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Suppose $u$ is a harmonic function of a domain $\Omega\subset \mathbb{R}^n$ and $u$ is continuous up to the boundary. If $\partial\Omega$ has an open smooth portion. Can , can $u$ be extended to a harmonic function outside this smooth portion?

I have a very vague claim that if this portion is analytic, then we can extend $u$ by schwarz reflection principle. But I don't know anything about the smooth case. Can anyone give me a hint?

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Suppose $u$ is a harmonic function of a domain $\Omega\subset \mathbb{R}^n$ and $\partial\Omega$ has a an open smooth portion. Can $u$ be extended to a harmonic function outside this smooth portion?

I have a very vague claim that if this portion is analytic, then we can extend $u$ by schwarz reflection principle. But I don't know anything about the smooth case. Can anyone give me a hint?

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# Extension of harmonic function

Suppose $u$ is a harmonic function of a domain $\Omega\subset \mathbb{R}^n$ and $\partial\Omega$ has a smooth portion. Can $u$ be extended to a harmonic function outside this smooth portion?

I have a very vague claim that if this portion is analytic, then we can extend $u$ by schwarz reflection principle. But I don't know anything about the smooth case. Can anyone give me a hint?