Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems
The authors investigate the asymptotic profiles of all eigenfunctions \(\varphi_j^n\) (\(j\in\mathbb{N}\cup\{0\}\), \(n\in\mathbb{N}\)) of the linearized eigenvalue problem \[ \begin{cases}\varepsilon^2\varphi_{xx}(x)+f_u(u_{n,\varepsilon}(x))\varphi(x)+\mu\varphi(x)=0,&\text{in }(0,1),\\ \varphi_x(0)=\varphi_x(1)=0,\end{cases} \] where \(\varepsilon\) is a positive parameter, \(f(u)=\sin u\) and \(u_{n,\varepsilon}\) (\(n\in\mathbb{N}\)) is the \(n\)-mode solution of the problem \[ \begin{cases}\varepsilon^2u_{xx}(x)+f(u(x))=0&\text{in }(0,1),\\ u_x(0)=u_x(1)=0.\end{cases} \] They show that two special eigenfunctions completely control the asymptotic profiles of the other eigenfunctions.
scientific article; zbMATH DE number 5711089