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

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