W1,p estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains

We study W1,p estimates in Lipschitz domains for second order elliptic equations and systems of divergence form with real-valued, bounded, measurable coefficients. For any fixed p>2, we prove that a weak reverse Hölder inequality implies the W1,p estimates for solutions with Neumann boundary conditions. As a result, we are able to show that if the coefficient matrix of elliptic equation is symmetric and in VMO(Rn), the W1,p estimate holds for if n≥3, and for if n=2, and for elliptic systems the W1,p estimate is true for .