arXiv research

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

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

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…

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.

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

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…

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

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

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 propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss functi…

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

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

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…

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

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

We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work in all dimensions and without variational assumptions on the Allen-Cahn solution…

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

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

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

Morse theory connects low energy submanifolds in 3-sphere.

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

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

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

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

Stable solutions to a specific equation are one-dimensional.

Study quantizes energy distribution in inhomogeneous phase transitions.

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

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

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

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…

Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.

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

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 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.

The Allen-Cahn system on manifolds yields multiple phase distributions.

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.

We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage…

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 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.

Proves existence of a single-valued minimal hypersurface in compact manifolds.

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

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 …

Deep learning model solves high-dimensional PDEs using Actor-Critic approach.

Study shows strong min-max principle for phase transitions.

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…

Develops a PDE approach to constructing nontrivial anisotropic surfaces.