Advances in Differential Equations

$\Gamma$-convergence of graph Ginzburg-Landau functionals

Andrea L. Bertozzi and Yves van Gennip

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 $\varepsilon$ 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.

Article information

Adv. Differential Equations Volume 17, Number 11/12 (2012), 1115-1180.

First available in Project Euclid: 17 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces) 35Q56: Ginzburg-Landau equations


Gennip, Yves van; Bertozzi, Andrea L. $\Gamma$-convergence of graph Ginzburg-Landau functionals. Adv. Differential Equations 17 (2012), no. 11/12, 1115--1180.

Export citation