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Given a $\mathcal{C}^\infty$ function $\varphi$ defined on a portion of a surface $\Sigma^-$ and let $\Sigma$ be a closed surface or union of surfaces bounding a compact volume $\Omega \subset \mathbb{R}^3$ such that $\Sigma^- \subset \Sigma$. When is it possible to extend the function $\varphi$ to the whole domain $\Omega$ such that the following condition is satisfied:

$\varphi$ is $\mathcal{C}^\infty$ on $\Omega$ and $\frac{\partial \varphi}{\partial {\mathbf n}}=0$ on $\Sigma$ with ${\mathbf n}$ the normal vector to $\Sigma$?

Of course I assume that $\Sigma^-$ and $\Sigma$ are $\mathcal{C}^\infty$ 2-dim real manifolds.

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The main issue is extending to $\Sigma$: if $\Sigma^-$ is relatively open in $\Sigma$, smooth functions do not necessarily extend smoothly. Once you extended to $\Sigma$ you can just take a tubular neighborhood of $\Sigma$ and smoothly cut off $\psi$ in some epsilon distance to $\Sigma$. – Willie Wong Jan 17 '13 at 9:18
@Willie: thank you for the comment. – user16974 Jan 17 '13 at 9:42
@Willie: If we know that $\Sigma^_$ is relatively closed in $\Sigma$ does this solve the problem? – user16974 Jan 17 '13 at 10:28

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