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

By Yves van Gennip, Andrea L. Bertozzi. Advances in Differential Equations, 2012

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/953b5ba6118ff9718d29de51d290d8aa23025b5e