CMMSE: Estimates of singular numbers ( s -numbers) and eigenvalues of a mixed elliptic-hyperbolic type operator with parabolic degeneration

Abstract

a) | k ( w ) | ≥ 0 is a piecewise continuous and bounded function in R = ( - ∞ , ∞ ) . The coefficients b ( w ) and q ( w ) are continuous functions in R and can be unbounded at infinity. The operator L admits closure in the space L 2 ( Ω ) and the closure is also denoted by L. Taking into consideration certain constraints on the coefficients b ( w ) q ( w ) , apart from the above-mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two-sided estimates for singular numbers ( s-numbers). Here we note that the estimate of singular numbers ( s-numbers) shows the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. It is given an example of how the obtained estimates for the s-numbers enable to identify the estimates for the eigenvalues of the operator L. We note that the above results are apparently obtained for the first time for a mixed-type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.