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On boundary behavior and Dirichlet problem for Beltrami equations D. Kovtonyuk, I. Petkov, V. Ryazanov and R. Salimov 2 ARXIV-260112-KPRS.tex 1 January 27, 2012 ( ) 0 2 n a Abstract J 6 1,1 2 We show that homeomorphic W solutions to the Beltrami equations ∂f = µ∂f satisfy certain loc moduli inequalities. On this basis, we develope the theory of the boundary behavior of such solutions ] V and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for C the Dirichlet problem to the Beltrami equations in Jordan domains and finitely connected domains. . h t 2000 Mathematics Subject Classification: Primary 30C65; Secondary 30C75 a m Key words: Dirichlet problem, Beltrami equation, moduli inequalities, boundary behavior, homeo- [ morphic, regular, pseudoregular, multi-valued solutions. 1 v 0 1 Introduction 7 5 5 C C Let D be a domain in the complex plane , i.e., a connected open subset of , and . 1 C 0 let µ : D → be a measurable function with |µ(z)| < 1 a.e. (almost everywhere) 2 in D. A Beltrami equation is an equation of the form 1 : v i f = µ(z)f (1.1) X z¯ z r a where f = ∂f = (f + if )/2, f = ∂f = (f − if )/2, z = x + iy, and f and f z¯ x y z x y x y are partial derivatives of f in x and y, correspondingly. The function µ is called the complex coefficient and 1 + |µ(z)| K (z) = (1.2) µ 1 − |µ(z)| the dilatation quotient of the equation (1.1). The Beltrami equation (1.1) is said to be degenerate if esssupK (z) = ∞. The existence of homeomorphic µ 1,1 W solutions was recently established to many degenerate Beltrami equations, loc see, e.g., related references in the recent monograph [24] and in the surveys [12] and [42]. Boundary value problems for the Beltrami equations are due to the well-known Riemann dissertation in the case of µ(z) = 0 and to the papers of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy–Riemann system. The Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., [1] and [45]. The Dirichlet problem for degenerate Beltrami equations in the unit disk was studied in [6]. However, the criteria for the existence of solutions for the Dirichlet problem in [6] are not invariant under conformal mappings of Riemann. Hence we give here the corresponding theorems on the existence of regular solutions in arbitraryJordan domainsas well as of pseudoregular and multi-valued solutionsin arbitrary multiply connected domains bounded by a finite collection of mutually disjoint Jordan curves. In comparison with the work [6], our approach is based on estimates of the modulus of dashed lines but not of paths under arbitrary 1,1 homeomorphic W solutions of the Beltrami equations. loc C Recall that every holomorphic(analytic)function f in a domain D ⊂ satisfies the simplest Beltrami equation f = 0 (1.3) z¯ D with µ(z) ≡ 0. If a holomorphic function f given in the unit disk is continuous in its closure, then by the Schwarz formula 1 ζ + z dζ f(z) = iImf(0) + Ref(ζ) · , (1.4) 2πi Z ζ − z ζ |ζ|=1 see, e.g., Section 8, Chapter III, Part 3 in [15]. Thus, the holomorphic function D f in the unit disk is determinated, up to a purely imaginary additive constant D ic, c = Imf(0), by its real part ϕ(ζ) = Ref(ζ) on the boundary of . Hence the Dirichlet problem for the Beltrami equation (1.1) in a domain C C D ⊂ is the problem on the existence of a continuous function f : D → having partial derivatives of the first order a.e., satisfying (1.1) a.e. and such that limRef(z) = ϕ(ζ) ∀ ζ ∈ ∂D (1.5) z→ζ R for a prescribed continuous function ϕ : ∂D → . It is obvious that if f is a R solution of this problem, then the function F(z) = f(z) + ic, c ∈ , is so. Here we show that the Dirichlet problem (1.5) has regular solutions in an arbitrary Jordan domain and pseudoregular and multi-valued solutionsin an arbitrary finitely connected domain for wide classes of degenerate Beltrami equations (1.1). Finally, it is necessary to note that the existence of solutions for the Dirichlet problem to the degenerate Beltrami equations with two characteristics f = µ(z)f + ν(z)f (1.6) z¯ z z remains open although the corresponding theorems on the existence of homeo- 1,1 morphic W solutions of (1.6) have been established in the series of the recent loc papers [2]–[4]. This problem is important because the Beltrami equations of the second type f = ν(z)f (1.7) z¯ z play a great role in many problems of mathematical physics, see, e.g., [21]. How- ever, the Dirichlet problem for the equation (1.7) demands an essential modifica- tion of our approach. 2 Preliminaries C D Throughout this paper, B(z ,r) = {z ∈ : |z −z| < r}, = B(0,1), S(z ,r) = 0 0 0 C C {z ∈ : |z − z| = r}, S(r) = S(0,r), R(z ,r ,r ) = {z ∈ : r < |z −z | < r }. 0 0 1 2 1 0 2 The class BMO was introduced by John and Nirenberg (1961) in the paper [16] and soon became an important concept in harmonic analysis, partial differential equations and related areas; see, e.g., [13] and [40]. C Recall that a real-valued function u in a domain D in is said to be of bounded mean oscillation in D, abbr. u ∈ BMO(D), if u ∈ L1 (D) and loc 1 kuk := sup |u(z) − u |dm(z) < ∞, (2.1) ∗ B |B| Z B B where the supremum is taken over all discs B in D, dm(z) corresponds to the C Lebesgue measure in and 1 u = u(z)dm(z). B |B| Z B We write u ∈ BMO (D) if u ∈ BMO(U) for every relatively compact subdomain loc U of D (we also write BMO or BMO if it is clear from the context what D is). loc Following the paper [17], see also [18] and [24], we say that a function ϕ : D → R has finite mean oscillation at a point z ∈ D if 0 lim − |ϕ(z) − ϕ (z )|dm(z) < ∞, (2.2) ε 0 ε→0 Z B(z0,ε) where e ϕ (z ) = − ϕ(z)dm(z) (2.3) ε 0 Z B(z0,ε) is the mean value of the feunction ϕ(z) over the disk B(z ,ε). Note that the 0 condition (2.2) includes the assumption that ϕ is integrable in some neighborhood R of the point z . We say also that a function ϕ : D → is of finite mean 0 oscillation in D, abbr. ϕ ∈ FMO(D) or simply ϕ ∈ FMO, if ϕ ∈ FMO(z ) for 0 C all points z ∈ D. We write ϕ ∈ FMO(D) if ϕ is given in a domain G in such 0 that D ⊂ G and ϕ ∈ FMO(z ) for all z ∈ D. 0 0 The following statement is obvious by the triangle inequality. R Proposition 2.1. If, for a collection of numbers ϕ ∈ , ε ∈ (0,ε ], ε 0 lim − |ϕ(z) − ϕ |dm(z) < ∞, (2.4) ε ε→0 Z B(z0,ε) then ϕ is of finite mean oscillation at z . 0 In particular choosing in Proposition 2.1, ϕ ≡ 0, ε ∈ (0,ε ], we obtain the ε 0 following. Corollary 2.1. If, for a point z ∈ D, 0 lim − |ϕ(z)|dm(z) < ∞, (2.5) ε→0 Z B(z0,ε) then ϕ has finite mean oscillation at z . 0 R Recall that a point z ∈ D is called a Lebesgue point of a function ϕ : D → 0 if ϕ is integrable in a neighborhood of z and 0 lim − |ϕ(z) − ϕ(z )|dm(z) = 0. (2.6) 0 ε→0 Z B(z0,ε) It is known that, almost every point in D is a Lebesgue point for every function ϕ ∈ L1(D). Thus we have by Proposition 2.1 the following corollary. R Corollary 2.2. Every locally integrable function ϕ : D → , has a 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 ∆, see, e.g., [40], p. 5, and hence also to FMO. However, ϕ (0) → ε ∞ as ε → 0, showing that condition (2.5) is only sufficient but not necessary e for a function ϕ to be of finite mean oscillation at z . Clearly, BMO(D) ⊂ 0 p BMO (D) ⊂ FMO(D) and as well-known BMO ⊂ L for all p ∈ [1,∞), see, loc loc loc e.g., [5], by FMO is not a subclass of Lp for any p > 1 but only L1 . Thus the loc loc class FMO is much more wide than BMO . loc Versions of the next lemma has been first proved for the class BMO in the planar case in [32], [33], and then in the space case in [22], [23]. For the FMO case, see the papers [17], [18], [31], [34], [35], and the monograph [24]. C R Lemma 2.1. Let D be a domain in and let ϕ : D → be a non-negative function of the class FMO(z ) for some z ∈ D. Then 0 0 ϕ(z)dm(z) 1 = O loglog as ε → 0 (2.7) Z 2 (cid:18) ε(cid:19) |z − z |log 1 ε<|z−z0|<ε0 0 |z−z0| (cid:16) (cid:17) for some ε ∈ (0,δ ) where δ = min(e−e,d ), d = sup |z − z |. 0 0 0 0 0 z∈D 0 Recall a connection between some integral conditions, see, e.g., [37]–[39]. D Theorem 2.1. Let Q : → [0,∞] be a measurable function such that Φ(Q(z))dm(z) < ∞ (2.8) Z D where Φ : [0,∞] → [0,∞] is a non-decreasing convex function such that ∞ dτ = ∞ (2.9) Z τΦ−1(τ) δ0 for some δ > Φ(0). Then 0 1 dr = ∞ (2.10) Z rq(r) 0 where q(r) is the average of the function Q(z) over the circle |z| = r. The following lemma is also useful, see Lemma 2.1 in [20] or Lemma 9.2 in [24]. Lemma 2.2. Let (X,µ) be a measure space with a finite measure µ, p ∈ (1,∞) and let ϕ : X → (0,∞) be a measurable function. Set I(ϕ,p) = inf ϕαpdµ (2.11) α Z X where the infimum is taken over all measurable functions α : X → [0,∞] such that αdµ = 1. (2.12) Z X Then 1 − λ I(ϕ,p) =  ϕ−λdµ (2.13) Z X  where q 1 1 λ = , + = 1, (2.14) p p q i.e. λ = 1/(p−1) ∈ (0,∞). Moreover, the infimum in (2.11) is attained only for the function α = C · ϕ−λ (2.15) 0 where −1 C =  ϕ−λdµ . (2.16) Z X  C Finally, recall that the (conformal) modulus of a family Γ of paths γ in is the quantity M(Γ) = inf ̺2(z)dm(z) (2.17) ̺∈admΓZ C C where a Borel function ̺ : → [0,∞] is admissible for Γ, write ̺ ∈ admΓ, if ̺ds > 1 ∀ γ ∈ Γ. (2.18) Z γ Here s is a natural parameter of the length on γ. 3 On regular domains C First of all, recall the following topological notion. A domain D ⊂ is said to be locally connected at a point z ∈ ∂D if, for every neighborhood U of the 0 point z , there is a neighborhood V ⊆ U of z such that V ∩ D is connected. If 0 0 this condition holds for all z ∈ ∂D, then D is said to be locally connected on 0 C ∂D. Note that every Jordan domain D in is locally connected on ∂D, see, e.g., [46], p. 66. U V z 0 D ¶D We say that ∂D is weakly flat at a point z ∈ ∂D if, for every neighborhood 0 U of the point z and every number P > 0, there is a neighborhood V ⊂ U of z 0 0 such that M(∆(E,F;D)) > P (3.1) 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 z ∈ ∂D. 0 U V z 0 ¶D E F D We also say that a point z ∈ ∂D is strongly accessible if, for every neigh- 0 borhood U of the point z , there exist a compactum E in D, a neighborhood 0 V ⊂ U of z and a number δ > 0 such that 0 M(∆(E,F;D)) > δ (3.2) for all continua F in D intersecting ∂U and ∂V. We say that ∂D is strongly accessible if each point z ∈ ∂D is strongly accessible. 0 Here, in the definitions of strongly accessible and weakly flat boundaries, one can take as neighborhoods U and V of a point z only balls (closed or open) 0 centered at z or only neighborhoods of z in another fundamental system of 0 0 neighborhoods of z . These conceptions can also be extended in a natural way to 0 C the case of and z = ∞. Then we must use the corresponding neighborhoods 0 of ∞. C It is easy to see that if a domain D in is weakly flat at a point z ∈ ∂D, then 0 the point z is strongly accessible from D. Moreover, it was proved by us that if 0 C a domain D in is weakly flat at a point z ∈ ∂D, then D is locally connected 0 at z , see, e.g., Lemma 5.1 in [20] or Lemma 3.15 in [24]. 0 The notions of strong accessibility and weak flatness at boundary points of a C domain in defined in [19], see also [20] and [31], are localizations and generalizations of the corresponding notions introduced in [22] and [23], cf. with the properties P and P by Väisäla¨ in [44] and also with the quasiconformal acces- 1 2 sibility and the quasiconformal flatness by Näkki in [29]. Many theorems on a 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 role for a continuous extension of the mappings to the boundary. C A domain D ⊂ is called a quasiextremal distance domain, abbr. QED- domain, see [10], if M(∆(E,F;C) 6 K · M(∆(E,F;D)) (3.3) for some K > 1 and all pairs of nonintersecting continua E and F in D. It is well known, see, e.g., Theorem 10.12 in [44], that 2 R M(∆(E,F;C)) > log (3.4) π r C for any sets E and F in intersecting all the circles S(z ,ρ), ρ ∈ (r,R). Hence 0 a QED-domain has a weakly flat boundary. One example in [24], Section 3.8, shows that the inverse conclusion is not true even in the case of simply connected C domains in . C A domain D ⊂ is called a uniform domain if each pair of points z and 1 z ∈ D can be joined with a rectifiable curve γ in D such that 2 s(γ) 6 a · |z − z | (3.5) 1 2 and min s(γ(z ,z)) 6 b · dist(z,∂D) (3.6) i i=1,2 for all z ∈ γ where γ(z ,z) is the portion of γ bounded by z and z, see [25]. i i It is known that every uniform domain is a QED-domain but there exist QED- domains that are not uniform, see [10]. 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. In the mapping theory and in the theory of differential equations, it is also often applied the so-called Lipschitz boundaries. C They say that a domain D in is Lipschitz if every point z ∈ ∂D has a 0 neighborhood U that can be mapped by a bi-Lipschitz homeomorphism ϕ onto D C D the unit disk ⊂ in such a way that ϕ(∂D ∩ U) is the intersection of with the real axis. Note that a bi-Lipschitz homeomorphism is quasiconformal and the modulus is a quasiinvariant under such mappings. Hence the Lipschitz domains have weakly flat boundaries. Recall that a map ϕ : U → C is said to be Lipcshitz provided |ϕ(z )−ϕ(z )| 6 1 2 M · |z − z | for some M < ∞ and for all z and z ∈ U, and bi-Lipcshitz if in 1 2 1 2 addition M∗|z −z | 6 |ϕ(z )−ϕ(z )| for some M∗ > 0 and for all z and z ∈ U. 1 2 1 2 1 2 4 On estimates of modulus of dashed lines R A continuous mapping γ of an open subset ∆ of the real axis or a circle into D is called a dashed line, see, e.g., 6.3 in [24]. Note that such a set ∆ consists of a R countable collection of mutually disjoint intervals in . This is the motivation for the term. The notion of the modulus of the family Γ of dashed lines γ is defined similarly to (2.17). We say that a property P holds for a.e. (almost every) γ ∈ Γ if a subfamily of all lines in Γ for which P fails has the modulus zero, cf. C [8]. Later on, we also say that a Lebesgue measurable function ̺ : → [0,∞] is extensively admissible for Γ, write ̺ ∈ extadmΓ, if (2.18) holds for a.e. γ ∈ Γ, see, e.g., Section 9.2 in [24]. 1,1 Theorem 4.1. Let f be a homeomorphic W solution of the Beltrami equa- loc C tion (1.1) in a domain D ⊆ . Then ̺2(z) M(fΣ ) > inf dm(z) (4.1) ε ̺∈extadmΣεZ Kµ(z) D for all z ∈ D, where ε ∈ (0,ε ), ε ∈ (0,d ), d = sup |z −z | and Σ denotes 0 0 0 0 0 z∈D 0 ε the family of dashed lines consisting of all intersections of the circles S(z ,r), 0 r ∈ (ε,ε ), with D. 0 Proof. Let B be a (Borel) set of all points z in D where f has a total differential with J (z) 6= 0. It is known that B is the union of a countable collection of Borel f sets B , l = 1,2,..., such that f = f| is a bi-Lipschitz homeomorphism, see, l l B l e.g., Lemma 3.2.2 in [7]. With no loss of generality, we may assume that the B l are mutually disjoint. Denote also by B the set of all points z ∈ D where f has ∗ a total differential with f′(z) = 0. C Note that the set B = D \ (B ∪ B ) has the Lebesgue measure zero in 0 ∗ by the well known Gehring–Lehto–Menchoff theorem, see [9] and [28]. Hence by Theorem 2.11 in [20], see also Lemma 9.1 in [24], length(γ∩B ) = 0 for a.e. paths 0 γ in D. Let us show that length(f(γ) ∩ f(B )) = 0 for a.e. circle γ centered at 0 z ∈ D. 0 The latter follows from absolute continuity of f on closed subarcs of γ ∩D for 1,1 a.e. such circle γ. Indeed, the class W is invariant with respect to local quasi- loc 1,1 isometries and the functions in W is absolutely continuous on lines, see, e.g., loc Theorems 1.1.7 and 1.1.3 in [27], respectively. Applying say the transformation of coordinates log(z − z ), we come to the absolute continuity on a.e. such circle 0 γ. Fix γ on which f is absolutely continuous and length(γ ∩ B ) = 0. Then 0 0 0 length(f(γ)∩f(B )) = lengthf(γ ∩B ) and for every ε > 0 there is an open set 0 0 0 ω in γ ∩D such that γ ∩B ⊂ ω with lengthω < ε, see, e.g., Theorem III(6.6) ε 0 0 0 ε ε in [41]. The open set ω consists of a countable collection of open arcs γ of the ε i circle γ . By the construction lengthγ < ε and by the absolute continuity of 0 i i P f on γ the sum δ := lengthf(γ ) is arbitrarily small for small enough ε > 0. 0 i i P Hence lengthf(γ ∩ B ) = 0. 0 0 Thus, length(γ ∩ f(B )) = 0 where γ = f(γ) for a.e. circle γ centered at z . ∗ 0 ∗ 0 Now, let ̺ ∈ admf(Γ) where Γ is the collection of all dashed lines γ∩D for such ∗ circles γ and ̺ ≡ 0 outside f(D). Set ̺ ≡ 0 outside D and on B and ∗ 0 ̺(z): = ̺ (f(z))(|f | + |f |) for z ∈ B ∪ B . ∗ z z¯ ∗ Arguing piecewise on B , we have by Theorem 3.2.5 under m = 1 in [7] that l ̺ds > ̺ ds > 1 for a.e. γ ∈ Γ ∗ ∗ Z Z γ γ∗ because length(f(γ) ∩ f(B )) = 0 and length(f(γ) ∩ f(B )) = 0 for a.e. γ ∈ Γ, 0 ∗ consequently, ̺ ∈ extadmΓ. On the other hand, again arguing piecewise on B , we have the inequality l ̺2(z) dm(z) 6 ̺2(w)dm(w) Z K (z) Z ∗ µ D f(D) because ̺(z) = 0 on B . Thus, we obtain (4.1). ∗ 1,1 Theorem 4.2. Let f be a homeomorphic W solution of the Beltrami equa- loc C tion (1.1) in a domain D ⊆ .Then ε0 dr M(fΣ ) > , ∀ z ∈ D, ε ∈ (0,ε ), ε ∈ (0,d ), (4.2) ε 0 0 0 0 Z ||K || (z ,r) µ 1 0 ε where d = sup |z −z |, Σ denotes the family of dashed lines consisting of all 0 z∈D 0 ε the intersections of the circles S(z ,r), r ∈ (ε,ε ), with D and 0 0 ||K || (z ,r) := K (z)|dz| (4.3) µ 1 0 µ Z D(z0,r) is the norm in L of K over D(z ,r) = {z ∈ D : |z − z | = r} = D ∩ S(z ,r). 1 µ 0 0 0 Proof. Indeed, for every ̺ ∈ extadmΣ , ε A (r) = ̺(z)|dz| 6= 0 a.e. in r ∈ (ε,ε ) ̺ 0 Z D(z0,r) is a measurable function in the parameter r, say by the Fubini theorem. Thus, we may request the equality A (r) ≡ 1 a.e. in r ∈ (ε,ε ) and ̺ 0 ε0 ̺2(z) α2(z) inf dm(z) =  inf |dz|dr ̺∈extadmΣε Z Kµ(z) Z α∈I(r) Z Kµ(z) D∩Rε ε  D(z0,r)    where R = R(z ,ε,ε ) and I(r) denotes the set of all measurable functions α on ε 0 0 the dashed line D(z ,r) = S(z ,r) ∩ D such that 0 0 α(z)|dz| = 1. Z D(z0,r) Hence Theorem 4.2 follows by Lemma 2.2 with X = D(z ,r), the length as a 0 measure µ on D(x ,r), ϕ = 1 | and p = 2. 0 K D(z0,r) µ Later on, the following lemma will be useful, too. Here we use the standard conventions a/∞ = 0 for a 6= ∞ and a/0 = ∞ if a > 0 and a · ∞ = 0, see, e.g., [41], p. 6. C Lemma 4.1. Let Q : → (0,∞) be a locally integrable function. Set C ||Q|| (z ,r) = Q(z)|dz|, z ∈ , r ∈ (0,∞) , (4.4) 1 0 0 Z S(z0,r) r2 dr 1 I = , η (r) = , r ∈ (r ,r ), 0 < r < r < ∞. 0 1 2 1 2 Z ||Q|| (z ,r) I · kQk (z ,r) 1 0 1 0 r1 (4.5)