## Elliptic divergence operator isomorphism for Lp spaces p greater than 2

For the problem $Lu:=div(A\nabla u)=div(g)$,  we will show $L: W_{0}^{1,p}\to W^{-1,p}$ (cf. Meyers “An Lp-estimate for the gradient of solutions of second order elliptic divergence equations “).  Assume that we have coercivity: $\inf_{\phi\in L^{1/q'}}\sup_{\psi\in L^{1/q}}|B_{A}(\psi,\phi) |\geq \frac{1}{K}> 0$.

1. Suppose that $g\in L^{q'}$ and consider sequence $g_{k}\in L^{2}$ such that $|g_{k}-g|_{L^{q'}}\to 0$.
2. For $div(A\nabla u_{k})=div(g_{k})$  the solutions $u_{k}\in W^{1,2}_{0}$. From the coercivity assumption we get: $| \nabla u_k |_{q'}\leq K| g_k |_{q'}$.
3. Therefore, there exists $u\in W^{1,q'}$ s.t.  $\nabla u_{k}\to \nabla u$ weakly in $L^{q'}$.
4. Therefore, u solves the original problem $B_{A}(\psi, u)-\int_{\Omega} (\nabla \psi, g)dx=B_{A}(\psi, u-u_{k})+\int_{\Omega} (\nabla \psi, g_{k}-g)dx\to 0$.
5. and satisfies the estimate $|\nabla u |_{q'}\leq K |g |_{q'}$.
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