Higher order elliptic problems and positivity

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.