arXiv research

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

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

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 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)…

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

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

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

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

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.

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

Stable solutions to a specific equation are one-dimensional.

Study quantizes energy distribution in inhomogeneous phase transitions.

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…

We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularit…

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…

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

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.

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

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.

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.

New minimal surfaces found using Toda lattice and integrable systems.

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 …

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 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…