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Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. (English) Zbl 1099.35070

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.

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.)
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