The Neumann problem in the half-space
Consider the Neumann boundary value problem for the Laplace equation in the half space, \(-\Delta=f\) in \(\mathbb R^N_+\), \(\partial u/\partial x_N=g\) on \(\mathbb R^{N-1}\) with \(N\geq 2\). The Laplace equation has been studied by many authors either in the whole space \(\mathbb R^N\) or in an exterior domain. Several authors studied the present problem in the Hilbert case of \(p=2\) and without the critical cases corresponding to the logarithmic factor. The author studies the problem in the half-space giving an \(L^p\)-theory with \(1<p<\infty\). He introduces appropriate weighted Sobolev spaces and obtains the corresponding trace space and proves for them basic existence and uniqueness results. Note that similar results were obtained for the case of the Dirichlet boundary value problem by \textit{C. Amrouche} and \textit{Š. Nečasová} in [Math. Bohem. 126, 265-274 (2001; Zbl 0979.35038)].
scientific article; zbMATH DE number 1803918