Related papers: Bouncing Jacobi fields and the Allen-Cahn equation…
This paper is concerned with the motion of a time dependent hypersurface that evolves by mean curvature flow with a a volume constraint. Phase field approximation of this motion leads to the well known nonlocal Allen--Cahn equation. Here we…
A well-known conjecture of De Giorgi -- motivated by analogy with the Bernstein problem for minimal surfaces -- asserts the rigidity of monotone solutions to the Allen--Cahn equation in $\mathbb{R}^{d+1}$, with $d\leq 7$. We establish close…
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…
The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work…
The free boundary Allen--Cahn equation $\Delta u=0$ in $\{|u|<1\}$, $|\nabla u|=1/\varepsilon$ on $\partial\{|u|<1\}$ has recently attracted considerable attention because it retains the essential features of the classical Allen--Cahn…
We consider the isoperimetric problem defined on the whole $\mathbb{R}^n$ by the Allen--Cahn energy functional. For non-degenerate double well potentials, we prove sharp quantitative stability inequalities of quadratic type which are…
We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for…
We show that certain functions whose nodal sets lie near a fixed nondegenerate minimal hypersurface satisfy a strong min-max principle for the Allen--Cahn energy which is analogous to the strong min-max principle for non-degenerate minimal…
This paper establishes bounds on the homogenized surface tension for a heterogeneous Allen-Cahn energy functional in a periodic medium. The approach is based on relating the homogenized energy to a purely geometric variational problem…
We establish existence and non-existence results for entire solutions to the fractional Allen-Cahn equation in $\mathbb R^3$, which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces…
We consider the varifold associated to the Allen--Cahn phase transition problem in $\mathbb R^{n+1}$(or $n+1$-dimensional Riemannian manifolds with bounded curvature) with integral $L^{q_0}$ bounds on the Allen--Cahn mean curvature (first…
In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE)$_{\varepsilon}$ for $ \varepsilon \geq 0 $. For each $ \varepsilon \geq 0 $, the system (ACE)$_{\varepsilon}$ consists of an Allen-Cahn type equation in a…
The convex-concave splitting discretization of the Allen-Cahn is easy to implement and guaranteed to be energy decreasing even for large time-steps. We analyze the time-stepping scheme for a large class of potentials which includes the…
In this article we study the second variation of the energy functional associated to the Allen-Cahn equation on closed manifolds. Extending well known analogies between the gradient theory of phase transitions and the theory of minimal…
We study the solutions of a generalized Allen-Cahn equation deduced from a Landau energy functional, endowed with a non-constant higher order stiffness. We analytically solve the stationary problem and deduce the existence of so-called…
The goal of this paper is to study the slow motion of solutions of the nonlocal Allen-Cahn equation in a bounded domain $\Omega \subset \mathbb{R}^n$, for $n > 1$. The initial data is assumed to be close to a configuration whose interface…
This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $\Omega\subset\mathbb{R}^n$, with or without a source term in the right hand side of the equation…
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…
The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics (or variational) model for a nonconserved order parameter field. After revising main literature results for the occuring different types of moving fronts, we employ…
Boundary correlation functions provide insight into the emergence of an effective geometry in higher spin gravity duals of O(N) or U(N) symmetric field theories. On a compact manifold, the singlet constraint leads to nontrivial dynamics at…