Asymptotic distribution of eigenvalues for some elliptic operators with intermediate remainder estimate

Article type: Research Article

Authors: Zielinski, Lech

Affiliations: Institut de Mathématiques de Paris‐Jussieu UMR 9994, Université Paris 7 (D. Diderot), 2 Place Jussieu, 75252 Paris Cedex 05, Case Postale 7012, France

Abstract: We consider the Weyl formula for the asymptotic number of eigenvalues N(\lambda) for some elliptic operators. We prove remainder estimates of the form N(\lambda){\rm O}(\lambda^{-\mu}) with \mu depending on the regularity of coefficients and ranging between 0 and the optimal value in the standard situation of smooth coefficients, i.e., 0<\mu<2/m in the case of globally elliptic operators on {\NBbbR}^d, and 0<\mu<1/m in the case of a smooth compact manifold without boundary, where m is the degree of the considered operators.

Journal: Asymptotic Analysis, vol. 17, no. 2, pp. 93-120, 1998