Rolf Leis

Initial boundary value problems in mathematical physics

This book serves as an introduction both to classical scattering theory and to the time-dependent theory of linear equations in mathematical physics. Hibert space methods are used to develop the latter theory in such a way that the asymptotic behaviour of large time can be discussed; among other topics discussed are the proof of the existence of wave operators, some of the special equations of mathematical physics (Maxwell equations, the linear equations of elasticity and thermoelasticity, and the plate equation), exterior boundary value problems, radiation conditions, and limiting absorption principles. The background knowledge required for a full understanding of the concepts discussed here is presented in the second chapter; there is also an extensive reference list, which enables readers to increase their familiarity with topics which are not central to the subject under discussion. The material in this book is drawn from courses given by Professor Leis in West Germany, and in Great Britain where he was a visiting Professor. These courses were given to graduate students of Mathematics and Physics, and it is for these people in particular that the book is intended, although it will also prove useful to their teachers, and to those carrying out detailed research in the field.

A transmission problem for the plate equation

A scattering theory is developed for transmission problems associated with the plate equation. Asymptotic methods of solution for large time are examined as are questions concerning regularity of solution, nature of the associated spectrum and existence of appropriate wave operators. It is shown that in contrast to solutions of the wave equation, signals can propagate with an infinite dispersion velocity.

Initial value problems in elasticity

heat conduction phenomena like initial-boundary value problems in linear thermoelasticity are described. Except for the last example A is a selfadjoint operator. To solve such problems one has to give a solution concept first. Afterwards one tries to get more specific knowledge of the solutions obtained. For instance one inquires about their regularity, or about their asymptotic behaviour for large times and proves the existence of wave and scattering operators. Problems of inverse scattering theory are of great mathematical and practical interest (data like initial values, boundary or medium shall be recovered from the reflected signals). To treat such problems one has to know, for instance, how the scattering operator is determined by the boundary and to invert this mapping. In the high frequency limiting case one obtains relatively simple formulae. In the following, however, I shall not touch on such questions. Rather, in the first part of the lecture I shall only briefly treat linear boundary value problems for exterior domains, and also with the restriction that the underlying medium is homogeneous outside a sufficiently large ball. Of course boundary value problems for domains with unbounded boundaries are also of great importance.

The spectrum of A and boundary value problems

In the last chapter we proved existence and uniqueness of the solution of the Cauchy problem for the wave equation. To get more detailed knowledge of the behaviour of the solutions, we have to present a more careful description of the spectrum of the underlying operator A. This will be done in the present chapter.

More than Forty Years with Erhard Meister

Erhard Meister and I we first met in 1958 at a GAMM meeting in Saarbrucken. For both of us it was the first attendance at such a conference, as far as I remember. The Saar territory had only shortly before been integrated into the Federal Republic a milestone in the process of Franco-German reconciliation. Erhard Meister was moving to Saarbrucken at that time to be an assistant of H. Sohngen, who had just received a Chair there. As for me, my first visit to the USA was coming up.

A Schrödinger equation

Schrodinger equations are of great interest for both the physicist and the mathematician. A lot of papers have been written on them. We only want to treat one example in these lectures assuming G = ℝ3 and perturbing the Δ operator with a potential V. One usually distinguishes between ‘short-range’ potentials and ‘long-range’ potentials. Roughly speaking V is short-range if V ∈ ℒ 2 loc and |V(x)| = O (|x|−1−e ) for |x| → ∞ and e > 0. On the other hand the Coulomb potential V(x) = −q |x| −1, q > 0, is long-range.

Maxwell’s equations

Maxwell’s equations form an especially interesting example of the description of wave propagation phenomena. Thus we treat them in some detail. We start by formulating Maxwell’s equations, proving existence and uniqueness, and in section 8.3 treat the free space problem. Sections 8.1–8.3 are similar to sections 7.1–7.3; again the dimension of the null space of the underlying operator is infinite and the system is not elliptic. A consequence is the impossibility of generally estimating the first derivatives of the solutions up to the boundary. Thus we cannot use Rellich’s theorem to show compactness and therefore prove a selection theorem in section 8.5 after having discussed solutions in bounded domains in section 8.4. Finally, in section 8.6, we briefly treat exterior boundary value problems. Exterior boundary value problems and the existence of wave operators will be dealt with in Chapter 9 in more detail, where we shall present a unified approach to both Maxwell’s equations and the linearized system of acoustics.

The free space problem for the wave equation

In this chapter we want to discuss the free space problem for the wave equation, so we assume G = ℝ and a ik = δ ik . It is relatively easy to study the behaviour of the solutions in this simple case. In particular we are interested in getting the asymptotics for large time. This is important because, in the next chapter, we want to compare solutions of arbitrary exterior problems with free space solutions.

The wave equation continued: time-asymptotic behaviour of the solutions

In this chapter we want to describe the asymptotic behaviour for large t of the solutions in the general case. We start by treating solutions in bounded domains. In section 6.2 we deal with exterior domains and prove local energy decay. In section 6.3 we show the existence of wave operators using a result of A. L. Belopolskii and M.Sh. Birman (1968), which will be proved in section 6.4.

The plate equation

In this chapter we treat the plate equation, which is of second order in t and of fourth order in x. In contrast to linear elasticity the solutions show dispersion. The reduced plate equation involves a product of (Δ + 1) and (Δ − 1). Thus with respect to x we expect both a vibrating and a damping component. The procedure will be similar to that already used. Thus we start by formulating the problem and prove existence and uniqueness. In section 12.3 we treat the free space problem and in section 12.4 present some aspects of exterior boundary value problems.