| Abstract: |
| We are interested here in questions related to the {\bf regularity} of solutions of {\bf elliptic} problems
%$$
%\mathrm{div}\, (A \nabla\, u) = f \quad \mathrm{in}\quad \Omega
%$$
with {\bf Dirichlet} or {\bf Neumann} boundary condition (see (\cite{1}). For the last 30 years, many works have been concerned with questions
%when $A$ is a matrix or a function and
when $\Omega$ is a {\bf Lipschitz domain}.
We give here some complements for the case of the {\bf Laplacian} (see \cite{3}), the {\bf Bilaplacian} (\cite{2}, \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 or Bilaplacian.
Using the duality method, we will then revisit the work of Lions-Magenes \cite{4}, concerning the so-called {\bf very weak solutions}, when the data are less regular. Thanks to the {\bf interpolation theory}, it permits us to extend the classes of solutions and then to obtain new results of regularity.
\KEYWORDS{Elliptic problems, Lipschitz domains, regularity, Steklov Poincar\`e operator.}
\begin{thebibliography}{9}
\bibitem{1} \textsc{C. Amrouche, M. Moussaoui, H.H. Nguyen}. Laplace equation in smooth or non smooth domains. Work in Progress.
\bibitem{2} \textsc{B.E.J. Dahlberg, C.E. Kenig, J. Pipher, 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, 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, E. Magenes}. \emph{ Probl\`{e}mes aux limites non-homog\`{e}nes et applications}, Vol. 1, Dunod, Paris, (1969).
\bibitem{5}
\textsc{J. Necas}. \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} |
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