The free-boundary Brakke flow

@article{Edelen2016TheFB,
  title={The free-boundary Brakke flow},
  author={Nick Edelen},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={2016},
  volume={2020},
  pages={137 - 95}
}
  • Nick Edelen
  • Published 11 February 2016
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
Abstract We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must “pop” upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White [23] to the free-boundary setting. Using Ilmanen’s elliptic regularization procedure [10… Expand
Mean Convex Mean Curvature Flow with Free Boundary
In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is toExpand
Convergence of the Allen–Cahn equation with a zero Neumann boundary condition on non-convex domains
We study a singular limit problem of the Allen–Cahn equation with a homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data.Expand
CONTRACTING CONVEX SURFACES BY MEAN CURVATURE FLOW WITH FREE BOUNDARY ON CONVEX BARRIERS
We consider the mean curvature flow of compact convex surfaces in Euclidean 3-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface isExpand
Contracting convex surfaces by mean curvature flow with free boundary on convex barriers.
We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surfaceExpand
Gradient estimates for mean curvature flow with Neumann boundary conditions
We study the mean curvature flow of graphs both with Neumann boundary conditions and transport terms. We derive boundary gradient estimates for the mean curvature flow. As an application, theExpand
Singular Neumann Boundary Problems for a Class of Fully Nonlinear Parabolic Equations in One Dimension
TLDR
The singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension describes motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. Expand
A local regularity theorem for mean curvature flow with triple edges
Abstract Mean curvature flow of clusters of n-dimensional surfaces in ℝ n + k {\mathbb{R}^{n+k}} that meet in triples at equal angles along smooth edges and higher order junctions onExpand
Rectifiability of the free boundary and density set for varifolds.
We establish a partial rectifiability result for the free boundary of a $k$-varifold $V$ and an upper bound on the Hausdorff dimension of the set where the $k$-density of $V$ is infinite or does notExpand
Level set mean curvature flow with Neumann boundary conditions
where V is the normal velocity of Γt and H is the mean curvature of Γt. In the case of Ω = R N , i.e. without boundary, the problem has long been studied and several different approaches have beenExpand
RECTIFIABILITY OF THE FREE BOUNDARY FOR VARIFOLDS
Abstract. We establish a partial rectifiability result for the free boundary of a k-varifold V . Namely, we first refine a theorem of Grüter and Jost by showing that the first variation of a generalExpand

References

SHOWING 1-10 OF 33 REFERENCES
Mean curvature flow with free boundary on smooth hypersurfaces
Abstract The classical mean curvature flow of hypersurfaces with boundary satisfying a Neumann condition on an arbitrary, fixed, smooth hypersurface in Euclidean space is examined. In particular, theExpand
Convexity estimates for mean curvature flow with free boundary
We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier $S$. Here $S$ can be any properly embedded,Expand
Non-parametric radially symmetric mean curvature flow with a free boundary
We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The firstExpand
Convergence of solutions to the mean curvature flow with a Neumann boundary condition
This work continues our considerations in [15], where we discussed existence and regularity results for the mean curvature flow with homogenious Neumann boundary data. We study the long timeExpand
On the singular set of mean curvature flows with Neumann free boundary conditions
We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of theExpand
Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition
In this work we study the behaviour of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidian space. We thereby prescribe the Neumann boundaryExpand
Regularity of mean curvature flows with Neumann free boundary conditions
We consider n-dimensional hypersurfaces flowing by the mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. Under assumptions mirroring those for the caseExpand
Convergence of the Allen–Cahn equation with a zero Neumann boundary condition on non-convex domains
We study a singular limit problem of the Allen–Cahn equation with a homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data.Expand
Regularity results for minimizing currents with a free boundary.
The regularity question for minimal surfaces or minimizing integer multiplicity rectifiable currents in the context of geometric measure theory is naturally divided into three cases. In the classicalExpand
Convergence of the Allen-Cahn Equation with Neumann Boundary Conditions
TLDR
It is proved that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary. Expand
...
1
2
3
4
...