On Moments of Negative Eigenvalues of an Elliptic Operator
Yu. V. Egorov, V. А. Kondratiev · Birkhäuser Basel eBooks · 1995
Let Ω be a domain in ℝn, let $$ L = \sum\limits_{{\left| {\alpha \left| {m,\left| {\beta \left| {m} \right.} \right.} \right.} \right.}} {{{D}^{\alpha }}{{a}_{a}}_{\beta }} \left( \chi \right){{D}^{\beta }} $$ be an elliptic symmetric positive operator of order 2m with measurable coefficients, considered with zero Drichlet boundary conditions, such that $$ \int_{\Omega } {\sum\limits_{{\left| {a\left| { \leqslant m,\left| {\beta \left| \leqslant \right.} \right.} \right.} \right.}} {{{\alpha }_{\alpha }}} } \beta \left( x \right){{D}^{\alpha }}u\overline {{{D}^{\beta }}u} dx \geqslant {{a}_{0}}\int_{\Omega } {\sum\limits_{{\left| {\alpha \left| { = m} \right.} \right.}} {{{{\left| {{{D}^{\alpha }}u(x)} \right|}}^{2}}dx} } , $$ where α0 = const > 0 for all functions u ∈ C 0 ∞ (Ω). Let V (x)be a real-valued function defined in Ω.