Dilatation and the asymptotics of the eigenvalues of spectral problems with singularities. (English. Russian original) Zbl 0628.35077

Funct. Anal. Appl. 20, 277-281 (1986); translation from Funkts. Anal. Prilozh. 20, No. 4, 29-34 (1986).
The eigenvalue problem \(Au=\lambda^ 2\mu u\) is investigated providing that A is a uniformly elliptic selfadjoint operator of the 2nd order and \(\mu\) is a nonnegative weight function. The term singularity means either the nonregularity of \(\mu\) or of the coefficients of A, degeneration of \(\mu\), unboundedness of the domain or nonregularity of the boundary. It is shown how the method of dilatations facilitates to derive very precise asymptotics of eigenvalues of the spectral problem with sufficiently weak singularities.
Reviewer: P.Burda


35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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