In this talk I will discuss a recent paper on the mixed problem for the Lame system of elasticity in a bounded Lipschitz domain in two dimensions. In the mixed problem, we assume that the boundary is written as the union of two disjoint sets D and N. On the set D we prescribe Dirichlet data from a Sobolev space W^{1,p}(D) and on N we prescribe traction data from the space L^p(N). We seek a solution u of Lu=0 with the given boundary data, where L is the Lame operator. I will discuss existence and uniqueness of solutions for p near 1 under certain conditions on the set D. This is joint work with Russell Brown.