Partial regularity for mass-minimizing currents in Hilbert spaces

Lemma A.1 (Cartesian products).

Consider an (n-1)-current T∈Mn-1⁢(H). Then the specification

for (φ,ψ)∈Lipb(R×H)×Lip(R×H)n defines a metric functional T with

Moreover:

1. If T∈𝐍n-1⁢(ℋ) is normal, then 〚0,1〛×T∈𝐍n⁢(ℝ×ℋ) is a normal n-dimensional current with

   (A.2)∂⁡(〚0,1〛×T)=(i1)♯⁢T-(i0)♯⁢T-〚0,1〛×(∂⁡T).

2. If T is (integer-)rectifiable, then also 〚0,1〛×T is an (integer-)rectifiable current.

Proof.

Clearly, 〚0,1〛×T is linear in φ and each ψi and is thus a metric functional. To prove (A.1) we consider (φ,ψ)∈Lipb(ℝ×ℋ)×Lip(ℝ×ℋ)n with Lip⁡(ψi)≤1 for all i∈{1,2,…,n}. Then from the definitions of the Cartesian product, the pushforward, and the mass, combined with the Lipschitz bound Lip⁡(ψi∘it)≤1, we readily deduce

By multilinearity and again by the definition of mass this implies (A.1) and in particular 𝐌⁢(〚0,1〛×T)<∞.

If T is a normal current, following the proof of [5, Proposition 10.2] one can show that 〚0,1〛×T satisfies the continuity axiom and the formula (A.2). Indeed, we will not discuss the adaption of the respective arguments, as the required changes^[10] are mostly notational ones. By applying (A.1) with ∂⁡T in place of T we obtain 𝐌⁢(〚0,1〛×∂⁡T)<∞; then (A.2) allows to conclude 𝐌⁢(∂⁡(〚0,1〛×T))<∞. As the locality axiom is easily verified, we thus have 〚0,1〛×T∈𝐍n⁢(ℝ×ℋ).

Finally, for T∈ℛn-1⁢(ℋ) we denote by (ST,θT,T→) a corresponding triplet as in Section 2.4 with countably (n-1)-rectifiable ST, and we write T→=⋀i=1n-1Ti with Borel functions Ti:ST→ℋ. As candidates for the multiplicity and the orientation of 〚0,1〛×T we consider ϑ:[0,1]×ST→(0,∞) with ϑ⁢(t,z):=θT⁢(z) and τ:[0,1]×ST→Λn⁢(ℝ×ℋ) with τ⁢(t,z):=(1,0)∧⋀i=1n-1(0,Ti⁢(z)). We moreover fix (φ,ψ)∈Lipb(ℝ×ℋ)×Lip(ℝ×ℋ)n, and we recall that the inner product of n-vectors is given by the determinant in (2.1); by Laplace expansion of this determinant we then get

for all (t,z)∈[0,1]×ST. We multiply this equality with φ⁢(t,z) and ϑ⁢(t,z)=θT⁢(z) and integrate with respect to ℋn-1 in z. Using additionally the representation (2.9) for the (n-1)-current T, we infer

for all t∈[0,1]. Now we integrate the resulting equality also in t, on the left-hand side we involve the coarea formula [6, Theorem 9.4] with area factor 1 on the countably n-rectifiable set [0,1]×ST, and on the right-hand side we exploit the definition of 〚0,1〛×T. We then find

All in all, we have thus shown that 〚0,1〛×T is the current which is induced by the triplet ([0,1]×ST,ϑ,τ) in the sense of (2.9). By [5, Theorem 9.1] this current is rectifiable.

Finally, for T∈In-1⁢(ℋ) the multiplicity θT can be chosen ℕ-valued. Consequently, the above function ϑ is ℕ-valued, and in conclusion we get 〚0,1〛×T∈In⁢(ℝ×ℋ) in this situation. ∎

Remark A.2.

The following two observations concern the rectifiability part of the preceding argument. They are scarcely relevant for our purposes, but may still be worth pointing out:

1. For rectifiable currents T one can actually improve the estimate (A.1) – by similar arguments as above and the representation of mass in (2.11) – to the equality

   ∥〚0,1〛×T∥=(ℒ1⁢⌞⁢[0,1])⊗∥T∥.

   This situation may be compared to [22, 2.10.45, 3.2.23].

2. The rectifiability of 〚0,1〛×T can alternatively be proved by the following shorter and more elementary reasoning provided that T is normal and rectifiable: By definition T is concentrated on a countably (n-1)-rectifiable set S, and then (A.1) implies that 〚0,1〛×T is concentrated on the countably n-rectifiable set [0,1]×S. By [5, Theorem 3.9]^[11] this property already implies the rectifiability of 〚0,1〛×T.

Lemma A.3 (Homotopy retraction on a graph).

Consider a current T∈Nn-1⁢(H) with ∂⁡T≡0, an n-plane π in H, and a Lipschitz function f:pπ⁢(spt⁡T)→(Span⁡π)⊥. If we have

then there exists another current V∈Nn⁢(H) with spt⁡V⊂(pπ)-1⁢(pπ⁢(spt⁡T)) and

Additionally, if T is (integer-)rectifiable, then also V can be chosen (integer-)rectifiable.

Proof.

We set

where 〚0,1〛×T is defined in Lemma A.1, and where H:[0,1]×spt⁡T→ℋ is the Lipschitz homotopy given by

Then the image of H is contained in (pπ)-1⁢(pπ⁢(spt⁡T)), and the claimed inclusion of the support of V follows at once. From the interchangeability of ∂ and H♯, equation (A.2), and ∂⁡T≡0 we moreover deduce that V is normal with

To get the mass estimate in (A.4) we first notice Lip⁡(H∘it)≤1+Lip⁡(f) for t∈[0,1]. Moreover, we have

which implies that |Dt⁢(χ∘H)|≤|qπ-f∘pπ| holds (ℒ1⊗∥T∥)-a.e. on [0,1]×spt⁡T for every fixed χ∈Lip⁡(ℋ) with Lip⁡(χ)≤1. Next we employ the definitions of pushforward, product, and mass together with the preceding observations in the following estimate^[12] for (φ,ψ)∈Lipb(ℋ)×Lip(ℋ)n with Lip⁡(ψi)≤1 for all i∈{1,2,…,n}:

As qπ-f∘pπ vanishes on Graph⁡f and is elsewhere controlled by (A.3), we finally get

We have thus shown

and in particular the total mass estimate in (A.4) follows.

Finally, the remaining claim about conservation of (integer-)rectifiability follows from the fact that this property is preserved under both the product construction of Lemma A.1 and the pushforward operation. ∎

Proposition B.1 (First variation).

Let Ω⊂H be open, let T∈In⁢(H) be locally minimizing in Ω, and consider Φ∈C1⁢(H,H), with support at a positive distance from H∖Ω, such that D⁢Φ is bounded. Then we have

Proof.

Given L:ℋ→ℋ linear and continuous, we denote by Jn⁢(L,T→) the n-dimensional Jacobian in (2.4) with the n-plane π given by T→. We write Ψε:=Id+ε⁢Φ and notice that, for ε small enough, Ψε is injective; in addition, the area formula and the local minimality of T give

Using (2.5) (and the rule for the derivative of the square root), we can differentiate with respect to ε to obtain (B.1). ∎

Proof of (2.17).

For z∈Ω and 0<η<σ<dist⁡(z,∂⁡Ω) we will show that

We can assume, without loss of generality, ∥T∥⁢(z+∂⁡𝐁η)=∥T∥⁢(z+∂⁡𝐁σ)=0. Under this extra assumption, an easy approximation argument shows that (B.1) still holds for all vector fields Φ of the form Φ⁢(w)=χ⁢(|w-z|)⁢(w-z), with χ:[0,∞)→[0,∞) Lipschitz, with support in [0,dist⁡(z,∂⁡Ω)), whose derivative χ′⁢(t) has at most jump discontinuities at t=η and t=σ. Now we specifically insert in (B.1) the vector field

whose support is the closure of z+𝐁σ. Then, the claim follows at once, when we use divT→⁡Φ≡n⁢[η-n-σ-n] on z+𝐁η and divT→⁡Φ≥-n⁢σ-n on (z+𝐁σ)∖(z+𝐁η). ∎

Proposition B.2.

Consider T∈In⁢(H). Then, for ∥T∥-a.e. z∈H there hold

where Iz,r⁢(w):=(w-z)/r, Tz,r:=(Iz,r)♯⁢T and the latter convergence is understood in duality with bounded continuous functions with bounded support.

Proof.

Let us first reduce the proof to the case when T=F♯〚θ〛 for some Lipschitz map F:ℝn→ℋ and θ∈L1⁢(ℝn), with θ∈ℕ∪{0}ℒn-a.e. in ℝn and F bi-Lipschitz on θ-1⁢(ℕ). Indeed, thanks to [5, Theorem 4.5], we can write any integer-rectifiable T with finite mass as an 𝐌-convergent series of currents Ti of this form with pairwise disjoint measure-theoretic supports Si. Since for any i it holds

(for the second statement, see for instance [22, 2.10.18 (2)]), we see that both (B.2) and (B.3) for T at ℋn-a.e. point of Si follow from (B.2) and (B.3) for Si.

So, let us assume T=F♯〚θ〛 for some Lipschitz map F:ℝn→ℋ and some θ∈L1⁢(ℝn) such that θ∈ℕ∪{0} holds ℒn-a.e. in ℝn and such that F is bi-Lipschitz on θ-1⁢(ℕ). We denote by e1,…,en the canonical basis of ℝn. Using the area formula [6, Theorem 5.1], one can check by a straightforward computation that (B.2) holds at a point z=F⁢(x), with T→⁢(z) equal to the normalization of ⋀i=1nD⁢F⁢(x)⁢ei, provided that we have Θ*n⁢(∥T∥,z)<∞ and that x∈θ-1⁢(ℕ) satisfies

Here, D⁢F exists ℒn-a.e. on ℝn by the Rademacher theorem [9, Theorem 5.11.1], (B.4) holds for ∥〚θ〛∥-a.e. x∈ℝn, and all in all the preceding conditions are satisfied for ∥T∥-a.e. z∈ℋ; thus, we arrive at the claim about (B.2).

In connection with (B.3), we consider z=F⁢(x) with x∈θ-1⁢(ℕ) such that F is classically Fréchet-differentiable at x with Span⁡T→⁢(z) equal to the image of D⁢F⁢(x) and with

When we set Fr⁢(y):=(F⁢(x+r⁢y)-F⁢(x))/r and θr⁢(y):=θ⁢(x+r⁢y), then Fr⁢(y) converges to D⁢F⁢(x)⁢y locally uniformly in y∈ℝn, on the level of the Jacobians Jn⁢(D⁢Fr) converges to Jn⁢(D⁢F⁢(x)) in Lloc1⁢(ℝn), and also θr converges to θ⁢(x) in Lloc1⁢(ℝn). Furthermore, we have

and with the help of the preceding convergences we conclude

for every bounded continuous function φ:ℋ→ℝ with bounded support. As a side benefit of this convergence we also get Θn⁢(∥T∥,z)=limr↘0⁡∥Tz,r∥⁢(𝐁1)/ωn=θ⁢(x). Similar as above, the assumed conditions on z=F⁢(x) hold true for ℒn-a.e. x∈θ-1⁢(ℕ) and thus for ∥T∥-a.e. z∈ℋ, so that we arrive at the claim regarding (B.3). ∎