Some orthogonal decompositions of Sobolev spaces and applications (Q2759163)

Let \(G\) be a bounded Lipschitz domain of \(\mathbb R^n\). Let \(W^{1}_{2}(G)\) be the usual \(\mathbb C^n\)-valued Sobolev space on \(G\) and \(\overset{\circ}{W}^{1}_{2}(G)\) be the closure in \(W^{1}_{2}(G)\) of the space of complex-valued \(C^{\infty}\) functions compactly supported in \(G\). The authors give two series of orthogonal decomposition of \(\overset{\circ}{W}^{1}_{2}(G)\). NEWLINENEWLINENEWLINEThe first decomposition involves the so-called \(\Delta^k\)-solenoidal functions (\(k\geq 0\)). For any integer \(k\geq 0\), define NEWLINE\[NEWLINE \overset{\circ}{S}^1_{\Delta^k,2}=\left\{u\in \overset{\circ}{W}^{1}_{2}(G);\text{ div }\Delta^ku=0\text{ on }G\right\}. NEWLINE\]NEWLINE Write \(\overset{\circ}{S}^1_{2}\) instead of \(\overset{\circ}{S}^1_{\Delta^0,2}\). Define also \(\overset{\circ}{W}^{2k}_{\Delta^k,2}\) as the closure in \(W^{2k}_2(G)\) of the space of scalar \(C^{\infty}\) functions on \(\overline{G}\) such that \(\Delta^lp=0\) and \(\nabla\Delta^lp=0\) on \(\partial G\) for all \(0\leq l\leq k-1\). Then, for all \(k\geq 1\), NEWLINE\[NEWLINE\overset{\circ}{W}^{1}_{2}(G)=\overset{\circ}{S}^1_{\Delta^k,2} \overset{\perp}{\oplus} \nabla \Delta^{k-1}(\overset{\circ}{W}^{2k}_{\Delta^k,2}).{\text{ortho}}NEWLINE\]NEWLINE A version of this result for domains of \(\mathbb C^n\) is also given. NEWLINENEWLINENEWLINEWhen \(k=0\), \((\text{ref{ortho}})\) becomes NEWLINE\[NEWLINE \overset{\circ}{W}^{1}_{2}(G)=\overset{\circ}{S}^1_{2} \overset{\perp}{\oplus} \Delta_0^{-1}\nabla\left(L^2/\mathbb C\right), NEWLINE\]NEWLINE where \(\Delta_0^{-1}:W_2^{-1}(G)\rightarrow \overset{\circ}{W}^{1}_{2}(G)\) is the inverse operator of \(\Delta\). NEWLINENEWLINENEWLINEOrthogonal decomposition of spaces of \(\Delta^k\)-solenoidal functions are also given. Analogous decompositions for Sobolev spaces \(W^m_p\), which involve subspaces of harmonic functions, are proved. NEWLINENEWLINENEWLINEFinally, applications of the previous decompositions to variational problems and boundary value problems of Stokes type are given.