Spectrum of quasielliptic differential operators. (English. Russian original) Zbl 0807.35104
Math. Notes 53, No. 6, 661-663 (1993); translation from Mat. Zametki 53, No. 6, 145-148 (1993).
This article deals with the spectrum in \(L_ 2\) of the selfadjoint differential operator
\[
{\mathcal L} u = \sum_{(\alpha, \lambda) = 2} a_ \alpha D^ \alpha u + Q(x)u
\]
for which
\[
\gamma_ 1 \sum_{\alpha \in P} | \xi^ \alpha |^ 2 \leq \sum_{(\alpha, \lambda) = 2} a_ \alpha (i \xi)^ \alpha \leq \gamma_ 2 \sum_{\alpha \in P} | \xi^ \alpha |^ 2
\]
where \(\lambda = (p_ 1^{-1}, \dots, p_ n^{-1})\), \(P = \{(0, \dots, p_ i, \dots,0)\): \(i=1, \dots,n\}\) and \(Q(x)\) is measurable and locally bounded. Some statements on the structure of the negative part of \(sp({\mathcal L})\) are presented.
Reviewer: P.Zabreiko (Minsk)
MSC:
35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
35J62 | Quasilinear elliptic equations |