Infiltrations in immiscible fluids systems

In this paper we prove a certain regularity property of configurations of immiscible fluids, filling a bounded container Ω and locally minimizing the interface energy ∑i<jcij‖Sij‖, where Sij represents the interface between fluid i and fluid j, ‖·‖ stands for area or more general area-type functional, and cij is a positive coefficient. More precisely, we show that, under strict triangularity of the cij, no infiltrations of other fluids are allowed between two main ones. A remarkable consequence of this fact is the almost-everywhere regularity of the interfaces. Our analysis is performed in general dimension n ≥ 2 and with volume constraints on fluids.


Introduction
When a small quantity of oil is added to a glass of water, one sees at rst some small oil drops ®oating on the water surface, that progressively tend to aggregate into bigger drops. This is a simple example of immiscible ®uid system that tries to evolve towards a stable equilibrium con guration. In general, one could consider mixtures of m ®uids and ask whether some equilibrium con gurations, or even con gurations attaining the minimum of the total free energy, exist and (hopefully) have some regularity properties.
From a theoretical point of view, this corresponds to analysing models of ®uid systems, where the energies are essentially of interface type, i.e. depending upon the interfaces separating the various ®uids in the mixture. The problem can thus be viewed as follows: given a container « , a set of°uids ff 1 ; : : : ; f m g and prescribed volumes v i > 0 so that P v i equals the volume of the container « , nd an absolute minimizer of the interface energy subject to volume constraints, i.e. a partition of « into regions F i , i = 1; : : : ; m, such that the volume of F i equals v i and the energy of the interface set fS ij = @F i \ @F j \ « g is minimum.
Probably the easiest formulation of an energy model for immiscible ®uids is where c ij > 0 for all i < j. This represents an isotropic surface energy, where each interface S ij has a cost equal to c ij times its (Euclidean) area, and thus depends upon the pair (f i ; f j ) of ®uids touching along it. Of course, one could take into account other contributions to the total free energy of the system, such as gravity or other external forces; moreover, the energy could also depend upon surface orientation, as happens for crystals and polycrystals (see [6]). In this paper we prove a regularity property for minimizing con gurations of immiscible ®uids, under strict triangularity of the interface coe¯cients c ij (see theorem 3.1). It has been shown by Ambrosio and Braides in [2], and independently by White in [22], that the triangle inequality c ij 6 c ik + c kj is necessary and su¯cient for the lowersemicontinuity of the energy functional E (see also the paper by Morgan [18]). White announces in [22] some regularity results obtained under strict triangularity of the c ij , in particular, an elimination property saying that, if any locally minimizing con guration is weakly close, in a ball B r , to a con guration with only two ®uids separated by a ®at interface, then in a smaller ball it consists of exactly those two ®uids. This means that no in¯ltration is permitted between two ®uids that are shaped (at least locally) in an almost ®at con guration.
Here we prove a stronger elimination-type result (theorem 3.1), under the same hypothesis as in White's paper. This result points out that the ®atness of the con guration does not really matter, the absence of in ltrations between two ®uids being a pure consequence of energy minimization. The proof uses (1) a decay lemma (lemma 3.2) that incorporates the elimination result (the original technique was developed by Tamanini and Congedo in [21]), (2) a technical balancing lemma (lemma 3.3), where the main estimate needed to make (1) work is deduced from thè cooperation' of two weak energy estimates, and nally, (3) a representation of the immiscible ®uid con guration as a network (see [18]), on which we apply classical graph-theory results (`max®ow-mincut' and ®ow decomposition) to obtain one of the two weak energy estimates (the most hidden one). Thanks to the elimination property, one can then apply classical regularity results (see [15]), ensuring that the interface set is made of smooth surfaces with constant mean curvature, plus a singular set of zero (n 1)-dimensional Hausdor¬ measure.

Basic de¯nitions
By R n we denote the real Euclidean space of dimension n, and always take n > 2. B r (x) denotes the open Euclidean n-ball centred at x 2 R n with radius r > 0; B r is used in place of B r (0). We denote by ! n the volume (Lebesgue measure L n ) of the unit ball of R n . Then the volume of B r is jB r j = ! n r n (the notation jAj is preferred to L n (A)). We also denote by H n 1 the (n 1)-dimensional Hausdor¬ measure in R n .
Given two sets A and B, we de ne their symmetric di® erence by If « is an open set, we say that A is relatively compact in « (and write A b « ) if the closure of A is a compact subset of « ; we say that A is a compact variation of B in « if A ¢ B is relatively compact in « . Given a Borel set E » R n and ¬ 2 [0; 1], we de ne the set of points of density ¬ of E as follows: Clearly, E(¬ ) » @E for all ¬ 2 (0; 1). Moreover, we have that E(1) is the Lebesgue set of E, hence jE(1) ¢ Ej = 0 (see [20]). Another remarkable density set is E( 1 2 ), as will be better shown in the following.
We recall the notion of a Caccioppoli set, i.e. a set of (locally) nite perimeter, and of a Caccioppoli partition. For E; A » R n , with A open and E Borel, the perimeter of E in A is de ned as follows: It can be shown that P (E; ¢) extends to a Borel measure, by setting for all Borel set B » R n . When A = R n , we use P (E) instead of P (E; R n ). We say that E is a set of locally nite perimeter (or a Caccioppoli set) if P (E; A) < 1 for every bounded open set A » R n . We recall some properties of the perimeter of E in B.
For these and additional properties, we refer to [11,14]. Given a Borel set E, its characteristic function is denoted by À E (x), taking the value 1 if x 2 E and 0 otherwise. When E is a Caccioppoli set, we consider the traces (from outside and from inside) of E on @B r (see [14] for general de nitions and properties), denoted by À + E and À E , respectively. These are functions of L 1 (@B r ) and coincide for L 1 -almost all r > 0, in which case they are simply denoted by À E . Moreover, we have for all r > 0. We quote the well-known isoperimetric inequality. Let E » R n be of nite perimeter in R n , then min(jEj; jR n n Ej) (n 1)=n 6 c n P (E); where c n = (n! 1=n n ) 1 (for the proof, see [9] or [17]). Given a Caccioppoli set E, one can consider the so-called reduced boundary @ ¤ E of E, which is a subset of E( 1 2 ), where a certain measure-theoretical unit normal vector¸E exists (for the precise de nition, see [14]).
The following properties are of crucial importance (for the proof, see [10,14]).
of Borel subsets of « , such that (i) F i has locally nite perimeter in « ; (ii) jF i \ F j j = 0 whenever i 6 = j; and Proof. For the proof, see [8].
Summing up, according to theorem 2.1, a Caccioppoli partition of « yields a decomposition of « into`solid components' F i (1)\« , along with a set of`interfaces' , and a H n 1 -negligible set containing, for example,`multiple points' where three or more components meet. Moreover, we are allowed to rede ne the interface S ij to be @ ¤ F i \ @ ¤ F j \ « , and, owing to property (a) above, this becoming our default setting from now on. Finally, to simplify the notation, we shall write kSk instead of H n 1 (S).

Statement of the problem and preliminary lemmas
Let « » R n be a non-empty bounded open set and let F = (F i ) m+ 2 i= 1 be a Caccioppoli partition of « with (m + 2) components of nite perimeter in « (here, m > 1). Hence (recall x 2) the F i are Borel subsets of « , such that (i) F i has nite perimeter in « for all i; We shall refer to such F as a generic con¯guration of ®uids. We also prescribe the volume of each ®uid, that is, we x v 1 ; v 2 ; : : : We will be concerned about regularity properties of con gurations G minimizing the energy functional among all admissible con gurations F . Here, c ij > 0 represents the energy density associated with the interface S ij = @ ¤ F i \ @ ¤ F j \ « separating ®uids F i and F j . We also consider the localized energy where O is a generic open subset of « and S O ij = S ij \ O. In case O = B r , we write S r ij instead of S Br ij . Let us start by localizing within a xed ball B R » « and by considering a xed pair G 1 , G 2 of ®uids of a minimizing con guration. Our aim is to see if in¯ltrations can be excluded by imposing suitable hypotheses on the coe¯cients c ij . We would eventually like to prove the following result. If there exists a radius r < R such that G 1 and G 2 ll a su¯ciently high percentage of B r , then G 1 and G 2 completely ll 1 2 B r . This result can be viewed as a`two against m' version of an elimination-type theorem proved by Tamanini and Congedo in [21] (see also [16]). In that paper, the authors consider`general' Caccioppoli partitions (even with countably many components) minimizing the simple perimeter functional (c ij = 1 for all i 6 = j) plus higher-order volume-type terms. The technique of [21] fails when, as in our case, more general hypotheses on the c ij are assumed, and has to be integrated with additional results involving, among other things, some tools from graph theory.
A similar result, with extra assumptions on the minimizing con guration, has been announced by White in [22] for an immiscible ®uid energy like (3.1). More precisely, White considers the following property: if a minimizing con guration is weakly close, in a small ball B(x; r), to a con guration consisting of ®uid i and ®uid j separated by a hyperplane H through x, then, in a smaller ball B(x; 1 2 r), the con guration consists exactly of ®uid i and ®uid j separated by a smooth hypersurface. (ii) c ij = c ji .
Strict triangularity is simply the following c ij < c ih + c hj 8i; j; 8k 6 = i; j; or, equivalently, c ij 6 c ik + c kj¯8 i; j; 8k 6 = i; j; for a suitable constant¯> 0. Let us rst consider a simple example. Suppose that three ®uids f 1 , f 2 , f 3 ll some container « (we always assume absence of external forces and of surface tension with walls of « ) and that two of them (say, f 1 and f 2 ) meet along some ®at interface. If the tension coe¯cients c ij did not satisfy a strict triangle inequality and, in particular, if even c 12 > c 13 + c 23 , then it would be energetically advantageous to let a thin layer of ®uid f 3 ®ow between f 1 and f 2 , in such a way that these two ®uids do not touch anymore. Clearly, this corresponds to a loss of lower semicontinuity of the energy functional, which could be macroscopically observed as a`relaxation' of the energy densities c ij (i.e. the ®uid system behaves as if c 12 = c 13 + c 23 ). Absence of in ltrations appears to be quite related to lower semicontinuity of the energy functional. Indeed, it is shown in [2], as well as in [22], that the (simple) triangle inequality is necessary and su¯cient for the lowersemicontinuity of the ®uid energy (3.1) (for a more general and detailed study of the semicontinuity of ®uid-type energies, see [2,3,18]). This justi es, in some sense, the choice of (ST1) as a sharp condition to prevent in ltrations between pairs of ®uids.
White also announces in [22] results regarding both the regularity of the interfaces S ij and the estimate of the dimension of the`singular set' of a minimizing con guration of ®uids. Some other problems remain unsolved, like the validity of a lower volume-density estimate for each ®uid at every boundary point (see [7]).
Here we state our main result for a pair (G 1 ; G 2 ) of ®uids in a locally minimizing con guration (of course, any other pair will do the same); its proof will need some intermediate lemmas and, as anticipated before, the use of graph-theory techniques.
The need to perform compact variations of a given con guration of ®uids (i.e. variations that are localized in small balls and preserve boundary data) suggests us to de ne the concept of cut. A cut K (relative to the pair (F 1 ; F 2 )) is a bipartition (K 1 ; K 2 ) of the index set f1; : : : ; m + 2g such that i 2 K i , i = 1; 2. It gives a way of`re lling' « by using the rst two ®uids (F 1 ; F 2 ). Precisely, starting with some con guration F , some cut K and a xed radius 0 < r < R, we can de ne a new con guration F r ² F K ;r as follows: Clearly, F r is a compact variation of F inside B R . At this point, we introduce some more notation. By ¢E we mean the energy change E(F r ; B R ) E(F ; B R ) and by ¢ r E we mean E(F r ; B r ) E(F ; B r ). Finally, we de ne A K r (improperly called the area of K inside B r ) as follows where, as usual, S r ht = @ ¤ F h \ @ ¤ F t \ B r . Let G be an admissible ®uid con guration which is`locally minimizing' inside a ball B R , that is, E(G ) 6 E(F ) for all compact variations F of G inside B R . Actually, we should be more careful when considering compact variations of a ®uid con guration, since most of such variations will not preserve volumes and will thus not be admissible anymore. However, thanks to an argument originally due to Almgren (see [1,theorem VI.2(3)] and [19, lemma 13.5]), any (small) volume change ¢v can be adjusted at a cost proportional to ¢v itself (hence, in nitesimal of higher order when compared with an area-type change). This fact is extremely important, since it allows us to virtually ignore the problem of adjusting volumes after a small change of the ®uid con guration.
The following decay lemma incorporates the main engine that gives rise to the elimination property (EP).
i= 1 is locally minimizing inside B R and that there exists a constant C > 0 such that, for almost all 0 < r < R, there is at least one cut K for which where V = S j>2 G j is the in¯ltration. Then G has the elimination property (EP), i.e. there exists a positive constant ² and a radius r 0 < R such that, if 0 < » < r 0 and jV \ B » j 6 ² » n , then jV \ B » =2 j = 0. Moreover, we can choose for almost all r 2 (0; R). Since, for almost all r and all i, j we have k@B r \ S ij k = 0, we can write for almost all r 2 (0; R). Fix now r 0 = R (had we taken into account the small cost due to possible volume adjustments, r 0 should have been chosen su¯ciently small and ² changed a little) and take » 2 (0; r 0 ) such that ¬ (» ) 6 ² » n . By contradiction, suppose that ¬ (» =2) > 0, so that, in particular, ¬ (r) > 0 for all » =2 < r < » . Given r 2 ( 1 2 » ; » ) for which (3.3) and (3.4) are veri ed, we obtain thanks to the minimality of G . The isoperimetric inequality (2.1) gives P (V; B r ) = P (V \ B r ) ¬ 0 (r) > n! 1=n n ¬ (n 1)=n (r) ¬ 0 (r): Thus, by (3.2) and (3.5), for almost all r. Hence, by integration between 1 2 » and » , we get that is, a contradiction.
The next lemma shows how to get (3.2) starting from the weaker estimate (3.6).
where A K r is the`area of the cut inside B r ' (see above). Then there exists a positive constant C = C(¯; C 0 ; C 1 ), with C 0 = min i<j c ij , C 1 = max i<j c ij , such that kS r 2j k´: On the other hand, it is not di¯cult to check that, by theorem 2.1, Therefore, from (3.7), we deduce At this point we use the following inequality (its proof is straightforward). Let ; C 0 ; C 1 be positive constants, then, for any A; P 2 R, By combining (3.9), (3.8) and (3.6), we conclude that as was to be proved.

Network representation
In this section we will use graphs to represent ®uid con gurations and employ typical graph-theory results to deal with region-merging procedures and to get estimates on the energy changes. First, we recall some basic de nitions and results about directed graphs and networks. A directed graph G is a nite set of vertices (or nodes) v i , i = 1; : : : ; n, that are connected by oriented arcs. The arc going from node v i to node v j is represented by the ordered pair (v i ; v j ), or e ij for short: in this case, v i is called the tail and v j the head of e ij . The set E of all arcs of G is, in general, a subset of G £ G. A graph G is said to be weighted if a certain non-negative coe¯cient p ij is associated with each arc e ij , representing its capacity. This last kind of graph is quite often called a network.
We are particularly interested in networks where each node has no connection (arc) to itself and where each pair of nodes v i 6 = v j is connected by both arcs e ij and e ji . Therefore, the arc set E coincides with G £ G n ¢ (where ¢ is the diagonal of G £ G), and hence our networks are completely connected, that is, each pair of di¬erent nodes has exactly two connecting arcs (with opposite orientation ); on the other hand, the capacities p ij are chosen to be symmetric (p ij = p ji ) and are allowed to be zero. then ¿ (v) = 0 for each v 6 = s; t and ¿ (s) = ¿ (t) > 0. We will also denote by kf k the`intensity' of the ®ow f , that is, kf k = ¿ (s).
Definition 4.2. Let s and t be, respectively, the source and the sink in a network (G; p). A bipartition K = (K 1 ; K 2 ) of G is called a`cut' if the source s and the sink t are contained, respectively, in K 1 and K 2 (compare with the de nition given in x 3). The`size' of the cut is then de ned as Now some results follow (for their proof, see [5]). 3. If f is a°ow between s and t, and K is a cut with respect to s and t, then kf k 6 ¼ (K ): Let us say that f is a maximum°ow if there is no other ®ow f 0 such that kf 0 k > kf k. Symmetrically, let us say that K is a minimum cut if there is no other cut K 0 such that ¼ (K 0 ) < ¼ (K ). We recall that an algorithm has been developed by Ford and Fulkerson (see [12]) to nd a maximum ®ow in a network. This algorithm is essentially contained in the proof of the following fundamental result.
Theorem 4.4 (Max®ow-mincut). If f is a maximum°ow and K is a minimum cut, then kf k = ¼ (K ): We now recall a ®ow-decomposition result, better known as conformal decomposition (see [4]), saying that any ®ow can be decomposed as a`sum' of ®ows along paths. First of all, we give a de nition of`path'.  Sketch of proof. If kf k = 0, there is nothing to do, since any path ® from s to t together with the zero ®ow gives the required family, so that we suppose kfk > 0. Let be the set of all paths ® from s to t such that the minimum value of f over the arcs belonging to ® is greater than zero. If were empty, then one could show that the ®ow f is a`null ®ow', i.e. kf k = 0, but this contradicts our assumption, hence 6 = ;. By induction, if has only one element ® , then the pair (® ; min ® f ) necessarily gives the decomposition, otherwise we suppose that such a decomposition can be found whenever has at most d elements, and prove that this is also true when has d+ 1 elements (recall that, by the de nition of path, is always a nite set). Indeed, consider a path ® d+ 1 2 and de ne f d+ 1 as the minimum (positive) value of f over ® d+ 1 . By subtracting f d+ 1 from f over the arcs of ® d+ 1 , we obtain a new ®ow f 0 such that its corresponding set has at most d elements. Finally, the decomposition of f 0 plus the pair (® d+ 1 ; f d+ 1 ) gives the required family.

Proof of the main result
We can represent a con guration of ®uids as a network, with nodes corresponding to ®uids, arcs representing the separating interfaces and with capacity equal to interfacial area. Then a suitable use of the above results will let us prove the elimination theorem, under hypothesis (ST2) of strict triangularity of the coe¯cients c ij .
Proof of theorem 3.1. We only need to show that the hypotheses of lemma 3.3, and hence of lemma 3.2, are all satis ed. To do this, we represent the con guration inside B r as a network, where node v i corresponds to ®uid F i and arc capacity p ij equals kS r ij k (area of the interface S r ij separating ®uid F i from ®uid F j inside B r ). Choose v 1 as source and v 2 as sink, then take a minimum cut K of size (c) if ¿ is the Euclidean norm (or, more generally, a smooth, uniformly convex norm), then the interface set is made of smooth surfaces with constant mean curvature, plus an H n 1 -negligible singular set.
Proof. Part (a) follows immediately from the de nition of G i (1) and theorem 3.1. To prove (b), simply observe that, thanks to theorem 2.1, H n 1 -almost every`boundary' point x 0 belongs to the reduced boundaries of exactly two ®uids, hence x 0 is a`zero-density point' for all other ®uids. Therefore, by theorem 3.1, there exists a neighbourhood of x 0 with only those two ®uids inside. Finally, part (c) is a consequence of well-known results about the regularity of area-minimizing boundaries with volume constraint (see [1, ch. IV], [15] and [13]).