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

Journal article (2005)

Authors

A Dall'Acqua Analysis - EEMCS
C Meister External organisation
G.H. Sweers Analysis - EEMCS
Research Group
Analysis (EEMCS) (TU Delft)
Techniek
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Published Date

2005

Language

Undefined/Unknown

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

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.