Prof. Chérif Amrouche “Elliptic Problems in Lipschitz and in C^{1,1} Domains”

We are interested here in questions related to the maximal regularity of
solutions of elliptic problems with Dirichlet boundary condition. 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, the Bilaplacian and the
operator div (A∇ ), 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.

This is joint work with Mohand Moussaoui.