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

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
Full Text: