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

• * Corresponding author

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

• 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.

 Citation: • • 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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