# On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded

Chipot, M; Rougirel, A (2008). On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded. Transactions of the American Mathematical Society, 360(7):3579-3602.

## Abstract

The aim of this work is to analyze the asymptotic behaviour of the eigenmodes of some elliptic eigenvalue problems set on domains becoming unbounded in one or several directions. In particular, in the case of a linear elliptic operator in divergence form, we prove that the sequence of the $k$-th eigenvalues convergences to the first eigenvalue of an elliptic problems set on the section of the domain. Moreover, an optimal rate of convergence of this sequence is given.

## Abstract

The aim of this work is to analyze the asymptotic behaviour of the eigenmodes of some elliptic eigenvalue problems set on domains becoming unbounded in one or several directions. In particular, in the case of a linear elliptic operator in divergence form, we prove that the sequence of the $k$-th eigenvalues convergences to the first eigenvalue of an elliptic problems set on the section of the domain. Moreover, an optimal rate of convergence of this sequence is given.

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