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

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Jun Yang is supported by National Natural Science Foundation of China (Grant No. 11771167 and No. 11831009)

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  • We consider the nonlinear problem of inhomogeneous Allen-Cahn equation

    $ \epsilon^2\Delta u+V(y)\,(1-u^2)\,u = 0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu} = 0\quad \mbox{on}\ \partial \Omega, $

    where $ \Omega $ is a bounded domain in $ \mathbb R^2 $ with smooth boundary, $ \epsilon $ is a small positive parameter, $ \nu $ denotes the unit outward normal of $ \partial \Omega $, $ V $ is a positive smooth function on $ \bar\Omega $. Let $ \Gamma\subset\Omega $ be a smooth curve dividing $ \Omega $ into two disjoint regions and intersecting orthogonally with $ \partial\Omega $ at exactly two points $ P_1 $ and $ P_2 $. Moreover, by considering $ {\mathbb R}^2 $ as a Riemannian manifold with the metric $ g = V(y)\,({\mathrm d}{y}_1^2+{\mathrm d}{y}_2^2) $, we assume that: the curve $ \Gamma $ is a non-degenerate geodesic in the Riemannian manifold $ ({\mathbb R}^2, g) $, the Ricci curvature of the Riemannian manifold $ ({\mathbb R}^2, g) $ along the normal $ \mathbf{n} $ of $ \Gamma $ is positive at $ \Gamma $, the generalized mean curvature of the submanifold $ \partial\Omega $ in $ ({\mathbb R}^2, g) $ vanishes at $ P_1 $ and $ P_2 $. Then for any given integer $ N\geq 2 $, we construct a solution exhibiting $ N $-phase transition layers near $ \Gamma $ (the zero set of the solution has $ N $ components, which are curves connecting $ \partial\Omega $ and directed along the direction of $ \Gamma $) with mutual distance $ O(\epsilon|\log \epsilon|) $, provided that $ \epsilon $ stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.

    Mathematics Subject Classification: Primary: 35J60; Secondary: 58J20.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  $\begin{array}{*{20}{l}} {{ Curves}{\rm{: }}{C_1} = \left( {{y_1},{\varphi _1}\left( {{y_1}} \right)} \right),\;\;\;{\kern 1pt} {C_2} = \left( {{y_1},{\varphi _2}\left( {{y_1}} \right)} \right), - {\delta _0} < {y_1} < {\delta _0},}\\ {{ Points}{\rm{: }}{P_1} = (0,\varphi (0)),{P_2} = \left( {0,{\varphi _2}(0)} \right).} \end{array}$

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