The Laplacian and Some Natural Variants

Abstract

Our focus in this chapter is largely on the p-Laplacian. The theory of Chap. 1 concerning the representation of compact linear maps is used to establish the existence of a countable family of certain types of weak solutions of the Dirichlet eigenvalue problem for the p-Laplacian, with associated eigenvalues. When the underlying space domain is a bounded interval in the real linemore direct methods are available: we give an account of the work of [39] which leads to the representation in terms of p-trigonometric functions of the eigenfunctions of the one-dimensional p-Laplacian under a variety of initial or boundary conditions.