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Anna Dall'Acqua, Christian Meister, Guido Sweers
Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains
Keywords: biharmonic operator, Dirichlet boundary conditions, Green function estimates, positivity, maximum principle
The main result in this paper is that the solution operator for the bi-Laplace problem with zero Dirichlet boundary conditions on a bounded smooth 2d-domain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part contains the singularity and the negative part inherits the full regularity of the boundary. Such a splitting allows one to find a priori estimates for fourth order problems similar to the one proved via the maximum principle in second order elliptic boundary value problems. The proof depends on a careful approximative fill-up of the domain by a finite collection of limaçons. The limaçons involved are such that the Green function for the Dirichlet bi-Laplacian on each of these domains is strictly positive.
Analysis, Oldenbourg Wissenschaftsverlag
Print ISSN: 0174-4747
Volume: 25, 03/2005
Pages: 205 - 261