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 over(LA, b, -) 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 λ u (x) - LA, b u (x) = f (x), x ∈ [0, 1]d, for a sufficiently large class of functions f.