| In this dissertation we study the Lp Neumann boundary problem for Laplace's equation in convex domains and the W1,p estimates for the second order elliptic equations with Neumann boundary data in Lipschitz domains. We also study the uniform W1, p estimates for homogenization of elliptic systems. In the case of convex domains we are able to show that the Lp Neumann problem for Laplace's equation is uniquely solvable for 1 < p 2, we prove that a weak reverse Holder 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 for elliptic equation is symmetric and in VMO( Rn ), the W1,p estimates hold for 32 -- epsilon < p < 3 + epsilon if n ≥ 3, and for 43 -- epsilon < p < 4 + epsilon if n = 2. Finally, we show that the uniform W 1,p estimates for homogenization of elliptic systems hold when | 1p -- 1/2| < 12n + delta.;KEYWORDS: Lipschitz domains; convex domains; Neumann problem; Dirichlet problem; Homogenization problem. |
