## Illinois Journal of Mathematics

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

#### Abstract

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 $\lambda_k$ as a functional on 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.

As an application, we obtain a characterization of critical domains of the trace of the heat kernel under Dirichlet boundary conditions.

#### Article information

Source
Illinois J. Math., Volume 51, Number 2 (2007), 645-666.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138436

Digital Object Identifier
doi:10.1215/ijm/1258138436

Mathematical Reviews number (MathSciNet)
MR2342681

Zentralblatt MATH identifier
1124.49035

#### Citation

El Soufi, Ahmad; Ilias, Saïd. Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold. Illinois J. Math. 51 (2007), no. 2, 645--666. doi:10.1215/ijm/1258138436. https://projecteuclid.org/euclid.ijm/1258138436