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We consider the mixed problem for $L$ the Lam\'e system of elasticity in abounded Lipschitz domain $ \Omega\subset\reals ^2$. We suppose that theboundary is written as the union of two disjoint sets, $\partial\Omega =D\cupN$. We take traction data from the space $L^p(N)$ and Dirichlet data from aSobolev space $ W^{1,p}(D)$ and look for a solution $u$ of $Lu =0$ with thegiven boundary conditions. We give a scale invariant condition on $D$ and findan exponent $ p_0 >1$ so that for $1<p<p_0$, we have a unique solution of thisboundary value problem with the non-tangential maximal function of the gradientof the solution in $L^ p(\partial\Omega)$. We also establish the existence of aunique solution when the data is taken from Hardy spaces and Hardy-Sobolevspaces with $ p$ in $(p_1,1]$ for some $p_1 <1$.

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