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.