1 Introduction - Springer Link

ON THE DIRICHLET PROBLEM FOR THE BELTRAMI EQUATION By

D. KOVTONYUK, I. PETKOV, V. RYAZANOV, AND R. SALIMOV 1,1 Abstract. We show that homeomorphic Wloc solutions of the Beltrami equations ∂ f = µ ∂ f satisfy certain modular inequalities. On this basis, we develop the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to the Beltrami equation in Jordan domains and finitely connected domains, respectively. These results have important applications to various problems of mathematical physics.

1

Introduction

It is well known that the Beltrami equation has found a great number of significant applications in the theory of stationary flows, hydrodynamics, magneto- and electrostatics of inhomogeneous media and elasticity. Most applications are based on its close relation to quasiconformal mappings and their generalizations. The Beltrami equation has also turned out to be useful in the study of Riemann surfaces, Teichmüller spaces, Kleinian groups, meromorphic functions, low dimensional topology, holomorphic motion, complex dynamics, Clifford analysis, control theory, robotics etc. In this connection, note that the existence of homeo1,1 morphic Wloc solutions was recently established for many degenerate Beltrami equations; see, e.g., the monographs [1], [3], [16] and [29] and the surveys [15] and [49] for the history and further references. Let D be a domain in the complex plane C, i.e., a connected open subset of C, and let µ : D → C be a measurable function with |µ(z)| < 1 a.e. (almost everywhere) in D. A Beltrami equation is an equation of the form (1.1)

f z¯ = µ(z) f z ,

where f z¯ = ∂ f = ( f x + i f y )/2, f z = ∂ f = ( f x − i f y )/2, z = x + iy, and f x and f y are partial derivatives of f in x and y, respectively. The function µ is called the 113 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 122 (2014) DOI 10.1007/s11854-014-0005-x

114

D. KOVTONYUK, I. PETKOV, V. RYAZANOV, AND R. SALIMOV

complex coefficient and (1.2)

Kµ (z) =

1 + |µ(z)| 1 − |µ(z)|

the dilatation quotient of (1.1). Equation (1.1) is said to be degenerate if ess sup Kµ (z) = ∞. Boundary value problems for Beltrami equations first appeared in Riemann’s dissertation in the case of µ(z) ≡ 0 and were studied in the papers of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy–Riemann system. The Dirichlet problem for uniformly elliptic systems has been studied extensively; see, e.g., [2] and [52]. The Dirichlet problem for degenerate Beltrami equations in the unit disk was studied in [9]. However, the criteria for the existence of solutions of the Dirichlet problem in [9] are not invariant under conformal mappings. Here we give the corresponding theorems on the existence of regular solutions in arbitrary Jordan domains as well as of pseudoregular and multi-valued solutions in arbitrary multiply connected domains bounded by a finite collection of mutually disjoint Jordan curves. In comparison with the work [9], our approach is based on estimates of the modulus of dashed lines but not of paths under arbitrary homeo1,1 morphic Wloc solutions of the Beltrami equations. Recall that every holomorphic (analytic) function f in a domain D ⊂ C satisfies the simplest Beltrami equation (1.3)

f z¯ = 0

with µ(z) ≡ 0. If a holomorphic function f given in the unit disk D is continuous in its closure, then by the Schwarz formula, Z ζ + z dζ 1 (1.4) f (z) = i Im f (0) + Re f (ζ ) · ; 2πi |ζ | =1 ζ −z ζ see, e.g., [19, Section 8, Chapter III, Part 3]. Thus, f is determined, up to a purely imaginary additive constant ic, c = Im f (0), by its real part ϕ(ζ ) = Re f (ζ ) on the boundary of D. Hence the Dirichlet problem for the Beltrami equation (1.1) in a domain D ⊂ C is the problem of finding a continuous function f : D → C having partial derivatives of the first order a.e., satisfying (1.1) a.e., and such that (1.5)

lim Re f (z) = ϕ(ζ ) for all ζ ∈ ∂D

z→ζ

for a prescribed continuous function ϕ : ∂D → R. It is obvious that if f is a solution of this problem, then so is the function F (z) = f (z) + ic, c ∈ R.

DIRICHLET PROBLEM FOR THE BELTRAMI EQUATION

115

Here, we show that the Dirichlet problem (1.5) has regular solutions in an arbitrary Jordan domain and pseudoregular and multi-valued solutions in an arbitrary multiply connected domain bounded by a finite number of mutually disjoint Jordan curves for wide classes of degenerate Beltrami equations; see, especially, the precise results in Theorems 6.3, 7.3, and 8.3. To stress the importance of these results, let us recall the relevance of the Beltrami equations and the second order elliptic equations in divergence form which are governing equations in many problems of applied mathematics; see, e.g., [1], [17], [26], and [52]. Namely, let us consider general linear elliptic equations (1.6)

div A∇u = 0

with measurable coefficient matrix A(z) = {ai j }, det (I + A) > 0, where I is the 1,1 identity matrix. By a weak solution u of (1.6) we mean a function u ∈ Wloc (D) such that Z (1.7) hA(z)∇u, ∇ϕi dm(z) = 0 for all ϕ ∈ C0∞ (D). D

A function v ∈ of u) if

1,1 Wloc (D) is

called the stream function (A-harmonic conjugate

∇v = J A(z)∇u,

(1.8) where J is the Hodge operator (1.9)

J =

! 0 −1 , 1 0

i.e., u and v satisfy a linear system of the form vy = a11 ux + a12 uy

(1.10)

−vx = a21 ux + a22 uy

We see that f = u + iv satisfies the Beltrami equation with two characteristics f z = µ(z) · f z + ν(z) · f z ,

(1.11) where (1.12)

µ =

a22 − a11 − i(a12 + a21 ) , 1 + Tr A + det A

ν=

1 − det A + i(a12 − a21 ) . 1 + Tr A + det A

In terms of µ and ν, the ellipticity condition is written as (1.13)

|µ(z)| + |ν(z)| < 1 a.e. in D.

116


Conversely, given measurable complex valued functions µ, ν satisfying (1.13), one can invert the algebraic system (1.12) and obtain (1.14) ! (|1 − µ|2 − |ν|2 )/(|1 + ν|2 − |µ|2 ) 2Im (ν − µ)/(|1 + ν|2 − |µ|2 ) A = . −2Im (ν + µ)/(|1 + ν|2 − |µ|2 ) (|1 + µ|2 − |ν|2 )/(|1 + ν|2 − |µ|2 ) Thus, (1.11) is, in fact, a complex form of one of the main equations of mathematical physics (1.6) in anisotropic and inhomogeneous media. It is also clear that the Beltrami equations of the first type (1.1) correspond to the equation (1.6) with symmetric coefficients a12 = a21 and the Beltrami equations of the second type f z¯ = ν(z) f z

(1.15)

correspond to the equations (1.6) with anti-symmetric coefficients a11 = a22 and a12 = −a21 . In this connection, it is worth noting that although the corresponding theorems 1,1 on the existence of homeomorphic Wloc solutions of (1.11) have been established in the series of the recent papers [4]–[6] (see also the monograph [16]), the Dirichlet problem for (1.11) was solved only quite recently for Jordan domains in the paper [7]. The approach in [7] is essentially different from that in the present paper and cannot be extended to the case of multiply connected domains.

2

Preliminaries

Throughout this paper, B(z0 , r) = {z ∈ C : |z0 − z| < r},

D = B(0, 1),

S(z0 , r) = {z ∈ C : |z0 − z| = r},

S(r) = S(0, r),

R(z0 , r1 , r2 ) = {z ∈ C : r1 < |z − z0 | < r2 }. The well-known class of BMO functions was introduced by John and Nirenberg in [20] and soon began to play an important role in harmonic analysis, partial differential equations, and related areas; see, e.g., [17] and [39]. Recall that a realvalued function u in a domain D in C is said to be of bounded mean oscillation in D (abbreviated u ∈ BMO(D), or simply u ∈ BMO) if u ∈ L1loc (D) and Z 1 (2.1) kuk∗ : = sup |u(z) − uB | dm(z) < ∞, B |B| B where the supremum is taken over all discs B in D, dm(z) denotes the Lebesgue measure in C, and uB is the average of u over B. We say u ∈ BMOloc (D) if


117

u ∈ BMO(U) for every relatively compact subdomain U of D (we also write BMO or BMOloc if it is clear from the context what D is). A function ϕ ∈ BMO(D) is said to have vanishing mean oscillation (abbreviated ϕ ∈ VMO) if the supremum in (2.1) taken over all disks B in D with |B| < ε converges to 0 as ε converges to 0. VMO was introduced by Sarason in [48]. Many papers have been devoted to the existence, uniqueness, and properties of solutions of various kinds of differential equations and, in particular, of elliptic type with coefficients of the class VMO; see, e.g., [8, 22, 30, 36, 37]. Following [21] (see also [29]), we say that a function ϕ : D → R has finite mean oscillation at a point z0 ∈ D if Z (2.2) lim − |ϕ(z) − ϕeε (z0 )| dm(z) < ∞, ε→0

B(z0 ,ε)

where

Z e ϕε (z0 ) = −

(2.3)

ϕ(z) dm(z)

B(z0 ,ε)

is the mean value of ϕ(z) over the disk B(z0 , ε). Note that the condition (2.2) includes the assumption that ϕ is integrable in some neighborhood of the point z0 . We say also that a function ϕ : D → R has finite mean oscillation in D (and write ϕ ∈ FMO(D) or simply ϕ ∈ FMO) if ϕ ∈ FMO(z0 ) for all z0 ∈ D. If ϕ is given in a domain G in C containing D and ϕ ∈ FMO(z0 ) for all z0 ∈ D, we write ϕ ∈ FMO(D). The following statement is obvious from the triangle inequality. Proposition 2.1. If (2.4)

Z lim −

ε→0

|ϕ(z) − ϕε | dm(z) < ∞

B(z0 ,ε)

for a collection of numbers ϕε ∈ R, ε ∈ (0, ε0 ], then ϕ is of finite mean oscillation at z0 . In particular, choosing ϕε ≡ 0, ε ∈ (0, ε0 ] in Proposition 2.1, we obtain the following. Corollary 2.1. If (2.5)

Z lim −

ε→0

|ϕ(z)| dm(z) < ∞

B(z0 ,ε)

for z0 ∈ D, ϕ has finite mean oscillation at z0 .

118


Recall that a point z0 ∈ D is called a Lebesgue point of a function ϕ : D → R if ϕ is integrable in a neighborhood of z0 and Z (2.6) lim − |ϕ(z) − ϕ(z0 )| dm(z) = 0. ε→0

B(z0 ,ε)

It is well known that almost every point in D is a Lebesgue point for a function ϕ ∈ L1 (D). Thus, by Proposition 2.1, we have the following corollary, which shows that the FMO condition is natural. Corollary 2.2. Each locally integrable function ϕ : D → R has finite mean oscillation at almost every point in D. Remark 2.1. Note that the function ϕ(z) = log (1/|z|) belongs to BMO in the unit disk 1 (see, e.g., [39, p. 5]), and hence also to FMO. However, ϕeε (0) → ∞ as ε → 0; thus condition (2.5), while sufficient, is not necessary for ϕ to be of finite mean oscillation at z0 . Clearly, BMO(D) ⊂ BMOloc (D) ⊂ FMO(D) and, as is well known (see, e.g., [20]), BMOloc ⊂ Lploc for all p ∈ [1, ∞). However, FMO is not a subclass of Lploc for any p > 1 but only L1loc ; see the examples in [29, p. 211]. Thus, the class FMO is essentially wider than BMOloc . Versions of the next lemma have been proved for the class BMO, first in the planar case in [41] and then in the space case in [28]. For the class FMO, see the papers [21], [40], [42], [43], and the monograph [29]. Lemma 2.1. Let D be a domain in C and let ϕ : D → R be a non-negative function of class FMO(z0 ) for some z0 ∈ D. Then Z ϕ(z) dm(z) 1 (2.7) as ε → 0 2 = O log log ε 1 ε 0, there is a neighborhood V ⊂ U of z0 such that (3.1)

M (1(E, F ; D)) ≥ P

for all continua E and F in D intersecting ∂U and ∂V . We say that ∂D is weakly flat if it is weakly flat at each point z0 ∈ ∂D.

U V z0

¶D

E

F D

We also say that a point z0 ∈ ∂D is strongly accessible if, for every neighborhood U of the point z0 , there exist a compactum E in D, a neighborhood V ⊂ U of z0 and a number δ > 0 such that (3.2)

M (1(E, F ; D)) ≥ δ

for all continua F in D intersecting ∂U and ∂V . We say that ∂D is strongly accessible if each point z0 ∈ ∂D is strongly accessible. In the definitions of strongly accessible and weakly flat boundaries, one can take as neighborhoods U and V of a point z0 only balls (closed or open) centered at z0 or only neighborhoods of z0 in another fundamental system of neighborhoods of z0 . These concepts can also be extended in a natural way to the case of C and z0 = ∞. There, we must use the corresponding neighborhoods of ∞.


121

It is easy to see that if a domain D in C is weakly flat at a point z0 ∈ ∂D, then the point z0 is strongly accessible from D. Moreover, if a domain D in C is weakly flat at a point z0 ∈ ∂D, then D is locally connected at z0 ; see, e.g., [24, Lemma 5.1] or [29, Lemma 3.15]. The notions of strong accessibility and weak flatness at boundary points of a domain in C defined in [23] (see also [24] and [40]) are localizations and generalizations of the corresponding notions introduced in [27] and [28]; cf. the properties P1 and P2 by Väisälä in [51] and also quasiconformal accessibility and quasiconformal flatness used by Näkki in [35]. Many theorems on the homeomorphic extension to the boundary of quasiconformal mappings and their generalizations are valid under the condition of weak flatness of boundaries. The condition of strong accessibility plays a similar part for the continuous extension of mappings to the boundary. A domain D ⊂ C is called a quasiextremal distance domain, or, briefly, a QED-domain (see [13]) if (3.3)

M (1(E, F ; C)) ≤ K · M (1(E, F ; D))

for some K ≥ 1 and all pairs of nonintersecting continua E and F in D. It is well known (see, e.g., [51, Theorem 10.12]) that (3.4)

M (1(E, F ; C)) ≥

R 2 log π r

for any sets E and F in C intersecting all the circles S(z0 , ρ), ρ ∈ (r, R). Hence a QED-domain has a weakly flat boundary. An example in [29, Section 3.8] shows that the converse is false even in the case of simply connected domains in C. A domain D ⊂ C is called a uniform domain if each pair of points z1 and z2 ∈ D can be joined with a rectifiable curve γ in D such that (3.5)

s(γ) ≤ a · |z1 − z2 |

and (3.6)

min s(γ(zi , z)) ≤ b · dist(z, ∂D)

i =1,2

for all z ∈ γ, where γ(zi , z) is the portion of γ bounded by zi and z [31]. It is known that every uniform domain is a QED-domain but there exist QED-domains that are not uniform [13]. Bounded convex domains and bounded domains with smooth boundaries are simple examples of uniform domains and, consequently, QED-domains as well as domains with weakly flat boundaries.

122


Finally, we require the notions of Lipschitz maps and Lipshitz domains. Recall first of all that a map ϕ : U → C is said to be Lipschitz provided |ϕ(z1 )−ϕ(z2 )| ≤ M · |z1 − z2 | for some M < ∞ and for all z1 , z2 ∈ U; it is bi-Lipschitz if, in addition, M ∗ |z1 − z2 | ≤ |ϕ(z1 ) − ϕ(z2 )| for some M ∗ > 0 and for all z1 , z2 ∈ U. A domain D in C is Lipschitz if every point z0 ∈ ∂D has a neighborhood U that can be mapped by a bi-Lipschitz homeomorphism ϕ onto the unit disk D ⊂ C in such a way that ϕ(∂D ∩ U) is the intersection of D with the real axis. Note that a biLipschitz homeomorphism is quasiconformal and the modulus is a quasiinvariant under such mappings. Hence Lipschitz domains have weakly flat boundaries.

4

Estimates of modulus of dashed lines

A continuous mapping γ of an open subset 1 of the real axis R or of a circle into D is called a dashed line in D, see, e.g., [29, 6.3]. Such a set 1 consists of a countable collection of mutually disjoint intervals in R. The modulus of the family Ŵ of dashed lines γ is defined similarly to (2.12). We say that a property P holds for a.e. (almost every) γ ∈ Ŵ if the subfamily of all lines in Ŵ for which P fails has modulus zero [11]. Later on, we also say that a Lebesgue measurable function ̺ : C → [0, ∞] is extensively admissible for Ŵ, and write ̺ ∈ ext adm Ŵ, if (2.13) holds for a.e. γ ∈ Ŵ; see, e.g., [29, Section 9.2]. 1,1 Theorem 4.1. Let f be a homeomorphic Wloc solution of the Beltrami equation (1.1) in a domain D ⊆ C. Then Z ̺2 (z) (4.1) M ( f 6ε ) ≥ inf dm(z) ̺∈ext adm 6ε D Kµ (z)

for all z0 ∈ D, where ε ∈ (0, ε0 ), ε0 ∈ (0, d0 ), d0 = supz∈D |z − z0 |, and 6ε denotes the family of dashed lines consisting of all intersections of the circles S(z0 , r), r ∈ (ε, ε0 ), and D. Proof. Let B be the (Borel) set of all points z in D for which f has a total differential with J f (z) 6 = 0. It is known that B is the union of a countable collection of Borel sets B l , l = 1, 2, . . ., such that f l = f |Bl is a bi-Lipschitz homeomorphism [10, Lemma 3.2.2]. With no loss of generality, we may assume that the B l are mutually disjoint. Denote by B∗ the set of all points z ∈ D for which f has a total differential with f ′ (z) = 0. The set B0 = D \ (B ∪ B∗ ) has Lebesgue measure zero in C by the well-known Gehring–Lehto–Menchoff Theorem; see [12] and [34]. Hence, by [24, Theorem 2.11] (see also [29, Lemma 9.1]), length(γ ∩ B0 ) = 0 for a.e. path γ in D.


123

Let us show that length( f (γ) ∩ f (B0 )) = 0 for a.e. circle γ centered at a point z0 ∈ D. This follows from absolute continuity of f on closed subarcs of γ ∩ D 1,1 for a.e. such circle γ. Indeed, the class Wloc is invariant with respect to local 1,1 quasi-isometries, and the functions in Wloc are absolutely continuous on lines; see, e.g., [33, Theorems 1.1.7 and 1.1.3], respectively. Applying the transformation of coordinates log(z − z0 ), we obtain the absolute continuity on a.e. such circle γ. Fix γ0 on which f is absolutely continuous and length(γ0 ∩ B0 ) = 0. Then length( f (γ) ∩ f (B0 )) = length f (γ0 ∩ B0 ), and for every ε > 0, there is an open set ωε in γ0 ∩ D such that γ0 ∩ B0 ⊂ ωε with length ωε < ε, see, e.g., [47, Theorem III(6.6)]. The open set ωε consists of a countable collection of open arcs γi of the P circle γ0 . By the construction, i length γi < ε; and by the absolute continuity of P f on γ0 , the sum δ : = i length f (γi ) can be made arbitrarily small by choosing small enough ε > 0. Hence length f (γ0 ∩ B0 ) = 0. Thus, length(γ∗ ∩ f (B0 )) = 0, where γ∗ = f (γ) for a.e. circle γ centered at z0 . Now, let ̺∗ ∈ adm f (Ŵ), where Ŵ is the collection of all dashed lines γ ∩ D for such circles γ and ̺∗ ≡ 0 outside of f (D). Set ̺ ≡ 0 outside of D and on B0 and ̺(z) : = ̺∗ ( f (z)) (| f z | + | f z¯ |)

for z ∈ B ∪ B∗ .

Arguing piecewise on B l , we have by Theorem 3.2.5 under m = 1 in [10] that Z Z ̺ ds ≥ ̺∗ ds∗ ≥ 1 for a.e. γ ∈ Ŵ, γ

γ∗

because length( f (γ) ∩ f (B0 )) = 0 and length( f (γ) ∩ f (B∗ )) = 0 for a.e. γ ∈ Ŵ. Consequently, ̺ ∈ ext adm Ŵ. On the other hand, again arguing piecewise on B l , we have Z Z ̺2 (z) dm(z) ≤ ̺2∗ (w) dm(w), K (z) D µ f (D) because ̺(z) = 0 on B∗ . Thus, we obtain (4.1).

1,1 Theorem 4.2. Let f be a homeomorphic Wloc solution of the Beltrami equation (1.1) in a domain D ⊆ C. Then Z ε0 dr for all z0 ∈ D, ε ∈ (0, ε0 ), ε0 ∈ (0, d0 ), (4.2) M ( f 6ε ) ≥ kKµ k1 (z0 , r) ε

where d0 = supz∈D |z − z0 |, 6ε denotes the family of dashed lines consisting of all the intersections of the circles S(z0 , r), r ∈ (ε, ε0 ), and D, and Z Kµ (z)| dz| (4.3) kKµ k1 (z0 , r) : = D(z0 ,r)

is the L1 norm of Kµ over D(z0 , r) = {z ∈ D : |z − z0 | = r} = D ∩ S(z0 , r).

124


Proof. It follows from Fubini’s theorem that for every ̺ ∈ ext adm 6ε , Z A̺ (r) = ̺(z) |dz| 6 = 0 a.e. in r ∈ (ε, ε0 ) D(z0 ,r)

is a measurable function of r, e.g. by the Fubini Theorem. Thus, we may assume the equality A̺ (r) ≡ 1 a.e. in r ∈ (ε, ε0 ) and Z Z ε0 Z ̺2 (z) α2 (z) inf dm(z) = inf | dz| dr, ̺∈ext adm 6ε D∩Rε Kµ (z) α∈I(r) D(z0 ,r) Kµ (z) ε where Rε = R(z0 , ε, ε0 ) and I (r) denotes the set of all measurable functions α R on the dashed line D(z0 , r) = S(z0 , r) ∩ D such that D(z0 ,r) α(z) |dz| = 1. Hence Theorem 4.2 follows by Lemma 2.2 with X = D(z0 , r), arclength as the measure µ on D(x0 , r), ϕ = (1/Kµ )|D(z0 ,r) , and p = 2. In the sequel, the following lemma is useful. Here, we use the standard conventions a/∞ = 0 for a 6 = ∞, a/0 = ∞ if a > 0, and a · ∞ = 0; see, e.g., [47, p. 6]. Lemma 4.1. Let Q : C → (0, ∞) be a locally integrable function, and set Z (4.4) kQk1 (z0 , r) = Q(z)| dz|, z0 ∈ C, r ∈ (0, ∞), S(z0 ,r)

(4.5) I =

Z

r2

r1

dr 1 , η0 (r) = , kQk1 (z0 , r) I · kQk1 (z0 , r) r ∈ (r1 , r2 ), 0 < r1 < r2 < ∞.

Then I

−1

=

Z

r1 H (+0). Here, the integral in (6.10) is understood as the Lebesgue– Stieltjes integral and the integrals in (6.8) and (6.11)–(6.13) as ordinary Lebesgue integrals. Corollary 6.4. The conclusion of Theorem 6.3 holds if, for some α > 0, Z (6.14) eαKµ (z) dm(z) < ∞. D

7

Pseudoregular solutions in multiply connected domains

As was first noted by Bojarski (see, e.g., [52, Section 6 of Chapter 4]), in the case of multiply connected domains, the Dirichlet problem for the Beltrami equation

134


generally fails to be solvable in the class of continuous (single-valued) functions. Hence one may ask whether there exist solutions for the Dirichlet problem belonging to some wider class of functions. It is turns out that such solutions exist in the class of functions having a certain number of poles at prescribed points in D. For ϕ(ζ ) 6≡ const, a pseudoregular solution of the problem is a continuous 1,1 (in C = C ∪ {∞}) discrete open mapping f : D → C in the class Wloc (outside of these poles) with the Jacobian J f (z) = | f z |2 − | f z¯ |2 6 = 0 a.e. satisfying (1.1) a.e. and the condition (1.5). Lemma 7.1. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves. Let p ≥ n − 1 be an integer, and µ : D → C a measurable function such that |µ(z)| < 1 almost everywhere. Suppose that Kµ (z) ≤ Q(z) a.e. in D, where Q : C → (0, ∞) is locally integrable and satisfies the condition (5.12) for all z0 ∈ D. Then the Beltrami equation (1.1) has a pseudoregular solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R, ϕ(ζ ) 6≡ const, with poles at p prescribed points zi ∈ D, i = 1, . . . , p. Proof. Let F be a regular homeomorphic solution of the Beltrami equation 1,1 (1.1) of the class Wloc that exists by [44, Lemma 4.1]; see also [16, Lemma 7.1]. ∗ Let D = f (D). Then ∂D ∗ has n connected components Ŵi , i = 1, . . . , n, which correspond in a natural way to the connected components of ∂D, the Jordan curves γi ; see, e.g., [21, Lemma 5.3] or [29, Lemma 6.5]. Thus, by [14, Theorem V.6.2], the domain D∗ can be mapped conformally by a map R onto a circular domain D∗ whose boundary consists of n circles or points, so D∗ has a weakly flat boundary. Note that the mapping g : = R ◦ F is a regular homeomorphic solution of the 1,1 Beltrami equation in the class Wloc admitting a homeomorphic extension g∗ : D → D∗ by Lemma 5.4. Moreover, components of ∂D∗ cannot be singletons simply because of the extension is a homeomorphism. Let us seek a solution of the Dirichlet problem (1.5) of the form f = h ◦ g, where h is a meromorphic function with n poles at the prescribed points wi = g(zi ), i = 1, . . . , p in D∗ with the boundary condition lim Re h(w) = ϕ(g−1 ∗ (ζ ))

w→ζ

for all ζ ∈ ∂D∗ .

Such a function h exists by [52, Theorem 4.14]. We see that the function f = h ◦ g is the desired pseudoregular solution of the Dirichlet problem (1.5) for the Beltrami equation (1.1) with p poles at just these prescribed points zi , i = 1, . . . , p.


135

Arguing similarly to the last section, by special choices of the functional parameter ψ of condition (5.12) in Lemma 7.1, we obtain the following results. Theorem 7.1. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves, p ≥ n − 1 an integer, and µ : D → C a measurable function such that |µ(z)| < 1 a.e. and Kµ (z) ≤ Q(z) a.e. in D for a function Q : C → [0, ∞] in the class FMO(D). Then the Beltrami equation (1.1) has a pseudoregular solution of the Dirichlet problem (1.5) for every continuous function ϕ : ∂D → R, ϕ(ζ ) 6≡ const, with poles at p prescribed points in D. Corollary 7.1. The conclusion of Theorem 7.1 holds if every point z0 ∈ D is a Lebesgue point of some locally integrable function Q : C → [0, ∞] such that Kµ (z) ≤ Q(z) a.e. in D. Corollary 7.2. The conclusion of Theorem 7.1 holds if Z lim − (7.1) Kµ (z) dm(z) < ∞ for all z0 ∈ D. ε→0

B(z0 ,ε)

As above, here we assume that Kµ is extended by zero outside of D. Theorem 7.2. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves, p ≥ n − 1 an integer, and µ : D → C a measurable function with |µ(z)| < 1 almost everywhere. If Kµ ∈ L1loc (D) and satisfies the condition (7.2)

Z

0

δ (z0 )

dr = ∞ for all z0 ∈ D kKµ k1 (z0 , r)

for some δ (z0 ) ∈ (0, d (z0)), where d (z0 ) = supz∈D |z − z0 | and Z Kµ | dz|, (7.3) kKµ k1 (z0 , r) = D∩S(z0 ,r)

then the Beltrami equation (1.1) has a pseudoregular solution of the Dirichlet problem (1.5) for every continuous function ϕ : ∂D → R, ϕ(ζ ) 6≡ const, with poles at p prescribed points in D. Corollary 7.3. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves, p ≥ n − 1 an integer, and µ : D → C a measurable function such that 1 as ε → 0 for all z0 ∈ D, (7.4) kz0 (ε) = O log ε

136


where kz0 (ε) is the average of the function Kµ (z) over S(z0 , ε). Then the Beltrami equation (1.1) has a pseudoregular solution of the Dirichlet problem (1.5) for every continuous function ϕ : ∂D → R, ϕ(ζ ) 6≡ const, with poles at p prescribed points in D. Remark 7.1. The conclusion of Corollary 7.3 holds if 1 as z → z0 for all z0 ∈ D. (7.5) Kµ (z) = O log |z − z0 | Theorem 7.3. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves, p ≥ n − 1 an integer, and µ : D → C a measurable function with |µ(z)| < 1 a.e. such that Z (7.6) 8 Kµ (z) dm(z) < ∞ D

for a convex non-decreasing function 8 : [0, ∞] → [0, ∞]. If Z ∞ dτ (7.7) =∞ τ8−1 (τ) δ

for some δ > 8(0), the Beltrami equation (1.1) has a pseudoregular solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R, ϕ(ζ ) 6≡ const, with poles at p prescribed inner points in D. Corollary 7.4. The conclusion of Theorem 7.3 holds if, for some α > 0, Z (7.8) eαKµ (z) dm(z) < ∞. D

8

Multi-valued solutions in multiply connected domains

In finitely connected domains D in C, in addition to pseudoregular solutions, the Dirichlet problem (1.5) for the Beltrami equation (1.1) admits multi-valued solutions in the spirit of the theory of multi-valued analytic functions. We say that a discrete open mapping f : B(z0 , ε0 ) → C, where B(z0 , ε0 ) ⊆ D, is a local regu1,1 lar solution of the equation (1.1) if f ∈ Wloc , J f (z) 6 = 0 and f satisfies (1.1) a.e. in B(z0 , ε0 ). The local regular solutions f : B(z0, ε0 ) → C and f ∗ : B(z∗, ε∗ ) → C of the equation (1.1) are called extensions of each to other if there is a finite chain of such solutions f i : B(zi , εi ) → C, i = 1, . . . , m, such that f 1 = f 0 , f m = f ∗ and f i (z) ≡ f i+1 (z) for z ∈ Ei : = B(zi , εi ) ∩ B(zi+1, εi+1 ) 6 = ∅, i = 1, . . . , m − 1. A collection of local regular solutions f j : B(z j , ε j ) → C, j ∈ J, is called a


137

multi-valued solution of the equation (1.1) in D if the disks B(z j , ε j ) cover the whole domain D and f j are extensions of each to other through the collection. A multi-valued solution of the equation (1.1) is called a multi-valued solution of the Dirichlet problem (1.5) if u(z) = Re f (z) = Re f j (z), z ∈ B(z j , ε j ), j ∈ J, is a single-valued function in D satisfying the condition limz∈ζ u(z) = ϕ(ζ ) for all ζ ∈ D. Lemma 8.1. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves and let µ : D → C be a measurable function such that |µ(z)| < 1 almost everywhere. Suppose that Kµ (z) ≤ Q(z) a.e. in D, where Q : C → (0, ∞) is locally integrable and satisfies (5.12) for all z0 ∈ D. Then the Beltrami equation (1.1) has a multi-valued solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R. Proof. Let F be the regular homeomorphic solution of the Beltrami equation 1,1 (1.1) in the class Wloc , which exists by [44, Lemma 4.1]. As was shown in the proof of Lemma 7.1, we may assume that D∗ : = F (D) is a circular domain and that F can be extended to a homeomorphism F∗ : D¯ → D∗ . Let u : D∗ → R be a harmonic function such that lim u(w) = ϕ(F∗−1 (ζ )) for all ζ ∈ ∂D ∗ .

w→ζ

The existence of such a function is well known; see, e.g., [14, Section 3 of Chapter VI]. Let z0 ∈ D, B0 : = B(z0, ε0 ) ⊆ D for some ε0 > 0. Then the domain D0 = F (B0 ) is simply connected, and hence there is a harmonic function v(w) such that h(w) = u(w) + iv(w) is a holomorphic function which is unique up to an additive constant. Then f 0 : = h ◦ F |B0 is a local regular solution of the Beltrami equation (1.1). Since the function h can be extended to a multi-valued analytic function H in the domain D∗ , H ◦ F gives the desired multi-valued solution of the Dirichlet problem (1.5) for the Beltrami equation (1.1). In particular, from Lemmas 2.1 and 8.1, we obtain the following results. Theorem 8.1. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves and µ : D → C a measurable function with |µ(z)| < 1 a.e. and such that Kµ (z) ≤ Q(z) a.e. in D for a function Q : C → [0, ∞] in FMO(D). Then the Beltrami equation (1.1) has a multi-valued solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R. Corollary 8.1. The conclusion of Theorem 8.1 holds if every point z0 ∈ D is a Lebesgue point of some locally integrable function Q : C → [0, ∞] such that Kµ (z) ≤ Q(z) a.e. in D.

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In the following results, we assume that Kµ is extended by zero outside of D. Corollary 8.2. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves and µ : D → C a measurable function such that Z (8.1) lim − Kµ (z) dm(z) < ∞ for all z0 ∈ D. ε→0

B(z0 ,ε)

Then the Beltrami equation (1.1) has a multi-valued solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R. Theorem 8.2. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves and µ : D → C a measurable function with |µ(z)| < 1 a.e. If Kµ ∈ L1loc (D) and satisfies the condition Z δ (z0 ) dr (8.2) =∞ kKµ k1 (z0 , r) 0 for some δ (z0 ) ∈ (0, d (z0)), where d (z0 ) = supz∈D |z − z0 | and Z (8.3) kKµ k1 (z0 , r) = Kµ (z)| dz|, D∩S(z0 ,r)

at each point z0 ∈ D, then the Beltrami equation (1.1) has a multi-valued solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R. Corollary 8.3. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves and µ : D → C a measurable function such that 1 as ε → 0 for all z0 ∈ D, (8.4) kz0 (ε) = O log ε where kz0 (ε) is the average of the function Kµ (z) over S(z0 , ε). Then the Beltrami equation (1.1) has a multi-valued solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R. Remark 8.1. In particular, the conclusion holds if 1 (8.5) Kµ (z) = O log as z → z0 for all z0 ∈ D. |z − z0 | Theorem 8.3. Let D be a bounded domain in C whose boundary consists of n ≥ 2 mutually disjoint Jordan curves and µ : D → C a measurable function such that |µ(z)| < 1 a.e. and Z 8 Kµ (z) dm(z) < ∞ (8.6) D


139

for a convex non-decreasing function 8 : [0, ∞] → [0, ∞]. If Z ∞ dτ =∞ (8.7) −1 (τ) τ8 δ for some δ > 8(0), the Beltrami equation (1.1) has a multi-valued solution of the Dirichlet problem (1.5) for each continuous function ϕ : ∂D → R. Corollary 8.4. The conclusion of Theorem 8.3 holds if, for some α > 0, Z (8.8) eαKµ (z) dm(z) < ∞. D

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Denis Kovtonyuk INSTITUTE OF A PPLIED MATHEMATICS AND MECHANICS NATIONAL ACADEMY OF S CIENCES OF U KRAINE 74 ROZE L UXEMBURG STR ., 83114 D ONETSK , U KRAINE email: denis [email protected]

Igor Petkov INSTITUTE OF A PPLIED MATHEMATICS AND MECHANICS NATIONAL ACADEMY OF S CIENCES OF U KRAINE 74 ROZE L UXEMBURG STR ., 83114 D ONETSK , U KRAINE email: [email protected]

Vladimir Ryazanov INSTITUTE OF A PPLIED MATHEMATICS AND MECHANICS NATIONAL ACADEMY OF S CIENCES OF U KRAINE 74 ROZE L UXEMBURG STR ., 83114 D ONETSK , U KRAINE email: vl [email protected],[email protected]

Ruslan Salimov INSTITUTE OF A PPLIED MATHEMATICS AND MECHANICS NATIONAL ACADEMY OF S CIENCES OF U KRAINE 74 ROZE L UXEMBURG STR ., 83114 D ONETSK , U KRAINE email: [email protected]

(Received June 12, 2012 and in revised form May 21, 2013)