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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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems