Nodal set openings on perturbed rectangular domains
The authors investigate how the nodal set of eigenfunctions of the Dirichlet Laplacian changes with respect to a perturbation of domain's boundary. More precisely, let \(\mathcal{R} = (0,N) \times (0,1)\) be a planar rectangle such that the corresponding eigenvalue \(\lambda_{2,2}\) is simple. Let \(\Omega\) be the perturbation of \(\mathcal{R}\) defined as \((\eta \phi_L(\cdot),N) \times (0,1)\), where \(\eta>0\) is sufficiently small and the function \(\phi_L\) is sufficiently regular, \(-1 \leq \phi_L \leq 0\), and \N\[\N\int_0^1 \phi_L(y) \sin(2\pi y) \sin(\pi y) \,dy \neq 0. \N\]\NUnder these assumptions, the authors study an eigenfunction \(v\) in \(\Omega\) which is a perturbation of the eigenfunction \(\varphi_{2,2}\) in \(\mathcal{R}\). In the main results, it is shown that the nodal set of \(v\) develops (with respect to \(\eta\)) an opening near the point of crossing of nodal lines of \(\varphi_{2,2}\), and this opening is analyzed both qualitatively and quantitatively. In particular, the orientation, size, shape, and regularity of the nodal set of \(v\) are investigated. Apart from many other tools, the authors carefully work with the Fourier expansion of \(v\). Numerical simulations are also given.
scientific article; zbMATH DE number 7936712
scientific article