Linear Problems

Abstract

Linear polyharmonic problems and their features are essential in order to achieve the main tasks of this monograph, namely the study of positivity and nonlinear problems. With no hope of being exhaustive, in this chapter we outline the main tools and results, which will be needed subsequently.We start by introducing higher order Sobolev spaces and relevant boundary conditions for polyharmonic problems. Then using a suitable Hilbert space, we show solvability of a wide class of boundary value problems. The subsequent part of the chapter is devoted to regularity results and a priori estimates both in Schauder and Lpsetting, including also maximum modulus estimates. These regularity results are particularly meaningful when writing explicitly the solution of the boundary value problem in terms of the data by means of a suitable kernel. Focusing on the Dirichlet problem for the polyharmonic operator, weintroduce Green’s functions and the fundamental formula by Boggio in balls. We conclude with a study of a biharmonic problem in nonsmooth domains explaining two paradoxes which are important in particular when approximating solutions numerically.