## Hardy spaces and the Neumann problem in $$L^ p$$ for Laplace’s equation in Lipschitz domains.(English)Zbl 0658.35027

From 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
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