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As the title says, the question is to prove the solutions to the Neumann problem of the reduced Helmholtz equation

$$\Delta u - ku = 0 ,k>0$$

in a bounded domain $D$ , are unique.

I was able to show using one of Green's identities that solutions to the problem is unique inside the domain, but I'm not sure what claims I can make about the uniqueness of the solutions on the boundary of $D$. .

I know that if $u$ and $v$ are two solutions the problem, then the Neumann boundary conditions tell me

$$\mathbf{n}\cdot\nabla u=g(x,y)=\mathbf{n}\cdot\nabla v$$

for $(x,y)$ on the boundary of $D$. .

I feel like I'm missing something really obvious, but my question is: is there a way to infer from this that $u=v$ on the boundary of $D$ , given that $u=v$ on the interior of $D$?

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up vote 1 down vote accepted

First, you should know the maximum principle for elliptic equation.

It says

$\sum a_{ij}u_{ij} + b\cdot \nabla u + cu = 0$

which is uniform elliptic in $\Omega$, and $c\le0$, then

the maximum can be achieved on the boundary.

And if your boundary condition satisfies Hopf boundary point condition, then the equation will give you strong maximum principle.

if $w$ and $v$ both satisfy your problem setting, take $u = w-v$, $u$ satisfies Helmholtz equation also.

Now you have Neumann data $\dfrac{\partial u}{\partial n} = 0$, but according to maximum principle, if $u\neq \mathrm{const}$, then the exterior norm derivative should be positive at the maximum point.

So $u= \mathrm{const}$. Thus $u = 0$, $w=v$.

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