Privat, Yannick; Sigalotti, Mario The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent. (English) Zbl 1203.35005 ESAIM, Control Optim. Calc. Var. 16, No. 3, 794-805 (2010); erratum ibid. 16, No. 3, 806-807 (2010). The authors study the Laplace-Dirichlet operator \[ -\Delta:\; H^2(\Omega)\cap H^1_0(\Omega)\to L^2(\Omega) \] on a domain \(\Omega\subset{\mathbb R}^d\). Let \(\Sigma_m\) be the space of domains in \({\mathbb R}^d\) with boundaries of class \(C^m\); the topology on \(\Sigma_m\) is induced by the topology of the space of diffeomorphisms of \({\mathbb R}^d\). The authors study properties of the spectrum and eigenfunctions of the Laplace-Dirichlet operator for a generic \(\Omega\subset\Sigma_m\). In particular, their results imply that for a generic \(\Omega\subset\Sigma_m\), the squares of eigenfunctions are linearly independent on \(\Omega\). The results are applied to several problems in shape optimization and control theory. Reviewer: Sergei Yu. Pilyugin (St. Petersburg) Cited in 1 ReviewCited in 8 Documents MSC: 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35P05 General topics in linear spectral theory for PDEs 93C20 Control/observation systems governed by partial differential equations 49K20 Optimality conditions for problems involving partial differential equations Keywords:genericity; Laplacian-Dirichlet eigenfunctions; non-resonant spectrum; shape optimization; control PDFBibTeX XMLCite \textit{Y. Privat} and \textit{M. Sigalotti}, ESAIM, Control Optim. Calc. Var. 16, No. 3, 794--805 (2010; Zbl 1203.35005) Full Text: DOI arXiv EuDML