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

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.