Related papers: Brakke Regularity for the Allen-Cahn Flow

We give a new proof of Brakke's partial regularity theorem up to C^{1,\varsigma} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The…

The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean…

We prove convergence of solutions to the parabolic Allen-Cahn equation to Brakke's motion by mean curvature in space forms, generalizing previous results from [15] in Euclidean space. We show that a sequence of measures, associated to…

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…

In this paper we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution…

In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral…

We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our…

We study a singular limit problem of the Allen-Cahn equation with the homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the…

We establish the $\varepsilon$-regularity theorem for $k$-dimensional, possibly forced, Brakke flows near a static, multiplicity-one triple junction. This result provides the parabolic analogue to L. Simon's foundational work on the…

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given \alpha-H\"{o}lder…

We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are…

We prove that, if a Brakke flow with boundary is close enough to a stationary half-plane with density one, then it is $C^{1,\alpha}$. Our approach is based on viscosity techniques introduced by Savin in the context of elliptic equations.…

We study Brakke's mean curvature flow with obstacles and with a right-angle boundary condition. Assuming that the obstacles have $C^{1,1}$-boundaries we prove that a weak solution exists globally in time. To show the existence we apply the…

It is known that there is a strong relation between the parabolic Allen--Cahn equation and the mean curvature flow, in the sense that the parabolic Allen--Cahn equation can be considered as a ``diffused" mean curvature flow. In this work,…

We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the…

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean…

We prove convergence of the nonlocal Allen-Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface…

The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the…

In this paper we establish an improvement of tilt-excess decay estimate for the Allen-Cahn equation, and use this to give a new proof of Savin's theorem on the uniform $C^{1,\alpha}$ regularity of flat level sets, which then implies the one…

We consider the obstacle problem of the weak solution for the mean curvature flow, in the sense of Brakke's mean curvature flow. We prove the global existence of the weak solution with obstacles which have $C^{1,1}$ boundaries, in two and…