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

Stefano Amato , Giovanni Bellettini and Maurizio Paolini

Published/Copyright: October 21, 2015

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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.

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Received: 2015-5-18

Revised: 2015-8-24

Accepted: 2015-9-14

Published Online: 2015-10-21

Published in Print: 2017-1-1

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Articles in the same Issue

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

Keywords for this article

Constrained BV functions; Plateau’s problem; coverings