Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$? Where $\omega\subset\subset \Omega$ with $\Omega$ is a $C^2$ bounded domain of $\mathbb{R}^d, d\geq 2$ and $H(\Omega)=\{u\in (H^1(\Omega))^d | \ \nabla\cdot u=0 \mbox{ and }u\cdot N=0 \mbox{ on }\partial\Omega\}$, here $N(x)$ being the outward unit normal to $\Omega$ at the point $x\in \partial \Omega$. .
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2$\begingroup$ Take the eigenfunctions of the Stokes operator. They are analytic in $\Omega$, so if any linear combination vanishes in $\omega$, then it also vanishes in $\Omega$. $\endgroup$– Michael RenardyCommented Apr 22, 2014 at 17:57
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$\begingroup$ Thank you for your suggest. Can you explain the eigenfunctions of the Stokes operator. In my knowledge, the eigenfunctions of the stokes operator have to take with the Dirichlet boundary condition!? $\endgroup$– Tran LamCommented Apr 23, 2014 at 16:55
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$\begingroup$ You can use the Stokes operator with slip conditions instead. $\endgroup$– Michael RenardyCommented Apr 24, 2014 at 13:00
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