Chérif Amrouche Thursday 16th June at 11:30
We are interested here in questions related to the maximal regularity of solutions to elliptic problems with Dirichlet or Neumann boundary conditions (see (). For the last 40 years, many works have been concerned with questions when Ω 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 ), the Bilaplacian ( and ) and the operator div (A∇) (see () 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 , concerning the so-called very weak solutions, when the data are less regular.
 C. Amrouche and M. Moussaoui. Laplace equation in smooth or non smooth do- mains. Work in Progress.
 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).
 D. Jerison and C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161–219, (1995).
 J.L. Lions and E. Magenes. Probl`emes aux limites non-homog`enes et applications, Vol. 1, Dunod, Paris, (1969).
 J. Necas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, (2012).
 G.C. Verchota. The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194-2, 217–279, (2005).