Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows

Abstract

We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales.Such results will be used in [40] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in space forms.

Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates

“…For a general class of initial conditions we show nonpositivity of the limiting energy discrepancy. This in turn allows us to prove an almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales.These results will be used in [41] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in Riemannian manifolds.…”

“…These results will be used in [41] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in Riemannian manifolds.…”

“…Finally, Brakke's inequality for dµ t is obtained from the corresponding approximate ones valid for dµ " t . The aim of this paper and of [41] is to generalize the results in [31] recalled above, to the case of solutions u " to problem (1.1)-(1.2) on Riemannian manifolds. We always assume that there exists 2 R such that…”

“…in [16,39] and [40]. Now, we outline results that will be shown in the present paper and we briefly mention the content of [41]. For any " > 0, define the energy density…”

Min–max for phase transitions and the existence of embedded minimal hypersurfaces

“…A typical example of such a non-linearity is W (u) = 1 4 (1 − u 2 ) 2 . From a geometrical point of view solutions to equation (1) have a remarkable feature: roughly speaking, as ε → 0 the level sets of u concentrate around a hypersurface (called the limit-interface) that is evolving in time under the action of the mean curvature flow (see for example [16,23,28]). This suggests that for the particular case of stationary states of (1), i.e.…”