# New extremal domains for the first eigenvalue of the Laplacian in flat tori

@article{Sicbaldi2010NewED,
title={New extremal domains for the first eigenvalue of the Laplacian in flat tori},
author={Pieralberto Sicbaldi},
journal={Calculus of Variations and Partial Differential Equations},
year={2010},
volume={37},
pages={329-344}
}
We prove the existence of nontrivial compact extremal domains for the first eigenvalue of the Laplacian in manifolds $${\mathbb{R}^{n}\times \mathbb{R}{/}T\, \mathbb{Z}}$$ with flat metric, for some T > 0. These domains are close to the cylinder-type domain $${B_1 \times \mathbb{R}{/}T\, \mathbb{Z}}$$, where B1 is the unit ball in $${\mathbb{R}^{n}}$$, they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations. Such domains can be extended by… CONTINUE READING

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