Let $G = (V, E)$ denote the $n \times n$ integer grid, with the natural boundary $\partial V$. If $f$ is any real valued function defined on $\partial V$, then it is well known that $f$ can be extended to a harmonic function $g$ on the entire grid, in the sense that for every $v \in V \setminus \partial V$, $$g(v) = \sum_{u|(u,v)\in E} \frac{g(u)}{4},$$ and for $v \in \partial V$ $$g(v) = f(v).$$ I'm asking about a possible inverse to this.
Suppose that $f$ is defined on $U \subset V$, is there a function $g$ such that $g$ is harmonic on $V \setminus \partial V$ and $g = f$ when restricted to $U$?
(Note that this is not the same as solving the Dirichlet problem for the set $U$.)
Clearly, this is impossible in the general case. Suppose that $U$ contains a 'cross', i.e. an inner vertex as well as all of its neighbors. If $f$ is not harmonic on this cross, then surely it cannot be extended.
Another bad case could be when $\partial V \subsetneq U$. Since the values of any harmonic extension of $f$ depend exclusively on the values of $f$ on $\partial V$, if the values of the possible extension would not agree with the value of $f$ on an inner vertex, then $f$ cannot be extended.
Thus, we make the following restrictions on $U$:
- $|U| \leq |\partial V|$.
- $U$ is simply connected, in the sense that it does not contain any cycle (except maybe if $U = \partial V$).
Notice that solving the problem amounts to solving a system of linear equations with $|V \setminus U|$ variables and $|V \setminus \partial V|$ equations.
Intuitively, the first restriction tells us that the number of variables is not less than the number of equations and the second restriction tells us that the equations are independent, so we should expect to find a solution.
I'm looking for any references on these types of problems, beyond the integer grid.
Also, if anything could be said about the continuous case, that should be very interesting, what type of restrictions would be required there?