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

Abstract

This paper consists of two parts. The first part is devoted to the study of the Beltrami equation with a polar singularity in a circle centred at the origin, with a cut along the positive semiaxis. The coefficients of the equation have a first-order pole at the origin and do not belong to the class $L_2\left(G\right).$ Therefore, despite having a unique form, this equation is not covered in the analysis of I.N.Vekua [Generalized analytic functions. Nauka: Moscow; 1988. Russian] and needs to be independently studied. 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. 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.

Keywords:

• Beltrami equation
• equation with a polar singularity
• analytic functions
• variety of continuous solutions

AMS Subject Classification:

• 30A99

Acknowledgements

We would like to acknowledge the contributions of the late professor A. Tungaratov.

Disclosure statement

No potential conflict of interest was reported by the author(s).