I'm reading the following paper on applying mean curvature flow to gerrymandering:
- Matt Jacobs, Olivia Walch, A partial differential equations approach to defeating partisan gerrymandering.
The setting is that we have a weighted graph of vertices, $x\in \mathbb{R}^2$ where each $x$ is a "small blob" of a larger subset $D$ and we're forming a partition of $D$ into districts $D_i$ by basically sorting these $x$ into districts based on some criteria.
And I'm having a lot of trouble calculating the Gradient flow (wrt the partition $\mathbf{u}$) of the compactness energy defined to be $$ E_\alpha(\mathbf{u},\mathbf{c})= \frac{1}{2}\sum_{i=1}^N\sum_{x,y\in V} u_i(x) A(x,y)(1-u_i(y)) + \alpha \sum_{i=1}^N\sum_{x\in V}u_i(x)\lVert c_i-c(x)\rVert^2$$
where $A(x,y)$ is a weight matrix transposed and multiplied to itself. $\mathbf{u}$ is a vector of characteristic equations.
The paper shows later on that the gradient (for a single partition) is $$ \psi_i^{n+1}(x) = \alpha\lVert c_i^n -c(x)\rVert ^2- \sum_{y\in V}W(x,y)u_i^n(y)$$
Is this the Gradient flow of E? To a greater extent, I'm just having a lot of issues understanding where the PDE plays in. I know that Mean Curvature flow is a sort of nonlinear heat equation and that you can derive it from taking this Energy and computing gradient flow, so my professor told me to try calculating it to derive the explicit PDE, but it doesn't seem like the paper reaches any type of PDE.
Thank you for any help and advice and intuition if you're experienced with the content of this paper.