Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

Abstract

The spectral problem in a bounded domain Ω⊂Rⁿ is considered for the equation Δu= λu in Ω, −u=λ∂υ/∂ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ ⁰ _j } ^∞ _j=1 and {λ ^∞ _j } ^∞ _j=1 , converging respectively to 0 and +∞. It is also established that

$$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$

The constants are explicitly calculated.