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Ordinary Differential Equations 193
Proof - We start by considering ψ,φ ∈ P 0 n ⊂ H 1 0,w(I), for some n ≥ 2. Then, the
first inequality is obtained by applying the Schwarz inequality (2.1.7) to the integral
I (ψ′√ w)[(φw) ′ / √ w]dx. Recalling that (w ′ ) 2 /w ≤ w/(1 − x 2 ) 2 , we conclude using
the results of lemma 8.6.1. The second inequality is proven in lemma 8.2.1. The third
one comes from the Schwarz inequality and (5.7.5). The proof of (9.3.10), (9.3.11),
(9.3.12) in H 1 0,w(I) requires a little care. In short, one approximates φ ∈ H 1 0,w(I)
by a sequence of polynomials φn ∈ P 0 n, n ∈ N, converging to φ in such a way that
limn→+∞ φ − φn H 1 0,w (I) = 0. This also implies: limn→+∞ φn H 1 0,w (I) = φ H 1 0,w (I).
The proposition follows by a limit process which is standard in Lebesgue integration
theory (see kolmogorov and fomin(1961)). The details are omitted.
Thus, (9.3.4) is a well-posed problem. In the special case when f ∈ C 0 ( Ī) ⊂ L2 w(I),
the solution U of (9.3.4) is the classical strong solution satisfying (9.1.4).
Generalizations can be examined with little modification. For example, let us
consider the boundary-value problem
(9.3.13)
⎧
⎨
−U ′′ + A1U ′ + A2U = f in I,
⎩
U(±1) = 0,
where A1 : Ī → R and A2 : Ī → R are given bounded integrable functions. Here, we
can replace (9.3.2) by the bilinear form
(9.3.14) Bw(ψ,φ) =
ψ
I
′ (φw) ′ dx +
I
[A1ψ ′ φ + A2ψφ]w dx, ∀ψ,φ ∈ X.
One checks that, if the inequality A2w − 1
2 (A1w) ′ ≥ 0 holds in I, then the hypotheses
of theorem 9.3.1 are satisfied. Therefore, the variational formulation of (9.3.13) admits
a unique weak solution.