On mixed Dirichlet-Neumann eigenvalues of triangles (Q2796717)

Let \(T\) be a right triangle with sides of length \(L \geq M \geq S \). Then the Dirichlet-Neumann eigenvalue problem is the followingNEWLINENEWLINENEWLINE\[NEWLINE\begin{aligned} \Delta u &= \lambda^{D} u, \text{ on } T, \\ u &= 0 \text{ on } D \subset \{ L,M,S \}, \\ \partial_\nu u &= \text{ on } \partial T \setminus D. \end{aligned}NEWLINE\]NEWLINENEWLINENEWLINEIt is well known that an orthonormal sequence of eigenfunctions exists, and NEWLINE\[NEWLINE 0 < \lambda_1^D<\lambda_2^D \leq \lambda_3^D \leq ... \to \infty . NEWLINE\]NEWLINE The purely Neumann eigenvalues \((D = \emptyset )\) are denoted by \(\mu\) and the purely Dirichlet eigenvalues by \(\lambda\). \newline It is generally evident that for the purely Dirichlet eigenvalues the Dirichlet conditions on a superset leads to larger eigenvalues, but it is not easy to compare the eigenvalues of the mixed case with the eigenvalues of the purely Dirichlet case. The aim of the paper is the following theorem:NEWLINENEWLINEFor any right triangle with smallest angle satisfying \( \pi / 6<\alpha<\pi / 4 \), NEWLINE\[NEWLINE 0 = \mu_1<\lambda_1^S< \lambda_1^M< \mu_2<\lambda_1^L<\lambda_1^{MS}<\lambda_1^{LS}< \lambda_1^{LM}<\lambda_1. NEWLINE\]NEWLINE If \(\alpha = \pi/6 \) then \(\lambda_1^M = \mu_2 \), and for \(\alpha = \pi / 4 \), \( \lambda_1^L = \mu_2 \). All other inequalities stay sharp in these cases. Furthermore, for arbitrary triangles, NEWLINE\[NEWLINE \min\{\lambda_1^S,\lambda_1^M,\lambda_1^L\}<\mu_2 \leq \lambda_1^{MS}< \lambda_1 ^{LS}<\lambda_1^{LM}, NEWLINE\]NEWLINE as long as the appropriate sides have different lengths. However, it is possible that \(\mu_2> \lambda_1^L \) or \( \mu_2< \lambda_1^M.\)NEWLINENEWLINEIn contrast to the purely Dirichlet case it is not always the case that for arbitrary polygonal domains a longer restriction leads to a higher eigenvalue.