A spectral asymptotics of an elliptic operator in a bounded domain
Let \(P_ 0\) be the symmetric uniformly elliptic operator of 2m-th order with its coefficients in \(C^{\infty}({\bar \Omega})\), (\(\Omega\) a bounded domain in \(R_ n)\), P the Friedrichs extension of \(P_ 0\). In the note the authors prove the asymptotic formula for N(\(\lambda)\) - the number of eigenvalues of the operator not exceeding prescribed value. Namely, for \(\lambda \to +\infty\), \(N(\lambda)=const \lambda^{n/2m}+O(1)\lambda^{(n-1)/2m}\int_{\Omega \lambda}dist^{- 1}(x\), \(\partial \Omega)dx+O(1)\lambda^{n/2m}mes\Omega '_{\lambda}\), \(\Omega_{\lambda}=\{x\in \Omega\); \(dist(x,\partial \Omega)>\lambda^{- 1/2m}\}\), \(\Omega '_{\lambda}=\Omega \setminus {\bar \Omega}_{\lambda}\).
scientific article; zbMATH DE number 3953643
scientific article