BELTRAMI EQUATIONS WITH DEGENERATE ON ARCS
In the recent paper [5] were obtained some conditions for the existence and uniqueness of solutions with singularity of the associated equation with the Beltrami equation ????(??) = ??(??)????(??). (*) Here we gave geometric interpretation this results. The main results are as follows. Let ?? ? C be a simply connected domain divided by curve ?? ? ?? of class C(3) into two subdomain ??1, ??2. Theorem 1. Suppose that ??(??) can be represented in the form ??(??) = (1 + ? ?? (??)??(????(??))) e2????(??), where: 1) function ??(??) is continuous on [0,+?), and ??(0) = 0 and ??(??) > 0 for ?? ?= 0; 2) function ??(??) ? C(1)(??) is such that everywhere on ?? ???? + e2????(??) ???? ?= 0, at ???? tangential to ??; 3) complex-valued function ? ?? (??) is measurable and almost everywhere in ?? 1 ?? ? |Re ? ?? (??)| ? ??, |Im ???(??)| ? ?? (?? ? const). Then in some neighborhood of ??(??), there exists an solution with a singularity ?? of the equation associated with the (*). Theorem 2. Suppose that 1) ??(??) ? C1(??), ???(??) ?= 0 in ?? and ?? defined by the equation ??(??) = 0; 2) the function ??(??) can be written as: ??(??) = ??? ??? +??*(??)??(|??(??)|), where: 1) the function ??(??) is continuous on [0,+?); 2) ??(0) = 0 and ??(??) > > 0 for ?? ?= 0, and 1 ??(??) has an integrable singularity at zero; 3) ??*(??) is complex-valued measurable function in ??, and ??? Re (????(??) ???(??) ??*(??))???? ? 1 ??1, |??*(??)| ? ??2 (??1, ??2 ? const). Then in some neighborhood of ??(??), there exists an solution with a singularity ?? of the equation associated with the (*). Corollary. Assume that ??(??) = ?????(??) ?????(??) +??*(??)??(????(??)), where: 1) the function ??(??) is continuous on [0,+?); 2) ??(0) = 0 and ??(??) > > 0 for ?? ?= 0, and 1 ??(??) has an integrable singularity at zero; 3) ??*(??) is complex-valued measurable function in ??, and ??? Re (??????(??) ?????(??) ??*(??))???? ? 1 ??1, |??*(??)| ? ??2 (??1, ??2 ? const). Then in some neighborhood of ??(??), there exists an solution with a singularity ?? of the equation associated with the (*).