arXiv research

New insights on solutions to Allen-Cahn equation with degenerate minimal hypersurfaces.

Improved convergence and curvature estimate for parabolic Allen-Cahn equation.

Paper introduces a modified Allen-Cahn equation for better energy equipartition.

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 converges to the classical solution of the mean curvature flow under the suitable c…

Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.

Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.

The paper constructs solutions to the Allen-Cahn equation using special minimal hypersurfaces.

The paper proves smoothness of transition layers in the Allen-Cahn equation.

Stable solutions to a specific equation are one-dimensional.

Study approximates Plateau's laws using the Allen-Cahn equation.

Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.

Study compares nodal sets of solutions to the Allen-Cahn equation.

Strictly stable Allen-Cahn hypersurfaces have multiplicity one.

Rigidity theorem for critical points of Allen-Cahn equation on S³.

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 ambient manifold is a compact manifold with non-negative sectional curvature and pa…

Survey on Allen-Cahn equations and systems, focusing on multiplicity results and geometric interpretation.

Study of Dirichlet minimizers on manifolds with boundary and their asymptotic behavior.

Study on phase transitions on surfaces using Allen-Cahn equation.

Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.

We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and ap…

Proves generic nondegeneracy for solutions under volume constraint in closed manifolds.

We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation to show that, given any compact hypersurface $Σ$ of R^d, with $d\geq 4$, there is a bounded entire solution of the Allen-Cahn equation on R^d whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a re…

Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.

In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature.

In this paper we find solutions $u_ε$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $ε\to 0$ develops as interface a given triod evolving under curve shortening flow.

In this short note, we prove that on the three-sphere with any bumpy metric there exist at least four solutions of the Allen-Cahn equation with spherical interface and index at most two. The proof combines several recent results from the literature.

The paper proves existence of solutions to the Allen-Cahn equation on certain Riemannian manifolds.

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $\frac{\partial f}{\partial t}=\triangle f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of M…

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 curvature flow with a generalized right angle condition on the boundary.

Stable saddle solutions found for specific dimensions of the Allen-Cahn equation.

This research proves that two min-max theories for hypersurfaces are equivalent.

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 of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are ab…

We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological …

Study introduces a new Allen-Cahn energy on hypersurfaces and analyzes its properties.

In this paper we consider the Allen-Cahn equation $$ -Δu = u-u^3 \ \mbox{in} \ {\mathbb R}^3 $$ We prove that for each $k\in\left( \sqrt{2},+\infty\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $\| u-H(\cdot -k \ln r + c_k) \|_{L^\infty} \to 0$ The main ingredients of our proof con…

We prove the existence of global minimizers of Allen-Cahn equation in dimensions $8$ and above. More precisely, given any strictly area-minimizing Lawson's cones, there are global minimizers whose nodal sets are asymptotic to the cones. As a consequence of Jerison-Monneau's program we establish the existence of many co…

In this paper, we will address to the following parabolic equation $u_t=Δ_fu + F(u)$ on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here $F$ is a differentiable function defined in $\mathbb{R}$. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equ…

We prove that given a minimal hypersurface $Γ$ in a compact Riemannian manifold $M$ without boundary, if all the Jacobi fields of $Γ$ are generated by ambient isometries, then we can find solutions of the Allen-Cahn equation $-\varepsilon^2Δu +W'(u)=0$ on $M$, for sufficiently small $\varepsilon>0$, whose nodal sets co…

We construct new families of two-ended $O(m)\times O(n)$-invariant solutions to the Allen- Cahn equation Δu+u-u3=0 in $\mathbb{R}^{N+1}$, with $N\ge 7$, whose zero level sets diverge logarithmically from the Lawson cone at infinity. The construction is based on a careful study of the Jacobi-Toda system on a given $O(m)…

Study boundary behavior of limit interfaces in Riemannian manifolds without convexity assumptions.

The paper proves the existence of hypersurfaces with prescribed mean curvature.

New non-canonical flows found via parabolic Allen-Cahn equations.

New varifold solutions for mean curvature flow converge and are unique.

In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $α^2 Δu + u(1-u^2)=0, \quad \hbox{in }Ω\subset \R^N$ where $N=3$, $Ω$ is a smooth bounded domain and $\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as $α\to 0$, the level sets of the soluti…

Proves existence of multiple solutions to a multiphasic equation on manifolds.

In this paper we establish a uniform $C^{2,θ}$ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions $n\leq 10$ (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the $C^{2,θ}$ estimate …

We consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $Δu + f(u) = 0 \hbox{in} \R^3$. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. We assume that …

We show that the Morse index of every 2k-ended solution of the Allen-Cahn equation in R^2 is >= k-1. This bound is expected to be sharp.