Constrained BV functions on covering spaces
for minimal networks and Plateau’s type problems

Stefano Amato; Giovanni Bellettini; Maurizio Paolini

doi:10.1515/acv-2015-0021

Article

Constrained BV functions on covering spaces for minimal networks and Plateau’s type problems

, and

Published/Copyright: October 21, 2015

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Calculus of Variations Volume 10 Issue 1

Abstract

We link covering spaces with the theory of functions of bounded variation, in order to study minimal networks in the plane and Plateau’s problem without fixing a priori the topology of solutions. We solve the minimization problem in the class of (possibly vector-valued) BV functions defined on a covering space of the complement of an (n-2)-dimensional compact embedded Lipschitz manifold S without boundary. This approach has several similarities with Brakke’s “soap films” covering construction. The main novelty of our method stands in the presence of a suitable constraint on the fibers, which couples together the covering sheets. In the case of networks, the constraint is defined using a suitable subset of transpositions of m elements, m being the number of points of S. The model avoids all issues concerning the presence of the boundary S, which is automatically attained. The constraint is lifted in a natural way to Sobolev spaces, allowing also an approach based on Γ-convergence.

Keywords: Constrained BV functions; Plateau’s problem; coverings

MSC 2010: 49Q05; 49Q20; 57M10

Communicated by Frank Duzaar

A Appendix: An abstract covering construction

In this appendix we give an alternative construction of the covering of M built up in Section 1. The construction is standard (see, e.g., [14, 17]), and has the advantage to avoid all issues about the definition of admissible cuts. Setting up the minimization problem on the covering space MH below could have an independent interest; we have preferred to use the “cut and paste” construction (and next proving independence of the cuts) in order to deal with a more “handy” formula (like (2.12)) for the total variation of a BV function defined on the covering space.

Let Ω, S, M, and m be as in Section 1. Fix x0∈M, and set Cx0⁢([0,1];M):={γ∈C⁢([0,1];M):γ⁢(0)=x0}. For γ∈Cx0⁢([0,1];M), let [γ] be the class of paths in Cx0⁢([0,1];M) which are homotopic to γ with fixed endpoints. We recall that the universal covering of M is the pair (M~,𝔭), where M~:={[γ]:γ∈Cx0⁢([0,1];M)} and 𝔭:[γ]∈M~↦𝔭⁢([γ]):=γ⁢(1)∈M. A basis for the topology of M~ is given by the family

{[γ⁢λ]:[γ]∈M~,γ⁢(1)∈B⁢ open ball,λ∈C⁢([0,1];B),λ⁢(0)=γ⁢(1)}.

Let π1⁢(M,x0) be the first fundamental group of M with base point x0∈M, and let

H:={[ρ]∈π1⁢(M,x0):link⁢(ρ;S)≡0⁢(mod⁢m)}.

Remark A.1

The set H is a (normal) subgroup of π1⁢(M,x0) of index m.

For γ∈Cx0⁢([0,1];M), set γ¯⁢(t):=γ⁢(1-t) for all t∈[0,1]. Associated with H, we can consider the following equivalence relation ∼H on M~: for [γ],[λ]∈M~,

[γ]∼H[λ]⇔γ⁢(1)=λ⁢(1),link⁢(γ⁢λ¯;S)≡0⁢(mod⁢m).

We denote by [γ]H the equivalence class of [γ]∈M~ induced by ∼H, and we set

MH:=M~/∼H.

Letting 𝔭~H:M~→MH be the projection induced by ∼H, we endow MH with the corresponding quotient topology. We set 𝔭H,M:[γ]H∈MH↦γ⁢(1)∈M, so that we have the commutative diagram

(A.1)

and the pair (MH,𝔭H,M) is a covering of M, see [14, Proposition 1.36].

Let (Y,πY) be a covering of M, and let y0∈πY-1⁢(x0). By (πY)*:π1⁢(Y,y0)→π1⁢(M,x0) we denote the homomorphism defined as (πY)*⁢([ϱ]):=[πY∘ϱ]. By [14, Proposition 1.36], we have

(A.2)(𝔭H,M)*⁢(π1⁢(MH,[x0]H))=H.

Proposition A.2

Let Σ∈Cuts⁡(Ω,S). Then YΣ and MH are homeomorphic.

Proof.

Recall the notation in Section 1.1. By [14, p. 28], it is not restrictive to assume that x0∈O. Let y0∈π𝚺,M-1⁢(x0), and let [ϱ]∈π1⁢(Y𝚺,y0). By [14, Proposition 1.36], since H and (π𝚺,M)*⁢(π1⁢(Y𝚺,y0)) have the same index, the statement follows if we are able to prove that (π𝚺,M)*⁢([ϱ])∈H, or equivalently that

(A.3)link⁢(π𝚺,M∘ϱ;S)≡0⁢(mod⁢m),

where m is given in (1.2).

Let us first consider the case n=2. Notice that

(A.4)link⁢(π𝚺,M∘ϱ;S)=∑j=1mlink⁢(π𝚺,M∘ϱ;pj),

and, for any j=1,…,m, link⁢(π𝚺,M∘ϱ;pj) equals the number of times that π𝚺,M∘ϱ turns around pj, a counterclockwise (resp. clockwise) turn around pj being counted with positive (resp. negative) sign. By construction (see for instance Figure 3 when m=3), any counterclockwise (resp. clockwise) turn of π𝚺,M∘ϱ around a point in S corresponds to moving one sheet forward (resp. backward) in Y𝚺. Thus, the sum in the right-hand side of (A.4) is equal to the number of sheets visited by the loop ϱ until it comes back to y0. It is now clear that this number can be only a multiple of m, proving (A.3).

The case n>2 is even simpler, since we have m=2, and (A.3) follows noticing that [ϱ] can change sheet in Y𝚺 just an even number of times. ∎

Let 𝚺, 𝚺^∈𝐂𝐮𝐭𝐬⁢(Ω,S). By Proposition A.2, and by general results in coverings theory [14], there exists a homeomorphism f:Y𝚺→Y𝚺^ such that

(A.5)π𝚺,M=π𝚺^,M∘f.

The map f is defined by path-lifting. More precisely, fix x0∈M, and let y0∈Y𝚺, y^0∈Y𝚺^ be such that π𝚺,M⁢(y0)=x0=π𝚺^,M⁢(y^0). Let y∈Y𝚺, and let γ∈C⁢([0,1];Y𝚺) be such that γ⁢(0)=y0, γ⁢(1)=y. Then, f⁢(y)∈Y𝚺^ is defined as the ending point of the lift of π𝚺,M∘γ to Y𝚺^, starting at y^0.

References

[1] Almgren F. J., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 165 (1976). 10.1090/memo/0165Search in Google Scholar

[2] Ambrosio L., Fusco N. and Pallara D., Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Clarendon Press, Oxford, 2000. Search in Google Scholar

[3] Baldo S., Minimal interface criterion for phase transitions in mixtures of Cahn–Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 37–65. 10.1016/s0294-1449(16)30304-3Search in Google Scholar

[4] Bellettini G., An almost everywhere regularity result for minimal partitions, Boll. Unione Mat. Ital. (7) Ser. A 4 (1990), 57–63. Search in Google Scholar

[5] Brakke K., Soap films and covering spaces, J. Geom. Anal. 5 (1995), 445–514. 10.1007/BF02921771Search in Google Scholar

[6] Dal Maso G., An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993. 10.1007/978-1-4612-0327-8Search in Google Scholar

[7] David G., Should we solve Plateau’s problem again?, Advances in Analysis: The Legacy of Elias M. Stein, Princeton University Press, Princeton (2014). 10.23943/princeton/9780691159416.003.0006Search in Google Scholar

[8] De Lellis C., Ghilardin F. and Maggi F., A direct approach to Plateau’s problem, preprint 2014, http://arxiv.org/abs/1408.4047v2. 10.4171/JEMS/716Search in Google Scholar

[9] Dierkes U., Hildebrandt S. and Sauvigny F., Minimal Surfaces, Springer, Berlin, 2010. 10.1007/978-3-642-11698-8Search in Google Scholar

[10] Federer H., Geometric Measure Theory, Springer, Berlin, 1969. Search in Google Scholar

[11] Gilbert E. N. and Pollak H. O., Steiner minimal trees, SIAM J. Appl. Math. 16 (1969), 1–29. 10.1137/0116001Search in Google Scholar

[12] Hardt R. and Simon L., Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. of Math. (2) 110 (1979), 439–486. 10.2307/1971233Search in Google Scholar

[13] Harrison J., Soap film solutions to Plateau’s problem, J. Geom. Anal. 24 (2014), 271–297. 10.1007/s12220-012-9337-xSearch in Google Scholar

[14] Hatcher A. E., Algebraic Topology, Cambridge University Press, Cambridge, 2001. Search in Google Scholar

[15] Hirsch M. W., Differential Topology, Springer, New York, 1976. 10.1007/978-1-4684-9449-5Search in Google Scholar

[16] Maggi F., Sets of Finite Perimeter and Geometrical Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Stud. Adv. Math. 135, Cambridge University Press, Cambridge, 2012. 10.1017/CBO9781139108133Search in Google Scholar

[17] Massey W. S., Algebraic Topology. An Introduction, Springer, New York, 1987. Search in Google Scholar

[18] Modica L., The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal. 98 (1987), 123–142. 10.1007/BF00251230Search in Google Scholar

[19] Morgan F., Geometric Measure Theory: A Beginner’s Guide, Academic Press, Burlington, 2009. Search in Google Scholar

[20] Nitsche J. C. C., Lectures on Minimal Surfaces. Vol. 1: Introduction, Fundamentals, Geometry and Basic Boundary Problems, Cambridge University Press, Cambridge, 1989. Search in Google Scholar

[21] Osserman R., A Survey on Minimal Surfaces, Van Nostrand, New York, 1969. Search in Google Scholar

[22] Rolfsen D., Knots and Links, Mathematics Lecture Series, Houston, 1990. Search in Google Scholar

[23] Simon L., Survey lectures on minimal submanifolds, Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton University Press, Princeton (1983), 3–52. 10.1515/9781400881437-002Search in Google Scholar

[24] Simons J., Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. 10.2307/1970556Search in Google Scholar

[25] Taylor J. E., The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), 489–539. 10.2307/1970949Search in Google Scholar

[26] Wall C. T. C., Determination of the cobordism ring, Ann.of Math. (2) 72 (1960), 292–311. 10.2307/1970136Search in Google Scholar

Received: 2015-5-18

Revised: 2015-8-24

Accepted: 2015-9-14

Published Online: 2015-10-21

Published in Print: 2017-1-1

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/acv-2015-0021

Keywords for this article

Constrained BV functions; Plateau’s problem; coverings