Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube [0,1]d, with d 1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C2([0,1]d) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C0-semigroup of contractions on C([0,1]d) and C2([0,1]d) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x [0,1]d, for a sufficiently large class of functions f.