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

By Ahmad El Soufi, S. Ilias. 2007

58 citations recorded by the source corpus.

For any bounded regular domain $\Omega$ of a real analytic Riemannian manifold $M$, we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $\lambda_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c76ed5075747fc1baf55c3e8fb4e53e4c6fe2282

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

New extremal domains for the first eigenvalue of the Laplacian in flat tori (2010, background): “1 of [9], also proved in [4] and in [3], it is clear that such a domain Ωj is extremal with respect to the first eigenvalue of the Laplacian in R×R/Tj Z for the fixed volume Tj vol(B1), in the sens that for any volume preserving deformation {Ωs}s∈(j− ,j+ ) of Ω0, we have dλs ds |s=0 = 0 , where λs i”
On an overdetermined elliptic problem (2011, background): “Smooth extremal domains are characterized by the property that the eigenfunctions associated with the first eigenvalue of the Laplace–Beltrami operator have constant Neumann boundary data [Soufi 2007].”
BIFURCATING EXTREMAL DOMAINS FOR THE FIRST EIGENVALUE OF THE LAPLACIAN (2011, background): “has a positive solution (here ν is the outward unit normal vector to ∂Ω with respect to g), see [7] and [16].”
Extremal domains for the first eigenvalue of the Laplace-Beltrami operator (2009, methodology): “This formula has been obtained by P. R. Garabedian and M. Schier in [3] when the underlying manifold is the euclidean space and by A. El Sou and S. Ilias [ 2 ] (see Corollary 2.1) when the underlying manifold is a Riemannian manifold.”