Spectral approximations of strongly degenerate elliptic differential operators

Abstract

We establish analytical estimates of spectral approximations errors for strongly degenerate elliptic differential operators in the Lebesgue space $L_q(\Omega)$ on a bounded domain $\Omega$. Elliptic operators have coefficients with strong degeneration near boundary. Their spectrum consists of isolated eigenvalues of finite multiplicity and the linear span of the associated eigenvectors is dense in $L_q(\Omega)$. The received results are based on an appropriate generalization of Bernstein-Jackson inequalities with explicitly calculated constants for quasi-normalized Besov-type approximation spaces which are associated with the given elliptic operator. The approximation spaces are determined by the functional $E\left(t,u\right)$, which characterizes the shortest distance from an arbitrary function ${u\in L_q(\Omega)}$ to the closed linear span of spectral subspaces of the given operator, corresponding to the eigenvalues such that not larger than fixed ${t>0}$. Such linear span of spectral subspaces coincides with the subspace of entire analytic functions of exponential type not larger than ${t>0}$. The approximation functional $E\left(t,u\right)$ in our cases plays a similar role as the modulus of smoothness in the functions theory.