I would appreciate any help with the following problem:

Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two connected components $M_{+}$ and $M_{-}$. Is it possible to construct a harmonic function such that it is zero on $\Gamma$ and it is non-zero away from it everywhere in the manifold? As a follow up question is it possible to construct a harmonic function which is zero on $\Gamma$ and monotone in normal direction to $\Gamma$?

Two comments are due: it is clear that by setting $v=0$ on $\Gamma$ and $ dv \neq 0$ we can achieve the first question locally but not globally in M. second comment: one can use maximum principle in following way; put dirichlet data 1 on $\partial M_{+}$ and zero on $\Gamma$. put Dirichlet data -1 on $\partial M_{-}$ in this way we are guaranteed that the harmonic function would never be zero away from the surface but the first normal derivatives on $\Gamma$ will not necessarily match.

cheers,