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

Abstract

Let ${(\mathcal{M}, \tilde{g})}$ be an N-dimensional smooth compact Riemannian manifold. We consider the problem ${\varepsilon^2 \triangle_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0\; {\rm in}\; \mathcal{M}}$, where ${\varepsilon > 0}$ is a small parameter and V is a positive, smooth function in ${\mathcal{M}}$. Let ${\kappa \subset \mathcal{M}}$ be an (N − 1)-dimensional smooth submanifold that divides ${\mathcal{M}}$ into two disjoint components ${\mathcal{M}_{\pm}}$. We assume κ is stationary and non-degenerate relative to the weighted area functional ${\int_{\kappa}V^{\frac{1}{2}}}$. For each integer m ≥ 2, we prove the existence of a sequence ${\varepsilon = \varepsilon_\ell \rightarrow 0}$, and two opposite directional solutions with m-transition layers near κ, whose mutual distance is ${{\rm O}(\varepsilon | \log \varepsilon | )}$. Moreover, the interaction between neighboring layers is governed by a type of Jacobi–Toda system.