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 Mat. Zametki, 1977, Volume 22, Issue 5, Pages 699–710 (Mi mz8093)

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

A. N. Kozhevnikov

Moscow Aviation Institute

Abstract: The spectral problem in a bounded domain $\Omega\subset R^n$ is considered for the equation $-\Delta u=\lambda u$ in $\Omega$, $-u=\lambda \partial u/\partial\nu$ on the boundary of $\Omega$ ($\nu$ the interior normal to the boundary, $\Delta$, 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 $\{\lambda_j^0\}_{j=1}^\infty$ and $\{\lambda_j^\infty\}_{j=1}^\infty$, converging respectively to 0 and $+\infty$. It is also established that
\begin{gather*} N^0(\lambda)=\sum_{\operatorname{Re}\lambda_j^0\ge1/\lambda}1\approx\mathrm{const} \lambda^{b-1},
N^\infty(\lambda)\equiv\sum_{\operatorname{Re}\lambda_j^\infty\le\lambda}1\approx\mathrm{const} \lambda^{n/2}, \end{gather*}
The constants are explicitly calculated.

Full text: PDF file (890 kB)

English version:
Mathematical Notes, 1977, 22:5, 882–888

Bibliographic databases:

UDC: 517.4

Citation: A. N. Kozhevnikov, “Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem”, Mat. Zametki, 22:5 (1977), 699–710; Math. Notes, 22:5 (1977), 882–888

Citation in format AMSBIB
\Bibitem{Koz77} \by A.~N.~Kozhevnikov \paper Separate asymptotics of two series of eigenvalues for a~single elliptic boundary-value problem \jour Mat. Zametki \yr 1977 \vol 22 \issue 5 \pages 699--710 \mathnet{http://mi.mathnet.ru/mz8093} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=499823} \zmath{https://zbmath.org/?q=an:0372.35065} \transl \jour Math. Notes \yr 1977 \vol 22 \issue 5 \pages 882--888 \crossref{https://doi.org/10.1007/BF01098353}