Brakke’s inequality for the thresholding scheme

Abstract

We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke's notion of mean curvature flow. Our proof is based on a localized version of the minimizing movements interpretation of Esedoglu and the second author. We apply De Giorgi's variational interpolation to the thresholding scheme and pass to the limit in the resulting energy-dissipation inequality. The result is conditional in the sense that we assume the time-integrated energies of the approximations to converge to those of the limit.

De Giorgi’s inequality for the thresholding scheme with arbitrary mobilities and surface tensions

“…The present work fits into the theory of general gradient flows even better than the two previous ones [19,20] and crucially depends on De Giorgi's abstract framework, cf. [2].…”

“…where we used the semigroup property (22) and the symmetry (20) to derive the last equality. We also point out that since…”

“…We exploit the gradient-flow structure and show that under the natural assumption of energy convergence, any limit of thresholding satisfies De Giorgi's inequality, a weak notion of multiphase mean curvature flow. This assumption is inspired by the fundamental work of Luckhaus-Sturzenhecker [25] and has appeared in the context of thresholding in [19,20]. We expect it to hold true before the onset of singularities such as the vanishing of grains.…”

“…The variational interpretation of the thresholding scheme has of course implications for the analysis of the algorithm as well. It allowed Otto and one of the authors to prove convergence results in the multiphase setting [19,20], which lies beyond the reach of the more classical viscosity approach based on the comparison principle implemented in [3,9,13]. Also in different frameworks, this variational viewpoint turned out to be useful, such as MCF in higher codimension [23] or the Muskat problem [14].…”

Mullins-Sekerka as the Wasserstein flow of the perimeter

“…Glasner [12] introduced a phase-field approximation to the one-phase Mullins-Sekerka equation and studied its convergence by formal asymptotic expansions. Recently, also the computationally efficient thresholding scheme by Merriman, Bence, and Osher [22,23] has been reinterpreted as a minimizing movements scheme by Esedoglu and Otto [8], which allowed one of the author together with Otto to prove conditional convergence results to multiphase mean curvature flow [18,17]. Most recently, Jacobs, Kim, and Mészáros [13] introduced an interesting thresholding-type approximation for the Muskat problem and proved a similar (conditional) convergence result for their scheme.…”