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The mixed problem for the Lamé system in two dimensions

We consider the mixed problem for $L$ the Lamé system of elasticity in a bounded Lipschitz domain $ Ω\subset\reals ^2$. We suppose that the boundary is written as the union of two disjoint sets, $\partialΩ=D\cup N$. We take traction data from the space $L^p(N)$ 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 $L^ p(\partialΩ)$. 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 $p_1 <1$.