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 $u$ 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?



Extension of harmonic function
Certainly not: think of $\Omega$ the unit disk, and $u$ the harmonic extension to $\Omega$ of any continuous, nowhere differentiable function on $\partial \Omega$.


