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

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 limac¸ons. The limac¸ons involved are such that the Green function for the Dirichlet bi-laplacian on each of these domains is strictly positive. 2000 Mathematics Subject Classification: Primary 35J30; Secondary 31A30; 35B50. Key words: Biharmonic Operator, Dirichlet Boundary Conditions, Green function estimates, Positivity, Maximum Principle.