We are interested here in questions related to the maximal regularity of solutions of elliptic problems with Dirichlet boundary condition (see [1]). For the last 40 years, many works have been concerned with questions 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 (see [3]), the Bilaplacian ([2] and [6]) and the operator div \( A\nabla \) (see [5]), when \( A \) 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 [4], concerning the so-called very weak solutions, when the data are less regular. [1] C. Amrouche and M. Moussaoui. The Dirichlet problem for the Laplacian in Lipschitz domains. Submitted. See also the abstract in https://arxiv.org/pdf/2204.02831.pdf [2] B.E.J. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota. Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier, 47-5, 1425–1461, (1997). [3] D. Jerison and C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161–219, (1995). [4] J.L. Lions and E. Magenes. Problèmes aux limites non-homogènes et applications, Vol. 1, Dunod, Paris, (1969). [5] J. Necas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, (2012). [6] G.C. Verchota. The biharmonic Neumann problem in Lipschitz domains. Acta Math., 194-2, 217–279, (2005).