An existence theorem for Brakke flow with fixed boundary conditions

Abstract

Consider an arbitrary closed, countably n-rectifiable set in a strictly convex $$(n+1)$$ ( n + 1 ) -dimensional domain, and suppose that the set has finite n-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As $$t \uparrow \infty $$ t ↑ ∞ , the flow sequentially converges to non-trivial solutions of Plateau’s problem in the setting of stationary varifolds.

On the existence of canonical multi-phase Brakke flows

“…The grains in [18] move continuously with respect to the Lebesgue measure, but the problem concerning the validity of an exact identity involving the volume change was not addressed in there. Previous works by the authors of the present paper (see [32,31]), in which certain Brakke flows are constructed with an approximation scheme analogous to that introduced in [18], could be reworked so that the additional conclusions concerning the interplay between the flow of the grains and the Brakke flow can be drawn in those contexts as well: in particular, it is possible to have the Brakke flow with prescribed boundary constructed in [32] satisfy the formulae (1.1) and (1.3) (see Section 7.2).…”

“…Hence, the proof of the present paper works with no essential change away from ∂U , and (2.14) holds for φ ∈ C 1 c (U × R + ); since the formula does not involve ∇φ, by approximation, the same formula holds even for φ ∈ C 1 (clos U × [0, T ]) for arbitrary T > 0. Since the existence results in [31] are based on [32], the same applies to the solutions discussed in [31]. ˆT 0 H n (∂ * E i (t)) dt…”

“…MCF with fixed boundary conditions. In [32], given a strictly convex bounded domain U ⊂ R n+1 with C 2 boundary ∂U , a countably n-rectifiable set Γ 0 ⊂ U with H n (Γ 0 ) < ∞ and an open partition E 0,1 , . .…”

“…The construction method is along the lines of [18], and we may also carry it out using the volume-controlled Lipschitz maps. If one compares the construction in [32] with that in [18], one sees that differences occur only near the boundary ∂U : more precisely, the approximate smoothed mean curvature vector is damped near the portion of Γ 0 close to ∂U , and there is another step in each epoch -a Lipschitz retraction step (see [32,Section 2.6]). Hence, the proof of the present paper works with no essential change away from ∂U , and (2.14) holds for φ ∈ C 1 c (U × R + ); since the formula does not involve ∇φ, by approximation, the same formula holds even for φ ∈ C 1 (clos U × [0, T ]) for arbitrary T > 0.…”

Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities

“…For instance, in the case of the volume-preserving mean curvature flow, a characterization of equilibrium states requires the generalization of the classical Alexandrov's theorem (smooth boundaries with constant mean curvature enclosing finite volumes are spheres [Ale62]) to the class of sets of finite perimeter and finite volume with constant distributional mean curvature; see [DM19]. In a similar vein, Theorem 1.2 could be used to understand the long time behavior of (singular) mean curvature flows with fixed boundary given by two parallel convex curves; see [ST19].…”