Estimates of singular numbers (s$$ s $$‐numbers) and eigenvalues of a mixed elliptic‐hyperbolic type operator with parabolic degeneration

Abstract

This paper is concerned with a mixed type differential operator which is initially defined with , where and is a set of infinitely differentiable functions with compact support with respect to the variable and satisfying the conditions: Regarding the coefficient , with supposition that satisfies the condition: is a piecewise continuous and bounded function in . The coefficients and are continuous functions in and can be unbounded at infinity. The operator admits closure in the space , and the closure is also denoted by . Taking into consideration certain constraints on the coefficients , 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 ( ‐numbers). Here, we note that the estimate of singular numbers ( ‐numbers) shows the rate of approximation of the resolvent of the operator by linear finite‐dimensional operators. It is given an example of how the obtained estimates for the ‐numbers enable to identify the estimates for the eigenvalues of the operator . 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.