The partially free boundary problem for parametric double integrals

STEFAN HILDEBRANDT AND HEIKO VON DER MOSEL Mathematisches Institut der Universität Bonn

Abstract We prove the existence of conformally parametrized minimizers for parametric two-dimensional variational problems subject to partially free boundary conditions. We establish regularity of class H_{loc}^{2,2} ∩ C^{1,α}, 0 < α < 1, up to the free boundary under the assumption that there exists a perfect dominance function in the sense of C.B. Morrey.

Mathematics Subject Classification (2000): 49J45, 49Q10, 49N60, 53A10

1 Introduction and main results

In ℝ^{n}, n ≥ 3, we consider a boundary configuration ⟨Γ, S⟩ consisting of a closed Jordan arc Γ and a support surface S. First we merely assume that S is a closed, arcwise connected set such that any two points on S can be connected by some rectifiable Jordan arc contained in S, and that Γ is a rectifiable Jordan arc whose endpoints P_{1} and P_{2} lie on S. We suppose also that the open subarc Γ^{∘} := Γ − {P_{1}, P_{2}} does not meet S. Denote by B the parameter domain

B := {w = (u, v) ∈ ℝ^{2} : u^{2} + v^{2} < 1, v > 0}

whose boundary ∂B consists of the open interval I on the u-axis and the closed semicircle C with the endpoints w_{1} := (−1, 0), w_{2} := (1, 0) :

I := {(u, 0) ∈ ℝ^{2} : |u| < 1}, C := {(u, v) ∈ ℝ^{2} : u^{2} + v^{2} = 1, v ≥ 0}.

Let C(Γ, S) be the class of surfaces X ∈ H^{1,2}(B, ℝ^{n}) whose traces X|_{C} and X|_{I} satisfy the following conditions:

2 S. HILDEBRANDT, H. VON DER MOSEL

(i) X|_{C} is a continuous, weakly monotonic mapping of C onto Γ such that X(w_{1}) = P_{1} and X(w_{2}) = P_{2}.

(ii) X(u, 0) ∈ S for L^{1}-almost all u ∈ I. We note that C(Γ, S) is nonempty.

In this paper we consider the variational problem F(X) → min in C(Γ, S), (P) where F is a general parametric double integral of the form

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

The Lagrangian F (x, z) is defined for (x, z) ∈ ℝ^{n} × 𝔹^{n}, where 𝔹^{n} is the space of bivectors ζ = ξ ∧ η with ξ, η ∈ ℝ^{n}, i.e. ζ = (ζ^{jk})_{j<k}, where ζ^{jk} := ξ^{j}η^{k} − ξ^{k}η^{j} for ξ = (ξ^{1}, ξ^{2}, …, ξ^{n}) and η = (η^{1}, η^{2}, …, η^{n}). We can identify 𝔹^{n} with ℝ^{N}, N := n(n − 1)/2. The Lagrange identity |ξ ∧ η|^{2} = |ξ|^{2}|η|^{2} − (ξ · η)^{2} implies

|ξ ∧ η| ≤ \frac{1}{2}(|ξ|^{2} + |η|^{2}),

and we have equality if and only if |ξ|^{2} = |η|^{2} and ξ · η = 0. Definition 1.1. A function F : ℝ^{n} × ℝ^{N} → ℝ is said to be a parametric Lagrangian if it is of class C^{0}(ℝ^{n} × ℝ^{N}) and satisfies F (x, tz) = tF (x, z) for all t > 0, (x, z) ∈ ℝ^{n} × ℝ^{N}. (H) The associated Lagrangian f : ℝ^{n} × ℝ^{2n} → ℝ of F is defined as f(x, p) := F (x, p_{1} ∧ p_{2}) for p = (p_{1}, p_{2}) ∈ ℝ^{n} × ℝ^{n} ≅ ℝ^{2n}. (1.1) Definition 1.2. A parametric Lagrangian F is called definite, if there are numbers m_{1} and 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) and it is said to be semi-elliptic if (C) F (x, z) is convex with respect to z.

Our first result establishes existence of solutions to the partially free boundary problem (P) for parametric double integrals; it will be proved in Section 2 using an approximation argument developed in [7].

Theorem 1.3. If F ∈ C^{0}(ℝ^{n} × ℝ^{N}) satisfies (H),(D), and (C), then there exists a minimizer X of F in C(Γ, S) which is conformally parametrized, i.e. X ∈ C(Γ, S) satisfies F(X) = inf_{C(Γ,S)} F, and

|X_{u}|^{2} = |X_{v}|^{2}, X_{u} · X_{v} = 0 a.e. on B. (1.2)

Free boundary problems for parametric double integrals were treated by Lipkin [11], who proved the existence of minimizers, modifying a method of Morrey [13] and using ideas of Courant [2]. However he did not prove that there exist conformally parametrized minimizers.

In Section 3 we prove Hölder continuity up to the boundary of the parameter domain B if the configuration ⟨Γ, S⟩ satisfies a chord-arc condition (cf. e.g. [4], §7.5).

Definition 1.4. A set S in ℝ^{n} satisfies a chord-arc condition if there are numbers M ≥ 1 and δ > 0 such that any two points P, Q ∈ S with |P −Q| ≤ δ can be connected in S by a rectifiable arc Γ^{∗} whose length L(Γ^{∗}) satisfies L(Γ^{∗}) ≤ M|P − Q|. More precisely, we speak in this case of an (M, δ)- chord-arc condition.

Theorem 1.5. Suppose that F ∈ C^{0}(ℝ^{n} × ℝ^{N}) satisfies (H),(D), and let X ∈ C(Γ, S) be a conformally parametrized minimizer of F in C(Γ, S). If S satisfies an (M, δ)-chord-arc condition, then, for any w_{0} ∈ B ∪ I and any r > 0 we obtain

∫_{B∩B_{r}(w_{0})} |∇X|^{2} dudv ≤ (\frac{2r}{R} )^{2σ} ∫_{B} |∇X|^{2} dudv, (1.3)

where 0 < R ≤ dist (w_{0}, C) and where σ ∈ (0, 1/2] is defined by

σ := min{m_{1}m_{2}^{−1}(1 + M^{2})^{−1}, [2πF(X)]^{−1}m_{1}δ^{2}}. (1.4)

It follows that X ∈ C^{0,σ}(\overline{Ω}, ℝ^{n}) for every domain Ω ⊂⊂ B ∪ I. In particular for Ω_{ρ_{0}} := B∩B_{ρ_{0}}(0) we obtain the following a priori estimate for the Hölder seminorm [X]_{σ,\overline{Ω}_{ρ_{0}}} of X on \overline{Ω}_{ρ_{0}} if 0 < ρ_{0} < 1 :

[X]_{σ,\overline{Ω}_{ρ_{0}}} ≤ c(σ)(1 − ρ_{0})^{−σ}\sqrt{∫_{B} |∇X|^{2} dudv} , (1.5)

where c(σ) > 0 depends only on σ.

Note that this result ensures Hölder continuity of any conformally parametrized minimizer X of F in B and up to the “free part” I of the boundary ∂B, but does not even provide continuity in the corners w_{1} = (−1, 0) and w_{2} = (1, 0). This, however, can be obtained by requiring a chord-arc condition on ⟨Γ, S⟩ (and not on S alone).

Theorem 1.6. Suppose that F ∈ C^{0}(ℝ^{n} × ℝ^{N}) satisfies (H),(D) and let X ∈ C(Γ, S) be a conformally parametrized minimizer of F in C(Γ, S). If S ∪ Γ satisfies an (M, δ)-chord-arc condition, then, for σ ∈ (0, 1/2] defined by (1.4) we obtain for arbitrary w_{0} ∈ ℝ^{2} that

∫_{B∩B_{r}(w_{0})} |∇X|^{2} dudv ≤ cr^{2σ} ∫_{B} |∇X|^{2} dudv for all r > 0. (1.6)

This implies X ∈ C^{0,σ}(\overline{B}, ℝ^{n}) and

[X]_{σ,\overline{B}} ≤ c(σ)\sqrt{∫_{B} |∇X|^{2} dudv} . (1.7)

Remark 1.7. Since X(C) = Γ, inequality (1.7) provides an L^{∞}-bound for X of the following kind:

sup\overline{B} |X| ≤ c(σ, Γ, D(X)), (1.8)

where

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

denotes the Dirichlet integral of X. Let Y be a minimal surface minimizing D in C(Γ, S), i.e., D(Y) = inf_{C(Γ,S)} D. Then, for any conformally parametrized minimizer of F in C(Γ, S) we have by (D)

m_{1}D(X) = m_{1}A(X) ≤ F(X) ≤ F(Y) ≤ m_{2}A(Y) ≤ m_{2}D(Y),

and therefore D(X) ≤ (m_{2}/m_{1}) inf_{C(Γ,S)} D. Thus we obtain the a priori bound

sup\overline{B} |X| ≤ c(σ, Γ, S). (1.9)

Following Morrey ([14], pp. 390–394) we introduce the useful notion of a dominance function. For the convenience of the reader we repeat the terminology introduced in [8].

Definition 1.8. Let F (x, z) be a parametric Lagrangian with the associated Lagrangian f(x, p) = F (x, p_{1} ∧ p_{2}).

(i) A function G : ℝ^{n} × ℝ^{2n} → ℝ is called a dominance function for F if it is continuous and satisfies the following two conditions:

f(x, p) ≤ G(x, p) for any (x, p) ∈ ℝ^{n} × ℝ^{2n}, (D1)

(D2) f(x, p) = G(x, p) if and only if p ∈ Π_{0},

where Π_{0} denotes the algebraic surface in ℝ^{2n} defined by

Π_{0} := {p = (p_{1}, p_{2}) ∈ ℝ^{2n} : |p_{1}|^{2} = |p_{2}|^{2}, p_{1} · p_{2} = 0}.

(ii) A dominance function G of the parametric Lagrangian F is said to be quadratic if

(D3) G(x, tp) = t^{2}G(x, p) for all t > 0, (x, p) ∈ ℝ^{n} × ℝ^{2n},

and it is called positive definite if there are two constants μ_{1}, μ_{2} such that 0 < μ_{1} ≤ μ_{2} and

(D4) μ_{1}|p|^{2} ≤ G(x, p) ≤ μ_{2}|p|^{2} for any (x, p) ∈ ℝ^{n} × ℝ^{2n}.

(iii) A function G ∈ C^{0}(ℝ^{n} × ℝ^{2n}) ∩ C^{2}(ℝ^{n} × (ℝ^{2n} − {0})) is called a perfect dominance function for the parametric Lagrangian F if it satisfies (D1)–(D4) and the ellipticity condition

π · G_{pp}(x, p)π ≥ λ(R_{0})|π|^{2} for |x| ≤ R_{0} and p, π ∈ ℝ^{2n}, p ≠ 0, (E)

and any R_{0} > 0 where λ(R_{0}) > 0 is a number depending only on R_{0}.

Condition (E) means that

G_{p_{α}^{j}p_{β}^{k}}(x, p)π_{α}^{j}π_{β}^{k} ≥ λ(R_{0})π_{α}^{j}π_{α}^{j}.

(Greek indices run from 1 to 2, Latin indices from 1 to n; repeated indices are to be summed from 1 to 2, or n, respectively.)

The associated Lagrangian f(x, p) of a parametric Lagrangian F (x, z) fails to be a dominance function for F since (D2) is not satisfied, as one can see for the area integrand A(z) := |z| with the associated Lagrangian

a(p) := A(p_{1} ∧ p_{2}) = \sqrt{|p_{1}|^{2}|p_{2}|^{2} − (p_{1} · p_{2})^{2}} .

The Lagrangian A(z) has the perfect dominance function

D(p) := \frac{1}{2}|p|^{2} = \frac{1}{2}(|p_{1}|^{2} + |p_{2}|^{2}), p = (p_{1}, p_{2}).

Note that the associated function f(x, p) of a parametric Lagrangian F ≢ 0 can only be differentiable on the set ℝ^{2n} − Π, where

Π := {p = (p_{1}, p_{2}) ∈ ℝ^{2n} : p_{1} ∧ p_{2} = 0}.

A perfect dominance function G(x, p) has at most p = 0 as singular point; but this point will truly be singular, except for some special cases such as D(p); cf. [8], Proposition 1.7 (Grüter’s result).

For a dominance function G of F we introduce the functional

G(X) := ∫_{B} G(X, ∇X) dudv.

Then F(X) ≤ G(X) for X ∈ H^{1,2}(B, ℝ^{n}), and F(X) = G(X) if and only if (1.2) holds true. Correspondingly we set

A(X) := ∫_{B} A(X_{u} ∧ X_{v}) dudv,

and we have A(X) ≤ D(X), with equality if and only if (1.2) is fulfilled.

The next result is the key in proving higher regularity of conformally parametrized minimizers, see Theorem 1.12 below.

Corollary 1.9. Suppose that F is continuous and satisfies (H),(D),(C), and let G be an arbitrary dominance function of F. Then we have for the integrals F and G corresponding to F and G, respectively:

(i) inf_{C(Γ,S)} F = inf_{C(Γ,S)} G.

(ii) Any minimizer of G in C(Γ, S) is a conformally parametrized minimizer of F in C(Γ, S).

(iii) Conversely, any conformally parametrized minimizer of F in C(Γ, S) is a minimizer of G in C(Γ, S).

Proof: (i) By Theorem 1.3 there exists a conformally parametrized minimizer X of F in C(Γ, S). Then we obtain

inf_{C(Γ,S)} G ≤ G(X) = F(X) = inf_{C(Γ,S)} F ≤ inf_{C(Γ,S)} G,

and the first assertion is proved. In the same way we obtain (iii). (ii) If X minimizes G in C(Γ, S), we get

inf_{C(Γ,S)} F ≤ F(X) ≤ G(X) = inf_{C(Γ,S)} G =_{(i)} inf_{C(Γ,S)} F,

whence F(X) = inf_{C(Γ,S)} F and F(X) = G(X), and this implies (1.2). □

In particular we have inf_{C(Γ,S)} A = inf_{C(Γ,S)} D. We can prove higher regularity if the parametric Lagrangian F possesses a perfect dominance function G. In general we do not know for which F such a G exists; but we have at least one modest result in store:

Theorem 1.10. Let F^{∗} ∈ C^{2}(ℝ^{n} × (ℝ^{N} − {0})) satisfy (H),(D) and the ellipticity condition

|z| ζ · F_{zz}(x, z)ζ ≥ λ^{∗}|P_{z}^{⊥}ζ|^{2} for x ∈ ℝ^{n}, z, ζ ∈ ℝ^{N}, z ≠ 0,

for some λ^{∗} > 0 and P_{z}^{⊥}ζ := ζ − |z|^{−2}(z · ζ)^{2}. Then for any k > k_{0} := max{2(m_{2} − λ^{∗}), −m_{1}/2} the parametric Lagrangian F_{k} defined by

F_{k}(x, z) := kA(z) + F^{∗}(x, z)

possesses a perfect dominance function.

The proof is based on Morrey’s construction in [14], we refer the reader to [10] and also to [8, Proof of Theorem 1.10].

The arguments used in [8] to show higher regularity in the interior were based on the weak Euler equations, a variant of which we will also employ in the present case choosing admissible test functions.

Proposition 1.11. Let G ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − {0})) be an arbitrary function satisfying (D3). Then for any surface X ∈ H^{1,2}(B, ℝ^{n}) ∩ L_{loc}^{∞}(B, ℝ^{n}) and any ϕ ∈ H^{1,2}(B, ℝ^{n}) ∩ L_{loc}^{∞}(B, ℝ^{n}) with supp ϕ ⊂⊂ B we have

lim_{ϵ→0} ϵ^{−1}[G(X + ϵϕ) − G(X)] = δG(X, ϕ),

where

δG(X, ϕ) := ∫_{B}[G_{p}(X, ∇X) · ∇ϕ + G_{x}(X, ∇X) · ϕ] dudv. (1.10)

If, in addition, X is a minimizer of G in C(Γ, S), then

(1.11) δG(X, ϕ) = 0 for any ϕ ∈ H^{∘}^{1,2}(B, ℝ^{n}) ∩ L_{loc}^{∞}(B, ℝ^{n}).

According to Morrey (see [14], and for details, [10],[16]) every parametric Lagrangian F (x, z) of class C^{2}(ℝ^{n}×(ℝ^{N} −{0})) has a dominance function G_{0} ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − {0})) satisfying (D1)–(D4). If F satisfies (H),(D),(C), every conformally parametrized minimizer X of F in C(Γ, S) is a minimizer of G_{0} in C(Γ, S) with X ∈ L_{loc}^{∞}(B, ℝ^{n}). Thus we have δG_{0}(X, ϕ) = 0 for all ϕ ∈

H^{∘}^{1,2}(B, ℝ^{n}) ∩ L_{loc}^{∞}(B, ℝ^{n}). By virtue of Lemma 3.5 of [8] on the other hand, we have for any dominance function G ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − Π)) for F with (D1)–(D3) the identities G_{x}(x, p) = f_{x}(x, p) and G_{p}(x, p) = f_{p}(x, p) for (x, p) ∈ ℝ^{n} × Π_{0}, where f(x, p) = F (x, p_{1} ∧ p_{2}) is the associated Lagrangian of F. In particular these derivatives exist, and so we obtain (1.11) for arbitrary dominance functions G ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − Π)) with (D1)–(D3), in particular for f itself, if X is a conformally parametrized minimizer of F.

Finally we state our main result on higher regularity of solutions X to the minimum problem (P), to be proved in Section 4.

Theorem 1.12. Suppose that F has the properties (H),(D),(C), and possesses a perfect dominance function. Assume also that S is an m-dimensional submanifold in ℝ^{n}, 1 ≤ m ≤ n − 1, which is of class C^{3} and satisfies a chord-arc condition. Then any conformally parametrized minimizer of F in C(Γ, S) is of class H_{loc}^{2,2}(B ∪ I, ℝ^{n}) ∩ C^{1,α}(B ∪ I, ℝ^{n}) for some α ∈ (0, 1).

Note that already S ∈ C^{1} implies a chord-arc condition for S if S is compact, but a noncompact S might not satisfy a chord-arc condition, even if it is of class C^{∞}. If S is complete as a Riemannian submanifold of ℝ^{n}, but not compact, we might view the chord-arc condition as a regularity assumption on S at infinity.

Concerning the regularity of minimizers at the “Plateau boundary” C corresponding to Γ we refer to [9].

2 Existence of minimizers

In this section we prove the existence of conformally parametrized minimizers for the problem (P) as stated in Theorem 1.3.

We choose some point P_{3} ∈ Γ^{∘} and set w_{3} := (0, 1). Then we define C(Γ, S, P_{3}) as the subclass of surfaces X ∈ C(Γ, S) satisfying X(w_{3}) = P_{3}. Note that the surfaces X ∈ C(Γ, S, P_{3}) respect the 3-point-condition

X(w_{ν}) = P_{ν}, ν = 1, 2, 3. (2.1)

Let us recall the following variant of the Courant–Lebesgue Lemma (cf. e.g., [4, vol. I, p. 259]) which is essential for proving existence of “partially free” minimizers:

Proposition 2.1. From any sequence {X_{j}} of surfaces X_{j} ∈ C(Γ, S, P_{3}) with uniformly bounded Dirichlet integrals D_{B}(X_{j}) we can select a subsequence {Y_{j}} such that:

(i) {Y_{j}} converges weakly in H^{1,2}(B, ℝ^{n}) to some X ∈ H^{1,2}(B, ℝ^{n});

(ii) the traces Y_{j}|_{C} converge uniformly on C to X|_{C};

(iii) the weak limit X is of class C(Γ, S, P_{3}).

A consequence of this proposition is

Corollary 2.2. The class C(Γ, S, P_{3}) is closed with respect to weak convergence of sequences in H^{1,2}(B, ℝ^{n}).

Proof of Theorem 1.3: We can essentially argue as in [7] or in [8], Section 2, replacing the Courant-Lebesgue Lemma by Proposition 2.1 above. For the convenience of the reader we present the line of reasoning.

(i) For ϵ > 0 we consider the functional F^{ϵ} : H^{1,2}(B, ℝ^{n}) → ℝ defined by F^{ϵ}(X) := F(X)+ϵD(X). Introducing the nonparametric integrand f^{ϵ}(x, p) := f(x, p) + ϵ|p|^{2}/2 we have F^{ϵ}(X) = ∫_{B} f^{ϵ}(X, ∇X) dudv. By (C), the Lagrangian f^{ϵ}(x, p) is polyconvex and therefore quasiconvex in p and satisfies

\frac{1}{2}ϵ|p|^{2} ≤ f^{ϵ}(x, p) ≤ \frac{1}{2}(m_{2} + ϵ)|p|^{2}.

Thus F^{ϵ} is sequentially weakly lower semicontinuous on H^{1,2}(B, ℝ^{n}), cf. [1], and satisfies

ϵD(X) ≤ F^{ϵ}(X) ≤ (m_{2} + ϵ)D(X) for any X ∈ H^{1,2}(B, ℝ^{n}).

Let {X_{j}} be a sequence of surfaces X_{j} ∈ C(Γ, S) such that

lim_{j→∞} F^{ϵ}(X_{j}) = inf_{C(Γ,S)} F^{ϵ} =: d(ϵ).

Since F^{ϵ} is conformally invariant we can assume that X_{j} ∈ C(Γ, S, P_{3}) for all j ∈ ℕ, and we also have that the sequence of numbers D(X_{j}) is bounded by some constant c(ϵ) depending on ϵ. By Proposition 2.1 we can assume that X_{j} ⇀ X^{ϵ} in H^{1,2}(B, ℝ^{n}) for some surface X^{ϵ} ∈ C(Γ, S, P_{3}), and so we obtain

d(ϵ) ≤ F^{ϵ}(X^{ϵ}) ≤ lim inf_{j→∞} F^{ϵ}(X_{j}) = d(ϵ),

whence F^{ϵ}(X^{ϵ}) = d(ϵ). Thus X^{ϵ} minimizes F^{ϵ} in C(Γ, S), and we obtain ∂F(X^{ϵ}, η) = 0 for every η ∈ C^{1}(\overline{B}, ℝ^{2}), where ∂F^{ϵ}(X^{ϵ}, η) is the inner variation of F^{ϵ} at X^{ϵ} in direction of η. Since F is parameter invariant we have ∂F(X^{ϵ}, η) = 0, whence

∂D(X^{ϵ}, η) = 0 for any η ∈ C^{1}(\overline{B}, ℝ^{2}).

This implies the conformality relations

|X_{u}^{ϵ}|^{2} = |X_{v}^{ϵ}|^{2}, X_{u}^{ϵ} · X_{v}^{ϵ} = 0 a.e. on B, (2.2)

cf., e.g., [2] or [4], and so we have A(X^{ϵ}) = D(X^{ϵ}). Assumption (D) implies (m_{1} + ϵ)D(X^{ϵ}) ≤ F^{ϵ}(X^{ϵ}), and for any Z ∈ C(Γ, S) we have

F^{ϵ}(X^{ϵ}) = d(ϵ) ≤ F^{ϵ}(Z) ≤ m_{2}A(Z) + ϵD(Z) ≤ (m_{2} + ϵ)D(Z).

Since (m_{2}+ϵ)/(m_{1}+ϵ) ≤ m_{2}/m_{1} for ϵ ≥ 0, we see that D(X^{ϵ}) ≤ (m_{2}/m_{1})D(Z) for any Z ∈ C(Γ, S). If we choose Z as a minimal surface Y ∈ C(Γ, S) with D(Y) = inf_{C(Γ,S)} D, we obtain

∥X^{ϵ}∥_{H^{1,2}(B,ℝ^{n})} ≤ c(m_{1}, m_{2}, Γ, S) (2.3)

for some number c independent of ϵ > 0 by virtue of a suitable Poincaré inequality.

(ii) By (2.3) and Proposition 2.1 there is a sequence of numbers ϵ_{j} > 0 with ϵ_{j} → 0 and a surface X ∈ C(Γ, S, P_{3}) such that X^{ϵ_{j}} ⇀ X in H^{1,2}(B, ℝ^{n}) as j → ∞. Setting d(0) := inf_{C(Γ,S)} F we obtain

d(0) ≤ F(X) ≤ lim inf_{j→∞} F(X^{ϵ_{j}})

since F is sequentially weakly lower semicontinuous on H^{1,2}(B, ℝ^{n}) (cf. [1], and note that f(x, p) is polyconvex with respect to p and satisfies 0 ≤ f(x, p) ≤ m_{2}|p|^{2}/2).

Clearly d : [0, ∞) → ℝ_{+} is nondecreasing; therefore lim_{ϵ→+0} d(ϵ) exists and satisfies

d(0) ≤ lim_{ϵ→+0} d(ϵ) = lim_{ϵ→+0} F^{ϵ}(X^{ϵ}),

taking (2.3) into account.

On the other hand, we have F^{ϵ}(X^{ϵ}) ≤ F^{ϵ}(Z) for any Z ∈ C(Γ, S) whence lim_{ϵ→+0} F^{ϵ}(X^{ϵ}) ≤ F(Z), and therefore lim_{ϵ→+0} F^{ϵ}(X^{ϵ}) = d(0). We conclude that

d(0) = inf_{C(Γ,S)} F = F(X) = lim_{ϵ→+0} F^{ϵ}(X^{ϵ}),

and hence X is a minimizer of F in C(Γ, S).

(iii) It remains to be seen that X is conformally parametrized. The minimum properties of X^{ϵ_{j}} and X imply

F(X^{ϵ_{j}})+ϵ_{j}D(X^{ϵ_{j}}) = F^{ϵ_{j}}(X^{ϵ_{j}}) ≤ F^{ϵ_{j}}(X) = F(X)+ϵ_{j}D(X) ≤ F(X^{ϵ_{j}})+ϵ_{j}D(X),

whence D(X^{ϵ_{j}}) ≤ D(X) and thus lim sup_{j→∞} D(X^{ϵ_{j}}) ≤ D(X). On the other hand, X^{ϵ_{j}} ⇀ X in H^{1,2}(B, ℝ^{n}) implies D(X) ≤ lim inf_{j→∞} D(X^{ϵ_{j}}); therefore D(X^{ϵ_{j}}) → D(X) and finally ∥X − X^{ϵ_{j}}∥_{H^{1,2}(B,ℝ^{n})} → 0. On account of (2.2) we arrive at

|X_{u}|^{2} = |X_{v}|^{2}, X_{u} · X_{v} = 0 a.e. on B.
□

3 Hölder continuity

First we state a result which is proved in the same way as Theorem 1.5 in [8], and so we can omit its proof.

Theorem 3.1. If F ∈ C^{0}(ℝ^{n} × ℝ^{N}) satisfies (H),(D),(C), then every conformally parametrized minimizer X of F in C(Γ, S) satisfies the Morrey condition

∫_{B_{r}(w_{0})} |∇X|^{2} dudv ≤ (\frac{r}{R})^{2γ} ∫_{B_{R}(w_{0})} |∇X|^{2} dudv, γ := \frac{m_{1}}{m_{2}} , (3.1)

for any w_{0} = (u_{0}, v_{0}) ∈ B and 0 < r ≤ R ≤ dist (w_{0}, ∂B). Therefore X is of class C^{0,γ}(B, ℝ^{n}).

Proof of Theorem 1.5: For any w_{0} = (u_{0}, v_{0}) ∈ B ∪ I we define Φ(r, w_{0}) := ∫_{S_{r}(w_{0})} |∇X|^{2} dudv where S_{r}(w_{0}) := B ∩ B_{r}(w_{0}) and d_{0} := dist (w_{0}, C).

(i) If ρ ∈ (0, d_{0}), and v_{0} ≥ ρ, then B_{r}(w_{0}) ⊂ B for any r ∈ [0, ρ], and Theorem 3.1 with R := ρ yields

Φ(r, w_{0}) ≤ (r/ρ)^{2γ}Φ(R, w_{0}) if 0 < r ≤ ρ. (3.2)

(ii) Let w_{0} ∈ I. For 0 < r ≤ ρ ≤ d_{0} set B_{r} := B_{r}(w_{0}), S_{r} := S_{r}(w_{0}), and Φ(r) := Φ(r, w_{0}). Introducing polar coordinates ρ, θ around w_{0} = (u_{0}, v_{0}) ≅ u_{0} + iv_{0} by w ≅ w_{0} + ρe^{iθ} and writing X(ρ, θ) = X(w_{0} + ρe^{iθ}), we have |X_{ρ}|^{2} = ρ^{−2}|X_{θ}|^{2}, X_{ρ} · X_{θ} = 0 and

Φ(r) = 2 ∫_{0}^{r} ρ^{−1} ∫_{0}^{π} |X_{θ}(ρ, θ)|^{2} dθdρ.

For almost all r ∈ (0, d_{0}) we have Φ′(r) = 2r^{−1} ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ, and the limits Q_{1}(r) := lim_{θ→π−0} X(r, θ) and Q_{2}(r) := lim_{θ→+0} X(r, θ) exist.

Suppose first that ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ ≤ δ^{2}/π. Then

|Q_{1}(r) − Q_{2}(r)| ≤ ∫_{0}^{π} |X_{θ}(r, θ)| dθ ≤ \sqrt{π}[ ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ]^{1/2} ≤ δ.

Hence by the (M, δ)-chord-arc condition on S there is a curve Γ^{∗} in S of length L(Γ^{∗}) = l^{∗}, connecting Q_{1}(r) and Q_{2}(r) and satisfying l^{∗} ≤ M|Q_{1}(r)−Q_{2}(r)|. We can assume that Γ^{∗} is given by a Lipschitz parametrization ξ : [0, l^{∗}] → ℝ^{n} such that Γ^{∗} = ξ([0, 1]) and |ξ′(s)| = 1 a.e. on [0, l^{∗}]. For the representation ζ : [π, 2π] → ℝ^{n} of Γ^{∗} defined by ζ(θ) := ξ((θ − π)l^{∗}/π) we obtain |ζ_{θ}(θ)| = l^{∗}/π a.e. and l^{∗} = ∫_{π}^{2π} |ζ_{θ}| dθ and ∫_{π}^{2π} |ζ_{θ}|^{2} dθ = l^{∗}^{2}/π. Therefore,

∫_{π}^{2π} |ζ_{θ}|^{2} dθ ≤ M^{2} ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ.

We may assume that ζ(2π) = Q_{2}(r) and ζ(π) = Q_{1}(r), and we form the harmonic map H : B_{r} → ℝ^{n} with the absolutely continuous boundary values η(θ) = H(w_{0} + re^{iθ}) defined by η(θ) := H(r, θ) on [0, π] and η(θ) := ζ(θ) on [π, 2π]. It follows that

∫_{0}^{2π} |η_{θ}|^{2} dθ ≤ (1 + M^{2}) ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ.

On the other hand, we have the well–known inequality ∫_{B_{r}} |∇H|^{2} dudv ≤ ∫_{0}^{2π} |η_{θ}|^{2} dθ, and so we arrive at

∫_{B_{r}} |∇H|^{2} dudv ≤ \frac{1}{2}(1 + M^{2})rΦ′(r).

Now we consider the mapping Y : B ∪ B_{r} → ℝ^{n} which is defined by Y (w) := X(w) for w ∈ B − B_{r} and Y (w) := H(w) for w ∈ B_{r}. Let τ be the homeomorphism of \overline{B} onto \overline{B ∪ B_{r}}, which maps B conformally onto B ∪ B_{r} keeping w_{1}, w_{2}, w_{3} fixed. Then the reparametrization Z := Y ∘ τ lies in C(Γ, S), and so the minimum property of X implies F(X) ≤ F(Z).

For Ω ⊂ ℝ^{2} and V ∈ H^{1,2}(Ω, ℝ^{n}) we set F_{Ω}(V) := ∫_{Ω} f(V, ∇V) dudv. It follows that F(Z) = F_{B∪B_{r}}(Y), because F is parameter invariant, and so we obtain F_{S_{r}}(X) ≤ F_{B_{r}}(H). Since X is conformally parametrized we have m_{1}D_{S_{r}}(X) = m_{1}A_{S_{r}}(X) ≤ F_{S_{r}}(X), and trivially F_{B_{r}}(H) ≤ m_{2}A_{B_{r}}(H) ≤ m_{2}D_{B_{r}}(H). Hence

Φ(r) = 2D_{S_{r}}(X) ≤ (m_{2}/m_{1})2D_{B_{r}}(H),

and so we arrive at

Φ(r) ≤ 2^{−1}m_{1}^{−1}m_{2}(1 + M^{2})rΦ′(r) (3.3)

provided that ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ ≤ δ^{2}/π. If, however, ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ > δ^{2}/π, we immediately have

Φ(r) ≤ 2D(X) ≤ 2m_{1}^{−1}F(X) < δ^{−2}m_{1}^{−1}[2πF(X)] ∫_{0}^{π} |X_{θ}(r, θ)|^{2} dθ,

whence

Φ(r) < δ^{−2}m_{1}^{−1}πF(X)rΦ′(r). (3.4)

Set σ := min{m_{1}m_{2}^{−1}(1 + M^{2})^{−1}, [2πF(X)]^{−1}m_{1}δ^{2}}, and note that σ satisfies 0 < σ ≤ min{γ, 1/2}. We infer from (3.3) and (3.4) that

2σΦ(r) ≤ rΦ′(r) for a.e. r ∈ (0, ρ). (3.5)

Integrating we arrive at Φ(r) ≤ (r/ρ)^{2σ}Φ(ρ) for all r ∈ (0, ρ], and this means

(3.6) Φ(r, w_{0}) ≤ (r/ρ)^{2σ}Φ(ρ, w_{0}) for all r ∈ (0, ρ],

where 0 < ρ ≤ d_{0} = dist (w_{0}, C).

(iii) Now we fix some R ∈ (0, 1) and choose an arbitrary point w_{0} = (u_{0}, v_{0}) ∈ B ∪ I with R ≤ dist (w_{0}, C).

Case 1: v_{0} ≥ R/2.

Choosing ρ := R/2 in (i) we infer from (3.2) and the fact that 0 < σ ≤ γ that

Φ(r, w_{0}) ≤ (2r/R)^{2σ} ∫_{B} |∇X|^{2} dudv for 0 < r ≤ R/2. (3.7)

Case 2: 0 ≤ v_{0} ≤ R/2 and v_{0} ≤ r ≤ R/2.

Then for w_{0}^{∗} := (u_{0}, 0) ∈ I we have B_{r}(w_{0}) ⊂ B_{2r}(w_{0}^{∗}), and therefore Φ(r, w_{0}) ≤ Φ(2r, w_{0}^{∗}). By (ii) for ρ := R, in particular (3.6), it follows that Φ(2r, w_{0}^{∗}) ≤ (2r/R)^{2σ}Φ(R, w_{0}^{∗}) and thus

Φ(r, w_{0}) ≤ (2r/R)^{2σ} ∫_{B} |∇X|^{2} dudv. (3.8)

In particular, we obtain

Φ(v_{0}, w_{0}) ≤ (2v_{0}/R)^{2σ} ∫_{B} |∇X|^{2} dudv. (3.9)

Case 3: 0 ≤ v_{0} ≤ R/2 and 0 < r ≤ v_{0}.

By the choice ρ := v_{0} in inequality (3.2) of (i) we obtain Φ(r, w_{0}) ≤ (r/v_{0})^{2γ}Φ(v_{0}, w_{0}), and by inequality (3.9) we conclude that

Φ(r, w_{0}) ≤ (r/v_{0})^{2γ}(2v_{0}/R)^{2σ} ∫_{B} |∇X|^{2} dudv,

whence by 0 ≤ σ ≤ γ :

Φ(r, w_{0}) ≤ (2r/R)^{2σ} ∫_{B} |∇X|^{2} dudv. (3.10)

With (3.7),(3.8),(3.10) we have verified in all possible cases that

Φ(r, w_{0}) ≤ (2r/R)^{2σ} ∫_{B} |∇X|^{2} dudv for all 0 < r ≤ R/2,

for w_{0} ∈ B ∪ I and R ∈ (0, dist (w_{0}, C)]. If r > R/2, this inequality holds for obvious reasons, and so inequality (1.3) is proved. The final statements follow from Morrey’s Dirichlet growth theorem. □

Theorem 1.6 will be obtained by a straightforward modification of the preceding reasoning so that we may leave the details to the reader.

4 Higher regularity at the free boundary

Now we turn to the

Proof of Theorem 1.12: Step 1. Interior regularity. Let X be a conformally parametrized minimizer of F in C(Γ, S), and assume that G is a perfect dominance function of F. Then by Proposition 1.11 we have the weak Euler equation

∫_{B}[G_{p}(X, ∇X) · ∇ϕ + G_{x}(X, ∇X) · ϕ] dudv = 0, (4.1)

which holds for all test functions ϕ ∈ H^{∘}^{1,2}(B, ℝ^{n}) ∩ L_{loc}^{∞}(B, ℝ^{n}). By Theorem 1.5 we obtain X ∈ C^{0,σ}(B ∪ I, ℝ^{n}) for some σ ∈ (0, 1), and

∫_{B∩B_{r}(w_{0})} |∇X|^{2} dudv ≤ (\frac{2}{R})^{2σ} 2D(X)r^{2σ} for all r > 0, (4.2)

where w_{0} ∈ B ∪ I with 0 < R ≤ d_{0} := dist (w_{0}, C).

Suppose that ζ_{0} ∈ I, 0 < r_{0} < dist (ζ_{0}, C)/2, and Ω = Ω(ζ_{0}, 2r_{0}) := B ∩ B_{2r_{0}}(ζ_{0}). By a straightforward reasoning we infer from (4.2) for Ω_{ρ}(w_{0}) := Ω ∩ B_{ρ}(w_{0}) = B ∩ B_{2r_{0}}(ζ_{0}) ∩ B_{ρ}(w_{0}) that

∫_{Ω_{ρ}(w_{0})} |∇X|^{2} dudv ≤ M_{0}ρ^{2σ} for w_{0} ∈ ℝ^{2} and ρ > 0, (4.3)

where M_{0} = M(d_{0}, D(X)) does not depend on r_{0} and ρ. By the reasoning of [8], Section 4, we obtain that

(4.4) X ∈ H_{loc}^{2,2}(B, ℝ^{n}) ∩ C^{1,ν}(B, ℝ^{n}) for some ν ∈ (0, 1).

Step 2. Linearizing the free boundary condition. Pick any ζ_{0} ∈ I and set x_{0} := X(ζ_{0}). Then there is a C^{3}-diffeomorphism g of ℝ^{n} onto itself and an open neighbourhood U of x_{0} such that

g(x_{0}) = 0, g(U) = K := {y ∈ ℝ^{n} : |y| < 1}

and

g(S) = {y ∈ K : y^{j} = 0 for m + 1 ≤ j ≤ n}.

For sufficiently small r_{0} > 0 and w ∈ \overline{Ω} = \overline{Ω(ζ_{0}, 2r_{0})} we obtain X(w) ∈ U, and so Y := g ∘ X ∈ H^{1,2}(B, ℝ^{n}) ∩ C^{0,σ}(B ∪ I, ℝ^{n}) satisfies Y (w) ∈ K for w ∈ \overline{Ω}. Since X(I) ⊂ S, it follows that

Y^{j}(w) = 0 for w ∈ \overline{Ω} ∩ I, m + 1 ≤ j ≤ n, (4.5)

and the chain rule implies Y ∈ H_{loc}^{2,2}(B, ℝ^{n}) ∩ C^{1,ν}(B, ℝ^{n}). Let h := g^{−1} be the inverse of g and H := h′ be the Jacobian matrix of h. Then we can write X = h ∘ Y with D_{u}X = H(Y)D_{u}Y and D_{v}X = H(Y)D_{v}Y. We introduce the new Lagrangian G̃ ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − {0})) by

G̃(y, q) := G̃(y, q_{1}, q_{2}) := G(h(y), H(y)q_{1}, H(y)q_{2}) (4.6)

for q = (q_{1}, q_{2}) ∈ ℝ^{2n}, and set

G̃(Z) := ∫_{B} G̃(Z, ∇Z) dudv for any Z ∈ H^{1,2}(B, ℝ^{n}). (4.7)

Then we have G(h ∘ Z) = G̃(Z), and in particular G(X) = G̃(Y). Set Γ̃ := g(Γ), S̃ := g(S), and define C(Γ̃, S̃) accordingly. Suppose now that ϕ ∈ H^{1,2}(B, ℝ^{n}) ∩ L^{∞}(Ω, ℝ^{n}) satisfies

ϕ^{m+1}(w) = 0, …, ϕ^{n}(w) = 0 for w ∈ \overline{Ω} ∩ I,

(4.8)

ϕ(w) = 0 for w ∈ B − Ω.

Then, for |ϵ| < ϵ_{0} ≪ 1, we have Y^{ϵ} := Y + ϵϕ ∈ C(Γ̃, S̃) and therefore X^{ϵ} := h(Y^{ϵ}) ∈ C(Γ, S), whence

0 ≤ G(X^{ϵ}) − G(X) = G̃(Y^{ϵ}) − G̃(Y) for |ϵ| < ϵ_{0} ≪ 1.

We note that G̃(y, q) is positively homogeneous of degree two with respect to q, and Y satisfies the generalized conformality relations

Y_{u} · Ξ(Y)Y_{u} = Y_{v} · Ξ(Y)Y_{v}, Y_{u} · Ξ(Y)Y_{v} = 0 a.e. on B, (4.9)

where Ξ : ℝ^{n} → ℝ^{n×n} is a positive definite, symmetric matrix function of class C^{2}, defined by Ξ := H^{T} · H. By Proposition 1.11 which is still applicable, since G̃ satisfies (D3), we then obtain

∫_{B}[G̃_{q}(Y, ∇Y) · ∇ϕ + G̃_{y}(Y, ∇Y) · ϕ dudv = 0 (4.10)

for all test functions ϕ with the properties (4.8). Note that

|∇X|^{2} = Y_{u} · Ξ(Y)Y_{u} + Y_{v} · Ξ(Y)Y_{v}, (4.11)

and for suitable numbers λ_{1}, λ_{2} with 0 < λ_{1} ≤ λ_{2} we have

λ_{1}|ξ|^{2} ≤ ξ · Ξ(y)ξ ≤ λ_{2}|ξ|^{2} (4.12) for ξ ∈ ℝ^{n} and y ∈ K. From (4.3) it follows that

∫_{Ω_{ρ}(w_{0})} |∇Y |^{2} dudv ≤ M′_{0} ρ^{2σ} for w_{0} ∈ ℝ^{2} and ρ > 0 (4.13)

and M′_{0} := λ_{1}^{−1}M_{0}. Finally, the ellipticity condition (E) for G is transformed into an analogous ellipticity condition for G̃ :

π · G̃_{qq}(y, q)π ≥ λ̃(R_{0})|π|^{2} for |y| ≤ R_{0}, π, q ∈ ℝ^{2n}, q ≠ 0, (4.14)

where λ̃(R_{0}) > 0 depends only on R_{0} > 0.

Summarizing, we see that Y has analogous properties as X, but the corresponding variational equation (4.10) is true for a larger class of test functions ϕ. This we shall now exploit.

Step 3. H^{2,2}-estimates of Y on Ω = Ω(ζ_{0}, 2r_{0}) for 0 < r_{0} ≪ 1.

Let us introduce the set Σ_{0} of singular points of X defined by

Σ_{0} := {w_{0} ∈ B : X_{u}(w) ∧ X_{v}(w) = 0},

which is relatively closed in B since X ∈ C^{1}(B, ℝ^{n}); thus B − Σ_{0} is an open set of ℝ^{2}. From (1.2) and Π ∩ Π_{0} = {0} we infer that

Σ_{0} = {w ∈ B : |∇X(w)| = 0},

and therefore also

Σ_{0} = {w ∈ B : |∇Y (w)| = 0}.

From (4.10) and Y ∈ H_{loc}^{2,2}(B, ℝ^{n}) we obtain

−D_{β}[G̃_{q_{β}^{j}}(Y, ∇Y)] + G̃_{y}(Y, ∇Y) = 0 a.e. on B − Σ_{0},

where u^{1} := u, u^{2} := v, D_{1} := D_{u}, D_{2} := D_{v}. This is symbolically written as

−∇[G̃_{q}(Y, ∇Y)] + G̃_{y}(Y, ∇Y) = 0 a.e. on B − Σ_{0}.

On B − Σ_{0} we can apply the chain rule, inferring

∇[G̃_{q}(Y, ∇Y)] = G̃_{qq}(Y, ∇Y)∇^{2}Y + G̃_{qy}(Y, ∇Y)∇Y.

This implies

G̃_{qq}(Y, ∇Y)∇^{2}Y = −G̃_{qy}(Y, ∇Y)∇Y + G̃_{y}(Y, ∇Y) a.e. on B − Σ_{0}.

Since G̃_{y}(y, q) and G̃_{qy}(y, q) are positively homogeneous in q of degree two and one, respectively, we have

|G̃_{y}(y, q)| ≤ c_{0}(R_{0})|q|^{2}, |G̃_{qy}(y, q)| ≤ c_{1}(R_{0})|q| if |y| ≤ R_{0}

for some constants c_{0}(R_{0}), c_{1}(R_{0}) depending on R_{0}. Thus we arrive at

|G̃_{qq}(Y, ∇Y)∇^{2}Y | ≤ const |∇Y |^{2} a.e. on B − Σ_{0}.

Furthermore, G̃_{qq}(Y, ∇Y)∇^{2}Y = G̃_{q_{k}q_{s}}(Y, ∇Y)D_{k}D_{s}Y, and we have

|G̃_{qq}(y, q)| ≤ c_{2}(R_{0}) if |y| ≤ R_{0}, q ≠ 0,

for some constant c_{2}(R_{0}) depending on R_{0}, since G̃_{qq}(y, q) is positively homogeneous in q of degree zero. Thus

G̃_{q_{2}q_{2}}(Y, ∇Y)D_{v}D_{v}Y = G̃_{qq}(Y, ∇Y)∇^{2}Y + R(Y, ∇Y, ∇^{2}Y),

where the remainder term

R(Y, ∇Y, ∇^{2}Y) = − ∑_{(k,s)≠(2,2)} G̃_{q_{k}q_{s}}(Y, ∇Y)D_{k}D_{s}Y

can be estimated by

|R(Y, ∇Y, ∇^{2}Y)| ≤ const |∇D_{u}Y | a.e. on B − Σ_{0}.

From (4.14) it follows that G̃_{q_{2}q_{2}}(Y (w), ∇Y (w)) is invertible for all w ∈ B − Σ_{0}, and its inverse is uniformly bounded in norm on B − Σ_{0}. Hence we arrive at

(4.15) |D_{v}D_{v}Y |^{2} ≤ const (|∇D_{u}Y |^{2} + |∇Y |^{4}) a.e. on B − Σ_{0},

and on Σ_{0} this estimate is trivially satisfied, since ∇Y = 0 on Σ_{0} and Y ∈ H_{loc}^{2,2}(B, ℝ^{n}) imply ∇^{2}Y = 0 a.e. on Σ_{0}. Consequently, inequality (4.15) holds a.e. on B. Now we proceed in a similar way as in [5], pp. 64 – 68, using some refinements developed for the proof of interior regularity; we refer the reader to [8], Section 4.

First some notation: For 0 < |k| ≪ 1 and z(u, v) let △_{k}z(u, v) := k^{−1}[z(u + k, v) − z(u, v)] be the difference quotient in tangential direction; symbolically we have D_{u} = lim_{k→0} △_{k}. Let η be some “friend” on Ω(ζ_{0}, 2r) for 0 < r ≤ r_{0}, i.e. η ∈ C^{∞}(\overline{B}), 0 ≤ η ≤ 1, η ≡ 1 on B_{r}(ζ_{0}), η ≡ 0 on \overline{B} − B_{2r}(ζ_{0}), and |∇η| ≤ 2/r on the semi-annulus

T_{2r} := {w ∈ B : r < |w − ζ_{0}| < 2r} = Ω(ζ_{0}, 2r) − Ω(ζ_{0}, r),

and ∇η ≡ 0 on \overline{B} − T_{2r}.

Then

ϕ := −△_{−k}(η^{2}△_{k}Y), 0 < |k| ≪ 1, (4.16)

is an admissible test function for the variational equation (4.10) since Y ∈ H^{1,2}(B, ℝ^{n}) ∩ C^{0}(B ∪ I, ℝ^{n}) satisfies the boundary conditions (4.5), which are not violated by taking tangential difference quotients as in (4.16), and so we have ϕ^{m+1}(w) = 0, …, ϕ^{n}(w) = 0 for w ∈ Ω ∪ I. Now the same kind of manipulations as in [8, specifically (4.17)–(4.22)] lead first to

∫_{B} η^{2}|∇△_{k}Y |^{2} dudv ≤ const [ ∫_{B} η^{2}|△_{k}Y |^{2}(|∇Y |^{2}+|∇Y_{k}|^{2}) dudv+r^{−2}D(Y)],

where Y_{k}(u, v) := Y (u + k, v), and then to

∫_{B} η^{2}|∇△_{k}Y |^{2} dudv ≤ c(r)D(Y) for |k| ≪ 1 and 0 < r ≪ 1, (4.17)

where the number c(r) does not depend on k. (The main difficulty to be overcome is that G̃_{qq}(y, q) is not defined for q = 0, and this is achieved by an appropriate perturbation argument carried out in [8].) As k → 0 we arrive at

∫_{B} η^{2}|∇D_{u}Y |^{2} dudv ≤ c(r)D(Y) for 0 < r ≪ 1. (4.18)

Similarly we see as in [8, (4.20)] that

∫_{B} η^{2}|△_{k}Y |^{2}|∇Y |^{2} dudv ≤ c(r)[ ∫_{B} η^{2}|∇△_{k}Y |^{2} dudv + D(Y)],

which in conjunction with (4.17) implies

∫_{B} η^{2}|△_{k}Y |^{2}|∇Y |^{2} dudv ≤ c(r)D(Y) for |k| ≪ 1 and 0 < r ≪ 1,

and as k → 0 we obtain

∫_{B} η^{2}|D_{u}Y |^{2}|∇Y |^{2} dudv ≤ c(r)D(Y) for 0 < r ≪ 1, (4.19)

in particular

∫_{B} η^{2}|D_{u}Y |^{4} dudv ≤ c(r)D(Y) for 0 < r ≪ 1.

By (4.9) and (4.12) we have

|D_{v}Y |^{2} ≤ (λ_{2}/λ_{1})|D_{u}Y |^{2}

whence

∫_{B} η^{2}|∇Y |^{4} dudv ≤ c(r)D(Y) for 0 < r ≪ 1. (4.20)

Combining (4.15),(4.18), and (4.20) we find that

∫_{B} η^{2}|∇^{2}Y |^{2} dudv ≤ c(r)D(Y) for 0 < r ≪ 1, (4.21)

and thus

∫_{Ω(ζ_{0},r)} |∇^{2}Y |^{2} dudv + ∫_{Ω(ζ_{0},r)} |∇Y |^{4} dudv ≤ c(r)D(Y) (4.22)

for 0 < r ≪ 1, since η ≡ 1 on Ω(ζ_{0}, r). From X = h(Y) we obtain

∇X = h′(Y)∇Y and ∇^{2}X = h′(Y)∇^{2}Y + h′′(Y)∇Y ∇Y,

and on account of (4.22) it follows that

∫_{Ω(ζ_{0},r)} |∇^{2}X|^{2} dudv ≤ c(r)D(X) for 0 < r ≪ 1.

A covering argument leads to

∫_{B∩B_{r}(0)} |∇^{2}X|^{2} dudv ≤ const for 0 < r < 1

if we also use that X ∈ H_{loc}^{2,2}(B, ℝ^{n}).

Step 4. Hölder continuity of ∇X on B ∪ I. Now we take instead of (4.16) the test function

ϕ := −η^{2}△_{−k}△_{k}Y, 0 < |k| ≪ 1,

which is also admissible in (4.10). By the same manipulations that lead to (4.25) in [8], and by (4.15) above, we arrive at

∫_{Ω(ζ_{0},r)} |∇^{2}Y |^{2} dudv

(4.23) ≤ const [ ∫_{Ω(ζ_{0},2r)} |∇Y |^{4} dudv + r^{−2} ∫_{T_{2r}} |D_{u}Y − C|^{2} dudv],

where C can be an arbitrary vector in ℝ^{n}. We choose C as the mean value ∫ −_{T_{2r}} D_{u}Y dudv of D_{u}Y over T_{2r}. By Poincaré’s inequality there is a constant K_{p} such that

∫_{T_{2r}} |D_{u}Y − C|^{2} dudv ≤ K_{p} r^{2} ∫_{T_{2r}} |∇D_{u}Y |^{2} dudv,

and so we infer from (4.23) that

∫_{Ω(ζ_{0},r)} |∇^{2}Y |^{2} dudv ≤ const [ ∫_{T_{2r}} |∇^{2}Y |^{2} dudv + ∫_{Ω(ζ_{0},2r)} |∇Y |^{4} dudv].

Recall that T_{2r} = Ω(ζ_{0}, 2r) − Ω(ζ_{0}, r). Hole filling and Sobolev’s inequality lead to

∫_{Ω(ζ_{0},r)} |∇^{2}Y |^{2} dudv ≤ θ_{0}[ ∫_{Ω(ζ_{0},2r)} |∇^{2}Y |^{2} dudv + k(δ)r^{2−2δ}]

for 0 < r ≤ r_{0} ≪ 1, δ ∈ (0, 1/2), and some constant k(δ). A standard iteration procedure yields

(4.24) ∫_{Ω(ζ_{0},r)} |∇^{2}Y |^{2} dudv ≤ const (r/r_{1})^{2α} for 0 < r ≤ r_{1}

where ζ_{0} ∈ I, α := −(log θ)/(2 log 2), θ := max{θ_{0}, 2^{−2+2τ} } ∈ (0, 1), τ ∈ (2δ, 1), r_{1} := min{r_{0}, r^{∗}}, r^{∗} := [θ^{−1}k(δ)^{−1}(2^{τ} − 1)]^{1/(τ−2δ)}. The interior estimates of [8] give

(4.25) ∫_{B_{r}(ζ_{0})} |∇^{2}Y |^{2} dudv ≤ const (r/r_{2})^{2α} for 0 < r ≤ r_{2} and ζ_{0} ∈ B,

where r_{2} := min{r′_{0}, r^{∗}} and r′_{0} := 4^{−1} dist (w_{0}, ∂B). Combining the two estimates (4.24) and (4.25) a similar reasoning as in the proof of Theorem 1.5, Section 3, we obtain that for ζ_{0} ∈ I and w_{0} ∈ Ω(ζ_{0}, ρ_{0}) we have

∫_{B_{r}(w_{0})∩Ω(ζ_{0},ρ_{0})} |∇^{2}Y |^{2} dudv ≤ c(ρ_{0})r^{2α} for all r > 0 (4.26)

where 0 < ρ_{0} ≪ 1 (in particular ρ_{0} < 2^{−1}(1 − |ζ_{0}|)) and c(ρ_{0}) → ∞ as ρ_{0} → +0. By Morrey’s Dirichlet growth theorem we infer from (4.26) that ∇Y is of class C^{0,α} on Ω(ζ_{0}, ρ_{0}) for 0 < ρ_{0} ≪ 1, and by ∇X = h′(Y)∇Y and Theorem 1.5 the same is true for ∇X if w.l.o.g. we assume that 0 < α ≤ σ. A covering argument finally implies that X ∈ C^{1,α}(B ∪ I, ℝ^{n}), as we have claimed. □

22 S. HILDEBRANDT, H. VON DER MOSEL

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[16] H. von der Mosel, Regularität von Lösungen parametrischer Variationsprobleme. Diplom thesis, University of Bonn 1992.

Stefan Hildebrandt Mathematisches Institut Universität Bonn Beringstraße 1 D-53115 Bonn GERMANY E-mail: beate@

math.uni-bonn.de

Heiko von der Mosel Mathematisches Institut Universität Bonn Beringstraße 1 D-53115 Bonn GERMANY E-mail: heiko@

                               math.uni-bonn.de