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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 Hp1,(partial D)(Hp (partial D)), p>2/3-E, where D C R2 and E is a (small) number depending on the Lipschitz nature of D. This in turn implies that solutions to the Dirichlet problem with data in the Holder class C1/2+E(partial D) are themselves in C1/2+E(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 H2/3-E and C1/2+E solvability result for the regularity and Dirichlet problem for the biharmonic equation on Lipschitz domains.

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