Untitled - Cdm.unimo.it

Ordinary Differential Equations 191
well-posed. This means that it admits a unique weak solution U ∈ X. We remark that
two solutions of (9.3.1) which differ on a set of measure zero are in the same class of
equivalence in X.
We first introduce some notation. Consider the bilinear form Bw : X ×X → R defined
as follows:
(9.3.2) Bw(ψ,φ) :=
ψ
I
′ (φw) ′ dx, ∀ψ,φ ∈ X.
The mapping Bw is linear in each argument (this justifies the term bilinear). We also
define the linear operator Fw : X → R according to
(9.3.3) Fw(φ) :=
I
fφw dx, ∀φ ∈ X.
With the above definitions we restate (9.3.1). We are concerned with finding U ∈ X
such that
(9.3.4) Bw(U,φ) = Fw(φ), ∀φ ∈ X.
The answer to our question relies on the following general result.
Theorem 9.3.1 (Lax & Milgram) - Let X be a Hilbert space. Let B : X × X → R
be a bilinear form and F : X → R be a linear operator. Let us assume that there exist
three positive constants C1,C2,C3 such that
(9.3.5) |B(ψ,φ)| ≤ C1 ψX φX, ∀ψ,φ ∈ X,
(9.3.6) B(ψ,ψ) ≥ C2 ψ 2 X, ∀ψ ∈ X,
(9.3.7) |F(φ)| ≤ C3 φX, ∀φ ∈ X.