# Harmonic/Subharmonic lifting of functions on an annulus

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 $\Omega_2 \setminus \Omega_1$?

By the maximum principle for subharmonic function, a necessary condition is that the max of $f_1$ on $\partial\Omega_1$ is less than the max of $f_2$ on $\partial\Omega_2$. But even in this case, there is no reason, in general, that the function $h$ exists. Indeed, by (c), it should be equal on the annulus $\Omega_2\setminus\Omega_1$ to the unique solution $u$ of the Dirichlet problem with data $f_1$ and $f_2$ on the boundaries of $\Omega_2\setminus\Omega_1$. Then, the question is to extend the (sub-)harmonic function $u$ to $\Omega_1$. The only general result, which I know, ensuring that one can extend a subharmonic function is given by the Removable Singularity Theorem (see Ransford, Potential Theory in the Complex Plane, Theorem 3.6.1.). It only applies for very small set of singularities, namely closed polar sets, in particular, it cannot be applied in the present situation where the set of singularities is the open set $\Omega_1$.
Of course, for very special choice of $f_1$, like being equal on $\partial\Omega_1$ to the solution of the Dirichlet problem in $\Omega_2$ with data $f_2$, the function $h$ would exist (and then be harmonic in the whole of $\Omega_2$)