We are interested here in questions related to the maximal regularity of solutions of elliptic problems div $(A \nabla\, u) = f$ in $\Omega$ with Dirichlet boundary condition. For the last 40 years, many works have been concerned with questions when $A$ is a matrix or a function and when $\Omega$ is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work.

We give here new proofs and some complements for the case of the Laplacian, the Bilaplacian and the operator $\mathrm{div}\, (A \nabla)$, when ${\bf A}$ is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the {Dirichlet-to-Neumann operator for Laplacian and Bilaplacian.

Using the duality method, we can then revisit the work of Lions-Magenes, concerning the so-called very weak solutions, when the data are less regular.

Thanks to the interpolation theory, it permits us to extend the classes of solutions and then to obtain new results of regularity.