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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.

1 Answers 1

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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.