Initial Boundary Value Problems in Mathematical Physics

1

Introduction

In this introductory chapter we intend to lead the reader up to the complex of problems to be treated in this book, essentially by means of examples. In doing so, we do not want to go into too much detail and especially do not bother about exact assumptions on regularity of the data.

Let us start by discussing the well known problem of the undamped vibrating string. Let G be the interval (0, l) ⊂ ¹ and let the string be fixed at the points x = 0 and x = l. We assume that the string is vibrating in a plane only, and let u(t, x) be its displacement (cf. Figure 1.1). Furthermore let us assume that we know the initial displacement, say at t = 0, and the initial velocity. Thus we have the boundary condition

Figure 1.1

and the initial condition

But in order to determine u(t, x) for all t ≥ 0, we have to know more about the underlying physics. The kinetic energy is given by

m > 0 being the mass density. For the moment we define the potential energy to be proportional to the alteration of the length of the string

μ > 0 being the factor of proportionality.

Giving U really defines the vibrating string. If we had taken U to be proportional to the curvature, for example, we would have got the problem of the ‘vibrating beam’ leading to the ‘plate equation’, a fourth-order equation, while the problem of the vibrating string is governed by the ‘wave equation’, a second-order equation, as we shall immediately see.

Having defined the energy, we get the displacement u(t, x) by applying Hamilton’s principle, which states that we have to make the functional

stationary. This means that u is characterized by

for all test functions Thus, at this point variational principles are used.

In doing so we arrive at the differential equation

which is nonlinear and difficult to handle.

In order to make things easier, one treats small displacements only, replacing (1 + )¹/² in U by 1 + /2. Thus we redefine the potential energy by

yielding the ‘wave equation’

Therefore our problem of finding the displacement u is reduced to that of solving equations (1.1)–(1.3).

Of course we have to say at first what we really mean by a solution. There are several concepts, classical, weak or even distributional solutions; all of them will be defined later on. For the moment let us look for sufficiently smooth solutions with finite energy

and for simplicity let us furthermore assume

It is then easy to show E = constant via E′(t) = 0, thus proving uniqueness. In order to prove existence, let us use a classical physically motivated approach by first looking for ‘standing waves’, that is for solutions of the form w(t, x) = a(t)v(x) (separation of variables), and afterwards obtaining a solution of the initial–boundary value problem by superposition of standing waves.

Inserting w = av into the wave equation and boundary condition we get (λ is a constant)

Equation (1.5) is solved by

and the boundary condition shows that only

is possible with

We call an ‘eigenvalue’ and vn an ‘eigenfunction’ of the underlying operator. The {vn} form an orthonormal sequence in 2(G), and the corresponding standing waves have the form

Next we want to determine u by superposition,

where the coefficients αn and βn have to be chosen such that

That this is possible can easily be seen by antisymmetrically extending u⁰ and u¹ and developing them into Fourier series. We assume u⁰ and u¹ to be sufficiently smooth such that the series converge.

These ideas may be generalized to similar equations of the form

in bounded domains G ⊂ n with variable coefficients. Although one cannot give eigenvalues and eigenfunctions for A explicitly, it is still possible to show that the eigenfunctions form a complete orthonormal sequence and to give asymptotic estimates for the λn e.g.

These asymptotic estimates go back to H. Weyl (1912) and were originally derived to prove Jean’s radiation formula.

Solutions of the wave equation in unbounded domains are of great interest also. Think, for example, of a signal being reflected by the Earth, or of the Schrödinger equation for an electron in the Coulomb field of a hydrogen-like atom. In this case the underlying operator has a continuous spectrum also and the discussion of the solution is not so easy. To understand what happens let us simplify slightly and treat the corresponding free space problem for the wave equation in ¹.

Thus let us consider

in G = ¹, again with the initial condition (1.2). In this case an explicit solution, the d’Alembert solution, is well known:

Once more we immediately notice that different solution concepts are possible. If u⁰ is not differentiable, for example, then u is not and we do not get a classical solution. But equation (1.7) yields a generalized solution. By the way, the vibrating string problem may be solved via the d’Alembert solution also. One has only to extend u⁰ and u¹ to ¹ antisymmetrically and periodically.

Let us return to the free space problem and introduce the Huygens operator

Then the d’Alembert solution reads

In order to understand how signals move, let us discuss Iv. For (t, x) fixed, (I(t)v)(x) depends on v in [x – t, x + t] only. If v is known in (x0 – r, x0 + r) then we know (I(t)v)(x) in {(t, x) | |x – x0| < r – t} (cf. Figure 1.2).

Figure 1.2

To make it clearer, choose a δ-impulse as initial value, v(x) = δ(x – x0) (the concept of distributional solutions will be introduced in Chapter 5), and let t be positive. Then

Here and in the following

is the Heaviside function. Let (cf. Figure 1.3)

Figure 1.3

be the future cone. Then for t > 0

On the other hand consider the first component of the d’Alembert solution ∂I(t)v/∂t with v(x) = δ(x – x0):

In this case

One says that u satisfies the Huygens principle if for initial values δ(x – x0)

In this case, δ impulses will be transmitted sharply, an observer at x1 (cf. Figure 1.3) receives the signal at t = x1 – x0 and has silence for all other t. We shall see later on that in n, n ≥ 3 and odd, the Huygens principle holds, and for n ≥ 2 and even, it does not. In ¹ we found a mixed behaviour. The first component ∂I(t)v/∂t does but the second I(t)v does not satisfy the Huygens principle.

We are interested in learning more about the behaviour of the solution for large t (positive or negative). To simplify matters we assume u⁰ and u¹ to have finite support and define

Then and for x ≠ 0

Now u± (t, x) are functions of t |x| and x0 = x/|x| alone, say

and by integration we find

Thus for t → + ∞, u behaves like the ‘outgoing’ wave u+, and for t → – ∞, u behaves like the ‘incoming’ wave u–.

This splitting into incoming and outgoing components is essential in the following. Sommerfeld’s radiation condition will be derived from it. If u is outgoing then

Thus for t = 0

Similarly if u is incoming, then

On the other hand these conditions on the initial values give rise to outgoing or incoming solutions.

Treating initial–boundary value problems for exterior domains (domains with bounded complement), we first solve the problem using methods of spectral or semi-group theory. Afterwards we want to discuss the solution, especially its time dependence, and characterise the time-asymptotic behaviour. To do so we first study the corresponding free space problem and subsequently show that for large t the solution of the exterior initial-boundary value problem behaves similarly.

2

Linear operators

In the first two sections of this chapter, we want to explain certain notation and put together some results on linear operators which will be needed later. The reader may consult them when needed. The chapter really starts with the discussion of the Δ operator in the third section; simple boundary value problems are treated afterwards.

2.1 Function spaces

In the introduction we have already noticed the connection between partial differential equations (wave equation) and variational calculus (Hamilton’s principle). Historically, the best known example of this connection probably is Dirichlet’s principle.

Let G be a bounded domain (an open and connected set) in n and g a continuous function on We are looking for the solution of the Dirichlet problem in potential theory, a with

Uniqueness is clear by the maximum principle. Dirichlet’s principle says that one may get a solution by minimizing the Dirichlet integral

for

Let v be a minimum of D(u). Then from

for all test functions ϕ and all , and by using a regularity result, it can easily be seen that v solves the boundary value problem. But it is in no way clear that such a minimum exists.

Dirichlet’s principle was invented in the middle of the last century. The integral, which is bounded from below, was to be minimized and one made the mistake of confusing the concepts of a greatest lower bound and a minimum value. B. Riemann founded some of his famous existence theorems on it and K. Weierstrass gave the first counter-example. Only at the turn of the century was D. Hilbert able to prove the validity of the principle under appropriate restrictions.

A large part of the modern treatment of initial–boundary value problems is based on Hilbert’s methods for solving the Dirichlet problem. For brevity we call them Hilbert space methods. Assuming that the reader is familiar with the concept of Hilbert spaces, the first theorem we cite is the basis for correctly applying Dirichlet’s principle. It gives the connection to variational calculus and can be proved very easily by actually constructing a minimizing sequence and showing that the limit exists.

The approximation theorem Let be a closed subspace of the Hilbert space

Then

There are two corollaries of the approximation theorem:

The projection theorem Let be a closed subspace of the Hilbert space and

Then

The Riesz representation theorem Let be a Hilbert space and F a bounded linear functional on . Then

There is a slight generalization of the Riesz representation theorem using sesquilinear forms instead of the scalar product:

Lax–Milgram’s representation theorem Let be a Hilbert space, c, p > 0 and £(· , ·) a sesquilinear form on with

Let F be a bounded linear functional on . Then

We remark that instead of (ii) it is enough to demand, for example,

Next we want to list some function spaces for G ⊂ n:

, the space of k times continuously differentiable functions.

, the space of Holder continuous functions in G.

functions of having compact support in G.

the space of test functions.

1(G) the space of Lebesgue integrable functions.

2(G) the space of Lebesgue square-integrable functions.

In case G is unbounded, an upper index f means having finite support in n, e.g.

We also use the following norms and semi-norms:

Now we can define

Functions in k(G) have ‘strong’ derivatives up to the kth order. And f 2(G) is called a ‘weak’ derivative of u 2(G), f = ∂iu or briefly ∂iu 2(G), if

k(G) is the subspace of 2(G) of functions having weak derivatives up to the kth order. k(G) is a Hilbert space with the scalar product (u, v)k.

The following theorem going back to T. Kasuga (1957) is essential. A proof can also be found in the paper of N. G. Meyers and J. Serrin (1964).

Theorem 2.1

To indicate let Then and 1 = l if we can show

To do so let . Then

Thus u is a solution of (– Δ + 1)u = 0 and by local regularity (compare Weyl’s lemma which will be formulated at the end of this section) . Therefore

We also notice the following:

Theorem 2.2 Functions u m(G) having compact support in n are dense in m(G).

This theorem is of interest if G is unbounded. A proof may be found in Agmon’s book (1965, p. 10).

Remark 2.3 Some authors define In this case ′k ⊂ k = k and ′k = k only for domains having the ‘segment property’, which will be defined at the end of this section. A proof is given in Agmon (1965, p. 11).

We also use spaces of functions or vector fields having special derivatives, for example

and for G ⊂ ³

where we have used ‘weak’ equals ‘strong’ and

We are specially interested in subspaces of functions satisfying a homogeneous boundary condition. Let us think of the Dirichlet problem and look at

If the boundary and u, F are sufficiently smooth, implies (n is the normal vector on ∂G)

and so u|∂G = 0. Again we define strongly

and have the following theorem.

Theorem 2.4

Let us again indicate a short proof. ⊂ is clear. is a closed subspace of , and so by the projection theorem

Let . Then

yielding Δu