Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 253–264

THE LIMIT OF MAPPINGS WITH FINITE DISTORTION

F. W. Gehring and T. Iwaniec

University of Michigan, Department of Mathematics Ann Arbor, MI 48109, U.S.A.; fgehring@math.lsa.umich.edu Syracuse University, Department of Mathematics Syracuse, NY 13244, U.S.A.; tiwaniec@mailbox.syr.edu

Abstract. We show here that the limit mapping f of a weakly convergent sequence of mappings f_{ν} with finite distortion also has finite distortion and give several dimension free estimates for the dilatation of f. Our arguments are based on the weak continuity of the Jacobian determinants and the concept of polyconvexity.

1. Introduction

Let f: Ω → R^{n} be a mapping in the Sobolev space W_{loc}^{1,n}(Ω, R^{n}) where Ω is a domain in R^{n}. Then the differential matrix Df(x) ∈ R^{n×n} and its determinant J(x, f) = det Df(x) are well defined at almost every point x ∈ Ω. Here R^{n×n} denotes the space of all n × n-matrices, where n > 1, equipped with the operator norm

|A| = max{|Aξ| : ξ ∈ S^{n−1}}.

We assume most of the time that J(x, f) ≥ 0 a.e. and refer to such mappings f as orientation preserving. We let Rn×n_{+} denote the set of matrices with positive determinant and write Rn×n_{+} ∪ {0} when the zero matrix is included.

Definition 1.1. A mapping f ∈ W_{loc}^{1,n}(Ω, R^{n}) is said to be of finite distortion if

Df(x) ∈ Rn×n_{+} ∪ {0}

for almost every x ∈ Ω.

In what follows it is vital that the Sobolev exponent is at least the dimension of Ω so that we can integrate the Jacobian. In this case the mappings of finite distortion are actually continuous [18].

Definition 1.1 asserts that

(1.2)  |Df(x)|^{n} ≤ K_{O}(x)J(x, f)

1991 Mathematics Subject Classification: Primary 30F40.

Research supported in part by grants from the U. S. National Science Foundation.

where 1 ≤ K_{O}(x) < ∞ a.e. The smallest such function defined by

(1.3)  K_{O}(x, f) = \frac{|Df(x)|^{n}}{J(x, f)}

if J(x, f) ≠ 0 and 1 otherwise is called the outer dilatation function of f.

We shall establish the following limit theorem.

Theorem 1.4. Suppose that f_{ν}: Ω → R^{n} is a sequence of mappings of finite distortion which converges weakly in W_{loc}^{1,n}(Ω, R^{n}) to f and suppose that

(1.5)  K_{O}(x, f_{ν}) ≤ M(x) < ∞  for ν = 1, 2, …

a.e. in Ω. Then f has finite distortion and

(1.6)  K_{O}(x, f) ≤ M(x)

a.e. in Ω.

Theorem 1.4 is a refinement of Reshetnyak’s theorem [15] concerning mappings f_{ν} of bounded distortion, that is mappings which satisfy (1.5) with M(x) ≤ K where K is a constant. In this case, weak convergence in W_{loc}^{1,n}(Ω, R^{n}) implies uniform convergence on compact sets and hence, by Reshetnyak’s theorem, that the limit mapping f satisfies K_{O}(x, f) ≤ K instead of the pointwise bound given in (1.6).

Remark 1.7. The hypotheses of Theorem 1.4 imply a stronger conclusion than (1.6), namely the existence of a subsequence {f_{ν_{k}}} such that

(1.8)  K_{O}(x, f) ≤ b∗ lim_{k→∞} K_{O}(x, f_{ν_{k}})

in Ω.

The limit in (1.8) is to be understood in the sense of biting convergence defined in Section 2; see [1], [3] and [6]. The basic ingredient of our proof is the higher integrability of nonnegative Jacobians. For a discussion of this property for mappings with bounded distortion see [5], [7], [8], [11] and [16].

The outer dilatation function K_{O}(x, f) has a simple geometric interpretation. If f: Ω → R^{n} has a differential Df(x) ≠ 0, then Df(x) maps the unit sphere onto an ellipsoid E and

(1.9)  K_{O}(x, f) = \frac{vol(B_{O})}{vol(E)},

where B_{O} is the smallest ball circumscribed about E. In the same way, we may define the inner dilatation of f at x by

(1.10)  K_{I}(x, f) = \frac{vol(E)}{vol(B_{I})},

where B_{I} is the largest ball inscribed in E. We set K_{I}(x, f) = 1 at degenerate points where Df(x) = 0 and we call

(1.11)  K(x, f) = max{K_{O}(x, f), K_{I}(x, f)}

the maximal dilatation,

(1.12)  K_{M}(x, f) = \frac{1}{2}(K_{O}(x, f) + K_{I}(x, f))

the mean dilatation and

(1.13)  H(x, f) = (K_{O}(x, f) K_{I}(x, f))^{1/n}

the linear dilatation for f at x. The linear dilatation has the following dimension free representation

(1.14)  H(x, f) = \frac{max\{|Df(x)ξ| : ξ ∈ S^{n−1}\}}{min\{|Df(x)ξ| : ξ ∈ S^{n−1}\}}

at points where Df(x) ≠ 0.

All of these dilatation functions coincide when n = 2; this is not the case when n > 2. However, the functions K_{I}, K_{M} and K have the same lower semicontinuity property as K_{O} when n > 2.

Theorem 1.15. Theorem 1.4 and Remark 1.7 remain valid with K_{I}(x, f), K_{M}(x, f) and K(x, f) in place of K_{O}(x, f).

This is not true of the geometrically appealing linear dilatation H(x, f) when n > 2. Indeed a striking example in [10] exhibits for each K > 1 a sequence of mappings f_{ν} ∈ W_{loc}^{1,n}(R^{n}, R^{n}) such that

1. H(x, f_{ν}) ≡ K,

2. f_{ν} converges uniformly to a linear map f: R^{n} → R^{n},

3. H(x, f) ≡ K′ > K where K′ is a constant. In light of this anomaly it is desirable to see what one can say about the linear dilatation of the limit f of a sequence of mappings. The following analogue of Theorems 1.4 and 1.15 answers a question raised at the Saariselkä Conference in June 1997. See also Section 14 in [17].

Theorem 1.16. Suppose that f_{ν}: Ω → R^{n} is a sequence of mappings of finite distortion which converges weakly in W_{loc}^{1,n}(Ω, R^{n}) to f and suppose that

(1.17) H(x, f_{ν}) ≤ M(x) < ∞ for ν = 1, 2, …

a.e. in Ω. Then f has finite distortion and

(1.18)  H(x, f) ≤ \frac{1}{2}(M(x) + M(x)^{n−1})^{2/n} ≤ M(x)^{2−(2/n)}

a.e. in Ω.

2. Biting convergence

We shall make use of some ideas of Brooks and Chacon [6], in particular, the notion of biting convergence or weak convergence in measure.

Suppose that h and h_{ν}, ν = 1, 2, …, are Lebesgue measurable functions on E ⊂ R^{n} with values in a finite dimensional normed space (V, ∥ · ∥). In our applications we shall assume that V = R or V = R^{n×n}. We say that h_{ν} converges to h in the biting sense in E if there exist an increasing sequence of measurable subsets E_{k} of E with

⋃_{k} E_{k} = E

such that for each k, the functions h and h_{ν} are in L^{1}(E_{k}, V) for all ν and

(2.1)  lim_{ν→∞} ∫_{E_{k}} ϕ h_{ν} dx = ∫_{E_{k}} ϕ h dx

whenever ϕ ∈ L^{∞}(E_{k}). In other words, the sequence h_{ν} converges weakly to h outside arbitrarily small bites from E, that is outside E \ E_{k} for k = 1, 2, …. We shall call h the biting limit of the sequence h_{ν} and write

(2.2)  h_{ν} →^{b} h  or h = b∗ lim_{ν→∞} h_{ν}.

It is immaterial which increasing sequence of subsets E_{k} of E we choose to define h as long as the weak limits on these sets exist; different bites yield the same limit. We leave it to the reader to verify the following two simple properties of biting convergence.

Lemma 2.3. If h_{ν} →^{b} h in E and if λ is finite and measurable in E, then

λ h_{ν} →^{b} λ h in E.

Lemma 2.4. If h_{ν} are measurable functions in E, ν = 1, 2, …, and if

sup_{ν} ∥h_{ν}(x)∥ < ∞

a.e. in E, then h_{ν} contains a subsequence which converges in the biting sense in E.

We shall require the following lemma.

Lemma 2.5. If A_{ν} converges weakly to A in L_{loc}^{p}(E, R^{n×n}) where 1 ≤ p < ∞, then there is a subsequence {ν_{k}} such that

(2.6)  |A|^{s} ≤ b∗ lim_{k→∞} |A_{ν_{k}}|^{s}

for 1 ≤ s ≤ p.

Proof. Choose measurable unit vectors ξ = ξ(x) and ζ = ζ(x) such that

|A(x)| = ⟨A(x)ξ(x), ζ(x)⟩  whence  |A_{ν}(x)| ≥ ⟨A_{ν}(x)ξ(x), ζ(x)⟩

in E. Since t^{s} is convex in 0 < t < ∞,

(2.7)  |A_{ν}|^{s} − |A|^{s} ≥ s|A|^{s−1}(|A_{ν}| − |A|) ≥ s|A|^{s−1}(⟨A_{ν}ξ, ζ⟩ − ⟨Aξ, ζ⟩)

in E, the right hand side of (2.7) converges to 0 in the biting sense as ν → ∞ by Lemma 2.3 and we obtain (2.6) for some subsequence {ν_{k}} by Lemma 2.4.

3. Weak continuity of minors

Suppose that {f_{ν}} is a sequence of orientation preserving mappings

(3.1)  f_{ν} = (f_{ν}^{1}, …, f_{ν}^{n}): Ω → R^{n},  ν = 1, 2, …,

which converge weakly in W_{loc}^{1,n}(Ω, R^{n}) to a mapping f = (f^{1}, …, f^{n}). This simply means that for each i, j = 1, 2, …, n we have

(3.2)  lim_{ν→∞} ∫_{Ω} ϕ \frac{∂f_{ν}^{i}}{∂x_{j}} dx = ∫_{Ω} ϕ \frac{∂f^{i}}{∂x_{j}} dx

for each ϕ in L_{0}^{n/(n−1)}(Ω), the space of test functions in L^{n/(n−1)} with compact support in Ω.

A similar conclusion can be drawn for arbitrary minors of the differential matrix Df. Given l-tuples 1 ≤ i_{1} < ⋯ < i_{l} ≤ n and 1 ≤ j_{1} < ⋯ < j_{l} ≤ n we let

\frac{∂(f^{i_{1}}, …, f^{i_{l}})}{∂(x_{j_{1}}, …, x_{j_{l}})}

denote the corresponding l × l minor of Df. We then have the following counterpart for (3.2).

Lemma 3.3 (Weak continuity). The above hypotheses imply that

(3.4)  lim_{ν→∞} ∫_{Ω} ϕ \frac{∂(f_{ν}^{i_{1}}, …, f_{ν}^{i_{l}})}{∂(x_{j_{1}}, …, x_{j_{l}})} dx = ∫_{Ω} ϕ \frac{∂(f^{i_{1}}, …, f^{i_{l}})}{∂(x_{j_{1}}, …, x_{j_{l}})} dx

for each ϕ in L_{0}^{n/(n−l)}(Ω) and corresponding l × l minors of Df_{ν} and Df, l = 1, 2, …, n.

Proof. The convergence of the minors in the case where ϕ is in C_{0}^{∞}(Ω) follows from integration by parts and the compactness of the Sobolev imbedding. The result in this case can be traced back at least as far as [2], [12] and [14]. The extension to arbitrary ϕ in L_{0}^{n/(n−l)}(Ω) with 1 ≤ l < n poses no problem because C_{0}^{∞}(Ω) is dense in L_{0}^{n/(n−l)}(Ω). It is the case where l = n and ϕ is in L_{0}^{∞}(Ω) that requires our mappings f_{ν} to be orientation preserving; for this see Corollary 1.2 in [13].

The case where the mappings f_{ν} are K -quasiregular can be handled due to the higher degree of integrability of the Jacobians [5] and [8]. For yet another approach see the biting theorem for Jacobians, Corollary 2.3 in [4] and Corollary 2.2 in [19].

Corollary 3.5 (Biting convergence). The above hypotheses imply that

(3.6)  b∗ lim_{ν→∞} \frac{∂(f_{ν}^{i_{1}}, …, f_{ν}^{i_{l}})}{∂(x_{j_{1}}, …, x_{j_{l}})} = \frac{∂(f^{i_{1}}, …, f^{i_{l}})}{∂(x_{j_{1}}, …, x_{j_{l}})}

for corresponding l × l minors of Df_{ν} and Df, l = 1, 2, …, n.

4. Dilatation functions

We introduce as in Section 1 the following quantities for a matrix A in Rn×n_{+} :

(4.1)  Outer dilatation Inner dilatation Mean dilatation Maximal dilatation Linear dilatation  \begin{aligned}K_{O}(A) & = |A|^{n}/ det(A), \\ K_{I}(A) & = K_{O}(A^{−1}), \\ K_{M}(A) & = \frac{1}{2}(K_{O}(A) + K_{I}(A)), \\ K(A) & = max\{K_{O}(A), K_{I}(A)\}, \\ H(A) & = (K_{O}(A) K_{I}(A))^{1/n}. \end{aligned}

By Cramer’s rule, we can express K_{I}(A) in terms of the minors of order (n − 1) and the determinant of A as follows:

(4.2)  K_{I}(A) = \frac{|A^{#}|^{n}}{det(A)^{n−1}} .

Here A^{#} is the matrix in R^{n×n} whose entries are co-factors of A,

A_{jk}^{#} = (−1)^{j+k} det(M_{jk}),

where M_{jk} is a submatrix of A obtained by deleting the j th row and k th column.

The above definitions and an elementary analysis of the eigenvalues of AA^{T} yield the following estimates

(4.3)  K_{O} K_{I} = H^{n},  K_{O} ≤ H^{n−1} ≤ K_{I}^{n−1},  K_{I} ≤ H^{n−1} ≤ K_{O}^{n−1}.

From this and the arithmetic-geometric mean inequality we obtain the following estimates for the linear dilatation H:

(4.4)  H^{n/2} ≤ \frac{1}{2}(K_{O} + K_{I}) = K_{M} = \frac{1}{2}(K_{O} + \frac{H^{n}}{K_{O}} ) ≤ \frac{1}{2}(H + H^{n−1}).

Equality holds in the first inequality only when K_{O} = K_{I} and in the second inequality only when K_{O} = K_{I}^{n−1} or K_{I} = K_{O}^{n−1} .

5. Polyconvexity

We show here that the dilatation functions K_{O}, K_{I}, K_{M} and K are polyconvex on the set Rn×n_{+} , that is, that they can be expressed as convex functions of minors of the matrix A ∈ Rn×n_{+} .

Lemma 5.1. The function

(5.2)  F (x, y) = F_{p,q}(x, y) = \frac{x^{p}}{y^{q}}

is convex on R_{+} × R_{+} whenever p ≥ q + 1 ≥ 1.

Proof. We must show that

(5.3)  F (x, y) − F (a, b) ≥ A(x − a) + B(y − b)

for all x, y, a, b ∈ R_{+} where

A = \frac{∂F}{∂x} (a, b) = \frac{pa^{p−1}}{b^{q}} , B = \frac{∂F}{∂y} (a, b) = −\frac{qa^{p}}{b^{q+1}} .

Inequality (5.3) is an immediate consequence of the arithmetic-geometric mean inequality

(5.4)  u_{1}^{r_{1}} u_{2}^{r_{2}} u_{3}^{r_{3}} ≤ r_{1}u_{1} + r_{2}u_{2} + r_{3}u_{3}

which holds whenever r_{j}, u_{j} are nonnegative for j = 1, 2, 3 with r_{1} + r_{2} + r_{3} = 1. See, for example, Section 2.5 in [9] In particular if we set

r_{1} = \frac{1}{p}, r_{2} = \frac{q}{p}, r_{3} = \frac{p − q − 1}{p}, u_{1} = \frac{x^{p}}{y^{q}} , u_{2} = \frac{a^{p}y}{b^{q+1}} , u_{3} = \frac{a^{p}}{b^{q}} ,

then we obtain

\frac{xa^{p−1}}{b^{q}} = u_{1}^{r_{1}} u_{2}^{r_{2}} u_{3}^{r_{3}} ≤ r_{1}u_{1} + r_{2}u_{2} + r_{3}u_{3} = \frac{1}{p} \frac{x^{p}}{y^{q}} + \frac{q}{p} \frac{a^{p}y}{b^{q+1}} + \frac{p − q − 1}{p} \frac{a^{p}}{b^{q}}

whence

(5.5)  \frac{x^{p}}{y^{q}} − \frac{a^{p}}{b^{q}} ≥ \frac{pa^{p−1}}{b^{q}} (x − a) − \frac{qa^{p}}{b^{q+1}} (y − b)

which is (5.3).

We see from (4.1) and (4.2) that

(5.6)

K_{O}(A) = F (|A|, det(A))  with p = n and q = 1,
K_{I}(A) = F (|A^{#}|, det(A))  with p = n and q = n − 1.

We observe next that the function F (x, y) is increasing in the variable x and that x is a convex function of the minors of A, x = |A| or x = |A^{#}|, respectively, in (5.6). This implies that both K_{O} and K_{I} are polyconvex and hence so are the mean and maximal dilatations K_{M} and K. On the other hand, the linear dilatation H fails to be even rank-one convex [10].

We conclude by recording from (5.5) and (5.6) what polyconvexity means for the outer and inner dilatations:

(5.7) K_{O}(X) − K_{O}(A) ≥ \frac{n|A|^{n−1}}{det(A)} (|X| − |A|) − \frac{|A|^{n}}{det(A)^{2}} (det(X) − det(A))

and (5.8)

K_{I}(X) − K_{I}(A) ≥ \frac{n|A^{#}|^{n−1}}{det(A)^{n−1}} (|X^{#}| − |A^{#}|) − \frac{(n − 1)|A^{#}|^{n}}{det(A)^{n}} (det(X) − det(A)).

6. Lower semicontinuity

For simplicity of notation we will use the symbol 𝒦 (f) = 𝒦 (x, f) to denote any one of the dilatations K_{O}(x, f), K_{I}(x, f), K_{M}(x, f) or K(x, f). Then

𝒦 (x, f) = 𝒦 (Df(x))

whenever J(x, f) > 0, where 𝒦 (Df) denotes the corresponding dilatation function of matrices defined in (4.1).

Theorem 6.1. Suppose that f_{ν}: Ω → R^{n} is a sequence of mappings of finite distortion which converge weakly in W_{loc}^{1,n}(Ω, R^{n}) to f and suppose that

(6.2)  𝒦 (x, f_{ν}) ≤ M(x) < ∞  for ν = 1, 2, …

a.e. in Ω. Then f has finite distortion and there exists a subsequence {f_{ν_{k}}} such that

(6.3)  𝒦 (x, f) ≤ b∗ lim_{k→∞} 𝒦 (x, f_{ν_{k}})

in Ω. In particular

(6.4)  𝒦 (x, f) ≤ M(x)

a.e. in Ω.

Proof. We consider first the case where 𝒦 (f) is the outer dilatation K_{O}(f). Then (6.2) implies that

|Df_{ν}(x)|^{n} ≤ M(x)J(x, f_{ν})

a.e. in Ω while

(6.5)  b∗ lim_{ν→∞} det(Df_{ν}) = det(Df),  b∗ lim_{ν→∞} M det(Df_{ν}) = M det(Df)

by Corollary 3.5 and Lemma 2.3. Next by Lemma 2.5 we can choose a subsequence {f_{ν_{k}}} such that

(6.6)  |Df|^{n} ≤ b∗ lim_{k→∞} |Df_{ν_{k}}|^{n} ≤ b∗ lim_{k→∞} M det(Df_{ν_{k}}) = M det(Df)

and

(6.7)  |Df(x)|^{n} ≤ M(x)J(x, f)

a.e. in Ω. Thus f has finite distortion and (6.4) holds a.e. in Ω.

Finally in order to establish (6.3) we apply (5.7) to the matrices X = Df_{ν} and A = Df to obtain

(6.8)  K_{O}(f_{ν}) − K_{O}(f) ≥ M_{1}(|Df_{ν}| − |Df|) − M_{2}(det(Df_{ν}) − det(Df)),

where

(6.9)  M_{1} = \frac{n|Df|^{n−1}}{det(Df)} ,  M_{2} = \frac{|Df|^{n}}{det(Df)^{2}} ,

a.e. in the set E ⊂ Ω where det(Df) ≠ 0. Next if we restrict our attention to the set E, then by Lemma 2.5, Corollary 3.5 and Lemma 2.4 we can choose a subsequence {f_{ν_{k}}} such that

(6.10)  |Df| ≤ b∗ lim_{k→∞} |Df_{ν_{k}}|,  det(Df) = b∗ lim_{k→∞} det(Df_{ν_{k}})

and such that K_{O}(f_{ν_{k}}) converges in the biting sense. Then we obtain

(6.11)  K_{O}(f) ≤ b∗ lim_{k→∞} K_{O}(f_{ν_{k}})

in E from (6.8), (6.9) and (6.10). By our convention

(6.12)  K_{O}(f) = 1 ≤ b∗ lim_{k→∞} K_{O}(f_{ν_{k}})

a.e. in Ω \ E, completing the proof of Theorem 6.1 for the case where 𝒦 = K_{O}.

Suppose next that 𝒦 = K_{I}. Then (4.3) yields the rough estimate

(6.13)  K_{O}(x, f_{ν}) ≤ K_{I}^{n−1}(x, f_{ν}) ≤ M(x)^{n−1}

and f has finite distortion by what was proved above. Next (4.2) and (5.8) applied to the matrices X = Df_{ν} and A = Df yield

(6.14) K_{I}(f_{ν}) − K_{I}(f) ≥ M_{3}(|(Df_{ν})^{#}| − |(Df)^{#}|) − M_{4}(det(Df_{ν}) − det(Df)),

where

(6.15)  M_{3} = \frac{n|(Df)^{#}|^{n−1}}{det(Df)^{n−1}} ,  M_{4} = \frac{(n − 1)|(Df)^{#}|^{n}}{det(Df)^{n}} ,

a.e. in E. Then (Df_{ν})^{#} converges weakly to (Df)^{#} in L_{loc}^{n/(n−1)}(Ω, R^{n×n}), there is a subsequence {f_{ν_{k}}} such that the K_{I}(f_{ν_{k}}) converges in the biting sense and

(6.16)  |(Df)^{#}| ≤ b∗ lim_{k→∞} |(Df_{ν_{k}})^{#}|,  det(Df) = b∗ lim_{k→∞} det(Df_{ν_{k}})

again by Lemma 2.5 and Corollary 3.5. Then

(6.17)  K_{I}(f) ≤ b∗ lim_{k→∞} K_{I}(f_{ν_{k}})

in E as above while

(6.18)  K_{I}(f) = 1 ≤ b∗lim inf_{k→∞} K_{I}(f_{ν_{k}})

in Ω \ E since f has finite distortion. This completes the proof of Theorem 6.1 for the case where 𝒦 = K_{I}.

If 𝒦 = K_{M}, then
max{K_{O}(x, f_{ν}), K_{I}(x, f_{ν})} ≤ 2K_{M}(x, f_{ν}) ≤ 2M(x)

and f has finite distortion. Next there exists a subsequence {f_{ν_{k}}} such that K_{O}(f_{ν_{k}}), K_{I}(f_{ν_{k}}) and K_{M}(f_{ν_{k}}) converge in the biting sense, and we obtain

2K_{M}(f) = K_{O}(f) + K_{I}(f)

≤ b∗ lim_{k→∞} K_{O}(f_{ν_{k}}) + b∗ lim_{k→∞} K_{I}(f_{ν_{k}}) = 2 b∗ lim_{k→∞} K_{M}(f_{ν_{k}})

from (6.3) with 𝒦 = K_{O} and 𝒦 = K_{I}.

The limit of mappings with finite distortion 263

Finally if 𝒦 = K, then

max{K_{O}(x, f_{ν}), K_{I}(x, f_{ν})} ≤ K(x, f_{ν}) ≤ M(x)

and we can choose a subsequence {f_{ν_{k}}} such that

K_{O}(f) ≤ b∗ lim_{k→∞} K_{O}(f_{ν_{k}}) ≤ b∗ lim_{k→∞} K(f_{ν_{k}}),
K_{I}(f) ≤ b∗ lim_{k→∞} K_{I}(f_{ν_{k}}) ≤ b∗ lim_{k→∞} K(f_{ν_{k}}).

Hence

(6.19)  K(f) ≤ b∗ lim_{k→∞} K(f_{ν_{k}})

completing the proof of Theorem 6.1.

7. Conclusions

Theorem 1.4, Remark 1.7 and Theorem 1.15 of Section 1 follow from Theorem 6.1. For the proof of Theorem 1.16, (1.17) and (4.4) imply that

K_{M}(x, f_{ν}) ≤ \frac{1}{2}(H(x, f_{ν}) + H(x, f_{ν})^{n−1}) ≤ \frac{1}{2}(M(x) + M(x)^{n−1}) for ν = 1, 2, …

a.e. in Ω. Hence f has finite distortion and

H(x, f) ≤ K_{M}(x, f)^{2/n} ≤ (\frac{1}{2}(M(x) + M(x)^{n−1}))^{2/n} ≤ M(x)^{2−(2/n)}

a.e. in Ω by (4.4) and Theorem 1.15.

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Received 13 March 1998