Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation

TL;DR

This paper develops a Hardy space theory for the conjugate Beltrami equation on smooth domains, applying it to solve Dirichlet and Neumann boundary value problems for the conductivity equation with new analytical tools.

Contribution

It introduces a Hardy space framework for the conjugate Beltrami equation and applies it to boundary value problems in conductivity theory, extending existing methods.

Findings

Established Hardy space $H^p_ u$ for conjugate Beltrami equation

Solved Dirichlet and Neumann problems for conductivity equation

Analyzed boundary trace density and approximation properties

Abstract

We study Hardy spaces $H^p_\nu$ of the conjugate Beltrami equation $\bar{\partial} f=\nu\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $\nu\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We develop a theory of conjugate functions and apply it to solve Dirichlet and Neumann problems for the conductivity equation $\nabla.(\sigma \nabla u)=0$ where $\sigma=(1-\nu)/(1+\nu)$. In particular situations, we also consider some density properties of traces of solutions together with boundary approximation issues.