an:04214514
Zbl 0734.35063
Egorov, Yu. V.; Kondrat'ev, V. A.
On the negative spectrum of an elliptic operator
EN
Math. USSR, Sb. 69, No. 1, 155-177 (1991); translation from Mat. Sb. 181, No. 2, 147-166 (1990).
00180406
1991
j
35P20 35J30
even order elliptic operator; embedding theorems; negative spectrum; degenerate elliptic operators
The authors deal with the even order elliptic operator
\[
L=\sum_{| \alpha | \leq m,\quad | \beta | \leq m}D^{\alpha}(a_{\alpha \beta}(x)D^{\beta}u)-V(x)
\]
with measurable coefficients for which \(a_{\alpha \beta}=\bar a_{\beta \alpha}\) and
\[
\sum_{| \alpha | \leq m,\quad | \beta | \leq m}\int a_{\alpha \beta}(x) D^{\alpha}u \overline{D^{\beta}u} dx\geq c_ 0\int \sum_{| \alpha | \leq m}| D^{\alpha}u|^ 2 dx
\]
for \(u\in C_ 0^{\infty}\), \(c_ 0>0\). The operator L is considered in the whole space \({\mathbb{R}}^ n\) or in the domain \(\Omega\) with weak zero Dirichlet conditions on the boundary.
Using the dimensionless embedding theorems, the authors derive new sharp estimations for the number of points in the negative spectrum for the operator L. These results concern degenerate elliptic operators, unbounded domains \(\Omega\) and the case in which the spectrum of L has an infinite number of real negative points. The present theorems generalize previous results of the authors, G. V. Rosenbloom, E. Lieb and others.
V.??urikovi?? (Bratislava)