For the problem , we will show (cf. Meyers “An Lp-estimate for the gradient of solutions of second order elliptic divergence equations “). Assume that we have coercivity: .
- Suppose that and consider sequence such that .
- For the solutions . From the coercivity assumption we get: .
- Therefore, there exists s.t. weakly in .
- Therefore, u solves the original problem .
- and satisfies the estimate .