-
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
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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
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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
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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).
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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].
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
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B. V. Bojarski, V. V. Gutlyanski and V. I. Ryazanov 317
|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
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318 General Beltrami equations...
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
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B. V. Bojarski, V. V. Gutlyanski and V. I. Ryazanov 319
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
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320 General Beltrami equations...
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
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B. V. Bojarski, V. V. Gutlyanski and V. I. Ryazanov 321
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|
-
B. V. Bojarski, V. V. Gutlyanski and V. I. Ryazanov 323
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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324 General Beltrami equations...
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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]
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326 General Beltrami equations...
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]