On First and Second Order Planar Elliptic Equations with Degeneracies

TL;DR

This paper investigates elliptic equations in the plane with degeneracies along a curve, constructing kernels and analyzing solution properties near degeneracies, with applications to equations with point singularities.

Contribution

It introduces new methods for constructing kernels and analyzing solutions for elliptic equations with degeneracies along a curve.

Findings

Kernels for degenerate elliptic equations are explicitly constructed.

Solution properties near degeneracy curves are characterized.

Applications to equations with point singularities are demonstrated.

Abstract

This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degeneracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.