On the theory of the Beltrami equation - Springer Link

Ukrainian Mathematical Journal, Vol. 58, No. 11, 2006

ON THE THEORY OF THE BELTRAMI EQUATION V. Ryazanov,1 U. Srebro,2 and E. Yakubov3

UDC 517.5

We study ring homeomorphisms and, on this basis, obtain a series of theorems on the existence of the socalled ring solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami equations that extends earlier results is formulated. In particular, we give new existence 1,1 1,2 with f −1 ∈ Wloc in terms of tangential criteria for homeomorphic solutions f of the class Wloc dilatations and functions of finite mean oscillation. The ring solutions also satisfy additional capacity inequalities.

1. Introduction As is known, the Beltrami equation plays an important role in the mapping theory. The main goal of this paper is to present general principles that allow one to obtain variety of conditions for the existence of homeomorphic ACL solutions in the degenerate case. Our existence theorems are proved by an approximation method. Let D be a domain in the complex plane C, i.e., an open connected subset of C, and let μ : D → C be a measurable function with |μ(z)| < 1 a.e. The Beltrami equation is an equation of the form fz = μ(z) · fz ,

(1)

where fz = ∂f = (fx + ify )/2, fz = ∂f = (fx − ify )/2, z = x + iy, and fx and fy are partial derivatives of f with respect to x and y, respectively. The function μ is called the complex coefficient and Kμ (z) =

1 + |μ(z)| 1 − |μ(z)|

(2)

is called the maximal dilatation, or, briefly, the dilatation, of Eq. (1). The Beltrami equation (1) is said to be degenerate if ess sup Kμ (z) = ∞.

(3)

An ACL homeomorphism f : D → C is called a ring solution of the Beltrami equation (1) if f satisfies (1) 1,2 , and f is a ring Q-homeomorphism at every point z0 ∈ D with Q(z) = KμT (z, z0 ), where a.e., f −1 ∈ Wloc the function 2 1 − z − z0 μ(z) z − z0 (4) KμT (z, z0 ) = 1 − |μ(z)|2 1

Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Technion Israel Institute of Technology, Haifa, Israel. 3 Holon Academic Institute of Technology, Holon, Israel. 2

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1571–1583, November, 2006. Original article submitted June 29, 2006. 1786

0041–5995/06/5811–1786

c 2006

Springer Science+Business Media, Inc.

O N THE T HEORY OF THE B ELTRAMI E QUATION

1787

is called the tangential dilatation with respect to z0 (cf. [1–3]). For the definition of ring Q-homeomorphisms, see Sec. 2. These terms were first introduced in [4]. Recall that a mapping f : D → C is absolutely continuous on lines (or, briefly, f ∈ ACL) if, for every closed rectangle R in D whose sides are parallel to the coordinate axes, f |R is absolutely continuous on almost all line segments in R that are parallel to the sides of R. In particular, f ∈ ACL if it belongs to the Sobolev class 1,1 Wloc (see, e.g., [5, p. 8]). Note that if f ∈ ACL, then f has partial derivatives fx and fy a.e., and, thus, by the well-known Gehring–Lehto theorem, every ACL homeomorphism f : D → C is totally differentiable a.e. (see [6] or [7, p. 128]). For a sense-preserving ACL homeomorphism f : D → C, the Jacobian Jf (z) = |fz |2 − |fz |2 is nonnegative a.e. (see [7, p. 10]). In this case, the complex dilatation μf of f is the ratio μ(z) = fz /fz if fz = 0 and μ(z) = 0 otherwise, and the dilatation Kf of f is Kμ (z) [see (2)], and, thus, |μ(z)| ≤ 1 and Kμ (z) ≥ 1 a.e. Note that every homeomorphic ACL solution f of the Beltrami equation with Kμ ∈ L1loc belongs to the class 1,1 Wloc as in all our theorems. Indeed, 1/2

|fz | + |fz | = Kμ1/2 (z)Jf (z) and, by the H¨older inequality, on every compact set C ⊂ D we have 1/2

fz 1 ≤ fz 1 ≤ Kμ 1 A(f (C))1/2 , 1,1 (see, e.g., [5, p. 8]). Similarly, it is easy to show where A(f (C)) is the area of the set f (C). Hence, f ∈ Wloc 1,s , where s = 2p/(1 + p) ∈ [1, 2]. that if Kμ ∈ Lploc , p ∈ [1, ∞], then f ∈ Wloc 1,2 Also note that the condition f −1 ∈ Wloc given in the definition of ring solution implies that almost every point z is a regular point for the mapping f, i.e., f is differentiable at z and Jf (z) = 0 (see Remark 2). Finally, note that the condition Kμ ∈ L1loc is necessary for a homeomorphic ACL solution f of (1) to have the property 1,2 g = f −1 ∈ Wloc because this property implies that

Kμ (z) dxdy ≤ 4

C

C

dxdy =4 1 − |μ(z)|2

|gw |2 dudv < ∞

f (C)

for every compact set C ⊂ D. Recall that a real-valued function ϕ ∈ L1loc (D) is said to be of bounded mean oscillation in D (ϕ ∈ BMO(D), or, simply, ϕ ∈ BMO) if ϕ ∗ = sup − ϕ(z) − ϕB dxdy < ∞, B⊂D

B

where the supremum is taken over all disks B in D and 1 ϕ(z) dxdy ϕB = − ϕ(z)dxdy = |B| B

B

is the mean value of the function ϕ over B. It is well known that L∞ (D) ⊂ BMO(D) ⊂ Lploc (D) for all 1 ≤ p < ∞ (see, e.g., [8]). A function ϕ ∈ BMO is said to have vanishing mean oscillation (ϕ ∈ VMO) if the above supremum taken over all disks B in D with |B| < ε converges to 0 as ε → 0.

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V. RYAZANOV, U. S REBRO ,

AND

E. YAKUBOV

The BMO space was introduced by John and Nirenberg (see [9]) and soon became one of the main concepts in harmonic analysis, complex analysis, and partial differential equations. BMO functions are related in many ways to quasiconformal and quasiregular mappings (see, e.g., [10–15]) and to mappings with finite distortion (see, e.g., [16–19]). VMO has been introduced by Sarason (see [20]). A large number of papers are devoted to the study of the existence, uniqueness, and properties of solutions for various kind of differential equations, in particular of elliptic type with VMO coefficients (see, e.g., [21–24]). In this connection, it should be noted that, according to a 1,2 recent result of Brezis and Nirenberg [25], the Sobolev class Wloc is a subclass of VMO (see also [26]). Let D be a domain in the complex plane C. We say that a function ϕ : D → R has finite mean oscillation at a point z0 ∈ D (f is of FMO at z0 ) if −

dϕ (z0 ) = lim sup ε→0

|ϕ(z) − ϕε (z0 )|dxdy < ∞,

(5)

ϕ(z)dxdy < ∞

(6)

D(z0 ,ε)

where

ϕε (z0 ) = − D(z0 ,ε)

is the mean value of the function ϕ(z) over the disk D(z0 , ε) with small ε > 0. Thus, this notion includes the assumption that ϕ is integrable in some neighborhood of the point z0 . We call dϕ (z0 ) the dispersion of the function ϕ at the point z0 . We say that a function ϕ : D → R is of finite mean oscillation in D (ϕ ∈ FMO(D), or, simply, ϕ ∈ FMO) if ϕ has a finite dispersion at every point z ∈ D. We also call the number dϕ (z0 , ε0 ) = sup ε∈(0,ε0 ]

−

ϕ(z) − ϕε (z0 ) dxdy < ∞

(7)

D(z0 ,ε)

the maximal dispersion of the function ϕ in the disk D(z0 , ε0 ). Below we use the notation D(r) = D(0, r) = {z ∈ C : |z| < r} and A(ε, ε0 ) = {z ∈ C : ε < |z| < ε0 }.

(8)

Lemma 1. Let D ⊂ C be a domain such that D(e−1 ) ⊂ D and let ϕ : D → R be a nonnegative function. If ϕ is integrable in D(e−1 ) and of FMO at 0, then A(ε,e−1 )

ϕ(z) dxdy 1 2 ≤ C log log ε 1 |z| log |z|

(9)

for ε ∈ (0, e−e ), where C = 2π 2ϕ0 + 3e2 d0 ,

(10)

ϕ0 is the mean value of ϕ over the disk D(e−1 ), and d0 is the maximal dispersion of ϕ in D(e−1 ). Versions of this lemma were first established for BMO functions in [27, 28] and then for FMO functions in [29] and [30, p. 251] (Lemma 2.15).


1789

Conditions for the existence and uniqueness of ACL homeomorphic solutions for the Beltrami equation can be given in terms of the maximal dilatation Kμ (z). It was shown, e.g., that if Kμ (z) has a BMO or an FMO majorant, then the Beltrami equation (1) has a homeomorphic ACL solution (see, e.g., [27, 28, 30, 31]; cf. [32, 33]). Various conditions for the existence of solutions for the Beltrami equation were formulated in terms of integral and measure constraints on Kμ (see, e.g., [32–39]). These conditions assume either the exponential integrability or at least the high local integrability of the dilatation Kμ . Here, as in [2–4], the existence criteria are expressed in terms of the tangential dilatations KμT (z, z0 ). In [30], we proved that if Kμ (z) has a majorant Q(z) of finite mean oscillation in D, then (1) has a homeomorphic ACL solution. We now prove a stronger result, namely, the existence of a ring solution of (1) where the assumption on an FMO majorant for Kμ in D is replaced by the weaker condition that every point z0 ∈ D has a neighborhood Uz0 and a function Qz0 : Uz0 → [0, ∞] of finite mean oscillation at z0 such that KμT (z, z0 ) ≤ Qz0 (z) for all z ∈ Uz0 (see Theorem 4 in Sec. 3 below). This and other new existence theorems established here are based on a general existence principle (Lemma 3). Some of these existence theorems are expressed in terms of the mean and the logarithmic mean of the tangential dilatation KμT (z, z0 ) over infinitesimal disks and annuli centered at z0 (see, e.g., Theorems 5 and 6). We also use Lemma 3 in a new proof of an extension of the Lehto theorem (Theorem 7), which we first established in [4]. Finally, note that a series of interesting distortion theorems have been formulated in various terms in Sec. 2 (see Remark 3 in the end of the work). 2. Distortion Estimates for Ring Q-Homeomorphisms Below we use the following standard conventions: a/∞ = 0 for a = ∞ and a/0 = ∞ if a > 0 and 0 · ∞ = 0 (see, e.g., [40, p. 6]). Given a measurable function Q : D → [1, ∞], we say that a homeomorphism f : D → C is a Q-homeomorphism if (11) M (f Γ) ≤ Q(z) · ρ2 (z)dxdy D

for every path family Γ in D and every ρ ∈ adm Γ. This term was introduced in [41–43]. Recall that, given a family of paths Γ in C, a Borel function ρ : C → [0, ∞] is called admissible for Γ (ρ ∈ adm Γ) if ρ(z) |dz| ≥ 1 γ

for every γ ∈ Γ. The modulus of Γ is defined as follows: M (Γ) =

ρ2 (z)dxdy.

inf

ρ∈adm Γ

C

We say that a property P holds for almost every (a.e.) path γ in a family Γ if the subfamily of all paths in Γ for which P fails has modulus zero. In particular, almost all paths in C are rectifiable. Given a domain D and two sets E and F in C, Γ(E, F, D) denotes the family of all paths γ : [a, b] → C that join E and F in D, i.e., γ(a) ∈ E, γ(b) ∈ F, and γ(t) ∈ D for a < t < b. We set Γ(E, F ) = Γ(E, F, C) if D = C. A ring domain (or, briefly, a ring) in C is a doubly connected domain R in C. Let R be a ring in C. If C1 and C2 are the connected components of C \ R, then we write R = R(C1 , C2 ). The capacity of R can be defined as follows (see, e.g., [44]):

1790


cap R(C1 , C2 ) = M (Γ(C1 , C2 , R)).

AND

E. YAKUBOV

(12)

Note that (see, e.g., Theorem 11.3 in [45]) M (Γ(C1 , C2 , R)) = M (Γ(C1 , C2 )).

(13)

Motivated by the ring definition of quasiconformality in [46], we introduce the following notion, which localizes and extends the notion of Q-homeomorphism. Let D be a domain in C, let z0 ∈ D, let r0 ≤ dist(z0 , ∂D), and let Q : D(z0 , r0 ) → [0, ∞] be a measurable function in the disk D(z0 , r0 ) = {z ∈ C : |z − z0 | < r0 }. We set A(r1 , r2 , z0 ) = {z ∈ C : r1 < |z − z0 | < r2 }, C(z0 , ri ) = {z ∈ C : |z − z0 | = ri },

i = 1, 2.

We say that a homeomorphism f : D → C is a ring Q-homeomorphism at the point z0 if capR(f C1 , f C2 ) ≤

Q(z) · η 2 |z − z0 | dxdy

(14)

A

for every annulus A = A(r1 , r2 , z0 ), 0 < r1 < r2 < r0 , and every measurable function η : (r1 , r2 ) → [0, ∞] such that r2 η(r)dr = 1.

(15)

r1

Note that every Q-homeomorphism f : D → C is a ring Q-homeomorphism at every point z0 ∈ D. The proposition below gives other conditions on f that force it to be a ring Q-homeomorphism (see Theorem 2.17 in [4]). 1,2 Proposition 1. Let f : D → C be a sense-preserving homeomorphism of the class Wloc such that f −1 ∈ 1,2 . Then, at every point z0 ∈ D, the mapping f is a ring Q-homeomorphism with Q(z) = KμT (z, z0 ), where Wloc μ(z) = μf (z). 1,2 1,2 If f is a plane Wloc homeomorphism with locally integrable Kf (z), then f −1 ∈ Wloc (see, e.g., [47]). Hence we obtain the following consequence of Proposition 1: 1,2 and suppose that Corollary 1. Let f : D → C be a sense-preserving homeomorphism of the class Wloc Kf (z) is integrable in a disk D(z0 , r0 ) ⊂ D for some z0 ∈ D and r0 > 0. Then f is a ring Q-homeomorphism at the point z0 ∈ D with Q(z) = KμT (z, z0 ), where μ(z) = μf (z).

Also note the convergence theorem that plays an important role in our scheme for deducing existence theorems for the Beltrami equation (see Theorem 2.22 in [4]).


1791

Proposition 2. Let fn : D → C, n = 1, 2, . . . , be a sequence of ring Q-homeomorphisms at a point z0 ∈ D. If fn converge locally uniformly to a homeomorphism f : D → C, then f is also a ring Q-homeomorphism at the point z0 . For points z, ζ ∈ C, the spherical (chordal) distance s(z, ζ) between z and ζ is given by the relations s(z, ζ) =

|z − ζ| 1

if z = ∞ = ζ,

1

(1 + |z|2 ) 2 (1 + |ζ|2 ) 2

s(z, ∞) =

1 1

(1 + |z|2 ) 2

if z = ∞.

Given a set E ⊂ C, δ(E) denotes the spherical diameter of E, i.e., δ(E) = sup s(z1 , z2 ). z1 ,z2 ∈E

Given a number Δ ∈ (0, 1), domain D ⊂ C, point z0 ∈ D, number r0 ≤ dist (z0 , ∂D), and measurable function Q : D(z0 , r0 ) → [0, ∞], let RΔ Q denote the class of all ring Q-homeomorphisms f : D → C at z0 such that δ C \ f (D) ≥ Δ.

(16)

Δ Δ Next, we introduce classes BΔ Q and FQ of certain quasiconformal mappings. Let BQ denote the class of all quasiconformal mappings f : D → C satisfying (16) and such that

2 1 − z − z0 μ(z) z − z0 ≤ Q(z) KμT (z, z0 ) = 2 1 − |μ(z)|

a.e. in D(z0 , r0 ),

(17)

where μ = μf . Similarly, let FΔ Q denote the class of all quasiconformal mappings f : D → C satisfying (16) and such that Kμ (z) =

1 + |μ(z)| ≤ Q(z) a.e. in D(z0 , r0 ) . 1 − |μ(z)|

(18)

Remark 1. Note that Δ Δ FΔ Q ⊂ BQ ⊂ RQ .

(19)

Lemma 2. Let f ∈ RΔ Q and let ψε : (0, ∞) → [0, ∞], 0 < ε < r0 , be a one-parameter family of measurable functions such that r0 ψε (t)dt < ∞,

0 < I(ε) = ε

ε ∈ (0, r0 ).

(20)

1792


AND

E. YAKUBOV

Then, for all ζ ∈ D(z0 , r0 ), one has

32 2π , s f (ζ), f (z0 ) ≤ exp − Δ ω(|ζ − z0 |)

(21)

where

1 ω(ε) = 2 I (ε)

Q(z)ψε2 |z − z0 | dxdy

(22)

A(ε)

and A(ε) = A(ε, r0 , z0 ) = z ∈ C : ε < |z − z0 | < r0 . Proof. We set C = {z ∈ C : |z − z0 | = |ζ − z0 |} and C0 = {z ∈ C : |z − z0 | = r0 }. Then

32 2π s f (ζ), f (z0 ) ≤ exp − Δ capR(f C, f C0 ) for every f ∈ RΔ Q by Lemma 3.21 in [4]. Thus, choosing η(r) = ψε (r)/I(ε), r ∈ (ε, r0 ), in (14), we come to (21). Theorem 1. Let f ∈ RΔ Q and let ψ : [0, ∞] → [0, ∞] be a measurable function such that r0 ψ(t) dt < ∞,

0
0 because E = fμ−1 (fμ (E)) and fμ−1 preserves null sets. Clearly, fμ−1 is not differentiable at any point of fμ (E), contradicting the fact that fμ−1 is differentiable a.e. Corollary 3. Let μ : D → C be a measurable function such that |μ(z)| < 1 a.e. and Kμ ∈ L1loc and let ψ : (0, ∞) → (0, ∞) be a measurable function such that, for all 0 < t1 < t2 < ∞, one has t2

t2 ψ(t)dt < ∞,

0< t1

ψ(t)dt = ∞.

(34)

0

Suppose that, for every z0 ∈ D, there is ε0 < dist(z0 , ∂D) such that ⎛ε ⎞ 0 Kμ (z)ψ |z − z0 | dxdy ≤ O ⎝ ψ(t)dt⎠

2

ε 1: Corollary 6. If Kμ ∈ Lploc for p ≥ 1 and (41) holds for Kμ (z) instead of KμT (z, z0 ) for every point 1,s z0 ∈ D, then (1) has a Wloc homeomorphic solution with s = 2p/(p + 1). Since the maximal dilatation dominates the tangential dilatation, all the above results obviously imply similar existence theorems in terms of conditions on the maximal dilatation established earlier in [30], which are important particular cases (see also the survey [39]). Remark 3. Note that all distortion estimates for Q-homeomorphisms given in Sec. 2 can be applied to ring solutions from the above existence theorems. Indeed, in these theorems, for every domain D ⊂ C and every z1 , z2 ∈ D and w1 , w2 ∈ C, there exist ring solutions with normalization f (z1 ) = w1 and f (z2 ) = w2 . The 1/2 estimates hold for such solutions with Q(z) = KμT (z, z0 ) and Δ = min 1/(1 + |w1 |2 , 1/(1 + |w2 |2 )1/2 ), 2 1/2 . For ring solutions with f (z ) and and, in particular, if w = 0 and w = w , then Δ = 1/ 1 + |w | 1 2 0 0 1 f (z2 ) ≤ R < ∞, one has Δ = 1/(1 + R2 )1/2 . REFERENCES 1. E. Reich and H. Walczak, “On the behavior of quasiconformal mappings at a point,” Trans. Amer. Math. Soc., 117, 338–351 (1965). 2. O. Lehto, “Homeomorphisms with a prescribed dilatation,” Lect. Notes Math., 118, 58–73 (1968). 3. V. Gutlyanskii, O. Martio, T. Sugawa, and M. Vuorinen, “On the degenerate Beltrami equation,” Trans. Amer. Math. Soc., 357, 875–900 (2005). 4. V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solutions of Beltrami equation,” J. Anal. Math., 96, 117–150 (2005). 5. V. Maz’ya, Sobolev Classes, Springer, Berlin–New York (1985).

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