Partial Differential Equations - Modelling and ... - ResearchGate

80 J.Ch. Gilbert and P. Joly
Therefore, since ψ m is an affine function, for any t ∈ [0, 1], there holds
∀x ≤ α m,k , 0 ≤ [ ψ m
(
tR1 +(1−t)R 2
)]
(x) ≤ 4.
Hence
( )
α m tR1 +(1−t)R 2 = αm,k .
In other words, any point of the segment [R 1 ,R 2 ] is a solution of (20), i.e.,
the set of solutions of (20) is convex.
⊓⊔
As a consequence of Lemmas 2 and 3, we know that any solution R of (20)
is such that
T R ≡{τ ∈ ]0,α m,k [ | [ψ m (R)] (τ) =0or4}
is nonempty. Let us call tangent point an element of T R . Theorem 4 below is
more precise, since it claims that there is at least M ≥ k tangent points τ j at
which ψ m (R) takesalternatively the values 0 and 4. For any R, itisconvenient
to construct and enumerate these tangent points in decreasing order:
τ M+1 =0