On an orthogonal decomposition of Sobolev spaces and a Stokes-type boundary value problem. (Q1432342)

Let \(\Omega\) be a bounded Lipschitz domain in \(\mathbb R^n\) and let \(\overset\circ {W}^1_2\) be the space of vector-functions \(u(x)= (u_1 (x), \dots, u_n (x))\) belonging to the scalar spaces \(\overset\circ {W}^1_2 (\Omega)\). Let \[ \overset\circ {S}^1_{\Delta, 2} = \left\{ u \in \overset\circ {W}^1_2: \text{div} \Delta u(x) =0 \right\}. \] The main aim of the paper is to determine the orthogonal complement \[ \overset\circ {W}^1_2 =\overset\circ {S}^1_{\Delta,2} \oplus \left( \overset\circ {S}^1_{\Delta,2} \right)^\perp \] with the outcome \[ \overset\circ {W}^1_2 = \overset\circ {S}^1_{\Delta, 2} \oplus \nabla\overset\circ {W}^2_2 (\Omega), \] where \(\overset\circ {W}^2_2 (\Omega)\) is the respective scalar space. This extends to decompositions with \(\overset\circ {W}^1_p\) and \(\overset\circ {W}^{-1}_{p'}\) in place of \(\overset\circ {W}^1_2\) (\(1<p< \infty\)). The results are applied to Stokes type boundary value problems.