Carleman estimates for a class of degenerate parabolic operators

Given α ∈ [0, 2) and f ∈ L2((0, T) × (0, 1)), we derive new Carleman estimates for the degenerate parabolic problem wt + (xαwx)x = f, where (t, x) ∈ (0, T) × (0, 1), associated to the boundary conditions w(t, 1) = 0 and w(t, 0) = 0 if 0 ≤ α < 1 or (xαwx)(t, 0) = 0 if 1 ≤ α < 2. The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all 0 ≤ α < 2 and ω ⊂⊂ (0, 1), we deduce null controllability results for the degenerate one-dimensional heat equation ut − (xαux)x = hχω with the same boundary conditions as above.