In [BF12, BF16] the graph Ginzburg–Landau functional was introduced. Here u is a real-valued function on the node set V of a simple1, undirected graph (with ui its value at node i), ωij ≥ 0 are edge weights which are assumed to be positive on all edges in the graph and zero between non-neighbouring nodes i and j, ε is a positive parameter, and W is a double well potential with wells of equal depth. A typical choice is the quartic polynomial W(x) = x2(x − 1)2 which has wells of depth 0 at x = 0 and x = 1, but we will encounter some situations where other choices are useful or even necessary.
|Title of host publication||Proceedings of 44th Sapporo Symposium on Partial Differential Equations|
|Editors||S.I. Ei, Y. Giga, N. Hamamuki, S. Jimbo, H. Kubo, H. Kuroda, T. Ozawa, T. Sakajo, K. Tsutaya|
|Publication status||Published - 2019|