Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let
$$ S\_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{on $U^{\textrm{int}}$}\},$$
and
$$ S\_B= \{u \in C^{\infty}(B)\,:\, \Delta u =0 \quad\text{on $B^{\textrm{int}}$}\}.$$


Is the following statement true? Given any $\epsilon>0$ and any $u \in S\_U$, there exists an element $v \in S\_B$ such that $\|v-u\|\_{L^2(U)} \leq \epsilon$.



Density of restrictions of harmonic functions inside a ball
No. If $\varphi \in C^\infty\_c(B)$ is a bump function equal to $1$ in $\lvert x\rvert \leq 1/2$ then from Green's theorem we have
$$ \int\_B u \Delta \varphi = 0$$
for all $u \in S\_B$, but the same is not true in general for typical $u \in S\_U$, which by the Cauchy–Schwarz inequality implies that $u$ is a positive distance away from $S\_B$ in the $L^2(U)$ norm. For instance, if we take $u = K\rvert\_U \in S\_U$ where $K(x) = \frac{-1}{4\pi \lvert x\rvert}$ is the Newton potential (the fundamental solution to $\Delta K = \delta$) then
$$ \int\_B u \Delta \varphi = \int (\Delta K) \varphi = 1 \neq 0$$
and hence $u$ is a positive distance from $S\_B$.


One can create similar obstructions using functions $\varphi$ which behave like a specified spherical harmonic in the angular variable (instead of being constant in the angular variable, which is basically what is being done here).


