tailieunhanh - The Riemann Hilbert problem for generalized Q-holomorphic functions

We clearly do not investigate the conditions under which there are continuous solution in G even when the boundary vector family has positive index. In this work, the classical Riemann Hilbert boundary value problem is extended to generalized Q-holomorphic functions. | Turk J Math 34 (2010) , 167 – 180. ¨ ITAK ˙ c TUB doi: The Riemann Hilbert problem for generalized Q-holomorphic functions Sezayi Hızlıyel and Mehmet C ¸ a˘glıyan Abstract In this work, the classical Riemann Hilbert boundary value problem is extended to generalized Qholomorphic functions. Key Words: Generalized Beltrami systems, Q-Holomorphic functions, Riemann Hilbert problem. 1. Introduction In [6] A. Douglis developed an analogue of analytic functions theory for more general elliptic systems in the plane of the form wx + iwy + aEwx + bEwy = 0, (1) where E is an m × m constant matrix, w is an m × 1 vector, and a and b are complex valued functions of x and y . Subsequently in [5] B. Bojarski˘ı extended the function theory of Douglis to a system which he wrote in the form (2) wz = qwz . He assumed that the variable m × m matrix q is “lower diagonal with all eigenvalues of q having magnitude less than 1 . The systems (1) and (2) are natural ones to consider because they arise from the reduction of general elliptic systems of first order in the plane to a standard canonical form. Douglis and Bojarski˘ı theory has been used to study the elliptic systems of more general form: wz − qwz = aw + bw. Solutions of this equation were called generalized (or pseudo) hyperanalytic functions. Works in this direction appear in [7, 8, 10, 11]. These results extend the generalized (or “pseudo”) analytic function theory of Bers [4] and Vekua [17]. Also, the classical boundary value problems for analytic functions were extended to the generalized hyperanalytic functions. A good survey of the methods encountered in the hyperanalytic case may be found in [3, 9], see also [1, 2]. AMS Mathematics Subject Classification: 30G20, 30G35, 35J55. 167 ˘ HIZLIYEL, C ¸ AGLIYAN In [13], Hile noticed that what appears to be the essential property of the elliptic systems in the plane for which one can obtain a useful extension of analytic function theory is the self .

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