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International mathematical journal of analysis and its applications

Editor-in-Chief: Schulz, Friedmar

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Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains

Anna Dall'Acqua / Christian Meister / Guido Sweers

Citation Information: Analysis. Volume 25, Issue 3/2005, Pages 205–261, ISSN (Print) 0174-4747, DOI: 10.1524/anly.2005.25.3.205, September 2009

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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.

Keywords: biharmonic operator; Dirichlet boundary conditions; Green function estimates; positivity; maximum principle

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