A celebrated conjecture due to De Giorgi states that any bounded solution of the equation
Δu+(1–u²) u= 0 in ℝ N with∂ yN u> 0 must be such that its level sets u–λ are all …

Abstract In 1979, De Giorgi conjectured that the only bounded monotone solutions to the
Allen-Cahn equation Δ u+ uu^ 3= 0\; in\; R^ N Δ u+ u− u 3= 0 in RN are one-dimensional …

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of
its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of …

A new nonlocal, gradient based damage model is proposed for isotropic elastic damage
using the phase field method in order to show the evolution of damage in brittle materials …

We prove that finite Morse index solutions to the Allen‐Cahn equation in ℝ2 have finitely
many ends and linear energy growth. The main tool is a curvature decay estimate on level …

This paper is chiefly concerned with qualitative properties of some reaction–diffusion fronts.
The recently defined notions of transition fronts generalize the standard notions of traveling …

For all n⩾ 1, we are interested in bounded solutions of the Allen–Cahn equation Δu+ u− u3=
0 which are defined in all Rn+ 1 and whose zero set is asymptotic to a given minimal cone …

Elliptic Systems of Phase Transition Type Page 1 Progress in Nonlinear Differential Equations
and Their Applications 91 Nicholas D. Alikakos Giorgio Fusco Panayotis Smyrnelis Elliptic …

We study the qualitative properties of a limiting elliptic system arising in phase separation for
Bose–Einstein condensates with multiple states: When n= 1, we prove uniqueness of the …

Let (M, ̃ g) be an N-dimensional smooth compact Riemannian manifold. We consider the
singularly perturbed Allen–Cahn equation ε^ 2 Δ _ ̃ gu\,+\,(1-u^ 2) u\,=\, 0 in\, M, where ε is a …