The mixed problem for the Lam\'e system in two dimensions

We consider the mixed problem for $L$ the Lam'e system of elasticity in a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$. We suppose that the boundary is written as the union of two disjoint sets, $\partial\Omega =D\cup N$\mathrm{.}Wetaketractiondatafromthespace$L^p(N)$ and Dirichlet data from a Sobolev space $W^{1,p}(D)$ and look for a solution $u$ of $Lu=0$ with the given boundary conditions. We give a scale invariant condition on $D$ and find an exponent $p_0>1$ so that for $1<p<p_0$, we have a unique solution of this boundary value problem with the non-tangential maximal function of the gradient of the solution in $L^p(\mathrm{\partial }\mathrm{\Omega })$. We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with $p$ in $(p_1,1]$ for some $p_1<1$.