Harmonic Functions on Connected Riemannian Manifold with Matching Restrictions to Boundary are Identical

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Theorem

Let $\struct {M, g}$ be a connected Riemannian manifold with the boundary $\partial M \ne \empty$.

Let $u, v \in \map {C^\infty} M$ be smooth harmonic functions on $M$.

Suppose restrictions of $u$ and $v$ to $\partial M$ agree.


Then $u = v$ identically.


Proof




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