Operator-valued pseudodifferential operators and the resolvent of a degenerate elliptic operator

A. I. Karol'

Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 553-567

Résumé

In this paper the author constructs an asymptotic expansion of the resolvent of the operator of the Dirichlet problem for an elliptic equation of divergence form with a power degeneracy on the boundary. To construct the expansion a variant of the technique of pseudodifferential operators ($\Psi$DO's) with operator-valued symbols is used, in combination with the technique of “ordinary” scalar $\Psi$DO's. The difference between the resolvent and the approximation thus obtained is an integral operator whose kernel decreases at infinity faster than any power of the spectral parameter. In a neighborhood of the boundary this operator smooths only in directions tangent to the boundary. Bibliography: 16 titles.