## Eigenvalues of Laplacian with Boundary Condition

Given a bounded domain with the homogeneous Neumann boundary condition, show that the Laplacian has an eigenvalue equal to zero (show that there is a nonzero function u such that ∆u = 0, with the homogeneous Neumann B.C.).

I said: ∇•(u∇u)=u∆u+∇u2, since ∆u = 0, we have ∇•(u∇u)=∇u2

∫ ∇•(u∇u) dV = ∫ ∇u2 dV, and by divergence theorem on the left hand side, we get ∫ (u∇u)•dS=∫ ∇u2 dV, but since ∇u is zero on the boundary, we have

∫ ∇u2 dV=0. Since the integrand is non-negative, ∇u=0, which means u is constant.

This can't be the right answer, could it?! It seems way to trivial, I could have just said if u is constant, then it satisfies the question trivially...Did I do something wrong? am I misunderstanding the question?
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 Tags eigenvalue, laplacian, neumann

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