  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 H^p, p>2/3, where This in turn implies that solutions to the
Dirichlet problem with data in the Holder class C^{1/2}(\partial D) are
themselves in C^{1/2}(D). Both of these results are sharp. In fact, we prove a
more general statement regarding the H^p 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.
