Fixed-energy harmonic functions

Fixed-energy harmonic functions, Discrete Analysis 2017:18, 21 pp.

The classical Dirichlet problem asks for a harmonic function in the interior of a region that takes specified values on the boundary. One can formulate a discrete version of the problem as follows. Let $G$ be a finite graph. Associate with each edge $e$ a positive real number $c_e$ called its conductance, and define a subset $B$ of the vertices, of size at least 2, to be the boundary. Given a function $f$ defined on the vertices, we say that it is harmonic if $\sum_{y\sim x}c_{xy}(f(x)-f(y))=0$ for every vertex $x\notin B$, where we have written $\sim$ for the relation “is a neighbour of”. That is, the value at each non-boundary vertex $x$ is an average, weighted by the conductances, of the values at its neighbours.

One of the methods of solving the classical Dirichlet problem is an energy-minimization argument, and that works for the discrete version as well. First, one defines the energy of $f$ to be $\sum_{x\sim y}c_{xy}(f(x)-f(y))^2$. To understand why this weighting is appropriate, one should think of $f$ as a voltage. Then if $c_{xy}$ is large, meaning that the resistance of the edge is small, there should be a tendency for the values $f(x)$ and $f(y)$ not to be too different, whereas if $c_{xy}$ is small and the resistance is large, then this tendency should be reduced.

An easy variational argument (just differentiate with respect to the value at $x$) shows that if this energy is minimized and $x\notin B$, then $f$ is indeed harmonic at $x$.

This paper turns the usual discrete Dirichlet problem on its head, in the following sense. We can regard the usual problem as providing for us a function that takes as its input a set of conductances and outputs the energies of each of the edges. Here this is reversed: one is given the edge energies and also directions on the edges (which have to satisfy some simple compatibility conditions), and one finds conductances such that the solution $f$ to the corresponding Dirichlet problem gives rise to those energies and such that if the edge $xy$ is directed from $x$ to $y$, then $f(x)>f(y)$. The main theorem of the paper is that the solution to this problem is unique, from which it follows that the number of solutions if one just wishes to obtain the given energies is obtained by enumerating the compatible sets of directions on the edges. These solutions turn out to be the local maxima of the expression $\prod_{x\sim y}|f(x)-f(y)|^{\mathcal E_{xy}}$, where $\mathcal E_{xy}$ is the energy associated with the edge $xy$, and if $f$ is such a solution, then it satisfies the equation $\sum_{y\sim x}\frac{\mathcal E_{xy}}{f(x)-f(y))}=0$ for every interior vertex $x$. The authors call such functions enharmonic.

The results of this study have some interesting consequences. For example, as is well known, there are close connections between electrical networks and rectangle tilings: one of the applications in this paper is that that certain polygons cannot be tiled by rectangles with rational areas. The authors also define an enharmonic conjugate function and obtain results analogous to the Cauchy-Riemann equations and the Riemann mapping theorem for the resulting “analytic functions”.