Bojarski, B.V.

Gutlyanskiĭ, V.Ya.

Ryazanov, V.I.

Reports of the National Academy of Sciences of Ukraine

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 517.5B.V. Bojarski,Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanski˘ı,V. I. RyazanovThe Dirichlet problem for a Beltrami equation of thesecond typeСriteria on the existence of regular solutions of the Dirichlet problem for the degenerate Beltramiequation ∂f = ν∂f in a Jordan domain of the complex plane C are given.Let D be a domain in the complex plane C. Throughout this paper, we use the notations z == x+ iy, B(z0, r) : = {z ∈ C : |z − z0| < r} for z0 ∈ C and r > 0, B(r) : = B(0, r), B : = B(1),and C : = C⋃∞.The purpose of this paper is to study the Dirichlet problem fz = ν(z) · fz, z ∈ D,limz→ζRef(z) = ϕ(ζ), ∀ ζ ∈ ∂D,(1)in a Jordan domain D of the complex plane C with continuous boundary data ϕ(ζ). Here, ν(z)stands for a measurable coeﬃcient satisfying the inequality |ν(z)| < 1 a. e. in D. The degeneracyof the ellipticity for Beltrami equations of the second type fz = ν(z) · fz is controlled by thedilatation coeﬃcientKν(z) :=1 + |ν(z)|1− |ν(z)|∈ L1loc. (2)Note that the Beltrami equations of the second type take a key part in many problems ofmathematical physics, see, e. g., [11].We will look for a solution as a continuous, discrete, and open mapping f : D → C of theSobolev classW 1,1loc and such that the Jacobian Jf (z) 6= 0 a. e. in D. Such a solution will be calleda regular solution of the Dirichlet problem (1) in a domain D. Recall that a mapping f : D → Cis called discrete if the preimage f−1(y) consists of isolated points for every y ∈ C and open if fmaps every open set U ⊆ D onto an open set in C.For the uniformly elliptic case, i. e. when Kν(z) 6 K < ∞ a. e. in D, the Dirichlet problemwas studied in [1]. The solvability of the Dirichlet problem for the degenerate Beltrami equationsof the ﬁrst typefz = µ(z) · fz (3)was studied in [5, 8], and [12]. Recall that the problem of the existence of homeomorphic solutionsfor equation (3) was solved in the uniformly elliptic case where ‖µ‖∞ < 1 long ago, see, e. g., [1].The existence problem for the degenerate Beltrami equations (3), whenKµ /∈ L∞, is currentlyan active area of research, see, e. g., survey [7] and monograph [8] and references therein. A seriesof criteria on the existence of regular solutions for a Beltrami equation of the second type were© B.V. Bojarski, V.Ya. Gutlyanski˘ı, V. I. Ryazanov, 2013ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №6 29given in our recent papers [3, 4]. There, a homeomorphism f ∈W 1,1loc (D) is called a regular solutionof the equation if f satisﬁes the equation a. e. in D, and Jf (z) = |fz|2 − |fz|2 6= 0 a. e. in D.To derive criteria of the existence of regular solutions for the Dirichlet problem (1) in a Jordandomain D ⊂ C, we make use of the approximate procedure based on the existence theorems inthe case Kν ∈ L∞ given in [1] and convergence theorems for the Beltrami equations of the secondtype when Kν ∈ L1loc established in [3]. The Schwarz formulaf(z) = i Im f(0) +12pii∫|ζ|=1Re f(ζ) ·ζ + zζ − zdζζ, (4)that allows one to recover an analytic function f in the unit disk B by its real part ϕ(ζ) = Ref(ζ)on the boundary of B up to a purely imaginary additive constant c = iImf(0), see, e. g., Section 8,Chapter III, Part 3 in [6], and the Arzela–Ascoli theorem combined with moduli techniques arealso used.2. On BMO, VMO, and FMO functions. Recall that a real-valued function u in adomain D in C is said to be a bounded mean oscillation in D, abbr. u ∈ BMO(D), if u ∈ L1loc(D)and‖u‖∗ := supB1|B|∫B|u(z)− uB | dxdy <∞, (5)where the supremum is taken over all disks B in D anduB =1|B|∫Bu(z) dxdy.We write u ∈ BMOloc(D) if u ∈ BMO(U) for every relatively compact subdomain U of D (wealso write BMO or BMOloc if it is clear from the context what D is).The class BMO was introduced by John and Nirenberg (1961) in [10] and soon became animportant concept in harmonic analysis, the theory of partial diﬀerential equations, and relatedareas.A function u in BMO is said to have a vanishing mean oscillation, abbr. u ∈ VMO, if thesupremum in (5) taken over all balls B in D with |B| < ε converges to 0 as ε→ 0. VMO has beenintroduced by Sarason in [15]. There exist a number of papers devoted to the study of partialdiﬀerential equations with coeﬃcients of the class VMO.Following [9], we say that a function u : D → R has a finite mean oscillation at a pointz0 ∈ D iflimε→0−∫B(z0,ε)|u(z)− u˜ε(z0)| dxdy <∞, (6)where u˜ε(z0) u˜ε(z0) = −∫B(z0,ε)u(z) dxdy is the mean value of the function u(z) over the diskB(z0, ε) with small ε > 0. We also say that a function u : D → R has a finite mean oscillationin D, abbr. u ∈ FMO(D) or simply u ∈ FMO, if relation (6) holds at every point z0 ∈ D.Remark 1. Clearly, BMO ⊂ FMO. There exist the examples showing that FMO is not BMOloc,see, e. g., [8]. By deﬁnition, FMO ⊂ L1loc, but FMO is not a subset of Lploc for any p > 1 incomparison with BMOloc ⊂ Lploc for all p ∈ [1,∞).30 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №6Proposition 1. If, for some collection of numbers uε ∈ R, ε ∈ (0, ε0],limε→0−∫B(z0,ε)|u(z)− uε| dxdy <∞, (7)then u has a finite mean oscillation at z0.Corollary 1. If, for a point z0 ∈ D,limε→0−∫B(z0,ε)|u(z)| dxdy <∞ (8)then u has a finite mean oscillation at z0.Remark 2. Note that the function u(z) = log(1/|z|) belongs to BMO in the unit disk B andhence also to FMO. However, u˜ε(0)→∞ as ε→ 0, showing that condition (8) is only suﬃcient,but not necessary for a function u to have a ﬁnite mean oscillation at z0.Lemma 1. Let u : D → R be a nonnegative function with finite mean oscillation at 0 ∈ D,and let u be integrable in B(0, e−1) ⊂ D. Then∫A(ε,e−1)u(z) dxdy(|z| log 1|z|)2 6 C · log log 1ε , ∀ ε ∈ (0, e−e). (9)Here, we use the notation A(ε, ε0) = {z ∈ C : ε < |z| < ε0}.3. The main lemma. The following lemma is the main tool for deriving criteria on theexistence of regular solutions for the Dirichlt problem with degenerate Beltrami equations of thesecond type in a Jordan domain D ⊂ C.Lemma 2. Let D be a Jordan domain in C with 0 ∈ D and let ν : D → C be a measurablefunction with Kν ∈ L1(D). Suppose that, for every z0 ∈ D, there exist ε0 = ε(z0) > 0 and afamily of measurable functions ψz0,ε : (0,∞) → (0,∞), ε ∈ (0, ε0), such that0 < Iz0(ε) :=ε0∫εψz0,ε(t) dt <∞, (10)and such that∫ε<|z−z0|<ε0Kν(z) · ψ2z0,ε(|z − z0|) dxdy = o(I2z0(ε)) (11)as ε → 0. Then the Dirichlet problem (1) has a regular solution f with Im f(0) = 0 for eachnonconstant continuous function ϕ : ∂D → R.Here, we assume that ν is extended by zero outside the domain D.Corollary 2. Let D be a Jordan domain in C with 0 ∈ D and let ν : B→ C be a measurablefunction with Kν ∈ L1(B). Suppose that, for every z0 ∈ B and some ε0 > 0,∫ε<|z−z0|<ε0Kν(z) · ψ2(|z − z0|) dxdy 6 O( ε0∫εψ(t) dt)(12)ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №6 31as ε → 0, where ψ : (0,∞) → (0,∞) is a measurable function such thatε0∫0ψ(t) dt =∞, 0 <ε0∫εψ(t) dt <∞, ∀ ε ∈ (0, ε0). (13)Then the Dirichlet problem (1) has a regular solution f with Im f(0) = 0 for each nonconstantcontinuous function ϕ : ∂D → R.4. Existence theorems. Everywhere further, we assume that the function ν : D → C isextended by zero outside the domain D. In particular, by Lemmas 1 and 2 with ψz0,ε(t) ≡≡ 1/(t log 1/t), we have the following result.Theorem 1. Let D be a Jordan domain in C with 0 ∈ D and let ν : D → B be a measurablefunction such that Kν(z) 6 Q(z) ∈ FMO. Then the Dirichlet problem (1) has a regular solution fwith Im f(0) = 0 for each nonconstant continuous function ϕ : ∂D → R.Now, Theorem 1 and Corollary 1 yield the following conclusion.Corollary 3. In particular, iflimε→0−∫B(z0,ε)1 + |ν(z)|1− |ν(z)|dxdy <∞, ∀ z0 ∈ D, (14)then the Dirichlet problem (1) in a Jordan domain D, 0 ∈ D, has a regular solution f withIm f(0) = 0 for each nonconstant continuous function ϕ : ∂D → R.Similarly, choosing the function ψz0,ε(t) ≡ 1/t, in Lemma 2, we come to the followingstatement.Theorem 2. Let D be a Jordan domain in C with 0 ∈ D and let ν : D → B be a measurablefunction such that Kν ∈ L1loc(D). Suppose that∫ε<|z−z0|<ε0Kν(z)dxdy|z − z0|2= o([log1ε]2), ∀ z0 ∈ D, (15)as ε → 0 for some ε0 = δ(z0). Then the Dirichlet problem (1) has a regular solution f withIm f(0) = 0 for each nonconstant continuous function ϕ : ∂D → R.Remark 3. Choosing the function ψ(t) = 1/(t log 1/t) instead of ψ(t) = 1/t in Lemma 2, weare able to replace (15) by∫ε<|z−z0|<ε0Kν(z) dxdy(|z − z0| log1|z − z0|)2 = o([log log 1ε]2). (16)In general, we are able to give here the whole scale of the corresponding conditions in log, usingfunctions ψ(t) of the form 1/(t log 1/t · log log 1/t · · · · · log . . . log 1/t).Theorem 3. Let D be a Jordan domain in C with 0 ∈ D, let ν : D → B be a measurablefunction, Kν ∈ L1(D), and let kz0(r) be the mean value of Kν(z) over the circle |z − z0| = r.Suppose thatδ(z0)∫0drrkz0(r)=∞, ∀ z0 ∈ D. (17)32 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №6Then the Dirichlet problem (1) has a regular solution f with Imf(0) = 0 for each nonconstantcontinuous function ϕ : ∂D → R.Theorem 3 also follows from Lemma 2 by a special choice of the functional parameterψz0,ε(t) ≡ 1/[tkz0(t)].Corollary 4. In particular, the conclusion of Theorem 3 holds ifkz0(r) = O(log1r)as r→ 0, ∀ z0 ∈ D. (18)In fact, it is clear that condition (17) yields the whole scale of conditions in terms of log withthe use of functions of the form log 1/r ·log log 1/r · · · log · · · log 1/r on the right-hand side in (18).By Theorem 3.1 in [13], see also Theorem 3.17 in [14], it follows that conditions (19) and (20)below yield condition (17). Thus, by Theorem 3, we obtain the following signiﬁcant result.Theorem 4. Let D be a Jordan domain in C with 0 ∈ D, and let µ and ν : D → B bemeasurable functions such that∫DΦ(Kν(z)) dxdy <∞, (19)where Φ: [0,∞] → [0,∞] is a non-decreasing convex function satisfying the condition∞∫δdττΦ−1(τ)=∞ (20)for some δ > Φ(0). Then the Dirichlet problem (1) has a regular solution f with Im f(0) = 0 foreach nonconstant continuous function ϕ : ∂D → R.Remark 4. Note also that we may assume in Theorem 4 that the function Φ(t) is not convexand non-decreasing on the whole segment [0,∞], but only on a segment [T,∞] for some T ∈∈ (1,∞).Corollary 5. In particular, the conclusion of Theorem 4 holds if, for some α > 0,∫DeαKν(z)dxdy <∞. (21)Finally we would like to note that our approach makes it possible to study the Dirichletproblem for the degenerate Beltrami equations also in ﬁnitely connected domains of the complexplane.1. Bojarski B. Generalized solutions of a system of differential equations of the first order of the elliptic typewith discontinuous coefficients // Mat. Sb. – 1957. – 43 (85), No 4. – P. 451–503 [in Russian]; transl. inRep. Univ. Jyva¨skyla¨, Dept. Math. Stat. – 2009. – 118. – P. 1–64.2. Bojarski B., Gutlyanskii V., Ryazanov V. General Beltrami equations and BMO // Ukr. Mat. Visn. –2008. – 5, No 3. – P. 305–326; transl. in Ukrainian Math. Bull. – 2008. – 5, No 3. – P. 305–326.3. Bojarski B., Gutlyanskii V., Ryazanov V. On Beltrami equations with two characteristics // Comp. Var.Ell. Eq. – 2009. – 54, No 10. – P. 933–950.4. Bojarski B., Gutlyanskii V., Ryazanov V. On integral conditions for the general Beltrami equations //Comp. Anal. Oper. Theory. – 2011. – 5, No 3. – P. 835–845.5. Dybov Yu. On regular solutions of the Dirichlet problem for the Beltrami equations // Comp. Var. Ell.Equ. – 2010. – 55, No 12. – P. 1099–1116.ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №6 336. Hurwitz A., Courant R. Theory of functions. – New York: Chelsea. – 1958. – 1960. – 648 p.7. Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. On recent advances in the degenerate Beltrami equa-tions // Ukr. Mat. Visn. – 2010. – 7, No 4. – P. 467–515; transl. in J. Math. Sci. – 2011. – 175, No 4. –P. 413–449.8. Gutlyanskii V., Ryazanov V., Srebro U. Yakubov E. The Beltrami equation. A geometric approach /Developments in Mathematics. Vol. 26. – New York: Springer, 2012. – 301 p.9. Ignat’ev A., Ryazanov V. Finite mean oscillation in the mapping theory // Ukr. Mat. Visn. – 2005. – 2,No 3. – P. 395–417 [in Russian]; transl. in Ukrainian Math. Bull. – 2005. – 2, No 3. – P. 403–424.10. John F., Nirenberg L. On functions of bounded mean oscillation // Comm. Pure Appl. Math. – 1961. –14. – P. 415–426.11. Krushkal’ S. L., Ku¨hnau R. Quasiconformal mappings: new methods and applications. – Novosibirsk:Nauka, 1984. – 216 p. [in Russian].12. Kovtonyuk D.A., Petkov I.V., Ryazanov V. I. On the Dirichlet problem for the Beltrami equations infinitely connected domains // Ukr. Mat. Zh. – 2012. – 64. – P. 932–944.13. Ryazanov V., Srebro U., Yakubov E. Integral conditions in the mapping theory // Ukr. Mat. Visn. – 2010. –7, No 1. – P. 73–87; transl. in Math. Sci. J. – 2011. – 173, No 4. – P. 397–407.14. Ryazanov V., Srebro U., Yakubov E. Integral conditions in the theory of the Beltrami equations // Comp.Var. Ell. Equ. – 2012. – 57, No 12. – P. 1247–1270.15. Sarason D. Functions of vanishing mean oscillation // Trans. Amer. Math. Soc. – 1975. – 207. – P. 391–405.Received 25.12.2012Institute of Mathematics of Polish Academyof Sciences, Warsaw, PolandInstitute of Applied Mathematics and Mechanics,NAS of Sciences of Ukraine, Donetsk, UkraineБ.В. Боярський, член-кореспондент НАН України В.Я. Гутлянський,В.И. РязановЗадача Дiрiхле для рiвняння Бельтрамi другого родуДля виродженого рiвняння Бельтрамi ∂f = ν∂f доведено критерiї iснування регулярногорозв’язку задачи Дiрiхле у довiльнiй жордановiй областi комплексної площини C.Б.В. Боярский, член-корреспондент НАН Украины В.Я. Гутлянский,В.И. РязановЗадача Дирихле для уравнения Бельтрами второго родаДля вырожденного уравнения Бельтрами ∂f = ν∂f доказаны критерии существования ре-гулярного решения задачи Дирихле в произвольной жордановой области комплексной плос-кости C.34 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №6

The Dirichlet problem for a Beltrami equation of the second type

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The Dirichlet problem for a Beltrami equation of the second type

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)