A Boundary Value Problem for Conjugate Conductivity Equations

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 σ (x, y) = x p , p ∈ Z * . The representations are obtained via the so-called unified transform method or Fokas method, involving a RiemannHilbert problem on the complex plane when p is even and on a two-sheeted Riemann surface when p is odd. They are given in terms of the Dirichlet and Neumann data on the boundary of the domain. For even exponent p, we also show how to make the conversion from one type of conditions to the other by using the global relation that follows from the closedness of some differential form. 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.

Moutard transforms for the conductivity equation

“…In literature equation (5) is sometimes considered as the conjugate equation to (1); see, for example, [3].…”