## Unique continuation property on the boundary

I am writing a theorem proved by Jin Zhiren in his thesis.

Suppose is a smooth domain in , and is a harmonic function in . If there exists such that

for small, then . If , the same conclusion holds for the solutions of a general second order linear elliptic equation.

A borderline example for this theorem is be the real part of , . is harmonic in the right half plane and and consequently .

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