Untitled - Cdm.unimo.it

158 Polynomial Approximation of Differential Equations
Also for problem (8.2.1), we have eigenvalues with positive real part. We first prove
a preliminary result in the ultraspherical case.
Lemma 8.2.1 - Let ν := α = β with −1 < ν ≤ 1. Then we can find a constant
C > 0 such that, for any n ≥ 2 and p ∈ Pn satisfying p(±1) = 0, one has
(8.2.3)
1
p
−1
′ (pw) ′ 1
dx ≥ C [p
−1
′ ] 2 w dx.
Proof - We follow the guideline of the proof given in canuto and quarteroni(1981),
in the case ν = −1/2. First we have
(8.2.4)
1
p
−1
′ (pw) ′ dx =
1
−1
[p ′ ] 2 w dx − 1
1
p
2 −1
2 w ′′ dx,
where we integrated by parts, using the condition p(±1) = 0. If 0 ≤ ν ≤ 1, the proof
is complete with C = 1, since w ′′ ≤ 0. On the other hand, let −1 < ν < 0 and define
τ := ν−1
2ν
(8.2.5)
+ τ 2
=
> 0. Then, we have
1
Considering that
1
p
−1
′ (pw) ′ dx =
−1
1
(p
−1
′ w + τpw ′ ) 2 w −1 dx +
1
[p
−1
′ ] 2 1
w dx + 2τ pp
−1
′ w ′ dx
p 2 (w ′ ) 2 w −1 dx + ( 1
1
2 − τ) [p
−1
2 ] ′ w ′ dx − τ 2
1
p
−1
2 (w ′ ) 2 w −1 dx
1
(8.2.6) (τ − 1
2 )w′′ − τ 2 (w ′ ) 2 w −1 ≥ −
formula (8.2.5) yields
(8.2.7)
−1
1
p
−1
′ (pw) ′ dx ≥ −
p 2 (τ − 1
2 )w′′ − τ 2 (w ′ ) 2 w −1 dx.
1 + ν
4ν
1 + ν
4ν w′′ ,
1
p
−1
2 w ′′ dx.