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beltrami equations paper

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  • 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

  • 308 General Beltrami equations...

    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].

  • B. V. Bojarski, V. V. Gutlyanski and V. I. Ryazanov 309

    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).

  • 310 General Beltrami equations...

    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

  • B. V. Bojarski, V. V. Gutlyanski and V. I. Ryazanov 311

    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].