Gamma-convergence of graph Ginzburg-Landau functionals

Abstract

We study Gamma-convergence of graph based Ginzburg-Landau functionals, both the limit for zero diffusive interface parameter epsilon->0 and the limit for infinite nodes in the graph m -> infinity. For general graphs we prove that in the limit epsilon -> 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 epsilon -> 0 and m -> infinity, by expressing epsilon as a power of m and taking m -> infinity. Finally we investigate the continuum limit for a nonlocal means type functional on a completely connected graph.

version 1: 53 pages, 1 figure; version 2: some slight changes and corrections of minor details/typos, 54 pages; version 3: some minor, yet confusing, typos corrected in Lemma 3.3 and its proof; version 4: minor ambiguity corrected in proof of Lemma 3.3

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