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On General Degenerate Beltrami Equation

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On General Degenerate Beltrami Equation On general degenerate Beltrami equation B. Bojarski, V. Gutlyanski¸³ and V. Ryazanov We study the Beltrami equations @f = ¹(z)@f + º(z)@f under the assumption that the coe±cients ¹; º satisfy the inequality j¹j + jºj < 1 almost everywhere. Su±cient conditions for the existence of homeomorphic ACL solutions to the Beltrami equations are given in terms of the bounded mean oscillation by John and Nirenberg. 2000 Mathematics Subject Classi¯cation: Primary 30C65; Secondary 30C75 1. Introduction Let D be a domain in the complex plane C: We study the Beltrami equation @f = ¹(z)@f + º(z)@f a.e. in D (1.1) where @f = (fx + ify)=2 and @f = (fx ¡ ify)=2; z = x + iy; and ¹ and º are measurable functions in D with j¹(z)j + jº(z)j < 1 almost everywhere in D: Equation (1.1) arises, in particular, in the study of conformal mappings between two domains equipped with di®erent measurable Riemannian structures, see [22]. Equation (1.1) and second order PDE's of divergent form are also closely related. For instance, given a domain D; let σ be the class of symmetric matrices with measurable entries, satisfying 1 jhj2 · hσ(z)h; hi · K(z)jhj2; h 2 C: K(z) 1;2 Assume that u 2 Wloc (D) is a week solution of the equation div[σ(z)ru(z)] = 0: 1 2 On Degenerate Beltrami Equation Consider the mapping f = u + iv where rv(z) = Jf (z)σ(z)ru(z) and Jf (z) stands for the Jacobian determinant of f: It is easily to verify that f satis¯es the Beltrami equation (1.1). On the other hand, the above second order partial di®erential equation naturally appears in a number of problems of mathematical physics, see, e.g., [3]. In the case º(z) ´ 0 in (1.1) we recognize the classical Beltrami equation, which generates the quasiconformal mappings in the plane. Given an arbitrary measurable coe±cient ¹(z) in D ⊆ C; the well-known measurable Riemann map- ping theorem for the Beltrami equation @f = ¹(z)@f a.e. in D ⊆ C; (1.2) see, e.g., [1], Chapter 5, [24], Chapter 5, guarantees the existence of a homeomor- 1;2 phic solution f 2 Wloc (D) to the equation (1.2), which maps D onto an arbitrary conformally equivalent domain G: Moreover, the mapping f can be represented in the form f = F ± !; where F stands for an arbitrary conformal mapping of D onto G and ! is a quasiconformal self-mapping of D with complex dilatation ¹(z) a.e. in D: The corresponding measurable Riemann mapping theorem for the general Beltrami equation (1.1) was given in [5] and [6], Theorem 5.1 and Theorem 6.8, see also [38], Chapter 3, x17. Bellow we give its statement in the form, convenient for our application. 1.3. Theorem. Let B be the unit disk and G be a conformally equivalent simply connected domain. If ¹ and º are measurable functions in B with j¹(z)j+ jº(z)j · q < 1 a.e. in B; then there exists a quasiconformal mapping f : B ! G; satisfying the equation (1.1). The mapping f has the representation f = F ± ! where F stands for a conformal mapping of B onto G and the quasiconformal self-mapping ! of B can be normalized by !(0) = 0;!(1) = 1: If F (!) = !; then the normalized solution is unique. 1;2 Notice, that if f is a Wloc solution to (1.1), then f is also a solution to (1.2) where ¹ is replaced by¹ ~ = ¹ + º@f=@f if @f 6= 0 and¹ ~ = 0 if @f = 0: The case, when the assumption of strong ellipticity condition j¹(z)j+jº(z)j · q < 1 is replaced by the assumption j¹(z)j + jº(z)j < 1 a.e., the similar existence B. Bojarski, V. Gutlyanski¸³ and V. Ryazanov 3 and uniqueness problem was not studied so far. Let us consider a couple of illustrative examples. De¯ne the following Beltrami coe±cients µ ¶ jzj2 ¡ 1 z z ¹(z) = + jzj ¢ ; º(z) = ¡jzj ¢ ; 3jzj2 + 1 z z in the punctured unit disk 0 < jzj < 1: Since j¹(z)j ! 1 as z ! 0; we see that the equation fz ¡ ¹(z) fz ¡ º(z)fz = 0 degenerates near the origin. It is easy to verify that the radial stretching z f(z) = (1 + jzj2) ; 0 < jzj < 1: jzj satis¯es the above equation and is a homeomorphic mapping of the punctured unit disk onto the annulus 1 < jwj < 2: Thus, we observe the e®ect of cavitation. For the second example we choose i z i z 2 ¹(z) = ; º(z) = e2i log jzj : 2 z 2 z In this case j¹(z)j+jº(z)j = 1 holds for every z 2 C: In other words, we deal with "global" degeneration. However, the corresponding globally degenerate general Beltrami equation (1.1) admits the spiral mapping f(z) = zei log jzj2 as a quasiconformal solution. The above observation shows, that in order to obtain existence or uniqueness results, some extra constraints must be imposed on ¹ and º: In this paper we give su±cient conditions for the existence of a homeomorphic ACL solution to the Beltrami equation (1.1), assuming that the degeneration of ¹ and º is is controlled by a BMO functions. More precisely we assume that the maximal dilatation function 1 + j¹(z)j + jº(z)j K (z) = (1.4) ¹;º 1 ¡ j¹(z)j ¡ jº(z)j is dominated by a function Q(z) 2 BMO; where BMO stands for the class of functions with bounded mean oscillation in D; see [21]. Recall that, by John and Nirenberg in [21], a real-valued function u in a domain D in C is said to be of bounded mean oscillation in D; u 2 BMO(D); if 1 u 2 Lloc(D), and Z 1 kuk¤ := sup ju(z) ¡ uBjdxdy < 1 (1.5) B jBj B 4 On Degenerate Beltrami Equation where the supremum is taken over all discs B in D and Z 1 u = u(z)dxdy: B jBj B We also write u 2 BMO if D = C: If u 2 BMO and c is a constant, then u + c 2 BMO and kuk¤ = ku + ck¤. The space of BMO functions modulo constants with 1 p the norm given by (1.5) is a Banach space. Note that L ½ BMO ½ Lloc for all p 2 [1; 1); see e.g. [21] and [30]. Fe®erman and Stein [13] showed that BMO can be characterized as the dual space of the Hardy space H1. The space BMO has become an important concept in harmonic analysis, partial di®erential equations and related areas. The case, when º = 0 and the degeneration of ¹ is expressed in terms of j¹(z)j; has recently been extensively studied, see, e.g., [7] { [10], [16], [19], [20], [23], [25], [27], [32] { [33], [36], and the references therein. In this article, unless otherwise stated, by a solution to the Beltrami equation (1.1) in D we mean a sense-preserving homeomorphic mapping f : D ! C in 1;1 the Sobolev space Wloc (D); whose partial derivatives satisfy (1.1) a.e. in D: 1.6. Theorem. Let ¹; º be measurable functions in D ½ C; such that j¹j + jºj < 1 a.e. in D and 1 + j¹(z)j + jº(z)j · Q(z) (1.7) 1 ¡ j¹(z)j ¡ jº(z)j a.e. in D for some function Q(z) 2 BMO(C): Then the Beltrami equation (1.1) 1;s has a homeomorphic solution f : D ! C which belongs to the space Wloc (D) for all s 2 [1; 2): Moreover, this solution admits a homeomorphic extension to C ¡1 1;2 such that f is conformal in C n D: For the extended mapping f 2 Wloc ; and for every compact set E ½ C there are positive constants C; C0; a and b such that µ ¶ ¯ ¯ a ¯ 1 ¯¡b C exp ¡ · jf(z0) ¡ f(z00)j · C0 ¯log ¯ (1.8) jz0 ¡ z00j2 ¯ jz0 ¡ z00j¯ for every pair of points z0; z00 2 E provided that jz0 ¡ z00j is su±ciently small. 1.9. Remark. Note that C0 is an absolute constant, b depends only on E and Q and the right inequality in (1.8) holds whenever jz0 ¡ z00j < e¡2: B. Bojarski, V. Gutlyanski¸³ and V. Ryazanov 5 1.10. Remark. Prototypes of Theorem 1.6 when º(z) ´ 0 can be found in the pioneering papers on the degenerate Beltrami equation [27] and [10], see also [32] and [36]. In [2] it was shown that a necessary and su±cient condition for a measurable function K(z) ¸ 1 to be majorized in D ½ C by a function Q 2 BMO is that ZZ ´K(z) dxdy e 3 < 1 (1.11) D 1 + jzj for some positive number ´: Thus, the inequality (1.11) can be viewed as a test for K¹;º(z) to satisfy the assumptions of Theorem 1.6. 2. Auxiliary Lemmas For the proof of Theorem 1.6 we need the following lemmas. 2.1. Lemma. Let fn : D ! be a sequence of homeomorphic ACL solutions to the equation (1.1) converging locally uniformly in D to a homeomorphic limit function f: If p K¹n;ºn (z) · Q(z) 2 Lloc(D) (2.2) 1;s a.e. in D for some p > 1; then the limit function f belongs to Wloc where s s = 2p=(1 + p) and @fn and @fn converge weakly in Lloc(D) to the corresponding generalized derivatives of f: Proof.
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