Untitled - Cdm.unimo.it

192 Polynomial Approximation of Differential Equations
Then, there exists a unique solution U ∈ X of problem
(9.3.8) B(U,φ) = F(φ), ∀φ ∈ X.
Moreover, we can find a positive constant C4 > 0 such that
(9.3.9) UX ≤ C4 sup
φ∈X
φ≡0
|F(φ)|
.
φX
The Lax-Milgram theorem is a powerful tool for proving existence and uniqueness for a
large class of linear differential problems. Though the proof is not difficult, we omit it
for simplicity. Applications and extensions have been considered in many publications.
We suggest the following authors: lax and milgram(1954), brezis(1983), brezzi and
gilardi(1987). Inequality (9.3.6) plays a fundamental role in the proof of theorem
9.3.1. It is in general known as a coerciveness condition and indicates that (9.3.8) is an
elliptic problem.
The final step is to check wether the Lax-Milgram theorem can be applied to study
the solution of (9.3.4). It is quite easy to show that X ≡ H1 0,w(I) ⊂ C0 ( Ī) is a Hilbert
space (see section 5.3). In addition, inequalities (9.3.5), (9.3.6) and (9.3.7) are directly
obtained from the following result (we recall that the norm in H 1 0,w(I) is given by
(5.7.7)).
Theorem 9.3.2 - Let ν := α = β with −1 < ν < 1. Then, we can find three positive
constants C1,C2,C3 such that
(9.3.10)
ψ ′ (φw) ′
dx
≤ C1
(9.3.11)
(9.3.12)
I
I
[ψ
I
′ ] 2 w dx
ψ
I
′ (ψw) ′ dx ≥ C2
fφw dx
≤ C3 f L 2 w (I)
I
1
2
I
[φ ′ ] 2 1
w dx
2
, ∀ψ,φ ∈ H 1 0,w(I),
[ψ ′ ] 2 w dx, ∀ψ ∈ H 1 0,w(I),
[φ
I
′ ] 2 w dx
1
2
, ∀φ ∈ H 1 0,w(I).