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Some inequalities between Dirichlet and Neumann eigenvalues. (English) Zbl 0789.35124

The author considers the relationship between the eigenvalues of the Dirichlet problem \((\lambda_ 1<\lambda_ 2\leq\dots)\) and the Neumann problem \((0=\mu_ 1< \mu_ 2\leq\dots)\) for the Laplace operator in a domain \(\Omega\subset \mathbb{R}^ n\) with \(C^ 1\)-boundary \(\Gamma\). He establishes a conjecture of Payne, namely that \(\mu_{k+1}\leq \lambda_ k\), \(k=1,2,\dots\;\). Let \(u\) be a solution of the problem \[ (\Delta+\lambda)u(x)=0 \text{ in }\Omega, \qquad u(x)=\varphi(x) \text{ on } \Gamma. \] One introduces the Dirichlet-Neumann operator \(R(\lambda): \varphi\to \partial u/\partial\nu\). The operator \(R(\lambda)\) is a pseudo-differential operator on the boundary with symbol \((\xi')\), in coordinates \(\xi'\) dual to the local coordinates on the boundary. Let \(\sigma_ D\) and \(\sigma_ N\) be the spectra of the Dirichlet and Neumann Laplacians with \(N_ D(\lambda)\) and \(N_ N(\lambda)\) the corresponding distribution functions for the eigenvalues and let \(n(\lambda)\) be the number of negative eigenvalues of \(R(\lambda)\). Two central results are proved, namely: \[ n(\lambda)=N_ N(\lambda)- N_ D(\lambda) \text{ for } \lambda\not\in \sigma_ D\cup \sigma_ N \qquad \text{and} \qquad n(\lambda)\geq 1, \quad \lambda>0, \quad \lambda\not\in \sigma_ D\cup \sigma_ N. \] The first involves the properties of the “spectral flow” of the family \(R(\lambda)\). Modulo a few technical results this exceptionally elegant paper involves only elmentary and direct calculations.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35R30 Inverse problems for PDEs
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References:

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