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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