beltrami equations

i

5 (2008), 3, 305 326

General Beltrami equations and BMO

Bogdan V. Bojarski, Vladimir V. Gutlyanski,

Vladimir I. Ryazanov

Abstract. We study the Beltrami equations f = (z)f + (z)funder the assumption that the coefficients , satisfy the inequality

|| + || < 1 almost everywhere. Sufficient conditions for the existenceof homeomorphic ACL solutions to the Beltrami equations are given in

terms of the bounded mean oscillation by John and Nirenberg.

2000 MSC. 30C65, 30C75.

Key words and phrases. Degenerate Beltrami Equations, quasicon-formal mappings, bounded mean oscillation.

1. Introduction

Let D be a domain in the complex plane C. We study the Beltramiequation

f = (z)f + (z)f a.e. in D (1.1)

where f = (fx + ify)/2 and f = (fx ify)/2, z = x + iy, and and are measurable functions in D with |(z)| + |(z)| < 1 almosteverywhere in D. Equation (1.1) arises, in particular, in the study ofconformal mappings between two domains equipped with different mea-surable Riemannian structures, see [22]. Equation (1.1) and second orderPDEs of divergent form are also closely related. For instance, given a do-main D, let be the class of symmetric matrices with measurable entries,satisfying

1

K(z)|h|2 (z)h, h K(z)|h|2, h C.

Assume that u W 1,2loc (D) is a week solution of the equation

div[(z)u(z)] = 0.

Received 25.09.2008

ISSN 1810 3200. c I

306 General Beltrami equations...

Consider the mapping f = u + iv where v(z) = Jf (z)(z)u(z) andJf (z) stands for the Jacobian determinant of f . It is easily to verify thatf satisfies the Beltrami equation (1.1). On the other hand, the abovesecond order partial differential equation naturally appears in a numberof problems of mathematical physics, see, e.g., [3].

In the case (z) 0 in (1.1) we recognize the classical Beltrami equa-tion, which generates the quasiconformal mappings in the plane. Givenan arbitrary measurable coefficient (z) with

B. V. Bojarski, V. V. Gutlyanski and V. I. Ryazanov 307

that the equation fz (z) fz (z)fz = 0 degenerates near the origin.It is easy to verify that the radial stretching

f(z) = (1 + |z|2) z|z| , 0 < |z| < 1.

satisfies the above equation and is a homeomorphic mapping of the punc-tured unit disk onto the annulus 1 < |w| < 2. Thus, we observe the effectof cavitation. For the second example we choose

(z) =i

2

z

z, (z) =

i

2

z

ze2i log |z|

2

.

In this case |(z)| + |(z)| = 1 holds for every z C. In other words,we deal with global degeneration. However, the corresponding globallydegenerate general Beltrami equation (1.1) admits the spiral mapping

f(z) = zei log |z|2

as a quasiconformal solution. The above observation shows, that in orderto obtain existence or uniqueness results, some extra constraints must beimposed on and .

In this paper we give sufficient conditions for the existence of a home-omorphic ACL solution to the Beltrami equation (1.1), assuming thatthe degeneration of and is is controlled by a BMO function. Moreprecisely we assume that the maximal dilatation function

K,(z) =1 + |(z)|+ |(z)|1 |(z)| |(z)| (1.3)

is dominated by a function Q(z) BMO, where BMO stands for theclass of functions with bounded mean oscillation in D, see [21].

Recall that, by John and Nirenberg in [21], a real-valued function uin a domain D in C is said to be of bounded mean oscillation in D, u BMO(D), if u L1loc(D), and

u := supB

1

|B|

B

|u(z) uB| dx dy < (1.4)

where the supremum is taken over all discs B in D and


uB =1

|B|

B

u(z) dx dy.

We also write u BMO if D = C. If u BMO and c is a constant, thenu+ c BMO and u = u+ c. The space of BMO functions moduloconstants with the norm given by (1.4) is a Banach space. Note thatL BMO Lploc for all p [1,), see e.g. [21, 30]. Fefferman andStein [13] showed that BMO can be characterized as the dual space ofthe Hardy space H1. The space BMO has become an important conceptin harmonic analysis, partial differential equations and related areas.

The case, when = 0 and the degeneration of is expressed in termsof |(z)|, has recently been extensively studied, see, e.g., [710,16,19,20,23,25,27,32,33,36], and the references therein.

In this article, unless otherwise stated, by a solution to the Beltramiequation (1.1) in D we mean a sense-preserving homeomorphic mappingf : D C in the Sobolev space W 1,1loc (D), whose partial derivativessatisfy (1.1) a.e. in D.

Theorem 1.2. Let , be measurable functions in D C, such that||+ || < 1 a.e. in D and

1 + |(z)|+ |(z)|1 |(z)| |(z)| Q(z) (1.5)

a.e. in D for some function Q(z) BMO(C). Then the Beltrami equa-tion (1.1) has a homeomorphic solution f : D C which belongs to thespace W 1,sloc (D) for all s [1, 2). Moreover, this solution admits a homeo-morphic extension to C such that f is conformal in C\D and f() = .For the extended mapping f1 W 1,2loc , and for every compact set E Cthere are positive constants C,C , a and b such that

C exp

(

a|z z|2)

|f(z) f(z)| C

log1

|z z|

b

(1.6)

for every pair of points z, z E provided that |z z| is sufficientlysmall.

Remark 1.1. Note that C is an absolute constant, b depends only onE and Q.

Remark 1.2. Prototypes of Theorem 1.2 when (z) 0 can be found inthe pioneering papers on the degenerate Beltrami equation [27] and [10],see also [32] and [36].


In [2] it was shown that a necessary and sufficient condition for ameasurable function K(z) 1 to be majorized in D C by a functionQ BMO is that

D

eK(z)dx dy

1 + |z|3 < (1.7)

for some positive number . Thus, the inequality (1.7) can be viewed asa test for K,(z) to satisfy the hypothesis of Theorem 1.2.

2. Auxiliary lemmas

For the proof of Theorem 1.2 we need the following lemmas.

Lemma 2.1. Let fn : D C be a sequence of homeomorphic ACLsolutions to the equation (1.1) converging locally uniformly in D to ahomeomorphic limit function f . If

Kn,n(z) Q(z) Lploc(D) (2.1)

a.e. in D for some p > 1, then the limit function f belongs to W 1,slocwhere s = 2p/(1+ p) and fn and fn converge weakly in L

sloc(D) to the

corresponding generalized derivatives of f .

Proof. First, let us show that the partial derivatives of the sequence fnare bounded by the norm in Ls over every disk B with B D. Indeed,

|fn| |fn| |fn|+ |fn| Q1/2(z) J1/2n (z)

a.e. in B and by the Holder inequality and Lemma 3.3 of Chapter IIIin [24]

fns Q1/2p |fn(B)|1/2

where s = 2p/(1 + p), Jn is the Jacobian of fn and p denotes theLpnorm in B.

By the uniform convergence of fn to f in B, for some > 1 and largen, |fn(B)| |f(B)| and, consequently,

fns Q1/2p |f(B)|1/2.

Hence fn W 1,sloc , see e.g. Theorem 2.7.1 and Theorem 2.7.2 in [26].On the other hand, by the known criterion of the weak compactnessin the space Ls, s (1,), see [12, Corollary IV.8.4], fn f andfn f weakly in Lsloc for such s. Thus, f belongs to W

1,sloc where

s = 2p/(1 + p).


Lemma 2.2. Under assumptions of Lemma 2.1, if n(z) (z) andn(z) (z) a.e. in D, then the limit function f is a W 1,sloc solution tothe equation (1.1) with s = 2p/(1 + p).

Proof. We set (z) = f(z)(z) f(z)(z) f(z) and, assuming thatn(z) (z) and n(z) (z) a.e. in D, we will show that (z) = 0 a.e.in D. Indeed, for every disk B with B D, by the triangle inequality

B

(z) dx dy

I1(n) + I2(n) + I3(n)

where

I1(n) =

B

(

f(z) fn(z))

dx dy

,

I2(n) =

B

((z) f(z) n(z) fn(z)) dx dy

,

I3(n) =

B

(

(z) f(z) n(z) fn(z))

dx dy

.

By Lemma 2.1, fn and fn converge weakly in Lsloc(D) to the corre-

sponding generalized derivatives of f . Hence, by the result on the repre-sentation of linear continuous functionals in Lp, p [1,), in terms offunctions in Lq, 1/p + 1/q = 1, see [12, IV.8.1 and IV.8.5], we see thatI1(n) 0 as n . Note that I2(n) I 2(n) + I 2 (n), where

I 2(n) =

B

(z)(f(z) fn(z)) dx dy

and

I 2 (n) =

B

((z) n(z))fn(z) dx dy

,

and we see that I 2(n) 0 as n because L. In order toestimate the second term, we make use of the fact that the sequence|fn| is weakly compact in Lsloc, see e.g. [12, IV.8.10], and hence |fn| isabsolutely equicontinuous in L1loc, see e.g. [12, IV.8.11]. Thus, for every > 0 there is > 0 such that


E

|fn(z)| dx dy < , n = 1, 2, . . . ,

whenever E is measurable set in B with |E| < . On the other hand, bythe Egoroff theorem, see e.g. [12, III.6.12], n(z) (z) uniformly onsome set S B such that |E| < where E = B\S. Now |n(z)(z)| < on S for large n and consequently

I 2 (n)

S

|(z) n(z)| |fn(z)| dx dy

+

E

|(z) n(z)| |fn(z)| dx dy

B

|fn(z)| dx dy + 2

E

|fn(z)| dx dy

(

Q1/2 |f(B)|1/2 + 2)

for large enough n, i.e. I 2 (n) 0 as n because > 0 is arbi-trary. The fact that I3(n) 0 as n is handled similarly. Thus,

B (z) dx dy = 0 for all disks B with B D. By the Lebesgue theo-rem on differentiability of the indefinite integral, see e.g. [34, IV(6.3)],(z) = 0 a.e. in D.

Remark 2.1. Lemma 2.1 and Lemma 2.2 extend the well known con-vergence theorem where Q(z) L, see Lemma 4.2 in [6], and [4].

Recall that a doubly-connected domain in the complex plane is calleda ring domain and the modulus of a ring domain E is the number mod Esuch that E is conformally equivalent to the annulus {1 < |z| < e mod E}.We write A = A(r,R; z0), 0 < r < R < , for the annulus r < |z z0| n, it follows that

|S1| N

n=1

An

|Q(z)Qn||z|2(log |z|)2 dx dy

N

n=1

e2

n2

(

1

|Bn|

Bn

|Q(z)Qn| dx dy)

.

Hence,

|S1| 2e2Q. (2.16)

Now, note that


|Qk Qk1| =1

|Bk|

Bk

(Q(z)Qk1) dx dy

1|Bk|

Bk

|Q(z)Qk1| dx dy

=e2

|Bk1|

Bk

|Q(z)Qk1| dx dy

e2

|Bk1|

Bk1

|Q(z)Qk1| dx dy e2Q.

Thus, by the triangle inequality,

Qn Q1 +n

k=2

|Qk Qk1| Q1 + ne2Q, (2.17)

and, since

An

dx dy

|z|2(log |z|)2 1

n2

An

dx dy

|z|2 =2

n2,

it follows by (2.15), that

S2 2N

n=1

Qnn2

2Q1N

n=1

1

n2+ 2e2Q

N

1

1

n. (2.18)

Finally,N

n=1 1/n2 is bounded, and

Nn=1 1/n < 1 + logN < 1 +

log log 1/t, and, thus, (2.12) follows from (2.13), (2.16) and (2.18).

Lemma 2.5. Let f : D C be a quasiconformal mapping with complexdilatation (z) = fz(z)/fz(z), such that Kf (z) = K,0(z) Q(z) BMO a.e. in D. Then for every annulus A(r,Re1; z0), r < Re

2,contained in D,

M(f()) clog log(R/r)

(2.19)

where stands for the family of curves joining the boundary componentsof A(r,Re1; z0) in A(r,Re

1; z0) and c is the constant in Lemma 2.4associated with the function Q(Rz + z0).

Proof. Since

ds

|z z0| log(R/|z z0|)

R/e

r

dt

t logR /t= log logR /r = a


we see that the function (|z z0|) = 1/a(|z z0| logR/|z z0|) isadmissible for the family . By Lemma 2.3, and the inequality

D,z0(z) K,0(z) Q(z),

we get that

M(f()) 1a2

r


Proof of Theorem 1.2. We split the proof of Theorem 1.2 into threeparts. Given , , we first generate a sequence of quasiconformal map-pings, corresponding to a suitable truncation of the above Beltrami co-efficients, and show, making use of Lemma 2.5, that the chosen sequenceis normal with respect to the locally uniform convergence. Then weprove that the limit mappings are univalent, belong to the Sobolev spaceW 1,sloc (D), s [1, 2), and satisfy the differential equation (1.1) a.e. in D.Finally we deduce the regularity properties of the required solution tothe equation (1.1).

n01. Let , , be Beltrami coefficients defined in D with ||+ || < 1a.e. in D. For n = 1, 2, . . . , we set in Dn = D

B(n)

n(z) = (z), if |(z)| 1 1/n, (3.1)

n(z) = (z), if |(z)| 1 1/n, (3.2)

and n(z) = n(z) = 0 otherwise, including the points z B(n) \ Dn.Here B(n) stands for the disk |z| < n. The coefficients n, n noware defined in the disk B(n) and satisfy the strong ellipticity condition|n(z)| + |n(z)| qn < 1. Therefore, by Theorem 1.1, there exists aquasiconformal mapping fn(z) = n(z/n)/|n(1/n)| of B(n) onto B(Rn)for some Rn = 1/|n(1/n)| > 1 satisfying a.e in B(n) the equation

fnz n(z)fnz n(z)fnz = 0 (3.3)

and normalized by fn(0) = 0, |fn(1)| = 1. We extend fn over B(n) tothe complex plane C by the symmetry principle. It implies, in particular,that fn() = . We will call such fn the canonical approximating se-quence. It follows from (3.3) and the symmetry principle that fn satisfiesa.e. in C the Beltrami equation

fnz = n(z)fnz

where


n(z) =

{

n(z), if z B(n),n(n2/z)z

2/z2, if z C \B(n),

and

n(z) = n(z) + n(z) fnzfnz

.

Note that K,(z) Q(z) a.e. in B(n).Our immediate task now is to show that the canonical approximating

sequence of quasiconformal mappings fn : C C forms a normal familyof mappings with respect to the locally uniform convergence in C. To thisend, we first prove that the family is equicontinuous locally uniformly inC. More precisely, we show that for every compact set E C

|fn(z) fn(z)| C(

log1

|z z|

)

, (3.4)

for every n N and z, z E such that |z z| is small enough. HereC is an absolute positive constant and > 0 depends only on E and Q.

Indeed, let E be a compact set of C and z, z E be a pair of pointssatisfying |z z| < e4. If we choose N such that dist(E, B(N)) > 1,then we see that the annulus

A = {z C : |z z| < |z z| < |z z|1/2 e1}

is contained in B(N). Moreover, at least one of the points 0 or 1 liesoutside of the annulus A and belongs to the unbounded component ofits complement.

Let be the family of curves joining the circles |zz| = |zz| = rand |z z| = |z z|1/2e1 = Re1 in A. The complement of the ringdomain fn(A) to the complex plane has the bounded and unboundedcomponents n and n, respectively. Then, by the well-known Gehringslemma, see [14],

M(fn()) 2

log(/nn)

where n and n stand for the spherical diameters of n and n. Since

for small enough |z z| n 1/2, we get that

n 2e2/M(fn())

where is an absolute constant. On the other hand, by Lemma 2.5, wehave


M(fn()) c

log log(1/|z z|1/2) (3.5)

where the positive constant c depends only on E and Q. If |z z| issmall enough, then 2n |fn(z) fn(z)| and hence

|fn(z) fn(z)| 22e2/M(fn()). (3.6)

Combining the estimate (3.6) with the inequality (3.5), we arrive at (3.4).

The required normality of the family {fn} with respect to the spher-ical metric in C now follows by the AscoliArzela theorem, see e.g. [37,20.4]. Thus, we complete the first part of the proof.

n02. Now we show that the limit mapping f is injective. To thisend, without loss of generality, we may assume that the sequence fnconverges locally uniformly in C to a limit mapping f which is not aconstant because of the chosen normalization. Since the mapping degreeis preserved under uniform convergence, f has degree 1, see e.g., [15].We now consider the open set V = {z C : f is locally constant at z}.First we show that if z0 C \ V, then f(z) 6= f(z0) for z C \ {z0}.Picking a point z 6= z0, we choose a small positive number R so that|z z0| > R/e. Then, by Lemma 2.5

mod fn(A(r,R/e; z0) =2

M(fn()) 2

clog log(R/r) > C0

for sufficiently small 0 < r < R/e2, where C0 is the constant in Lem-ma 2.6. By virtue of Lemma 2.6, we can find an annulus An = {w : rn 1 and n = 1, 2, . . . , andn(z) (z) and n(z) (z) a.e. in D and to 0 in C \D as n .Then, by Lemma 2.2, we arrive at the conclusion that the limit mappingf is homeomorphic solution for the equation f = (z)f + f in D ofthe class W 1,sloc (D), s = 2p/(1 + p), and moreover this solution f admitsa conformal extension to C \D. Furthermore, the infinity is the remov-able singularity for the limit mapping by Theorem 6.3 in [32]. Thus, themapping f admits extension to a self homeomorphism of C, f() = ,which is conformal in C \D.

n03. The mappings fn, n = 1, 2, . . . , are homeomorphic and thereforegn := f

1n g := f1 as n locally uniformly in C, see [11, p. 268].

By the change of variables, that is correct because fn and gn W 1,2loc , weobtain under large n

D

|gn|2 du dv =

gn(D)

dx dy

1 |n(z)|2

D

Q(z) dx dy <

for bounded domains D C and relatively compact sets D C withg(D) D. The latter estimate means that the sequence gn is boundedin W 1,2(D) for large n and hence g W 1,2loc (C). Moreover, gn gand gn g weakly in L2loc, see e.g. [31, III.3.5]. The homeomorphismg has (N)property because g W 1,2loc , see e.g. [24, Theorem 6.1 ofChapter III], and hence Jf (z) 6= 0 a.e., see [28].

Finally, the right inequality in (1.6) follows from (3.4). In order toget the left inequality we make use of the length-area argument, see,e.g. [35], p. 75. Let E be a compact set in C and E = f(E). Next, letw, w be a pair of points in E with |w w| < 1. Consider the familyof circles {S(w, r)} centered at w of radius r,

r1 = |w w| < r < r2 = |w w|1/2.

Since g = f1(w) W 1,2loc (C), we can apply the standard oscillationestimate

r2

r1

osc2(g, S(w, r)) drr

c

|ww|


infr(r1,r2)

osc(g, S(w, r)) c1 log1/21

|w w| .

The mapping g is a homeomorphism, so osc(g,B(w, r))osc(gn, S(w, r))for every r (r1, r2) where B(w, r) = {w : |w w| < r}. Thus, we getthe inequality

|g(w) g(w)| < C1 log1/21

|w w| . (3.7)

Setting w = f(z) and w = f(z), we arrive at the required estimate

|f(z) f(z)| > Cea/|zz|2 . (3.8)

The last result can be deduced from Gehrings oscillation inequality, see,e.g., [14].

Remark 3.1. The first two parts of the proof for Theorem 1.2 are basedon Lemma 2.1, Lemma 2.2, the modulus estimate

mod fn(A(r,R/e; z0)) C log log(R/r), (3.9)

as well as on the fact that the right hand side of (3.9) approaches asr 0. Recall that the proof of inequality (3.9) is based on Lemma 2.3and the estimate (1.5). More refined results, based on Lemma 2.3, canbe obtained for the degenerate Beltrami equation (1.1) if we replace thebasic assumption (1.5) by another one, say, by the inequality

(

1 (z) zz0zz0

+ |(z)|

)2

1 (|(z)|+ |(z)|)2 Qz0(z) (3.10)

where Qz0(z) BMO for every z0 D. We also can replace (1.5) by theinequality

D

eH(K,(z)) dx dy

(1 + |z|2)2 < M (3.11)

where H stands for a dominating factor of divergence type, see for details[16]. Notice, that typical choices for H(x) are x and x/(1+ log+ x) fora positive constant . However, we will not pursue these directions hereand have an intention to publish the corresponding results elsewhere.

Acknowledgments. The research of the third author was partiallysupported by the Ukrainian State Foundation of Fundamental Investiga-tions (FFI), Grant number F25.1/055.


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Contact information

Bogdan V. Bojarski Institute of MathematicsPolish Academy of Sciences,ul. Sniadeckich 8, P.O. Box 21,00-956 Warszawa,PolandE-Mail: [email protected]


Vladimir Ya.

Gutlyanski,

Vladimir I.

Ryazanov

Institute of Applied Mathematicsand Mechanics, NAS of Ukraine,ul. Roze Luxemburg 74,83114, Donetsk,UkraineE-Mail: [email protected],

[email protected]

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