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 $\|v-u\|_{L^2(U)} \leq \epsilon$.
