ABSTRACT
In this chapter we study certain nonlinear operators which arise in applications and we also discuss the so-called Young measures, which roughly speaking capture the limits of minimizing sequences in variational problems which do not have a solution. For some cases we also develop the corresponding linear theory in order to have a complete picture of the theory, see the similarities and differences of the two and appreciate the limitations of the nonlinear theory. In Section 3.1, we consider compact operators. Compactness was intro-
duced as a first attempt to deal with infinite dimensional nonlinear operator equations. By its nature, compactness approximates infinite objects by finite ones. We see that in the context of compact operators (linear and nonlinear alike) this principle is in general true. We also discuss proper maps, the spectral theory of linear, compact, self-adjoint operators on a Hilbert space and Fredholm operators. A broader framework for the analysis of infinite dimensional problems is provided by monotone operators, which extend to an infinite dimensional context, the simple notion of an increasing real function. In Section 3.2 we examine monotone operators from a Banach space into
its dual, with special emphasis on maximal monotone operators, which are a generalization of a continuous increasing real function. Maximal monotone operators have remarkable surjectivity properties. We point out that surjectivity results are important because they correspond to existence results for certain classes of nonlinear operator equations. At the end of the section we also discuss generalizations of the notion of monotonicity. These are the so-called operators of monotone type, the most important of which are the pseudomonotone operators. Monotone operators map a Banach space to its dual. If instead we want to consider nonlinear operators mapping a Banach space to itself, we need to consider accretive and m-accretive operators. Their importance comes from the fact that they are the generators of linear and nonlinear semigroups, which, roughly speaking, are an abstraction of the trajectories of a given differential equation. In Section 3.3 first we examine accretive operators and then we look at
semigroups of operators generated by certain accretive operators. We present in detail both the linear and nonlinear theories. Undoubtedly the most common nonlinear operator is the so-called Nemytskii operator (or superposition)
In Section 3.4 we examine this operator and we have a first look at integral
functionals corresponding to normal integrands. In a variational problem, when the objective functional is not inf-compact, a solution does not exist. Nevertheless, the minimizing sequences (or appropriate subsequences of them) have a limit behaviour (usually more and more oscillating), which is captured by embedding the original functions to the space of Young measures (or parametrized probabilities). This embedding leads to a larger inf-compact problem which has a solution (relaxation). In Section 3.5 we discuss the theory of the Young measures and obtain additional lower semicontinuity results for integral functionals. Some of the topics of this chapter will be revisited in the course of the next chapter.
