Resistance form

Remarks

Example: let $X$ be the vertex set of a finite connected graph, with a conductance $c_{xy} = c_{yx} > 0$ on each edge $\set{x, y}$, let $\mathcal{F} = \ell(X)$, and let $\mathcal{E}(u) = \sum_{\set{x, y}} c_{xy} (u(x) - u(y))^2$, the sum over edges. Clause 1 holds because $\mathcal{E}(u) = 0$ forces $u$ to be equal at the two ends of every edge, hence constant on the connected graph. Clauses 2 and 4 hold because $\mathcal{F} / {\sim}$ is finite-dimensional and $\mathcal{E}$ is positive definite on it. Clause 3 holds with the indicator function of $\set{x}$. Clause 5 holds because $t \mapsto \min(\max(t, 0), 1)$ does not increase distances, so no term $(u(x) - u(y))^2$ grows. Here $R(x, y)$ is the effective resistance of the electrical network.

Non-example: the same formula on a graph with two connected components fails clause 1. The function equal to $0$ on one component and $1$ on the other has energy $0$ but is not constant.

On $X = [0, 1]$, the form $\mathcal{E}(u, v) = \int_0^1 u'(s) v'(s) \, ds$ on the absolutely continuous functions $u$ with $u' \in L^2$ is a resistance form with $R(x, y) = \abs{x - y}$. On a domain of $\R^n$ with $n \ge 2$, the Dirichlet energy $\int \abs{\nabla u}^2$ does not give a resistance form. There are smooth functions with arbitrarily small energy that equal $1$ at a given point $x$ and vanish at another point $y$, so clause 4 fails. The definition and the main theory of resistance forms are due to Kigami.