Optimal solvability for the Dirichlet and Neumann problem in dimension two

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We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces Hᵖ, p>2/3, where This in turn implies that solutions to the Dirichlet problem with data in the Holder class C¹/²(∂ D) are themselves in C¹/²(D). Both of these results are sharp. In fact, we prove a more general statement regarding the Hᵖ solvability for divergence form elliptic equations with bounded measurable coefficients. We also prove similar solvability result for the regularity and Dirichlet problem for the biharmonic equation on Lipschitz domains.