To the Study of the Negative Spectrum of an Elliptic Operator
Yu. V. Egorov, Владимир Александрович Кондратьев · Teubner-Texte zur Mathematik · 1992
This paper is a continuation of our work [1], in which some estimates were obtained for the negative spectrum of the elliptic operator 1 L= L 0 −V( x ), L 0 u= ∑ | α |≤m,| β |≤m D α ( α αβ ( x ) D β u ), $$ L = L_0 - V\left( x \right),\quad L_0 u = \sum\limits_{\left| \alpha \right| \leqslant m,\left| \beta \right| \leqslant m} {D^\alpha \left( {\alpha _{\alpha \beta } \left( x \right)D^\beta u} \right),} $$ , with measurable coefficients a αβ such that a αβ = α βα ¯ <![CDATA[$$ a_{\alpha \beta } = \overline {\alpha _{\beta \alpha } } $$ and Re( L 0 u,u )≡Re ∫ ∑ | α |≤m,| β |≤m α αβ ( x ) D α u( x ) D β u( x ) ¯ dx ≥ c 0 | u | m 2 , $$ \operatorname{Re} \left( {L_0 u,u} \right) \equiv \operatorname{Re} \int {\quad \sum\limits_{\left| \alpha \right| \leqslant m,\left| \beta \right| \leqslant m} {\alpha _{\alpha \beta } \left( x \right)D^\alpha u\left( x \right)D^\beta \overline {u\left( x \right)} dx} } \geqslant c_0 \left| u \right|_m^2 ,$$ , where c 0 is a positive constant and | u | m 2 =∫ ∑ | α |=m | D α u | 2 dx. $$ \left| u \right|_m^2 = \smallint \quad \sum\limits_{\left| \alpha \right| = m} {\left| {D^\alpha u} \right|^2 dx.} $$