Clustering layers for the Fife-Greenlee problem in \({\mathbb R}^n\) (Q2799612)

In this paper, the following Fife-Greenlee problem is studied: NEWLINE\[NEWLINE\begin{aligned} &\epsilon^2\Delta u + (u-a(x))(1-u^2) = 0 \quad\text{in }\Omega, \\ &\frac{\partial u}{\partial\nu} = 0 \quad \text{on }\partial\Omega. \end{aligned} NEWLINE\]NEWLINE Here \(a:\overline{\Omega} \to (-1,1)\) is smooth and such that \(\nabla a \neq 0\) on \(K=\{x : a(x) = 0\}\). It is shown that, for each odd integer \(m \geq 3\), there exists a sequence \(\{\epsilon_j\}\) converging to \(0\) and such that, for each \(\epsilon_j\), the associated problem has a solution \(u_{\epsilon_j}\) with \(m\) transition layers near \(K\) whose mutual distance is \(O(\epsilon\log(1/\epsilon))\). Moreover, \(\{u_{\epsilon_j}\}\) converges uniformly to \(\pm1\) on compact subsets of \(\Omega_\pm = \{x : \pm a(x) > 0\}\) as \(j\to \infty\).NEWLINENEWLINEA theorem due to Kato plays a key role in the proof of this result. It permits to estimate the derivatives with respect to \(\epsilon\) of the functions associating to \(\epsilon\) the eigenvalues of the corresponding linearized problem.