Nonlinear Functional Analysis and its Applications

In the previous chapter we considered the concept of particles. This chapter we begin by introducing a number of basic concepts, which are essential for an understanding of wave phenomena in all parts of physics. We then will discuss the relation between waves and particles, which has played an important role in the development of modern physics. In 1925, Werner Heisenberg formulated his matrix mechanics. This is a quantum mechanics, which is derived from classical mechanics by introducing particle quantization. This theory has already been discussed in Section 58.21. Independently, in 1926, Erwin Schrödinger formulated an equivalent wave mechanics which is derived from wave quantization. The main objective of this chapter is to present a survey. This, together with the previous chapter, might help the reader to better understand many of the problems discussed later on. We thereby follow the fascinating line of development, which leads to the central problem of modern physics—the creation of a unified theory for all four interactions in nature. Quantum theory will be discussed in greater detail in Part V. Only a minimal program is presented here. Some interesting problems that we consider are:

In this chapter we consider:

With regard to the general nonlinear model of Section 62.3 we now consider boundary-value problems, for which the elastic body is supported on parts of its boundary. The boundary conditions thereby have the form of inequalities. In functional-analytic terms, this leads to convex variational problems on convex sets. The corresponding Euler equations are variational inequalities. We shall use Theorem 46.A of Part III in order to obtain a general existence and uniqueness theorem. Throughout, the same notation as in the previous chapter will be employed.

In Theorem 43.B we proved the existence of a bifurcation point for equations with potential operators. Thereby we used a maximum principle.

In this chapter we consider a plate which is clamped at the boundary. Our method of proof, however, can also be applied to other boundary conditions. We use the following tools:

In this chapter we generalize the results of Chapter 60 about the wire to three-dimensional bodies.

The main objectives of phenomenological thermodynamics are:

During the study of the Big Bang in Section 58.15 we already made essential use of Planck’s radiation law. In order to find this law, Planck formulated his famous hypothesis about the quantization of energy for the harmonic oscillator. This was the hour of birth of quantum theory. Planck’s radiation law implies the Stefan-Boltzmann radiation law, which will be used during the following chapter in the discussion of Carleman’s radiation problem. In the present chapter we want to show how these important physical laws can be derived from general principles of statistical physics. The development of statistical physics is mainly connected with the names of Maxwell (1831–1879), Boltzmann (1844–1906), Gibbs (1839–1903), Planck (1858–1947), and Einstein (1879–1955).

As we shall see, the basic equations of hydrodynamics for liquids and gases are obtained by modifying the basic equations of elastodynamics.

In this chapter we apply, step-by-step, the following functional-analytical results:

Typical examples of manifolds are sufficiently smooth curves and surfaces in ℝ ⁿ which have a tangent space (tangent line, tangent plane) at each point. Manifolds will always be manifolds without boundary. One may think, for example, of the surface of a ball. Manifolds with boundary, such as the ball itself, will be considered in Section 73.19.

In this and the following two chapters we consider three central applications of the theory of manifolds:

In this and the following chapters we shall discuss the basic ideas of the general theory of relativity, explain its connection with the theory of manifolds, and give applications in the form of three interesting problems:

We use the notations of Section 75.1. The basic equations of the general theory of relativity which determine the metric tensors g _ij of Einstein’s four-dimensional space-time manifold E ₄ are with the universal constant

In the following two chapters, we present two ways of introducing fixed-point theory which have been developed intensely during the last years and provide an effective method for solving nonlinear equations on computers: In (i) we use triangulations and suitable labelings. The starting point is the lemma of Sperner of Section 77.1. In (ii), curves, i.e., one-dimensional manifolds are being followed. Thereby differential topological methods, especially Sard’s theorem are employed. The disadvantage of (i) over (ii), in view of computer usage, is that with an increasing number of variables in the nonlinear equations, the necessary storage place increases very rapidly.

In this chapter, Sard’s theorem (Proposition 4.55 of Part I) plays a central role. Before reading this chapter, one should look again at this theorem as well as Definition 4.52 about regular values. For didactical reasons, we use a parametrized version of Sard’s theorem already in Section 78.2, and present the proof afterwards in Section 78.7. The definition of the fixed-point index and the mapping degree of Section 78.6, however, only requires Sard’s theorem and not the parametrized version. Sard’s theorem is one of the most important theorems in modern mathematics. It gives a precise formulation of the following philosophy: Most situations in nature are generic, i.e., not degenerate.

Because of its great importance for science and numerical analysis, stability questions have been discussed already in a number of chapters of this volume and the three previous ones. In the present chapter we examine the following two important principles: