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Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$.
Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$?

Wikipedia says the solution is unique when the boundary is sufficiently smooth.
But, an application of the maximum principle always gives uniqueness of the solution.
No matter how smooth the domain is.
What am I missing?

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The wikipedia page you link to says that a solution both exists and is unique when the boundary is smooth enough. I think the smoothness of the boundary is relevant for the existence part more than uniqueness. Indeed, the wikipedia page say that when the boundary is $C^{1,\alpha}$, a solution exists. – froggie Nov 25 '12 at 23:22
up vote 3 down vote accepted

Dirichlet problem for what differential operator? By default I assume the Laplacian. With what kind of boundary data? By default I assume continuous.

Then smoothness is not needed for uniqueness: if two harmonic functions on a bounded open set have the same continuous extension to the boundary, then they are identical. The reason is indeed the maximum principle.

$C^{1,\alpha}$ is not needed for existence either. It would be odd to rule out rectangular boxes when considering boundary value problems. A sufficient condition for existence is the exterior cone condition: every point of $\partial D$ is the vertex of some (finite) circular cone which is disjoint from $D$. (This condition is not necessary, but it's good enough for practical purposes. The necessary and sufficient condition for existence is something you would not want to check in practice.)

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