QUASIMINIMAL SETS FOR HAUSDORFF MEASURES

Guy David

Résumé. On cherche à expliquer comment l’utilisation des ensembles quasiminimaux d’Almgren et un lemme de semicontinuité de Dal Maso, Morel, and Solimini pourrait permettre d’obtenir des résultats d’existence d’ensembles minimisant la quantité J_{h}(E) = ∫_{E} h(x) dH^{d}(x) sous certaines contraintes topologiques.

Abstract. We discuss potential existence results for sets that minimize a functional like J_{h}(E) = ∫_{E} h(x) dH^{d}(x) under some topological constraint. The approach would use Almgren quasiminimal sets and a lowersemicontinuity lemma of Dal Maso, Morel, and Solimini.

AMS classification. 49K99, 49Q20.

Key words. Minimal sets under topological constraints, Almgren restricted or quasiminimal sets, Hausdorff measure.

The main goal of this text is to present a type of “minimal set problem”, and then give a vague scheme for proving existence results in some cases. The main tools that will be suggested are the use of Almgren quasiminimal sets, and a very nice lower-semicontinuity lemma of Dal Maso, Morel, and Solimini that applies to them.

A good part of the text will concern rather old results with S. Semmes [DS3,4], while another big portion will consist of speculations on results to come. The purpose of these notes is thus at the same time to advertise for quasiminimal sets, clear up the author’s mind, and convince the reader that some of the problems mentioned below are worth studying, or else give him an opportunity to correct the author’s point of view.

The author wishes to thank Donatella Danielli for the kind invitation and warm welcome at the symposium, and Thierry De Pauw for useful comments on these notes.

1. A minimization problem

Here is a typical problem that we would like to consider. Let Ω ⊂ ℝ^{n} be a nice “domain”. For reasons that will be clear later, we often prefer Ω to be closed. Then let F be a class of closed subsets of Ω; examples will be given soon. We also give ourselves an integer d < n and a lower-semicontinuous function h : Ω → ℝ^{+}, and then set

(1.1)  J_{h}(E) = ∫_{E} h(x) dH^{d}(x)

for E ∈ F. Here H^{d} denotes the d-dimensional Hausdorff measure. So H^{d} would coincide with the d-dimensional surface measure on E if E were contained in a nice d-dimensional submanifold, but we do not want to assume this a priori.

We shall make sure to choose F so that J_{h}(E) < +∞ for some E ∈ F, and then we want to study minimizers for J_{h} in the class F. As we shall see, the role of F is to record various constraints on the set E, often of topological nature.

We do not want to insist too much on the choice of function h. We require lowersemicontinuity because otherwise we cannot seriously hope for the existence of minimizers. But it is good to allow nonconstant functions h, because it makes some of our examples less ridiculously simple, and more seriously because we expect most practical applications to use nonconstant functions h. We shall even see in Section 4 that allowing h to have jumps can be useful. But we shall systematically assume that there is a constant M ≥ 1 such that

(1.2)  1 ≤ h(x) ≤ M for x ∈ Ω.

In fact, most of what shall be said here works also with nonisotropic functions h; see 1.20 below for definitions.

As the reader expects, many things should depend on our choice of class F. Let us give a few simple examples.

Example 1.3. Take d = n − 1, Ω = \overline{B}(0, 2) \ B(0, 1) ⊂ ℝ^{n}, and

(1.4)  F = {E ⊂ Ω ; E is closed and separates 0 from ∞ in ℝ^{n}}.

Naturally the separation condition means that 0 and ℝ^{n} \ \overline{B}(0, 2) lie in different connected components of ℝ^{n} \ E, and the precise choice of domain Ω does not really matter.

If we take h = 1, there will be essentially one minimizer, which is E = ∂B(0, 1). [See Remark 1.6 below for a proof.] “Essentially” is needed because we may take E = ∂B(0, 1) ∪ Z, where Z is closed and H^{n−1}(Z) = 0. In general, we may have to clean up competitors or minimizers to avoid mentioning the useless set Z.

This example also shows why it is usually a good idea to work in a closed domain Ω. If we did not include ∂B(0, 1) in Ω, we would have no minimizer for J_{h}, just because minimizers want to sit on a piece of ∂Ω.

If h is not constant, it becomes harder to guess what minimizers should be, but we can still prove the existence of minimizers rather easily, because we work in codimension 1. That is, first observe that if E ∈ F and U is the connected component of ℝ^{n} \ E that contains the origin, then ∂U ⊂ E. But ∂U ∈ F because it is closed and separates U from ∞, and J_{h}(∂U) ≤ J_{h}(E) because ∂U ⊂ E. Thus it is equivalent to minimize J_{h} in the smaller class

(1.5)  F_{1} = {∂U ; U is open and B(0, 1) ⊂ U ⊂ B(0, 2)}.

If U is as in (1.5) and J_{h}(∂U) < +∞, then U has finite perimeter. That is, the distribution gradient of the function 1_{U} is a finite measure, which is supported in ∂U, and even of the form ν(x)dH^{d}(x) for some vector-valued function ν on ∂U such that |ν(x)| ≤ 1. Said with a different vocabulary, 1_{U} has bounded variation (for short, 1_{U} ∈ BV), or equivalently U is a set of finite perimeter.

It turns out that, due to the good compactness properties of BV, it is rather easy to find minimizers for the functional ∫ h(x)|d∇1_{U}|(x) in the class of sets of finite perimeter U such that B(0, 1) ⊂ U ⊂ B(0, 2). In addition, if U is a minimizer for this new functional, it is a little more regular than the average set of finite perimeter, and one can check that it coincides almost-everywhere with an open set which minimizes J_{h} in the class F_{1}.

So in the present case, there are minimizers for J_{h} in the classes F and F_{1}. See [Gi] or [AFP] for properties of BV functions and sets of finite perimeter, [DS3] for the existence of minimizers and their regularity properties, and [DS1,2] for a rapid presentation of [DS3]. As far as I know, this example is about the only one for which a complete answer to our questions is written down.

Remark 1.6. Our topological constraint gives a lower bound on H^{n−1}(E) for E ∈ F, so the infimum of J_{h} on F is not zero, and our problem is not stupid. Indeed, if E ∈ F, every ray from the origin meets E somewhere between ∂B(0, 1) and ∂B(0, 2) (we go from one connected component to another one, so we have to cross the boundary). This means that the radial projection π : x → x/|x| maps E onto ∂B(0, 1). But E ⊂ \overline{B}(0, 2) \ B(0, 1), where π is 1-Lipschitz, so

(1.7)  H^{n−1}(∂B(0, 1)) = H^{n−1}(π(E)) ≤ H^{n−1}(E).

This is also the simplest proof of minimality of ∂B(0, 1) when h = 1.

Example 1.8. Now take 0 < d < n, and call S the unit sphere of dimension n − d − 1 in ℝ^{n−d} ⊂ ℝ^{n}. Then pick a small tubular open neighborhood of S in ℝ^{n}, call it V, and set Ω = \overline{B}(0, 2) \ V. Consider the set

(1.9)

F = {E ⊂ Ω ; E is closed and cannot be deformed

into a set of vanishing H^{d}-measure inside Ω}.

This last means that we cannot find a continuous function F : [0, 1] × E → Ω, such that F (0, x) = x for x ∈ E, and the image of E by F (1, ·) has vanishing Hausdorff measure of dimension d. One can think of other definitions of deformations, but in the present simple situation they would probably be equivalent anyway. Also, it turns out that we can replace sets of zero d-dimensional Hausdorff measure with sets of Hausdorff dimension ≤ d−1 without changing F. See the proof of (1.10) below in [DS4] (look for Theorem 0.10 or Proposition 12.46 there).

So F consists of closed sets E ⊂ Ω that are somehow linked with S and V. To get an idea of the general situation, the reader may want to think about the case when n = 3, d = 1, S is a circle, and F corresponds to sets that are linked to the circle (and don’t get close to it). Of course d = 1 is a little too easy to yield serious results, but at least we can make a picture.

Let us see why our minimization problem on F is not trivial. First we need to find E ∈ F, with H^{d}(E) < +∞. Pick any point ξ ∈ S, and call P the (d + 1)-dimensional vector plane that contains ξ and the orthogonal complement of ℝ^{n−d}. Then set E = P ∩ ∂B(ξ, 10^{−1}). It is clear that H^{d}(E) < +∞, but we also need to check that E ∈ F. Let us assume that d < n − 1; the case when d = n − 1 is easier and also the proof of (1.14) below applies. We shall use a little bit of degree theory.

Define φ : S × E → 𝕊^{n−1} by φ(x, y) = \frac{x−y}{|x−y|}. Observe that S × E is a nice (n − 1)- dimensional manifold and φ is smooth. To compute its degree, we evaluate the number of inverse images of a = ξ/|ξ|. If φ(x, y) = a, then both x and y have to lie on P ∩ ℝ^{n−d}; this leaves only two choices for x (namely, ξ and −ξ) and two choices for y (namely, ξ − ξ/10 and ξ + ξ/10). Only one combination gives φ(x, y) = a, we leave it to the reader to verify that φ is regular at that point, and we get that φ has degree ±1.

Then consider a deformation F : [0, 1] × E → Ω; we can define φ_{t} : S × E → 𝕊^{n−1} by φ_{t}(x, y) = \frac{x−F (t,y)}{|x−F (t,y)|} because F (t, E) never meets S; then the degree of φ_{1} is still ±1. Set E_{1} = F (1, E). We need to check that H^{d}(E_{1}) > 0.

So assume that H^{d}(E_{1}) = 0. By a Federer-Flemming argument, we can deform E_{1} inside Ω onto a subset of some (d−1)-dimensional grid G. Here we rush a little because the argument is a little technical and not very original; rather detailed constructions for this and the similar estimate (1.10) below can be found in [DS4]. We can further deform that grid into a point inside \overline{B}(0, 2) (the map (t, y) → (1 − t)y would do the job). Moreover, since dim(S) + dim(G) = n − 2, a general position argument allows us to arrange the deformation so that the image of the grid never crosses S. [First deform G to a correctly chosen point of B(0, 10^{−1}) as suggested above, and then move that point to the origin.] Call F_{t}, 0 ≤ t ≤ 2, the deformation that we get when we put all these deformations together; then F_{t}(E) never meets S and F_{2} ≡ 0 on E. As before, φ_{t}(x, y) = \frac{x−F_{t}(y)}{|x−F_{t}(y)|} defines a one-parameter family of continuous functions on S × E, and so the degree of φ_{2} is ±1. But φ_{2}(x, y) = x/|x|, and the degree of this map is clearly 0 (it is not even surjective!). This contradiction shows that H^{d}(E_{1}) > 0 and E ∈ F, as promised.

To complete the verification that our minimization problem on F is not trivial, let us observe that there is a constant c > 0, that depends on n, d, and the width of our neighborhood V around S, such that

(1.10)  H^{d}(E) ≥ c for E ∈ F.

The simplest proof is a little similar to the projection argument for (1.7); see [Da1], page 107. But the most instructive proof is perhaps to observe that if H^{d}(E) is too small (depending on the geometry of Ω), then it is possible to deform E onto some (d − 1)- dimensional dyadic grid, because when we do the standard Federer-Flemming projection of E onto a d-dimensional grid (of fixed size, depending on Ω), we end up with a set with too small a H^{d}-measure to fill up a single face. This proof has the big advantage of applying to Example 1.12 below. See Theorem 0.10 or Proposition 12.46 in [DS4] for details.

We continue our list of examples with two variants. For the first one, we keep Ω and F as in Example 1.8, but we fix E_{0} ∈ F and set

(1.11)  F(E_{0}) = {E ∈ F; E is a deformation of E_{0} in Ω}.

By deformation of E_{0} in Ω, we still mean a set E = F_{1}(E_{0}), where F_{1}(x) = F (1, x), for some continuous function F : [0, 1] × E → Ω such that F (0, x) = x for x ∈ E. Note that if E is a deformation of E_{0} in Ω, then any deformation of E in Ω is also a deformation of E_{0} in Ω, hence E ∈ F, with F as in (1.9), because E_{0} ∈ F.

General Example 1.12. This time Ω is any reasonably simple closed domain in ℝ^{n} (with some nontrivial topology if we want something to happen). For instance, we may take for Ω a finite union of dyadic cubes of the same size, or a bilipschitz image of such a set. Compact manifolds of dimension n would do as well. Then we can still define F as in (1.9), of fix any E_{0} ∈ F and define F(E_{0}) as in (1.11), and try to minimize J_{h} in the class F or a more precise class F(E_{0}).

If we are a little lucky, F will contain some sets of finite measure, and then our problem will not be trivial. (In this case again (1.10) holds; see [DS4].) Probably F and F(E_{0}) will be hard to characterize by algebraic geometry, but we still can try to understand minimizers for J_{h} on F or F(E_{0}).

There seems to be overwhelmingly many different types of topological conditions to consider. In the next sections we shall restrict our attention to Examples 1.8 and 1.12, but let us rapidly discuss other types of topological conditions, mostly to convince the reader that it is not so easy to make a nice list.

1.13. Topological conditions in codimension 1. We start with a discussion of the simpler case when d = n−1, to give an idea of what we have in mind. With the notation of Example 1.8, S consists of two points, but it will be a little cleaner to work with S = {0}, and take Ω = \overline{B}(0, 2) \ B(0, 1) as in Example 1.3. Let us try to check that in this case, the class F of (1.9) coincides with the class of Example 1.3 (which we shall call F_{0} to avoid any confusion). We need to know that

(1.14)  F_{0} is stable under deformations inside Ω.

That is, if E ∈ F_{0} and F : [0, 1] × E → Ω is continuous, with F (0, x) = x for x ∈ E, then E_{1} = F (1, E) lies in F_{0}. To prove this, pick any point y ∈ ℝ^{n} \ \overline{B}(0, 2). By definition of F_{0}, E separates 0 from y. In addition, none of the intermediate sets F (t, E) ever contain 0 or y (because 0 and y lie out of Ω); then all the F (t, E), including E_{1}, separate 0 from y. See for instance 4.3 in Chapter XVII of [Du], page 360, where this is derived from Borsuk’s theorem on extensions of maps into spheres, and Brouwer’s theorem that says that the identity map on the sphere is not nullhomotopic.

If E ∈ F_{0} and E_{1} is (the end result of) a deformation of E inside Ω, (1.14) says that E_{1} ∈ F_{0}, and then (1.7) says that H^{d}(E_{1}) > 0. Thus F_{0} ⊂ F.

Let us now discuss the opposite inclusion. The author wishes to warn the reader that he did not check the argument below too seriously.

Let E ⊂ Ω be closed, and suppose that E ∉ F_{0}. Pick a nice simple path γ ⊂ Ω that goes from ∂B(0, 1) to ∂B(0, 2). Let us first deform B(0, 2) by a map that preserves the spheres ∂B(0, r), so that (after our first deformation) the two endpoints of γ lie on a same line segment L through the origin. Then deform γ inside the interior of Ω, without moving its endpoints and while keeping it simple, so that at the end of the deformation γ is just the line segment L ∩ Ω. If n = 3, this involves getting rid of the apparent knots of γ, by passing them around B(0, 1). We give no proof here, but just say that this is possible because one could also undo the knots by retracting γ along itself (starting from its extremity that lies on ∂B(0, 1)).

We need a little bit more than we just said, we also want the deformation of γ to come from a one-parameter family of homeomorphisms of Ω, or at least mappings of Ω such that the complement of some neighborhood of γ never gets too close to (the deformed) γ. If we manage this, we are in good shape because the final image of E through the corresponding deformation does not meet a small tube around L. We can deform it once more, by expanding brutally around L, by a one-parameter family of mappings that preserve the spheres ∂B(0, r). After this is done, the final image of E is contained in the line segment −L. If n > 2, we are already happy because the line segment is less than (n − 1)- dimensional; if n = 2 we finish the job by projecting radially on ∂B(0, 1). In both cases, E ∉ F.

So our simplified version of Example 1.8 in codimension 1 seems equivalent to Example 1.3. The original version where S has two points should not be much more complicated; F in (1.9) should be the set of closed subsets of Ω that separate at least one of the three components of ℝ^{n} \ Ω from another one; the deformation argument above should be more painful to write down because we have to keep an additional ball untouched, but the author does not expect significant complications. Even in the case of Example 1.12, it seems reasonable to expect characterizations of the classes F by separation properties, even though this may require some work.

1.15. Other linking conditions. Return to Examples 1.8 and 1.12 in codimensions larger than 1. We decided to use the absence of deformation of E into a set of vanishing Hausdorff measure as a natural linking condition, but other definitions would make sense.

First, we could have used the slightly larger class

(1.16)  F′ = {E ⊂ Ω ; E is closed and cannot be deformed into a point inside Ω}

instead of F. In the precise setting of Example 1.8, this does not make any difference. Indeed let E ∉ F be given. Then E can be deformed onto E_{1}, with H^{d}(E_{1}) = 0. We can further deform E_{1} into a subset E_{2} of a dyadic grid of dimension d−1, as in our proof that the small d-sphere centered on ξ lies in F (in the discussion of Example 1.8). Then E_{2} itself can be deformed into a point, without crossing S (by a general position argument, as above). Now the deformation of E_{2} could cross V, but since it avoids some neighborhood W of S, we can conjugate by a homeomorphism of \overline{B}(0, 2) \ W onto \overline{B}(0, 2) \ V and get a deformation that avoids V. Thus F′ = F when Ω is as in Example 1.8.

In the more general situation of Example 1.12, F′ could be strictly larger than F. For instance, if n = 3, d = 2, and Ω = \overline{B}(0, 2) \ V, where V is a solid torus, some circles around V lie in F′ \ F, and it is a good idea to exclude them by sticking to F. In general, it seems more reasonable to use F rather than F′; the point is that this way we get (1.10) and our problem is less likely to be degenerate.

Now we may also decide to code the fact that S and E are linked in terms of deformations of S in the complement of E. For instance, we could work with

(1.17)

F′′ ={E ⊂ Ω ; E is closed, H^{d}(E) < +∞, and
S cannot be deformed into a point inside \overline{B}(0, 2) \ E\}.

We could also have talked of deformations of S into a set of dimension less than n − d − 1 (instead of a point), but hopefully the restriction to H^{d}(E) < +∞ makes the two definition equivalent, by the same sort of general position argument as suggested above.

When d = n−1, this is a separation condition, and we discussed in 1.13 how it relates with conditions of deformations for E.

When n = 3 and d = 1, it looks like F′′ could be the same as {E ∈ F; H^{d}(E) < +∞} (notice that the constraint H^{d}(E) < +∞ allows us to get rid of two-dimensional sets that belong to F because they separate S from ∂B(0, 2)). But the author does not know enough topology to decide whether this is trivial or false. It is even possible that both classes are characterized by a simple condition of algebraic topology.

When we work with any dimension and codimension, and especially in the general case of Example 1.12, it seems quite unlikely to the author that the condition in (1.17) will have a simple translation in terms of sets like F, or that any of the two classes can be characterized in terms of algebraic topology.

It is not clear either that F′′ is stable under deformations in Ω; we know that this is the case when d = n − 1, by the proof of (1.14), but the situation in higher codimensions could be much more complicated. And if F′′ is not stable under deformations, the methods described in this paper will not apply to it.

A third way to code linking conditions would be to use algebraic topology directly. The author expects such conditions to be stable under deformations of E in Ω (and then the approach suggested below has a chance to work), but our Example 1.12 is probably richer, in the sense that some classes F or F(E_{0}) will be impossible to characterize algebraically.

We end this section with a few remarks of a slightly more technical nature.

1.18. Boundary conditions. The reader probably noticed that our list of examples does not include the celebrated Plateau problem, where one looks for a surface of minimal area that is bounded by a given curve. In fact, in the present context of sets, it is fairly hard to give a nice definition of “E is bounded by the curve Γ”. Currents are a very nice way to do this, but there are problems in this setting too. We shall briefly return to this question in Section 6, but for the major part of this text we shall leave boundary conditions alone. The case of domains or manifolds (even with complicated topology) is simpler, and in the author’s opinion should be treated first.

Incidentally, we should say that in this text we are trying very hard to avoid using currents; we are also, to some extent, trying to talk about problems for which currents do not give an easier solution. But the reader should be aware of the existence of interesting work on similar problems, involving and using currents. See Section 6 for a just few words about this.

1.19. Free boundaries. These can be somewhat different from boundary conditions, because in some cases they can be reduced to a problem on a manifold, by a symmetry argument. For instance, call H^{+} the upper half space in ℝ^{n}, and suppose we want to minimize J_{h}(E) = ∫_{E} h(x) dH^{n−1}(x) among closed subsets E of H^{+} ∩ [\overline{B}(0, 2) \ B(0, 1)] that separate H^{+} ∩ B(0, 1) from H^{+} \ \overline{B}(0, 2) in H^{+}. This looks like a problem with a free boundary on the hyperplane ∂H^{+}, but the simplest way to deal with it is to extend the function h by symmetry and reduce to Example 1.3. Manifolds, or changes of variables, are likely to show up in such problems: think about the case when our free boundary ∂H^{+} is no longer flat.

1.20. Non-isotropic functions h. It is probably important to allow fairly general functions h in (1.1), not only because it makes our Example 1.3 less trivial, but because we may need the flexibility, for instance if we want to work on a manifold, or if we want to set up a symmetry argument like the one suggested above. In all these cases, we want to know that a piecewise smooth change of variables leads to another functional like J_{h}. In fact, for nonconformal changes of variables, the resulting function h is no longer isotropic, so we need to use measures h(x)dH^{d}(x) that also depend on the direction of the tangent d-plane to E (assuming it exists). Here is a rapid suggestion of how one could define such measures without assuming a priori that E is rectifiable. Probably there are cleaner, more invariant, or more general ways to do this, but at least the effect of changes of variables will be included and easy to compute. Also, we should perhaps not spend too much time on this, as we expect all minimizers, if they exist, to be rectifiable. Anyway, we shall take h with values in M_{n}, the set of n × n matrices. We start with the case when h is a finite sum, h = ∑_{i} 1_{F_{i}}A_{i}, where the F_{i} form a (measurable) partition of Ω, and A_{i} ∈ M_{n}. Then we set

(1.21)  J_{h}(E) = ∑_{i} H^{d}(A_{i}(E ∩ F_{i})),

where we use the same notation for A_{i} and the linear mapping that it defines. One simple way to make sure that J_{h}(E) is comparable with H^{d}(E), as in (1.2), is to require that

(1.22)  h(x) is invertible, with ||h(x)|| ≤ M and ||h(x)^{−1}|| ≤ M, for every x ∈ Ω.

If h is lowersemicontinuous and satisfies (1.22), it is easy to define J_{h}(E) for E ⊂ Ω closed and such that H^{d}(E) < +∞: we simply approximate h by finite linear combinations of characteristic functions, use (1.21) for them, and take limits. Most of what will be said in this text is still valid for such matrix-valued functions h.

1.23. Additional terms. We could easily add extra terms to J_{h}, provided that they have appropriate lower semicontinuity properties and do not interfere much with the surface terms at small scales. This should not change the problems seriously.

1.24. Stability under small deformations; retractions on Ω. As we shall see soon, it is pleasant to work with classes F that are stable under small deformations. First observe that if F is as in Examples 1.3, 1.8, or 1.12, and if E ∈ F and E′ is a deformation of E inside Ω, then E′ ∈ F. [For Example 1.3, this is (1.14).] The same thing happens with F(E_{0}) (again, by definition).

Often, as in the case of Example 1.12, we can say a little more. Indeed, assume that there is a retraction f, defined on a neighborhood Ω_{δ} = {x ∈ ℝ^{n}; dist(x, Ω) ≤ δ}, and such that f(Ω_{δ}) ⊂ Ω and f(x) = x for x ∈ Ω. Then F and F(E_{0}) are still stable under small deformations, even if intermediate sets get out of Ω. That is, if F : [0, 1] × E → ℝ^{n} is as above, except that it does not necessarily map into Ω, but into the slightly larger Ω_{δ}, we can still say that E′ = F_{1}(E), if it lies in Ω, is a deformation of E in Ω (and hence lies in F or the same class F(E_{0}) as E). This is easy: we just use the deformation by f ∘ F. Notice also that F will automatically map [0, 1] × E into Ω_{δ} if it does not move points too far.

The existence of a retraction f may also be a nice tool for getting rid of ∂Ω in some arguments. That is, we can extend the function h to Ω_{δ} by giving some enormous value to h(x) out of Ω; then the corresponding functional on Ω_{δ} could be slightly easier to study, because you know in advance that no good competitor will ever get out of Ω (or close to ∂Ω_{δ}). Incidentally, allowing h to be merely lower semicontinuous makes such arguments easier.

We shall also use Lipschitz retractions in the next section, to show that minimizers for J_{h} are Almgren quasiminimal sets.

2. Almgren quasiminimal sets

Suppose that E ∈ F minimizes J_{h} in some class F; how regular does E have to be? If we only know that 1 ≤ h(x) ≤ M, as in (1.2), we cannot hope E to be better

than a Lipschitz graph, because it is easy to arrange problems for which the minimizers are Lipschitz graphs. For instance, if D is a bounded domain in ℝ^{n−1} and f : D → ℝ is a Lipschitz function, we can choose a and b ∈ ℝ such that a < f(x) < b for x ∈ D, set Ω = D × [a, b], call F the class of closed sets in Ω that separate D × {a} from D × {b}, and set h(x) = 1 on the graph Γ of f, and h(x) = M on Ω \ Γ. It is easy to see that Γ minimizes J_{h} in F if M large enough compared to the Lipschitz constant for f.

Put in another way, if we do not know h too well or just want to use (1.2), the best information we can get for a minimizer E is that

(2.1)  H^{d}(E \ Ẽ) ≤ MH^{d}(Ẽ \ E) for every competitor Ẽ ∈ F,

because after all h could be equal to 1 on E \ Ẽ and to M on Ẽ \ E.

Here we shall assume that F is stable under deformations; then (2.1) will say that E is an Almgren-quasiminimal set, and this in turn will give regularity properties for E (see the next section).

We start with the definition of quasiminimal sets. This is almost exactly the same thing as what F. Almgren called restricted sets. We start with an open set U ⊂ ℝ^{n}, a quasiminimality constant M ≥ 1, and a diameter δ ∈ (0, +∞].

Then we consider a closed subset E of U. To avoid difficulties with the definitions, we assume that H^{d}(E ∩ B) < +∞ for every compact ball contained in U. We want to compare E with deformations of E in U. That is, we consider functions φ : [0, 1] × U → U with the following properties:

(2.2)  φ(0, x) = x for x ∈ U,

(2.3)  φ is continuous on [0, 1] × U,

(2.4)  φ(1, ·) is Lipschitz,

and, if we set φ_{t}(x) = φ(t, x),

(2.5)  W_{t} = {x ∈ U; φ_{t}(x) ≠ x}, and Z = ⋃_{t∈[0,1]} [W_{t} ∪ φ_{t}(W_{t})],

then

(2.6)  Z is relatively compact in U and diam(Z) ≤ δ.

These are the conditions on φ under which we may compare E to φ_{1}(E). We shall comment on them after the next definition.

Definition 2.7. We say that the set E is an (M, δ)-quasiminimal set in U when E is closed in U, H^{d}(E ∩ B) < +∞ for every compact ball contained in U, and

(2.8)  H^{d}(E ∩ W_{1}) ≤ MH^{d}(φ_{1}(E ∩ W_{1}))

for every φ such that (2.2)-(2.6) hold.

The requirement (2.4) is not too important; it is just here so that (2.8) makes sense. But no Lipschitz bound for φ_{1} is required. Also, it is very important that in this definition, we do not require φ_{1} to be injective. That is, we are allowed to crush E locally.

Almgren did not require (2.2) or (2.3); in other words, he only considered mappings φ_{1} : U → U and did not care whether they come from real deformations or not. Our definition of quasiminimal sets is slightly less restrictive (because we add constraints on the deformations φ), but this is all right, because the regularity results are the same. In fact, in most arguments one uses functions φ_{1} such that W_{1} ∪ φ_{1}(W_{1}) is contained in a compact ball B contained in U and with diameter at most δ. Then we can use the obvious deformation φ given by φ(t, x) = (1 − t)x + tφ_{1}(x), because then Z is contained in B by convexity.

We may take U = ℝ^{n} and δ = +∞, and then the only requirement in (2.6) is merely that φ_{1}(x) = x out of some compact set. The role of δ is to allow more flexibility, for instance by allowing to keep some global topological restrictions; thus the unit sphere is an (1+ε, δ)-quasiminimizer in ℝ^{n} if δ is small enough, and catenoids are (1, δ)-quasiminimizers in ℝ^{3}, but only for δ small enough. Similarly, the mention of U corresponds to a boundary constraint (we are not allowed to modify E near ∂U).

Next we want to show that if Ω is as in Example 1.12 and there is a Lipschitz retraction from a neighborhood of Ω onto Ω, then minimizers for J_{h} are Almgren quasiminimizers.

So let Ω ⊂ ℝ^{n} be a closed set, and assume that there is a neighborhood Ω_{δ} = {x ∈ ℝ^{n}; dist(x, Ω) ≤ δ} of Ω, and a Lipschitz function f : Ω_{δ} → Ω such that f(x) = x for x ∈ Ω. Also let d < n and h : Ω → [1, M] be given, and define J_{h} by (1.1). Finally define F by (1.9).

Lemma 2.9. Suppose in addition that E ∈ F minimizes J_{h} in F, or let E_{0} ∈ F be given, and suppose that E ∈ F(E_{0}) minimizes J_{h} in F(E_{0}). Then E is an (ML^{d}, δ)-quasiminimal set in ℝ^{n}, where L is the Lipschitz constant for f.

Note that this is only interesting if H^{d}(E) > 0, because sets of vanishing H^{d}-measure are always quasiminimal by definition. But H^{d}(E) > 0, for instance, if Ω is a bilipschitz image of a finite union of (closed) dyadic cubes, by (1.10).

To prove the lemma, let φ : [0, 1] × ℝ^{n} → ℝ^{n} be as in the definition of quasiminimal sets; we want to use it to construct a deformation of E in Ω. If W_{1} does not meet Ω, (2.8) is trivial because E ∩ W_{1} is empty. Otherwise, (2.6) says that Z is contained in Ω_{δ}. This means that φ(t, x) ∈ Ω_{δ} for x ∈ E, because either x ∈ W_{t} and φ(t, x) ∈ Z by (2.5), or else φ(t, x) = x ∈ Ω by definition of W_{t}. Thus F = f ∘ φ : [0, 1] × E → Ω defines a deformation of E in Ω, and Ẽ = F_{1}(E) is a competitor for E. Then (2.1) applies, and

(2.10)  H^{d}(E \ f ∘ φ_{1}(E)) ≤ MH^{d}(f ∘ φ_{1}(E) \ E).

We still need to check that this implies (2.8). Set E_{1} = E ∩ W_{1}. Notice that f ∘ φ_{1}(x) = x on E \ E_{1}, so f ∘ φ_{1}(E) \ E = f ∘ φ_{1}(E_{1}) \ E and

(2.11)  \begin{aligned}H^{d}(E_{1} \ f ∘ φ_{1}(E)) & ≤ H^{d}(E \ f ∘ φ_{1}(E)) \\ & ≤ MH^{d}(f ∘ φ_{1}(E) \ E) = MH^{d}(f ∘ φ_{1}(E_{1}) \ E). \end{aligned}

by trivial manipulations and (2.10). Then

(2.12)  \begin{aligned}H^{d}(E ∩ W_{1}) & = H^{d}(E_{1}) = H^{d}(E_{1} \ f ∘ φ_{1}(E)) + H^{d}(E_{1} ∩ f ∘ φ_{1}(E)) \\ & ≤ MH^{d}(f ∘ φ_{1}(E_{1}) \ E) + H^{d}(E_{1} ∩ f ∘ φ_{1}(E)) \\ & ≤ MH^{d}(f ∘ φ_{1}(E_{1}) \ E_{1}) + H^{d}(E_{1} ∩ f ∘ φ_{1}(E)) \\ & = MH^{d}(f ∘ φ_{1}(E_{1}) \ E_{1}) + H^{d}(E_{1} ∩ f ∘ φ_{1}(E_{1})) \\ & ≤ MH^{d}(f ∘ φ_{1}(E_{1})) ≤ ML^{d}H^{d}(φ_{1}(E_{1})) \end{aligned}

by (2.11), because E_{1} ∩f ∘φ_{1}(E) = E_{1} ∩f ∘φ_{1}(E_{1}) (indeed if x ∈ E \E_{1}, then f ∘φ_{1}(x) = x and this point lies out of E_{1}), and because f is L-Lipschitz. This proves (2.8) with the constant ML^{d}, and Lemma 2.9 follows. □

Remark 2.13. Lemma 2.9 can be extended to classes F different from the ones above, and that may include boundary conditions. Suppose for instance that X is a closed set, and that F is stable under small deformations away from X. That is, φ_{1}(E) ∈ F if E ∈ F and φ : [0, 1] × Ω → Ω is continuous and satisfies (2.2)-(2.6) with U = Ω \ X. Then the proof above yields that minimizers for J_{h} in the class F are (ML^{d}, δ)-quasiminimal sets in ℝ^{n} \ X. This will give some regularity as below, but only far from X.

Similarly, if we do not have the Lipschitz retraction on Ω, we can still say that minimizers for J_{h} in the class F or F(E_{0}) of General Example 1.12 are (ML^{d}, δ)-quasiminimal sets in the interior of Ω.

3. Regularity properties of quasiminimal sets

Obviously Lemma 2.9 would not be interesting if we could not prove some regularity properties for Almgren quasiminimal sets. We rapidly review some of them in this section. The main references are [Al] and [DS4].

The main result is the local Ahlfors-regularity and uniform rectifiability of E^{∗} when E is a quasiminimal set. Here E^{∗} denotes the closed support of Hausdorff measure on E, defined by

(3.1)  E^{∗} = {x ∈ E; H^{d}(E ∩ B(x, r)) > 0 for every r > 0}.

It is easy to see that H^{d}(E \ E^{∗}) = 0. However, E \ E^{∗} could play an important role in the definition of E, because in general E^{∗} does not lie in F as soon as E does. When n = 3 and d = 2, for instance, we can think about E^{∗} as a fairly nice surface, and then E \ E^{∗} could be a bunch of one-dimensional wires that keep track of some topology. See Section 7 for a few additional comments.

The fact that H^{d}(E \ E^{∗}) = 0 implies that we have little control on E \ E^{∗} when E minimizes J_{h}, and indeed our regularity results concern E^{∗}. The next one is a combination of results of Almgren [Al] and from [DS4].

Theorem 3.2. If E is an (M, δ)-quasiminimal set in U, x ∈ E^{∗}, and 0 < r < δ/2 is also such that B(x, 2r) ⊂ U, then

(3.3)  C^{−1}r^{d} ≤ H^{d}(E ∩ B(x, r)) ≤ Cr^{d}

and there is a C-Lipschitz graph Γ of dimension d (that is, the graph of a C-Lipschitz function defined on some d-plane and with values in the orthogonal (n − d)-plane) such that

(3.3)  H^{d}(E ∩ Γ ∩ B(x, r)) ≥ C^{−1}r^{d}.

The constant C depends only on n, d, and M.

In fact, there is even a set F = F (x, r), such that E ∩ B(x, r) ⊂ F ⊂ E ∩ B(x, 2r), which is Ahlfors-regular, uniformly rectifiable, and with big pieces of Lipschitz graphs. The reader may consult [DS4] for a precise definition and a proof. What we want to say here with Theorem 3.2 is just that E^{∗} has some nontrivial amount of regularity inside U, which is not so far from what we would expect from Lipschitz graphs.

See the first lines of Section 2, which say that we cannot expect much more, and also recall that we should not hope for a good control on E \E^{∗} (except that it has H^{d}-measure zero), or on E^{∗} near ∂U (because we do not have enough choice for the competitors there).

Remark 3.4. In the special case of Example 1.3 (with the separation property in codimension 1), we can say a little more about minimizers for J_{h}. It turns out that if E minimizes J_{h} in the class F of (1.4), then E^{∗} is the boundary of some open connected set O such that B(0, 1) ⊂ O ⊂ B(0, 2), O is a John domain centered at the origin, Õ = ℝ^{n} \ \overline{O} is also connected, E^{∗} = ∂Õ, and Õ is a John domain centered at infinity. In particular, E^{∗} satisfies S. Semmes’ Condition B from [Se1]. See [DS3].

Quasiminimizers have another very nice regularity property that we want to mention here: if {E_{k}} is a sequence of closed sets in U, and each E_{k} is an (M, δ)-quasiminimal set in U, with constants M and δ that do not depend on k, then the sequence {E_{k}^{∗}} satisfies the \underline{uniform concentration} property of Dal Maso, Morel, and Solimini [DMS]. Let us not say what this means here, but just give an important consequence: if in addition the sets E_{k}^{∗} converge to a limit E^{∗} (for the Hausdorff distance, locally in U) then

(3.5)  H^{d}(E^{∗} ∩ O) ≤ lim inf_{k→+∞} H^{d}(E_{k}^{∗} ∩ O) for every open set O ⊂ U.

Of course (3.5) would not hold for any convergent sequence of closed sets (think about dotted lines converging to a line segment), and the semicontinuity property (3.5) is quite useful.

This property is mostly a consequence of Theorem 3.2. See [Da2] for a proof. It is also proved in [Da2] that the limit E^{∗} is a quasiminimal set in U, with the same constants as the E_{k}; of course the proof relies heavily on (3.5).

Perhaps we should stress the fact that the reason why we do not get better regularity than Lipschitz is because we refuse to take into account the regularity of h. If h is smooth, then E will be quite regular near almost every point. But even in this case, we should still expect some mild singularities, as already occur in soap films.

The author’s impression is that Almgren saw quasiminimality as a first regularity step, and then additional regularity was provided, for instance once we knew that E is (M, δ)-quasiminimal with an M that tends to 1 when δ tends to 0. Here we would like to see it as a potential gate to lower semicontinuity and existence results.

4. A (not so recent) application

The goal of this slightly digressive section is to give at least one example of result for which we needed to minimize J_{h} (in (1.1)) with a discontinuous function h. The point was to prove the following.

Theorem 4.1. [DS3], [JKV]. Set Ω = \overline{B}(0, 2) \ B(0, 1) ⊂ ℝ^{n}, and let F ⊂ Ω be closed. Suppose that F separates 0 from ∞ in ℝ^{n}, and at the same time that H^{n−1}(F) ≤ C_{0} < +∞. Then there a C-Lipschitz graph Γ (of codimension 1) such that H^{n−1}(F ∩ Γ) ≥ C^{−1}. The constant C depends only on n and C_{0}.

This answered a question of Semmes [Se2]. The idea is that there is a tension between our information on the size of F and our topological constraint of separation (which itself implies a lower bound on the size of F, see (1.7)). Form this tension results a low regularity result on some part of F. One could not expect the whole F to be nice (add an ugly fractal set of finite measure to F), and the size condition is needed (otherwise F could be a snowflake). The fact that we get quantitative bounds is an important part of the result; the mere rectifiability of a piece of F is much easier to get.

The proof of [JKV] is very nice (combinatorics with checkerboards), but let us just say a few words about the proof of [DS3].

We start with the same setting as in Example 1.3, with Ω = \overline{B}(0, 2) \ B(0, 1) ⊂ ℝ^{n} and F ∈ F, and we define h by

(4.2)  h(x) = 1 for x ∈ F and h(x) = M otherwise,

where the constant M will be chosen soon, depending on C_{0}.

Then we find E ∈ F such that

(4.3)  J_{h}(E) = inf{J_{h}(A) ; A ∈ F}.

The existence is not too hard to prove, because we are in codimension 1 and we can use sets of finite perimeter and compactness in BV (see [DS3] or the rapid discussion in Section 1). We expect E to have some regularity, because it minimizes a functional J_{h}, but before we get to that let us say why it has a nontrivial intersection with F. Simply observe that

(4.4)  MH^{n−1}(E \ F) ≤ J_{h}(E) ≤ J_{h}(F) = H^{n−1}(F) ≤ C_{0}

by (1.1) and (4.2), (4.3) and the fact that F ∈ F, (4.2) again, and the definition of F.

On the other hand, set α = H^{n−1}(∂B(0, 1)) and choose M = 2C_{0}/α; (4.4) says that H^{n−1}(E \ F) ≤ α/2, but we know from (1.7) that H^{n−1}(E) ≥ α, so

(4.5)  H^{n−1}(E^{∗} ∩ F) = H^{n−1}(E ∩ F) ≥ H^{n−1}(E) − H^{n−1}(E \ F) ≥ α/2,

because H^{n−1}(E \ E^{∗}) = 0.

Thus F has a large intersection with E. We also know from Lemma 2.9 and Theorem 3.2 that E^{∗} is locally Ahlfors-regular, with big pieces of Lipschitz graphs. In the present case of codimension 1, Theorem 3.2 is somewhat simpler to prove, so the argument is not too painful.

If E^{∗} was a Lipschitz graph, we would be finished. Here we just know that E^{∗} is fairly nice; this almost what we wanted, but a little bit of additional work is needed to get the precise conclusion of the theorem. We refer to [DS3] for details.

There is an analogue of Theorem 4.1 in the situation of Examples 1.8 and 1.12. Take a simple set Ω ⊂ ℝ^{n} (a finite union of dyadic cubes, for instance), and a closed set F ⊂ Ω such that H^{d}(F) < +∞ and that cannot be deformed into a set of H^{d}-measure zero inside Ω. Then for each η > 0 we can find an Ahlfors-regular, uniformly rectifiable set Z of dimension d, with big pieces of Lipschitz graphs (and all this with constants that depend only on Ω, and η), such that H^{d}(Z \ F) ≤ η and H^{d}(Z) ≥ θ. Here θ is a positive constant that depends only on Ω, so we are allowed to take η << θ and get that H^{d}(Z ∩ F) ≥ θ/2.

See Theorem 0.10 in [DS4]. The general idea for the proof is the same, but there are complications because of the larger codimension, and more importantly because we do not want to prove the existence of minimizers for J_{h}!

What saves us is the fact that we do not need to find minimizers for J_{h}, but only quasiminimizers. That is, we do not care if we lose a geometric constant in the minimality of E or in (4.4). So the way we get around the issue of existence is by using discrete approximations of our problem. We pick any very small dyadic scale τ > 0, replace F by the union F_{τ} of the dyadic cubes of size τ that meet F, and then define a function h_{τ} as in (4.2). We minimize the corresponding functional J_{h_{τ}}, in the class F_{τ} of finite unions E of faces of dimension d or lower, of dyadic cubes of size τ contained in Ω, and for which E cannot be deformed into a set of dimension < d inside Ω. The point of the discretization is that there always are minimizers E_{τ} of J_{h_{τ}} on F_{τ}.

Then we prove that the E_{τ} are Almgren-quasiminimal sets, with bounds that do not depend on τ. This allows us to take limits, and eventually find a uniformly rectifiable set Z as in the statement. Roughly, the fact that H^{d}(Z) ≥ θ is a remainder of (1.10) for the sets E_{τ}, while H^{d}(Z \ F) ≤ η by an argument similar to (4.4). We need to be a little careful with the estimates, but at the end we get a proof.

5. Existence of minimizers?

It is time to discuss the existence of minimizers for our problem. As was alluded to above, Example 1.3 is more or less the only case that was explicitly treated, and for this one we could use semicontinuity and compactness in BV (the space of distributions whose gradient is a finite measure).

Here we just want to discuss a general plan to get existence results, and leave the difficulties for the future. So let us take the most naive approach and try to get a minimizer as a limit of a minimizing sequence. That is, take our class F, assume that m = inf{J_{h}(E) ; E ∈ F} is both finite and positive, and pick a sequence {E_{k}} such that lim_{k→+∞} J_{h}(E_{k}) = m. Is is easy to see that there is a subsequence that converges to some limit E. Notice that we do not have much choice on the notion of convergence for sets, because if it is not weak enough, we won’t be able to find convergent subsequences. So in the examples of Section 1, we use the Hausdorff distance on Ω. If there was a boundary condition on a set X, we should use convergence for Hausdorff distance on compact subsets of Ω \ X, with the risk of making (5.2) below even harder to verify. But let us concentrate on Examples 1.8 and 1.12.

Once we have a convergent minimizing sequence, we want to check that

(5.1)  J_{h}(E) ≤ lim inf_{k→+∞} J_{h}(E_{k}),

but also that

(5.2)  E ∈ F.

The main claim of this section is that even though (5.1) looks ominous, there is a good chance that it can be taken care of. Then (5.2) will be the main point of the discussion, and the outcome will depend on which classes F we consider.

It is not true that (5.1) holds for any convergent minimizing sequence. If we are not careful, E_{k} will have a useful part (that we want to keep), and lots of thin hair with essentially no contribution to J_{h}(E_{k}), but that may converge to anything (like the whole domain Ω). So we want to cut the hair first, and then only take limits. Notice that this is the approach that was taken by Reifenberg [Re], except perhaps that we would want an automatic way to cut the hair.

So here is a way to proceed. We start with a set E_{k} such that J_{h}(E_{k}) is almost minimal. Here let us only consider rectifiable sets; otherwise another argument would be needed to show that unrectifiability does not help. We first replace E_{k} with a polyhedral approximation E′_{k}. A brutal way to do it would be to use a dyadic grid of very small mesh size and use the Federer-Flemming construction to deform E_{k} onto a finite collection of faces of cubes (of dimensions at most d). This is what was done for the generalization of Theorem 4.1, but it is not good enough here, because when we do this, J_{h}(E_{k}) roughly gets multiplied by a constant, and we want a true minimizing sequence. So we should design an adapted (no longer dyadic) grid that often stays close to E_{k}, so that when we do the Federer-Flemming projection, we get a set E′_{k} such that J_{h}(E′_{k}) is very close to the minimum too. The usual polyhedral approximation lemma is not good enough, because we want our grid to be sufficiently regular (i.e., we do not want faces to make small angles, with uniform bounds). The construction of adapted grids is under way [Fv].

Now assume that we have E′_{k} and the adapted grid. Notice that if our class F is stable under deformations as in our examples, E′_{k} ∈ F too. We replace E′_{k} with a new set E′′_{k} ∈ F such that E′′_{k} is a finite union of faces of dimension ≤ d in the grid, and which minimizes J_{h}(E′′_{k}) under these constraints. Notice that E′_{k} is a competitor in this definition, and since there is a finite number of faces altogether, the minimizer E′′_{k} exists. This is the set with cut hair that we want to use.

Now there is an additional verification to perform. Due to the regularity of the grid, E′′_{k} should be a quasiminimal set, with estimates that do not depend on k. We checked something similar in [DS4], but with dyadic grids. The spirit is a little like in Section 2, but we use Federer-Flemming projections instead of our Lipschitz retraction. The point of all this is that once we know that E′′_{k} is a quasiminimal set, we can apply Theorem 3.2 and the uniform concentration property; the result is that the sets E_{k}^{′′∗} satisfy (3.5), and then (5.1).

Incidentally, this part of the argument should even tolerate boundary conditions, because (3.5) only requires quasiminimality locally.

An additional ad hoc argument will then be needed to take care of the lower dimensional set E′′_{k} \ E_{k}^{′′∗}, to make sure that it does not converge to a large set. Once this is done, we will get (5.1) (under quite general assumptions) and be happy for a short time.

The main point of this section is to say that the serious problem is with (5.2), which should be expected to depend on the specific situation. I expect (but did not check seriously) that there are ways to define classes F with linking conditions coming from algebraic topology, and that should be stable under small perturbations in Hausdorff distance. For these classes, if they exist, existence of minimizers would be reasonably easy to get (modulo the discussion above).

We’ll try to give in Section 7 at least one other case where we expect to have a solution (already with some work), that could be hard to recover by other means.

6. Currents and size-minimizers

At this stage, the reader probably wonders why we do not want to use currents like everyone else. A first, but not too good answer, is that the author knows almost nothing about currents mistakes and carefully avoided in the examples of Section 1 the cases where using currents would give a clear advantage.

A slightly better answer is that in many cases, the difficulties should be roughly the same. For instance, if we work in the classes F(E_{0}) of deformations of a given object, the main difficulty with any approach should be that we have to prove that limits of minimizing sequences lie in the same class, and that it seems hard to characterize F(E_{0}) algebraically. What Section 5 tries to do is ameliorate the most obvious difficulty with sets, i.e., the lack of lower semicontinuity for the Hausdorff measure. Note that by contrast, very good lower semicontinuity theorems are already available for currents.

One of the advantages of currents seems to be that it is much easier to define boundary conditions (as in the Plateau problem, for instance) in terms of integral currents. Incidentally, this does not mean that it is impossible to define boundary problems in terms of sets only: see [Re] and the last lines of this section.

So let us say a few words about currents. The reader is kindly asked to excuse the concentration of the gross mistakes that probably can be found in the next few lines. The most basic example of integral current of dimension d is the current of integration on a smooth oriented d-dimensional surface A without boundary, where for every d-form φ, T (φ) is the integral of φ on A. For general integral currents T of dimension d, there is a rectifiable set A of dimension d T) and a measurable function m : A → ℤ, such that ∫_{A} |m|dH^{d} < +∞ and

(6.1)  T (φ) = ∫_{A} m(x)φ(x) for every d-form φ.

The integral of m(x)φ(x) (and the hidden measurable orientation) on A can be defined, for instance, with disjoint coverings of A by smooth pieces with an orientation. In the case above of a smooth surface, we can take m = 1 in (6.1).

And there is a similar representation of the boundary ∂T [which, as we know, is defined by the duality formula ∂T (φ) = T (∂φ), where the second ∂ is the exterior derivative.] In fact, if the author did not forget something vital, the two properties above could be used as a definition of integral currents.

Return to the Plateau problem. A simple way to say that the surface σ is bounded by a given curve γ is to require that ∂S = Γ, where Γ is the current of integration on γ and S is the current of integration on σ. More generally, one may consider equations of the form

(6.2)  ∂S = ∂T,

where T is a given integral current of dimension d, and we look for integral currents S that minimize something. In the case of the curve γ, we just rewrote Γ = ∂T for some T; in general it is more convenient to write the equation as in (6.1), because it is a way to make sure that the boundary of the right-hand side is zero, and that there is at least one candidate S.

The most standard problem concerns integral currents S that minimize the mass

(6.3)  ||S|| = ∫_{A} |m(x)|dH^{d}(x),

where the rectifiable set A and the multiplicity m are as in (6.1). This is what is most often referred to as the Plateau problem, and there are very nice and comprehensive existence and regularity results.

But the setting of Section 1 rather corresponds to size minimizers, where the size of S is just the Hausdorff measure of A′ = {x ∈ A; m(x) ≠ 0} in the description above. In fact, we are using a slightly distorted version of size, since we are interested in J_{h}(A′) = ∫_{A′} h(x)dH^{d}(x).

Recall that size minimizers correspond to soap films, while mass minimizers correspond to the standard notion of minimal surface. As we have all seen one day or another, soap films can have mild singularities already in dimensions d = 1 and n = 2 (three half lines making 120^{∘} angles) or d = 2 and n = 3 (three walls making 120^{∘} angles or six equal angular sectors meeting at a point).

As far as the author knows, not much is known about existence for size minimizers. One of the main difficulties in the context of (6.2) seems to be that we need to find minimizing sequences {S_{k}} for which the masses do not tend to +∞, so that we can apply the standard compactness results for currents. Apparently new results on size minimizers were recently obtained by T. De Pauw and R. Hardt; they also use currents to generalize Reifenberg’s theorem on the existence of minimal sets of dimension 2 “algebraically freely” bounded by a given smooth one-dimensional manifold in ℝ^{3}, where the algebraic free boundary condition is a variant of the algebraic boundary condition of Adams and Reifenberg, but defined in terms of singular homology. Please ask them directly for details.

7. Guesses about 2-dimensional sets in ℝ^{3}

We conclude these notes with a rapid description of what could be needed for a solution to the existence problem when d = 2, n = 3, Ω is as in Example 1.12, and we try to minimize J_{h} in F(E_{0}) for some E_{0} ∈ F (see (1.9) and (1.11)). The point of choosing a class F(E_{0}) is that it is probably difficult to define or deal with algebraically, so the corresponding existence result would probably be new. We seem to have to take d = 2 and n = 3 because otherwise we do not know the local structure of minimal sets; see below.

First we need to find a minimizing sequence {E′′_{k}} as in Section 5. Suppose this is done. Recall that E′′_{k} is the union of a set E_{k}^{′′∗} of 2-faces, plus a union F_{k} of 1-faces (line segments). We extract a subsequence so that the E_{k}^{′′∗} converge to some limit E^{∗}, and we use the uniform concentration property to show that J_{h}(E^{∗}) = inf_{A∈F(E_{0})} J_{h}(A).

Now E^{∗} itself probably does not lie in F(E_{0}). For instance, take Ω = \overline{B}(0, 2) \ V, where V is an open solid torus contained in unit ball, and take E_{0} = ∂V ∈ F. Even with h = 1, we expect two types of minimizers for J_{h} in F(E_{0}): E_{0} itself (if V is thin enough), or else (if V is too fat) a stranger set composed of the boundary of the convex hull of V, plus a little wire that connects the two flat faces. The wire is part of the definition of F(E_{0}); we cannot get rid of it by deformation. But in the argument above, we expect E^{∗} to miss the wire. So we should not expect E^{∗} to be a deformation of E_{0} or the E′′_{k} in general.

Now we still can try to show that for k large, E^{∗} is a deformation of E_{k}^{′′∗}. This needs a proof too; the point could be that since it is the limit of the E_{k}^{′′∗}, E^{∗} is locally Almgrenminimizing (i.e., with M = 1 in Definition 2.7). Then, by a theorem of J. Taylor’s [Ta], we would get a good local description of E^{∗} and use it show that there is a local Lipschitz retraction f of a neighborhood of E^{∗} onto E^{∗}. This would be enough to deform E_{k}^{′′∗} onto E^{∗} for k large.

Finally, one should also take care of the wires F_{k} = E′′_{k} \ E_{k}^{′′∗}. But since they can be taken to have finite length, we just need to show that our nice retraction has a Lipschitz extension that leaves Ω alone, because then we can set E = f(E′′_{k}), and J_{h}(E) = J_{h}(E^{∗}) because f(F_{k}) is so small. We leave the details of the proof and the extension to more general situations as a long exercise for the reader. Remark 7.1 We expect less trouble, and even perhaps a solution in higher dimensions, with the class F itself. The point is that if E^{∗} can be deformed into a set of H^{d}-measure zero (or equivalently of dimension < d) inside Ω, the same deformation should apply to some E_{k}^{′′∗} close to E^{∗}, and (by extending its domain of definition and making it Lipschitz, if we can) send E′′_{k} \ E_{k}^{′′∗} to a small set too. This would eventually contradict the fact that E′′_{k} ∈ F.

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Guy David, Mathématiques, Bâtiment 425, Université de Paris-Sud, 91405 Orsay Cedex, France guy.david@math.u-psud.fr