Joint French-Czech mathematics meeting

INSA-Lyon, November 29-30, 2018

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)