Second order smoothness of weak solutions of degenerate linear equations

Let m and n be positive integers and {Yk(x)}k=1m be a collection of m first order Lipschitz vector field derivatives in a bounded open set Ω⊂Rn. Given a real-valued function b in Ω, we find conditions guaranteeing that a weak solution u of ∑k=1mYk′Yku=b in Ω has some second order smoothness. More precisely, if Z is a first order vector field derivative, we study when the iterated weak derivatives ZYku exist in Ω and belong to Lloc2(Ω). The main theorem extends a result of the author for the special case when m=n and Z=Yi for some i. Corollaries include the existence of ZXju, j=1,…,n, for weak solutions u of divergence form equations div(Q∇u)=b if Q(x) is a nonnegative definite, symmetric n×n matrix and {Xj}j=1n are the derivatives corresponding to the rows of Q. The reverse orders YkZu and XjZu are also considered.