The eigenvalue distribution of elliptic operators with Hölder continuous coefficients. II
The article is the continuation of the previous paper [\textit{Y. Miyazaki}, Osaka J. Math. 28, No. 4, 935-973 (1991; Zbl 0754.35101)]. An improvement of the remainder estimate for the eigenvalue distribution of an elliptic operator of order \(2m\) with Hölder continuous coefficients of top order is studied. A symmetric integro-differential form \[ B[u,v]= \int_ \Omega\;\sum_{|\alpha |,| \beta|\leq m} a_{\alpha\beta} (x) D^ \alpha u(x) \overline {D^ \beta v(x)}dx \] and a closed subspace \(V\) of the Sobolev space \(H^ m(\Omega)\) are assumed. The asymptotic behavior of the number of eigenvalues of the selfadjoint operator \({\mathcal A}\) associated with the variational triple \(B\), \(V\), \(L_ 2(\Omega)\) is analyzed. This number satisfies (WF1) \(N(t)= \mu_{\mathcal A} (\Omega) t^{n/2m}+ O(t^{(n-\theta)/ 2m})\) as \(t\to\infty\) with an appropriate constant \(\theta\), \(0<\theta\leq 1\), or (WF2) \(N(t)= \mu_{\mathcal A}(\Omega) t^{n/2m}+ O(t^{(n-1)/ 2m} \log t)\), as \(t\to\infty\). Conditions are derived under which \(\theta= 3\tau/ (2\tau+3)\) when \(0<\tau<3\) for (WF1) and \(\tau=3\) for (WF2). Furthermore (WF1) holds for \(\theta=\tau\) when \(0<\tau< 1\) and (WF2) holds when \(\tau=1\).
scientific article; zbMATH DE number 579223