Nonhomogeneous biharmonic problem in the half-space, \(L^{p}\) theory and generalized solutions (Q879374)

The aim of this pape is to solve of the biharmonic problem with nonhomogeneous boundary conditions \[ \Delta^2 u=f\quad\text{in }\mathbb{R}^N_+, \qquad u= g_0\quad\text{on }\Gamma= \mathbb{R}^{N-1}, \qquad \partial_N u= g_1\quad\text{on }\Gamma, \] where \(N\geq 2\). Using \(L^p\) theory, \(1< p<\infty\), the authors prove existence and uniqueness results, taking data from weighted Sobolev spaces. The analysis of the authors is based on the isomorphism properties of the biharmonic operator in the whole space and the solution of the Dirichlet and Neumann problems for the Laplacian in the half-space.