact this condition is also sucient (x7). Now suppose we have a partition of  into N measurable sets U1  : : :  UN . Associated with this partition we have the at n-chain

T = Rn]xg where



(x 2 Ui ) 0 (x 2= ) (The K of x3 can be any compact convex set containing .) Note that if the Ui 's have piecewise smooth boundaries, then M(@T ) is precisely the energy of the partition. (More generally this holds whenever the Ui's are cacciopoli sets, i.e., whenever the at chain has

nite boundary mass.) Conversely, given any at n-chain T , we can represent T as

g(x) = fi

T = Rn]xg where g 2 L1 ( G). If g vanishes outside , takes only the values f1  : : :  fN in , and if

Ln fx : g(x) = fi g = Vi then we say that T is an admissible at chain. Of course an admissible at chain determines a partition of  (with the prescribed volumes) by setting Ui = fx : g(x) = fi g. Theorem 1. There is a least energy partition of . Equivalently, there is at n-chain T that minimizes M(@T ) among all admissible at chains. Proof. Let E be the in mum of M(@T ) among all admissible at chains T . Let Tk be a sequence of admissible at chains with lim M(@Tk ) = E k Note that the Tk 's all have the same ( nite) mass, namely M(Tk ) =

n X i=1

jfi j  Vi

Also the M(@Tk ) are bounded, and the Tk 's are all supported in a common compact set K. 5

Thus by the compactness theorem (x3) for at chains, there is a subsequence Tk(`) that converges in the at topology to a limit T :

F (Tk(`) ; T ) ! 0 Recall that the at norm of an n-chain in Rn is the same as the L1 norm of the multiplicity function. That is, if we write

Tk = Rn]xgk then

F (Tk(`) ; T ) =

T = Rn]xg

Z 

jgk(`)(x) ; g(x)j dx

From this we see immediately that T is admissible. Finally, since Tk(`) ! T , it follows that @Tk(`) ! @T and (by lower semicontinuity of mass) that M(@T )  lim M(@Tk(`)) = E  6. The Weaker Triangle Inequality

In the preceding section, we assumed that the triangle inequality (*)

eik  eij + ejk

held for all i, j , and k. Now suppose merely that (*) holds for j 6= 0. De ne new coecients e0ij by:

e0ij = eij e0i0 = ei0 + C

(i 6= 0 j 6= 0) (i 6= 0)

where C is a large constant. Note that for any partition X of  into the N uids,

e0 -energy(X ) = e-energy(X ) + CA where A is the surface area of @ . Since these two functionals dier by a constant, minimizing one is equivalent to minimizing the other. If we choose C large enough, say

C max e  i6=j ij then the e0ij s will satisfy the triangle inequality (*) for all i, j , and k, and hence we get existence of minimizers by the proof in the preceding section. 6

7. Existence of a Suitable Group Norm

Here we complete the proof of theorem 1 by showing Lemma. If eik  eij + ejk for 0  i j k  N , then G has a norm j  j such that 

eij = jfi ; fj j if 0 < i j jfi j if i = 0 and j 6= 0. Proof. We de ne jgj to be the minimum of

ei1 j1 + ei2 j2 + : : : eik jk

(1)

among all i1 j1 : : :  ik jk such that (2)

g = (fi1 ; fj1 ) + : : : (fik ; fjk )

where we set f0 = 0. Of course we could just as well write (3)

g = (fi1 + fj1 ) + : : : (fik + fjk )

since all elements of G have order 2. Clearly jj is a group norm. Now let us show, for example, that if g = f1, then jgj = e01. (The other cases are essentially the same.) Among all ways of expressing g as a sum as in (3), consider those which minimize (1). Among those ways, choose one for which k is a minimum. Note that if k > 1, then some generator, say f2, must occur more than once in the right hand side of (3). Hence the right hand side must include a term of the form (fi + f2) and another term of the form

(fj + f2) Now if we replace these two terms by fi + fj , the eect on (1) is to replace ei2 + ej2 by eij . That is, we have decreased the k in (3) without increasing (1), which contradicts the choice of k. Thus k = 1, which means that the decomposition (3) must be

g = f1 + f0 so that



jgj = e10 7

8. Other Groups

For the group G, instead of using the direct sum of N copies of Z2, one could use the direct sum of N copies of Zp or of Z or of R (or even a direct sum of N dierent groups). It does not seem to make much dierence, except that for calibration arguments (as in LM]) it is more natural to use Z's or R's. Of course the proof of the lemma in x7 needs to be modi ed slightly for these other groups. 9. Soap Bubble Clusters

A cluster of N soap bubbles in a container is equivalent to N immiscible uids for which the eij 's are all equal. In this case we can also let the group be ZN +1 (the integers mod N ), and the norm be  if g 6= 0 jgj = 10 if g = 0 10. Bulk Energies

So far gravity has been ignored. To model incompressible uids in the presence of gravity, one minimizes: X

i
eij  Aij +

XZ

x2Ui

i

a  i  h(x) dx

where a is the acceleration due to gravity, i is the density of uid i, and h(x) = xn is the height of the point x. In general, one can try to minimize an energy of the form (y)

X

i
eij  Aij +

Z

U1

h1 : : : 

Z

UN

hN



For incompressible uids, the volumes are prescribed, but for compressible uids they are not. (In the latter case, the expression for will typically include terms for the energies required to compress the various uids.) The proof of theorem 1 shows that minimizers to (y) exist (whether or not the volumes are prescribed) provided the hi are integrable (i.e., in L1 ()) and : RN ! R is lower-semicontinuous. 11. Regularity

The only published regularity theorem that I am aware of is Almgren's. He proves almost everywhere regularity of the interfaces under his rather strong hypothesis on the coecients. If one combines that result with a general strati cation of singularities theorem one sees that the singular set must have Hausdor dimension at most n ; 2. (The strati cation theorem was originally proved by Almgren Alm2,2.27] for mass-minimizing integral currents see W] for a formulation that includes other equilibrium and non-equilibrium plateau-type problems.) This n ; 2 is sharp: consider the set along which three uids meet. 8

For general coecients eij satisfying the triangle inequality, Allard's regularity theorem (All] or S, ch. 5]) implies that the regular points form an open dense subset of the interface set. (One associates to the interfaces a varifold with bounded rst variation.) However, almost everywhere regularity is not known. A sequel to this paper will prove almost everywhere regularity (indeed that the singular set has dimension at most n ; 2) under a slightly stronger hypothesis than the triangle inequality. In particular, if the coecients eij satisfy the strict triangle inequality (that is, if one has strict inequality in (*) of x2 whenever j is distint from i, k and 0), then the singular set has dimension at most n ; 2. Indeed, for the strongest kind of Allard-type regularity theorem to hold, this strong triangle inequality is both necessary and sucient. More precisely, consider the following property of a pair (i k) (i 6= k): Property P. Whenever a minimizing conguration is weakly close, in a small ball B(x r), to a congution consisting of uid i and uid k separated by a hyperplane H through x, then in a smaller ball B(x r=2) the conguration consists exactly of uid i and uid k separated by a smooth hypersurface (namely a graph over H ). We have the following theorem: Theorem 2. Propery P is true for the pair (i k) if and only if

eik < eij + ejk whenever j is distinct from i, k, and 0. Let us call a pair (i k) indecomposable if it satis es the condition given in theorem 2. Note that for any pair (i k) with i 6= k, there exists a sequence (j0 j1 : : :  jp) such that (1) j0 = i,Pjpp = k, and j` 6= 0 for 0 < ` < p. (2) eik = `=1 ej`;1 j` (3) For ` = 1 : : :  p, the pair (j`;1 j`) is indecomposable and j` 6= j`;1. Such a sequence is called a complete decomposition of the pair (i k). Under a weaker hypothesis than indecomposability, a weaker regularity property than property P holds: Property Q. Whenever a minimizing conguration is weakly close, in a small ball B(x r), to a congution consisting of uid i and uid k separated by a hyperplane H through x, then in a smaller ball B(x r=2) the interface set consists of one or more smooth hypersurfaces (namely graphs over H .) Theorem 3. Property Q holds for the pair (i k) if and only if (i k) has a unique complete decomposition. If the hypothesis of theorem 3 holds for all pairs (i k), then this implies that the singular set of the interface set has dimension at most n ; 2. Although the statement is more complicated, theorem 3 is a signi cant improvement over theorem 2 because the hypothesis of theorem 3 is generic in a sense that the hypothesis of theorem 2 is not. Recall from section x2 how, given any set of actual energy densities eij , we obtain eective energy densities eij that satisfy the triangle inequality. Note that 9

failure of the eij 's to satisfy the triangle inequality is an open condition, and that such failure implies failure of the eij 's to satisfy the strict triangle inequality. On the other hand, for an open dense set of values of the eij , the eij 's will satisfy the hypothesis of theorem 3. Finally we remark that theorems 2 and 3 have some analogs for general mass minimizing (or \almost mass minimizing") at chains over coecient groups (not just at chains associated to uid con gurations.) They will be explored in another paper. References All] W. K. Allard, On the rst variation of a varifold, Annals of Math. 95 (1972), 417{491. Alm1] F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs Amer. Math. Soc. 165 (1976). Alm2] , Q-valued functions minimizing dirichlet's integral and the regularity of area minimizing rectiable currents up to codimension two, preprint. F] W. Fleming, Flat chains over a coecient group, Trans. Amer. Math. Soc. 121 (1966), 160{186. LM] G. Lawlor and F. Morgan, Paired calibrations applied to soap lms, immiscible uids, and surfaces or networks minimizing other norms, Pacic J. Math. 166 (1994), 55{83.. S] L. Simon, Lectures on geometric measure theory, Australian National Univ., Canberra, 1983. W] B. White, Stratication of minimal surfaces, mean-curvature ows, and harmonic maps, preprint. Mathematics Department, Stanford University, Stanford CA 94305

E-mail address : [email protected]

10

existence of least-energy configurations of immiscible ...

M(Ti) and M(@Ti) uniformly bounded, there is an F-convergent subsequence. ..... Existence and regularity almost everywhere of solutions to elliptic variational.

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