Geodesic
Asymptotic Property of Eigenvalues and Eigenfunctions of the Laplace Operator in Domain with a Perturbed Boundary
Fractional calculus and applied analysis, Tome 8 (2005) no. 3, pp. 277-298.

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In this paper, we consider the variations of eigenvalues and eigenfunctions for the Laplace operator with homogeneous Dirichlet boundary conditions under deformation of the underlying domain of definition. We derive recursive formulas for the Taylor coefficients of the eigenvalues as functions of the shape-perturbation parameter and we establish the existence of a set of eigenfunctions that is jointly holomorphic in the spatial and boundary-variation variables. Using integral equations, we show that these eigenvalues are exactly built with the characteristic values of some meromorphic operator-valued functions.
Keywords: Eigenvalues, Eigenfunctions, Laplace Operator, Domain Perturbation, Integral Equation, Analyticity, 35J05, 35C15, 44P05 
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Khelifi, Abdessatar. Asymptotic Property of Eigenvalues and Eigenfunctions of the Laplace Operator in Domain with a Perturbed Boundary. Fractional calculus and applied analysis, Tome 8 (2005) no. 3, pp. 277-298. http://geodesic.mathdoc.fr/item/FCAA_2005_8_3_a3/