## 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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