Let $U_2,U_3$ be the open unit balls in $\mathbb{R}^2,\mathbb{R}^3$ respectively.
Fact 1: (From Rudin's real complex analysis) Let $u:\partial U_2\rightarrow \mathbb{R}$ be continuous, then there exists a unique continuous function $u_{*}:\overline{U_2}\rightarrow \mathbb{R}$ such that $u_{*}|U_2$ is harmonic. The function $u_{*}$ is given by: $$ u_{*}(r\cos\theta,r\sin\theta)=\begin{cases} \frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1-r^2}{1-2r\cos(\theta-t)+r^2}u(\cos(t),\sin(t)) dt & \textrm{if } 0\leq r<1 \textrm{ } \\ u(r\cos\theta,r\sin\theta) & \textrm{if } \,r=1{ } \end{cases} $$
Question 1: Let $v:\partial U_3\rightarrow \mathbb{R}$ be continuous, Is there a continuous function $u_{*}:\overline{U_3}\rightarrow \mathbb{R}$ such that $u_{*}|U_3$ is harmonic ?
Question 2: Is there a known way to construct the function $v_{*}$ in question 1 like how the function $u_{*}$ was constructed in fact 1 ?