In this talk I will discuss recent results on the mixed boundary value problem in Lipschitz domains. Consider a bounded Lipschitz domain with boundary decomposed into two disjoint sets. On one portion of the boundary Neumann data is prescribed. On the remainder of the boundary Dirichlet data is prescribed. I will discuss the existence and uniqueness of solutions of the mixed problem for the Laplacian and the Lame system of elastostatics with boundary data taken from L^p, where p is greater than or equal to 1. I will highlight how reverse Holder estimates play a key role in obtaining estimates on the non-tangential maximal function of the gradient of solutions to the mixed problem.