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

A. N. Kozhevnikov

Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 699-710.

Résumé

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.