Smoothness of the gradient of weak solutions of degenerate linear equations

Abstract

Let Q(x) be a nonnegative definite, symmetric matrix such that \(\sqrt {Q\left( x \right)} \) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q∇u) = b, we find sufficient conditions in order that \(\sqrt Q \nabla u\) has some first order smoothness. Specifically, if Ω is a bounded open set in Rn, we study when the components of \(\sqrt Q \nabla u\) belong to the first order Sobolev space WQ1,2(Ω) defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of Σi=1nXi′Xiu + b = 0. We do not assume that {Xi} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.