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

Abstract

We study Hardy spaces $H_{\nu }^p$ of the conjugate Beltrami equation $\bar{\partial }f=\nu \overline{\partial f}$ 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.

41 pages