Chérif Amrouche

Université de Pau et des Pays de l’Adour

Elliptic Problems in Smooth and Non Smooth Domains.

We are interested here in questions related to the regularity of solutions of elliptic problems with Dirichlet or Neumann boundary condition (see ([1]). For the last 20 years, lots of work has been concerned with questions when Ω is a Lipschitz domain .We give here some complements for the case of the Laplacian (see [3]), the Bilaplacian ([2],[6]) and the operator div (A ∇) (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. Using the duality method, we will then revisit the work of Lions-Magenes [4], concerning the so-called very weak solutions, when the data are less regular. Thanks to the inter-polation theory, it permits us to extend the classes of solutions and then to obtain new results of regularity. References [1] C. Amrouche, M. Moussaoui, H.H. Nguyen. Laplace equation in smooth or non smooth domains. Work in Progress. [2] B.E.J. Dahlberg, C.E. Kenig, J. Pipher, G.C. Verchota. Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier,47, no. 5, 1425–1461, (1997). [3] D. Jerison, C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161–219, (1995). [4] J.L. Lions, E. Magenes. Probl`emes aux limites non-homog`enes 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 no. 2, 217–279, (2005)