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.


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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