Abstract Hadamard claimed in 1907 that the clamped-plate equation is positivity preserving for domains which are bounded by a Limaçon de Pascal. We will show that this claim is false in its full generality. However, we will also prove that there are nonconvex limaçons for which the clamped-plate equation has the sign-preserving property. In fact we will give an explicit bound for the parameter of the limaçon where sign change may occur. Keywords clamped plate - biharmonic - positivity - nonconvex Mathematics Subject Classification (2000) 35J40, 35Q72, 35B50@en

Abstract The main subject of this thesis concerns positivity for fourth order elliptic problems.By positivity we mean that a positive source term in the differential equation leads to a positive solution.For second order elliptic partial differential equations such a result is known and referred to by the name maximum principle.It is also well known that such a maximum principle does not have a straightforward generalization to higher order elliptic equations.A fourth order elliptic equation describes the displacement of an elastic plate loaded by some weight.In general the displacement is not everywhere in one direction. However, the mechanical model seems to indicate that some positivity remains.The main result of the thesis is a splitting of the solution operator as the sum of two terms: a positive singular term and a sign-changing regular one.As a consequence, we prove that the sign preserving effects are much stronger than the opposite ones.@en

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.@en

Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence, one obtains optimal weighted Lp-Lq-regularity estimates for weights involving the distance function. Author Keywords: Polyharmonic Dirichlet problem; Kernel estimates; Weighted Lp-Lq-regularity 35J40; 35B45; 31B30@en

Hadamard claimed in 1907 that the clamped-plate equation is positivity preserving for domains which are bounded by a Limaçon de Pascal. We will show that this claim is false in its full generality. However, we will also prove that there are nonconvex limaçons for which the clamped-plate equation has the sign-preserving property. In fact we will give an explicit bound for the parameter of the limaçon where sign change may occur. Keywords clamped plate - biharmonic - positivity - nonconvex Mathematics Subject Classification (2000) 35J40, 35Q72, 35B50@en