Partial Differential Equations - Modelling and ... - ResearchGate

82 J.Ch. Gilbert and P. Joly
Our next result shows that the necessary optimality conditions of Theorem
4 are also sufficient. We shall need the following lemma on polynomials.
Lemma 4. If P ∈ P k−1 takes alternatively nonnegative and non-positive values
at k +1 successive distinct points, then P =0.
Proof. Without loss of generality, we can assume that, for points x 0 0.
On the other hand, since P ε → P uniformly on [x 0 ,x k ], there exists ε 0 > 0
such that
∀ε 0.
Therefore, for ε