The mixed boundary problems in \(L^p\) and Hardy spaces for Laplace's equation on a Lipschitz domain. (Q2767862)

Let \(\Omega\) be a domain in \(Bbb(R)^n, n\geq 3\); it is assumed that its Lipschitz boundary \(\partial \Omega\) is decomposed as \(\partial \Omega = N \cup D\) where \(N \cup D \empty\). There is studed the mixed problem which consists in finding a function \(u\) satisfying the condition: NEWLINE\[NEWLINE\begin{cases} \Delta u = 0&\text (in) \Omega,\\ u =b f_D & \text(on) D \\ \frac {\partial u} {\partial \nu} = f_N& \text (on) N, \end{cases}NEWLINE\]NEWLINE where \(\Delta\) stands for the Laplace operator, \(\frac {\partial u} {\partial \nu}\) represents the outer normal derivative on \( \partial \Omega \). For a subclass of these domains, it is shown that if the Neumann data is in \(L^p (N)\) while the Dirichlet data is in the Sobolev space \( L^{p,1} (D)\), for \(1<p<2\), then the above problem has a unique solution for which the non-tangential maximal function of its gradient is in \(L^p (_\partial \Omega)\).NEWLINENEWLINEFor the entire collection see [Zbl 0970.00020].