A mixed eigenvalue problem on domains tending to infinity in several directions

TL;DR

This work analyzes the asymptotics of mixed Neumann-Dirichlet eigenvalues for elliptic operators on cylindrical domains $\Omega_\ell=\ell\omega_1\times\omega_2$ as $\ell\to\infty$, i.e., domains unbounded in $m$ directions. The authors introduce a direction-indexed family of reduced problems on $(-\infty,0)\times\omega_2$ with matrices $A_\nu$, and prove that the limit of the first eigenvalue satisfies $\lim_{\ell\to\infty}\lambda_\ell = \inf_{\nu\in S^{m-1}} Z^\nu$, with $Z^\nu$ defined by a variational problem on the semi-infinite cylinder; higher eigenvalues $\lambda_\ell^k$ (for $k>1$ and $m\ge2$) share the same limit. A key feature is a gap phenomenon: under $A_{12}\nabla_\xi W\not\equiv0$, the limit is strictly below the cross-section Dirichlet eigenvalue $\mu_1$, otherwise equality can occur. The paper employs variational constructions, boundary-flattening changes of variables, and a Picone identity to establish matching upper and lower bounds, with extensions to higher-order eigenvalues and discussion of eigenfunction decay.

Abstract

The aim of this article is to analyze the asymptotic behaviour of the eigenvalues of elliptic operators in divergence form with mixed boundary type conditions for domains that become unbounded in several directions, while they stay bounded in some directions (cylindrical domains). The limiting behavior of such eigenvalues is shown to depend on an ensemble of eigenvalue problems defined on a domain that is unbounded only in one direction. The asymptotic behavior of the eigenfunctions are also discussed. This work is a continuation of the work done in [6].

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Remark 3
• Theorem 4
• Theorem 5
• Lemma 6
• Theorem 7
• Theorem 8
• Remark 9
• Lemma 10