PDE multiplicity problem I need help with the following question, any help is appreciated.
Show that $0$ is an eigenvalues of multiplicity $1$ for the problem
$-\triangle u=\lambda u$     in D
$\triangledown u*\eta^\rightarrow =0$ on 2D
where D is a smooth Bounded Domain in $R^3$
Thank you for any help.

Assuming you mean a Neumann problem this follows from Green's identities:
$$
\int_D \nabla u\cdot \nabla v = -\int_D v\Delta u  +\int_{\partial D} v\frac{\partial u}{\partial n}   
$$
where $\frac{\partial u}{\partial n} = \nabla u \cdot n$ is the partial derivative with respect to the exterior normal vector $n$. Now just put $u=v$ to conclude that $\nabla u=0$ in $D$, and so $u$ is constant. 
