New extremal domains for the first eigenvalue of the Laplacian in flat tori. (English) Zbl 1188.35122
Summary: 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 \(B _{1}\) is the unit ball in \({\mathbb{R}^{n}}\), they are invariant by rotation with respect to the vertical axis, and are not invariant by vertical translations. Such domains can be extended by periodicity to nontrivial and noncompact domains in Euclidean spaces whose first eigenfunction of the Laplacian with Dirichlet boundary condition has also constant Neumann data at the boundary.
MSC:
| 35N25 | Overdetermined boundary value problems for PDEs and systems of PDEs |
| 37G10 | Bifurcations of singular points in dynamical systems |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 58J05 | Elliptic equations on manifolds, general theory |
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