16th June – ‪Chérif Amrouche: Elliptic Problems in Lipschitz and in $C^{1,1}$ Domains

Chérif Amrouche Thursday 16th June at 11:30



We are interested here in questions related to the maximal regularity of solutions to elliptic problems with Dirichlet or Neumann boundary conditions (see ([1]). 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 (see [3]), the Bilaplacian ([2] and [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. 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 [4], concerning the so-called very weak solutions, when the data are less regular.

References :

[1]  C. Amrouche and M. Moussaoui. Laplace equation in smooth or non smooth do- mains. Work in Progress.

[2]  B.E.J. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota. Area integral estimates for higher-order elliptic equations and systems. Ann. Inst. Fourier, 47-5, 1425– 1461, (1997).

[3]  D. Jerison and C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161–219, (1995).

[4]  J.L. Lions and 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. 194-2, 217–279, (2005).

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