Spectral approximations of strongly degenerate elliptic differential operators

M. I. Dmytryshyn, O. V. Lopushansky


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.


elliptic operators, spectral approximations

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