The infinitesimal operator of a semigroup: differentiable properties, general properties, the resolvent.

Evolution Equations: introduction to semigroup theory. Motivations and the abstract Cauchy problem. Preliminaries on functions with values in Banach spaces (continuity, differentiability, integration). The case of bounded operators. Definition of semigroups, and contraction semigroups.

Dirichlet Eigenvalues and Eigenfunctions for the Laplace Operator with Dirichlet boundary conditions.