Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds

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

“…For the two-dimensional case, ifΓ is a closed curve in Ω satisfying stationary and non-degenerate conditions with respect to the metric Γ V 1/2 , Z. Du and C. Gui [12] constructed a layer nearΓ, see also [13,19,37]. Note that these results concerned the existence of interior phase transition phenomena (away from ∂Ω) for the inhomogeneous Allen-Cahn problem (1.1).…”

“…Assumption (1.5) has the same geometric meaning of (3.10), which is (1.5) in [10]. In other words, in order to construct the clustering phase transition layers connecting the boundary ∂Ω in Theorem 1.1, we use the same assumptions as those in [10] (see also the assumptions in [13] and [37]) except the assumptions in (1.6).…”

Layered solutions for a fractional inhomogeneous Allen–Cahn equation

“…The authors proved that (2) has an interior layer solution and this layer appears near a non-degenerate closed geodesic curve relative to the weighted arclength Γ V 1 2 ds. Existence of layer solutions and clustering layer solution of (2) in general dimension Euclidean spaces and Riemannian manifolds were also obtained in [10,11,15,17]. The case V ≡ 1 of the equation in (2) corresponds to the standard Allen-Cahn equation (see [1])…”

Clustering layers for the Fife—Greenlee problem in ℝn

“…The function u represents a continuous realization of the phase present in a material confined to the region at the point x which, except for a narrow region, is expected to take values close to +1 or −1. Of particular interest are of course non-trivial steady state configurations in which the antiphases coexist, see for instance [4,17,18,19,20,23,26,27,32,33,34,36,37,39,40,41,42,45,46]. There are also many known results for the general inhomogeneous case: smooth function a satisfies −1 < a(x) < 1 in Ω and ∇a = 0 on the smooth closed hypersurface K = {a(x) = 0}, which separates the domain into two disjoint components Ω = Ω − ∪ K ∪ Ω + , with a < 0 in Ω − , a > 0 in Ω + , a = 0 on K.…”