I am interested in the following problem:
Let $\Omega, \Sigma \subset \mathbb{R}^3$ be two Lipschitz domains such that $\Sigma \subset \Omega$ and $\Sigma$ is not dense in $\Omega$, and let $u \in H^1(\Sigma)$ be a function with the property that \begin{align*} \Delta u=0 \qquad \text{ in } \Sigma. \end{align*}
Does there exist a function $\tilde u \in H^1(\Omega)$ with the property that \begin{align*} \Delta \tilde u=0 \qquad \text{ in } \Omega \end{align*}
and such that $\tilde u$ coincides with $u$ on $\Sigma$?
Can anyone kindly recommend a good reference for this problem? Thanks in advance!
No. Take $u=\log|z|$ and $\Sigma$ to be any domain that does not contain zero. Then $u$ is harmonic on $\Sigma$, but it cannot be extended to any domain $\Omega$ that contains $0$.
Edit: that was for $\mathbb R^2 \simeq \mathbb C$: I did not notice that OP asked about $\mathbb R^3$. As Daniel pointed out, a similar example can be taken in $\mathbb R^3$ by $u(x,y,z)=\|(x,y,z)\|_2^{-1}$.