Transition layers for an inhomogeneous Allen-Cahnequation in Riemannian manifolds
Transition layers for an inhomogeneous Allen-Cahnequation in Riemannian manifolds
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DOI：
10.3934/dcds.2013.33.1407
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发表时间：
2012-10
影响因子：
1.1
通讯作者：
Zhuoran Du;Baishun Lai
中科院分区：
文献类型：
--
作者：
Zhuoran Du;Baishun Lai
Let $(\mathcal{M}, \tilde{g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the problem $$ \varepsilon^2 Δ_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0 in \mathcal{M}, $$ where $\varepsilon >0$ is a small parameter and $V$ is a positive, smooth function in $\mathcal{M}$. Let $ \mathcal{K}\subset \mathcal{M}$ be an $(N-1)$-dimensional smooth submanifold that divides $\mathcal{M}$ into two disjoint components $\mathcal{M}_{\pm}$. We assume $\mathcal{K}$ is stationary and non-degenerate relative to the weighted area functional $\int_{\mathcal{K}}V^{\frac{1}{2}}$. We prove that there exist two transition layer solutions $u_\varepsilon^{(1)}, u_\varepsilon^{(2)}$ when $\varepsilon$ is sufficiently small. The first layer solution $u_\varepsilon^{(1)}$ approaches $-1$ in $\mathcal{M}_{-}$ and $+1$ in $\mathcal{M}_{+}$ as $\varepsilon$ tends to 0, while the other solution $u_\varepsilon^{(2)}$ exhibits a transition layer in the opposite direction.
Let $(\mathcal{M}, \tilde{g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the problem $$ \varepsilon^2 Δ_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0 in \mathcal{M}, $$ where $\varepsilon >0$ is a small parameter and $V$ is a positive, smooth function in $\mathcal{M}$. Let $ \mathcal{K}\subset \mathcal{M}$ be an $(N-1)$-dimensional smooth submanifold that divides $\mathcal{M}$ into two disjoint components $\mathcal{M}_{\pm}$. We assume $\mathcal{K}$ is stationary and non-degenerate relative to the weighted area functional $\int_{\mathcal{K}}V^{\frac{1}{2}}$. We prove that there exist two transition layer solutions $u_\varepsilon^{(1)}, u_\varepsilon^{(2)}$ when $\varepsilon$ is sufficiently small. The first layer solution $u_\varepsilon^{(1)}$ approaches $-1$ in $\mathcal{M}_{-}$ and $+1$ in $\mathcal{M}_{+}$ as $\varepsilon$ tends to 0, while the other solution $u_\varepsilon^{(2)}$ exhibits a transition layer in the opposite direction.