Phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation

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

“…Fan, B. Xu and J. Yang [14] constructed a solution with single phase transition layer connecting ∂Ω. In [14], the authors showed that the inhomogeneous term V as well as the boundary of Ω will play an important role in the procedure of the construction. They found a suitable method to decompose the interaction among the phase transition layers, the boundary and the inhomogeneous term V .…”

“…They found a suitable method to decompose the interaction among the phase transition layers, the boundary and the inhomogeneous term V . More precisely, for the existence of a single phase transition layer, they proposed the following assumptions (A1)-(A3) in [14], see Figure 1:…”

“…Stationary and non-degenerate curves. In this section, we first recall some geometric notions for the stationary and non-degenerate curves with respect to the functional Γ V 1/2 from Section 2 of [14]. In (t, θ) coordinates, we denote V (F(t, θ)) as V (t, θ).…”

Stable transition layer for the Allen–Cahn equation when the spatial inhomogeneity vanishes on a nonsmooth hypersurface in ℝn

“…This result was extended to higher dimensions in [11]. Additionally, [12] studied the case where the curve $$$ \tilde{\gamma} $$$ connects two points of $$$ \mathrm{\partial \Omega } $$$. We also cite [13] where stable transition layers were obtained assuming certain geometric conditions that correlate $$$ \tilde{\gamma} $$$ and $$$ h $$$.…”

Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain

“…On the other hand, X. Fan, B. Xu and J. Yang [22] constructed a solution with single interior phase transition layer connecting ∂Ω near a curve Γ, which connects perpendicularly the boundary ∂Ω and is also stationary and non-degenerate with respect to Γ V 1 2 . Clustered interior phase transition layers connecting ∂Ω can be found in the paper by S. Wei and J. Yang [46].…”