Hello,

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.