Carleman Estimates for a Class of Degenerate Parabolic Operators

Given $\alpha \in [0,2)$ and $f \in L^2 ((0,T)\times(0,1))$, we derive new Carleman estimates for the degenerate parabolic problem $w_t + (x^\alpha w_x) _x =f$, where $(t,x) \in (0,T) \times (0,1)$, associated to the boundary conditions $w(t,1)=0$ and $w(t,0)=0$ if $0 \leq \alpha <1$ or $(x^\alpha w_x)(t,0)=0$ if $1\leq \alpha <2$. The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all $0 \leq \alpha <2$ and $\omega\subset\subset(0,1)$, we deduce null controllability results for the degenerate one-dimensional heat equation $u_t - (x^\alpha u_x)_x = h \chi _\omega$ with the same boundary conditions as above.