Bilateral (two-sided) estimates of S-numbers’ distributions of one class of the mixed type differential operators
by
M.B. Muratbekov
Eurasian National University named by L.N.Gumilev
Coauthors: M.M. Muratbekov

Consider the differential operator of mixed type

Lu = k(y)uxx - uyy + a(y)ux + c(y)u,

which is originally determined on C_{0, p}^¥(W), where

W = { (x, y): - p < x < p, - ¥ < y < ¥ }

C_{0, p}^¥(W) - the set, consisting of infinitely differentiable functions, satisfying conditions: u(-p, y) = u(p, y), u_x(-p, y) = u_x(p, y) and finite by variable y. k(y) - the sectionally continuous and limited function in R = (-¥, +¥) and k(0)=0, y ·k(y) > 0, as y ¹ 0.

It is not so difficult to show, that the operator allows the closure in the metric of L₂ (W) and the closure we also denote by L.

Further, assume, that a(y) and c(y) coefficients satisfy the conditions:

1. | a(y) | \geqslant d₀ > 0, c(y) \geqslant d > 0 - continuous functions in R = (-¥, +¥);

2. c(y) \leqslant c₀ a²(y), at y Î R, c₀ > 0- a constant;

3. m =
   sup
   |y-t| \leqslant 1 [a(y)/a(t)] < ¥, m =
   sup
   |y-t| \leqslant 1 [c(y)/c(t)] < ¥

The following theorems hold.

Theorem 1. Let the conditions (1) be fulfilled. Then the operator L+lE is continuously invertible under a sufficiently large l > 0.

Theorem 2. Let the conditions (1) be fulfilled. Then the resolvent of operator L is compact if and only if

lim |y|®¥   y+w ó õ y  c(t)dt = ¥

for any w > 0.

Nonzero s-numbers of operator L^-1 we number in order of their decrease in view of their multiplicity, so S_k (L ^{- 1}) = l((L ^{- 1}) ^* L^-1), k = 1, 2, ...

We enter the next function N(l) = å_{sk > l} 1 - the amount of s_k are greater than l > 0.

Theorem 3. Let the conditions (1)-(3) be fulfilled. Then the following estimate holds: c^-1 _j^ ^-1/2 mes(y R: - n^2+ina(y)+c(y) - c^-1^-1/2) N()
c _j^ ^-1 mes(y R: - ina(y)+c(y) - c^-1) , where c = c(m, m₁), i²=- 1.

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Date received: May 15, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqw-40.