Closed Geodesics on Oval Surfaces and Pattern Formation

Abstract

We study a singularly perturbed semilinear elliptic partial differential equation with a bistable potential on an oval surface. We show that the transition region of minimizers of the associated functional with a suitable constraint converges in the sense of varifolds to a minimal closed geodesic on the surface.

Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation

“…The non-constant function V represents the spatial inhomogeneity. The case V ≡ 1 corresponds to the standard Allen-Cahn equation [3]    2 u + u (1 − u 2 ) = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, (1.2) for which extensive literature on transition layer solutions is available, see for instance [1,2,4,15,16,17,18,20,21,22,25,26,27,28,29,31], and the references therein.…”

Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

“…Local elliptic regularity implies in particular that φ is bounded. Since for some t 0 > 0, the equation satisfied by φ is 11) and c(y) is bounded, then enlarging t 0 if necessary, we see that for σ < √ 2, a suitable barrier argument shows that |φ| ≤ Ce −σ|t| , hence φ p,σ < +∞. From (4.7) we obtain that…”

The Global Attractor of the Allen-Cahn Equation on the Sphere

“…The main result is the following in which the existence of a global attractor is shown for equation (4) subject to constraint (5).…”

“…A similar behavior occurs on an oval surface for non-trivial solutions of (1). Using results in Hutchinson and Tonewaga [1] and Padilla and Tonewaga [2], in Garza-Hume and Padilla [4] it is established that, when ǫ → 0, non-trivial minima of the corresponding energy function (with a suitable restriction) have a transition layer located at the shortest closed geodesic.…”