Null controllability properties of some degenerate parabolic equations.

How to obtain the Carleman estimate : the method and
the difficulties
◮ description of the method : the weighted function
z(t, x) := e −Rσ w satisfies the differential problem :
and so
P + R z + P− R z = fe−Rσ ,
‖P + R z‖2 + ‖P − R z‖2 + 2〈P + R z, P− R z〉 = ‖fe−Rσ ‖ 2 ,
and the Carleman estimates comes from a suitable lower
bound of the scalar product 〈P + R z, P− R z〉.
◮ the difficulties due to the degeneracy :
adapt all the weights to the degeneracy ; and use suitable
Hardy type inequalities ;
the solution has no enough global regularity : compute on
subdomains Ω δ := {x ∈ Ω, d(x, Γ) > δ}, and pass to the limit
Ω δ → Ω and Γ δ → Γ for the boundary integrals (using
properties of W 1,1 functions).
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