Null controllability properties of some degenerate parabolic equations.

Useful 2-D (N-D) Hardy type inequalities
◮ normal parametrization of the boundary by arclength :
Γ = γ([0, l(Γ)]), γ(0) = γ(l(Γ)), |γ ′ (t)| = 1 ;
◮ this gives a parametrization of the neighborhood C(Γ, η) of Γ :
ψ : [0, l(Γ)] × (0, η) → C(Γ, η), ψ(s, t) = γ(s) − tν(γ(s)) :
ψ is a C 1 -diffeomorphism between (0, l(Γ)) × (0, η) and
C(Γ, η) \ ψ({0} × (0, η)) ;
◮ then use the 1-D hardy type inequality on every normal
segment, and integrate over Γ :
∫∫
C(Γ,η)
∫ ( ∫
d Γ (x) α−2 z 2 η
=
≤ HARDY
∫
Γ
(
Γ
0
∫
4 η
(α − 1) 2
)
“d Γ (x) α−2 z 2 “
0
)
“d Γ (x) α (∇z · ν) 2 “
c
≤
(α − 1)
∫∫C(Γ,η)
2 d Γ (x) α (∇z · ν) 2
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