Let A denote a general second order differential operator in divergence form, with real coefficients satisfying only local boundedness conditions. Then, A can be viewed as an unbounded operator A(p) on L(p)(R(N)) with maximal domain D(A(p)). The main result of this paper is a criterion for the injectivity of A(p) when p is an element of (1, infinity). This criterion is next used to establish the denseness of C(0)(infinity) (R(N)) in D(A(p)) or, under more general conditions, in a smaller natural domain. When p = 2, this also yields a positive answer to a selfadjointness question raised by Kato in 1981.