On an orthogonal decomposition of the spaces \(\overset{\circ} W^1_2\) and \(W^{-1}_2\) and its application to the Stokes problem. (Q1432349)

Let \[ \overset\circ {S}^1_2= \{u(x)\in \overset\circ {W}^1_2(G) \mid\text{div}\,u(x)=0\}, \] and \[ \overset\circ {S}^1_{\Delta,2}= \{u(x)\in\overset \circ{W}^1_2(G) \mid\text{div}\,\Delta u(x)=0\}, \] \(\overset\circ {W}^1_2\) can be represented as a sum of subspaces: \(\overset\circ {W}_2^1= \overset \circ{S}^1_2\oplus (\overset\circ {S}^1_{\Delta,2} \ominus\overset \circ {S}^1_2)\oplus (\overset\circ {W}^1_2\ominus \overset\circ {S}^1_{\Delta, 2})\), where all three subspaces are orthogonal to each other. The goal of the author is to characterize these subspaces and to give applications to elliptic operators and the Stokes problem.