Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold

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

“…Then, if T j is the period of the function v j , such a domain arise to a compact domain homeomorphe to B 1 × R/T j Z in the manifold R n × R/T j Z with flat metric, where the problem (1), adapted to this new manifold, has a solution (naturally B 1 denotes the unit ball centered at 0). From the proposition 2.1 of [9], also proved in [4] and in [3], it is clear that such a domain j is extremal with respect to the first eigenvalue of the Laplacian in R n × R/T j Z for the fixed volume T j vol(B 1 ), in the sens that for any volume preserving deformation { s } s∈( j− , j+ ) of 0 , we have…”

Extremal domains for the first eigenvalue in a general compact Riemannian manifold

“…The outward unit normal vector field to ∂Ω t is denoted by ν t . We have the following result, whose proof can be found in [3] or in [15]:…”

“…This result has been proved in the Euclidean space by P.R. Garabedian and M. Schiffer in 1953 [5], and in a general Riemannian manifold by A. El Soufi and S. Ilias in 2007 [3]. Extremal domains are then domains where the elliptic overdetermined problem…”

“…where B n (Ω) is a ball of R n with the same volume as Ω, because equality holds in (3) if and only if Ω = B n (Ω), see [4] and [10]. Similar facts hold for the hyperbolic space H n and the round sphere S n .…”

Generic Spectrum of Warped Products and G-Manifolds

“…The use of this technique is common in the works of some authors, see e.g. Albert [1], Berger [4], El Soufi and Ilias [7], Henry [9] and Pereira [12].…”