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November/December 2012 $\Gamma$-convergence of graph Ginzburg-Landau functionals
Andrea L. Bertozzi, Yves van Gennip
Adv. Differential Equations 17(11/12): 1115-1180 (November/December 2012).

Abstract

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.

Citation

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Andrea L. Bertozzi. Yves van Gennip. "$\Gamma$-convergence of graph Ginzburg-Landau functionals." Adv. Differential Equations 17 (11/12) 1115 - 1180, November/December 2012.

Information

Published: November/December 2012
First available in Project Euclid: 17 December 2012

zbMATH: 06120663
MathSciNet: MR3013414

Subjects:
Primary: 35Q56, 35R02

Rights: Copyright © 2012 Khayyam Publishing, Inc.

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Vol.17 • No. 11/12 • November/December 2012
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