Hardy spaces and the Neumann problem in \(L^ p\) for Laplace’s equation in Lipschitz domains.

*(English)*Zbl 0658.35027From the introduction: “Our main theorem asserts that if \(D\subset R^ n,\) \(n\geq 3\), is a bounded Lipschitz domain with connected boundary, then there exists \(\epsilon =\epsilon (D)>0\) such that, for all \(f\in L^ p(\partial D,d\sigma),\) with \(1<p<2+\epsilon\), and \(\int_{\partial D}f d\sigma =0\), there is a unique (modulo constants) harmonic function u in D with \(\| M(\nabla u)\|_{L^ p(d\sigma)}\leq C_ p(D)\| f\|_{L^ p(d\sigma)}\) and \(\partial u/\partial n=f\) on \(\partial D\). Here \(\sigma\) is the surface measure on \(\partial D\), and M(\(\nabla u)\) the non-tangential maximal function of \(\nabla u''\) and \(\partial u/\partial n\) is the generalized normal derivative of u which exists \(\sigma\)-a.e. on \(\partial D\). The authors first consider the case where D is an epigraph of a Lipschitz function. The case \(1<p<2\) is treated by interpolation, using the known result for \(p=2\) and establishing a similar result in the case where f belongs to the atomic \(H^ 1\) space on \(\partial D\). The case \(2<p<2+\epsilon\) is proved by using a variant of ‘good \(\lambda\) ’ inequalities. Analogous results for Dirichlet boundary condition are also obtained. [Reviewer’s remark: The proof after Theorem 4.12 appears to be not the proof of this theorem, but a proof of the main theorem whose statement is missing there.]

Reviewer: F.-Y.Maeda

##### MSC:

35J25 | Boundary value problems for second-order elliptic equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |