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

We study Hardy spaces $H^p_ u$ of the conjugate Beltrami equation $bar{partial} f= ubar{partial f}$ over Dini-smooth finitely connected domains, for real contractive $ uin 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 $ abla.(sigma abla u)=0$ where $sigma=(1- u)/(1+ u)$. In particular situations, we also consider some density properties of traces of solutions together with boundary approximation issues.