On some boundary value problems for a system of Poisson equations in a three-dimensional domain (Q376375)

The Poisson equation with the Laplace \(\Delta\) acting component-wise on vector fields \(u\) in a domain \(\Omega\subseteq\mathbb{R}^{3}\) with piece-wise smooth boundary is considered. Two different sets of unusual boundary condition resulting in a selfadjoint realization of the Laplacian \(\Delta\) are imposed. The two sets of boundary conditions require the vanishing of the normal -- or tangential -- component of \(u\) and of the tangential -- or, respectively, the normal -- component of the component-wise normal derivative of \(u\). Well-posedness, amounting here to showing that \(0\) is in the resolvent set of the resulting selfadjoint Laplacian, is shown based on assuming that \(\Omega\) is such that a compact embedding of the Sobolev space \(W_{2}^{1}\left(\Omega\right)\) in \(L^{2}\left(\Omega\right)\) and a Friedrichs type estimate is available.