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

Abstract

We provide a new convergence proof of the celebrated Merriman–Bence–Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoḡlu and Otto and De Giorgi’s general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

Weak-strong uniqueness for volume-preserving mean curvature flow

“…The optimal energy-dissipation rate in Item (iii) is the natural rate which is satisfied by any classical solution. We note that such a sharp inequality is at the heart of the definition of gradient flows [30,2] and has been verified for vanilla mean curvature flow by Otto and the author [22]; see also [20] for the case of multiple phases.…”

“…Since both terms on the right are non-negative (cf. ( 5)), the first estimate (30) then follows directly from the definition (20), and the second estimate (31) follows from the quantitative shortness condition (5) and the Lipschitz continuity of the weight function ϑ. Now we give the proof of Theorem 2, which partly follows the weak-strong uniqueness proof in the unconstrained case of vanilla mean curvature flow [7].…”

“…Furthermore, let χ be a distributional solution of volume-preserving mean curvature flow in the sense of Definition 3. Then, the relative entropy E(t) and the volume error F (t) given in (20) and (21), respectively, satisfy…”

Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

“…On a compact support the uniform continuity of the kernel is sufficient. Lastly, the convergence of quadratic surfaces is a simple consequence of (13), which is shown to hold in the proof of Theorem 4.4.…”

“…The convergence of the original scheme in the two-phase case to viscosity solutions has been established in the isotropic case by Evans [11], and Barles and Georgelin [4], and in the anisotropic case by Ishii, Pires and Souganidis [12]. More recently, in the multi-phase case, several conditional convergence results were established, see [14,15,16,13].…”