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Matthias Kurzke · Heiko von der Mosel

The Douglas problem for parametric double integrals

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Abstract. Let F be a parametric variational double integral and Γ ⊂ ℝ^{n} be a system of several distinct Jordan curves. We prove the existence of multiply connected, conformally parametrized minimizers of F spanned in Γ by solving the Douglas problem for parametric functionals on multiply connected schlicht domains. As a by-product we obtain a simple isoperimetric inequality for multiply connected F-minimizers, and we discuss regularity results up to the boundary which follow from corresponding results for the Plateau problem.

1. Introduction and main results

We consider parametric variational functionals F of the form

F_{B}(X) := ∫_{B} F (X, X_{u} ∧ X_{v}) dudv  (1.1)

where F is of class C^{0}(ℝ^{n} × ℝ^{N}), n ≥ 2, N = n(n − 1)/2, satisfying the homogeneity condition

F (x, tz) = tF (x, z) for all t > 0.  (H)

We assume that F is positive definite, that is, there are constants m_{1}, m_{2} with 0 < m_{1} ≤ m_{2} such that

m_{1}|z| ≤ F (x, z) ≤ m_{2}|z| for all (x, z) ∈ ℝ^{n} × ℝ^{N}.  (D)

Then F_{B}(X) is well-defined by (1.1) for any domain B ⊂ ℝ^{2} and any mapping X ∈ H^{1,2}(B, ℝ^{n}), where

a ∧ b := (a^{j}b^{l} − a^{l}b^{j})_{j<l} ∈ ℝ^{N}

denotes the bivector generated by a = (a^{1}, …, a^{n}) and b = (b^{1}, …, b^{n}) ∈ ℝ^{n}. Moreover, the homogeneity condition (H) implies that F is parameter invariant, i.e., invariant with respect to reparametrizations by means of C^{1}-diffeomorphisms

M. Kurzke: Mathematisches Institut der Universität Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany. e-mail: kurzke@mathematik.uni-leipzig.de H. von der Mosel: Mathematisches Institut der Universität Bonn, Beringstraße 1, D-53115 Bonn, Germany. e-mail: heiko@math.uni-bonn.de Mathematics Subject Classification (2000): 49J45, 49Q10, 53A07, 53A10

with positive Jacobian, and even under bi-Lipschitz transformations with positive Jacobian a.e.

The parametric integrand F is said to be semi-elliptic if and only if

F (x, ·) is convex on ℝ^{N} for any x ∈ ℝ^{n}.  (C)

A special Lagrangian of this kind is the area integrand A(z) := |z|, which leads to the area functional

A_{B}(X) := ∫_{B} A(X_{u} ∧ X_{v}) dudv for X ∈ H^{1,2}(B, ℝ^{n}).  (1.2)

We call a system Γ := ⟨Γ^{1}, Γ^{2}, …, Γ^{k}⟩ consisting of k disjoint rectifiable closed Jordan curves Γ^{j} ⊂ ℝ^{n}, j = 1, …, k, a Jordan system.

Our aim is to find a multiply connected surface spanning the contour Γ which minimizes the parametric functional (1.1). For the area functional this problem is known as the Douglas problem or general problem of Plateau, which was first attacked by Douglas [5] and then solved by Courant [2] for the genus zero case, and by Shiffman in [15] prescribing an arbitrary genus. Modern proofs and extensions can be found in the work of Tomi and Tromba [16] and Jost [12], and we refer to Chapter 11 in [4] for a more complete discussion of the Douglas problem. For a solution of this problem in the context of surfaces of constant or variable prescribed mean curvature we refer to Werner [17] and Luckhaus [14], respectively.

We are going to focus on the genus zero case, and will combine Courant’s classical approach for minimal surfaces as described in [3, Ch. IV] with the method of conformal approximation of parametric functionals, which was first introduced in [7] and improved in [8] for the Plateau problem, and recently used in [9] to treat partially free boundary value problems for F. This approximation procedure allows us to avoid the use of any conformal mapping theorem in our arguments.

Certain conditions on the solvability of the Douglas problem need to be imposed as may be illustrated by the particular boundary configuration consisting of two coaxial circles in parallel planes. Depending on the distance of the two planes either the (doubly connected) catenoid or the pair of spanning disks may have less area A; see, e.g., the discussion in [3, pp. 141-142]. This motivates the so-called Douglas condition to exclude degenerate solutions consisting of two or more surfaces bounded by complementary parts of Γ. Another technical problem arises from the fact that k-fold connected domains of the same topological type may be of different conformal type for k > 1, so that we are led to consider minimizing F over pairs of admissible surfaces and domains.

To be more precise, we will restrict our attention to domains of the type

B_{r_{1},... ,r_{k}}(p_{1}, …, p_{k}) := B_{r_{1}}(p_{1})\ ⋃_{j=2}^{k} \overline{B_{r_{j}}(p_{j})},

where B_{r}(p) denotes the open ball in ℝ^{2} of radius r > 0 centred at the point p ∈ ℝ^{2}. If \overline{B_{r_{j}}(p_{j})} ⊂ B_{r_{1}}(p_{1}) and \overline{B_{r_{j}}(p_{j})} ∩ \overline{B_{r_{i}}(p_{i})} = ∅ for all i, j ∈ \{2, …, k\}, i ≠ j, we call B_{r_{1},... ,r_{k}}(p_{1}, …, p_{k}) a k-circle domain. We denote the class of all such domains as K_{k}. A member of K_{k} with p_{1} = 0, r_{1} = 1 and p_{j} = 0 for some j ∈ {2, …, k}, is called a unit k-circle domain. The class of unit kcircle domains will be denoted by K_{k}^{1}. Note that K_{1}^{1} = {B_{1}(0)}, and that one can perform elementary Möbius transformations to see that every k-circle domain is conformally equivalent to a unit k-circle domain.

We define the class of domains and surfaces admissible for the given boundary contour Γ = ⟨Γ^{1}, …, Γ^{k}⟩ as

C(Γ) := {(B, X) : B ∈ K_{k}, X ∈ H^{1,2}(B, ℝ^{n}) ∩ C^{0}(∂B, ℝ^{n})
X|_{∂B} : ∂B →^{onto} Γ is weakly monotonic^{1}},

in contrast to the class of degenerate surfaces given by

C^{†}(Γ) := {(B, X) : B = ⋃_{j=1}^{s} B_{j}, B_{j} ∈ K_{k_{j}} disjoint, s > 1,

∑_{j=1}^{s} k_{j} = k, X ∈ H^{1,2}(B, ℝ^{n}) ∩ C^{0}(∂B, ℝ^{n}),

X|_{∂B} : ∂B →^{onto} Γ is weakly monotonic}.

Setting d(Γ) := inf_{(B,X)∈C(Γ)} F_{B}(X), and d^{†}(Γ) := inf_{(B,X)∈C^{†}(Γ)} F_{B}(X), we say that Γ satisfies the Douglas condition for F if and only if

d(Γ) < d^{†}(Γ).  (1.3)

Notice that C^{†}(Γ) ≠ ∅, since Γ = ⟨Γ^{1}, …, Γ^{k}⟩ consists of rectifiable components Γ^{j}, j = 1, …, k, each of which can be spanned by the harmonic extension of the respective boundary representation; compare [4, Vol. I, pp. 233–234]. Hence the right-hand side of (1.3) is finite.

For technical reasons^{2} we impose a chord-arc condition on Γ, i.e., there are numbers δ > 0, M ≥ 1 such that, for any two points P, Q ∈ Γ with |P − Q| < δ, the shorter arc on Γ connecting P and Q, denoted by △(P, Q), has length L(△(P, Q)) ≤ M|P − Q|.

Then we will prove the existence of conformally parametrized minimizers of F in the class C(Γ) :

Theorem 1.1 (Existence). Suppose that F satisfies properties (H), (D) and (C) and that the Douglas condition (1.3) holds. Assume also that Γ satisfies a chordarc condition. Then there is a pair (B, X) ∈ C(Γ) with

F_{B}(X) = inf_{C(Γ)} F.  (1.4)

Moreover, X is conformally parametrized, that is, X satisfies

|X_{u}|^{2} = |X_{v}|^{2}, and X_{u} · X_{v} = 0 a.e. on B.  (1.5)

1 See [4, Vol.I, p. 231] for the notion of weakly monotonic mappings on the boundary. 2 The chord-arc condition is not necessary for k = 1 in our arguments in agreement with the existence result proved in [7] and [8]. For k > 1 one could avoid this condition if one had an a priori L^{∞}-estimate for F-minimizers.

In principle, condition (1.3) can be verified for concrete boundary configurations. If, for example, Γ = ⟨Γ^{1}, Γ^{2}⟩ consists of two coaxial circles of radius R in parallel planes which are a distance λ > 0 apart, then one can compare the total area 2πR^{2} of the two spanning disks with the area 2πRλ of the cylinder surface bounded by Γ. Using (D) it is easy to check that λ < Rm_{1}/m_{2} is sufficient for (1.3) to hold.

As a by-product of the existence proof we obtain the following simple isoperimetric inequality for F-minimizing surfaces.

Theorem 1.2 (Isoperimetric Inequality). Let F satisfy (D). Then for any (B, X) of class C(Γ) satisfying (1.4) one has

A_{B}(X) ≤ \frac{m_{2}}{4πm_{1}} ∑_{l=1}^{k} L^{2}(Γ_{l}).  (1.6)

Since regularity considerations are of local nature we can apply the regularity results of [7] and [8] to obtain

Theorem 1.3 (Regularity). Let F satisfy (H), (D) and assume that Γ satisfies a chord-arc condition. Then every conformally parametrized minimizer of F in C(Γ) is of class C^{0,α}(\overline{B}, ℝ^{n}) ∩ H_{loc}^{1,q}(B, ℝ^{n}) for some α ∈ (0, 1/2] and some q > 2.

In addition, one can prove C^{1,α}-smoothness up to the boundary of any conformally parametrized F-minimizer in the presence of a perfect dominance function, if Γ ∈ C^{4}, and if F ∈ C^{2}(ℝ^{n} × (ℝ^{N} − {0})) is elliptic; we refer the reader to [8], [10] and [11] for the details.

The present paper is organized as follows.

In Section 2 we present suitable versions of the Courant-Lebesgue Lemma for sequences of multiply connected surfaces with boundary contours approximating a given contour in the Fréchet-sense. In addition, we adapt Courant’s compactness result for domains in K_{k}^{1} to this slightly more general situation. Guided by Courant’s ideas we work out different methods to manipulate multiply connected surfaces in Lemmas 3.1–3.3 of Section 3. In Section 4 we describe how to approximate the minimization problem for F on C(Γ) by introducing additional L^{∞}- bounds, and by adding a small multiple of the Dirichlet energy to the parametric functional F. The technical result Theorem 4.5 contains the key ingredient, which, in combination with the Douglas condition (1.3), is used later to exclude degenerate minimal sequences, since it implies Courant’s condition of cohesion. The existence proof in Section 5 proceeds in three steps: first we establish the existence of conformal minimizers for the perturbed functional under an additional L^{∞}-bound in Lemma 5.1, then we show how to remove the L^{∞}-bound in Lemmas 5.2 and 5.3, before we prove Theorem 1.1 approximating F by the perturbed functionals.

2. Definitions and preliminaries for the Douglas problem

A sequence of k-circle domains B_{m} = B_{r_{1,m},... ,r_{k,m}}(p_{1,m}, …, p_{k,m}) ∈ K_{k} is said to converge to B = B_{r_{1},... ,r_{k}}(p_{1}, …, p_{k}), i.e., B_{m} → B as m → ∞, if and only if there are permutations σ_{m} of the set {2, …, k} such that for j = 2, …, k,

lim_{m→∞} r_{1,m} = r_{1}, lim_{m→∞} p_{1,m} = p_{1}, lim_{m→∞} r_{σ_{m}(j)} = r_{j}, lim_{m→∞} p_{σ_{m}(j)} = p_{j}.

A sequence {(B_{m}, X_{m})} of domains B_{m} ∈ K_{k} and mappings X_{m} of class H^{1,2}(B_{m}, ℝ^{n}) is called separating if for all ϵ > 0 there is m_{0} = m_{0}(ϵ) ∈ ℕ such that for each m > m_{0} there is a domain D_{m} ∈ K_{k}, a bi-Lipschitz continuous mapping T_{m} : \overline{B_{m}} → \overline{D_{m}}, a closed Jordan curve c_{m} ⊂ \overline{D_{m}} not homotopic to zero in \overline{D_{m}} and bounding a Lipschitz domain in ℝ^{2}, a representative Z_{m} of X_{m} such that Z_{m} ∘ T_{m}^{−1}|_{c_{m}} is continuous and coincides with the boundary trace^{3} X_{m} ∘ T_{m}^{−1}|_{c_{m}} on c_{m} and satisfies diam(Z_{m} ∘ T_{m}^{−1}(c_{m})) < ϵ. If a sequence {(B_{m}, X_{m})} with B_{m} ∈ K_{k} and X ∈ H^{1,2}(B_{m}, ℝ^{n}) possesses no separating subsequence it is said to be cohesive.

For X ∈ H^{1,2}(B, ℝ^{n}) the Dirichlet energy D_{B}(X) is given by

D_{B}(X) := \frac{1}{2} ∫_{B} |∇X|^{2} dudv.

We will repeatedly use the following version of the Courant-Lebesgue Lemma for sequences of multiply connected surfaces with uniformly bounded Dirichlet energy, a proof of which can be modelled after the proof for k = 1 in [4, Ch. 4.4].

Lemma 2.1 (Courant-Lebesgue). Let B_{m} ∈ K_{k}^{1} and X_{m} ∈ H^{1,2}(B_{m}, ℝ^{n}), m ∈ ℕ, be a sequence of k-times connected surfaces such that there is a constant M with D_{B_{m}}(X_{m}) ≤ M < ∞ for all m ∈ ℕ. Let x_{0} ∈ ℝ^{2} and δ ∈ (0, 1) be fixed. Then for all m ∈ ℕ there exists a set I_{m} ⊂ (δ, \sqrt{δ}) with L^{1}(I_{m}) > 0 and a representative Z_{m} of X_{m} such that for all ρ ∈ I_{m} the mapping Z_{m} is absolutely continuous on ∂B_{ρ}(x_{0}) ∩ B_{m} and coincides^{4} with the boundary trace X_{m}|_{∂B_{ρ}(x_{0})∩B_{m}} and satisfies

|Z_{m}(x_{0} + ρe^{iθ_{1}}) − Z_{m}(x_{0} + ρe^{iθ_{2}})| ≤ \sqrt{\frac{4M}{log \frac{1}{δ}}} |θ_{1} − θ_{2}|^{1/2}  (2.1)

for all θ_{1}, θ_{2} ∈ {θ ∈ ℝ : x_{0} + ρe^{iθ} ∈ B_{m}}. In particular, for all ϵ > 0 there is some number δ_{1} = δ_{1}(ϵ) > 0 such that for all δ ∈ (0, δ_{1}) there exists a set I_{m} ⊂ (δ, \sqrt{δ}) with L^{1}(I_{m}) > 0 such that

diam(Z_{m}(∂B_{ρ}(x_{0}) ∩ B_{m})) < ϵ for all ρ ∈ I_{m}, m ∈ ℕ.  (2.2)

Denote w_{1} := (1, 0), w_{2} := (0, 1), w_{3} := (−1, 0) ∈ ∂B_{1}(0) ⊂ ℝ^{2}, and fix three distinct points P_{i} ∈ Γ, i = 1, 2, 3, where Γ ⊂ ℝ^{n} is a single (closed) Jordan curve. For a sequence of Jordan curves Γ_{m} converging to Γ in the Fréchetsense as m → ∞ we say that a sequence of mappings X_{m} : B_{1}(0) → ℝ^{n} with {(B_{1}(0), X_{m})} ⊂ C(Γ_{m}) satisfies a three-point condition for P_{i} ∈ Γ, i = 1, 2, 3, if and only if

lim_{m→∞} X_{m}(w_{i}) = P_{i} for i = 1, 2, 3.  (2.3)

As a consequence of Lemma 2.1 important for controlling the boundary values of sequences of surfaces we have

3 For the definition of the boundary trace operator see, e.g., [18, Ch. 4.4]. 4 For this statement we refer to [6, Vol. II, p. 232].

Corollary 2.2. Let {(B, X_{m})} ⊂ C(Γ_{m}) with Γ_{m} → Γ in the Fréchet-sense. Suppose that there is a constant M such that D_{B}(X_{m}) ≤ M for all m ∈ ℕ. Then, if

(i) {X_{m}} satisfies a three-point condition in the case k = 1, where B = B_{1}(0), or if

(ii) {(B, X_{m})} is cohesive in the case k > 1,

the sequence X_{m}|_{∂B} is uniformly equicontinuous and contains a subsequence converging uniformly to some vector-valued function X^{∗} ∈ C^{0}(∂B, ℝ^{n}) with X^{∗} : ∂B → Γ weakly monotonic.

Proof. For k = 1 see [4, Ch. 4.3, Thm. 3 & Ch. 4.2, Lemma 1], and for k > 1 one can adjust the arguments in [3, p. 148] appropriately to deal with the more general situation here, where Γ_{m} → Γ in the Fréchet-sense. ⊓ ⊔

In order to control the domains for minimal sequences with boundary contours converging in the Fréchet-sense one can modify the arguments in [3, pp. 146–148] to prove the following slightly generalized version of Courant’s compactness result for unit k-circle domains.

Lemma 2.3. Let Γ_{m} → Γ in the Fréchet-sense where Γ_{m}, Γ ⊂ ℝ^{n} are Jordan systems. Suppose that {(B_{m}, X_{m})} ⊂ C(Γ_{m}) with B_{m} ∈ K_{k}^{1} is a cohesive sequence with D_{B_{m}}(X_{m}) ≤ M for all m ∈ ℕ where 0 ≤ M < ∞ is a constant independent of m ∈ ℕ. Then there is a subsequence {B_{m_{ν}} } and a domain B ∈ K_{k}^{1} such that B_{m_{ν}} → B as ν → ∞.

3. Manipulating multiply connected surfaces

For comparison arguments it is important to work on a fixed domain B ∈ K_{k}^{1}, for which we will have the following lemma:

Lemma 3.1. Let B_{m} ∈ K_{k}^{1} and X_{m} ∈ H^{1,2}(B_{m}, ℝ^{n}), m ∈ ℕ, with B_{m} → B ∈ K_{k}^{1} and D_{B_{m}}(X_{m}) → d as m → ∞. Then there is m_{0} ∈ ℕ such that for m > m_{0} there exist bi-Lipschitz continuous mappings ζ_{m} : \overline{B} → \overline{B}_{m} with the properties

(i) Z_{m} := X_{m} ∘ ζ_{m} ∈ H^{1,2}(B, ℝ^{n}) for all m ∈ ℕ, (ii) D_{B}(Z_{m}) → d as m → ∞. (iii) {(B, Z_{m})} is cohesive if and only if {B_{m}, X_{m})} is cohesive.

Proof. We construct the mappings ζ_{m} as in [13, Lemma 2.10] by patching together the identity map and some simple diffeomorphisms B_{r,r_{1}}(p, p_{1}) → B_{r,r_{2}}(p, p_{2}) reducing to the identity on the outer boundary. This construction easily leads to bi-Lipschitz maps of the desired domains.

Statement (i) follows from the Lipschitz continuity of ζ_{m} and [18, Thm. 2.2.2], assertion (ii) from an explicit calculation of D_{B}(X_{m} ∘ ζ_{m}); see [13, Lemmas 2.8 and 2.10] for the details.

(iii) Any curve β ⊂ \overline{B} not homotopic to zero corresponds to a curve γ_{m} := ζ_{m} ∘ β ⊂ \overline{B_{m}} not homotopic to zero and vice versa. If X_{m} is not cohesive, then it must possess a separating subsequence X_{m_{i}}. This implies that the subsequence Z_{m_{i}} := X_{m_{i}} ∘ ζ_{m_{i}} is separating, so Z_{m} is not cohesive. We can argue in the same way starting with the assumption that Z_{m} is not cohesive. ⊓ ⊔

For technical reasons we introduce the sets

C_{K}(Γ) := {(B, X) ∈ C(Γ) : |X| ≤ K a.e. on B} for K ∈ (0, ∞].

Notice that C_{∞}(Γ) = C(Γ).

We are going to show that we can replace small parts of a surface by the constant map Z ≡ 0 without gaining too much energy. The lemma is formulated for general functionals with quadratic growth.

Lemma 3.2. Let G_{Ω}(Y) := ∫_{Ω} G(Y, ∇Y) dudv be a functional with Lagrangian G ∈ C^{0}(ℝ^{n} × ℝ^{2n}) satisfying G(x, p) ≤ (μ/2)|p|^{2} for some constant μ > 0, and let (B, X) ∈ C_{K}(Γ) for some K ∈ (0, ∞]. Then for any δ > 0 and p ∈ B there exists r_{0} ∈ (0, dist(p, ∂B)) depending on δ and on K such that for any r ∈ (0, r_{0}) there is (B, Z_{r}) ∈ C_{K}(Γ) with

(i) G_{B}(Z_{r}) < G_{B}(X) + δ, (ii) Z_{r}|_{B_{r}(p)} ≡ 0.

Proof. For p ∈ B there is R ∈ (0, dist(p, ∂B)) such that

D_{B_{ρ}(p)}(X) < \frac{δ}{6μ} =: δ_{0} for all ρ ∈ (0, R).  (3.1)

Take ρ ∈ (0, R) such that a representative of X (again denoted by X) is absolutely continuous and satisfies |X| ≤ K on ∂B_{ρ}(p).

Then choose H ∈ H^{1,2}(B_{ρ}(p), ℝ^{n}) such that H − X ∈ H^{∘}^{1,2}(B_{ρ}(p), ℝ^{n}) and ΔH = 0 in B_{ρ}(p). By the maximum principle for harmonic functions we obtain

sup_{\overline{B_{ρ}(p)}} |H| = sup_{∂B_{ρ}(p)} |H| = sup_{∂B_{ρ}(p)} |X| =: M,  (3.2)

(hence M ≤ K) and, using the Dirichlet principle,

D_{B_{ρ}(p)}(H) ≤ D_{B_{ρ}(p)}(X) < δ_{0}.  (3.3)

For some constant η < ρ to be chosen later set

h_{η^{2}}(s) := \begin{cases}1 & if s > η, \\ 1 + \frac{log η−log s}{log η} & if η^{2} ≤ s ≤ η, \\ 0 & if s < η^{2}. \end{cases}

Then the mapping

Z_{η^{2}}(w) := \begin{cases}X(w) & if |w − p| ≥ ρ, \\ h_{η^{2}}(|w − p|)H(w) & if |w − p| < ρ \end{cases}

is of class H^{1,2}(B, ℝ^{n}) and (B, Z_{η^{2}}) ∈ C(Γ), since we can estimate for the Dirichlet energy on B_{ρ}(p) (writing h_{u} for (h_{η^{2}}(|w − p|))_{u}) using Hölder’s inequality and the Dirichlet principle

D_{B_{ρ}(p)}(Z_{η^{2}}) = \frac{1}{2} ∫_{B_{ρ}(p)} |h_{u}H + hH_{u}|^{2} + |h_{v}H + hH_{v}|^{2} dudv

≤ M^{2}D_{B_{ρ}(p)}(h_{η^{2}}) + D_{B_{ρ}(p)}(H) + 2M\sqrt{2D_{B_{ρ}(p)}(h_{η^{2}})}\sqrt{2D_{B_{ρ}(p)}(H)} .

An easy calculation shows that 2D_{B_{ρ}(p)}(h_{η^{2}}) = −2π/(log η) =: δ_{1}(η); hence, by (3.3),

D_{B_{ρ}(p)}(Z_{η^{2}}) ≤ M^{2}δ_{1}/2 + δ_{0} + 2M\sqrt{δ_{1}}\sqrt{2δ_{0}}.

Now we choose η_{0} so small that M^{2}δ_{1}(η) < 2δ_{0} for all η ∈ (0, η_{0}), whence D_{B_{ρ}(p)}(Z_{η^{2}}) ≤ 6δ_{0} for all η ∈ (0, η_{0}). Setting r_{0} := η_{0}^{2} we obtain for any r = η^{2} < η_{0}^{2} = r_{0}

G_{B}(Z_{r}) = G_{B\B_{ρ}(p)}(X) + G_{B_{ρ}(p)}(Z_{r}) ≤ G_{B}(X) + μD_{B_{ρ}(p)}(Z_{r})
≤ G_{B}(X) + 6μδ_{0} = G_{B}(X) + δ,

and Z_{r}|_{B_{r}(p)} = Z_{η^{2}}|_{B_{η^{2}} (p)} ≡ 0 by construction. Moreover, Z_{r}(w) = X(w) for |w − p| ≥ ρ, which implies |Z_{r}(w)| ≤ K for a.e. w ∈ B\B_{ρ}(p). On the other hand, |Z_{r}(w)| ≤ |H(w)| for |w − p| < ρ; hence, by (3.2), |Z_{r}(w)| ≤ sup_{∂B_{ρ}(p)} |X| ≤ K. Thus (B, Z_{r}) ∈ C_{K}(Γ). ⊓ ⊔

Essential for our existence proof in the following section is a “pinching”- Lemma (modeled after Courant’s corresponding result for the Dirichlet energy) that allows us to contract a part of a surface contained in a small ball to the center of that ball without gaining too much energy.

Lemma 3.3. Let (B, X) ∈ C_{K}(Γ) and P ∈ \overline{B_{K}(0)} ⊂ ℝ^{n}, δ > 0, ϵ_{0} > 0 be given. Then there is η_{0} ∈ (0, min{K, 1}) depending on the modulus of continuity of F |_{\overline{B_{K}(0)}×S^{n−1}}, on Γ, δ, and on the value F_{B}^{ϵ_{0}}(X) such that for every η ∈ (0, η_{0}) there is a mapping Φ_{η} : ℝ^{n} → ℝ^{n} such that

(i) Φ_{η}(Γ) is a Jordan curve with ∥Φ_{η}(Γ) − Γ ∥_{C^{0}} < δ, (ii) Φ_{η} ∘ X ∈ C_{K}(Φ_{η}(Γ)), (iii) Φ_{η} = Id_{ℝ^{n}} on ℝ^{n}\B_{η}(P), (iv) Φ_{η} ≡ P on \overline{B_{η^{2}}(P)},

(v) F_{B}^{ϵ}(Φ_{η} ∘ X) < F_{B}^{ϵ}(X) + δ for all ϵ ∈ [0, ϵ_{0}].

Proof. Define

ω_{F} (σ) := max \{|F (x, z) − F (y, z)| : x, y ∈ \overline{B_{K}(0)}, z ∈ S^{n−1}, |x − y| ≤ σ\}

and choose η_{0} ∈ (0, min{K, 1}) so small that

max \{\frac{1}{m_{1}} [ω_{F} (η_{0}) + \frac{m_{2}}{| log η_{0}|}], \frac{3}{| log η_{0}|}\} < \frac{δ}{F_{B}^{ϵ_{0}}(X)}.  (3.4)

Then, for η ∈ (0, η_{0}) and x ∈ ℝ^{n}, set Φ_{η}(x) := P + h_{η}(|x − P |)(x − P) with

h_{η}(r) := \begin{cases}1 & if r > η, \\ 1 + \frac{log(η/r)}{log η} & if η^{2} ≤ r ≤ η, \\ 0 & if r < η^{2}. \end{cases}

Parts (iii) and (iv) follow immediately from this definition, and Part (i) and the weak monotonicity of Φ_{η} ∘ X on ∂B can be easily deduced from the facts that Γ consists of Jordan curves and Φ_{η} is a homeomorphism of ℝ^{n} \ \overline{B_{η^{2}}(P)} onto ℝ^{n} \{P }. By [18, Thm. 2.1.11] Φ_{η} ∘X ∈ H^{1,2}(B, ℝ^{n}) since Φ_{η} is Lipschitz continuous, and Φ_{η}(X(w)) ∈ [P, X(w)] for any w ∈ \overline{B}. This implies |Φ_{η}(X(w))| ≤ K for a.e. w ∈ \overline{B} since both P and X(w) are contained in \overline{B_{K}(0)} ⊂ ℝ^{n}. This together with (i) proves Part (ii). It remains to show Part (v). First notice that

|Φ_{η} ∘ X − X| ≤ η on B.  (3.5)

Then we calculate on B^{∗} := {w ∈ B : η^{2} < |X(w) − P | < η}

(Φ_{η} ∘ X)_{u} = h_{η}(|X − P |)X_{u} − \frac{1}{log η} [\frac{X − P}{|X − P |} · X_{u}] \frac{X − P}{|X − P |} ,

(3.6)

(Φ_{η} ∘ X)_{v} = h_{η}(|X − P |)X_{v} − \frac{1}{log η} [\frac{X − P}{|X − P |} · X_{v}] \frac{X − P}{|X − P |} ;

hence

(Φ_{η} ∘ X)_{u}^{2} = h_{η}^{2}(|X − P |)X_{u}^{2} + \frac{1}{(log η)^{2}} [1 − 2h_{η} log η] [\frac{X − P}{|X − P |} · X_{u}]^{2}
≤ h_{η}^{2}(|X − P |)X_{u}^{2} + \frac{3}{| log η|}X_{u}^{2} on B^{∗}, and
(Φ_{η} ∘ X)_{v}^{2} ≤ h_{η}^{2}(|X − P |)X_{v}^{2} + \frac{3}{| log η|}X_{v}^{2} on B^{∗}.

Thus, we can estimate the Dirichlet energy as

D_{B^{∗}}(Φ_{η} ∘ X) ≤ D_{B^{∗}}(X) + \frac{3}{| log η|}D_{B^{∗}}(X)
≤_{(3.4)} (1 + \frac{δ}{F_{B}^{ϵ_{0}}(X)})D_{B^{∗}}(X).  (3.7)

On the other hand, by (3.5) and (3.6),

(Φ_{η} ∘ X)_{u} ∧ (Φ_{η} ∘ X)_{v} = h_{η}^{2}(|X − P |)X_{u} ∧ X_{v}

−\frac{h_{η}(|X − P |)}{log η} [\frac{X − P}{|X − P |} · X_{v}] X_{u} ∧ \frac{X − P}{|X − P |}

−\frac{h_{η}(|X − P |)}{log η} [\frac{X − P}{|X − P |} · X_{u}] \frac{X − P}{|X − P |} ∧ X_{v}.

One has |(B · E)A ∧ E + (A · E)E ∧ B| ≤ |A ∧ B| for A, B, E ∈ ℝ^{n} with |E| = 1; hence

(Φ_{η} ∘ X)_{u} ∧ (Φ_{η} ∘ X)_{v} = h_{η}^{2}(|X − P |)X_{u} ∧ X_{v} + R  (3.8)

with |R| ≤ | log η|^{−1}|X_{u} ∧ X_{v}|. Using the assumptions (C), (D), and (H) on the parametric integrand F we estimate on B^{∗}

F (Φ_{η} ∘ X, (Φ_{η} ∘ X)_{u} ∧ (Φ_{η} ∘ X)_{v})

=_{(3.8)} F (Φ_{η} ∘ X, \frac{1}{2}(2h_{η}^{2}(|X − P |)X_{u} ∧ X_{v}) + \frac{1}{2}2R)
≤_{(C)}  \frac{1}{2}F (Φ_{η} ∘ X, 2h_{η}^{2}(|X − P |)X_{u} ∧ X_{v}) + \frac{1}{2}F (Φ_{η} ∘ X, 2R)
=_{(H)} h_{η}^{2}(|X − P |)F (Φ_{η} ∘ X, X_{u} ∧ X_{v}) + F (Φ_{η} ∘ X, R)
≤_{(H),(D)} h_{η}^{2}(|X − P |)F (Φ_{η} ∘ X, \frac{X_{u} ∧ X_{v}}{|X_{u} ∧ X_{v}|})|X_{u} ∧ X_{v}| + \frac{m_{2}}{| log η|}|X_{u} ∧ X_{v}|
≤_{(D)} h_{η}^{2}(|X −P |)[F (Φ_{η} ∘X, \frac{X_{u} ∧ X_{v}}{|X_{u} ∧ X_{v}|}) − F (X, \frac{X_{u} ∧ X_{v}}{|X_{u} ∧ X_{v}|})]|X_{u} ∧ X_{v}|

+ [h_{η}^{2}(|X − P |) + \frac{m_{2}}{m_{1}| log η|}] F (X, X_{u} ∧ X_{v})

≤_{h_{η}^{2}≤1,(D)} (1 + m_{1}^{−1} [ω_{F} (η) + \frac{m_{2}}{| log η|}]) F (X, X_{u} ∧ X_{v})
≤_{(3.4)} F (X, X_{u} ∧ X_{v})(1 + \frac{δ}{F_{B}^{ϵ_{0}}(X)}).

Integrating over B^{∗} we arrive at

F_{B^{∗}}(Φ_{η} ∘ X) ≤ F_{B^{∗}}(X) + δ(F_{B}^{ϵ_{0}}(X))^{−1}F_{B^{∗}}(X),

and together with (3.4) and (3.7) this leads to

F_{B^{∗}}^{ϵ}(Φ_{η} ∘ X) ≤_{(3.4),(3.7)} F_{B^{∗}}^{ϵ}(X) + δ \frac{F_{B^{∗}}^{ϵ}(X)}{F_{B}^{ϵ_{0}}(X)} ≤ F_{B^{∗}}^{ϵ}(X) + δ

for all ϵ ∈ [0, ϵ_{0}] since F_{B^{∗}}^{ϵ}(X) ≤ F_{B}^{ϵ}(X) ≤ F_{B}^{ϵ_{0}}(X) for all ϵ ∈ [0, ϵ_{0}]. ⊓ ⊔

4. Conformal approximation of parametric functionals

We consider the conformally invariant functionals

F_{B}^{ϵ}(X) := F_{B}(X) + ϵD_{B}(X) for ϵ ≥ 0 and X ∈ H^{1,2}(B, ℝ^{n}).

Recall from the previous section the definition of the class C_{K}(Γ) and define analogously

C_{K}^{†}(Γ) := {(B, X) ∈ C^{†}(Γ) : |X| ≤ K a.e. on B}.

Set for ϵ ≥ 0 and K ∈ (0, ∞]

d_{K}(Γ, ϵ) := inf_{(B,X)∈C_{K}(Γ)} F_{B}^{ϵ}(X) and d_{K}^{†}(Γ, ϵ) := inf_{(B,X)∈C_{K}^{†}(Γ)} F_{B}^{ϵ}(X).

Notice that d_{∞}(Γ, 0) = d(Γ) and d_{∞}^{†}(Γ, 0) = d^{†}(Γ) by definition.

Considering separating sequences we define the infimum of F^{ϵ} over all such sequences as

d_{K}^{∗}(Γ, ϵ) := inf_{\begin{matrix}(B_{m},X_{m})∈C_{K}(Γ) \\ (B_{m},X_{m}) sep. \end{matrix}} lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m})  (4.1)

for ϵ ∈ [0, ∞), K ∈ (0, ∞].

The following lemma can be easily proved by a contradiction argument and we omit the proof.

Lemma 4.1. The mappings ϵ ↦ d_{K}(Γ, ϵ), ϵ ↦ d_{K}^{†}(Γ, ϵ), and ϵ ↦ d_{K}^{∗}(Γ, ϵ) are nondecreasing. Moreover,

d_{K}(Γ, 0) = lim_{ϵ→+0} d_{K}(Γ, ϵ) and d_{K}^{†}(Γ, 0) = lim_{ϵ→+0} d_{K}^{†}(Γ, ϵ)  (4.2)

for all K ∈ (0, ∞].

Lemma 4.2. The mappings K ↦ d_{K}(Γ, ϵ), K ↦ d_{K}^{†}(Γ, ϵ) and K ↦ d_{K}^{∗}(Γ, ϵ) are monotonically decreasing, have respective limits as K → ∞, and we have

lim_{K→∞} d_{K}(Γ, ϵ) = d(Γ, ϵ) and lim_{K→∞} d_{K}^{†}(Γ, ϵ) = d^{†}(Γ, ϵ)  (4.3)

for all ϵ ≥ 0.

Proof. It suffices to show the claim for d_{K}(Γ, ϵ), the result for d_{K}^{†}(Γ, ϵ) follows by induction on the components. The function K ↦ d_{K}(Γ, ϵ) is decreasing and bounded from below for each ϵ ≥ 0. Moreover,

0 ≤ d(Γ) ≤ d(Γ, ϵ) ≤ d_{K}(Γ, ϵ) for all K ∈ (0, ∞], ϵ ≥ 0.

Hence the limits d_{∞}(ϵ) := lim_{K→∞} d_{K}(Γ, ϵ) exist and

d(Γ, ϵ) ≤ d_{∞}(ϵ) ≤ d_{K}(Γ, ϵ) for all ϵ ≥ 0, K ∈ (0, ∞].  (4.4)

To prove the claim we assume to the contrary that d_{∞}(ϵ) > d(Γ, ϵ) for some ϵ ≥ 0. Then we find (B, X) ∈ C(Γ) such that d_{∞}(ϵ) > F_{B}^{ϵ}(X) ≥ d(Γ, ϵ). For η ∈ (0, d_{∞}(ϵ) − F_{B}^{ϵ}(X)) we find δ > 0 such that

(m_{2} + ϵ)D_{E}(X) ≤ η for all E ⊂ B with L^{2}(E) < δ  (4.5)

since X ∈ H^{1,2}(B, ℝ^{n}). Moreover, we have for K ≥ 0

K^{2}L^{2}({w ∈ B : |X^{i}(w)| ≥ K}) ≤ ∫_{{|X^{i}(w)|≥K}} |X(w)|^{2} dudv ≤ ∥X∥_{L^{2}}^{2}.

Thus we may choose K_{0} > ∥Γ ∥_{∞} so large that the set

κ := ⋃_{i=1}^{n}{w ∈ B : |X^{i}(w)| ≥ K_{0}}

has measure

L^{2}(κ) < δ.  (4.6)

Then the truncated surface Y with the components

Y^{i}(w) := \begin{cases}sign(X^{i}(w))K_{0} & if |X^{i}(w)| ≥ K_{0}, \\ X^{i}(w) & if |X^{i}(w)| < K_{0} \end{cases}

is in the class C_{\sqrt{n}K_{0}}(Γ) and satisfies

F_{B}^{ϵ}(Y) = F_{B\κ}^{ϵ}(Y) + F_{κ}^{ϵ}(Y) ≤ F_{B\κ}^{ϵ}(X) + (m_{2} + ϵ)D_{κ}(Y)
≤ F_{B}^{ϵ}(X) + (m_{2} + ϵ)D_{κ}(X).

Thus we infer from (4.5) and (4.6) that

d_{\sqrt{n}K_{0}}(Γ, ϵ) ≤ F_{B}^{ϵ}(Y) ≤_{(4.5)} F_{B}^{ϵ}(X) + η < d_{∞}(ϵ)

by our choice of η, contradicting (4.4) for K := \sqrt{n}K_{0}. ⊓ ⊔

Lemma 4.3. Let ϵ ≥ 0 and K ∈ (0, ∞]. Suppose that {(B_{m}, X_{m})} ⊂ C_{K}(Γ_{m}) is cohesive with

D_{B_{m}}(X_{m}) ≤ C for all m ∈ ℕ  (4.7)

where C does not depend on m. Then there exist B ∈ K_{k}^{1}, X ∈ H^{1,2}(B, ℝ^{n}) such that

d_{K}(Γ, ϵ) ≤ F_{B}^{ϵ}(X) ≤ lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}).  (4.8)

Recall that for k = 1 one has K_{k}^{1} = {B_{1}(0)}; hence all sequences {(B_{m}, X_{m})} = {(B_{1}(0), X_{m})} ⊂ C_{K}(Γ_{m}) are cohesive for k = 1.

Proof. For k = 1 we may assume that the X_{m} satisfy a three-point condition for preassigned distinct points P_{i} ∈ Γ, i = 1, 2, 3. Thus, by (4.7), a suitable version of Poincaré’s inequality and Corollary 2.2, we find a surface X ∈ H^{1,2}(B_{1}(0), ℝ^{n})∩ C^{0}(∂B_{1}(0), ℝ^{n}) whose boundary mapping X|_{∂B_{1}(0)} : ∂B_{1}(0) → Γ is weakly monotonic and for which we have X_{m_{ν}} ⇀ X in H^{1,2}(B_{1}(0), ℝ^{n}) and also X_{m_{ν}} |_{∂B_{1}(0)} ⟶ X|_{∂B_{1}(0)} in C^{0}(∂B_{1}(0), ℝ^{n}) for a subsequence m_{ν} → ∞. Hence |X(w)| ≤ K for a.e. w ∈ B_{1}(0) and thus (B_{1}(0), X) ∈ C_{K}(Γ). This and the lower-semicontinuity result of Acerbi and Fusco in [1] applied to the functional F_{B}^{ϵ} imply (4.8).

For k > 1 we may assume that B_{m} ∈ K_{k}^{1} for all m ∈ ℕ by elementary Möbius transformations.

We deduce from (4.7) that

D_{B_{mν}} (X_{m_{ν}}) ⟶ D ∈ [0, C] as ν → ∞  (4.9)

for a subsequence, and taking a further subsequence we may assume that

lim_{ν→∞} F_{B_{mν}}^{ϵ} (X_{m_{ν}}) = lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}).  (4.10)

Thus by Lemma 2.3 we can find a unit k-circle domain B ∈ K_{k}^{1} and a subsequence {B_{l}} ⊂ {B_{m_{ν}} } such that B_{l} → B as l → ∞. Using the reparametrized surfaces Z_{l} ∈ H^{1,2}(B, ℝ^{n}) for l sufficiently large, constructed according to Lemma 3.1, we find

D_{B}(Z_{l}) → D as l → ∞.  (4.11)

A suitable version of Poincaré’s inequality yields a uniform bound on the H^{1,2}- norm of the Z_{l} independent of l so that we find a surface Z ∈ H^{1,2}(B, ℝ^{n}) such that (for a relabeled subsequence) Z_{l} ⇀ Z in H^{1,2}(B, ℝ^{n}) as l → ∞.

Moreover, we can apply Corollary 2.2 to get Z_{l}|_{∂B} → Z|_{∂B} in C^{0}(∂B, ℝ^{n}) as l → ∞ where Z|_{∂B} is weakly monotonic since the reparametrized sequence {(B, Z_{l})} remains cohesive according to Part (iii) of Lemma 3.1. In addition, we have |Z| ≤ K a.e. on B since |Z_{l}| ≤ K for all l ∈ ℕ. Consequently, Z ∈ C_{K}(Γ), and we may apply the lower-semicontinuity result of [1] valid for the functional F_{B}^{ϵ} to conclude

d_{K}(Γ, ϵ) ≤ F_{B}^{ϵ}(Z) ≤ lim inf_{l→∞} F_{B}^{ϵ}(Z_{l}).  (4.12)

The parametric invariance of F implies

F_{B}^{ϵ}(Z_{l}) = F_{B_{m_{l}}}^{ϵ} (X_{m_{l}}) + ϵ(D_{B}(Z_{l}) − D_{B_{m_{l}}} (X_{m_{l}})) for all l ∈ ℕ;

hence, by (4.9)–(4.11),

lim inf_{l→∞} F_{B}^{ϵ}(Z_{l}) = lim inf_{l→∞} F_{B_{m_{l}}}^{ϵ} (X_{m_{l}}) = lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}),

which proves (4.8) according to (4.12). ⊓ ⊔

Lemma 4.4. One has d_{K}(Γ, ϵ) ≤ d_{K}^{∗}(Γ, ϵ) ≤ d_{K}^{†}(Γ, ϵ) for all ϵ ≥ 0, K ∈ (0, ∞].

Proof. The first inequality is obvious since d_{K}(Γ, ϵ) ≤ F_{B_{m}}^{ϵ}(X_{m}) for any member (B_{m}, X_{m}) of a separating sequence in C_{K}(Γ).

For the second inequality it suffices to consider the case k > 1 since for k = 1 we have C_{K}^{†}(Γ) = ∅, whence d_{K}^{†}(Γ, ϵ) = ∞ for any K ∈ (0, ∞] and ϵ ≥ 0.

Moreover, it is enough to prove the second inequality for s = 2 components (see the definition of C_{K}^{†}(Γ) ) ; the rest follows from a simple inductive argument over the number of components. Hence we may assume that Γ = ⟨Γ^{1}, Γ^{2}⟩, and for δ > 0 we choose (B_{1}, X_{1}) ∈ C_{K}(Γ^{1}) and (B_{2}, X_{2}) ∈ C_{K}(Γ^{2}) with B_{1} ∈ K_{k_{1}} and B_{2} ∈ K_{k_{2}}, where k_{1} + k_{2} = k, and such that

F_{B_{1}}^{ϵ}(X_{1}) ≤ d_{K}(Γ^{1}, ϵ) + δ and F_{B_{1}}^{ϵ}(X_{1}) ≤ d_{K}(Γ^{1}, ϵ) + δ.

Applying Lemma 3.2 for G := F^{ϵ} we obtain new surfaces (B_{1}, Z_{1}) ∈ C_{K}(Γ^{1}), (B_{2}, Z_{2}) ∈ C_{K}(Γ^{2}) such that there are balls B_{r_{1}}(p_{1}) ⊂⊂ B_{1}, B_{r_{2}}(p_{2}) ⊂⊂ B_{2} with Z_{1}|_{B_{r_{1}} (p_{1})} ≡ 0, Z_{2}|_{B_{r_{2}} (p_{2})} ≡ 0, and such that

F_{B_{1}}^{ϵ}(Z_{1}) ≤ F_{B_{1}}^{ϵ}(X_{1}) + δ and F_{B_{2}}^{ϵ}(Z_{2}) ≤ F_{B_{2}}^{ϵ}(X_{2}) + δ.  (4.13)

Let B_{2}^{∗} be the reflection of B_{2} at ∂B_{r_{2}/2}(p_{2}), and Z_{2}^{∗} be the reflected surface defined on B_{2}^{∗}, i.e., Z_{2}^{∗}(w) := Z_{2}(τ^{−1}(w)) for w ∈ τ(B_{2}) where τ denotes the reflection mapping at ∂B_{r_{2}/2}(p_{2}). By dilation and translation which are conformal mappings we can assume that ∂B_{r_{2}/2}(p_{2}) = ∂B_{r_{1}/2}(p_{1}), and we introduce the k-circle domain B ∈ K_{k} by B := (B_{1}\B_{r_{1}/2}(p_{1})) ∪ B_{2}^{∗}. Now define a mapping Y ∈ H^{1,2}(B, ℝ^{n}) by

Y := \begin{cases}Z_{1} & on B_{1}\B_{r_{1}}(p_{1}), \\ 0 & on B_{r_{1}}(p_{1})\B_{2}^{∗}, \\ Z_{2}^{∗} & on B_{2}^{∗} \end{cases}

to obtain (B, Y) ∈ C_{K}(Γ). The conformal invariance of the functional F^{ϵ} and (4.13) imply

F_{B}^{ϵ}(Y) = F_{B_{1}\B_{r_{1}} (p_{1})}^{ϵ}(Y) + F_{B_{r_{1}} (p_{1})\B_{2}^{∗}}^{ϵ}(Y) + F_{B_{2}^{∗}}^{ϵ}(Y)
= F_{B_{1}\B_{r_{1}} (p_{1})}^{ϵ}(Z_{1}) + F_{B_{r_{1}} (p_{1})\B_{2}^{∗}}^{ϵ}(0) + F_{B_{2}^{∗}}^{ϵ}(Z_{2}^{∗})
≤_{(4.13)} F_{B_{1}}^{ϵ}(X_{1}) + δ + 0 + F_{B_{2}}^{ϵ}(X_{2}) + δ
≤ d(Γ^{1}, ϵ) + d(Γ^{2}, ϵ) + 4δ.

Since Y maps the reflected image τ(∂B_{ρ}(p_{1})) onto zero for all ρ ∈ (r_{1}/2, r_{1}), the constant sequence (B_{m}, Y_{m}) ≡ (B, Y) ∈ C_{K}(Γ) is separating^{5} by definition. Therefore

d_{K}^{∗}(Γ, ϵ) ≤ F_{B}^{ϵ}(Y) ≤ d_{K}(Γ^{1}, ϵ) + d_{K}(Γ^{2}, ϵ) + 4δ.

Since the partition Γ = ⟨Γ^{1}, Γ^{2}⟩ was arbitrary we obtain d_{K}^{∗}(Γ, ϵ) ≤ d_{K}^{†}(Γ, ϵ) for all ϵ ≥ 0. ⊓ ⊔

The central result for the existence proof in the next section is

Theorem 4.5. Let Γ_{m}, Γ be Jordan systems with Γ_{m} → Γ in the Fréchet-sense as m → ∞. Then

(i) d_{K}(Γ, ϵ) ≤ lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}), (ii) d_{K}^{†}(Γ, ϵ) = d_{K}^{∗}(Γ, ϵ)

for all ϵ > 0, 0 < K < ∞, and for all sequences {(B_{m}, X_{m})} ⊂ C_{K}(Γ_{m}).

5 The bi-Lipschitz transformations T_{m} required in the definition of separating sequences can be chosen to be the identity in the plane.

Proof. (i) We prove the claim by induction over k. In the case k = 1 we may take a sequence {(B_{1}(0), X_{m})} ⊂ C_{K}(Γ_{m}) with

lim_{m→∞} F_{B}^{ϵ}(X_{m}) = lim inf_{m→∞} F_{B}^{ϵ}(X_{m}) < ∞

(if the right-hand side is infinite there is nothing to prove). This implies for ϵ > 0 D_{B}(X_{m}) ≤ c(ϵ) for all m ∈ ℕ where c(ϵ) is a constant depending on ϵ. Now (i) follows from Lemma 4.3.

For the induction step k − 1 ↦ k we consider a sequence {(B_{m}, X_{m})} ⊂ C_{K}(Γ_{m}) with lim_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) = lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) and we may apply elementary Möbius transformations to get B_{m} ∈ K_{k}^{1} for all m ∈ ℕ.

We distinguish two cases:

Case A. Assume {(B_{m}, X_{m})} ⊂ C_{K}(Γ_{m}) is separating. Then (since |X_{m}| ≤ K a.e. on B_{m}) there are points Q_{m} ∈ \overline{B_{K}(0)} ⊂ ℝ^{n}, domains D_{m} ∈ K_{k} containing closed Jordan curves c_{m} ⊂ \overline{D_{m}} not homotopic to zero in \overline{D_{m}} and bounding a Lipschitz domain in ℝ^{2}, bi-Lipschitz mappings T_{m} : \overline{B_{m}} → \overline{D_{m}}, and constants η_{m} → 0 as m → ∞, such that a representative Z_{m} of X_{m} ∘ T_{m}^{−1} restricted to c_{m} is absolutely continuous and coincides with the boundary trace on c_{m} and satisfies ∥Z_{m} − Q_{m}∥_{∞,c_{m}} < η_{m}^{2}. Taking subsequences we may assume that Q_{m} → Q ∈ \overline{B_{K}(0)} and adjusting the constants η_{m} > 0 we get ∥Z_{m} − Q∥_{∞,c_{m}} < η_{m}^{2}.

Let δ_{i} → 0 be a given sequence of numbers. Then, according to Lemma 3.3, we find for fixed ϵ > 0 and for each i ∈ ℕ some number η_{0,i} > 0 and (taking a subsequence of the η_{m} above and relabeling) η_{i} < η_{0,i}, mappings Φ_{η_{i}} : ℝ^{n} → ℝ^{n}, such that

(i) Φ_{η_{i}}(Γ_{i}) is a Jordan curve with ∥Φ_{η_{i}}(Γ_{i}) − Γ_{i}∥_{C^{0}} < δ_{i}, (ii) Φ_{η_{i}} ∘ Z_{i} ∈ C_{K}(Φ_{η_{i}}(Γ_{i})), (iii) Φ_{η_{i}} = Id_{ℝ^{n}} on ℝ^{n}\B_{η_{i}}(Q), (iv) Φ_{η_{i}} ≡ Q on \overline{B_{η_{i}^{2}}(Q)},

(v) F_{D_{i}}^{ϵ}(Φ_{η_{i}} ∘ Z_{i}) < F_{B_{i}}^{ϵ}(X_{i}) + δ_{i}.

In particular, we have Φ_{η_{i}} ∘ Z_{i}|_{c_{i}} ≡ Q for all i ∈ ℕ. We define new surfaces^{6} Y_{i}^{1} ∈ H^{1,2}(E_{i}^{1}, ℝ^{n}), Y_{i}^{2} ∈ H^{1,2}(E_{i}^{2}, ℝ^{n}), by

Y_{i}^{1} := \begin{cases}Φ_{η_{i}} ∘ Z_{i} ∘ τ^{−1} & on τ(D_{i}^{1}) ⊂ E_{i}^{1}, \\ Q & elsewhere on E_{i}^{1} \end{cases}

and

Y_{i}^{2} := \begin{cases}Φ_{η_{i}} ∘ Z_{i} & on τ(D_{i}^{2}) ⊂ E_{i}^{2}, \\ Q & elsewhere on E_{i}^{2} \end{cases}

with D_{i}^{1} := int(c_{i}) ∩ D_{i}, D_{i}^{2} := D_{i}\D_{i}^{1}, E_{i}^{1} := int(τ(c_{i})) ∪ τ(D_{i}^{1}), and E_{i}^{2} := int(c_{i}) ∪ D_{i}^{2} where τ : ℝ^{2} → ℝ^{2} denotes the reflection at an arbitrary circle contained in the set \overline{int(c_{i})} ∩ \overline{D_{i}}.

6 Y_{i}^{1} and Y_{i}^{2} are in fact in the right Sobolev class since they are constructed from H^{1,2}- surfaces with matching boundary traces by the assumptions on the values on the curves c_{i}.

One can check that E_{i}^{1} and E_{i}^{2} are of lower topological type and that they satisfy (E_{i}^{1}, Y_{i}^{1}) ∈ C_{K}(Φ_{i}(Γ_{i}^{1})) and (E_{i}^{2}, Y_{i}^{2}) ∈ C_{K}(Φ_{i}(Γ_{i}^{2})) where Γ_{i}^{1} ˙∪Γ_{i}^{2} = Γ_{i} for all i ∈ ℕ by property (i), (ii) of the Φ_{η_{i}}. In addition, by property (i), we get Φ_{η_{i}}(Γ_{i}^{1}) → Γ_{i}^{1} and Φ_{η_{i}}(Γ_{i}^{2}) → Γ_{i}^{2} as i → ∞ where Γ^{1} ˙∪Γ^{2} = Γ. Moreover, we have by construction

F_{E_{i}^{1}}^{ϵ}(Y_{i}^{1}) + F_{E_{i}^{2}}^{ϵ}(Y_{i}^{2}) = F_{D_{i}}^{ϵ}(Φ_{η_{i}} ∘ Z_{i}).  (4.14)

Using the induction hypothesis

d_{K}(Γ^{1}, ϵ) ≤ lim inf_{i→∞} F_{E_{i}^{1}}^{ϵ}(Y_{i}^{1}) and d_{K}(Γ^{2}, ϵ) ≤ lim inf_{i→∞} F_{E_{i}^{2}}^{ϵ}(Y_{i}^{2})

we obtain by Lemma 4.4, (4.14) and the definition of d_{K}^{†}(Γ, ϵ)

d_{K}(Γ, ϵ) ≤_{L.4.4} d_{K}^{†}(Γ, ϵ) ≤ d_{K}(Γ^{1}, ϵ) + d_{K}(Γ^{2}, ϵ)
≤_{(4.14)} lim inf_{i→∞} F_{D_{i}}^{ϵ}(Φ_{η_{i}} ∘ Z_{i}) ≤_{(v)} lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) (4.15)

which proves (i) in this case.

Case B. Now we assume that {(X_{m}, B_{m})} ⊂ C_{K}(Γ_{m}) is cohesive. In the case where lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) is infinite, there is nothing to prove for Part (i). If, on the other hand, there is a constant C such that lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) ≤ C, then we find (for fixed ϵ) another constant c(ϵ) such that D_{B_{m}}(X_{m}) ≤ c(ϵ) for all m ∈ ℕ which allows us to apply Lemma 4.3 to prove (i) in Case B as well.

(ii) If d_{K}^{∗}(Γ, ϵ) is infinite, (which is the case, e.g., for k = 1) then so is d_{K}^{†}(Γ, ϵ) by Lemma 4.4, and there is nothing to prove. If d_{K}^{∗}(Γ, ϵ) is finite, then we choose for arbitrary δ > 0 a separating sequence {(B_{m}, X_{m})} ⊂ C_{K}(Γ_{m}) with lim_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) ≤ d_{K}^{∗}(Γ, ϵ) + δ. Now we apply the construction of Case A above to this sequence to arrive at (4.15), i.e.,

d_{K}(Γ, ϵ) ≤ d_{K}^{†}(Γ, ϵ) ≤ lim_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) ≤ d_{K}^{∗}(Γ, ϵ) + δ,

and let δ → 0. This together with Lemma 4.4 finishes the proof of (ii). ⊓ ⊔

5. Existence proof

Lemma 5.1. There exist ϵ_{1} > 0 and K_{1} > 0 such that for each ϵ ∈ (0, ϵ_{1}], K ∈ [K_{1}, ∞) there is a domain B_{K}^{ϵ} ∈ K_{k}^{1}, and (B_{K}^{ϵ}, X_{K}^{ϵ}) ∈ C_{K}(Γ) with

F_{B_{K}^{ϵ}}^{ϵ} (X_{K}^{ϵ}) = d_{K}(Γ, ϵ).  (5.1)

Moreover,

|(X_{K}^{ϵ})_{u}|^{2} = |(X_{K}^{ϵ})_{v}|^{2} and (X_{K}^{ϵ})_{u} · (X_{K}^{ϵ})_{v} = 0 a.e. on B_{K}.  (5.2)

Proof. For arbitrary ϵ > 0 we take a minimal sequence {(B_{m}, X_{m})} ⊂ C_{K}(Γ) with

lim_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) = d_{K}(Γ, ϵ).  (5.3)

Since C_{K}^{†}(Γ) ≠ ∅ for K sufficiently large, and since d_{K}(Γ, ϵ) ≤ d_{K}^{†}(Γ, ϵ) by Lemma 4.4, the right-hand side is finite, and we find a constant c(ϵ) independent of m such that D_{B_{m}}(X_{m}) ≤ c(ϵ) for all m ∈ ℕ. If {(B_{m}, X_{m})} is cohesive (which is automatically the case for k = 1) then we can apply Lemma 4.3 for fixed ϵ > 0 to obtain a pair (B_{K}^{ϵ}, X_{K}^{ϵ}) ∈ C_{K}(Γ) satisfying (4.8) and hence (5.1) by (5.3).

We claim that, for ϵ sufficiently small and K sufficiently large, there is no separating minimal sequence for F^{ϵ} in C_{K}(Γ). Indeed, the Douglas condition (1.3) combined with (4.2) for K = ∞ implies that there exists ϵ_{1} > 0 such that

d(Γ, ϵ) < d^{†}(Γ, ϵ) for all ϵ ∈ (0, ϵ_{1}].  (5.4)

This together with (4.3) yields a constant K_{1} ∈ (0, ∞) such that

d_{K}(Γ, ϵ) < d_{K}^{†}(Γ, ϵ) for all K ≥ K_{1}, ϵ ∈ (0, ϵ_{1}].  (5.5)

Consequently, any minimal sequence {(B_{m}, X_{m})} ⊂ C_{K}(Γ) with

lim_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) = d_{K}(Γ, ϵ)  (5.6)

must be cohesive for ϵ ∈ (0, ϵ_{1}], K ∈ [K_{1}, ∞), since otherwise, by (5.5) and Part (ii) of Theorem 4.5, we would have

d_{K}(Γ, ϵ) <_{(5.5)} d_{K}^{†}(Γ, ϵ) =_{Thm.4.5(ii)} d_{K}^{∗}(Γ, ϵ)
≤ lim inf_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) = lim_{m→∞} F_{B_{m}}^{ϵ}(X_{m}) =_{(5.6)} d_{K}(Γ, ϵ)

which is absurd.

Finally we have to show the conformality relations (5.2). The minimality (5.1) implies in particular, F_{B_{K}^{ϵ}}^{ϵ} (X_{K}^{ϵ}) ≤ F_{B_{K}^{ϵ}}^{ϵ} (Y) for all Y ∈ H^{1,2}(B_{K}^{ϵ}, ℝ^{n}) with (B_{K}^{ϵ}, Y) ∈ C_{K}(Γ). As in [7, p. 254] we obtain ∂F^{ϵ}(X_{K}^{ϵ}, η) = 0 for the inner variation of F_{B_{K}^{ϵ}}^{ϵ} at X_{K}^{ϵ} in the direction of an arbitrary vectorfield η ∈ C^{1}(\overline{B_{K}^{ϵ}}, ℝ^{2}). Since F is parameter invariant, we have ∂F(X_{K}^{ϵ}, η) = 0, and thus ∂D(X_{K}^{ϵ}, η) = 0 for any^{7} η ∈ C^{1}(\overline{B_{K}^{ϵ}}, ℝ^{2}) which implies (5.2); see [4, Vol. I, p. 246]. ⊓ ⊔

7 To avoid Riemann’s mapping theorem for multiply connected domains one can restrict the inner variations to a special class of vector fields (but still sufficiently large to obtain conformality), as done in [3, pp. 169–178], [13, Ch. 3]. These vector fields generate diffeomorphisms mapping \overline{B_{K}^{ϵ}} onto a k-circle domain.

Lemma 5.2. Assume Γ satisfies a chord-arc condition. Then there is a constant χ = χ(Γ) depending on Γ but independent of ϵ and K such that X_{K}^{ϵ} is of class C^{0,α}(\overline{B_{K}^{ϵ}}, ℝ^{n}) for some α ∈ (0, 1/2] independent of ϵ ∈ (0, ϵ_{1}] and K ∈ [K_{1}, ∞) with

∥X_{K}^{ϵ}∥_{C^{0,α}(\overline{B_{K}^{ϵ}},ℝ^{n})} ≤ χ for all ϵ ∈ (0, ϵ_{1}], K ∈ [K_{1}, ∞).  (5.7)

In particular, X_{K}^{ϵ} ∈ C_{χ}(Γ) for all ϵ ∈ (0, ϵ_{1}], K ∈ [K_{1}, ∞).

Proof. By assumption (D) on the parametric integrand F and by (5.1) and (5.2) we obtain

(m_{1} + ϵ)D_{B_{K}^{ϵ}} (X_{K}^{ϵ}) =_{(5.2)} m_{1}A_{B_{K}^{ϵ}} (X_{K}^{ϵ}) + ϵD_{B_{K}^{ϵ}} (X_{K}^{ϵ}) ≤_{(D)} F_{B_{K}^{ϵ}}^{ϵ} (X_{K}^{ϵ})
≤_{(5.1)} F_{B_{K}^{ϵ}}^{ϵ} (Z) ≤_{(D)} m_{2}A_{B_{K}^{ϵ}} (Z) + ϵD_{B_{K}^{ϵ}} (Z)
≤ (m_{2} + ϵ)D_{B_{K}^{ϵ}} (Z)

for any (B_{K}^{ϵ}, Z) ∈ C_{K}(Γ) and for every ϵ ∈ (0, ϵ_{1}], K ∈ [K_{1}, ∞). Using that (m_{2} + ϵ)(m_{1} + ϵ)^{−1} ≤ m_{2}/m_{1} for any ϵ > 0, we arrive at

D_{B_{K}^{ϵ}} (X_{K}^{ϵ}) ≤ \frac{m_{2}}{m_{1}} D_{B_{K}^{ϵ}} (Z) for all (B_{K}^{ϵ}, Z) ∈ C_{K}(Γ).  (5.8)

We can now use comparison arguments as in [7, pp. 261–263] by replacing X_{K}^{ϵ} locally by harmonic vectors (without leaving the class C_{K}(Γ) due to the maximum principle) to finish the proof. ⊓ ⊔

Lemma 5.3. For all ϵ ∈ (0, ϵ_{1}] there is a domain B^{ϵ} ∈ K_{k}^{1} and a pair (B^{ϵ}, X^{ϵ}) ∈ C(Γ) with

F_{B^{ϵ}}^{ϵ}(X^{ϵ}) = d(Γ, ϵ).  (5.9)

Moreover,

|X_{u}^{ϵ}|^{2} = |X_{v}^{ϵ}|^{2} and X_{u}^{ϵ} · X_{v}^{ϵ} = 0 a.e. on B^{ϵ}.  (5.10)

Proof. By Lemma 4.2 we know that

F_{B_{K}^{ϵ}}^{ϵ} (X_{K}^{ϵ}) = d_{K}(Γ, ϵ) ⟶ d(Γ, ϵ) as K → ∞.  (5.11)

The right-hand side is bounded by Lemma 4.4 for K := ∞ and by the fact that C^{†}(Γ) ≠ ∅. Thus there is a constant C independent of K ∈ [K_{1}, ∞) such that for fixed ϵ ∈ (0, ϵ_{1}] one has F_{B_{K}^{ϵ}}^{ϵ} (X_{K}^{ϵ}) ≤ C for all K ∈ [K_{1}, ∞). This yields a constant C(ϵ) depending on ϵ but not on K, such that

D_{B_{K}^{ϵ}} (X_{K}^{ϵ}) ≤ C(ϵ) for all K ∈ [K_{1}, ∞).  (5.12)

If {(B_{K}^{ϵ}, X_{K}^{ϵ})} ⊂ C(Γ) is cohesive (which is always true for k = 1) then we may use Lemma 4.3 (with K := ∞ there) to obtain a pair (B^{ϵ}, X^{ϵ}) ∈ C(Γ) with (4.8) for K = ∞, and thus by (5.11) the desired identity (5.9).

For k > 1 we claim that the sequence {(B_{K}^{ϵ}, X_{K}^{ϵ})} is cohesive. In fact, we infer from (5.7) in Lemma 5.2 that

(B_{K}^{ϵ}, X_{K}^{ϵ}) ∈ C_{χ}(Γ) for all K ∈ [K_{1}, ∞), ϵ ∈ (0, ϵ_{1}].  (5.13)

If {(B_{K}^{ϵ}, X_{K}^{ϵ})} were separating we would obtain from (5.11)

d_{χ}^{∗}(Γ, ϵ) ≤ lim inf_{K→∞} F_{B_{K}^{ϵ}}^{ϵ} (X_{K}^{ϵ}) =_{(5.11)} d(Γ, ϵ).  (5.14)

This together with (5.4), Lemma 4.2, and Part (i) of Theorem 4.5 leads to

d_{χ}^{∗}(Γ, ϵ) ≤_{(5.14)} d(Γ, ϵ) <_{(5.4)} d^{†}(Γ, ϵ) ≤_{L.4.2} d_{χ}^{†}(Γ, ϵ) =_{Thm.4.5} d_{χ}^{∗}(Γ, ϵ).

This is a contradiction and proves that {(B_{K}^{ϵ}, X_{K}^{ϵ})} is cohesive.

The conformality relations (5.10) follow by taking inner variations of F_{B^{ϵ}}^{ϵ} at X^{ϵ} as in the proof of (5.2) in Lemma 5.1. ⊓ ⊔

Proof of Theorem 1.1: By assumption (D), the conformality relations (5.10) and the minimality (5.9) we deduce for the (B^{ϵ}, X^{ϵ}) ∈ C(Γ) obtained in Lemma 5.3

(m_{1} + ϵ)D_{B^{ϵ}}(X^{ϵ}) =_{(5.10)} m_{1}A_{B^{ϵ}}(X^{ϵ}) + ϵD_{B^{ϵ}}(X^{ϵ}) ≤_{(D)} F_{B^{ϵ}}^{ϵ}(X^{ϵ})
≤_{(5.9)} F_{B^{ϵ}}^{ϵ}(Z) ≤ (m_{2} + ϵ)D_{B^{ϵ}}(Z)  (5.15)

for any (B^{ϵ}, Z) ∈ C(Γ) and every ϵ ∈ (0, ϵ_{1}]. Since (m_{2} + ϵ)(m_{1} + ϵ)^{−1} ≤ m_{2}/m_{1} for any ϵ > 0, we arrive at

D_{B^{ϵ}}(X^{ϵ}) ≤ \frac{m_{2}}{m_{1}} D_{B^{ϵ}}(Z) for all (B^{ϵ}, Z) ∈ C(Γ).  (5.16)

Using the fact F_{B^{ϵ}}^{ϵ}(X^{ϵ}) = d(Γ, ϵ) and Lemma 4.4 we can replace (5.15) by

(m_{1} + ϵ)D_{B^{ϵ}}(X^{ϵ}) ≤ d(Γ, ϵ) ≤_{L.4.4} d^{†}(Γ, ϵ)
≤ F_{Ω}^{ϵ}(Y) ≤ (m_{2} + ϵ)D_{Ω}(Y)

for any (Ω, Y) ∈ C^{†}(Γ), and thus,

D_{B^{ϵ}}(X^{ϵ}) ≤ \frac{m_{2}}{m_{1}} D_{Ω}(Y) for all (Ω, Y) ∈ C^{†}(Γ).  (5.17)

In particular, if Γ = ⟨Γ^{1} …, Γ^{k}⟩, we may take (Ω, Y) ∈ C^{†}(Γ) to be the collection of k disk-type minimal surfaces Y_{j} spanned in Γ^{j}, j = 1, …, k, defined on Ω := ⋃_{j=1}^{k} B_{j} where the disks B_{j} ∈ K_{1}, l = 1, …, k, are disjoint. More precisely,

Y := \begin{cases}Y_{1} & on B_{1}, \\ ⋮ & ⋮ \\ Y_{k} & on B_{k} \end{cases}  (5.18)

where Y_{j} ∈ H^{1,2}(B_{j}, ℝ^{n}) is a minimal surface such that (B_{j}, Y_{j}) ∈ C(Γ^{j}), j = 1, …, k. By the classical isoperimetric inequality for minimal surfaces (see e.g., [4, Vol. I, Ch. 6.3]) we infer from (5.17)

D_{B^{ϵ}}(X^{ϵ}) ≤ \frac{m_{2}}{4πm_{1}} ∑_{j=1}^{k} L^{2}(Γ^{j}) =: c(Γ, m_{1}, m_{2}) for all ϵ ∈ (0, ϵ_{1}] (5.19)

where L(Γ^{j}) denotes the length of the j-th component Γ^{j} of Γ. The right-hand side does not depend on ϵ ∈ (0, ϵ_{1}], so there is a sequence ϵ_{j} → 0 such that

D_{B^{ϵ_{j}}} (X^{ϵ_{j}}) ⟶ D ∈ [0, c(Γ, m_{1}, m_{2})].  (5.20)

As in the proofs of Lemma 5.1 and 5.3 we have to show that (for k > 1) the sequence {(B^{ϵ_{j}}, X^{ϵ_{j}})} ⊂ C(Γ) is cohesive, since then we may apply Lemma 4.3 (for K := ∞ and ϵ := 0) to obtain (B, X) ∈ C(Γ) such that

d(Γ) = d(Γ, 0) ≤ F_{B}(X) ≤ lim inf_{j→∞} F^{ϵ_{j}}(X^{ϵ_{j}}) = d(Γ)

which proves (1.4).

We claim that {(B^{ϵ_{j}}, X^{ϵ_{j}})} is cohesive. Indeed, otherwise we could infer from the Douglas condition, from (5.7), (5.20), (5.9), Part (ii) of Theorem 4.5, and from the Lemmas 4.1 and 4.2 the following:

d(Γ) <_{(1.3)} d^{†}(Γ) = d^{†}(Γ, 0) =_{L.4.2} lim_{K→∞} d_{K}^{†}(Γ, 0)
=_{L.4.1}  lim_{K→∞} lim_{j→∞} d_{K}^{†}(Γ, ϵ_{j}) ≤_{L.4.2} lim_{j→∞} d_{χ}^{†}(Γ, ϵ_{j})
=_{Thm.4.5(ii)} lim_{j→∞} d_{χ}^{∗}(Γ, ϵ_{j})
≤_{(5.7)}  lim_{j→∞} lim inf_{m→∞} F_{B^{ϵm}}^{ϵ_{j}} (X^{ϵ_{m}})
= lim_{j→∞} [ lim inf_{m→∞} F_{B^{ϵm}} (X^{ϵ_{m}}) + ϵ_{j} lim_{m→∞} D_{B^{ϵm}} (X^{ϵ_{m}})]
=_{(5.20)} lim_{j→∞} [ lim inf_{m→∞} F_{B^{ϵm}} (X^{ϵ_{m}}) + ϵ_{j}D]
≤ lim_{j→∞} [ lim inf_{m→∞} F_{B^{ϵm}}^{ϵ_{m}} (X^{ϵ_{m}}) + ϵ_{j}D]
=_{(5.9)}  lim_{j→∞} [ lim inf_{m→∞} d(Γ, ϵ_{m}) + ϵ_{j}D]
=_{L.4.1}  lim_{j→∞} [d(Γ) + ϵ_{j}D] = d(Γ)

which is absurd.

To prove (1.5) we argue as follows. By (5.9) and (1.4) and the fact that both (B^{ϵ_{j}}, X^{ϵ_{j}}) and (B, X) are contained in the class C(Γ) (where B = B^{ϵ_{j}} = B_{1}(0) for all j ∈ ℕ in the case k = 1) we obtain for all ϵ_{j} > 0

F_{B^{ϵ_{j}}} (X^{ϵ_{j}}) + ϵ_{j}D_{B^{ϵ_{j}}} (X^{ϵ_{j}}) = F_{B^{ϵ_{j}}}^{ϵ_{j}} (X^{ϵ_{j}})
≤_{(5.9)} F_{B}^{ϵ_{j}}(X) = F_{B}(X) + ϵ_{j}D_{B}(X)
≤_{(1.4)} F_{B^{ϵ_{j}}}^{ϵ_{j}} (X^{ϵ_{j}}) + ϵ_{j}D_{B}(X).

This implies

D_{B^{ϵ_{j}}} (X^{ϵ_{j}}) ≤ D_{B}(X) for all ϵ_{j} > 0.  (5.21)

Recall from the proof of Lemma 4.3 (applied to {(B^{ϵ_{j}}, X^{ϵ_{j}})} for K := ∞, and ϵ := 0) that there are reparametrizations Z_{j} ∈ H^{1,2}(B, ℝ^{n}) of X^{ϵ_{j}} such that by (5.20)

D_{B}(Z_{j}) ⟶ D as j → ∞,  (5.22)
Z_{j} ⇀ X in H^{1,2}(B, ℝ^{n}) as j → ∞.  (5.23)

This together with the weak lower semicontinuity of D_{B}(·) in H^{1,2}(B, ℝ^{n}) leads to

lim sup_{j→∞} D_{B}(Z_{j}) = D = lim_{j→∞} D_{B^{ϵ_{j}}} (X^{ϵ_{j}}) ≤ D_{B}(X) ≤ lim inf_{j→∞} D_{B}(Z_{j}).

Thus,

lim_{j→∞} D_{B}(Z_{j}) = D_{B}(X) = D,  (5.24)

and (5.23) then implies the strong convergence

Z_{j} → X in H^{1,2}(B, ℝ^{n}) as j → ∞.  (5.25)

Since the area functional A is parameter invariant and by (5.10) we have

A_{B}(Z_{j}) = A_{B^{ϵ_{j}}} (X^{ϵ_{j}}) =_{(5.10)} D_{B^{ϵ_{j}}} (X^{ϵ_{j}})

which implies by (5.24), (5.20) and (5.25)

D_{B}(X) =_{(5.24)} D =_{(5.20)} lim_{j→∞} D_{B^{ϵ_{j}}} (X^{ϵ_{j}}) = lim_{j→∞} A_{B}(Z_{j}) =_{(5.25)} A_{B}(X). (5.26)

This proves (1.5) and finishes the proof of Theorem 1.1. ⊓ ⊔

Proof of Theorem 1.2: By (1.4), (D) and Lemma 4.4 we have

m_{1}A_{B}(X) ≤_{(D)} F_{B}(X) =_{(1.4)} d(Γ) ≤_{L.4.4} d^{†}(Γ) ≤ F_{Ω}(Y)

for all (Ω, Y) ∈ C^{†}(Γ). Taking (Ω, Y) as in (5.18) we can use the classical isoperimetric inequality for minimal surfaces as in the previous proof to deduce (1.6). ⊓ ⊔

Acknowledgements. We would like to thank the Max-Planck Institute for Mathematics in the Sciences Leipzig, the Graduiertenkolleg “Analysis, Geometrie und ihre Verbindung zu den Naturwissenschaften” at the University of Leipzig, and the Sonderforschungsbereich 611 at the University of Bonn for their generous support.

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