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.
Stefanov, Atanas and Verchota, Gregory C., "Optimal Solvability for the Dirichlet and Neumann Problem in Dimension Two" (2000). Mathematics Faculty Scholarship. 135.
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