Solutions of the (free boundary) ReifenbergPlateau problem

Camille Labourie

Abstract

We solve two variants of the Reifenberg problem for all coefficientgroups. We carry out the direct method of the calculus of variationand search a solution as a weak limit of a minimizing sequence. Thisstrategy has been introduced by De Lellis, De Philippis, De Rosa, Ghi-raldin and Maggi in [DLGM1],[DPDRG2],[DLDRG4],[DPDRG5] andallowed them to solve the Reifenberg problem. We use an analogousstrategy proved in [L] which allows to take into account the free bound-ary. Moreover, we show that the Reifenberg class is closed under weakconvergence without restriction on the coefficient group.

Contents

1 Definitions and main results 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Reifenberg competitors . . . . . . . . . . . . . . . . . . . . . . 41.3 Deformations and energies . . . . . . . . . . . . . . . . . . . . 7

2 Operations on Reifenberg competitors 11

3 Existence of Plateau solutions 193.1 Direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1 Definitions and main results

1.1 Introduction

We present the Reifenberg approach to the Plateau problem [R] (1960). Wefix a (d − 1)-dimensional compact boundary Γ ⊂ Rn (the boundary). TheReifenberg competitors are the compact sets of Rn which contain and spanΓ in the sense of algebraic topology.

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Definition (Reifenberg competitors). Fix a subgroup L of the homologygroup Hd−1(Γ). A Reifenberg competitor is a compact subset E ⊂ Rn suchthat E contains Γ and the morphism induced by inclusion

Hd−1(Γ) Hd−1(E)

is zero on L.

Here the homology theory is the Čech homology theory with a compactabelian coefficient group. Reifenberg minimizes the (spherical) Hausdorffmeasure of dimension d. He searches a minimizer as a limit of a minimizingsequence in Hausdorff distance. The continuity property of the Čech theoryensures that such limit is a competitor. However, the area is not lowersemicontinuous with respect to convergence in Hausdorff distance. One canimagine a minimizing sequence which has more and more dense tentaclesso that the limit set is too large. Reifenberg works with a compact abeliancoefficient group so as to benefit from the Exactness Axiom and to be ableto cut out the tentacles and patch the holes. His construction leads to analternative minimizing sequence for which the area is lower semicontinuous.

In [A1] (1968), Almgren solves a variant of the Reifenberg problem wherethe area is the integral of a (bounded) elliptic integrand and the coefficientgroup is a finitely generated abelian group. His competitors only span asingle given cycle of the boundary. Almgren relies on the theory of currents,flat chains and integral varifolds instead of the arguments of Reifenberg.

In [N] (1984), Nakauchi solves a free boundary variant of the problemwhere the area is the Hausdorff measure of dimension d and the coefficientgroup is a compact abelian group. What we mean by "free boundary" isthat Γ is not necessarily (d− 1)-dimensional and that the intersection E ∩Γcan vary among the competitors E.

E

ΓFigure 1: Competitors having afixed intersection with Γ.

E

ΓFigure 2: Competitors having afree intersection with Γ. The inter-section E ∩Γ may not be Hd negli-gible. One can minimize Hd(E \Γ)or the whole set Hd(E).

2

Definition (Nakauchi competitors). Fix a subgroup L of the homologygroup Hd−1(Γ). The Nakauchi competitors are the compacts sets E ⊂ Rn

such that for all v ∈ L, there exists u ∈ Hd−1(E ∩ Γ) such that i∗(u) = vand i′∗(u) = 0 where i∗ and i′∗ are the morphisms induced by inclusions:

Hd−1(Γ)

Hd−1(E ∩ Γ)

Hd−1(E).

i∗

i′∗

Nakauchi minimizesHd(E\Γ) but it would be also interesting to minimizethe whole set Hd(E) so as take into account the free part E ∩ Γ.

In [DP] (2007), De Pauw solves simultaneously the Reifenberg and theFederer–Fleming problems in R3. In the Reifenberg formulation, the area isthe Hausdorff measure of dimension 2 and the coefficient group is Z. De Pauwproves a natural equivalence between the homology of integral currents andthe Čech homology on locally connected sets. He then builds a minimizingsequence of the Federer–Fleming problem whose supports are a minimizingsequence of the Reifenberg problem. This approach allows to exploit theproperties of the currents sequence.

In [Fa] (2015), Fang gives a new solution to the Reifenberg and Nakauchiproblems (the free part E ∩ Γ is not taken into account). The area is theintegral of an elliptic integrand and the coefficient group is an arbitraryabelian group. His proof takes advantage of the lower semicontinuity of thearea on quasiminimal sets. Thanks to a construction of Feuvrier [Fe]), Fangobtains an alternative minimizing sequence composed of such sets. In [FK](2018), Fang and Kolasiński obtain the same existence result by followingAlmgren’s original ideas.

In [DLGM1] and [DPDRG2] (2014), De Lellis, De Philippis, De Rosa,Ghiraldin and Maggi introduce a new direct method based on the weakconvergence of minimizing sequences in Rn \ Γ. The area is given by theHausdorff measure of dimension d in the two first articles, and then by theintegral of an elliptic integrand in [DLDRG4], [DPDRG5]. The authors alsoapply this technique to solve the problem of Reifenberg but E ∩ Γ is nottaken into account and a compact abelian coefficient group is required. In[L], this notion of weak limit is generalized to quasiminimizing sequences inany ambiant space (even containing Γ). Thus, we can follow this strategyand take into account the free boundary.

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1.2 Reifenberg competitors

Notation We fix an integer n ≥ 1, an integer d such that 1 ≤ d ≤ n and aclosed set Γ of Rn (the boundary). Given a topological space S, an integer kand an abelian groupG, we denote byHk(S;G) the kth Čech homology groupof S over G. We abbreviate this notation as Hk(S) because the coefficientgroup is not significant in this paper. We fix a subgroup L of Hd−1(Γ).

We work with a definition of Reifenberg competitors which allows freeboundaries as in Fig. 1.1.

Definition 1.1 (Reifenberg competitor). A Reifenberg competitor is a com-pact subset E ⊂ Rn such that the morphism induced by inclusion

Hd−1(Γ) Hd−1(E ∪ Γ) (1)

is zero on L.

Our goal is to prove the two following results. We omit the regularityof the boundary here (see Theorems 3.2 and 3.3 for the full statement).Admissible energies are precised in Definition 1.4.

Theorem (with the free boundary). Let I be an admissible energy. Weassume that

m := inf { I(E) | E Reifenberg competitor } <∞ (2)

and that there exists a compact set C ⊂ Rn such that

m = inf { I(E) | E Reifenberg competitor, E ⊂ C } . (3)

Then there exists a Reifenberg competitor E ⊂ C such that I(E) = m.

The next theorem is equivalent to [Fa, Theorem 1.3] (which is based onFeuvrier’s construction) and [DPDRG5, Theorem 3.4] (which is based onweak convergence of minimizing sequences).

Theorem (without the free boundary). Let I be an admissible energy. Weassume that

m := inf { I(E \ Γ) | E Reifenberg competitor } <∞ (4)

and that there exists a compact set C ⊂ Rn such that

m = inf { I(E \ Γ) | E Reifenberg competitor, E ⊂ C } . (5)

Then there exists a Reifenberg competitor E ⊂ C such that I(E \ Γ) = m.

Remark. If Γ is compact and I(Γ) < ∞, this amounts to minimizing I(E)among Reifenberg competitors containing Γ.

4

In the rest of this subsection, we compare Definition 1.1 with othersdefinition of Reifenberg competitors. The next lemma shows that Definition1.1 is equivalent to the original definition of Reifenberg when Γ is regularenough and L 6= 0. It has been suggested by Ulrich Menne.

Lemma 1.2. Assume that Γ is a compact connected C1 manifold of dimen-sion d − 1 imbedded in Rn. Let a compact subset E ⊂ Rn be such that themorphism induced by inclusion

Hd−1(Γ) Hd−1(E ∪ Γ) (6)

is non injective. Then Γ ⊂ E.

Proof. We proceed by contraposition and assume that there exists x ∈ Γ\E.Let us introduce some preliminary objects. According to the TriangulationTheorem [W, Chapter IV, Theorem 12A], there exists a simplicial complex1

K and a homeomorphic map π : |K| → Γ. Let p ∈ |K| be the point suchthat π(p) = x. We recall that the star of p,

St(p) :=⋃{ int(σ) | σ ∈ K, p ∈ σ } , (7)

is an open set of |K|. Moreover, L := {σ ∈ K | p /∈ σ } is a subcomplex ofK such that |L| = |K| \ St(p). Since π−1(Γ \ E) is an open set containingp, we can replace K by its barycentric subdivision until St(p) ⊂ π−1(Γ \E).In conclusion, U := π(St(p)) is a relative open set of Γ, U ⊂ Γ \ E and thepair (Γ,Γ \ U) is triangulated by (K,L).

Now, we consider the commutative diagram

Hd−1(Γ) Hd−1(Γ,Γ \ U)

Hd−1(E ∪ Γ) Hd−1(E ∪ Γ, (E ∪ Γ) \ U)

(8)

where the arrows are induced by inclusions. We are going to show that

Hd−1(Γ) Hd−1(Γ,Γ \ U) (9)

is injective and then that

Hd−1(Γ,Γ \ U) Hd−1(E ∪ Γ, (E ∪ Γ) \ U) (10)

is an isomorphism. It will follow that

Hd−1(Γ) Hd−1(E ∪ Γ) (11)

1We follow the formalism of [ES, Chapter II] where a simplicial complex K is a finitecollection of faces and |K| is the set of points which belong to simplexes of K.

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is injective.The set Γ is a connected manifold so, according to [H, Theorem 3.26],

the morphismHd−1(Γ) Hd−1(Γ,Γ \ x) (12)

is injective with respect to the singular homology. The factorisation Γ ⊂(Γ,Γ \ U) ⊂ (Γ,Γ \ x) implies that

Hd−1(Γ) Hd−1(Γ,Γ \ U) (13)

is also injective with respect to the singular homology. Let us justify thatthis result can be passed on to the Čech homology. According to [ES, Chap-ter IX, Corollary 9.4], the Čech homology is an (exact) homology theoryon the category of triangulable pairs and their simplicial maps. Then, theuniqueness Theorem [ES, Chapter III, Theorem 10.1] implies that the Čechhomology and the singular homology are isomorphic in a natural way onthis category. As the pair (Γ,Γ \ U) is triangulated, we conclude that themorphism (13) is also injective with respect to the Čech homology.

The second part relies on the Excision axiom and the fact that U ⊂ Γ\E.The covering

E ∪ Γ = (E ∪ Γ \ U) ∪ (Γ \ E) (14)

is open in E ∪ Γ so E ∪ Γ is covered by the relative interiors of (E ∪ Γ) \ Uand Γ. We can apply the Excision axiom [ES, Chapter IX, Theorem 6.1] tosee that

Hd−1(Γ,Γ \ U) Hd−1(E ∪ Γ, (E ∪ Γ) \ U) (15)

is an isomorphism.

Next, we compare Definition 1.1 to the definition of Nakauchi seen inintroduction (or [N, Definition 1]). Let E be a compact subset of Rn andconsider the commutative diagram

Hd−1(Γ)

Hd−1(E ∩ Γ) Hd−1(E ∪ Γ).

Hd−1(E)

j∗i∗

i′∗ j′∗

Observe that E satisfies Definition 1.1 if and only if all elements of the form(v, 0) ∈ L ⊗ Hd−1(E) are in the kernel of j∗ − j′∗. And E is a Nakauchicompetitor if and only if all elements of the form (v, 0) ∈ L ⊗Hd−1(E) are

6

in the image of (i∗, i′∗). Assuming that the Mayer Vietoris sequence holds

for the sets Γ, E in E ∪ Γ, the following sequence is exact:

Hd−1(E ∩ Γ) Hd−1(Γ)⊗Hd−1(E) Hd−1(E ∪ Γ).(i∗,i′∗) j∗−j′∗

Thus, the Mayer Vietoris sequence implies that Definition 1.1 is equivalentto the definition of Nakauchi. In that sense, we consider these definitionsto be essentially equivalent. We favor Definition 1.1 because we can provethat it is closed under weak convergence without restriction on the coefficientgroup (see Lemma 2.5).

1.3 Deformations and energies

Notation Here the ambiant space is an open setX ofRn and Γ is a relativelyclosed set of X. The interval [0, 1] is denoted by the capital letter I. Givena set E ⊂ X and a function F : I × E → X, the notation Ft means F (t, ·).Given two sets A,B ⊂ Rn, the notation A ⊂⊂ B means that there exists acompact set K ⊂ Rn such that A ⊂ K ⊂ B.

Sliding deformations have been introduced by David [D, Definition 1.3] tostudy the restricted sets of Almgren with an additional boundary constraint.

Definition 1.3 (Sliding deformation along a boundary). Let E be a closed,Hd locally finite subset of X. A sliding deformation of E in an open setU ⊂ X is a Lipschitz map f : E → X such that there exists a continuoushomotopy F : I × E → X satisfying the following conditions:

F0 = id (16a)F1 = f (16b)∀t ∈ I, Ft(E ∩ Γ) ⊂ Γ (16c)∀t ∈ I, Ft(E ∩ U) ⊂ U (16d)∀t ∈ I, Ft = id in E \K, (16e)

where K is some compact subset of E ∩U . Alternatively, the last axiom canbe stated as

{ x ∈ E | ∃ t ∈ I, Ft(x) 6= x } ⊂⊂ E ∩ U. (17)

The next definition comes from [D, Definition 25.3 and Remark 25.87].It is a slight generalization of [A1, Definition 1.6(2)] and [A2, DefinitionIV.1(7)]. After the statement, we will give a few explanations and compareit to other definitions.

Definition 1.4 (Admissible energy). An admissible energy in X is a Borelregular measure I in X which satisfies the following axioms:

1. There exists Λ ≥ 1 such that Λ−1Hd ≤ I ≤ ΛHd.

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2. Let x ∈ X, let a d-plane V passing through x, let a C1 map f : V → Rn

be such that f(x) = x and Df(x) is the inclusion map−→V ↪→ Rn. Then

limr→0

I(f(V ∩B(x, r)))

I(V ∩B(x, r))= 1. (18)

3. For each x ∈ X, there exists R > 0 and ε : ]0, R] → R+ such thatB(x,R) ⊂ X, limr→0 ε(r) = 0 and

I(V ∩B(x, r)) ≤ I(S ∩B(x, r)) + ε(r)rd (19)

whenever 0 < r ≤ R, V is a d-plane passing through x and S ⊂ B(x, r)is a compact Hd finite set which cannot be mapped into V ∩ ∂B(x, r)by a Lipschitz mapping ψ : B(x, r) → B(x, r) such that ψ = id onV ∩ ∂B(x, r).

The third axiom is important to establish the lower semicontinuity. Letus say that (Ek) is a minimizing sequence of competitors such that I Ek → µwhere µ is a d-rectifiable Radon measure. The third axiom is the mainargument to show that

I E∞ ≤ µ, (20)

where E∞ = spt(µ). This yields in particular I(E∞) ≤ limk I(Ek). Thereader can find an example below Definition 25.3 in [D] where I(E∞) is toolarge.

Here is an example of admissible energy. We consider two Borel functions,

i : X ×G(d, n)→ ]0,∞[ and j : X → ]0,∞[ (21)

called the integrands and we define the corresponding energy by the formula

I(S) =

∫Sr

i(y, TyS) dHd(y) +

∫Su

j(y) dHd(y) (22)

where S ⊂ X is a BorelHd finite set and Sr, Su are its d-rectifiable and purelyd-unrectifiable parts. We assume Λ−1 ≤ i, j ≤ Λ so that Λ−1Hd ≤ I ≤ ΛHdon such sets S. We define of course I(S) = ∞ on Borel sets S ⊂ X ofinfinite Hd measure and we extend I on all subsets of X by Borel regularity.Next, we assume that i is continuous on X ×G(d, n) and we show that thisimplies the second axiom. Let x, V and f be as in (ii). To shorten a bitthe notations, we assume x = 0 and we denote i(0, V ) by i0. For r > 0, wedenote B(0, r) by Br and f(V ∩B(0, r)) by Sr. Fix ε > 0. According to theinverse function theorem, there exists R > 0 such that f induces a (1 + ε)-bilipschitz diffeomorphism from V ∩ BR to S = SR. Note that the functiony 7→ (y, TyS) is continuous on S and that at y = 0, (y, TyS) = (0, V ). As iis continuous, there exists R′ > 0 such that for all y ∈ S ∩BR′

(1 + ε)−1i0 ≤ i(y, TyS) ≤ (1 + ε)i0. (23)

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If r > 0 is small enough so that Br ⊂ BR ∩ f−1(BR′), we have Sr ⊂ S ∩BR′and thus

(1 + ε)−1i0 ≤I(Sr)

Hd(Sr)≤ (1 + ε)i0. (24)

The function f is (1 + ε)-bilipschitz on V ∩Br so this simplifies to

(1 + ε)d−1i0 ≤I(Sr)

Hd(V ∩Br)≤ (1 + ε)d+1i0. (25)

We deduce thatlimr→0

I(Sr)

Hd(V ∩Br)= i0 (26)

and by the same reasoning,

limr→0

I(V ∩Br)Hd(V ∩Br)

= i0. (27)

The second axiom follows. It is more difficult to construct integrands thatyield the third axiom. A first example is when I = Hd or when i dependson x alone with i = j (we will detail this later). Let us try to relate ourintegrands to other definitions. The energy introduced by Almgren in [A1,Definition 1.6(2)] is like above with X = Rn and i of class C3 but (iii) isreplaced by a stronger condition called ellipticity bound. We present thiscondition in more detail. For x ∈ Rn, let

Ix(S) =

∫Sr

i(x, TyS) dHd(y) + j(x)Hd(Su) (28)

be defined on Borel Hd finite sets S ⊂ Rn. Almgren requires that thereexists a continuous function c : Rn →]0,∞[ such that for all x ∈ Rn,

Ix(S ∩B(x, r))− Ix(V ∩B(x, r))

≥ c(x)[Hd(S ∩B(x, r))−Hd(V ∩B(x, r))

](29)

whenever r > 0, V is a d-plane passing through x and S ⊂ B(x, r) is acompact, Hd rectifiable, Hd finite set which spans V ∩ ∂B(x, r). In [A2,Definition IV.1(7)], Almgren restricts this definition to the functions c whichare positive constants. The energy introduced by De Lellis, De Philippis, DeRosa, Ghiraldin and Maggi in [DPDRG5, Definitions 1.3, 1.5 and Remark1.8] is also like above with X = Rn and i of class C1 but (iii) is replaced by anew axiom called atomic condition. This axiom characterizes the integrandsfor which Allard’s rectifiability theorem holds ([DPDRG3]). In codimensionone, it is equivalent to a convexity condition on the integrand. In the general

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case, [DRK] shows that the atomic condition implies an ellipticity conditionof the form[Hd(S ∩B(x, r))−Hd(V ∩B(x, r))

]> 0

=⇒ [Ix(S ∩B(x, r))− Ix(V ∩B(x, r))] > 0 (30)

whenever r > 0, V is a d-plane passing through x and S ⊂ B(x, r) is acompact, Hd finite set which spans V ∩ ∂B(x, r).

In this paragraph, we compare (iii) to Almgren’s ellipticity bound. Weallow d-dimensional sets S which have a purely unrectifiable part becausewe allow competitors which have a purely unrectifiable part in the directmethod (Proposition 3.1). Almgren only tests the ellipticity bound againstrectifiable sets but [DRK, Corollary 5.13] justifies that this is not a loss ofgenerality. Next, we show that an inequality such as (29) implies (19). Wework at the point x = 0 and we fix a constant c > 0. Let r > 0 (to be chosensmall enough), let V be a d-plane passing through 0, let S ⊂ B(0, r) be acompact Hd finite set such that V ∩B(0, r) ⊂ pV (S) and let us assume that

I0(S ∩B(0, r))− I0(V ∩B(0, r))

≥ c[Hd(S ∩B(0, r))−Hd(V ∩B(0, r))

]. (31)

Note that Hd(V ∩ B(0, r)) ≤ Hd(S ∩ B(0, r)) because V ∩ B(0, r) ⊂ pV (S)and pV is 1-Lipschitz. We define

ε(r) = sup { |i(y,W )− i(0,W )| | y ∈ B(0, r), W ∈ G(d, n) }+ sup { |j(y)− j(0)| | y ∈ B(0, r) } . (32)

The function i is uniformly continuous on the compact set B(0, 1)×G(d, n)and j as well on B(0, 1) so limr→0 ε(r) = 0. We assume r small enough sothat ε(r) ≤ c. According to the definition of ε(r),∣∣I(S ∩B(0, r))− I0(S ∩B(0, r))

∣∣ ≤ ε(r)Hd(S ∩B(0, r)) (33)

and similarly∣∣I(V ∩B(0, r))− I0(V ∩B(0, r))∣∣ ≤ ε(r)Hd(V ∩B(0, r)). (34)

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Plugging this into (31), we get

I(S ∩B(0, r))− I(V ∩B(0, r))

≥ c[Hd(S ∩B(0, r))−Hd(V ∩B(0, r))

]− ε(r)Hd(S ∩B(0, r))− ε(r)Hd(V ∩B(0, r))

(35)

≥ (c− ε(r))[Hd(S ∩B(0, r))−Hd(V ∩B(0, r))

]− 2ε(r)Hd(V ∩B(0, r))

(36)

≥ −2ε(r)Hd(V ∩ B(0, r)) (37)

≥ −2ε(r)ωdrd (38)

where ωd is the Hd measure of the d-dimensional unit disk. This is (19) aspromised. We should not forget to mention that if i depends on x alone withi = j, then we have I0 = i(0)Hd so (31) is clearly verified.

2 Operations on Reifenberg competitors

We come back to Reifenberg competitors and their properties. We presentthe main operations that preserve these competitors. Unless otherwise indi-cated, Γ is just a closed set of Rn.

Lemma 2.1. Let E be a Reifenberg competitor. Let F be a compact subsetof Rn containing E. Then F is a Reifenberg competitor.

Proof. This follows from the following commutative diagram

Hd−1(Γ) Hd−1(E ∪ Γ)

Hd−1(F ∪ Γ)

where the arrows correspond to the morphisms induced by inclusion.

Lemma 2.2. Let E be a Reifenberg competitor. Let f : E ∪ Γ → Rn bea continuous map such that there exists a continuous map F : I × Γ → Γsatisfying F0 = id and F1 = f . Then f(E) is a Reifenberg competitor.

Proof. Consider the following commutative diagram

Hd−1(Γ) Hd−1(E ∪ Γ)

Hd−1(Γ) Hd−1(f(E) ∪ Γ)

f∗ f∗

where the unlabeled arrows are the morphisms induced by inclusion. Asf : Γ→ Γ is homotopic to id, f∗ = id on Hd−1(Γ).

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This lemma assumed f to be defined on E∪Γ but the image f(E) dependsonly on the values of f on E. The two following remarks complete the lemmaby justify that it is generally enough for f to be defined on E. The secondremark applies to sliding deformations when Γ is regular enough.

Remark 2.3. Let f : E → Rn be a continuous map such that f = id onE ∩ Γ. As E and Γ are closed sets of Rn, the gluing

g =

{f in Eid in Γ

(39)

is continuous. Then Gt = (1− t)id + tg is a continuous homotopy from id tog and Gt = id on Γ. We deduce that f(E) is a Reifenberg competitor.

Remark 2.4. Let f : E → Rn be a continuous map such that there exists acontinuous map F : I × (E ∩ Γ)→ Γ satisfying F0 = id and F1 = f . Let usassume that Γ is a neighborhood retract i.e. there exists an open set O ⊂ Rn

and a continuous map r : O → Γ such that r = id on Γ. According to theHomotopy Extension Lemma, F extends as a continuous map F : I×Γ→ Γ.Moreover, the gluing

g =

{f in EF1 in Γ

(40)

is continous because E and Γ are closed sets of Rn. We deduce that f(E) isa Reifenberg competitor.

Lemma 2.5. Let (Ek) ⊂ Rn be a sequence of Reifenberg competitors. LetE be a compact subset of Rn. We assume that

1. there exists a compact set C ⊂ Rn such that for all k, Ek ⊂ C;

2. for all open sets V containing E ∪ Γ,

limkHd(Ek \ V ) = 0. (41)

Then E is a Reifenberg competitor.

The proof requires a preliminary lemma about the general position ofspheres. For x ∈ Rn and r > 0, let S(x, r) denote the Euclidean sphere ofcenter x and radius r of Rn. Given an integer k, a k-sphere is an Euclideansphere of positive radius relative to a (k + 1)-affine plane. We extend thisdefinition to k < 0, by calling k-sphere the empty set.

Lemma 2.6. Let Sk be a k-sphere in Rn and let x be a point in Rn. Thenfor all r > 0 (except for at most one value), Sk ∩ S(x, r) is a subset of a(k − 1)-sphere.

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Proof. We assume k ≥ 1. The proof is based on the observation that theintersection of a (n−1)-sphere with a k-affine plane is either empty, a point,or a (k− 1)-sphere. In all cases, this intersection is part of a (k− 1)-sphere.Let P0 be the (k+ 1)-affine plane associated to Sk, let x0 ∈ P0 be the centerof Sk and r0 > 0 be its radius so Sk = P0 ∩ S(x0, r0). For r > 0, a pointy ∈ Sk ∩ S(x, r) is characterized by the system

y ∈ P0

|y − x| = r

|y − x0| = r0

(42a)(42b)(42c)

or equivalently y ∈ P0

|y − x| = r

|y − x|2 − |y − x0|2 = r2 − r20.

(43a)(43b)

(43c)

Assume x = x0. If r 6= r0 (this excludes one value of r), equation (43c) hasno solutions so Sk ∩ S(x, r) is empty so is part of a (k − 1)-sphere. Assumex 6= x0. Equation (43c) defines an hyperplane and, if |x− x0|2 6= r2 − r2

0

(this excludes at most one value of r), this hyperplane does not contain x0.Then, the intersection of the two planes (43a) and (43c) is included in ak-affine plane. The intersection of this plane with the sphere (43b) is partof a (k − 1)-sphere as seen in introduction.

The following proof makes use of the notion of complex (as in [L, Sub-section 2.1]) and the Federer–Fleming projection (as in [DS, Propositon 3.1]or [L, Lemma 2.9]).

Proof of Lemma 2.5. By Lemma 2.1, the sequence (Ek ∪ E)k also satisfiesthe lemma assumptions. So without loss of generality, we assume that forall k, E ⊂ Ek. We also assume that Γ is non-empty so that the coveringsconsidered in the proof are also non-empty. We define a general covering asan open family γ = (γj)j∈Vγ of Rn satisfying the following properties:

1. there exists k such that Ek ∪ Γ ⊂⋃j∈Vγ γj ;

2. for every subset S ⊂ Vγ of cardinal d+ 1,⋂S

γj 6= ∅ =⇒ (E ∪ Γ) ∩⋂S

γj 6= ∅. (44)

The main goal of the proof is to show that for any open covering α = (αi)i ofE ∪Γ, there exists a general covering γ = (γj)j∈Vγ such that ((E ∪Γ)∩ γj)jis a refinement of α. Let us explain how to conclude from there. Let γ be

13

a general covering and let k be an index such that Ek ∪ Γ ⊂⋃j∈Vγ γj . The

inclusions Γ ⊂ E ∪ Γ ⊂ Ek ∪ Γ induce morphisms i∗ and j∗:

Hd−1(Γ) Hd−1(E ∪ Γ) Hd−1(Ek ∪ Γ).i∗ j∗

The covering γ induces simplicial complexes

K(Γ) = {S ⊂ Vγ finite | Γ ∩⋂S

γj 6= ∅ } , (45a)

K(E ∪ Γ) = {S ⊂ Vγ finite | (E ∪ Γ) ∩⋂S

γj 6= ∅ } , (45b)

K(Ek ∪ Γ) = {S ⊂ Vγ finite | (Ek ∪ Γ) ∩⋂S

γj 6= ∅ } (45c)

and the inclusions K(Γ) ⊂ K(E ∪ Γ) ⊂ K(Ek ∪ Γ) induce morphisms iγ∗and jγ∗:

Hd−1(K(Γ)) Hd−1(K(E ∪ Γ)) Hd−1(K(Ek ∪ Γ)).iγ∗ jγ∗

Finally, there exists projection morphisms πΓ, π, πk such that the followingdiagramm commutes

Hd−1(Γ) Hd−1(E ∪ Γ) Hd−1(Ek ∪ Γ)

Hd−1(K(Γ)) Hd−1(K(E ∪ Γ)) Hd−1(K(Ek ∪ Γ)).

i∗

πΓ

j∗

π πk

iγ∗ jγ∗

As Ek is a Reifenberg set, we have j∗ ◦ i∗ = 0 on L. The second axiomof general coverings implies that the simplicial complexes K(E ∪ Γ) andK(Ek ∪ Γ) have the same d-simplexes. Hence the d-chains of K(E ∪ Γ) andK(Ek ∪ Γ) are identical and they induce the same boundaries. We deducethat jγ∗ is injective and then that iγ∗ = 0 on πΓ(L). Since every opencovering α of E ∪ Γ is refined by such covering γ, we conclude that i∗ is nulon L.

Step 1. We fix a relative open covering α = (αi)i of E ∪ Γ and we builda locally finite open sequence β = (βj)j∈N in Rn such that

1. β cover E ∪ Γ and ((E ∪ Γ) ∩ βj)j is a refinement of α;

2. for every non-empty finite set S ⊂ N, the intersection of boundaries⋂S ∂βi is included in a finite union of (n−m)-spheres, where m is the

cardinal of S;

3. for every non-empty finite set S ⊂ N,⋂S

βj 6= ∅ =⇒ (E ∪ Γ) ∩⋂S

βj 6= ∅. (46)

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We work with the closed set F := E ∪ Γ. For all x ∈ F , there exists i suchthat x ∈ αi so there exists an open ball B centred at x such that F∩2B ⊂ αi.We extract a sequence of open ball (Bj)j∈N covering F such that (2Bj)j islocally finite in Rn and (F ∩ 2Bj)j is a refinement of α. Next, we build byinduction an open sequence (βj)j∈N such that for all j,

1. F ∩Bj ⊂ βj and there exists i such that F ∩ βj ⊂ αi.

2. for every non-empty set S ⊂ { 1, . . . , j }, the intersection of boundaries⋂S ∂βi is included in a finite union of (n−m)-spheres, where m is the

cardinal of S;

3. for every non-empty set S ⊂ { 1, . . . , j },⋂S

βi 6= ∅ =⇒ F ∩⋂S

βi 6= ∅. (47)

We start with β0 = 2B0. Let us assume that β0, . . . , βj−1 has been built.For all x ∈ F ∩Bj , there exists an open ball B centered at x such that

1. B ⊂ 2Bj ;

2. for every set S ⊂ { 1, . . . , j − 1 }, the intersection of boundaries ∂B ∩⋂S ∂βi is included in a finite union of (n−m− 1)-spheres, where m is

the cardinal of S;

3. for every set S ⊂ { 1, . . . , j − 1 } such that (F ∩Bj) ⊂ Rn \⋂S βj , we

have B ⊂ Rn \⋂S βj . Equivalently,

B ∩⋂S

βj 6= ∅ =⇒ (F ∩Bj) ∩⋂S

βj 6= ∅. (48)

The first and third condition mean that the radius of B is small enough.The second conditions holds for almost all radius of B according to Lemma2.6. Extract a finite covering of F ∩Bj by such balls B and denote βj theirunion. Then βj solves the next step of the induction.

Step 2. We complete the family β with an open set β∞ to obtain a coveringof one of the Ek. We take care not to introduce new d-simplexes in thesense of (45). We want to reduce the problem to the case where for some k,Ek \

⋃j βj is a (d−1)-dimensional grid. Using a Federer–Fleming projection,

we are going to project Ek in a (d − 1)-dimensional grid away from E ∪ Γ.Let ` > 0 and consider a complex K describing a uniform grid of sidelength(√n)−1` in Rn. Note that

Rn =⋃{A | A ∈ K } =

⋃{ int(A) | A ∈ K } (49)

15

and the cells of K have a diameter ≤ `. We select the cells in which we wantto perform the Federer–Fleming projection. Let B0 be an open ball suchthat for all k, Ek ⊂ B0. Let L be the subcomplex of K defined by

L = {A ∈ K | ∃x ∈ A, x ∈ 2B0 and d(x,E ∪ Γ) ≥ 2` } . (50)

Since Rn is covered by the interiors of cells A ∈ K, it is clear that

{x ∈ 2B0 | d(x,E ∪ Γ) ≥ 2` } ⊂⋃{ int(A) | A ∈ L } . (51)

The set E∪Γ is a closed set included in⋃j βj so the function x 7→ d(x,E∪Γ)

is positive on Rn \⋃j βj . In particular, it has a positive minimum on the

compact set 2B0 \⋃j βj . This minimum does not depend on ` so we can

assume ` small enough so that for all x ∈ 2B0 \⋃j βj , d(x,E ∪ Γ) > 4`. By

contraposition,

{x ∈ 2B0 | d(x,E ∪ Γ) ≤ 4` } ⊂⋃j

βj . (52)

Next, we introduce the Federer–Fleming projection of Ek ∩ |L| in L. First,we justify that limkHd(Ek ∩ |L|) = 0. By local finitness of K, |L| is a closedset of Rn. Since the cells of K have a diameter ≤ `, the definition of Limplies that the cells of L cannot meet E ∪ Γ. The set V = Rn \ |L| is openand contains E ∪ Γ so according to the lemma assumptions,

limkHd(Ek ∩ |L|) = 0. (53)

Let an index k be such that Hd(Ek∩|L|) <∞ (it will be precised later). Weapply [L, Lemma 2.9] or [DS, Propositon 3.1] and we obtain a continuousmap φ : |L| → |L| such that

1. for all A ∈ L, φ(A) ⊂ A;

2. φ(Ek ∩ |L|) ⊂ |L| \⋃{ int(A) | A ∈ L, dim(A) > d };

3. for all A ∈ Ld,

Hd(φ(Ek ∩ |L|) ∩A) ≤ CHd(Ek ∩ |L|) (54)

where C is a positive constant that depends only on n. When k is big enough(depending on `), Hd(Ek ∩ |L|) becomes sufficiently small so that one canperform additional projections in the d-dimensional cells of L. The secondaxiom becomes

φ(Ek ∩ |L|) ⊂ |L| \⋃{ int(A) | A ∈ L, dim(A) ≥ d } ; (55)

16

and thus

φ(Ek ∩ |L|) ∩⋃{ int(A) | A ∈ L }

⊂⋃{ int(A) | A ∈ L, dim(A) ≤ d− 1 } . (56)

The sets E∪Γ and |L| are disjoint and closed so we can extend φ continuouslyon E ∪ Γ by φ = id. Observe that |φ− id| ≤ ` because φ preserves the cellsof L. We can extend φ continuously on Rn in such a way that |φ− id| ≤ `.Now, let us show that

φ(Ek) ⊂⋃{A | A ∈ L, dim(A) ≤ d− 1 } ∪

⋃j∈N

βj . (57)

Remember that Ek ⊂ B0. We assume that ` is less than the radius of B0

whence φ(Ek) ⊂ 2B0. For x ∈ Ek, we distinguish two cases. If d(x,E ∪Γ) ≤ 3`, then d(φ(x), E ∪ Γ) ≤ 4` and we have φ(x) ∈

⋃j βj by (52). If

d(x,E∪Γ) ≥ 3`, then we have both d(x,E∪Γ) ≥ 2` and d(φ(x), E∪Γ) ≥ 2`so (51) shows that x and φ(x) belongs to

⋃{ int(A) | A ∈ L }. Then by (56),

we have φ(x) ∈⋃{A | A ∈ L, dim(A) ≤ d− 1 }.

We are all set to introduce the set

β∞ = Rn \ (E ∪ Γ ∪⋃|S|=d

⋂S

βj). (58)

First, we justify that the set β∞ is open. It suffices to show that the family(⋂S βj

)|S|=d is locally finite in Rn. In step 1, we have built the family

(βj)j∈N in such a way that it is locally finite in Rn so for all x ∈ Rn, thereexists an open set U containing x such that

S0 := { j ∈ N | U ∩ βj 6= ∅ } (59)

is finite. Let S be any subset of N with cardinal d such that U meets⋂S βj .

In fact, U meets also⋂S βj because U is open so S ⊂ S0. We deduce that

there exists only a finite number of sets S ⊂ N of cardinal d such that Umeets

⋂S βj 6= ∅. We have proved that

(⋂S βj

)|S|=d is locally finite in Rn.

Next, we justify that β∞ does not add new d-simplexes. The definition ofβ∞ shows that for all S ⊂ N of cardinal d,

β∞ ∩⋂S

βj = ∅ (60)

so for all set S ⊂ N∪{∞} of cardinal d+1, the condition⋂S βj 6= ∅ implies

S ⊂ N. Finally, we would like to have the covering

φ(Ek) ⊂ β∞ ∪⋃j∈N

βj . (61)

17

This is where the projection of Ek in a (d−1)-dimensional grid outside⋃j βj

is useful. According to (57), the condition (61) holds if⋃{A | A ∈ L, dim(A) ≤ d− 1 } \ β∞ ⊂

⋃j

βj . (62)

As E ∪ Γ ⊂⋃j βj , this amounts to say that for all S ⊂ N of cardinal d,⋃{A | A ∈ L, dim(A) ≤ d− 1 } ∩

⋂S

βj ⊂⋃j

βj . (63)

We are going to see that for after a suitable translation of K, the set

|Kd−1| :=⋃{A | A ∈ K, dim(A) ≤ d− 1 } (64)

is disjoint from⋂S ∂βj for all S ⊂ N of cardinal d. Fix such a set S. The set⋂

S ∂βj is included in a finite union of (n−d)-spheres so for all (d−1)-linearplane P ,

Ln(⋂

S

∂βj + P

)= 0 (65)

and in particular,

Ln(⋂

S

∂βj + (−|Kd−1| )

)= 0. (66)

This means that for almost every x ∈ Rn, x+|Kd−1| is disjoint from⋂S ∂βj .

There are only a countable number of sets S ⊂ N of cardinal d so we canfind x such that this is true for all of them. We replace K by x+K so that(61) holds.

Step 3. We build the general covering γ. We consider the domain Vγ =N∪{∞} and for j ∈ Vγ , we consider the open set γj = φ−1(βj). Rememberthat φ = id on E ∪ Γ so for all j ∈ Vγ ,

(E ∪ Γ) ∩ γj = (E ∪ Γ) ∩ βj (67)

and in particular, (E ∪ Γ) ∩ γ∞ = ∅. By definition of (βj)j∈N, the family((E∪Γ)∩γj)Vγ is a refinement of α. The family γ also covers Ek∪Γ because

φ(Ek ∪ Γ) ⊂ φ(Ek) ∪ Γ ⊂⋃j∈Vγ

βj . (68)

Finally, we show that for all S ⊂ Vγ of cardinal d+ 1,⋂S

γj 6= ∅ =⇒ (E ∪ Γ) ∩⋂S

γj 6= ∅. (69)

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If⋂S γj 6= ∅, then

⋂S βj 6= ∅ and thus S ⊂ N because β∞ does not add new

d-simplexes. By definition of (βj)j∈N, this implies (E ∪ Γ) ∩⋂S βj 6= ∅ or

equivalently,(E ∪ Γ) ∩

⋂S

γj 6= ∅ (70)

because γj coincides with βj on E ∪ Γ.

3 Existence of Plateau solutions

We solve two formulations of the Reifenberg Plateau problem. In the firstone, we work in X = Rn and minimize Id(E) among Reifenberg competitorsE. In the second one, we work in X = Rn \ Γ (that is, away from theboundary) and minimize Id(E \ Γ) among Reifenberg competitors E. Inthis case, we do not require regularity on the boundary.

3.1 Direct method

We are going to recall the direct method [L, Corollary 4.1]. This is the samedirect method as in [DLGM1] but this version allows any ambient space(even containing the boundary).

The ambient space is an open set X of Rn. See [L, Definition 1.5] for thedefinition of a minimal set in X. See [L, Definition 1.8] for the definition ofa Whitney subset of X. It includes smooth compact manifolds imbedded inX.

Proposition 3.1 (Direct method). Fix a Whitney subset Γ of X. Fix anadmissible energy I in X. Fix C a class of closed subsets of X such that

m := inf { I(E) | E ∈ C } <∞ (71)

and assume that for all E ∈ C, for all sliding deformations f of E in X,

m ≤ I(f(E)). (72)

Let (Ek) be a minimizing sequence for I in C. Up to a subsequence, thereexists a coral2 minimal set E∞ with respect to I in X such that

I Ek ⇀ I E∞. (73)

where the arrow ⇀ denotes the weak convergence of Radon measures in X.In particular, I(E∞) ≤ m.

In the works of Reifenberg, the Hausdorff distance limit of a minimizingsequence is a competitor but the area is not lower semicontinuous. Withweak limits, the lower semicontinuity follows from the previous proposition.Moreover, we also proved in the last section that the weak limit is a com-petitor. There is not much work to do.

2A set E ⊂ X is coral in X if E is the support of Hd E in X. Equivalently, E isclosed in X and for all x ∈ E and for all r > 0, Hd(E ∩B(x, r)) > 0.

19

3.2 Applications

Theorem 3.2 (with the free boundary). Fix a Whitney subset Γ of Rn andfix a subgroup L of Hd−1(Γ). We assume that

m := inf { I(E) | E Reifenberg competitor } <∞ (74)

and that there exists a compact set C ⊂ Rn such that

m = inf { I(E) | E Reifenberg competitor, E ⊂ C } . (75)

Then there exists a Reifenberg competitor E ⊂ C such that I(E) = m.

Proof. We work in X = Rn and we consider the class

C = {E | E is a Reifenberg competitor } . (76)

By Lemma 2.2, the class C is closed under sliding deformations in Rn so itsatisfies the requirement of Proposition 3.1. Let (Ek) ⊂ C be a minimizingsequence of C. According to Proposition 3.1, there exists a coral set E∞ ofRn such that

I Ek ⇀ I E∞. (77)

We prove that E∞ is a Reifenberg competitor. First, we show that E∞ is acompact subset of C. Observe that Rn \C is an open set and that by lowersemicontinuity,

I(E∞ \ C) ≤ lim infkI(Ek \ C) = 0. (78)

This proves that the support of I E∞ is included in C. As E∞ is coral, E∞is included in C and in particular compact. We are going to apply Lemma2.5 to the set E∞. For all open set V containing E∞ ∪ Γ,

lim supkI(Ek \ V ) = lim sup

kI(Ek ∩ C \ V ) (79)

≤ I(E∞ ∩ C \ V ) (80)≤ 0. (81)

We conclude that E∞ is a Reifenberg competitor. Finally, we show thatI(E∞) = m. As E∞ is a Reifenberg competitor, we have of course I(E∞) ≥m. The fact that I(E∞) ≤ m has already been established in Proposition3.1.

The next theorem is equivalent to [Fa, Theorem 1.3] (which is based onFeuvrier’s construction) and [DPDRG5, Theorem 3.4] (which is based onweak limits of minimizing sequences).

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Theorem 3.3 (without the free boundary). Fix a closed set Γ of Rn and asubgroup L of Hd−1(Γ). We assume that

m := inf { I(E \ Γ) | E Reifenberg competitor } <∞ (82)

and that there exists a compact set C ⊂ Rn such that

m = inf { I(E \ Γ) | E Reifenberg competitor, E ⊂ C } . (83)

Then there exists a Reifenberg competitor E ⊂ C such that I(E \ Γ) = m.

Remark 3.4. If Γ is compact and I(Γ) < ∞, this amounts to minimizingI(E) among Reifenberg competitors containing Γ.

Proof. We work in X = Rn \ Γ (away from the boundary) and we considerthe class

C = {E \ Γ | E is a Reifenberg competitor } . (84)

By Lemma 2.2, the class C is closed under sliding deformations in X (theboundary is empty in X) so it satisfies the requirement of Proposition 3.1.Let (Ek) ⊂ C be a sequence of Reifenberg competitor such that (Ek \Γ) is aminimizing sequence of C. According to Proposition 3.1, there exists a coralset S∞ of X such that

I (Ek \ Γ) ⇀ I S∞ in X. (85)

We prove that there exists a Reifenberg competitor E∞ ⊂ C such thatS∞ = E∞ \Γ. First, we justify that S∞ ⊂ C. Observe that X \C is an openset of X and that by lower semicontinuity,

I(S∞ \ C) ≤ lim infkI((Ek \ Γ) \ C) = 0. (86)

As a consequence, the support of I S∞ in X is included in C. As S∞ iscoral in X, S∞ is included in C. Now, let

E∞ = (S∞ ∪ Γ) ∩ C. (87)

The set S∞ is closed in X so S∞ ∪ Γ is closed in Rn so E∞ is a compactsubset of C. Note also that E∞ \ Γ = S∞. We are going to apply Lemma2.5 to the set E∞. For all open set V containing E∞ ∪ Γ,

lim supkI(Ek \ V ) = lim sup

kI(Ek ∩ C \ V ) (88)

= lim supkI((Ek \ Γ) ∩ C \ V ) (89)

≤ I(E∞ ∩ C \ V ) (90)≤ 0. (91)

Thus, E∞ is a Reifenberg competitor and S∞ = E∞ \ Γ ∈ C. Finally, weshow that I(S∞) = m. As S∞ ∈ C, we have of course I(S∞) ≥ m. The factthat I(S∞) ≤ m has already been established in Proposition 3.1.

21

Acknowledgement: I would like to thank Guy David for his warm andhelpful discussions. I thank the anonymous reviewer which has improvedthis paper.

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Université Paris-Saclay, CNRS, Laboratoire de mathématiquesd’Orsay, 91405, Orsay, France.

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