| Abstract: |
| We are interested here in questions related to the {\bf maximal regularity} of solutions of {\bf elliptic} problems with {\bf Dirichlet} boundary condition (see (\cite{1}). For the last 40 years, many works have been concerned with questions when $\Omega$ is a {\bf 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 \bf Laplacian} (see \cite{3}), the {\bf Bilaplacian} (\cite{2} and \cite{6}) and the operator $\mathrm{div}\, (A \nabla)$ (see (\cite{5}), 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 {\bf Dirichlet-to-Neumann} operator for Laplacian and Bilaplacian.
Using the duality method, we can then revisit the work of Lions-Magenes \cite{4}, concerning the so-called {\bf very weak solutions}, when the data are less regular.
\KEYWORDS{Elliptic problems, Lipschitz domains, maximal regularity, Steklov Poincar\'e operator.}
\AMSCLASSIFICATION{35C15, 35J25, 35J40}
\begin{thebibliography}{9}
\bibitem{1} \textsc{C. Amrouche and M. Moussaoui}. The Dirichlet problem in Lipschitz and in $\mathcal{C}^{1, 1}$ domains. Submitted. See also the abstract in
https://arxiv.org/pdf/2204.02831.pdf
\bibitem{2} \textsc{B.E.J. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota}. Area integral estimates for higher order elliptic equations and systems. \emph{Ann. Inst. Fourier}, {\bf 47-5}, 1425--1461, (1997).
\bibitem{3} \textsc{D. Jerison and C.E. Kenig}. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, \emph{J. Funct. Anal.} {\bf 130}, 161--219, (1995).
\bibitem{4}
\textsc{J.L. Lions and E. Magenes}. \emph{ Probl\`{e}mes aux limites non-homog\`{e}nes et applications}, Vol. 1, Dunod, Paris, (1969).
\bibitem{5}
\textsc{J. Ne$\mathrm{\check{c}}$as}. \emph{ Direct methods in the theory of elliptic equations}. Springer Monographs in Mathematics. Springer, Heidelberg, (2012).
\bibitem{6} \textsc{G.C. Verchota}. The biharmonic Neumann problem in Lipschitz domains. \emph {Acta Math.} {\bf 194-2}, 217--279, (2005).
\end{thebibliography}
\vspace{3ex}
\ADDRESS{1}{
Laboratoire de Math\'ematiques et Leurs Applications, UMR CNRS 5142 \
Universit\'e de Pau et des Pays de l'Adour \
email: \texttt{cherif.amrouche@univ-pau.fr}}
\ADDRESS{2}{
Lab. des EDP Non Lin\'eaires et
Histoire des Math\'ematiques\
Ecole Normale Sup\'erieure de Kouba, Alger \
email: \texttt{mmohand47@gmail.comy}} |
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