Advances in Differential Equations

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

Andrea L. Bertozzi and Yves van Gennip

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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.

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