A boundary Problem for conjugate conductivity equations

TL;DR

This paper derives explicit integral formulas for solutions to planar conjugate conductivity equations with power-law conductivity in a circular domain, using Riemann-Hilbert problems and boundary data transformations.

Contribution

It provides novel integral representations for solutions in specific conductivity models, applicable via Riemann-Hilbert problems and boundary condition conversions.

Findings

Explicit integral formulas for solutions with even p

Explicit integral formulas for solutions with odd p

Method applicable to general smooth boundary domains

Abstract

We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half-plane with conductivity $\sigma(x,y)=x^p$, $p\in\mathbb{Z}$. The representations are obtained via a Riemann-Hilbert problem on the complex plane when $p$ is even and on a two-sheeted Riemann surface when $p$ is odd. They involve the Dirichlet and Neumann data on the boundary of the domain. We also show how to make the conversion from one type of conditions to the other by using the so-called global relation. The method used to derive our integral representations could be applied in any bounded simply-connected domain of the right half-plane with a smooth boundary.