I have the following question and I can't figure out how to do the proof. Could you give me some hints in both directions of the equivalence?
Suppose $A$ is a bounded subset of $\mathbb{Z}^d$. Then $\bar{A} = A \cup \partial A $ where $\partial A$ is the outer boundary defined by $ \partial A = \{y \in A^c : |x - y| = 1 $ for some $ x \in A\}$. We call $ \{x; y\} $ an edge of $ \bar{A} $ if $ x, y \in \bar{A}, |x - y| = 1 $ and at least one of $ x, y $ is in $ A $. If $ F : \bar{A} \rightarrow \mathbb{R} $ is a function, we define its energy by \begin{equation*} E(f) = \sum_{\{x,y\} \text{ edge of } \bar{A}} (f(x) - f(y))^2. \end{equation*} For any $ F : \partial A \rightarrow \mathbb{R} $, define $ E(F) $ to be the infimum of $ E(f) $ where the infimum is over all $ f $ on $ \bar{A} $ that agree with $ F $ on $ \partial A $. Show that if $ f $ agrees with $ F $ on $ \partial A $, then $ E(f) = E(F) $ if and only if $ f $ is harmonic in $ A $.
Update: I believe this can be proved using the Discrete Divergence Theorem, but I still don't know how.