Null controllability properties of some degenerate parabolic equations.

A useful improved version (J. Vancostenoble) :
Lemma
Given L > 0, α ∈ [0, 1), β > 0, n > 0, there exists
C β,n = C(L, β, n) > 0 and x β,n = x(L, β, n) ∈ (0, L) such that the
following inequality holds : for all z ∈ D((0, L])
(1 − α) 2
4
∫ L
0
x α−2 z(x) 2 dx + n
≤
∫ L
0
∫ L
0
x α−2+β z(x) 2 dx
x α z x (x) 2 dx + C β,n
∫ L
x β,n
z(x) 2 dx. (1)
◮ it allows us to estimate “non critical” terms of the form
∫ L
0 xα−2+β z(x) 2 dx uniformly with respect to α ∈ [0, 1) ;
◮ it remains valid for α ∈ [1, 2), in particular for α = 1 ;
◮ it is the key to obtain the observability cost estimates.
19/26