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 + (x^α wₓ) 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 (x^α wₓ)(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 - (x^α uₓ)ₓ = h χ _ω with the same boundary conditions as above.