Dominance functions for parametric Lagrangians

Stefan Hildebrandt and Heiko von der Mosel

Mathematisches Institut, Universität Bonn, Beringstraße 1, D-53115 Bonn heiko@math.uni-bonn.de Summary. We discuss the concept of dominance functions for parametric Lagrangians together with important examples and various applications to the existence and regularity theory for minimizers of parametric functionals, and for the construction of unstable stationary surfaces. The focus lies on the construction of a perfect dominance function based on ideas of Morrey.

Mathematics Subject Classification (2000): 49N60, 49Q10, 53A07, 53A10, 53B40, 58E12

1 Introduction

Parametric Lagrangians and the Plateau problem. Let F be a parametric double integral of the form

F_{Ω}(X) := ∫_{Ω} F (X, X_{u} ∧ X_{v}) dudv,  (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, x ∈ ℝ^{n}, z ∈ ℝ^{N}.  (H)

In addition, 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_{Ω}(X) is well-defined by (1) for any domain Ω ⊂ ℝ^{2} and any mapping X ∈ H^{1,2}(Ω, ℝ^{n}), where

p_{1} ∧ p_{2} := (p_{1}^{j}p_{2}^{k} − p_{1}^{k}p_{2}^{j})_{j<k} ∈ ℝ^{N}

denotes the bivector generated by the two vectors p_{1} = (p_{1}^{1}, …, p_{1}^{n}), p_{2} = (p_{2}^{1}, …, p_{2}^{n}) ∈ ℝ^{n}. Notice that the Lagrange identity

|p_{1} ∧ p_{2}|^{2} = |p_{1}|^{2}|p_{2}|^{2} − (p_{1} · p_{2})^{2}  (2)

leads to

|p_{1} ∧ p_{2}| ≤ \frac{1}{2}(|p_{1}|^{2} + |p_{2}|^{2})

with equality if and only if

|p_{1}|^{2} = |p_{2}|^{2} and p_{1} · p_{2} = 0.  (3)

The homogeneity condition (H) implies that F_{Ω} is parameter invariant, i.e., invariant with respect to reparametrizations by means of C^{1}-diffeomorphisms with positive Jacobian.

A special Lagrangian of this kind is the area integrand A(z) := |z| satisfying (D) with m_{1} = m_{2} = 1, which leads to the area functional

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

The parametric integrand F is said to be semi-elliptic on K ×ℝ^{N}, K a closed set in ℝ^{n}, if

F (x, z) is convex on in z ∈ ℝ^{N} for any x ∈ K.  (C)

Admitting arbitrary closed sets K ⊂ ℝ^{n} in condition (C) is useful for the treatment of parametric variational problems in Riemannian manifolds. In the sequel, however, we assume for simplicity that K = ℝ^{n}.

We call F elliptic if for every R_{0} > 0 there is some λ_{F} (R_{0}) > 0 such that F − λ_{F} (R_{0})A is semi-elliptic on \overline{B_{R_{0}}}(0) × ℝ^{N}. If F ∈ C^{2}(ℝ^{n} × (ℝ^{N} − \{0\})), then condition (C) is equivalent to the assumption that F_{zz}(x, z) is positive semi-definite for z ≠ 0, whereas F is elliptic if and only if for any R_{0} > 0 there is some λ_{F} (R_{0}) such that F_{zz}(x, z) − λ_{F} (R_{0})A_{zz}(z) is positive semi-definite for all (x, z) ∈ \overline{B_{R_{0}}}(0) × (ℝ^{N} − \{0\}).

Let

P_{z}^{⊥}ζ := ζ − |z|^{−2}(z · ζ)z for z ≠ 0  (4)

denote the projection of ζ ∈ ℝ^{N} onto the hyperplane perpendicular to z in ℝ^{N}. Then we can express the ellipticity of F as

|z|ζ · F_{zz}(x, z)ζ ≥ λ_{F} (R_{0})|P_{z}^{⊥}ζ|^{2} = λ_{F} (R_{0})|z|^{−2}|ζ ∧ z|^{2}  (5)

for some λ_{F} (R_{0}) > 0 and any x ∈ \overline{B_{R_{0}}}(0) ⊂ ℝ^{n}, z, ζ ∈ ℝ^{N}, z ≠ 0, since

|z|ζ · A_{zz}(z)ζ = |ζ|^{2} − |z|^{−2}(z · ζ)^{2} = |P_{z}^{⊥}ζ|^{2} = |z|^{−2}|ζ ∧ z|^{2}  (6)

by virtue of (2). The identity (6) implies in particular that the area integrand A is elliptic, where we may choose λ_{F} (R_{0}) = 1 for all R_{0} > 0.

Conformally parametrized extremals of the area functional, i.e., surfaces X ∈ H^{1,2}(Ω, ℝ^{n}) satisfying

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

which are stationary points of A, are minimal surfaces. More generally, surfaces of prescribed mean curvature H in ℝ^{3} may be regarded as conformally parametrized extremals of the parametric functional

E_{Ω}(X) := ∫_{Ω} E(X, X_{u} ∧ X_{v}) dudv

with the Lagrangian

E(x, z) := |z| + Q(x) · z, x, z ∈ ℝ^{3},  (7)

where Q is a differentiable vector field on ℝ^{3} with 2div Q = H. Again, E is elliptic satisfying (5) with λ_{F} (R_{0}) = 1 for all R_{0} > 0. However, E is positive definite only if

Q_{0} := sup_{x∈ℝ^{3}} |Q(x)| < 1.  (8)

The parametric Lagrangian given by the discrete l^{r}-norm, 1 ≤ r ≤ ∞,

I(z) := (|z^{1}|^{r} + |z^{2}|^{r} + |z^{3}|^{r})^{1/r}, z = (z^{1}, z^{2}, z^{3}) ∈ ℝ^{3},  (9)

leads to an anisotropic interface energy

I_{Ω}(X) := ∫_{Ω} I(X_{u} ∧ X_{v}) dudv

used in mathematical image processing to detect and sharpen edges [2]. Note, however, that I is not even semi-elliptic in general; a regularized version of (9), e.g., for r = 1 is given by

I_{ϵ}(z) := ∑_{i=1}^{3} \sqrt{ϵ|z|^{2} + (z^{i})^{2}},  (10)

which is elliptic for each ϵ > 0. We also refer to the work of J. Taylor on “crystalline structures” (see [29]–[35]), where the surface W := {ζ ∈ ℝ^{3} : ζ · z ≤ F (z) for all z ∈ ℝ^{3} with |z| = 1} known as Wulff shape is supposed to be a polyhedron.

Various geometric boundary value problems for general parametric functionals of the type (1) have been studied. For the sake of brevity we will restrict our attention to the Plateau problem. For a closed rectifiable Jordan curve Γ ⊂ ℝ^{n} and the fixed domain B := {(u, v) : u^{2} + v^{2} < 1} we consider the nonempty class C(Γ) of competing surfaces X ∈ H^{1,2}(B, ℝ^{n}) whose trace X|_{∂B} on ∂B is a continuous, weakly monotonic mapping of ∂B onto Γ. Setting F := F_{B}, A := A_{B}, we have the following existence result for the minimization problem “F → min in C(Γ)” proved in [14],[15]:

Theorem 1. If F ∈ C^{0}(ℝ^{n} × ℝ^{N}) satisfies (H),(D), and (C), then there exists an F-minimizing surface X ∈ C(Γ) of class C^{0}(\overline{B}, ℝ^{n})∩C^{0,α}(B, ℝ^{n})∩ H_{loc}^{1,q}(B, ℝ^{n}) for α = m_{1}/m_{2} ∈ (0, 1] and some q > 2, which is conformally parametrized, i.e. X satisfies F(X) = inf_{C(Γ)} F, and

|X_{u}|^{2} = |X_{v}|^{2}, X_{u} · X_{v} = 0 a.e. on B.  (11)

If Γ satisfies a chord-arc condition, any minimizer X of F in C(Γ) is of class C^{β}(\overline{B}, ℝ^{n}) for some β ∈ (0, 1).

We note that the existence of continuous minimizers for F in C(Γ) was proved by Sigalov, Cesari, and Danskin in the years 1951–1952 (cf. the references in [22], [23]). In 1961, Morrey [22] gave a new, greatly simplified existence proof, adding Hölder continuity of minimizers in the interior. His new approach consisted in minimizing a modified functional F^{∗} := ∫_{B} f^{∗}(X, ∇X) dudv with the Lagrangian

f^{∗}(x, p_{1}, p_{2}) := {F^{2}(x, p_{1} ∧ p_{2}) + c[4^{−1}(|p_{1}|^{2} + |p_{2}|^{2})^{2} + (p_{1} · p_{2})^{2}]}^{1/2}

under the subsidiary condition ∫_{B} |∇^{2}X|^{2} dudv ≤ R, and then letting R → ∞. In all of this work it was not shown that there exists an a.e. conformally parametrized minimizer. Later on, in his treatise [23], Morrey presented a modification of his method replacing f^{∗} by a very interesting dominance function f^{∗∗} thereby obtaining a conformally parametrized minimizer of F in C(Γ). However, there seems to be a gap in his proof since f^{∗∗} is only shown to be rank-one convex, and so the corresponding functional F^{∗∗}(X) := ∫_{B} f^{∗∗}(X, X_{u}, X_{v}) dudv need not be weakly lower semicontinuous in H^{1,2}(B, ℝ^{n}) since this property would require quasiconvexity of f^{∗∗}.

Another existence proof is due to Y.G. Reshetnyak [24] who in 1962 obtained the same result as Morrey [22] but with a very different method. On p. 767 the author states without proof that “the parameters u and v of the function yielding the minimum of the functional I(x, F, Q) and obtained by the above proof is isothermal”.

The result stated in Theorem 1, in particular the existence of conformally parametrized minimizers, was proved by the present authors, first in a slightly weaker from (cf. [14]), and then in the final form of [15] using an idea provided by S. Müller. The basic idea of our approach was to minimize first the new functional F^{ϵ} := F + ϵD penalized by Dirichlet’s integral D thus obtaining “approximate” minimizers X^{ϵ} of the original problem which are conformally parametrized. Passing to the limit ϵ → +0, one obtains the desired minimizer of F in conformal parameters. It seems to us that this is crucial for proving higher regularity results because an arbitrary continuous minimizer of F cannot even be expected to be of class C^{1}. First steps of proving higher regularity were carried out in [15] and [16], using a modification of Morrey’s perfect dominance function. For details we refer to the discussion below; cf. Theorems 2–5.

An existence result for a free boundary value problem for the functional (1) can be found in [20]; the semi-free boundary problem with a given boundary curve and a support surface was solved in [16], and the Douglas problem, where one seeks multiply connected F-minimizing surfaces bounded by several Jordan curves, was treated in [18] and, more generally, in [19].

Dominance functions. In the sequel we call f(x, p) := F (x, p_{1} ∧ p_{2}) for p = (p_{1}, p_{2}) ∈ ℝ^{n} × ℝ^{n} ≅ ℝ^{2n} the associated Lagrangian of the parametric integrand F.

Definition 1. (i) Let F (x, z) be a parametric Lagrangian with the associated Lagrangian f(x, p). Then a function G : ℝ^{n} × ℝ^{2n} → ℝ is said to be 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)

f(x, p) = G(x, p) if and only if |p_{1}|^{2} = |p_{2}|^{2}, p_{1} · p_{2} = 0.  (D2)

(ii) A dominance function G of the parametric Lagrangian F is called quadratic if

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

and it is said to be positive definite if there are two numbers μ_{1}, μ_{2} with 0 < μ_{1} ≤ μ_{2}, such that

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

(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 if for any R_{0} > 0 there is a constant λ_{G}(R_{0}) > 0 such that

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

Condition (E) means that

G_{p_{α}^{j}p_{β}^{k}}(x, p)ξ_{α}^{j}ξ_{β}^{k} ≥ λ_{G}(R_{0})ξ_{α}^{j}ξ_{α}^{j}, λ_{G}(R_{0}) > 0.

Here and in the sequel we use the following convention: Greek indices α, β, … run from 1 to 2, and Latin indices j, k, … from 1 to n. Correspondingly, repeated indices α, β, …, j, k, …, are to be summed from 1 to 2, or n, respectively.

Let us consider some examples. 1 The integrand

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

of the Dirichlet energy D is a perfect dominance function for the area integrand A(z) = |z| with the associated Lagrangian

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

2 The integrand E defined in (7) with n = 3 possesses the dominance function

G(x, p) := \frac{1}{2}|p|^{2} + Q(x) · (p_{1} ∧ p_{2}),  (12)

satisfying (D1)–(D3). If (8) holds, and if Q ∈ C^{2}(ℝ^{n}), then G(x, p) is a perfect dominance function where one can choose λ_{G}(R_{0}) = 1 − Q_{0} > 0 in the ellipticity condition (E) for all R_{0} > 0.

3 The preceeding example can be generalized from n = 3 to all n ≥ 2 replacing (7) by

E(x, z) := |z| + ∑_{i<k} ω_{ik}(x)z^{ik},  (13)

where z = (z^{ik})_{i<k}, and both z^{ik} and ω_{ik} are thought to be skew symmetric (no summation convention here). The associated Lagrangian is

e(x, p) := |p_{1} ∧ p_{2}| + ∑_{i<k} ω_{ik}(x)(p_{1}^{i}p_{2}^{k} − p_{2}^{i}p_{1}^{k}),  (14)

and

G(x, p) := D(p) + V (x, p) with V (x, p) := ∑_{i<k} ω_{ik}(x)(p_{1}^{i}p_{2}^{k} − p_{2}^{i}p_{1}^{k}) (15)

satisfies all properties of a perfect dominance function for E with the possible exception of (D4) which will be satisfied for (x, p) ∈ ℝ^{n} × ℝ^{2n} if

sup_{ℝ^{n}} ∑_{i<k} |ω_{ik}|^{2} < 1.

The Euler equations corresponding to G are

ΔX^{l} = ∑_{{i<k \atop i≠l}} q_{ikl}(x)(X_{u}^{i}X_{v}^{k} − X_{v}^{i}X_{u}^{k}),  (16)

where q_{ikl} := ω_{ik,l} + ω_{kl,i} + ω_{li,k}, and the sum is to be extended over all pairs (i, k) with i < k and i, k ≠ l. For n = 3 these equations reduce to

ΔX = 2 div Q(x)X_{u} ∧ X_{v}  (17)

if we set Q_{1} = ω_{23}, Q_{2} = ω_{31}, Q_{3} = ω_{12}, because in this case the admissible triples i k l are permutations of 1 2 3.

The whole setting can be carried over to a Riemannian metric ds^{2} = g_{ik}(x)dx^{i}dx^{k}. For instance, e(x, p) and G(x, p) in the above example become

e(x, p) := (∥p_{1}∥^{2}∥p_{2}∥^{2} − ⟨p_{1}, p_{2}⟩)^{1/2} + V (x, p),
G(x, p) := \frac{1}{2}(∥p_{1}∥^{2} + ∥p_{2}∥^{2}) + V (x, p),

where ⟨p_{α}, p_{β}⟩ := g_{ik}(x)p_{α}^{i}p_{β}^{k}, ∥p_{1}∥ := ⟨p_{1}, p_{1}⟩^{1/2}, ∥p_{2}∥ := ⟨p_{2}, p_{2}⟩^{1/2}, and the conformality relations (11) are to be replaced by

∥X_{u}∥^{2} = ∥X_{v}∥^{2}, ⟨X_{u}, X_{v}⟩ = 0.

Here we want to stay in the Euclidean case, but it should be mentioned that one is automatically forced to pass to the Riemannian case if one wants to study the boundary behaviour of minimizers.

4 For general parametric integrands F (x, z) with the properties (H),(D) and with the associated Lagrangians f(x, p), Morrey introduced in [22, pp. 571–572] the dominance function

G(x, p) :=  (18)

\{f^{2}(x, p) + \frac{1}{4}(m_{1} + m_{2})^{2} [\frac{1}{4}(|p_{1}|^{2} − |p_{2}|^{2})^{2} + (p_{1} · p_{2})^{2}]\}^{1/2} ,

satisfying (D1)–(D4) with μ_{1} := m_{1}/2, μ_{2} := m_{2}/2. A simpler choice is

G(x, p) := g(x, p)D(p),  (19)

where

g(x, p) := \begin{cases}m_{1} & for z = 0 \\ F (x, z/|z|) & for z ≠ 0 \end{cases} , z := p_{1} ∧ p_{2}.

Notice, however, that in general these dominance functions are far from being perfect, since their singular set

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

coincides with the singular set of the associated Lagrangian f(x, p), for which we cannot expect C^{2}-smoothness across Π due to the homogeneity condition (H) on the corresponding parametric integrand F (x, z). Only when F is a weighted area integrand, i.e., F (x, z) = γ(x)|z| with γ ∈ C^{2}(ℝ^{n}), m_{1} ≤ γ(x) ≤ m_{2} for all x ∈ ℝ^{n}, one obtains a perfect dominance function of the form (19) with g(x, p) := γ(x).

In Section 2 we will give the detailed construction of a rank-one convex dominance function G ∈ C^{2,α}(ℝ^{n}×(ℝ^{N} −{0})) for an elliptic positive definite parametric integrand F of class C^{2}(ℝ^{n} × (ℝ^{N} − {0})) following the ideas of Morrey sketched in [23, pp. 391–394]. This in turn will enable us to prove the existence of a perfect dominance function for a large class of parametric integrands. In fact, we have Theorem 2. Let F^{∗} ∈ C^{0}(ℝ^{n} × ℝ^{N}) ∩ C^{2}(ℝ^{n} × (ℝ^{N} − {0})) satisfy (H),(D) with constants m_{1}^{∗}, m_{2}^{∗}, and (5) with

λ^{∗} := inf_{R_{0}∈(0,∞]} λ_{F^{∗}}(R_{0}) > 0.  (20)

Then for

k > k_{0} := 2 [m_{2}^{∗} − min{λ^{∗}, m_{1}^{∗}/2}]

the parametric Lagrangian F defined by

F (x, z) := kA(z) + F^{∗}(x, z)  (21)

possesses a perfect dominance function. In particular, any conformally parametrized minimizer of

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

where F is of the form (21), is of class H_{loc}^{2,2}(B, ℝ^{n}) ∩ C^{1,σ}(B, ℝ^{n}).

Remark 1. In the applications regarding higher regularity of conformally parametrized F-minimizers one generally has control over the size of the minimizer X, that is, an L^{∞}-estimate of the form

sup_{w∈\overline{B}} |X(w)| ≤ R < ∞

is known. Then one can relax condition (20) to

λ^{∗} := inf_{R_{0}∈(0,R]} λ_{F^{∗}}(R_{0}) > 0.

Applications. Existence problems. Dominance functions were used to prove the existence of minimal surfaces and, more generally, of surfaces with prescribed mean curvature in Euclidean space and in Riemannian manifolds satisfying geometric boundary conditions. We refer to the more recent papers by K. Steffen [27] and F. Duzaar & K. Steffen [6], where this topic is surveyed from a general point of view. The gist of the method is to minimize the functional

G(X) := ∫_{B} G(X, ∇X) dudv  (22)

instead of F where G is a perfect dominance function for F. This approach leads to conformally parametrized minimizers which then can be proved to be regular, whereas minimization of the original functional F does not necessarily yield regular extremals. The use of G has a smoothing effect as it suppresses the badly parametrized extremals. However, it is not a priori clear that F and G have the same infima. To prove this one has to apply fairly general results on conformal or quasiconformal mappings as was pointed out by Courant [4, pp. 116–118], and Morrey [21]. Using this “mapping idea” one verifies that

inf_{C} F = inf_{C} G  (23)

provided that the class C of admissible surfaces X is invariant with respect to reparametrizations, which is the case for “geometric” boundary conditions. In principle it is clear how this should be done, but it will be quite tedious to carry out the details, and for every class C the discussion is to be repeated in a varied form. However, as soon as one has established relation (23), one can pass from the minimum problem “F → min in C” to the equivalent problem “G → min in C” to establish the existence of a conformally parametrized minimizer of F in C. The new problem can be solved if its Lagrangian G is quasiconvex in p and satisfies (D4) because these two properties imply (sequential) weak lower semicontinuity of G in H^{1,2}(B, ℝ^{n}). Yet it is not clear whether the semi-ellipticity condition (C) on F guarantees that F possesses a dominance function G(x, p) which is quasiconvex in p. For instance, we do not know if the two dominance functions (18) and (19) are polyconvex and therefore quasiconvex. These remarks explain why it is useful to work with the penalized Lagrangian F^{ϵ} := F + ϵD as we have done in our existence proofs (cf. [14]–[16] and also [18], [19]). First, (C) implies that F is quasiconvex in p, and adding the convex function ϵD does not destroy this property. Thus we can minimize the corresponding functional F^{ϵ}, and letting ϵ → +0 we obtain a minimizer of F which is seen to be conformally parametrized. As we do not use (23) in this procedure, we obtain this important relation as a by-product.

In the examples 1 – 3 , however, the passage from “F → min in C” to “G → min in C” works very well since the dominance functions (12) and (15) are even convex if sup_{ℝ^{3}} |Q| < 1, or sup_{ℝ^{n}} ∑_{i<k} ω_{ik}^{2} < 1, respectively.

Higher regularity of conformally parametrized minimizers. The classical approach to Plateau’s problem where one minimizes the Dirichlet integral D instead of the area functional A does not only lead to a minimal surface as a solution but also provides the simple Euler equation ΔX = 0 from which all regularity properties of minimal surfaces can be deduced.

In a similar way Morrey proceeded in his solution of Plateau’s problem in a Riemannian manifold, and also the work on surfaces with prescribed mean curvature uses the idea to derive regularity from the dominating functional G and not from F.

The key to proving higher regularity of conformally parametrized Fminimizers X ∈ C(Γ) is the analogous observation that any such surface is also the solution of the minimum problem “G → min in C(Γ),” where G is the functional belonging to any dominance function G(x, p) for F (x, z). To be more precise, we have proved in [14, pp. 265–266] and [15, Thm. 1.8] Theorem 3. Suppose that F ∈ C^{0}(ℝ^{n} ×ℝ^{N}) satisfies (H),(D),(C), and let G be an arbitrary dominance function of F with the corresponding variational integral

G(X) := G_{B}(X) = ∫_{B} G(X, ∇X) dudv.

Then we have:

(i) inf_{C(Γ)} F = inf_{C(Γ)} G. (ii) Any minimizer of G in C(Γ) is a conformally parametrized minimizer of

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

minimizer of G in C(Γ).

Notice that Theorem 3 implies the by no means trivial result

inf_{C(Γ)} A = inf_{C(Γ)} D,

which follows here, in contrast to the classical approach, without any kind of sophisticated conformal mapping theorem, as was already pointed out before.

Using Part (iii) of Theorem 3 we can exploit the weak Euler equation of the functional G,

δG(X, ϕ) = 0 for all ϕ ∈ H^{∘}^{1,2}(B, ℝ^{n}) ∩ L^{∞}(B, ℝ^{n})  (24)

with

δG(X, ϕ) := ∫_{B}[G_{p}(X, ∇X) · ∇ϕ + G_{x}(X, ∇X) · ϕ] dudv  (25)

to prove

Theorem 4. ([15, Thm. 1.9]) Suppose that F satisfies (H),(D),(C), and that F possesses a perfect dominance function G. Then any conformally parametrized minimizer X of F in C(Γ) is of class H_{loc}^{2,2}(B, ℝ^{n})∩C^{1,σ}(B, ℝ^{n}) for some σ > 0.

Notice that the Euler equation (24) is preferable to the corresponding Euler equation

δF(X, ϕ) = 0 for all ϕ ∈ H^{∘}^{1,2}(B, ℝ^{n}) ∩ L^{∞}(B, ℝ^{n})  (26)

of the original parametric functional F, where

δF(X, ϕ) := ∫_{B}[f_{p}(X, ∇X) · ∇ϕ + f_{x}(X, ∇X) · ϕ] dudv.  (27)

Indeed, working with (24) one can use the much better regularity and convexity properties of G(x, p) to prove Theorem 4 in case G(x, p) is a perfect dominance function for F (x, z).

One should note also that a priori it is not obvious at all whether the terms in (27) make sense, since f_{p}(x, p) does not exist if p ∈ Π. But with a perfect^{1} dominance function G(x, p) at hand, the expressions in (25) are well-defined and one can show that the respective gradients f_{x}, G_{x}, and f_{p}, G_{p} coincide for x ∈ ℝ^{n} and p ∈ ℝ^{2n} satisfying (3). Then one can use the conformality relations (11) proved in Theorem 1 to show that (27) is well-defined and that

δF(X, ϕ) = δG(X, ϕ) for all ϕ ∈ H^{∘}^{1,2}(B, ℝ^{n}) ∩ L^{∞}(B, ℝ^{n}).  (28)

This is carried out in the present paper in Corollary 1 in Section 2.

If the boundary contour Γ is of class C^{3} the regularity result Theorem 4 can be extended^{2} up to the boundary, see [17] for the details:

Theorem 5. Let Γ ⊂ ℝ^{n} be of class C^{3}. Suppose that F satisfies (H),(D), and (C), and that F possesses a perfect dominance function G. Then any conformally parametrized minimizer X of F in C(Γ) is of class H^{2,2}(B, ℝ^{n})∩ C^{1,σ}(\overline{B}, ℝ^{n}) for some σ > 0.

According to Theorem 2 these regularity results are valid for parametric functionals with integrands of the type (21) which indeed possess a perfect dominance function. It is an open problem at present whether or not any elliptic, positive definite parametric Lagrangian of class C^{2}(ℝ^{n} × (ℝ^{N} − {0}) possesses a perfect dominance function. It also remains open if a conformally parametrized minimizer of F has only isolated branch points in B or even in \overline{B}. U. Clarenz and the second author, however, use the Hopf maximum principle, to show in [3] how to exclude boundary branch points for F-stationary surfaces X that have no interior branch points under the hypothesis that Γ lies on the boundary of a convex body. For such a boundary configuration and even elliptic parametric integrands B. White [37] has shown the existence of smooth embedded minimizing disks. On the other hand, our regularity results Theorems 4 and 5 might be close to optimal, as J.E. Taylor [31] has proved the following theorem:

If F ∈ C^{0}(ℝ^{3})∩C^{3}(ℝ^{3}−{0}) is an essentially noneven elliptic parametric Lagrangian independent of the spatial variable, then there exists an oriented closed analytic Jordan curve Γ on the sphere S^{2} and a Lipschitz immersion X of the oriented disk (which is not an embedding) having Γ as boundary such that the value F(X) of the corresponding parametric functional F is less than the value F(Z) of any Lipschitz embedding Z of the oriented disk having Γ as boundary.

Here, a parametric Lagrangian F (x, z) is called essentially noneven, if it cannot be written in the form F (x, z) = cF̃(x, z) + Q(x) · z, where c > 0, F̃ is

1 Actually, any dominance function of class C^{1}(ℝ^{n} × (ℝ^{N} − {0})) satisfying (D3)

could be used to prove (28). 2 Since regularity considerations are of local nature Theorems 4 and 5 carry over

to solutions of the Douglas problem treated in [18],[19], and also to the partially free boundary problem as carried out in [16, Thm. 1.12].

even in z, i.e., F̃(x, z) = F̃(x, −z), and where div Q = 0. Because of this result one cannot necessarily expect that minimizers in the class of immersions, and even more so minimizers in the more general class considered in Theorems 4 and 5, are as well-behaved as minimizers in the class of embeddings, even if they are of the type of the disk. F.J. Almgren, R. Schoen and L. Simon [1], [25] showed that the F-minimizing integral current obtained by methods of geometric measure theory is a smooth embedded surface away from its boundary, and R. Hardt [8] showed that if the boundary curve Γ lies on the boundary of a convex body then it is also smooth at the boundary. However, these theorems say nothing about minimizing F among surfaces of specified topological type in contrast to our results and the result of White [37] mentioned above.

Unstable stationary surfaces. E. Heinz used the dominance function G defined in (12) with Q(x) := 2H ·x/3 for H ∈ ℝ with |H| < 1/2, to construct unstable surfaces of constant mean curvature H contained in the unit ball around the origin of ℝ^{3}, and bounded by a closed Jordan curve , see [10],[11]. Starting with polygonal boundaries he used Courant’s approach to work with a differentiable function of M variables whose critical points are in one-to-one correspondence with conformally parametrized solutions of the mean curvature system (17), which map M + 3 prescribed points on ∂B onto the M + 3 vertices of the given polygonal boundary contour. Essential ingredients for this method are certain convexity and minimum properties of the functional G defined in (22), which lead to the existence of unique G-minimizers subject to Dirichlet boundary data. Moreover, the regularity properties of G are crucial for proving that, for a given convergent sequence of boundary data, the corresponding G-minimizers converge in the C^{1}-norm.

There is work in progress by R. Jakob adapting Heinz’s method to parametric integrands F of the type considered in Theorem 2. The regularity properties of G might provide an alternative existence proof for unstable Fstationary surfaces compared to the contribution of Shiffman [26]. Shiffman considered parametric integrands of the form F̃(z) := kA(z) + F (z) with k sufficiently large, where F is a general parametric integrand of class C^{1} which does not depend on x. To obtain the necessary convexity and continuity properties Shiffman replaced F̃ by the dominance function G̃(p) := kD(p) + f(p), f(p) := F (p_{1} ∧ p_{2}). But, unfortunately, his arguments regarding the actual construction of the unstable stationary surface towards the end of his work do not seem to be complete, in contrast to Heinz’s detailed proof for H-surfaces in [9],[10], or the contributions regarding more general variational problems by G. Strömer, see, e.g., [28].

2 Construction of a perfect dominance function

The following result due to Morrey [23, pp. 391–394] guarantees the existence of a quadratic, positive definite dominance function with the remarkable property that the singular set Π of f(x, .) in ℝ^{2n} reduces to the singleton {0} in ℝ^{2n} for G(x, .).

Theorem 6. Let F (x, z) ∈ C^{0}(ℝ^{n} × ℝ^{N}) ∩ C_{loc}^{k,μ}(ℝ^{n} × (ℝ^{N} − {0})) be a parametric Lagrangian satisfying (H) and (D). Then there exists a function G(x, p) for F (x, z) satisfying (D1)–(D4) with μ_{1} = m_{1}/4, μ_{2} = m_{2}/2. In addition,

G ∈ C^{0}(ℝ^{n} × ℝ^{N}) ∩ C_{loc}^{k,ν}(ℝ^{n} × (ℝ^{N} − {0})).  (29)

Before proving this result we list various properties of this dominance function G, which follow from the homogeneity condition (D3) on F (x, z) and from the improved regularity of G(x, p) reflected in (29), see [15, Lemmas 3.1–3.3] for details. Part (iii) below in particular justifies the weak Euler equation for G, (24),(25) mentioned in the introduction.

Corollary 1. (i) G_{x}(x, p), G_{xx}(x, p) are positively homogeneous in p of degree two, G_{p}(x, p), G_{px}(x, p) are positively homogeneous in p of degree one, and G_{pp}(x, p) is positively homogeneous in p ≠ 0 of degree zero. Consequently, G, G_{x}, G_{xx}, G_{p}, G_{px}, are continuous in ℝ^{n} × ℝ^{2n} with

G(x, 0) = 0, G_{x}(x, 0) = 0, G_{xx}(x, 0) = 0,

(30)

G_{p}(x, 0) = 0, G_{px}(x, 0) = 0

for all x ∈ ℝ^{n}, and G_{pp} is bounded and continuous on ℝ^{n} × (ℝ^{2n} − {0}).

(ii) There are constants c_{0}(R_{0}), c_{1}(R_{0}), c_{2}(R_{0}) > 0, such that for |x| ≤ R_{0} and for p ∈ ℝ^{n} × ℝ^{2n} we have

|G(x, p)| + |G_{x}(x, p)| + |G_{xx}(x, p)| ≤ c_{0}(R_{0})|p|^{2},
|G_{p}(x, p)| + |G_{px}(x, p)| ≤ c_{1}(R_{0})|p|,

and if p ≠ 0, then

|G_{pp}(x, p)| ≤ c_{2}(R_{0}).

(iii) For any surface X ∈ H^{1,2}(B, ℝ^{n}) ∩ L^{∞}(B, ℝ^{n}) and any ϕ ∈

H^{∘}^{1,2}(B, ℝ^{n}) ∩ L^{∞}(B, ℝ^{n}) we have

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

where δG(X, ϕ) is defined by

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

If, in addition, X is a minimizer of G in C(Γ), then δG(X, ϕ) = 0.

Using these properties as well as (D1) and (D2) we can also justify the validity of the Euler equation (26),(27) for F. For notational convenience we introduce the algebraic surface

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

Lemma 1. Suppose that G ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − Π)) is a dominance function of a parametric Lagrangian F of class C^{0}(ℝ^{n} × ℝ^{N}) ∩ C^{2}(ℝ^{n} × (ℝ^{N} − {0})) with the associated Lagrangian f ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − Π)). Then we have

G_{x}(x, p) = f_{x}(x, p), G_{p}(x, p) = f_{p}(x, p), G_{px}(x, p) = f_{px}(x, p) (31)

for (x, p) ∈ ℝ^{n} × Π_{0}, and

ξ · G_{pp}(x, p)ξ ≥ ξ · f_{pp}(x, p)ξ for (x, p) ∈ ℝ^{n} × (Π_{0} − {0}), ξ ∈ ℝ^{2n}.

In particular, if G ∈ C^{2}(ℝ^{n} × (ℝ^{2n} − {0})) and if X ∈ H^{1,2}(B, ℝ^{n}) ∩ L^{∞}(B, ℝ^{n}) is conformally parametrized then

δF(X, ϕ) = δG(X, ϕ) for all ϕ ∈ H^{∘}^{1,2}(B, ℝ^{n}) ∩ L^{∞}(B, ℝ^{n}),

where δG(X, ϕ) and δF(X, ϕ) are given by (25) and (27), respectively.

Proof. Since G is a dominance function of F we have

G(x, p) − f(x, p) ≥ 0 on ℝ^{n} × ℝ^{2n}, and  (32)
G(x, p) − f(x, p) = 0 on ℝ^{n} × Π_{0}.  (33)

For for fixed x ∈ ℝ^{n} and p = (p_{1}, p_{2}) ∈ Π_{0} − {0} one uses (H) to calculate

f_{x}(x, p) = F_{x}(x, p_{1} ∧ p_{2})=F_{x}(x, \frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|})|p_{1} ∧ p_{2}|,
f_{p}(x, p) = F_{z}(x, p_{1} ∧ p_{2})(p_{1} ∧ p_{2})_{p}=F_{z}(x, p_{1} ∧ p_{2})(p_{1} ∧ p_{2})_{p},
f_{px}(x, p) = F_{zx}(x, p_{1} ∧ p_{2})(p_{1} ∧ p_{2})_{p}=F_{zx}(x, p_{1} ∧ p_{2})(p_{1} ∧ p_{2})_{p},

hence

|f_{x}(x, p)| ≤ sup_{|z|=1} |F_{x}(x, z)||p|^{2},
|f_{p}(x, p)| ≤ \sqrt{n} sup_{|z|=1} |F_{z}(x, z)||p|,
|f_{px}(x, p)| ≤ \sqrt{n} sup_{|z|=1} |F_{zx}(x, z)||p|,

which allows us to extend f_{x}, f_{p}, and f_{px} continuously to ℝ^{n}×[(ℝ^{2n}−Π)∪{0}] by setting

f_{x}(x, 0) = 0, f_{p}(x, 0) = 0, f_{px}(x, 0) = 0.

This and property (33) together with (30) in Lemma 1 and Π ∩ Π_{0} = {0} now imply

G_{x}(x, p) = f_{x}(x, p) on ℝ^{n} × Π_{0},

and (32) together with (33) yield G_{p}(x, p) − f_{p}(x, p) = 0 for all (x, p) ∈ ℝ^{n} × Π_{0}, which leads to G_{px}(x, p) − f_{px}(x, p) = 0 for all (x, p) ∈ ℝ^{n} × Π_{0}. Finally we get ξ · (G_{pp}(x, p) − f_{pp}(x, p))ξ ≥ 0 for (x, p) ∈ ℝ^{n} × (Π_{0} − {0}), again by (32) and (33), since Π ∩ Π_{0} = {0}. The last assertion follows from (31). ⊓ ⊔

Proof of Theorem 6 . We introduce two functions τ : ℝ^{2n} → ℝ and σ : ℝ^{n} × ℝ^{2n} → ℝ by

τ(p) := \frac{a(p)}{D(p)} for p ≠ 0, τ(0) := 1,  (34)

and

σ(x, p) := \frac{f(x, p)}{a(p)} for p ∉ Π, σ(x, p) := m_{1} for p ∈ Π.  (35)

Then we have

0 ≤ τ(p) ≤ 1 and m_{1} ≤ σ(x, p) ≤ m_{2}.  (36)

Set

G(x, p) := D(p)[(m_{1}/2) + ω(τ(p), σ(x, p))],  (37)

where ω : [0, 1] × [m_{1}, ∞) → ℝ is a function to be chosen later on. From (34) and (35) we infer that

τ(γp) = τ(p) and σ(x, γp) = σ(x, p) for any γ > 0

which implies property (D3). Now we set

u(t, s) := \frac{(s − (m_{1}/2))t}{(m_{1}/2) + (s − m_{1})t} for t ∈ [0, 1], s ∈ [m_{1}, ∞),  (38)

and notice first that u(t, m_{1}) = t for all t ∈ [0, 1]. Then we compute the first and second derivative of u(t, s) with respect to t:

u_{t}(t, s) = \frac{(m_{1}/2)(s − (m_{1}/2))}{((m_{1}/2) + (s − m_{1})t)^{2}} ≥ \frac{(m_{1}/2)(s − (m_{1}/2))}{(s − (m_{1}/2))^{2}}
= \frac{m_{1}}{2(s − (m_{1}/2))} > 0  (39)

and

u_{tt}(t, s) = −2\frac{(m_{1}/2)(s − (m_{1}/2))(s − m_{1})}{((m_{1}/2) + (s − m_{1})t)^{3}} ≤ 0.

Hence

u_{t}(t, s) > 0 for s ≥ m_{1} and u_{tt}(t, s) < 0 for s > m_{1}.  (40)

Therefore u(., s) is increasing and strictly concave for s > m_{1}. Moreover we have

u(0, s) = 0 and u(1, s) = 1 for s ≥ m_{1}  (41)

as well as

lim_{s→∞} u(t, s) = \begin{cases}0 & for t = 0 \\ 1 & for 0 < t ≤ 1. \end{cases}

Note also that

u(t, s_{1}) < u(t, s_{2}) for 0 < t < 1, m_{1} ≤ s_{1} < s)2,  (42)

since the derivative of u(t, s) with respect to s may be calculated as

u_{s}=\frac{m_{1}(t − t^{2})}{2[(m_{1}/2) + (s − m_{1})t]^{2}} ,  (43)

which is positive for t ∈ (0, 1). For two values u_{0} and u_{1} with 0 < u_{0} < u_{1} < 1 we choose a smooth cut-off function η : [0, 1] → ℝ such that

η ∈ C_{0}^{∞}((u_{0}, 1]), 0 < η(r) < 1 for u_{0} < r < 1, η(1) = 1,

(44)

\frac{1}{1 − u_{0}} < η′(1) ≤ \frac{1}{1 − u_{1}} , η′′(r) > 0 for r ∈ (u_{0}, 1].

Using the functions u(s, t) and η(r) we now define the function ω(t, s) appearing in (37) as

ω(t, s) := [(m_{1}/2) + (s − m_{1})t]η(u(t, s)) for (t, s) ∈ [0, 1] × [m_{1}, ∞). (45)

This leads to

G(x, p) = D(p) \{\frac{m_{1}}{2} + [(m_{1}/2) + (σ − m_{1})τ]η(u(τ, σ))\} ,  (46)

where τ := τ(p), σ := σ(x, p). By (36) and (38) we have 0 ≤ u(τ, σ) ≤ 1. Thus η(u(τ, σ)) is well-defined, and 0 ≤ η(u(τ, σ)) ≤ 1. We finally obtain

\frac{m_{1}}{2} ≤ \frac{m_{1}}{2} + [(m_{1}/2) + (σ − m_{1})τ]η(u(τ, σ))
≤ \frac{m_{1}}{2} + \frac{m_{1}}{2} + (σ − m_{1}) ≤_{(36)} m_{2},

whence

(m_{1}/2)D(p) ≤ G(x, p) ≤ m_{2}D(p),

that is, property (D4) holds with μ_{1} = (m_{1}/4) and μ_{2} = (m_{2}/2).

To prove (29) we note that G is of class C^{k,ν} in a sufficiently small neighbourhood of any point (x, p_{0}) ∈ ℝ^{n} ×(ℝ^{2n} −Π). If p_{0} ∈ Π −{0} on the other hand, then 0 ≤ τ(p) ≪ 1 for |p − p_{0}| ≪ 1, whence

0 ≤ u(τ(p), m_{2}) < u_{0} for |p − p_{0}| ≪ 1.

On account of (36) and (42) we also have

0 ≤ u(τ(p), σ(x, p)) ≤ u(τ(p), m_{2}).

Therefore, η(u(τ(p), σ(x, p))) = 0 if |p − p_{0}| ≪ 1, and consequently,

G(x, p) = (m_{1}/2)D(p) for |p − p_{0}| ≪ 1, p_{0} ∈ Π − {0} and x ∈ ℝ^{n}. (47)

This shows that G is of class C^{∞} in a neighbourhood of any point p_{0} in Π − {0}.

It remains to be shown that G(x, p) satisfies (D1) and (D2).

Case 1. p = 0. Then p ∈ Π and G(x, p) = 0 = f(x, p), by (34),(35),(44) and (46).

Case 2. p ∈ Π −{0}. Then f(x, p) = F (x, 0) = 0 and G(x, p) ≥ μ_{1}|p|^{2} > 0 by (D4) which was shown above, whence f(x, p) < G(x, p). Since Π ∩ Π_{0} = {0} we also have p ∉ Π_{0}.

Case 3. p ∉ Π. Then p_{1} ∧ p_{2} ≠ 0, and so a(p), D(p) and f(x, p) are all positive. Thus we can write

f(x, p) = \frac{f(x, p)}{a(p)} \frac{a(p)}{D(p)} D(p) = σ(x, p)τ(p)D(p).

If we compare this with (37) we obtain:

G(x, p) is a dominance function for F (x, z) if the following property holds true:

(⋆) For any (t, s) ∈ [0, 1] × [m_{1}, m_{2}] we have ts ≤ (m_{1}/2) + ω(t, s) with equality if and only if t = 1.

To verify (⋆) we investigate the function ω(t, s), which is given by (45). Differentiating ω with respect to t we get

ω_{t}(t, s) =_{(45)} (s − m_{1})η(u) + [(m_{1}/2) + (s − m_{1})t]η′(u)u_{t}(t, s)
=_{(39)} (s − m_{1})η(u) + η′(u)\frac{[(m_{1}/2) + (s − m_{1})t](m_{1}/2)(s − (m_{1}/2))}{[(m_{1}/2) + (s − m_{1})t]^{2}}
= (s − m_{1})η(u) + η′(u)\frac{m_{1}(s − (m_{1}/2))/2}{(m_{1}/2) + (s − m_{1})t} ,

where we have set u = u(t, s). Notice that by (40) u = 0 if and only if t = 0 in which case one finds by the properties (44) of the cut-off function η that ω_{t}(0, s) = 0.

We infer from (38) that

((m_{1}/2) + (s − m_{1})t) u = (s − (m_{1}/2))t,

which leads to

\frac{u}{t} = \frac{2}{m_{1}} [s − (m_{1}/2) − (s − m_{1})u]  (48)

for t ≠ 0 (or equivalently, for u ≠ 0), and

t = \frac{m_{1}u}{2[s − (m_{1}/2) − (s − m_{1})u]}  (49)

for t ≠ 0 (i.e., u = 0). Consequently we obtain

ω_{t}(t, s) = (s − m_{1})η(u) + η′(u)\frac{m_{1}u}{2t}
=_{(48)} (s − m_{1})η(u) + η′(u)[s − (m_{1}/2) − (s − m_{1})u],  (50)

which now also holds for t = 0 (i.e. u = 0). Thus, fixing s ∈ [m_{1}, m_{2}], we can express ω_{t} as a function of u, since the mapping t ↦ u(t, s) is invertible. By differentiating ω_{t} with respect to u we obtain

\frac{∂}{∂u}[ω_{t}_{s fixed}] = [s − (m_{1}/2) − (s − m_{1})u]η′′(u)
≥ [s − (m_{1}/2) − (s − m_{1})]η′′(u) = (m_{1}/2)η′′(u) ≥ 0, (51)

if we take 0 ≤ u ≤ 1, s ≥ m_{1} and η′′(r) ≥ 0 for r ∈ [0, 1] into account. Hence, for fixed s, the function ω_{t}(t(r, s), s) is nondecreasing with respect to r ∈ [0, 1], and so it assumes its maximum at r = 1. By (50) and η(1) = 1 we obtain

ω_{t}(t(1, s), s) = (s − m_{1})η(1) + [s − (m_{1}/2) − (s − m_{1})]η′(1)
= s − m_{1} + (m_{1}/2)η′(1) ≤ s − m_{1} + (m_{1}/2)(1 − u_{1})^{−1} (52)

for all r ∈ [0, 1] by (44). Because of

u_{1} ∈ (0, 1/2)  (53)

we have (m_{1}/2)(1 − u_{1})^{−1} < m_{1}, and we arrive at ω_{t}(t(r, s), s) < s for all r in [0, 1], i.e., ω_{t}(t, s) < s for 0 ≤ t ≤ 1, m_{1} ≤ s ≤ m_{2}. Then the function χ : [0, 1] × [m_{1}, m_{2}] → ℝ defined by χ(t, s) := (m_{1}/2) + ω(t, s) − ts satisfies χ_{t}(t, s) = ω_{t}(t, s)−s < 0, hence χ(., s) is a strictly decreasing function on [0, 1] for any fixed value of s ∈ [m_{1}, m_{2}]. We have χ(1, s) = (m_{1}/2)+ω(1, s)−s, and from (45),(41) and (44) ω(1, s) = s − (m_{1}/2), i.e., χ(1, s) = 0 and χ(t, s) > 0 for 0 ≤ t < 1. This implies

\frac{m_{1}}{2} + ω(t, s) > ts for 0 ≤ t < 1, and \frac{m_{1}}{2} + ω(1, s) = s,

which establishes the proof of (⋆), and thus of Theorem 6. ⊓ ⊔

Before we show how to refine the construction in Theorem 6 to obtain a perfect dominance function satisfying the ellipticity condition (E) we derive a couple of simple estimates for the gradient F_{z} of a parametric integrand F (x, z), and for the Hessian of the associated Lagrangian f(x, p).

Lemma 2. Let F ∈ C^{0}(ℝ^{n} × ℝ^{N}) ∩ C^{1}(ℝ^{n} × (ℝ^{N} − {0})) be a parametric Lagrangian satisfying (H),(D) and (C). Then we have

(i)

F (x, z) − z · F_{z}(x, z_{0}) ≥ 0 for all x ∈ ℝ^{n}, z, z_{0} ∈ ℝ^{N}, z_{0} ≠ 0. (54)

In particular,

|F_{z}(x, z_{0})| ≤ m_{2} for all (x, z_{0}) ∈ ℝ^{n} × (ℝ^{N} − {0}).  (55)

(ii) If, in addition, F ∈ C^{2}(ℝ^{n} × (ℝ^{N} − {0})) satisfies (5), then for f(x, p) = F (x, p_{1} ∧ p_{2}) one has

ξ · f_{pp}(x, p) ξ ≥ \frac{λ_{F} (R_{0})}{a(p)} |P_{p_{1}∧p_{2}}^{⊥}[(ζ ∧ p_{2}) − (η ∧ p_{1})]|^{2}  (56)
+2F_{z}(x, p_{1} ∧ p_{2}) · (ζ ∧ η)

for all ξ = (ζ, η) ∈ ℝ^{n} × ℝ^{n} ≅ ℝ^{2n}, (x, p) ∈ \overline{B_{R_{0}}}(0) × (ℝ^{2n} − Π), where λ_{F} (R_{0}) is the constant appearing in (5). In particular, we have

ξ · a_{pp}(x, p) ξ = \frac{1}{a(p)} |P_{p_{1}∧p_{2}}^{⊥}[(ζ ∧ p_{2}) − (η ∧ p_{1})]|^{2}  (57)

+ \frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|} · (ζ ∧ η).

Remark 2. In the rank-one case, i.e., if ξ = ζ ⊗ Λ for some ζ ∈ ℝ^{n}, Λ = (λ_{1}, λ_{2}) ∈ ℝ^{2}, the estimate (56) reduces to

ξ · f_{pp}(x, p) ξ ≥ \frac{λ_{F} (R_{0})}{a(p)} |P_{p_{1}∧p_{2}}^{⊥}[ζ ∧ (λ_{1}p_{2} − λ_{2}p_{1})]|^{2}

which implies that rank f_{pp}(x, p) = n − 2 for all Λ ≠ 0, since p ∈ ℝ^{2n} − Π has linearly independent column vectors p_{1}, p_{2}. In the language of Morrey this means: f is the integrand of a parametric regular problem, or f satisfies the parametric Legendre-condition, see [23, p. 392].

Proof of Lemma 2 . (i) By (H) and (C) we know that for z_{0} ≠ 0

F (x, z) − z · F_{z}(x, z_{0}) = F (x, z) − F (x, z_{0}) − (z − z_{0})F_{z}(x, z_{0}) ≥ 0,

which proves (54). If F_{z}(x, z_{0}) = 0 then (55) is trivially satisfied. For F_{z}(x, z_{0}) ≠ 0 we may choose z := F_{z}(x, z_{0}) in (54) to get

F (x, F_{z}(x, z_{0})) − |F_{z}(x, z_{0})|^{2} ≥ 0,

hence by (D) and (H),

|F_{z}(x, z_{0})|^{2} ≤ F (x, \frac{F_{z}(x, z_{0})}{|F_{z}(x, z_{0})|})|F_{z}(x, z_{0})| ≤ m_{2}|F_{z}(x, z_{0})| for all z_{0} ≠ 0.

(ii) We omit the arguments (x, p) ∈ \overline{B_{R_{0}}}(0)×(ℝ^{2n}−Π) of f and (x, p_{1}∧p_{2}) ∈ \overline{B_{R_{0}}}(0) × (ℝ^{N} − \{0\}) of F and write for ξ = (ζ, η) ∈ ℝ^{n} × ℝ^{n} ≅ ℝ^{2n}:

ξ · f_{pp}(x, p) ξ = ζ · f_{p_{1}p_{1}}ζ + ζ · f_{p_{1}p_{2}}η + η · f_{p_{2}p_{1}}ζ + η · f_{p_{2}p_{2}}η.

In order to calculate the derivatives of f we use the identity

(p_{1} ∧ p_{2})^{(j,k)} = ε_{st}^{(j,k)}p_{1}^{s}p_{2}^{t}

for the (j, k)-th component of the bivector p_{1} ∧ p_{2} ∈ ℝ^{N}, where (j, k) denotes a double index with components j, k ∈ {1, …, n}, j < k, and where ε is the permutation tensor defined as

ε_{st}^{(j,k)} := \begin{cases}1 & if j = s and k = t and j < k, \\ −1 & if j = t and k = s and j < k, \\ 0 & otherwise. \end{cases}

We obtain

f_{p_{1}^{i}} = F_{z^{(j,k)}} · (p_{1} ∧ p_{2})_{p_{1}^{i}}^{(j,k)} = F_{z^{(j,k)}}ε_{it}^{(j,k)}p_{2}^{t},
f_{p_{2}^{i}} = F_{z^{(j,k)}} · (p_{1} ∧ p_{2})_{p_{2}^{i}}^{(j,k)} = F_{z^{(j,k)}}ε_{si}^{(j,k)}p_{1}^{s},

where we used the convention to sum over all repeated ordered double indices, and for the second derivatives we arrive at

f_{p_{1}^{i}p_{1}^{k}} = F_{z^{(j,k)}z^{(j′,k′)}}ε_{it}^{(j,k)}p_{2}^{t}ε_{kτ}^{(j′,k′)}p_{2}^{τ},
f_{p_{1}^{i}p_{2}^{k}} = F_{z^{(j,k)}z^{(j′,k′)}}ε_{it}^{(j,k)}p_{2}^{t}ε_{σk}^{(j′,k′)}p_{1}^{σ} + F_{z^{(j,k)}}ε_{ik}^{(j,k)},
f_{p_{2}^{i}p_{1}^{k}} = F_{z^{(j,k)}z^{(j′,k′)}}ε_{si}^{(j,k)}p_{1}^{s}ε_{kτ}^{(j′,k′)}p_{2}^{τ} + F_{z^{(j,k)}}ε_{ik}^{(j,k)},
f_{p_{2}^{i}p_{2}^{k}} = F_{z^{(j,k)}z^{(j′,k′)}}ε_{si}^{(j,k)}p_{1}^{s}ε_{σk}^{(j′,k′)}p_{1}^{σ}.

Thus we obtain

ξ · f_{pp} ξ = F_{z^{(j,k)}z^{(j′,k′)}} [ε_{it}^{(j,k)}ζ^{i}p_{2}^{t}ε_{kτ}^{(j′,k′)}ζ^{k}p_{2}^{τ} + ε_{it}^{(j,k)}ζ^{i}p_{2}^{t}ε_{σk}^{(j′,k′)}η^{k}p_{1}^{σ}

+ ε_{si}^{(j,k)}η^{i}p_{1}^{s}ε_{kτ}^{(j′,k′)}ζ^{k}p_{2}^{τ} + ε_{si}^{(j,k)}η^{i}p_{1}^{s}ε_{σk}^{(j′,k′)}η^{k}p_{1}^{σ}]

+F_{z^{(j,k)}}[ε_{ik}^{(j,k)}ζ^{i}η^{k} + ε_{ki}^{(j,k)}ζ^{k}η^{i}]
= (ζ ∧ p_{2}) · F_{zz}(ζ ∧ p_{2}) + (ζ ∧ p_{2}) · F_{zz}(p_{1} ∧ η)
+(p_{1} ∧ η) · F_{zz}(ζ ∧ p_{2}) + (p_{1} ∧ η) · F_{zz}(p_{1} ∧ η)
+2F_{z} · (ζ ∧ η)
= [(ζ ∧ p_{2}) − (η ∧ p_{1})] · F_{zz}[(ζ ∧ p_{2}) − (η ∧ p_{1})]
+2F_{z} · (ζ ∧ η)

≥_{(5)} \frac{λ_{F} (R_{0})}{a(p)} |P_{p_{1}∧p_{2}}^{⊥}[(ζ ∧ p_{2}) − (η ∧ p_{1})]|^{2}
+2F_{z}(x, p_{1} ∧ p_{2}) · (ζ ∧ η). ⊓ ⊔

The quadratic form of the Hessian matrix of the dominance function G constructed in Theorem 6 can be estimated from below as follows:

Theorem 7. Under the assumptions of Theorem 6 with the additional hypothesis that the parametric Lagrangian F is elliptic, satisfying (5), the dominance function G given in Theorem 6 satisfies

ξ · G_{pp}(x, p) ξ ≥ ((m_{1}/2) + ω − τω_{t})|ξ|^{2}

+(ω_{s}λ_{F} (R_{0}) + τω_{t} − σω_{s}) |P_{p_{1}∧p_{2}}^{⊥}[(ζ ∧ p_{2}) − (η ∧ p_{1})]|^{2} /τa

+2 [ω_{s}F_{z}(x, p_{1} ∧ p_{2}) + (τω_{t} − σω_{s})\frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|}] · (ζ ∧ η)/τ (58)

+D [ \begin{matrix}τ_{p} · ξ \\ σ_{p} · ξ \end{matrix} ] · ( \begin{matrix}ω_{tt} & ω_{ts} − (ω_{s}/τ) \\ ω_{st} − (ω_{s}/τ) & ω_{ss} \end{matrix} ) [ \begin{matrix}τ_{p} · ξ \\ σ_{p} · ξ \end{matrix} ]

for all (x, p) ∈ \overline{B_{R_{0}}}(0) × (ℝ^{2n} − Π), ξ = (ζ, η) ∈ ℝ^{2n}, where ω = ω(τ, σ), τ, and σ are defined in (45),(34),(35) in the proof of Theorem 6. For (x, p) ∈ ℝ^{n} × (Π − {0}) we have

ξ · G_{pp}(x, p) ξ = m_{1}|ξ|^{2}/2 for all ξ ∈ ℝ^{2n}.  (59)

Proof. The identity (59) follows immediately from (47) derived in the proof of Theorem 6.

In the following computations we will frequently omit the argument (x, p) ∈ \overline{B_{R_{0}}}(0) × (ℝ^{2n} − \{0\}) for simplicity. We use the notation

d ⊗ e = (d_{α}^{k}e_{β}^{i}), 1 ≤ k, i ≤ n, 1 ≤ α, β ≤ 2,

for the dyadic product of two vectors d, e ∈ ℝ^{2n}. Differentiating (37) with respect to p one obtains

G_{p} = D_{p}[(m_{1}/2) + ω] + D[ω_{t}τ_{p} + ω_{s}σ_{p}], D_{p} = p,

and

ξ · G_{pp} ξ = ((m_{1}/2) + ω)|ξ|^{2} + D[ω_{t}ξ · τ_{pp} ξ + ω_{s}ξ · σ_{pp} ξ]
+ξ · [ω_{t}D_{p} ⊗ τ_{p} + ω_{s}D_{p} ⊗ σ_{p} + ω_{t}τ_{p} ⊗ D_{p} + ω_{s}σ_{p} ⊗ D_{p}] ξ(60)
+ξ · D[ω_{tt}τ_{p} ⊗ τ_{p} + ω_{ts}τ_{p} ⊗ σ_{p} + ω_{st}σ_{p} ⊗ τ_{p} + ω_{ss}σ_{p} ⊗ σ_{p}] ξ.

By (34) one has

τ = \frac{a}{D} , τ_{p} = \frac{a_{p}}{D} − \frac{aD_{p}}{D^{2}} = \frac{a_{p}}{D} − τ \frac{D_{p}}{D} ,

(61)

τ_{pp} = \frac{a_{pp}}{D} − \frac{aD_{pp}}{D^{2}} − \frac{a_{p} ⊗ D_{p}}{D^{2}} − \frac{D_{p} ⊗ a_{p}}{D^{2}} − \frac{2aD_{p} ⊗ D_{p}}{D^{3}} ,

and we infer

ξ · τ_{pp} ξ = \frac{ξ · a_{pp} ξ}{D} − \frac{aξ · D_{pp} ξ}{D^{2}} − ξ · [a_{p} ⊗ D_{p} + D_{p} ⊗ a_{p} − 2aD_{p} ⊗ D_{p}/D] ξ/D^{2}
= \frac{ξ · a_{pp} ξ}{D} − \frac{a|ξ|^{2}}{D^{2}} − ξ · [a_{p} ⊗ D_{p} + D_{p} ⊗ a_{p} − 2τD_{p} ⊗ D_{p}] ξ/D^{2}
=_{(2)} \frac{ξ · a_{pp} ξ}{D} − \frac{a|ξ|^{2}}{D^{2}}

−ξ · [(Dτ_{p} + τD_{p}) ⊗ D_{p} + D_{p} ⊗ (Dτ_{p} + τD_{p}) − 2τD_{p} ⊗ D_{p}] ξ/D^{2}

= \frac{ξ · a_{pp} ξ}{D} − \frac{a|ξ|^{2}}{D^{2}} − ξ · [Dτ_{p} ⊗ D_{p} + DD_{p} ⊗ τ_{p}] ξ/D^{2}.

Furthermore, with f = aσ one obtains

f_{p} = aσ_{p} + σa_{p} and f_{pp} = aσ_{pp} + σa_{pp} + a_{p} ⊗ σ_{p} + σ_{p} ⊗ a_{p},

whence

ξ · f_{pp} ξ = aξ · σ_{pp} ξ + σξ · a_{pp} ξ + ξ · [a_{p} ⊗ σ_{p} + σ_{p} ⊗ a_{p}) ξ
= aξ · σ_{pp} ξ + σξ · a_{pp} ξ
+ξ · [Dτ_{p} ⊗ σ_{p} + τD_{p} ⊗ σ_{p} + Dσ_{p} ⊗ τ_{p} + τσ_{p} ⊗ D_{p}] ξ,

since a_{p} = Dτ_{p} + τD_{p}. Hence

ξ · f_{pp} ξ − σξ · a_{pp} ξ = aξ · σ_{pp} ξ +ξ · [Dτ_{p} ⊗ σ_{p} + τD_{p} ⊗ σ_{p} + Dσ_{p} ⊗ τ_{p} + τσ_{p} ⊗ D_{p}] ξ.

Inserting this information into (60) we arrive at

ξ · G_{pp} ξ = ((m_{1}/2) + ω − τω_{t})|ξ|^{2} + ω_{s}(ξ · f_{pp} ξ − σξ · a_{pp} ξ)/τ
−ω_{s}ξ · [Dτ_{p} ⊗ σ_{p} + τD_{p} ⊗ σ_{p} + Dσ_{p} ⊗ τ_{p} + τσ_{p} ⊗ D_{p}] ξ/τ
+ω_{t}ξ · a_{pp} ξ − ω_{t}ξ · [τ_{p} ⊗ D_{p} + D_{p} ⊗ τ_{p}] ξ
+ω_{t}ξ · [D_{p} ⊗ τ_{p} + τ_{p} ⊗ D_{p}] ξ + ω_{s}ξ · [D_{p} ⊗ σ_{p} + σ_{p} ⊗ D_{p}] ξ

+Dξ · [ω_{tt}τ_{p} ⊗ τ_{p} + ω_{ts}τ_{p} ⊗ σ_{p} + ω_{st}σ_{p} ⊗ τ_{p} + ω_{ss}σ_{p} ⊗ σ_{p}] ξ

= ((m_{1}/2) + ω − τω_{t})|ξ|^{2} + ω_{s}ξ · f_{pp} ξ/τ + (ω_{t} − (σω_{s}/τ))ξ · a_{pp} ξ −ω_{s}Dξ · [τ_{p} ⊗ σ_{p} + σ_{p} ⊗ τ_{p})ξ/τ

+Dξ · [ω_{tt}τ_{p} ⊗ τ_{p} + ω_{ts}τ_{p} ⊗ σ_{p} + ω_{st}σ_{p} ⊗ τ_{p} + ω_{ss}σ_{p} ⊗ σ_{p}] ξ

= ((m_{1}/2) + ω − τω_{t})|ξ|^{2} + ω_{s}ξ · f_{pp} ξ/τ + (ω_{t} − (σω_{s}/τ))ξ · a_{pp} ξ +D[ω_{tt}(τ_{p} · ξ)^{2} + 2(ω_{ts} − (ω_{s}/τ))(τ_{p} · ξ)(σ_{p} · ξ) + ω_{ss}(σ_{p} · ξ)^{2}].

By Part (ii) of Lemma 2 it follows that

ξ · G_{pp} ξ ≥ ((m_{1}/2) + ω − τω_{t})|ξ|^{2}

+ω_{s}{λ_{F} (R_{0}) |P_{p_{1}∧p_{2}}^{⊥}[(ζ ∧ p_{2}) − (η ∧ p_{1})]|^{2} /a + 2F_{z}(x, p_{1} ∧ p_{2}) · (ζ ∧ η)}/τ

+(ω_{t} − (σω_{s}/τ)) \{\frac{1}{a} |P_{p_{1}∧p_{2}}^{⊥}[(ζ ∧ p_{2}) − (η ∧ p_{1})]|^{2} + 2\frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|} · (ζ ∧ η)\}

+D [ \begin{matrix}τ_{p} · ξ \\ σ_{p} · ξ \end{matrix} ] · ( \begin{matrix}ω_{tt} & ω_{ts} − (ω_{s}/τ) \\ ω_{st} − (ω_{s}/τ) & ω_{ss} \end{matrix} ) [ \begin{matrix}τ_{p} · ξ \\ σ_{p} · ξ \end{matrix} ]

= ((m_{1}/2) + ω − τω_{t})|ξ|^{2}

+(ω_{s}λ_{F} (R_{0}) + τω_{t} − σω_{s}) |P_{p_{1}∧p_{2}}^{⊥}[(ζ ∧ p_{2}) − (η ∧ p_{1})]|^{2} /τa

+2 [ω_{s}F_{z}(x, p_{1} ∧ p_{2}) + (τω_{t} − σω_{s})\frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|}] · (ζ ∧ η)/τ

+D [ \begin{matrix}τ_{p} · ξ \\ σ_{p} · ξ \end{matrix} ] · ( \begin{matrix}ω_{tt} & ω_{ts} − (ω_{s}/τ) \\ ω_{st} − (ω_{s}/τ) & ω_{ss} \end{matrix} ) [ \begin{matrix}τ_{p} · ξ \\ σ_{p} · ξ \end{matrix} ] . ⊓ ⊔

Theorem 8. Let F (x, z) ∈ C^{0}(ℝ^{n} × ℝ^{N}) ∩ C^{k,ν}(ℝ^{n} × (ℝ^{N} − {0})) for some k ∈ ℕ, k ≥ 2, ν ∈ [0, 1] be a parametric Lagrangian satisfying (H),(D) and (5) with

λ := inf_{R_{0}∈(0,∞]} λ_{F} (R_{0}) > 0.  (62)

Then for any ϵ > 0 there exists a dominance function G^{ϵ}(x, p) of F satisfying (D1)–(D4) with μ_{1} = (m_{1}/4), μ_{2} = (m_{2}/2). In addition G^{ϵ}(x, p) is of class C^{1}(ℝ^{n} × ℝ^{N}) ∩ C^{k,ν}(ℝ^{n} × (ℝ^{N} − {0})) and satisfies

ξ · G_{pp}^{ϵ}(x, p) ξ ≥ \frac{2}{τ} [ω_{s}P_{p_{1}∧p_{2}}^{⊥}(F_{z}(x, p_{1} ∧ p_{2})) + (τ/a)ω_{t}(p_{1} ∧ p_{2})] · (ζ ∧ η)
+[ min{λ, m_{1}/2} − ϵ] |ξ|^{2}  (63)

for all ξ = (ζ, η) ∈ ℝ^{2n}, (x, p) ∈ ℝ^{n} × (ℝ^{2n} − Π), and

(ζ̄ ⊗ Λ) · G_{pp}^{ϵ}(x, p)(ζ̄ ⊗ Λ) ≥ [ min{λ, m_{1}/2} − ϵ] |ζ̄|^{2}|Λ|^{2},  (64)

for all ζ̄ ∈ ℝ^{n}, Λ ∈ ℝ^{2}, (x, p) ∈ ℝ^{n} × (ℝ^{2n} − Π), where ω, τ, σ are defined as in the proof of Theorem 6. Moreover, for (x, p) ∈ ℝ^{n} × (Π − {0}) we have

ξ · G_{pp}^{ϵ}(x, p) ξ = m_{1}|ξ|^{2}/2 for all ξ ∈ ℝ^{2n}.  (65)

If λ ≥ m_{1}/2 then the full statement (including the estimates (63)–(65)) holds true for ϵ = 0 as well.

Remark 3. The second inequality (64) is known as the strict rank-one-condition, or strict Legendre-Hadamard-condition, see [7, vol. I, p.231].

Proof of Theorem 8 . We will choose a particular cut-off function η = η^{ϵ} which is admissible in the proof of Theorem 7 to prove the estimate (63). Then the particular choice ξ := ζ̄ ⊗ Λ in (63) leads to (64). Equation (65) follows directly from (59) in Theorem 7, since η^{ϵ} is admissible for the construction in the proof of that theorem. All the other statements have been proven in Theorem 6.

Hence it suffices to prove (63). For that we recall (50) and calculate suppressing the argument (t, s) ∈ [0, 1] × [m_{1}, m_{2}]

ω_{tt} = (s − m_{1})η′(u)u_{t} + η′′(u)u_{t}[s − (m_{1}/2) − (s − m_{1})u]
−η′(u)(s − m_{1})u_{t}  (66)
= η′′(u)u_{t}[s − (m_{1}/2) − (s − m_{1})u] ≥ η′′(u)u_{t}[m_{1}/2] ≥ 0,

by (36), 0 ≤ u ≤ 1, and since η′′(u) ≥ 0 and u_{t} ≥ 0, by (44) and (39). Differentiating ω with respect to s we get by (45)

ω_{s}(t, s) = tη(u(t, s)) + [(m_{1}/2) + (s − m_{1})t]η′(u(t, s))u_{s}(t, s)

and therefore we obtain from (43) at (t, s) ∈ [0, 1] × [m_{1}, m_{2}]

ω_{s} = tη(u) + [(m_{1}/2) + (s − m_{1})t]η′(u)\frac{m_{1}(t − t^{2})}{2[(m_{1}/2) + (s − m_{1})t]^{2}} (67)
= tη(u) + η′(u)t(1 − u),

because

\frac{m_{1}(t − t^{2})}{2[(m_{1}/2) + (s − m_{1})t]} = t(1 − u),  (68)

where we used (38) to deduce

1 − u = \frac{[(m_{1}/2) + (s − m_{1})t] − (s − (m_{1}/2))t}{(m_{1}/2) + (s − m_{1})t} = \frac{m_{1}(1 − t)}{2[(m_{1}/2) + (s − m_{1})t]} .

Differentiating (67) we obtain

ω_{ss} = tη′(u)u_{s} + η′′(u)u_{s}t(1 − u) − η′(u)tu_{s}
= η′′(u)u_{s}t(1 − u) ≥ 0,  (69)

again since η′′(u) ≥ 0, u_{s} ≥ 0, by (44) and (43) for 0 ≤ u, t ≤ 1. Observe that by (49) t = m_{1}u{2[s − (m_{1}/2) + (m_{1} − s)u]}^{−1}, and

u_{s}(1 − u)u =_{(43)} \frac{m_{1}(t − t^{2})}{2[(m_{1}/2) + (s − m_{1})t]^{2}} (1 − u)u
=_{(68)} \frac{t(1 − u)^{2}u}{(m_{1}/2) + (s − m_{1})t}  (70)
=_{(38)} \frac{t^{2}(1 − u)^{2}(s − (m_{1}/2))}{[(m_{1}/2) + (s − m_{1})t]^{2}}
=_{(39)} 2t^{2}(1 − u)^{2}u_{t}/m_{1}.

Using (70) one obtains from (69)

ω_{ss} = 2η′′(u)t^{2}(1 − u)^{2}u_{t}t(m_{1}u)^{−1}
=_{(49)} \frac{η′′(u)t^{2}(1 − u)^{2}u_{t}}{s − (m_{1}/2) + (m_{1} − s)u} ,

hence

ω_{tt}ω_{ss} = (η′′(u))^{2}u_{t}^{2}t^{2}(1 − u)^{2}.

Moreover, from (67) we infer

ω_{ts} = ω_{st} = η(u) + tη′(u)u_{t} + η′′(u)u_{t}t(1 − u) + η′(u)(1 − u) − η′(u)tu_{t}
= η(u) + η′(u)(1 − u) + t(1 − u)η′′(u)u_{t} .

Consequently, again by (67),

[ω_{ts} − (ω_{s}/t)]^{2} = [η(u) + η′(u)(1 − u) + t(1 − u)η′′(u)u_{t}
−η(u) − η′(u)(1 − u)]^{2}
= t^{2}(1 − u)^{2}(η′′(u))^{2}u_{t}^{2} = ω_{tt}ω_{ss}.  (71)

According to (66),(69) and (71) evaluated at the point (t, s) := (τ, σ) ∈ [0, 1] × [m_{1}, m_{2}] the matrix

( \begin{matrix}ω_{tt} & ω_{ts} − (ω_{s}/τ) \\ ω_{ts} − (ω_{s}/τ) & ω_{ss} \end{matrix} )

in (58) is positive semi-definite, hence the last term in (58) is nonnegative.

The third expression on the right-hand side of (58) coincides with the first on the right-hand side of (63), because one can use (4) to write

P_{p_{1}∧p_{2}}^{⊥}(F_{z}(x, p_{1} ∧ p_{2}))
= F_{z}(x, p_{1} ∧ p_{2}) − (F_{z}(x, p_{1} ∧ p_{2}) · (p_{1} ∧ p_{2}))\frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|^{2}}
=_{(H)} F_{z}(x, p_{1} ∧ p_{2}) − \frac{F (x, p_{1} ∧ p_{2})}{|p_{1} ∧ p_{2}|} \frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|}
=_{(35)} F_{z}(x, p_{1} ∧ p_{2}) − σ(x, p)\frac{p_{1} ∧ p_{2}}{|p_{1} ∧ p_{2}|} .

Now we look at the coefficient of the second term on the right-hand side of (58) in Theorem 7:

τω_{t} − (σ − λ_{F} (R_{0}))ω_{s}

=_{(50),(67)} τ{(σ − m_{1})η(u) + η′(u)[σ − (m_{1}/2) − (σ − m_{1})u]}
−(σ − λ_{F} (R_{0}))[τη(u) + η′(u)τ(1 − u)]
= τ{[σ − (m_{1}/2) − (σ − m_{1})u − (σ − λ_{F} (R_{0}))(1 − u)]η′(u)
+[σ − m_{1} − (σ − λ_{F} (R_{0}))]η(u)}
= τ{[λ_{F} (R_{0}) − (m_{1}/2) + (m_{1} − λ_{F} (R_{0}))u]η′(u) − (m_{1} − λ_{F} (R_{0}))η(u)}
=: τϑ(u).

By (39) τ is a nonnegative monotonically increasing function of u, and so is ϑ if we choose η appropriately. In fact, we have

ϑ_{u} = (m_{1} − λ_{F} (R_{0}))η′(u) + [λ_{F} (R_{0}) − (m_{1}/2) + (m_{1} − λ_{F} (R_{0}))u]η′′(u)
−(m_{1} − λ_{F} (R_{0}))η′(u) = α(u),

where α(u) := λ_{F} (R_{0}) − (m_{1}/2) + (m_{1} − λ_{F} (R_{0}))u satisfies α_{u}(u) = m_{1} − λ_{F} (R_{0}), which means that for λ_{F} (R_{0}) ≥ m_{1} the function α(u) is decreasing in u, and for λ_{F} (R_{0}) ∈ [0, m_{1}] it is nondecreasing in u. This implies for u ∈ [0, 1]

α(u) ≥ \begin{cases}α(1) = m_{1}/2 & if λ_{F} (R_{0}) ∈ [m_{1}, ∞], \\ α(0) = λ_{F} (R_{0}) − m_{1}/2 & if λ_{F} (R_{0}) ∈ [0, m_{1}]. \end{cases}

In other words,

ϑ_{u}(u) ≥ 0 for all u ∈ [0, 1] if λ_{F} (R_{0}) ≥ m_{1}/2.  (72)

If, on the other hand, λ_{F} (R_{0}) < m_{1}/2, we set

u_{0} := \frac{(m_{1}/2) − λ}{m_{1} − λ},  (73)

where λ > 0 is the quantity defined in our assumption (62), and where u_{0} is the constant appearing in the definition of the cut-off function η in (44).

One easily checks with (62) that 0 < u_{0} < 1/2 so that our previous choice of u_{1} ∈ (0, 1/2) (see (53)) can be modified by taking u_{1} sufficiently close to 1/2 to guarantee the condition 0 < u_{0} < u_{1} assumed in (44). Moreover we have ϑ_{u}(u) ≥ 0 for all u ∈ [0, 1] by (73) and (44), since, for λ_{F} (R_{0}) ∈ [λ, m_{1}/2) and for

0 ≤ u < \frac{(m_{1}/2) − λ_{F} (R_{0})}{m_{1} − λ_{F} (R_{0})} ≤ u_{0},

one has η′′(u) = 0. Hence 0 ≤ ϑ(u) ≤ ϑ(1) for all u ∈ [0, 1], τ(1) = 1, and therefore

0 = τ(0)ϑ(0) ≤ τ(u)ϑ(u) ≤ τ(1)ϑ(1)
= [λ_{F} (R_{0}) − (m_{1}/2) + m_{1} − λ_{F} (R_{0})]η′(1) − (m_{1} − λ_{F} (R_{0}))η(1)
=_{(44)} m_{1}η′(1)/2 − m_{1} + λ_{F} (R_{0}),

which proves that the coefficient of the second term on the right-hand side of (58) is nonnegative.

Recall from (44) and (53) that u_{1} ∈ (u_{0}, 1/2) is the only restriction on the parameter u_{1} so far, and if λ ≥ m_{1}/2, the lower bound u_{0} is no restriction at all, since then u_{0} > 0 can be chosen arbitrarily small.

Finally we are going to investigate the coefficient of the first term in (58),

φ(t) := (m_{1}/2) + ω(t, s) − tω_{t},

regarded as a function of t for fixed s ∈ [m_{1}, m_{2}] with

φ_{t}(t) = ω_{t} − ω_{t} − tω_{tt} ≤ 0,

by (66). That means, for τ ∈ [0, 1], σ ∈ [m_{1}, m_{2}],

φ(0) ≥ φ(τ) ≥ φ(1) =_{(45),(41),(44)} (m_{1}/2) + σ − (m_{1}/2) − ω_{t}(1, σ)
=_{(41),(52)} σ − (σ − m_{1}) − η′(1)m_{1}/2
= m_{1} − η′(1)m_{1}/2 = m_{1}(1 − (η′(1)/2)).

Without loss of generality we may assume that

0 < ϵ < min{λ, m_{1}/2}.  (74)

Recall (44), in particular

\frac{1}{1 − u_{0}} < η′(1) ≤ \frac{1}{1 − u_{1}} .  (75)

If λ ≥ m_{1}/2 the parameter u_{0} > 0 may be chosen arbitrarily small as mentioned above, hence we can choose

u_{0}^{ϵ} := \frac{ϵ}{m_{1} + 2ϵ} and u_{1}^{ϵ} := \frac{2ϵ}{m_{1} + 2ϵ} ∈_{(74)} (u_{0}, 1/2),

which, according to (75), leads to

η′(1) ≤ \frac{m_{1} + 2ϵ}{m_{1}} ,  (76)

and thus,

(m_{1}/2) + ω − τω_{t} = φ(τ) ≥ φ(1) = m_{1}(1 − η′(1)/2)
≥ m_{1} (1 − \frac{1}{2(1 − u_{1}^{ϵ})})
= m_{1} (1 − \frac{m_{1} + 2ϵ}{2m_{1}}) = \frac{m_{1}}{2} − ϵ > 0. (77)

If λ < m_{1}/2 we choose

u_{1}^{ϵ} := \frac{(m_{1}/2) − λ + ϵ}{m_{1} − λ + ϵ} ∈ (u_{0}, 1/2) for ϵ > 0,

where now u_{0} is given by (73), so that by (75)

η′(1) ≤ 2m_{1}^{−1}(m_{1} − λ + ϵ).  (78)

Thus we get

(m_{1}/2) + ω − τω_{t} ≥ λ − ϵ > 0.  (79)

With these choices the cut-off function η = η^{ϵ} depends on ϵ > 0, and hence also G = G^{ϵ}, and the construction is finished. The estimates (77) and (79) show that

(m_{1}/2) + ω − τω_{t} ≥ min{λ, m_{1}/2} − ϵ > 0,

which concludes the proof of Theorem 8. ⊓ ⊔

As a consequence we obtain

Corollary 2. The dominance function G^{ϵ}(x, p) of Theorem 8 satisfies

ξ · G_{pp}^{ϵ}(x, p) ξ ≥ 2 [min{λ, m_{1}/2} − m_{2} − ϵ] |ξ|^{2}

for all ξ ∈ ℝ^{2n}, (x, p) ∈ ℝ^{n} × (ℝ^{2n} − {0}).

Proof. Recalling (49)–(52) we find

0 =_{(50)} ω_{t}(t(0, s), s) ≤_{(51)} ω_{t}(t(r, s), s) ≤_{(51)} ω_{t}(t(1, s), s) =_{(52)} s−m_{1} +(m_{1}/2)η′(1),

which implies by (76) in the case λ ≥ m_{1}/2, and by (78) in the case λ < m_{1}/2, and by (36),

|ω_{t}| ≤ m_{2} + ϵ − min{λ, m_{1}/2}.

Consequently,

|2a^{−1}ω_{t}(p_{1} ∧ p_{2}) · (ζ ∧ η)| ≤ [m_{2} + ϵ − min{λ, m_{1}/2}] |ξ|^{2}.  (80)

On the other hand, by (55) of Lemma 2, we have |P_{p_{1}∧p_{2}}^{⊥}(F_{z}(x, p_{1}∧p_{2}))| ≤ m_{2}, and by (67), ω_{s}/τ = η(u) + η′(u)(1 − u), hence (ω_{s}/τ)_{u} = η′′(u)(1 − u) ≥ 0 according to (44). This implies

0 = η(0) + η′(0) = (ω_{s}/τ)|_{u=0} ≤ (ω_{s}/τ) ≤ (ω_{s}/τ)|_{u=1} = η(1) = 1.

Therefore,

|\frac{2ω_{s}}{τ} P_{p_{1}∧p_{2}}^{⊥}(F_{z}(x, p_{1} ∧ p_{2}))(ζ ∧ η)| ≤ m_{2}|ξ|^{2},

which together with (80) and (63) implies our assertion. ⊓ ⊔

Proof of Theorem 2 . Take ϵ = ϵ(k) := (k − k_{0})/4 and the dominance function G^{∗ϵ}(x, p) for F := F^{∗} constructed in Theorem 8. Then according to Corollary 2

ξ · G_{pp}^{∗ϵ}(x, p) ξ ≥ 2[min{λ^{∗}, m_{1}^{∗}/2} − m_{2}^{∗} − ϵ] |ξ|^{2}
= −[k_{0} + 2ϵ] |ξ|^{2}  (81)

for all ξ ∈ ℝ^{2n}, (x, p) ∈ ℝ^{n} × (ℝ^{2n} − {0}). Set G(x, p) := kD(p) + G^{∗ϵ}(x, p). Then G is of class C^{2}(ℝ^{n}×(ℝ^{2n}−{0})) and satisfies (D1)–(D4). Furthermore, for p ≠ 0 we have by (81)

ξ · G_{pp}(x, p) ξ = k|ξ|^{2} + ξ · G_{pp}^{∗ϵ}(x, p)ξ
≥ [k − k_{0} − 2ϵ]|ξ|^{2} = 2^{−1}(k − k_{0})|ξ|^{2},

so that also the ellipticity condition (E) holds for G(x, p) with a uniform constant λ_{G}(R_{0}) := (k − k_{0})/2 for all R_{0} > 0. ⊓ ⊔

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