Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. (English) Zbl 1099.35070 J. Math. Soc. Japan 58, No. 1, 231-245 (2006). In \(\mathbb R^n\), among the domains with the same volume, the ball minimizes the first Dirichlet eigenvalue of the Laplace operator. This is well known as the Rayleigh-Krahn-Faber theorem. By this token, the ball is a critical point of the functional eigenvalue with a volume constraint as described in [D. Henry, Topics in nonlinear analysis, Trabalho de Matemática, 192, Univ. Brasilia, Marco (1982)]: Let \(\phi (\Omega)\) denote a deformation of \(\Omega\), \(\lambda_j\) the \(j\)-th Dirichlet eigenvalue (counting multiplicity) on \(\phi (\Omega)\). If \(\Omega\) and \(\tilde \phi\) are regular enough, then any simple eigenvalue \(\lambda_j [ \cdot] \) is critical at \(\tilde \phi\) on those \(\phi(\Omega)\) having the same volume.It is also known, through the Hadamard variational formulas for \(\lambda_j [ \cdot ]\), that this statement can be rewritten as: let \(\nu\) denote the exterior unit normal, for the overdetermined problem: \(- \bigtriangleup v = \lambda_j [\tilde \phi] v\) in \(\tilde \phi (\Omega)\), \(v \equiv 0\) on \(\partial \tilde \phi (\Omega)\), \((\frac {\partial v}{\partial \nu})^2\) is constant on \(\partial \tilde \phi (\Omega)\), if \(\tilde \phi (\Omega)\) is bounded, then the problem has a solution for \(j = 1\) if and only if \(\tilde \phi (\Omega)\) is a ball.In this article, the authors generalize the above mentioned assertions in three aspects. (1) Both Dirichlet and Neumann eigenvalues are considered. (2) Eigenvalues of higher multiplicity are considered, whereby the elementary symmetric functionals of the eigenvalues replace the role of a single eigenvalue. (3) The regularity requirement of the variation of \(\Omega\) is relaxed. Additionally, overdetermined boundary value problems of the type of the Schiffer conjecture are formulated. Reviewer: Chie-Ping Chu (Taipei) Cited in 10 Documents MSC: 35P15 Estimates of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35N05 Overdetermined systems of PDEs with constant coefficients 35P05 General topics in linear spectral theory for PDEs 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) Keywords:Dirichlet and Neumann eigenvalues and eigenfunctions; Laplace operator; overdetermined problems; domain perturbation; special nonlinear operators PDFBibTeX XMLCite \textit{P. D. Lamberti} and \textit{M. Lanza de Cristoforis}, J. Math. Soc. Japan 58, No. 1, 231--245 (2006; Zbl 1099.35070) Full Text: DOI