Spectral approximations of strongly degenerate elliptic differential operators

M. I. Dmytryshyn, O. V. Lopushansky

Abstract


We establish analytical estimates of spectral approximationserrors for strongly degenerate elliptic differential operators inthe Lebesgue space $L_q(\Omega)$ on a bounded domain $\Omega$.Elliptic operators have coefficients with strong degenerationnear boundary. Their spectrum consistsof isolated eigenvalues of finite multiplicity and thelinear span of the associated eigenvectors is dense in$L_q(\Omega)$. The received results are based on an appropriategeneralization of Bernstein-Jackson inequalities with explicitlycalculated constants for quasi-normalized Besov-type approximationspaces which are associated with the given elliptic operator. Theapproximation spaces are determined by the functional$E\left(t,u\right)$, which characterizes the shortest distancefrom an arbitrary function ${u\in L_q(\Omega)}$ to the closedlinear 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 subspacescoincides with the subspace of entire analytic functions of exponential typenot 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.

Keywords


elliptic operators, spectral approximations

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