Eigenvalue Distribution of Semi-Elliptic Operators in Anisotropic Sobolev Spaces

Abstract

We study the spectral properties of the compact non-negative self-adjoint operator $T=A^{-1}\circ {\mathrm{tr}}^{\Gamma }$ acting in the anisotropic Sobolev space \( H^{s,a}_2(\rn) \) and give two-sided estimates for the asymptotic behaviour of its eigenvalues ${\lambda }_k(T)$, where $A$ is a semi-elliptic differential operator of type

and ${\mathrm{tr}}^{\Gamma }$ a special trace operator on an anisotropic $d$-set $\Gamma$.