\(\Gamma\)-convergence of graph Ginzburg-Landau functionals. (English) Zbl 1388.35200

van Gennip, Yves; Bertozzi, Andrea L.

Adv. Differ. Equ. 17, No. 11-12, 1115-1180 (2012).

Summary: We study Gamma-convergence of graph based Ginzburg-Landau functionals, both the limit for zero diffusive interface parameter \(\varepsilon\to 0\) and the limit for infinite nodes in the graph \(m\to \infty\). For general graphs we prove that in the limit \(\varepsilon\to 0\) the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4-regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg-Landau functional. For both functionals we also study the simultaneous limit \(\varepsilon\to 0\) and \(m\to\infty\), by expressing epsilon as a power of \(m\) and taking \(m\to\infty\). Finally we investigate the continuum limit for a nonlocal means type functional on a completely connected graph.

MSC:

35R02	PDEs on graphs and networks (ramified or polygonal spaces)
35Q56	Ginzburg-Landau equations
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
49J45	Methods involving semicontinuity and convergence; relaxation