Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation w(t) + (a(x)w(x))(x) = f, (t, x) is an element of (0, T) x (0, 1), where the function a mainly satisfies a is an element of C-o([0, 1]) boolean AND C-1((0, 1)), a > 0 on (0, 1) and 1/root a is an element of L-1(0, 1). We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x(alpha) (1 - x)(beta) with alpha, beta is an element of [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation u(t) - (a(x)u(x))(x) = h chi(omega), (t,x) is an element of (0,T) x (0,1), omega subset of subset of (0,1). The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x(alpha) with alpha is an element of[0, 2).