Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 423–438

ON THE Γ-CONVERGENCE OF

LAPLACE–BELTRAMI OPERATORS IN THE PLANE

Maria Rosaria Formica

Università di Napoli, Dipartimento di Matematica e Applicazioni via Cintia, IT-80126 Napoli, Italy; formica@matna2.dma.unina.it

Abstract. We show here that if f_{h} is a sequence of mappings of finite distortion K_{h}, uniformly bounded in some exponential norm, weakly converging to f in W^{1,2}(Ω), Ω ⊂ R^{2} , then the matrices A(x, f_{h}) in the Beltrami operators associated to each f_{h}, Γ-converge, in the sense of De Giorgi, to the matrix A(x, f) in the Beltrami operator associated to f.

1. Introduction

For Ω an open subset of R^{2} we shall study mappings f = (f^{1}, f^{2}): Ω → R^{2} in the Sobolev space W^{1,2}(Ω, R^{2}). We say that f has finite distortion if

(1.1)  |Df(x)|^{2} ≤ 𝒦 (x)J(x, f)  a.e.

Here |Df(x)| stands for the Hilbert–Schmidt norm of the differential matrix Df(x) ∈ R^{2×2} and J(x, f) = det Df(x). That is,

|Df(x)|^{2} = ∑_{i,j=1}^{2}|\frac{∂f^{i}}{∂x_{j}} |^{2}  and  J(x, f) = \frac{∂f^{1}}{∂x_{1}} \frac{∂f^{2}}{∂x_{2}} − \frac{∂f^{1}}{∂x_{2}} \frac{∂f^{2}}{∂x_{1}} .

The function 𝒦 = 𝒦 (x) is assumed to be measurable with values in the interval [2, ∞). It will be advantageous to write 𝒦 as

(1.2)  𝒦 (x) = K(x) + \frac{1}{K(x)}  where 1 ≤ K(x) < ∞.

We refer to the smallest such K(x) for which (1.1) holds as the distortion function of f. If K(x) is bounded by a constant, say 1 ≤ K(x) ≤ K a.e., then we say that f is K -quasiregular. An important quantity associated to a mapping with finite distortion is the so called distortion tensor G( · , f): Ω → R^{2×2} , defined by

(1.3)  G(x, f) = \{ \begin{matrix}\frac{D^{t}f(x)Df(x)}{J(x, f)} & if J(x, f) ≠ 0, \\ I & if J(x, f) = 0, \end{matrix}

1991 Mathematics Subject Classification: Primary 30C62; Secondary 35B40, 46E30.

Work performed as part of a National Research Project, MURST 40%(1997).

where D^{t}f(x) stands for the transposed differential.

The distortion inequality (1.1) reads as

(1.4)  \frac{|ξ|^{2}}{K(x)} ≤ ⟨G(x, f)ξ, ξ⟩ ≤ K(x)|ξ|^{2}

and we have det G(x, f) = 1 a.e.

The symmetric matrix function G( · , f) can be viewed as a Riemannian metric on Ω, the pullback of the Euclidean structure via the mapping f. It is obvious that f is conformal with respect to this new metric. This raises an important question: how does G( · , f) change with f? We are particularly concerned with the continuity property of the map f → G( · , f), since many constructions in quasiconformal geometry and elliptic PDE’s rely on limiting processes. The natural convergence of the mapping f_{h}: Ω → R^{2} with finite distortion is that of the weak topology in W^{1,2}(Ω, R^{2}). This, however, does not guarantee convergence of the matrices G(x, f_{h}) to G(x, f) in any familiar sense (compare with Example 6.1 here and also [LV]). Note that the condition det G(x, f_{h}) = 1 is not necessarily preserved under the weak convergence of G(x, f).

S. Spagnolo [S2] first realized that the proper way to overcome this difficulty is by considering the Γ-convergence of the inverse matrices

A(x, f) = G(x, f)^{−1}.

This matrix clearly verifies the bounds at (1.4) as well. See Section 3 for the definition of Γ-convergence.

Spagnolo’s result dealt with the special case of K -quasiregular mappings in which A(x, f) were bounded and uniformly elliptic matrices. In that case Γconvergence is equivalent to the L^{2} -convergence of solutions of the Dirichlet problem. More precisely, given a sequence {A_{h}} of 2 × 2 matrices satisfying

\frac{|ξ|^{2}}{K} ≤ ⟨A_{h}(x)ξ, ξ⟩ ≤ K|ξ|^{2}, K ≥ 1,

we consider the elliptic operators on a bounded open set Ω ⊂ R^{2}

ℒ_{h} = div [A_{h}(x)∇]: W_{0}^{1,2}(Ω) → W^{−1,2}(Ω).

They are certainly invertible. Following [S1], we say that {A_{h}} Γ-converges to A if for every φ ∈ W^{−1,2}(Ω), ℒ_{h}^{−1}(φ) ⇀ ℒ^{−1}(φ) in L^{2}(Ω), where ℒ = div [A(x)∇]. Later these results were generalized to the n-dimensional case by [DD].

In the present paper we extend Spagnolo’s result to sequences of mappings with pointwise unbounded distortion. Our only assumption will be that the distortion functions stay bounded in the EXP_{α} class for a certain α > 1, see Section 2, for the definitions.

The main result is as follows (see Section 5):

Theorem. Let f_{h} converge weakly in W^{1,2}(Ω, R^{2}) to a mapping f, and suppose that their distortion functions K_{h} converge to K weakly in L^{1}(Ω) and satisfy

∫_{Ω} exp(\frac{K_{h}(x)}{λ})^{α} dx ≤ c

for some α > 1, λ > 0 and c > 0. Then f has distortion K and

A(x, f_{h}) ⟶^{Γ_{α}} A(x, f).

For the notion of Γ_{α} -convergence, we refer to the definition in Section 3. In Section 6 we will relate our results to some known convergence theorems

for quasiregular mappings [GMRV], [IK], [Bo].

2. Some Orlicz spaces

Let Ω be a bounded open set in R^{n}. An Orlicz function is a nonnegative continuosly increasing function P: R_{+} → R_{+} , verifying P (0) = 0 and P (∞) = ∞. The Orlicz space L^{P} (Ω) consists of all measurable functions φ: Ω → R such that

∫_{Ω} P (λ^{−1}|φ|) < ∞

for some λ = λ(φ) > 0 (see [RR]).

For α > 1, we denote by EXP_{α}(Ω) the Orlicz space with the defining function P (t) = exp(t^{α}) − 1. It consists of all measurable functions φ on Ω such that

∥φ∥_{EXP_{α}(Ω)} = inf\{λ > 0 : ∫_{Ω} exp(\frac{|φ(x)|}{λ})^{α} dx ≤ 2\} < ∞.

Here

∫_{Ω} ψ = \frac{1}{|Ω|} ∫_{Ω} ψ = ψ_{Ω},

and ∥φ∥_{EXP_{α}(Ω)} provides a norm of φ. Another space of interest to us will be the Zygmund space L^{p} log^{β} L(Ω), with p ≥ 1 and β ≥ 0, with the defining function P (t) = t^{p} log^{β}(e + t). It consists of all measurable functions φ on Ω such that

∫_{Ω} |φ|^{p} log^{β}(e + \frac{|φ|}{|φ|_{Ω}} ) dx < ∞.

Observe that both are Banach spaces and EXP_{α}(Ω) is the dual to L^{1} log^{β} L, when β = 1/α.

The Luxemburg norm of a function φ ∈ L^{p} log^{β} L(Ω) is given by

∥φ∥_{L^{p} log^{β} L(Ω)} = inf\{λ > 0 : ∫_{Ω}(\frac{|φ|}{λ})^{p} log^{β}(e + \frac{|φ|}{λ}) dx ≤ 1\}.

Proposition 2.1 (Generalized Hölder inequality). Let α ≥ 1. Let K(x) ∈ EXP_{α}(Ω), φ ∈ L^{2} log^{1/α} L, and ψ ∈ L^{2} log^{1/α} L. Then

|∫_{Ω} K(x)φ(x)ψ(x)| dx ≤ c∥K∥_{EXP_{α}}∥φ∥_{L^{2} log^{1/α} L}∥ψ∥_{L^{2} log^{1/α} L}.

For P (t) = t^{2} log^{β}(e + t) we denote by W^{1,P} (Ω) the Orlicz–Sobolev space of functions φ ∈ L^{2} log^{β} L whose gradient belongs to the Zygmund space L^{2} log^{β} L. We supply this space with the norm

(2.1)  ∥φ∥_{W^{1,P} (Ω)} = ∥φ∥_{L^{2} log^{β} L(Ω)} + ∥∇φ∥_{L^{2} log^{β} L(Ω)}.

3. The Γ-convergence

We denote by R2×2_{+} the set of symmetric 2×2 matrices A, such that ⟨Aξ, ξ⟩ ≥ 0 for all ξ ∈ R^{2} . Consider measurable functions A: Ω → R2×2_{+} on Ω ⊂ R^{2} satisfying

(3.1)  \frac{|ξ|^{2}}{K(x)} ≤ ⟨A(x)ξ, ξ⟩ ≤ K(x)|ξ|^{2}

for some 1 ≤ K(x) < ∞ a.e. The smallest K(x), for which the above holds, denoted by K_{A}(x), is called the distortion function of A.

The present paper is concerned with mappings whose distortion belongs to the exponential class EXP_{α}(Ω), 1 < α ≤ ∞. For the purpose of this work, we adopt the following variant of De Giorgi’s notion of Γ-convergence ([DF]).

Definition 3.1. Let A and A_{h} (h = 1, 2, …) be matrix functions whose distortions K_{A} and K_{A_{h}} are uniformly bounded in the norm of EXP_{α}(Ω). We say that {A_{h}} Γ_{α} -converges to A if the following two conditions are verified:

(1) The inequality

(3.2)  ∫_{Ω}⟨A(x)∇u, ∇u⟩ dx ≤ lim inf_{h→∞} ∫_{Ω}⟨A_{h}(x)∇u_{h}, ∇u_{h}⟩ dx

holds whenever |∇u_{h}|, |∇u| ∈ L^{2} log^{1/α} L(Ω) and u_{h} → u in L^{2} log^{1/α} L.

(2) For every v ∈ L^{2} log^{1/α} L(Ω) with |∇v| ∈ L^{2} log^{1/α}(Ω) there exists a sequence v_{h} ∈ L^{2} log^{1/α} L(Ω) with |∇v_{h}| ∈ L^{2} log^{1/α} L such that v_{h} → v in L^{2} log^{1/α} L(Ω) and

(3.3)  ∫_{Ω}⟨A(x)∇v, ∇v⟩ = lim_{h} ∫_{Ω}⟨A_{h}∇v_{h}, ∇v_{h}⟩.

Remark. The assumption that K_{A} and K_{A_{h}} belong to EXP_{α}(Ω) is needed to guarantee that the above integrals are finite. This follows from the inequality

(3.4)

∫_{Ω}⟨A(x)∇u, ∇u⟩ dx ≤ ∫_{Ω} K_{A}(x)|∇u|^{2} dx
≤ c∥K_{A}∥_{EXP_{α}(Ω)}∥∇u∥_{L^{2} log^{1/α} L(Ω)}^{2}.

If one merely assumes that K_{A} and K_{A_{h}} ∈ L^{1} then one must be confined to Lipschitz functions. In this case we speak of Γ-convergence. We say that a sequence A_{h} of matrix functions A_{h} ∈ L^{1}(Ω, R2×2_{+}) Γ-converges to A if :

(1) Inequality (3.2) holds whenever u, u_{h} ∈ Lip (Ω) and u_{h} → u in L^{2}(Ω);

(2) For every v ∈ Lip (Ω) one can find a sequence v_{h} ∈ Lip (Ω) converging to v in L^{2}(Ω) satisfying (3.3).

Actually, by the general properties of Γ-convergence, conditions (1) and (2) remain true if we replace Ω by any of its open subsets.

We report here the fundamental compactness result concerning Γ-convergence [MS].

Theorem 3.1. Let A_{h} be a sequence of symmetric 2 × 2 matrices satisfying

0 ≤ ⟨A_{h}(x)ξ, ξ⟩ ≤ K_{h}(x)|ξ|^{2} for a.e. x ∈ Ω and ξ ∈ R^{2}.

Assume that K_{h} ⇀ K weakly in L^{1}(Ω). Then there exists a subsequence A_{h_{r}} Γ-converging to a symmetric matrix A. Moreover, this matrix A also satisfies

0 ≤ ⟨A(x)ξ, ξ⟩ ≤ K(x)|ξ|^{2}.

In this connection it is appropriate to mention another important notion of convergence of matrix functions A_{h}: Ω → R2×2_{+} , the so-called G-convergence. For simplicity we confine ourselves to bounded domains and to sequences such that

(3.5)  1 ≤ K_{A_{h}}(x) ≤ K  a.e.

for h = 1, 2, …, and

1 ≤ K_{A}(x) ≤ K a.e.

We recall from the introduction the elliptic operators and their inverse

ℒ_{h} = div [A_{h}(x)∇]: W_{0}^{1,2}(Ω) → W^{−1,2}(Ω), ℒ_{h}^{−1}: W^{−1,2}(Ω) → W_{0}^{1,2}(Ω),
ℒ = div [A(x)∇]: W_{0}^{1,2}(Ω) → W^{−1,2}(Ω), ℒ^{−1} : W^{−1,2}(Ω) → W_{0}^{1,2}(Ω).

Following Spagnolo [S1], {A_{h}} G-converges to A if ℒ_{h}^{−1}(φ) ⇀ ℒ^{−1}(φ) weakly in W_{0}^{1,2}(Ω), for every φ ∈ W^{−1,2}(Ω). We emphasize that under condition (3.5) all the above notions of convergence are equivalent, though we shall not pursue this matter here, see [MS].

4. Mappings of finite distortion and the Laplace–Beltrami operators

Let Ω be a bounded open set in R^{2} and f = (f^{1}, f^{2}) ∈ W^{1,2}(Ω, R^{2}) be a mapping of finite distortion K: Ω → [1, ∞), i.e. satisfying, for a.e. x ∈ Ω,

(4.1)  |Df(x)|^{2} ≤ [K(x) + K^{−1}(x)]J(x, f),

where J(x, f) is the Jacobian determinant of f. The distortion tensor G(x, f) of f at x is defined in (1.3). It is easy to check that G is a symmetric matrix with det G(x, f) = 1 and that (1.4) is equivalent to (4.1). In fact, for any 2 × 2-matrix F with det F > 0, we can consider

G = \frac{F^{t}F}{det F} .

Then, obviously

det G = 1.

Moreover, recalling the Hilbert–Schmidt norm of F,

|F |^{2} = tr F^{t}F

the distortion inequality

|F |^{2} ≤ (K + \frac{1}{K} ) det F

is equivalent to

tr G ≤ K + \frac{1}{K} .

Let λ and 1/λ be the eigenvalues of G. Then the last inequality means that

λ + \frac{1}{λ} ≤ K + \frac{1}{K};

hence 1/K ≤ λ ≤ K.

Now we consider the inverse matrix

A(x, f) = G(x, f)^{−1}

which obviously satisfies the ellipticity condition

\frac{|ξ|^{2}}{K(x)} ≤ ⟨A(x, f)ξ, ξ⟩ ≤ K(x)|ξ|^{2}.

Connections between mappings of finite distortion and PDEs are established via the Laplace–Beltrami operator ℒ = div [A(x, f)∇]. Note that the components f^{i} (i = 1, 2) solve the equations

(4.2)  \{ \begin{matrix}ℒ [f^{i}] = 0, \\ ⟨A(x, f)∇f^{i}, ∇f^{j}⟩ = δ_{ij}J(x, f), \end{matrix}

see for example [BI] and [HKM]. Planar mappings with unbounded distortion have been recently studied by [D], [IS̆] and most recently by [MM], [BJ], [RSY], [IS]. In particular in [MM] the following higher integrability result, which will be useful to us, was established.

Theorem 4.1. If f ∈ W^{1,2}(Ω) satisfies (4.1) with K ∈ EXP_{α}(Ω), for certain α > 1, then |Df| belongs to L^{2} log^{1/α} L(Ω_{1}) for any Ω_{1} ⊂⊂ Ω and the following inequality holds:

(4.3)  ∥Df∥_{L^{2} log^{1/α} L(Ω_{1})} ≤ c(Ω_{1})∥K∥_{EXP_{α}(Ω)}∥Df∥_{L^{2}(Ω)}.

This is true in all dimensions, provided the exponent 2 is replaced by the dimension n.

In view of Hadamard’s inequality

⟨A(x, f)∇f^{i}, ∇f^{i}⟩ = J(x, f) ≤ \frac{1}{2}|Df(x)|^{2},

we deduce by (4.3)

(4.4)  ∥⟨A(x, f)∇f^{i}, ∇f^{i}⟩∥_{L^{1} log^{1/α} L(Ω_{1})} ≤ c(Ω_{1})∥K∥_{EXP_{α}(Ω)} ∫_{Ω} |Df|^{2} dx.

We show here that the limit mapping f of a weakly convergent sequence of mappings f_{h} with finite distortion also has finite distortion. Our arguments are based on the weak continuity of the Jacobian determinant [R], [Mü] and the concept of polyconvexity. General n-dimensional results of this type have been recently obtained by F.W. Gehring and T. Iwaniec in [GI]. They adopted slightly different definition of the distortion, which for n = 2 reduces to

|Df(x)|^{2} ≤ 2K(x)J(x, f).

Theorem 4.2. Let f_{h}: Ω → R^{2} be mappings of finite distortion K_{h}(x):

(4.5)  |Df_{h}(x)|^{2} ≤ [K_{h}(x) + \frac{1}{K_{h}(x)}]J(x, f_{h}).

Assume that K_{h} are integrable and converge weakly to K in L^{1}(Ω), while f_{h} ⇀ f weakly in W^{1,2}(Ω, R^{2}). Then the above inequality remains valid for the limit map.

Proof. Let us first introduce some useful notation. Set F = (B, E) where the vectors B, E are defined by

E = ∇f^{1}, B = [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] ∇f^{2}

and let

F^{+} = \frac{1}{2}(E + B), F^{−} = \frac{1}{2}(E − B).

It is obvious that

\begin{aligned}J(x, f) & = ⟨B, E⟩ = |F^{+}|^{2} − |F^{−}|^{2} := J(F), \\ |F |^{2} & = 2(|F^{+}|^{2} + |F^{−}|^{2}). \end{aligned}

Hence the distortion inequality

|F |^{2} ≤ (K + \frac{1}{K} )J(F)

is easily seen to be equivalent to

|F^{−}| ≤ \frac{K − 1}{K + 1}|F^{+}|.

This, in turn, is equivalent to

(4.6)  ∥F ∥^{2} ≤ KJ(F),

where we have used another norm of F defined by ∥F ∥ = |F^{+}| + |F^{−}|. Now, assume that F_{h} ⇀ F weakly in L^{2} and

\frac{∥F_{h}∥^{2}}{J(F_{h})} ≤ K_{h}

with K_{h} ⇀ K weakly in L^{1} . The desired conclusion

(4.7)  \frac{∥F ∥^{2}}{J(F)} ≤ K

follows by applying the inequality

(4.8)  \frac{∥F ∥^{2}}{J(F)} ≤ \frac{∥F_{h}∥^{2}}{J(F_{h})} + \frac{2∥F ∥}{J(F)}(∥F ∥ − ∥F_{h}∥) − \frac{∥F ∥^{2}}{J(F)^{2}} [J(F) − J(F_{h})].

The latter is immediate from the convexity of the function (x, y) → x^{2}/y. The well-known weak continuity property of the Jacobians [R], together with the lower semicontinuity of the norm ∥ · ∥, imply (4.7). Here, for simplicity, we have assumed J(F) > 0 and J(F_{h}) > 0. To get rid of this redundant assumption one must replace J(F) by the expression J(F) + ε∥F ∥, J(F_{h}) and then pass to the limit as ε → 0.

5. The convergence theorem

In this section we consider a sequence f_{h} = (f_{h}^{1}, f_{h}^{2}) ∈ W^{1,2}(Ω, R^{2}) of nonconstant mappings with distortion 1 ≤ K_{h}(x) < ∞, that is

(5.1)  |Df_{h}(x)|^{2} ≤ [K_{h}(x) + K_{h}^{−1}(x)]J(x, f_{h}).

Our basic assumptions are:

(i) There exists α > 1 and c_{0} > 0 such that

∥K_{h}∥_{EXP_{α}(Ω)} ≤ c_{0} for h = 1, 2, ….

(ii) K_{h} ⇀ K weakly in L^{1}(Ω).

(iii) f_{h} ⇀ f = (f^{1}, f^{2}) weakly in W^{1,2}(Ω, R^{2}).

By virtue of Theorem 3.1 there exists a subsequence A_{r}(x) = A(x, f_{h_{r}}), r = 1, 2, …, such that

(5.2)  A(x, f_{h_{r}}) ⟶^{Γ} A(x)

where A(x) is a symmetric matrix field satisfying

(5.3)  0 ≤ ⟨A(x)ξ, ξ⟩ ≤ K(x)|ξ|^{2}.

Our aim here is to prove that A(x) can be identified with A(x, f), which is the inverse of the distortion tensor of f:

(5.4)  A(x, f) = [D^{t}f(x)Df(x)]^{−1}J(x, f).

As a byproduct of our proof, we improve the lower bound at (5.3)

K^{−1}(x)|ξ|^{2} ≤ ⟨A(x)ξ, ξ⟩

and show that actually the entire sequence {A(x, f_{h})} Γ-converges to A(x, f).

Theorem 5.1. Under the above assumptions

(5.5)  ∫_{Ω_{1}}⟨A(x)∇f^{i}, ∇f^{i}⟩ dx = lim_{r→∞} ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇f_{h_{r}}^{i}, ∇f_{h_{r}}^{i}⟩ dx

on compact subdomains Ω_{1} ⊂ Ω, for i = 1, 2.

Proof. In fact, we have

(5.6)

∫_{Ω}⟨A(x, f_{h})∇u, ∇u⟩ ≤ ∫_{Ω} K_{h}|∇u|^{2} dx ≤ c∥K_{h}∥_{EXP_{α}(Ω)}∥∇u∥_{L^{2} log^{1/α} L(Ω)}^{2}
≤ cc_{0}∥u∥_{W^{1,L^{2} log^{1/α} L}(Ω)}^{2}.

It then follows that the functionals (∫_{Ω}⟨A(x, f_{h})∇u, ∇u⟩ dx)^{1/2} are equilipschitz in W^{1,P} (Ω) with P (t) = t^{2} log^{1/α}(e + t), a legitimate reason for passing from Γ-convergence to the stronger one

(5.7)  A(x, f_{h_{r}}) ⟶^{Γ_{α}} A(x);

see [MS] for details.

For i = 1, 2 fixed, set for simplicity u_{r} = f_{h_{r}}^{i} and u = f^{i}. Note that u_{r} → u in L^{2} log^{1/α} L(Ω_{1}). Let now (v_{r}) be a sequence in W^{1,P} (Ω_{1}) such that v_{r} → u in L^{2} log^{1/α} L(Ω_{1}) and

lim_{r→∞} ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇v_{r}, ∇v_{r}⟩ dx = ∫_{Ω_{1}}⟨A(x)∇u, ∇u⟩ dx.

432 Maria Rosaria Formica Let Ω′ be an arbitrary compact subdomain of Ω_{1} and φ ∈ C_{0}^{∞}(Ω_{1}) be such that 0 ≤ φ ≤ 1, φ ≡ 1 in Ω′; then for every t ∈]0, 1[

∫Ω_{1}⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩ dx

≤ ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇(φv_{r} + (1 − φ)u_{r}), ∇(φv_{r} + (1 − φ)u_{r})⟩ dx

= ∫_{Ω_{1}}⟨A(x, f_{h_{r}})\{\frac{t}{t}(∇φ)(v_{r} − u_{r}) + \frac{1 − t}{1 − t}(φ∇v_{r} + (1 − φ)∇u_{r})\},

\{\frac{t}{t}(∇φ)(v_{r} − u_{r}) + \frac{1 − t}{1 − t}(φ∇v_{r} + (1 − φ)∇u_{r})\}⟩ dx

≤ t ∫_{Ω_{1}}⟨A(x, f_{h_{r}})\{\frac{1}{t}(∇φ)(v_{r} − u_{r})\}, \{\frac{1}{t}(∇φ)(v_{r} − u_{r})\}⟩ dx

+ (1 − t) ∫_{Ω_{1}}⟨A(x, f_{h_{r}})\{\frac{1}{1 − t}(φ∇v_{r} + (1 − φ)∇u_{r})\},

\{\frac{1}{1 − t}(φ∇v_{r} + (1 − φ)∇u_{r})\}⟩

≤ \frac{1}{t} ∫_{Ω_{1}} K|Dφ|^{2}|v_{r} − u_{r}|^{2} dx + \frac{1}{1 − t} ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇v_{r}, ∇v_{r}⟩φ dx

+ \frac{1}{1 − t} ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩(1 − φ) dx.

This yields

(1 − t) ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩ dx ≤ \frac{1 − t}{t}c∥v_{r} − u_{r}∥_{L^{2} log^{1/α} L}^{2} · ∥Dφ∥_{L^{∞}(Ω_{1})}^{2}
+ ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇v_{r}, ∇v_{r}⟩φ dx
+ ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩(1 − φ) dx.

The final estimate reads as

\begin{aligned}∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇v_{r}, ∇v_{r}⟩φ dx & ≥ ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩(1 − t − 1 + φ) dx \\ & − \frac{1 − t}{t}c∥Dφ∥_{L^{∞}(Ω_{1})}^{2} · ∥v_{r} − u_{r}∥_{L^{2} log^{1/α} L}^{2}. \end{aligned}

Now, passing to the limit as r → ∞, we obtain

∫_{Ω_{1}}⟨A(x)∇u, ∇u⟩ dx ≥ lim sup_{r→∞} ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩(φ − t) dx.

We let the parameter t go to zero

∫_{Ω_{1}}⟨A(x)∇u, ∇u⟩ ≥ lim sup_{r→∞} ∫_{Ω_{1}}⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩φ

≥ lim inf_{r→∞} ∫_{Ω′} ⟨A(x, f_{h_{r}})∇u_{r}, ∇u_{r}⟩ ≥ ∫_{Ω′} ⟨A(x)∇u, ∇u⟩.

Since Ω′ was arbitrary, we get (5.5).

Now we are in a position to rigorously state and prove our main result.

Theorem 5.2. Under the conditions (i), (ii), and (iii), the limit mapping f is either constant or, if not, has finite distortion K(x) and

(5.8)  A(x, f_{h}) ⟶^{Γ_{α}} A(x, f).

Proof. That f has finite distortion K(x) was already established in Section 4. Since we wish to identify the Γ_{α} -limit of A(x, f_{h}), we can assume that in (5.2) and (5.5) the convergence of the entire sequence holds.

As in the proof of Theorem 5.1, set u_{h} = f_{h}^{i}, u = f^{i}, for i = 1, 2 and A_{h}(x) = A(x, f_{h}).

For the compact subdomain Ω_{1} ⊂ Ω consider step functions

(5.9)  φ = ∑_{j=1}^{ν} λ_{j}χ_{B_{j}},  λ_{j} ≥ 0,

where B_{j} are pairwise disjoint open subsets of Ω_{1} such that |Ω_{1} \ ⋃_{j=1}^{ν} B_{j}| = 0. From (5.5) it follows that

(5.10)  lim inf_{h→∞} ∫_{Ω_{1}}⟨A_{h}(x)∇u_{h}, ∇u_{h}⟩φ dx ≥ ∫_{Ω_{1}}⟨A(x)∇u, ∇u⟩φ dx.

Moreover, by an approximation, this also holds if φ is a nonnegative continuous function on \overline{Ω}_{1} .

Let us now prove more, namely, that (5.10) holds as equality for every continuous function φ in \overline{Ω}_{1} , not necessarily nonnegative.

Applying (4.4), we infer that the sequence J(x, f_{h}) = ⟨A_{h}(x)∇u_{h}, ∇u_{h}⟩ admits a subsequence weakly converging in L^{1}(Ω_{1}) to a function E(x). Thus

(5.11)  lim_{r→∞} ∫_{Ω_{1}}⟨A_{h_{r}}(x)∇u_{h_{r}}(x), ∇u_{h_{r}}(x)⟩φ(x) dx = ∫_{Ω_{1}} E(x)φ(x) dx

for any φ ∈ C^{0}( \overline{Ω}_{1}). By (5.10) it follows

(5.12)  ∫_{Ω_{1}}⟨A(x)∇u, ∇u⟩φ(x) dx ≤ ∫_{Ω_{1}} E(x)φ(x) dx.

Let S be a measurable subset of Ω_{1} and let (φ_{k}) ⊂ C^{0}( \overline{Ω}_{1}) be such that φ_{k}(x) → χ_{s}(x) a.e. in Ω_{1} . Then from the previous relation and the Lebesgue theorem it follows that

(5.13)  ∫_{S}⟨A(x)∇u, ∇u⟩ ≤ ∫_{S} E(x) dx.

On the other hand we deduce from (5.11) and Theorem 5.1 that

(5.14)  ∫_{Ω_{1}}⟨A(x)∇u, ∇u⟩ dx = ∫_{Ω_{1}} E(x) dx.

Hence

E(x) = ⟨A(x)∇u, ∇u⟩ a.e. in Ω_{1}.

Therefore, we have for the whole sequence

(5.15)  lim_{h→∞} ∫_{Ω_{1}}⟨A(x, f_{h})∇u_{h}, ∇u_{h}⟩φ dx = ∫_{Ω_{1}}⟨A(x)∇u, ∇u⟩φ dx

for every φ ∈ C^{0}( \overline{Ω}_{1}).

Now we recall from (4.2) that

(5.16)  ⟨A(x, f_{h})∇f_{h}^{i}(x), ∇f_{h}^{j}(x)⟩ = J(x, f_{h})δ_{ij}  a.e. on Ω, i, j = 1, 2.

By the symmetry of the matrix A(x, f_{h}), (5.15), (5.16) and the weak continuity property of Jacobian ([R]) we have

(5.17)

∫_{Ω_{1}}⟨A(x)∇f^{i}, ∇f^{j}⟩φ dx = lim_{h→∞} ∫_{Ω_{1}}⟨A(x, f_{h})∇f_{h}^{i}, ∇f_{h}^{j}⟩φ dx

= lim_{h→∞} ∫_{Ω_{1}} J(x, f_{h})δ_{ij}φ dx = ∫_{Ω_{1}} J(x, f)δ_{ij}φ dx,

where φ ∈ C_{0}^{∞}(Ω_{1}), i, j = 1, 2. Since φ was arbitrary, it follows that

(5.18)  ⟨A(x)∇f^{i}(x), ∇f^{j}(x)⟩ = J(x, f)δ_{ij}  a.e. in Ω_{1}, i, j = 1, 2,

and consequently, as J(x, f) is a.e. positive,

(5.19)  A(x) = J(x, f)[Df(x)^{t} · Df(x)]^{−1}  a.e. in Ω_{1}.

Since Ω_{1} was arbitrary, (5.18) holds a.e. in Ω. Hence (5.8) holds .

6. The Bers–Bojarski theorem

For the sake of brevity we will now confine ourselves to the particular case K(x) = K ≥ 1 and relate our results to some classical convergence theorems for quasiregular mappings.

Let G(x, f) be defined as in (1.3). No natural continuity result can be traced for the map

(6.1)  f → G(x, f)

of the type obtained in the present paper for the map

f → A(x, f)

unless we consider a convergence f_{h} → f stronger than weak-W^{1,2} ; see also [LV], [D].

Example 6.1. Let ψ_{h}: R → R be a sequence of bounded measurable functions such that 0 < K^{−1} ≤ ψ_{h}(t) ≤ K and

ψ_{h} ⇀ 1, \frac{1}{ψ_{h}} ⇀ \frac{1}{c} (c ≠ 1),

in σ(L^{∞}, L^{1}); for example, let us choose

ψ_{h}(t) = 1 + δ \frac{sin ht}{| sin ht|} (0 < δ < 1).

Then, the sequence of K -quasiregular mappings

f_{h}(x_{1}, x_{2}) = (∫_{0}^{x_{1}} ψ_{h}(t) dt, x_{2})

converges locally uniformly to the identity mapping f(x_{1}, x_{2}) = (x_{1}, x_{2}). It is immediate that the distortion tensor of f_{h} is

G(x, f_{h}) = ( \begin{matrix}ψ_{h}(x_{1}) & 0 \\ 0 & (ψ_{h}(x_{1}))^{−1} \end{matrix} )

and the distortion tensor of the limit f is

G(x, f) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) .

The sequence G(x, f_{h}) does not converge weakly nor does it Γ-converge to the identity matrix G(x, f). Actually

G(x, f_{h}) ⇀ ( \begin{matrix}1 & 0 \\ 0 & c^{−1} \end{matrix} ) weakly in L^{1}(Ω, R^{2×2}).

Moreover it can be proved that

G(x, f_{h}) ⟶^{Γ} ( \begin{matrix}c & 0 \\ 0 & c^{−1} \end{matrix} ) .

Thus, of the two matrices A(x, f), G(x, f) only the first one exhibits a suitable continuity behaviour as a function of f.

In the following we deduce by our results a well-known theorem of Bers– Bojarski for planar K -quasiregular mappings whose n-dimensional version has been recently proved in [GMRV] (see also [IK]). The result states that if f_{h}: Ω ⊂ R^{2} → R^{2} verify a.e. in Ω (K ≥ 1)

|Df_{h}(x)|^{2} ≤ (K + \frac{1}{K} )J(x, f_{h});

if f_{h} → f locally uniformly and the distortion tensors G(x, f_{h}) defined as in (1.3) converge a.e. to G_{0}(x) then G_{0}(x) = G(x, f). Namely we have the following

Theorem 6.1. Let f_{h} be a sequence of mappings of finite distortion K ≥ 1 on Ω such that

(i) f_{h} ⇀ f in W^{1,2}(Ω),

(ii) G(x, f_{h}) → G_{0}(x) a.e. in Ω.

Then

G_{0}(x) = G(x, f) a.e. in Ω.

We start with

Lemma 6.1. Let A_{h} be a sequence of symmetric 2 × 2 matrices satisfying

\frac{|ξ|^{2}}{K} ≤ ⟨A_{h}(x)ξ, ξ⟩ ≤ K|ξ|^{2} for a.e. x ∈ Ω.

If

A_{h}^{−1} → A_{0}^{−1} in L^{1}(Ω, R^{2×2})

and

(6.2)  A_{h} ⟶^{Γ} A

then

A = A_{0}.

Proof. It is easy to check that

A_{h} − A_{0} = A_{h}(A_{0}^{−1} − A_{h}^{−1})A_{0}.

So by our assumptions we deduce

A_{h} → A_{0} in L^{1}(Ω, R^{2×2}).

Since it is well known that strong L^{1} convergence of coefficients matrices imply Γ-convergence [S1], we get

A_{h} ⟶^{Γ} A_{0}

and therefore, by (6.2)

A = A_{0}.

Proof of Theorem 6.1. Theorem 5.2 implies that A(x, f_{h}) ⟶^{Γ} A(x, f). By (ii) and Vitali’s theorem we deduce

G(x, f_{h}) = A(x, f_{h})^{−1} ⟶^{L^{1}} G_{0}(x) = A_{0}^{−1}(x)

so Lemma 6.1 implies A(x, f) = A_{0}(x) = G_{0}^{−1}(x) and this means A^{−1}(x, f) = G_{0}(x), that is G(x, f) = G_{0}(x).

Actually, L^{1} -convergence of the coefficient matrix A_{h} to A implies strong convergence in W_{loc}^{1,2} of local solutions u_{h} of the equation

div A_{h}(x)∇u_{h} = 0

to local solutions u of

div A(x)∇u = 0

(see [S1, Theorem 5]). So, in particular, under our assumptions we deduce f_{h}^{i} → f^{i} in W_{loc}^{1,2} , for i = 1, 2, due to the fact that div A_{h}(x, f_{h})∇f_{h}^{i} = 0.

Aknowledgments. I wish to thank Professors Tadeusz Iwaniec and Carlo Sbordone for their generous advice.

References

[Bo] Bojarski, B.: Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coefficients. - Mat. Sb. 43 (85), 1957, 451–503 (Russian).

[BI] Bojarski, B., and T. Iwaniec: Analytical foundations of the teory of quasiconformal mappings in R^{n}. - Ann. Acad. Sci. Fenn. Math. 8, 1983, 257–324.

[BJ] Brakalova, M.A., and J.A. Jenkins: On solutions of the Beltrami equation. - J. Anal. Math. 76, 1998, 67–92.

[D] David, G.: Solutions de l’équation de Beltrami avec ∥μ∥_{∞} = 1. - Ann. Acad. Sci. Fenn. Math. 13, 1988, 25–70.

438 Maria Rosaria Formica

[DD] De Arcangelis, R., and P. Donato: On the convergence of Laplace–Beltrami operators associated to quasiregular mappings. - Studia Math. LXXXVI, 1987, 189–204.

[DF] De Giorgi, E., and T. Franzoni: Su un tipo di convergenza variazionale. - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) 58, 1975, 842–850.

[ET] Ekeland, I., and R. Teman: Analyse convexe et probleme variationnels. - Dunod-Gauthier Villars, Paris, 1974.

[GI] Gehring, F.W., and T. Iwaniec: The limit of mappings with finite distortion. - Ann. Acad. Sci. Fenn. Math. 24, 1999, 253–264.

[GIM] Greco, L., T. Iwaniec, and G. Moscariello: Limits of the improved integrability of the volume forms. - Indiana Univ. Math. J. 44, 1995, 305–339.

[GMRV] Gutlyanskiĭ, V.Ya., O. Martio, V.I. Ryazanov, and M. Vuorinen: On convergence theorems for space quasiregular mappings. - Forum Math. 10, 1998, 353–375

[HKM] Heinonen, J., T. Kilpeläinen, and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Clarendon Press, Oxford, 1993.

[I] Iwaniec, T.: Geometric Function Theory and Partial Differential Equations. - Lectures in Seillac, France, May 27 – June 2, 1995, 1–41.

[IK] Iwaniec, T., and R. Kopiecki: Stability in the differential equations for quasi regular mappings. - In: Lecture Notes in Math. 798, 1980, 203–214.

[IS] Iwaniec, T., and C. Sbordone: Div-curl fields of finite distortion. - C. R. Acad. Sci. Paris Sér. I Math. 327, 1998, 729–734.

[IS̆] Iwaniec, T., and V. S̆verak: On mappings with integrable dilatation. - Proc. Amer.

Math. Soc. 118, 1993, 181–188.

[LV] Lehto, O., and K. I. Virtanen: Quasiconformal mappings in the plane. - Springer-Verlag, 1973.

[MM] Migliaccio, L., and G. Moscariello: Mappings with unbounded dilation. - Ricerche Mat. (to appear).

[MS] Marcellini, P., and C. Sbordone: An approach to the asymptotic behaviour of elliptic-parabolic operators. - J. Math. Pures Appl. 56, 1977, 157–182.

[Mü] Müller, S.: Higher integrability of determinants and weak convergence in L^{1} . - J.

Reine Angew. Math. 412, 1990, 20–34.

[R] Reshetnyak, G.Yu.: On the stability of conformal mappings in multidimensional spaces. - Sibirsk. Mat. Zh. 8, 1967, 91–114 (Russian).

[RR] Rao, M.M., and Z.D. Ren: Theory of Orlicz Spaces. - Pure and Applied Math. 146, Dekker, 1991.

[RSY] Ryazanov, V.I., U. Srebro, and E. Yakubov: BMO-quasiconformal mappings. - Preprint 155, University of Helsinki, August 1997.

[S1] Spagnolo, S.: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. - Ann. Scuola Norm. Sup. Pisa 22, 1968, 571–597.

[S2] Spagnolo, S.: Some convergence problems. - Sympos. Math. 18, 1976, 391–397.

Received 22 January 1999