Untitled - Cdm.unimo.it

Eigenvalue Analysis 177
(8.6.4)
1
−1
p ′ (pw) ′
dx
1
2
≤ C2 p ′ w.
Proof - Inequality (8.6.3) is a byproduct of (5.7.5). We provide however some details.
With the help of the Schwarz inequality, one obtains
(8.6.5)
≤
p
1 − x2
2
w
=
1
−1
x
−1
1 x
[p
−1 −1
′ (s)] 2 x
w(s)ds ·
−1
≤ p ′ 2 1 x
w
−1 −1
p ′ 2 w(x)
(s)ds
(1 − x2 dx
) 2
1
w(s) ds
1
w(s) ds
w(x)
w(x)
(1 − x2 dx
) 2
(1 − x2 dx.
) 2
The last integral in (8.6.5) is finite, by virtue of the hypotheses on α and β.
Concerning (8.6.4), the Schwarz inequality yields
(8.6.6)
1
p
−1
′ (pw) ′ dx = p ′ 2 w +
≤ p ′ 2 w +
1
−1
1
−1
p w′
√ (p
w
′√ w) dx
p 2 (w ′ ) 2 w −1
dx
1
2
p ′ w.
By noting that (w ′ ) 2 w −1 ≤ 4(|α| + |β|) 2 (1 − x 2 ) −2 w, we can use (8.6.3) to conclude.
Inequality (8.3.6) is now straightforward. Starting from relation (8.2.10), we use lemma
8.2.1, inequality (8.6.3) and theorem 3.8.2, to prove that |λn,m| ≥ c1. On the contrary,
from (8.6.4), (6.3.6) and theorem 3.8.2, we obtain |λn,m| ≤ c2n 4 .
As far as we know, a full analysis on the convergence of both the eigenvalues and the
relative eigenfunctions is not yet available. A general approach, based on the properties
of compact operators, is studied in vainikko(1964), vainikko(1967) and osborn(1975),
as well as in many other papers. A self-contained exposition with a comprehensive list