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Existence Theorem on Quasiconformal Mappings Seddik Gmira

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Existence Theorem on Quasiconformal Mappings Seddik Gmira Existence Theorem on Quasiconformal Mappings Seddik Gmira To cite this version: Seddik Gmira. Existence Theorem on Quasiconformal Mappings. 2016. hal-01285906 HAL Id: hal-01285906 https://hal.archives-ouvertes.fr/hal-01285906 Preprint submitted on 9 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Existence Theorem on Quasiconformal Mappings Seddik Gmira Quasiconformal mappings are, nowadays, recognized as a useful, impor- tant, and fundamental tool, applied not only in the theory of Teichmüller spaces, but also in various …elds of complex analysis of one variable such as the theories of Riemann surfaces, of Kleinian groups, of univalent functions. In this paper we prove the existence theorem of the solution of the Bel- trami di¤erential equation, and we give a fundamental variational formula for quasiconformal mappings, due to L.Ahlfors and L.Bers. 1 Quasiconformal Mapping We consider an orientation-preserving homeomorphism f, which is at least partially di¤erentiable almost every where on a domain in C, satisfying the Beltrami equation D fz = fz As a natural generalization of the notion of conformal mappings we con- sider the following 1.1 Analytic De…nition De…nition 1 Let f be an orientation-preserving homeomorphism of a do- main into C. f is quasiconformal (qc) on if D D 1. f is absolutly continuous on lines (ACL) 2. There exists a constant k, 0 k < 1 such that fz k fz j j j j almost every where on : D Setting K = (1 + k) = (1 k), we say that f is K-quasiconformal map- ping on . We call the in…mum of K > 1 such that f is qc, the maximal D dilatation of f, and denote it by Kf : Example 1 An a¢ ne mapping f (z) = az+bz+c; (a; b; c C; b < a ) 2 j j j j is qc: k = b = a and hence Kf = ( a + b ) =( a b ) j j j j j j j j j j j j 1 Example 2 Set f (z) = z , z . This f is an orientation- 1 z 2 2 preserving of the unit disk butj notj quasiconformal. In general, existence of the partial derivatives fz and fz is not enough to guarantee good properties of f applicable to further investigations. However in the case of homeomorphisms Gehring and Lheto obtained the following remrkable result. Proposition 1 If a homeomorphism f of a domain onto C has the D partial derevatives fx and fy almost.every where on , then f is totally di¤erentiable almost.every where on . D Proposition 2 Let f be a quasiconformalD mapping of a domain . Then D the partial derevatives fz and fz are totally square integrable on . Proof. Let (E) be the area of f (E) of a measurable Borel subsetD E of A , and Jf (z) the density of the set function (with respect to the Lebesgue measureD dxdy). The Lebesgue theorem impliesA Jf (z) dxdy (E) A ZE Then, f is totally di¤erentiable at almost every z and at each such 2 2 2 D a point z we have Jf (z) = fz fz : Since f is quasiconformal, then j j j j 2 2 1 fz fz Jf (z) a.e. on j j j j 1 k2 D Proposition 3 For every quasiconformal mapping f of a domain , D the partial derivatives fz and fz are coincident with those in the sens of distribution. Namely for every element ' 01 ( ), the set of all smooth functions on with compact supports it follows2 C thatD D fz:'dxdy = f:' dxdy z ZZD ZZD fz:'dxdy = f:' dxdy z ZZD ZZD Lemma 1 For a given p > 1, let f be a continuous function on a domain p whose distributional derivatives fz and fz are locally L on . Then for D D every compact subset F of , there is a sequence fn 1 in ( ) such D f g1 C01 D that fn converges to f uniformly on F , with p lim (fn) fz ddxdy = 0 n j z j !1 ZZF p lim (fn) fz ddxdy = 0 n j z j !1 ZZF 2 Proof. Fix 01 ( ) with = 1 in some neighbourhood of F . Then f 2 C D p has a compact support. Further (f)z and (f)z exist and belong to L ( ). Set D C: exp 1 ; z 1 z 2 ' (z) = j j 2 ( 0; z C 2 Here, is the unit disk. Choose a constant C so that ' (z) dxdy = 1 ZZ 2 Further, for every n > 0, set 'n (z) = n ' (nz) ; z C and 2 fn (w) = 'n (f)(w) = 'n (w z)(f)(z) dxdy; w C 2 ZZC Then for every n, we have fz = ' (f) and fz = ' (f) . Morever, n z n z fn ( ) for every su¢ ciently large n, and fn converges to f uniformly 2 C01 D on F as n . Since (f)z = fz and (f)z = fz on F and since we can show equalities:! 1 p p lim (fn) (f) dxdy = 0; lim (fn) (f) dxdy = 0 n j z zj n j z zj !1 ZZD !1 ZZD Lemma 2 (Weyl) Let f be a continous function on whose distrib- D utional derivative fz is locally integrable on . If fz = 0 in the sens of distributions on , then f is holomorphic on D. D D Proof. For an arbitrarily relatively subdomain D1of , we construct an 1 D L -smoothing sequence fn 1 for f with respect to D1 as in the proof of f g1 Lemma 1, we see that (f)z = 0 in some neighbourhood of D1, we see also that (fn)z = 0 on D1 for every su¢ ciently large n. Since fn converges to f uniformly on D1 as n , f is holomorphic on this arbitrary D1. ! 1 1.2 Geometric De…nition A quadrelateral (Q; q1; q2; q3; q4) is a pair of a Jordan closed domain Q and four points q1; q2; q3; q4 @Q, which are naturally distinct and located in this order with respect to2 the positive orientation of the boundary @Q. Proposition 4 For every quadrelateral (Q; q1; q2; q3; q4), there is a home- omorphism h of Q onto some rectangle = [0; a] [0; b](a; b > 0) which is conformal in the interior IntQ and satis…esR h (q1) = 0; h (q2) = a h (q3) = a + ib; h (q4) = ib 3 Morever a=b is independant of h. The value a=b is called the module of the quadrelateral (Q; q1; q2; q3; q4) and denotet by (Q). Proof. The Riemann mapping theorem impliesM the existence of a conformal mapping h1 : IntQ H (upper half-plane). By Carathéodory’sTheorem ! h1 can be extended to a homeomorphism of Q onto H R. Using a suitable Möbius transformation, we may assume that [ h1 (q1) = 1; h1 (q2) = 1 h1 (q3) = h1 (q4) > 1 Set k = 1=h1 (q3), and w dz h2 (w) = ; z H 2 2 2 0 (1 z ) (1 k z ) 2 Z Then, h2 is a conformal mappingp of H onto the interior of some rectangle [ K; K] [0;K ](K; K > 0). Hence we see that 0 0 h (z) = h2 h1 (z) + K; z Q 2 is a desired mapping. Now, let h : Q = [0; a] 0; d be another ! R mapping which satis…es the same conditions as the last proposition.h i Using the Schwarz re‡exion principale thee map e e e 1 h h can be extended to an element ofe Aut (C). Hence with suitable complex numbers c and d, we have h (z) = ch (z) + d; (z IntQ) 2 Since e h (q1) = h (q1) = 0; h (q1) > 0 and h (q2) > 0 we conclude that c > 0 and d = 0. Hearafter, for every quadrelateral e e (Q; q1; q2; q3; q4) and a homeomorphism f of Q onto C, we consider f (Q) as a quadrelateral with vertices f (q1) ; f (q2) ; f (q3) ; f (q4). Lemma 3 Every K-qc mapping f of a domain satis…es D 1 M (Q) M (f (Q)) KM (Q) K Proof. Fix mappings h : Q R = [0; a] [0; b] and h : f (Q) R = ! 1 ! [0; a] 0; b as before. Then F = h f h is a qc mapping of the IntR onto e e h i e e e 4 the IntR which maps 0; a, ib and a + ib to 0; a, ib and a + ib, respectively. In particular, F (z) is ALC on R, and hence for almost every y [0; b], we 2 have e e e e e a @F a a F (a + iy) F (iy) = (x + iy) dx ( Fz + Fz ) dx j j @x j j j j Z0 Z0 e Since JF dxdy R = ab; intgrating both sides of th above R A inequalityRR over [0; b] e ee 2 2 (ab) ( Fz + Fz ) dxdy j j j j ZZR Fz + Fz 2 2 e j j j jdxdy Fz Fz dxdy R Fz Fz R j j j j ZZ 0 j j j j ZZ 0 Kdxdy JF dxdy K (ab) ab R R ZZ 0 ZZ 0 e where R0 = w R : Fz (w) = 0 .
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