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

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

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