ABSTRACT
The functional-analytic approach to the solution of (partial) differential equations requires knowledge of the properties of spaces of functions of one or several real variables. A large class of infinite dimensional dynamical systems (evolution systems)
can be modelled as an abstract differential equation defined on a suitable Banach space or on a suitable manifold therein. The advantage of such an abstract formulation lies not only on its generality but also in the insight that can be gained about the many common unifying properties that tie together apparently diverse problems. It is clear that such a study relies on the knowledge of various spaces of vector valued functions (i.e., of Banach space valued functions). For this reason Section 2.1 deals with vector valued functions. We introduce
the various notions of measurability for such functions and then based on them we define the different integrals corresponding to them. The emphasis is on the so-called Bochner integral, which generalizes in a very natural way the classical Lebesgue integral to vector valued functions. In Section 2.2 we continue with vector valued functions and introduce the
so-called Lebesgue-Bochner spaces, which extend to vector valued functions the well known Lebesgue Lp-spaces. We also consider evolution triples and the function spaces associated with them. Evolution triples provide a suitable analytical framework for the study of a large class of linear and nonlinear evolution equations. In Section 2.3 we have compactness results for the spaces introduced and
studied in the previous section. The compactness results refer to both the strong and the weak topologies on the spaces under consideration. Thus far we are dealing with function spaces arising in evolutionary prob-
lems. In Section 2.4 we study Sobolev spaces, which are the main tools in the analysis of both stationary and nonstationary equations. Sobolev spaces play a central role in the modern theory of partial differential equations and they allow us to broaden significantly the notion of solution of a boundary value problem. They provide a natural functional analytical framework for the study of weak solutions of elliptic boundary value problems. No specific applications to problems in partial differential equations are discussed. Instead the section aims to serve as a concise introduction to the properties of
In Section 2.5 we present some fundamental inequalities associated with
Sobolev functions, the celebrated embedding theorems for the Sobolev spaces and some of their consequences. The embedding theorems are arguably the most important results in this theory and the reason why Sobolev spaces are so effective in dealing with boundary value problems. Finally in Section 2.6 we establish some fine properties of Sobolev spaces
and introduce functions of bounded variation (BV-functions). These are functions whose weak first partial derivatives are Radon measure and this is essentially the weakest measurable theoretic sense in which a function can be differentiable. They are particularly useful in theoretical mechanics.
