Spaces of Summable Functions and Partial Differential Equations

Abstract

This chapter aims at substantiating the abstract theory of Hilbert spaces developed in [GM3]. After introducing the Laplace, heat and wave equations we present the classical method of separation of variables in the study of partial differential equations. Then we introduce Lebesgue’s spaces of psummable functions and we continue with some elements of the theory of Sobolev spaces. Finally, we present some basic facts concerning the notion of weak solution, the Dirichlet principle and the alternative theorem.