Abstract
Many properties of solutions to Beltrami equations were established by I.N. Vekua [11], B. Bojarski [3], A. Dzhuraev [5], R.P. Gilbert [6], H. Begehr [1] and others. Based on Cauchy-type integrals for example Riemann-Hilbert as well as Riemann boundary value problems were studied. Because the kernel of the Cauchy-type integral in general is not known, only existence and uniqueness results were given. In this paper some special cases for the Beltrami equation are considered. A corresponding Cauchy formula and theorem are given where the Cauchy type integral can be represented explicitly as in the analytic case (q= 0). Riemann-Hilbert boundary value problems are solved explicitly for the half-plane and for an ellipse when q is constant. Besides the Riemann-Schwarz reflection principle is established. Using a Wiener-Hopf integral equation and convolution-type dual integral equations the solutions of some nonlocal problems are given in explicit form. For ellipticity of (1) the given function q must satisfy |q|≠1. Without loss of generality is assumed because if |q|q0>1 one can consider equation (1) for .
Published Version
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