Stokes' problems with non-standard boundary conditions (Q2716806)

The authors consider the linear system (1) \(-\Delta u+\text{grad} p=f\) with \(\text{div} u=0\) in a bounded and connected but non simply-connected domain \(\Omega\) of \(\mathbb{R}^3\), with \(\Gamma=\partial \Omega\) of \(C^\infty\) class. Here they consider the Hodge's decompositions of a vector field \(f\in(L^2 (\Omega))^3\), that is (2) \(f=\text{grad} p+ \text{curl} w\). Using this composition the authors show well-posedness for two boundary value problems involving normal or tangential components of the vector field \(u\).