Data evento: 
Wednesday, May 8, 2024 - 14:30

Prof. Chérif Amrouche (Pau University, France)

"Dirichlet problem for the Laplacian and the Bilaplacian in Lipschitz domains"

Aula Seminari (Palazzina RM004, via A. Scarpa 16)


Abstract: (si veda anche il file allegato)

We are interested here in questions related to the maximal regularity of solutions of elliptic problems with Dirichlet boundary condition (see [1]). For the last 40 years, many works have been concerned with questions when $\Omega$ 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 $div(A\nabla)$ (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.

[1] C. Amrouche and M. Moussaoui. The Dirichlet problem for the Laplacian in Lipschitz domains. Submitted. See also the abstract in

[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. Problèmes aux limites non-homogènes 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).








© Università degli Studi di Roma "La Sapienza" - Piazzale Aldo Moro 5, 00185 Roma