Regularity of solutions of the Neumann problem for the Laplace equation
Let u be a solution of the Neumann problem for the Laplace equation in G with the boundary condition g. It is shown that u ∈ L q (∂ G ) (equivalently, u ∈ Bq,21/q (G ) for 1 < q ≤ 2, u ∈ Lq1/q (G ) for 2 ≤ q < ∞) if and only if the single layer potential corresponding to the boundary condition g is in L q (∂ G ). As a consequence we give a regularity result for some nonlinear boundary value problem.
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