On continuous solutions of the model homogeneous Beltrami equation with a polar singularity
AbstractThis pap er consists of two parts. The first part is devoted to the study of the Beltrami mo delequation with a p olar singularity in a circle c entered at the ori gin , with a cut along the p ositivesemiaxis. The c o efficients of the equation have a first-order p ole at the origin and do not even b elongto the class L 2 ( G ) . For this reason, despite its specific form, this equation is not covered by theanalytical apparatus of I.N. Vekua  and needs to be independently studied. Using the techniquedeveloped by A.B. Tungatarov  in combination with the methods of the theory of functions of acomplex variable  and functional analysis , manifolds of continuous solutions of the Beltramimodel equation with a polar singularity are obtained. The theory of these equations has numerousapplications in mechanics and physics. In the second part of the article, the coefficients of theequation are chosen so that the resulting solutions are continuous in a circle without a cut .These results can be used in the theory of infinitesimal bendings of surfaces of positive curvaturewith a flat point and in constructing a conjugate isometric coordinate system on a surface ofpositive curvature with a planar point .
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