Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h$ such that

(a) $h=f_1$ on $\partial \Omega_1$ and $h=f_2$ on $\partial \Omega_2$,

(b) $-\Delta h \leq 0$ in $\Omega_2$,

(c) and $h$ is harmonic in $\Omega_2 \setminus \Omega_1$?

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\Omega_2)$ such that

(a) $h=f_1$ on $\partial \Omega_1$ and $h=f_2$ on $\partial \Omega_2$,

(b) $-\Delta h \leq 0$ in $\Omega_2$,

(c) and $h$ is harmonic in $\Omega_2 \setminus \Omega_1$?