On continuous solutions of the model homogeneous Beltrami equation with a polar singularity

Abstract

This pap er consists of two parts. The first part is devoted to the study of the Beltrami mo del
equation with a p olar singularity in a circle c entered at the ori gin , with a cut along the p ositive
semiaxis. The c o efficients of the equation have a first-order p ole at the origin and do not even b elong
to the class L 2 ( G ) . For this reason, despite its specific form, this equation is not covered by the
analytical apparatus of I.N. Vekua [1] and needs to be independently studied. Using the technique
developed by A.B. Tungatarov [2] in combination with the methods of the theory of functions of a
complex variable [3] and functional analysis [4], manifolds of continuous solutions of the Beltrami
model equation with a polar singularity are obtained. The theory of these equations has numerous
applications in mechanics and physics. In the second part of the article, the coefficients of the
equation are chosen so that the resulting solutions are continuous in a circle without a cut [5].
These results can be used in the theory of infinitesimal bendings of surfaces of positive curvature
with a flat point and in constructing a conjugate isometric coordinate system on a surface of
positive curvature with a planar point [6].