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 L-A,L-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 Holder regularity) we show that the operator L-A,L-b defined on C-2([0, 1](d)) is closable and its closure is m-dissipative. In particular, its closure (L-A,L-b) over bar is the generator of a C-0-semigroup of contractions on C([0, 1](d)) and C-2([0, 1](d)) is a core for it. The proof of such result is obtained by studying the solvability in Holder spaces of functions of the elliptic problem lambda u(x) - L(A,b)u(x) = f(x), x is an element of [0, 1](d), for a sufficiently large class of functions (c) 2007 Elsevier Inc. All rights reserved.