Untitled - Cdm.unimo.it

Eigenvalue Analysis 159
Combining with (8.2.4), we finally get
(8.2.8)
1
p
−1
′ (pw) ′ dx ≥
1
−1
Inequality (8.2.3) follows with C = 1+ν
1−ν .
[p ′ ] 2 w dx + 2ν
1
p
1 + ν −1
′ (pw) ′ dx.
Another proof of lemma 8.2.1 is given in bernardi and maday(1989).
When the polynomials are complex, the proof of lemma 8.2.1 is identical to that given
above. In this way, we can find a constant C > 0, such that
1
(8.2.9) Re p ′ (¯pw) ′ 1
dx ≥ C
−1
We can now state the next proposition.
|p
−1
′ | 2 w dx.
Theorem 8.2.2 - Let ν := α = β with −1 < ν ≤ 1. Then, for any n ≥ 2, the
eigenvalues of problem (8.2.1) satisfy Reλn,m > 0, 1 ≤ m ≤ n − 1.
Proof - We remark that formula (3.5.1) and theorem 3.5.1 also apply to complex
polynomials. It suffices to argue with the real and imaginary parts separately.
Recalling that pn,m(±1) = 0, we use integration by parts to obtain
(8.2.10)
= −
n
i=0
1
−1
p ′′ n,m(η (n)
i )¯pn,m(η (n)
i ) ˜w (n)
i
p ′ n,m(¯pn,mw) ′ 1
dx = − p
−1
′′ n,m¯pn,mw dx
= λn,m
n
i=0
[|pn,m|(η (n)
i )] 2 ˜w (n)
i , 1 ≤ m ≤ n − 1.
We first note that p ′ n,m ≡ 0 for all m. Otherwise, the vanishing conditions at the
boundaries would imply pn,m ≡ 0 for some m. Recalling (8.2.9), we easily conclude
the proof, since the real part of the left-hand side in (8.2.10) is strictly positive.
A straightforward and meaningful corollary is obtained by examining relation (8.2.10).