Variational Methods in Mathematical Physics: A Unified Approach (Theoretical and Mathematical Physics)

Apart from the natural assumption about the regularity of the point xo this theorem also contains the "technical" assumption that the kernel K = Ker¢'(xo) of ¢'(xo) : El - E2 has a topological complement in El. In a general situation it is not so easy to check this assumption. We want to show now that this assumption is quite adequate for the general situation of Theorem 4.2.1 by proving that it is automatically satisfied for three large and important classes of special cases:

4.2 Ljusternik's Theorems

69

Case 1: E1 = HI is a Hilbert space. Case 2: E2 is a finite-dimensional Banach space. Case 3: 0: U -+ E2 is a Fredholm mapping. Here, a differentiable mapping ¢ : U y E2 is called a Fredholm mapping of U into E2 if, for all x E U, DO(x) _- ¢'(x) is a Fredholm operator from E1 into E2. By definition, a continuous linear operator A from E1 into E2 is a Fredholm operator if and only if

(i) dim Ker A < oo, (ii) RanA = A(EI) C E2 is closed in E2, (iii) codim Ran A = dim E2/Ran A < oo.

The most important properties of Fredholm operators are discussed and proved in, for example [Ref. 4.4, pp. 107-112].

If Ei is a Hilbert space, then K = Ker0'(xo) is a subspace such that the projection theorem (Appendix A) guarantees the existence of the topological

complement L = K1; in fact, we then have a decomposition of El into an orthogonal direct sum: E1 = K ® K1. If E2 is a finite-dimensional Banach space and 0'(xo) : E1 -, E2 is a surjective continuous linear mapping, then there exist linearly independent vectors e1 , ... , em in El such that {fl , .. , fm}, fi = 0,(xo)(ei), is a basis of E2. Let V denote the m-dimensional subspace of E1 generated by {el , ... , em }. Simple considerations, or a general theorem [Ref. 4.2, p. 107, Theorem 32], show that V has a topological complement in El. (For every x E E1 one has ¢'(xo)(x) = E r A (x)fi E E2; define px E' 1 Ai(x)ei and qx := x - px

and show that p and q have the properties required; we can assume that fi , ... , fm is the standard basis of lR' _- E2.)

Corollary 4.2.2. Let El and E2 be two Banach spaces, U C E1 an open nonempty subset and ¢ : U -* E2 a differentiable mapping. In each of the three cases mentioned above, the tangent space of the level surface &-'(yo) at every regular point xo E 0-'(yo) is given by Tzu#-1(yo) = xo + Ker O'(xo).

The following theorem, due to Ljusternik, gives necessary conditions for the solution of the extreme value problem subject to constraints. It is primarily useful for explicit calculation of the extremal points once their existence has been established, say as a consequence of the results of Chap. 1. The theorem also establishes the famous method of Lagrange multipliers and one should thus regard it as a fundamental theorem.

Theorem 4.2.3. Let E1 and E2 be Banach spaces, U C E1 an open subset, 0:U

E2 a differentiable mapping, and f : U --* IR a differentiable function.

4. Constrained Minimisation Problems (Method of Lagrange Multipliers)

70

Let f have a local extremum at the point xo E U, subject to the constraint O(x) = yo = O(xo) E E2. If xo is a regular point of the mapping 0 and if Ker¢'(xo) has a topological complement in El, then there exists a continuous linear function I : E2 -, IR, such that xo is a critical point of the function

F=f-loo:U-.1R, that is, f'(xo) = Io c'(xo)

Proof. (a) Let L be the topological complement of K = Kee0'(xo) in E1; then H :_ ¢'(xo) ( L is an injective continuous linear mapping of the Banach space L onto the Banach space E2, since xo is a regular point of 0. The inverse mapping theorem (Appendix A) implies that H-1 : E2 -. L is also linear and

continuous. (b) The hypotheses of this theorem allow Theorem 4.2.1 to be applied. We can thus describe the points x of the level surface 0-'(yo) in the neighborhood

V of the point xo E 0-'(yo) by

x=xo+h+yo(h), hEKnBb (where b > 0) is as in the proof of Theorem 4.2.1). Let f have a local minimum

at, say, xo subject to the constraint ¢(x) = go. Then there exists a bo > 0, bo < 6, such that f(xo) <- f(xo + h + p(h)),

Vh E K n B6o.

It follows that

0
.

(*)

(c) If we denote by q the canonical projection from EI onto the topological complement L of K, then by virtue of the equation (*) we can define a continuous linear function f'(xo) : L -, IR by f1)(xo)(gx) = f'(xo)(x),

Vx E E1,

since q(x1) = q(x2) implies x1 - x2 E K and thus by (*) we have f'(xo)(xi) _

f'(xo)(x2)

Thus, by (a), 1

: ?(xo) o H-1 : E2 -. 1R

4.2 Ljusternik's Theorems

71

is a well-defined continuous linear function. If p : E1 -1 K denotes the canonical projection onto K we can write every x E E1 as px + qx and obtain the following chain of equations: 1 o O'(xo)(x) = I o '(xo)(gx) = 1 o H(qx)

= .f(xo)(gx) = f'(xo)(x)

so that 1 0 0'(xo) = f'(xo) follows; thus xo is indeed a critical point of the function F = f - I o 0, and we have proved Theorem 4.2.3. If E2 is an m-dimensional real Banach space, say E2 = 1R'", then every (continuous) linear function 1 : E2 -, IR is characterised uniquely by an mtuple of real numbers (a1 , ... , A,,,) and Theorem 4.2.3 immediately gives the well-known theorem on the existence of Lagrange Multipliers [4.5]. Theorem 4.2.3'. Let U C 1R" be open and f : U -1 1R a differentiable function; furthermore, let ¢ : U -+ IRm be a continuously differentiable mapping. Let the function f attain a local extremum at a regular point x° E U of the mapping 0 (rg0'(xo) = m) subject to the condition O(x) = y° E IRm. Then real numbers Al , ... , A. exist such that

of

ax; (xo) =

L Ai aoj (xo) , i=1

This theorem asserts that the set of extremal points of the function f subject to the constraint O(x) = y° is contained in the set of solutions (A, x°) E 1R' x U of the following system of equations:

Of (xo)= a10W1(x°)+...+Am aim (xo), 1

1

..........................................

af

0)

4901

L90.

yl =01(x ), Y. = Wm(x°). There are exactly n + in equations for n + in unknowns. The theorem on the existence of Lagrange multipliers has, in the form of Theorem 4.2.3', numerous important applications [4.5] and we mention here

4. Constrained Minimisation Problems (Method of Lagrange Multipliers)

72

a simple example which utilises, at least partially, the generality of Theorem 4.2.3.

Let A be a bounded self-adjoint linear operator in a Hilbert space 7{ and xo E 71, llxoll = 1, a minimising point of the function

f(x) = (x, Ax) O(x) = llxll2 = 1. Then Theorem 4.2.3' asserts the following: there exists a A E lit such that

f(x0) = AO'(xo). Now f'(xo)(h) = 2 Re(Axo,h)

and

'(xo)(h) = 2 Re(xo,h),

Vh E H.

This gives

2 Re(Axo, h) = 2A Re(xo, h) ,

Vh E 7{

and thus

Axo = Axo,

or, in other words, xo is an eigenvector of A with the real eigenvalue A. Here we have made use of the fact that every point x E 71, llxll = 1, is a regular point of the mapping ¢(x) = llxll2. This example indicates how one can prove the existence of eigenvalues of certain bounded self-adjoint operators in Hilbert spaces. We shall discuss this in more detail later (Chap. 6) in the proof of the spectral theorem for compact operators.

4.3 Necessary and Sufficient Conditions for Extrema Subject to Constraints Theorem 4.2.3 establishes those conditions necessary for the existence of a local minimum of a function f : U -* IR, subject to the constraint ¢ : U -, E2 where U C El is open, which are most often used in applications. However, it is also important to know of sufficient conditions. These can be obtained relatively easily from Theorem 4.2.1 and its proof. Let xo E Myo = #-'(yo) be a regular point of the mapping 0, for which K = Ker 0'(xo) has a topological complement L in El. According to Theorem 4.2.1 there exists a uniquely determined C' function

xEVCMya a x=xo+h+yo(h), hKf1Bb.

4.3 Necessary and Sufficient Conditions

73

If we now take ¢ti to be a C2 mapping of U into E2, it follows that cp too is a C2 mapping Kfl Bs -* L. We can easily calculate the first and second derivatives of `y from the equation yo = 95(xo + h + co(h)), Vh E K fl B6. Taking into account the fact that cp(O) = 0 and (0) = 0, we get

t L)-' o (D20)(xo), where, clearly, D20(0) must be regarded as a continuous bilinear form of K D2yy(0)

_

in L.

The extremal properties of a C2 function f : U -, lR subject to the constraint O(x) = yo are determined by the corresponding properties of the restriction fyp = f r Myo of f to the level surface Myo of ¢; in other words, in the neighborhood V C Myo of a regular point xo E Myo of ¢, at which K = Ker ¢'(xo) has a topological complement, these properties are determined by the equivalent extremal properties of the function F from K fl B6 into IR defined by F(h) = f(xo + h + Kp(h)).

Now F is a C2 function on the open subset K fl B6 = {h E KI IIhil < 6} of the Banach space K for which we can again use our old criteria, Theorems 3.2.3 and 3.3.1. To this end let us calculate the first and second Frechet derivatives of F at it = 0. It follows from ap(h) = 1(D2
F(h) = f(xo) + Df (xo)(h +y(h)) + 2i D2f(xo)(h +y(h), it + w(h))

+o(h+sp(h),h+yy(h)) = f(xo) + Df(xo)(h) + 1 [D2f(xo) + Df(xo) o

h)

+ o(h,h), and thus

DF(0) = Df(xo) t K, D2F(0) = (D2f(xo) +Df(xo) o D2(p(0)) t K x K. If we now substitute the above expression for D2y2(0) we obtain the following theorem from Theorems 3.2.3 and 3.3.1.

Theorem 4.3.1. Necessary and Sufficient Conditions for Extrema Subject to Constraints. Let E1 and E2 be real Banach spaces, U C E1 an open nonempty subset and 0 : U -* E2 a C2 mapping. For a yo E ¢(U), let the level surface Myo = 0-1(yo) C U of ¢ consist only of regular points x of ¢ at which the kernel Ifs C E1 of Dcb(x) : El - E2 has a topological complement L. in El. Then for every C2 function f : U -, IR the following holds:

(a) if f attains a local minimum at a point xo E Ms. subject to the constraint '4(x) = yo, then the following conditions hold:

(1) Df(xo) tIC = 0, (ii) [D2f(xo) - Df(xo) o (D'4(xo) t Lio)-' o D2'4(xo)](h, h) > 0, Vh E Kso.

74

4. Constrained Minimisation Problems (Method of Lagrange Multipliers)

(b) Conversely, if (i) and (ii) hold and if the bilinear form in (ii) is strictly positive in the sense of condition (ii) in Theorem 3.3.1, then the function f has a local minimum at xo subject to the constant 4'(x) = yo.

Remark .4.3.1. By the proof of Theorem 4.2.3 we know that condition (i) of Theorem 4.3.1 implies the existence of a Lagrange multiplier I : E2 lit satisfying f'(xo) = 1 o ¢'(xo). Using this relation, condition (ii) can be reformulated as (ii') (D2 f(xo) -10 D24'(xo))(h, h) > 0, Vh E K13. Theorems 4.2.3 and 4.3.1 are concerned with extremal points of real functions subject to constraints. The proofs of these results indicate that there is an immediate generalisation to "critical points".

Definition 4.3.2. Let Er, E2 be Banach spaces, U C El be open, 0 : U - E2 a differentiable mapping and f : U -' IR a differentiable function, xo E U is called a critical point of f subject to the constraint 4'(x) = yo E 4'(U) if and only if (i) 4'(xo) = yo and

(ii) xo is a critical point of the function fyo

f 1 4-1(yo) : 4-,(Yo) - 1R.

Then Theorem 4.3.3 follows easily.

Theorem 4.3.3. In the situation described in Definition 4.3.2, a regular point xo of the mapping 0 is a critical point of the function f subject to the constraint 4'(x) = yo if and only if (a) f'(xo) ( Ker 4'(xo) = 0 or (b) f'(xo) = I o O'(xo), where 1: E2 --' 1R is a continuous linear function. It is assumed here that the kernel K of 4"(x) : El -' E2 for x E 0-'(yo) has a topological complement. Proof. In the proof of Theorem 4.3.1 we showed, among other things, that at a regular point x E 4' `(yo) of the mapping 4' one has

fyo(x) = f'(x) I Ker0 (x). Thus condition (a) is necessary and sufficient. Condition (a) follows immediately from condition (b). Conversely, let (a) be true; this condition asserts that

Ker¢'(xo) C Kerf'(xo). We can conclude here, as in the proof of Theorem 4.2.3, that a "Lagrange multiplier" I E £(E2,1R) exists, so that condition (b) results.

4.4 A Special Case

75

4.4 A Special Case The method of Lagrange multipliers can be applied to good effect in the treatment of nonlinear eigenvalue problems (see Chap. 8). In such applications a C' function 0 : E -+ IR on a real Banach space E is given, and one wants to

investigate a C' function H : E -, IR on a level surface Mc = q-1(c), c E R. In particular, one wants to characterise the critical points of the restriction H, of H to Mc. Since in this case the Banach space E2 is simply equal to IR, we can reformulate our previous results in slightly more detail.

Theorem 4.4.1. Let 0 : E -. IR be a C' function on the real Banach space E and let H0 = H r M0 denote the restriction of the C' function H : E -. IR c E ¢(E) C IR, of ¢. Then for each regular point x of 0 there exists a point N(x) = N,(x) E E,

to the level surface MM =

which is not in the kernel K of D¢(x) - 0'(x) E £(E, 1R) = E', such that

(¢'(x),N(x)) = 1. With the help of the function x -a N(x) on the regular points of 0, we can establish the following relations between the differentials of H and H0:

DH0(x) = DH(x) C Ker ¢'(x)

.

(a) If we put eH(x) = (H'(x), N(x))

,

then

IIH'(x) - H(x)m'(x)IIE, < (1 + IIm'(x)IIE' IIN(x)II E)IIDHc(x)IIT=(M.)

.

(b) A regular point x of 0 is a critical point of H, if and only if

H'(x) = EH(xW(x)

Proof. If x E M0 is a regular point of 0, then 0'(x) E G(E,IR) = E' is a nonzero element of the dual space E' of E and thus it uniquely determines a hyperplane Kz = Ker *'(x) in E.

Furthermore, there exists a y(x) E E \ Ifx, such that (0'(x),y(x)) = I. The decomposition of E into a topological direct sum

E=Kz+IRN(x) results if we set N(x) = (id - p(x))y(x), where p(x) denotes the projector from E onto Kim. It follows that

(¢'(x), N(x)) = 1 and N(x) E E \ Ks

.

76

4. Constrained Minimisation Problems (Method of Lagrange Multipliers)

If we now set EH(x) = (H'(x), N(x)), then it follows for all it E E that (H'(x),Q(u)N(x)),

3(u) = (0'(x),u)

and thus (H'(x) - eH(x)O'(x),u) = (H'(x),u - P(u)N(x)) = (H'(x),p(x)u)

= (DH0(x),p(x)u).

Thus, because IIp(x)uIIT(M.) = IIu - Q(u)N(x)II E < (1 + II0'(x)IIE' IIN(x)IIE)IIUIIE we have

IIH'(x)-eH(x)O'(x)IIE' <- IIDHc(x)IIT,(M.)(1+II"'(x)IIE'IIN(x)IIE) which is precisely the assertion

If x is a critical point of H, then by definition DH,(x) = 0, and the relation H'(x) = eH(xW(x) thus follows from (a). Conversely, if this relation holds, then taking into account the fact that ,0(u) = 0, it follows for all it E T(MM) = K. that (DH,(x), u) _ (H'(x), u) _ (H'(x), u) - (H'(x), )3(u)N(x)) _ (H'(x),u) - (eH(x)'G'(x),u) _ (H'(x) - eH (x)O'(x), u) _ (0,u) = 0.

Thus DH,(x) = 0, or in other words x is a critical point of H..

Remark 4.4.1. Part (a) of this theorem will be important when we want to establish sufficient conditions for the so-called Palais-Smale condition (see Chap. 8). The Lagrange multiplier is explicitly determined in part (b).

5. Classical Variational Problems

5.1 General Remarks In many applications of the calculus of variations we have specific information

about the form of the functional f which we want to minimise. In practical terms, this means that the functional f has the form

f(w)=jF(t,w(t),w'(t),...,wl°>(t))dt, wlpl(t)= dep(t), where we have a certain function F : I x IRn}1 --* IR w I - IR, and a compact interval I = [a, b] or, alternatively, an equivalent generalisation to functions w of several variables and to functions yo with values in IRp, p > 1. We do not intend to treat explicitly the most general case in this chapter,

since it is technically rather involved and the crucial arguments can in any case be illustrated by simple special cases which have application in physics. We shall start with some classical examples, through which the calculus of variations evolved and which give some indication of the sort of problem which can typically be solved with the aid of variational calculus.

A. Minimum Surface of Revolution

Let two points in the plane, A and B, be given, with coordinates (a, a') and (b, b'), a c b. The problem of the minimum surface of revolution then consists in finding the curve C given by the equation y = w(t) and passing through A and B, for which the area b

f(w)=2v

w

l+w'Zdt

a

of the surface which results from revolving the curve C about the t-axis is a minimum. B. The Brachystochrone

We have to determine the curve C passing through two points on a plane perpendicular to the earth's surface, such that a point of mass m = 1, travelling

78

5. Classical Variational Problems

from A to B along C under the influence of gravity, reaches B in the shortest possible time. If we denote the coordinates of the points A and B by (0, 0) and (b, b') (where b > 0, b' > 0, say), then the magnitude of the velocity at an arbitrary point (x,q(x)) of C is ds v=dt=

29x O,

if the initial velocity at A is zero. The point mass needs a time n

T(q) = J

ra

ds V

=

1 + g'2(x)

dx

2yx

J

to travel along the curve C from A to B. Mathematically speaking, then, the brachystochrone problem proves to be a minimisation problem for the functional T(q). We can recognise the problem in this form as a special case of Fermat's Principle of Least Time which requires in its general case the minimisation of the functional

T(q) _

b ds JO

v

where v is the velocity of light in the medium being investigated. C. Geodesics

The problem of the geodesics consists in determining the shortest curve C connecting two points A and B on the plane or in space. Somewhat more generally, this problem consists in determining the shortest curve between two given points A and B on a given surface E in IRs, for instance. This case is obviously a variational problem subject to a constraint. D. The Isoperimetric Problem This problem requires to determine a curve C through two given points A = (0, 0) and B = (b, 0) on the plane, such that (i) the length of the curve C has a given value 1, and (ii) the curve C and the line AB enclose a surface with the largest possible area.

5.1 General Remarks

79

In other words, the functional

f(w)=

b

w(t)dt

0

is to be maximised, and at the same time the constraint b

l=J

1}yi2dt 0

must be satisfied. Problems of this kind, where, of two given functionals f, and f2, one has to take on an extreme value and the other a given value, are known as isoperimetric problems. The examples above show that one has to consider functionals of the form

f(w) =11 F(t, w(t), w(t)) dt, N E C'(I), 0 = dw/dt, with given boundary conditions. As we have already

pointed out several times, many problems in variational calculus can be regarded as infinite-dimensional analogues of the extreme value problem for functions of a real variable. However, additional phenomena now appear. Let us consider two examples, the first due to David Hilbert. Let 1

fl(w)=Jt21302(t)dl,

WEC'([O,1]),

w(a)=0, W(1)=1.

The Euler-Lagrange equation is 0,

dt

and its general solution is W(t)

= Ct113 +D.

The curve spo(t) = t1/3 passes through the given points, but it is not, however, in C1([0,1]). The solution of the Euler-Lagrange equation exists in this example; it is even uniquely determined and gives an absolute minimum, but it is not an element of the underlying function space C'([O, 1]). The second example is due to Karl WeierstraB; he used it as an argument against Riemann's explanation of the Dirichlet Principle. Consider the functional /r

f2(W) = J 1 t2tbl(t) dt 0

with the boundary conditions ,p(0) = 0, p(1) = 1. In this case the EulerLagrange equation is d dt (2t2W) = 0 ,

80

5. Classical Variational Problems

and the general solution is ap(t) = C/t + D. However, no curve in this family goes through the given points. Thus, no solution of the Euler-Lagrange equation satisfies the given boundary conditions. No absolutely continuous solution of the variational problem exists at all in this case. For absolutely continuous y where io $ 0, we have f2(c) > 0. The lower limit of the functional f2 has the value zero. To convince oneself of this, it is sufficient to consider the sequence

_ nt, 1 [0,1/n] , °(t)

1,

for then

t E [1/n,1] ,

11/n t2 n2 dt 0

= 3n

holds, and so limn+ f2('n) = 0-

5.2 Hamilton's Principle in Classical Mechanics In classical mechanics we learn that the dynamical behaviour of a physical system S can be described in terms of the Lagrangian

L=LE=T-V (where T is the kinetic energy and V is the potential energy of 5) assigned to the system. For this, we use Hamilton's Principle which states that the actual movement of the system S, starting at time t = a from a point qs and arriving

at time t = b > a at a point qE is such that the action functional W = WE, where

W=J

Ldt,

I=[a,b],

r is stationary. Consequently, determining the time behaviour of a classical me-

chanical system turns out to be a typical variational problem. Hamilton's Principle is also known as the Principle of Least Action, since there are cases in which the functional W is not only stationary, but attains a minimum. We want to investigate the naturally arising question as to whether, and when, the stationary value is a minimum. Many textbooks more or less maintain that the problem of determining the minimum of a functional is solved when all the critical points have been found, i.e. all stationary values of this functional. There is often a physical or geometrical reason for this mathematical nonsense: from the way in which the problem is put, one knows a priori that only (local) minima are possible as stationary points. In this section we want to show which assertions can be obtained in this special case from our general theorems. For the sake of a clear and transparent presentation of the basic arguments, we start with a detailed discussion of the case of mechanical systems with one degree of freedom. The more general case

5.2 Hamilton's Principle in Classical Mechanics

81

with N degrees of freedom can then easily be treated in an analogous fashion, but with more technical effort. We shall clarify later which modifications are necessary for the case N > 1. We shall discuss the problem of minimising action functionals of the form (5.2.1) (and the generalisations to several degrees of freedom and field theory) in great detail, because

(i) in many important applications we are indeed looking for a minimising point of W and not just a critical point, and (ii) the property of being a minimum point is often essential for the interpretation of the solution.

5.2.1 Systems with One Degree of Freedom Let L = L(t, q, q) be the Lagrangian of a physical system with one degree of freedom. To start with, our standard assumption will be only that L is twice continuously differentiable, i.e. L E C2(I X 1R.2). Later, with the aid of the fundamental theorems of variational calculus, we shall obtain further conditions on the Lagrangian which guarantee that the principle of least action can indeed be realised with a particular Lagrangian.

Let I = [a, b] be a compact interval and C' (I) = C'(1, R) be the Banach space of the continuously differentiable functions q I -. lit with norm II4II = 1g11',1= Sup{Iq(t)I, I4(t)I}, =EI

where 4 is the derivative of q. Further, let El (1) = El(I;gs,gE) denote the set of all continuously differentiable curves q E C'(I) from an initial point qs to an endpoint qE, El (1) = {q =C'(I)I4(a) = qs,q(b) = qE}. The elements of E, (I) can be regarded as potential trajectories of the system E. The action function W for the time interval I is then the function

W :E,(I) - R,

Wq=

L(t, q(t), 4(t)) dt ,

(5.2.1)

11

on the subset El(I) of the Banach space C'(I). If we now put E1(I) = {h E C'(I)Ih(a) = h(b) = 0),

then for arbitrary q c E1(I) and for arbitrary h E E°(I) it is true that q + h E Ei (I) too. We can thus investigate the action function's dependence on the trajectories. Now we shall start to discuss those conditions on the Lagrange function L which are necessary for a trajectory q E EL(I) to minimise locally the action

82

5. Classical Variational Problems

function W. Moreover, we include a fairly detailed discussion and proof of the necessary regularity conditions. But we shall not repeat these arguments for systems with several degrees of freedom or for field theories (Sect. 5.5). In both cases, we assume where necessary the required regularity for the calculation, and we hope that the discussion which follows, together with the discussion of regularity in Sect. 9.3, will provide enough information so that the reader will be able to fill in the gaps. If a trajectory q E E1(I) minimises the action functional W locally, then by Theorem 3.2.3, if D2W(q) # 0,

DW(q)(h) = 0, D2W(q)(h, h) > 0,

Vh E E°(I), Vh E EOM.

(5.2.2 a)

(5.2.2b)

Remark 5.2.1. One can show that E1(I) is a differentiable submanifold of the

Banach space C'(I). The tangent space of E1(I) at the point q E E,(I) is q+E°(I) (see Sects. 4.1, 5.2, 5.7). Since q+E°(I) is isomorphic to E10(1), one naturally considers the Frechet derivative of W : E1(I) -,IR at q E E1(I) as a continuous linear mapping DW(q) : E°(I) --* 111. The Frechet derivative of W was calculated in Example 2.2.6. Similarly, using the chain rule, the second Frechet derivative of W can be computed. We then have

[L9(t)h(t)+LQ(t)h(t)]dt

DW(q)(h) =

(5.2.3 a)

fr

and r

D2W(q)(h,h) = f [Lgq(t)h(t)2 +2L94(t)h(t)h(t)+L99(t)h(t)2Jdt, (5.2.3 b) r

where we have used the abbreviations

Lq(t) = 8q (t, q(t), 4(t)) ,92L t,q,4) Lgq(t) 8q2

with similar expressions for Ly, Lqq and L. In order to derive from (5.2.2, 3) conditions on q and L which are necessary for a local minimiser of the action functional W, we need a few basic lemmas.

Lemma 5.2.1 (Lagrange. Fundamental Lemma in the Calculus of Variations). If a continuous function g : I -. 1R, I = [a,b], satisfies the condition

g(t)h(t) dt = 0 JI

for all h E E°(I), then this function vanishes identically: g(t) = 0, Vt E I.

5.2 Hamilton's Principle in Classical Mechanics

83

proof. For the indirect proof we suppose that there is some to E I such that g(to) = 2A > 0. Then, by continuity of g, there is a 60 > 0 such that g(t) > A for all t E I n [to - 60, to + 6o] = I,. Now we construct a C°° function on IR which vanishes outside Il and use our hypothesis to derive a contradiction.

There are nonnegative C°° functions f with support in the interval [-1,+1], for example the function f defined by f(t) = exp[-(1 - t2)-I] for Iti < 1 and f(t) = 0 for Itl > 1. Using the centre point t1 and the width 261 of the interval I,, we can represent the interval ash = [t1 - 61,t1 + 61]. Then, for 0 < e < 61, define a function

hc(t)=Ef(t ti) This is again a nonnegative C°° function with support in [t1 - e, t1 + e] C h. Thus we get the contradiction

0 = f g(t)he(t) dt = f

dt

r

r

>J Ah,(t)dt=AJ f(r)dr>0 and it follows that g(t) < 0 for all t E I. Since our hypothesis is invariant under the transformation g - -g, it follows in the same way that (-g)(t) < 0 and therefore that g(t) = 0 for all t E I.

Lemma 5.2.1' (Du Bois-Reymond). If a continuous function g : I - IR, I = [a, b], satisfies for all h E Er°(I) the condition

g(t)h(t)dt = 0, Jr

then this function is constant: g(t) = g(a) for all t E I. Proof. Fix a nonnegative C°° function f on IR with support in I and normalise

it by f f(t) dt = 1. Then, with every C°° function y on IR with support in I, associate the functions

f

x =,P- M(w)f , M(w) = w(t) dt, r hw(t)= f e XP(T) dr. Since M(x,o) = 0, it follows that h,o has its support in I. The function h,, is easily seen to be C°° on IR and in particular it follows that h,, = xV, and thus by hypotheses

0 = f g(t)h,o(t) dt = f t r

M(w)f(t)] dt =

f

r

w(t)[g(t) - c] dt

84

5. Classical Variational Problems

where c=

f

g(t) f(t) dt

Since this equation holds for all cp E C°°(IR) with support in I, Lemma 5.2.1 implies that g(t) - c = 0 for all t E I, i.e. g = const.

Remark 5.2.2. Notice that we have actually proved stronger statements than those of Lemmas 5.2.1 and 5.2.1' since we have used only the assumption that the hypothesis holds for every C°° function on IR with support in I. These lemmas are used quite often in this form in the theory of distributions on IR, and in the language of this theory they say that (a) if a continuous function vanishes in the sense of distributions, it vanishes as a function; (b) if the derivative, in the sense of distributions, of a continuous function vanishes, then the function is constant.

We now prepare to investigate the implications of condition (5.2.2 b) by proving the following lemma on nonnegative quadratic forms on E°(I).

Lemma 5.2.2. If the quadratic form Q[h] = 1,[F3 (t)h(t)2 + F2(t)h(t)h(t) + F3(t)h(t)2] dt

on E°(I) with continuous functions Fj : I -i IR, I = [a, h], is nonnegative, i.e. Q[h] > 0 for all h E E°(I), then the function F3 is nonnegative: F3(t) > 0 for all t E I. Proof. As in the proof of Lemma 5.2.1 we choose a nonnegative C°° function f with support in [-1, 11 and introduce for 0 < E < co and to E lR the functions

he(t) =

f(tto). l

These are C°° functions on IR with support in [to - e, to + e]. If this interval is contained in I, the change of variables r (t - to)/E gives Q[he] = E J Fr(t0 + ET)f(T)2 dT + J F2(to + ET)f(T)f(T)dT

+ 1I F3(to + ET)f(T)2 dT. E

Suppose that there is a point ti E I such that F3(tr) = -2A < 0. Then by continuity of F3, a point to E I and a number eo > 0 exist such that

F3(t) < -A Vt E [to - E0, to + Eo] C I .

5.2 Hamilton's Principle in Classical Mechanics

85

We evaluate the quadratic form Q on the functions hE constructed above, with this point to, for all 0 < e < co and get

Q[h6] <eJ F1(to+67)f(T)2dT+ J F2(to+e7)f(T)f(7)dr-A J f(t)2dt and therefore Q[he] -, -co for e \ 0. This shows that Q cannot be nonnegative on E°(I) if the function F3 is negative at one point. Thus we conclude.

Theorem 5.2.3 (Euler's Equation and Legendre's Condition). For a Lagrangian L which satisfies the standard assumptions, define the action functional W on E1(I),

W(q) = J L(t, q(t), 4(t)) dt, and assume D2W(q) # 0. If a trajectory q E El (1) is a local minimum of W then the following conditions hold: (a) the function L9L

L4(t) -

194

(t, q(t), 4(t))

is continuously differentiable with derivative dtL4(t)=Lq(t)--

a4(t,q(t),4(t))

(5.2.4)

for all t E I (Euler's equation for a path of stationary action). (b) For all t E I, z

-(t, L44(t) = aqz

q(t), 4(t)? 0

(Legendre's condition).

Proof. We already know that for a local minimum the conditions (5.2.2) are satisfied. Now we introduce the function Lq(7)dT

O(t) =

J

into (5.2.3 a), the equation for the Frechet derivative DW(q). Then a is continuously differentiable and do =Lq. dt Hence integration by parts in 5.2.3) yields

DW(q)(h) = J[L4(t) - o(t)]h(t) dt , r

5. Classical Variational Problems

86

since ohla = 0 for all h E E°(I). Now we use (5.2.2 a) and Lemma 5.2.1' to conclude

L4(t) - o(t) = const. Vt E I and thus the function L4 is continuously differentiable. Euler's equation follows by differentiation. Similarly, we use (5.2.3 b) for the second Frechet derivative and Lemma 5.2.2 with Fr = Lqq, F2 = 2Lq4 and F3 = F44 to deduce Legendre's condition from (5.2.2 b). O

The proof of regularity of a local minimum of the action functional W will follow from part (a) of Theorem 5.2.3 and the following elementary lemma.

Lemma 5.2.4. If the Lagrangian L satisfies our standard assumptions and if q E E1(I) is a trajectory for which L4(t) =

L94

(t, q(t), 4(t))

is continuously differentiable with respect to t, then the function q is twice continuously differentiable at all points t E I where L49(t) # 0. Proof. By continuity of L99 we know that the set J = {t E I I L44(t) # 0} is open in I. For t, t' E J we calculate the difference

4(L4) = L4(t') - L4(t) = L4(t', q(t'), 4(t')) - L4(t, q(t), 4(t))

=J

L44(t,7,4(t'))dT

t

L4(r,q(t'),4(t'))dT+q(t)

4(t')

+

J4(t)

L44(t, q(t), r)) dr

4(t)I1(t') + 4(q)I2(t') + 4(4)I3(t') where

4(t) = t' - t , 4(q) = q(t') - q(t) , 4(4) = 4(t') - 4(t) By continuity of q and 4 and by the continuity assumption for the second derivatives of L, the integrals Ij(t') have limits for t' -t t:

I,(t')- f Lt4(t+s4(t),q(t'),4(t'))ds 1

0

-, Lt4(t,q(t),4(t))-- L24(t),

I2(t') --

f Lg4(t, q(t) + s4(q), 4(t'))ds -t Lq4(t),

I3(t') -

f

r

0

1

L44(t,q(t),4(t)+s4(4))ds

0

One also knows that

-2

L44(t)

5.2 Hamilton's Principle in Classical Mechanics

87

,*) - 4(t), a(t) t'-t and by assumption d

d(tj)

v-t TtL4(t)

The above identity can be written 1A(4)73(t')

a(t)

-

6(L4) _ 11 (t') _ a(q)12(t')

a(t)

a(t)

(*)

and for it E J we know that L94(t) # 0. Hence, for it E J, there is an e > 0 such that for fit' - tj < e, t' E J, I3(t') # 0. For these points t', the equation (*) implies 1

a(t)

a(L4) _ 11 (t') _ a(4)I2(t')

Is (t') l a(t)

a(t)

and according to our statements above, the right-hand side is known to have the limit 1

L44(t)

{ dtLi(t) - Lt4(t) - 4(t)L44(t)}

for t' -, it. Thus, the left-hand side too has such a limit. This proves that q E E3(I) is twice differentiable at t E J and we have the identity

4(t) = Lv4(t) { dtL4(t) - Lt4(t) - 4(t)La4(t)}

(5.2.5)

for the second derivative. The right-hand side of (5.2.5) is a continuous function on J, and hence 4(t) is continuous.

Theorem 5.2.5. If the Lagrangian L satisfies the standard assumptions and if q E Et (I) is a local minimiser of the action functional W associated with L on I such that D2W(q) # 0, then q is twice continuously differentiable at all points it E I for which L49(t) > 0. Proof. By Theorem 5.2.3 we know that, for a local minimiser q of the action functional W, the hypotheses of Lemma 5.2.4 are satisfied. So by this lemma we conclude.

Theorem 5.2.5 shows how, for a local minimiser, one gains one order of differentiability. Hence, this is a result on the improved regularity for a solution

of a minimisation problem. Actually, it is the first result of this kind in the calculus of variations.

In many typical application, the Lagrangian L not only belongs to class

C2, but also to class C3, and satisfies L44 > a > 0. Then for a trajectory q E Ei(I) which is a critical point of the action functional, we know that q actually belongs to the class C2(I). Therefore, the function

88

5. Classical Variational Problems 2

Los(t) = ag a4 (t, q(t), 4(t)) is continuously differentiable. `Integration by parts yields for all h E E°(I) 2LgQ(t)h(t)h(t) dt = Lq (t) dt h(t)2 dt = - ! h(t)2 ddt2 (t) dt . J J This allows the second Frechet derivative of the action functional W to be rewritten in the form

D2W(q)(h,h)=J [(L4q-dtL94)h(t)2+L4h(t)2]dt

(5.2.3 b')

and it is in this form that it is usually studied Remark 5.2.3. According to its derivation, Euler's condition (5.2.4) characterises the fact that a trajectory q E E1(I) is a stationary or critical point of the action functional, but not necessarily a point of extremal or even minimal action. In order to ensure that q is actually a trajectory of minimal action, at least locally, it is necessary that, in addition, Legendre's condition be satisfied. Thus, in order to realise the principle of least action, Euler's condition does not suffice.

Now we come to the analysis of the sufficient conditions, which are considerably more complicated. According to Theorem 3.3.1, the following conditions are sufficient for the curve q in E1 (I) to realise a strict local minimum of the action function W:

DW(q) = 0;

Euler's equation,

D2W(q) is strictly positive on E1(I) x E°(I).

(5.2.6 a) (5.2.6 b)

Our next objective is to translate these general conditions into actual requirements on the Lagrangian L. Legendre tried to show that

L44(t,q(t),4(t)) > 0,

Vt E I,

(5.2.7)

which is a simple and natural strengthening of the necessary condition (b) of Theorem 5.2.3, is also sufficient for obtaining condition (5.2.6 b) as long as Euler's equation (5.2.6 a), is valid. He started by considering a simple rearrangement of D2W(q)(h,h). According to (5.2.3b), for all h E E?(I),

J(h)=D2W(q)(h,h)= J[A(t)h2(t)+B(t)h2(t)]dt I

(5.2.8)

holds, where

A(t) = Lqq(t, q(t), 4(t)) - d Lg4(t, q(t), 4(t))

(5.2.9 a)

5.2 Hamilton's Principle in Classical Mechanics

B(t) = Lgq(t, q(t), ¢(t))

89

(5.2.9 b)

.

Legendre's conjecture was that, given the hypothesis (5.2.7), i.e. B(t) > 0, Vt E I, one can always diagonalise the quadratic form J on E°(I); in other words, J can be written as a sum of integrals over I of the squares of functions. Legendre used a second rearrangement of J to this end: for arbitrary w E C' (I)

and for all hEE°(I) 0=

J

dt [wh2] dt = J [ch2 + 2whh] dt

holds, and therefore

I

J(h) = f {B(t)h2(t) + 2w(t)h(t)h(t) + [A(t) +c(t)]h2(t)} dt

(5.2.8')

too. It follows that, for B(t) > 0 on I,

J(h) =

J

B(t) [ (h +

h2) 2

+ (B) 2 [B(A + w) - w2]] dt.

(5.2.10)

It follows from this representation that J(h) can be expressed as an integral over a product of nonnegative continuous functions as long as Ricatti's differential equation

B(A +

(5.2.11)

w2

has a solution win C' (I). Although this equation always has a local solution, it does not necessarily have a solution on the whole of I as simple examples show. Extra conditions on the functions A and B are needed in order to guarantee that this nonlinear equation is soluble on the whole of I. To this end, let us consider the following simple relationship between solutions w E C'(I) of equation (5.2.11) and solutions u of Euler's equation for

the functional J on E°(1), i.e. - dt [Bu]

+ Au = 0,

(5.2.12)

which do not vanish on I. If w E C'(1) is a solution of (5.2.11), then u( t ) = u(a)exp (

t

dr)

,

u(a) # 0

(5

.

2 13 ) .

J

is a solution of (5.2.12) which does not vanish on I, and conversely, if u is a solution of (5.2.12) which does not vanish on I then u --B u

(5.2.14)

is a solution of (5.2.11). Assuming B E C1(I), equation (5.2.12) is a secondorder linear differential equation whose coefficients are continuous functions

90

5. Classical Variational Problems

on I. Then the differential equation (5.2.12) always has nontrivial solutions on all of I. In other words, the problem of determining the existence of a solution of Ricatti's nonlinear differential equation (5.2.11) defined on the whole of I can be reduced to the problem of determining the existence of a solution, which vanishes nowhere on I, of the (linear) Euler equation (5.2.12) for the functional J(h). We thus obtain a simple sufficient condition (Theorem 5.2.6)

for the strict positivity of J(h). Theorem 5.2.6. Let A be a continuous, and B a continuously differentiable, real function on the interval I = [a, b]. Let

(a) B(t) > 0, Vt E I; (b) the differential equation (5.2.12) have a solution it E C2(I), which vanishes nowhere on I (Jacobi's condition).

Then the functional

J(h) = f [A(t)h2(t) + B(t)h2(t)] dt is strictly positive on E°(I). Proof. Let it E C2(I) be a solution of (5.2.12) which vanishes nowhere on I. Then w = -(u/u)B is a C' solution of (5.2.11) on I. We substitue w into the expression (5.2.10) for J(h) and we obtain

J(h) =

f B(t) [h(t) + B(t) h(t)] I

2

dt > 0

for all h E E°(I), which shows that J is nonnegative on E°(I). Now let J(h) = 0 for one h E E°(1). B(t) > 0 implies that h + (w/B)h = 0, so that It w(r)

h(t) = h(a)exp (-

dt)

.

B(T)

But h E E°(I) requires that h(a) = 0 and thus h - 0. Therefore J is strictly positive on E°(I). Before we start to discuss the converse of Theorem 5.2.6 we want to explain a particular simplification of Jacobi's condition which Jacobi himself proposed.

We shall do this with the aid of a few elementary facts from the theory of ordinary linear second-order differential equations.

Theorem 5.2.7. If the functions A and B satisfy the hypotheses of Theorem 5.2.6, then the following conditions are equivalent:

5.2 Hamilton's Principle in Classical Mechanics

91

(a) A and B satisfy Jacobi's condition on I = [a, b]. (b) The solution uo,1 of the differential equation (5.2.12) for the inital conditions uo,1(a) = 0, 'ho,1(a) = +1, has no zero in (a, b). Proof. If A and B satisfy Jacobi's condition on I, then there exists a solution v of (5.2.12) which has the property

0<8=infv(t). tEl Let ul,o denote the solution of (5.2.12) for the initial conditions ul,o(a) = 1; u1,o(a) = 0. The functions uo 1 and u1,o are then linearly independent and therefore generate the solution space of the differential equation (5.2.12). Thus there exist real numbers a and p such that v = au1,o + Quo,l

It follows that 0 < 6 < v(a) = a. We shall now show that the assumption that uo,1 has a zero at a point c E (a, b] leads to a contradiction. We can assume that c is the smallest of all numbers ry E (a, b] with uo,1(-y) = 0. The differential equation (5.2.12) implies

that the derivative of B[uo,lul,o -'ho,1u1,o] vanishes on I. It follows that

B[uo,lul,o -'ho,lul,o](t) = B[uo,1'h1,o - do,lul,o](a) = -B(a) and thus, since u,,1(c) = 0, that uo,1(c) -

B(a)

1

B(c) u1,o(c)

for t = c. On the other hand, 0 < 6 < v(c) = cru1,o(c), so that u1,o(c) > 0, which means vio,1(c) > 0. But by definition of c as the smallest zero of uo,1,

we know for some e > 0 that uo 1(t) > 0 for t E (a,c) and uo,1(t) < 0 for c < t < c + e. This implies J.o,1(c) < 0 and thus a contradiction.

Conversely, assume that the solution u0,1 of the differential equation (5.2.12) has no zero on (a, b]. Since the solution u1,o is continuous, there exists a c E (a, b) such that u1 o(t) > 1/2, Vt E [a, c]. If we now put -y

inf u1,0(t),

tE[c,b]

6= inf uo,1(t), IE(c,6)

then it follows that 6 > 0. For arbitrary a > 0 put P > la-y/61. It then follows for the solution v = aul,o + /3uo,i of (5.2.12), for t E [a, c], that v(t) = au1,o(t) + /fuo,1(t) > a/2 > 0,

92

5. Classical Variational Problems

and for t E [c, b] one has

v(t) > a# + fib > 0. Thus, v is a solution of the differential equation (5.2.12) which does not vanish on I. This implies that Jacobi's condition holds for the functions A and B on

the interval I. Remark 5.2.4. We should mention here the following notation which is often used. If the solution uo,1 of the differential equation (5.2.12) has a zero at c in (a, b], then c is called a conjugate point of a with respect to the differential equation (5.2.12). Lemma 5.2.7 can thus be formulated as follows.

Lemma 5.2.7'. The following conditions are equivalent: (a) A and B satisfy Jacobi's condition on I = [a, b]. (b) There is no conjugate point of a in (a, b] with respect to the differential equation (5.2.12).

We are now in a position to prove the converse of Theorem 5.2.6. Theorem 5.2.8. Suppose two functions A E C(I,IR) and B E C'(I,IR) where

I = [a, b] satisfy (a) B(t) > 0, Vt E I.

(b) The functional J(h) = ff[A(t)h2(t) + B(t)h2(t)] dt is strictly positive on E°(I), i.e. J(h) > 0, Vh E E°(I), h j4 0. Then these functions satisfy Jacobi's condition on I = [a, b].

Proof. The functional Jo(h) =

J

h2(t) dt

is obviously strictly positive on E°(I) and it follows that the functional JA, A E [0,1], defined by

JA(h) = AJ(h) + (1 - A) Jo(h) =

r

Jf

[Aah2 + BAh2I dt,

AA=AA, Ba-1-A+AB is also strictly positive on I. The corresponding Euler equation is

dt (Bau) + Aau = 0. Since the coefficients of this linear second-order differential equation depend analytically on A, the solution ua of this equation for the initial conditions

5.2 Hamilton's Principle in Classical Mechanics ua(a) = 0,

93

ita(a) = 1

depends differentiably on A. To prove the theorem it is sufficient to show that the solution ul has no zero in (a, b]. We show this indirectly by demonstrating the relationship between the possible zeros of ul and those of the solution no. But the solution no is known explicitly, i.e. it is the solution of the differential equation dt(uo) = 0

for the initial conditions uo(a) = 0, uo(a) _ +1, i.e.

uo(t)=t-a. The solution uo thus has no zeros in (a, b]. Let us assume that c E (a, b] is a zero of u1. The strict positivity of Jl = J

on E°(I) rules out the case c = b, since in this case u, E E°(I), ul # 0, and

J(ul)=f [

dt(Bii')+An,Iuldt=0

holds. It follows that necessarily c E (a, b). Thus the set Io = JA E [0,1] 3t a E (a, b), u a (t a) = 0 }

is not empty. Since ua depends continuously on A, then both A = 1 and a suitable neighborhood of A = 1 are contained in Io, which means that there exists a Ar E (0,1) with [Al, +1] C Io. Now let Ao E [0,1) denote the smallest number for which [Ao,1] C Io is true. If Ao > 0, then the implicit function theorem would produce a contradiction. It is true that uao(to) = 0 for to = t(Ao) E (a, b) and uaa(to) = (duap/di)(to) $ 0, since uao is a nonzero solution of a second-order differential equation. We can therefore solve the implicit equation UA(t) = 0 for t on a suitable neighborhood of (Ao,to) in (0,1) x (a, b) and in this way obtain a continuous function A - t(A) which

satisfies ua(t(A)) = 0. For a suitable S > 0, it follows that [Ao - 6, 1] C Io contradicting our assumption of Ao being the smallest number with this property. Thus, To = [0,1], which means no also has a zero, to = t(0), in (a, b). This is a contradiction, so it follows that ul has no zero in (a, b]. Corollary 5.2.9. Let A E C(I,IR) and B E C'(I,IR) where I = [a,b], and

assume the following properties:

(a) B(t) > 0, Vt E I. (b) J(h) is positive on E°(1), which means J(h) > 0, Vh E E°(I).

5. Classical Variational Problems

94

Then the functions A and B satisfy Jacobi's condition on the open interval (a, b).

Proof. We consider solutions uA, 0 < A < 1, just as in the proof of Theorem 5.2.8. Since Jr = J has now been assumed to be only positive and not strictly positive, we can no longer exclude the possibility that the point t = b is a zero of ul. The assumption that a zero c E (a, b) of u, exists can then be shown to lead to a contradiction, as it did above. In conclusion, let us sum up the results so far. Let L(t, q, 4) be a Lagrangian which satisfies the standard requirements, i.e. L E C2(I X IR2, IR). The action function associated with L for the time interval I = [a, b] is

W(q)=J L(t,q(t),4(t))dt,VgEE,(I). J,

We can calculate the functions 2

A(g)(t) = aqz (t, q(t), 4(t)) - d a-t (t, q(t), 4(t)) , B(q)(t)

= ga4(t,q(t),4(t))

with the aid of a "trajectory" q E E,(I) and the Lagrangian L. We can formulate the following theorem using this notation and these quantities.

Theorem 3.2.10 (Principle of Least Action).

(a) If q E E1(I) is a

trajectory for which the action W attains a local minimum, and if the second Frechet derivative of W does not vanish at the point q, then

(i) aiLQ(t,q(t),4(t))=Lg(t,q(t),4(t)); Euler's equation. (ii) L4 (t, q(t), 4(t)) > 0, Vt E I. (iii) Whenever A(q) E C(I, IR) and B(q) E C1(I, R), these functions satisfy Jacobi's condition on the (semiopen) interval [a, b). (b) Conversely, suppose that a trajectory qo E E,(I) satisfies the following:

(i) ajLy(t,go(t),4o(t)) = Lq(t,go(t),4o(t)), (ii) A(ga) EC(I,g;), B(go) EC'(I,1R),

(iii) B(q")(t) > 0, Vt E I, (iv) A(go) and BOO) satisfy Jacobi's condition on I = [a, b]. Then the action functional W(q) = ft L(t, q, 4) dt has a strict local minimum for the trajectory qo.

5.2 Hamilton's Principle in Classical Mechanics

95

5.2.2 Systems with Several Degrees of Freedom We now want to analyse systems with more than one degree of freedom using essentially the same ideas we have used up to now. The changes which will be necessary are primarily of a technical nature. Again, let C' (I, 1R") = C' (I, IR)" denote, for a time interval I = [a, b], the Banach space of continuously differentiable functions: q : I - 112" with norm

IIgII = IIgil',i = max "sup{I4i(t)I, I4i(t)I} where 4j

= ai

In general, the Lagrangian of a system with n degrees of freedom has the form L = L(t,q(t), 4(t)) and we again make the standard assumption that L = C2(I x lR2n,IR). Then the action function for a fixed starting point qs E IR.' and a fixed endpoint qE E IR", for the time interval I = [a, b], is a function on the subset

El(I)n = {q E C'(I, IR.")Iq(a) = qs, q(b) = qE} of the Banach space C'(I,IR"). The action associated with a trajectory q E El(I)" is

W(q) =

IJ

L(t, q(t), 4(t)) dt .

The first two Frdchet derivatives of this action function can be easily calculated

using Taylor's formula for functions of several variables. This yields for all h E Ei(I)n: E° (I)" = {h E C'(I,IR")Ih(a) = h(b) = 01,

DW(q)(h) = J

D2W(q)(h,h)= r

I i.i=1

, (Lqj

- d_L4,)hfdt,

(5. 2.15 a)

{Lqjgjhihj+2Lq,4jhihl+Lij4jhihi}dt (5.2.15b)

where the functions

Lq; =

a4

,

L4, = a4 ,

02L Lq;q; = 8q aq. ,

a2L Lq:4i = aqi 91

and

52L L4145 ='94>

ia4i

have the argument (t, q(t), 4(t)) while the functions hi and h, are evaluated at the point t.

5. Classical Variational Problems

96

We can write this in a more compact and convenient form by using the following notation. Let L. : I -. lR" denote the vector-valued function on I whose components at the points t E I are just L., (t, q(t), 4(t)), j = 1 , ... , it. Let L4 be defined accordingly. Moreover, let

Lqq : I - M"(lR) denote the matrix-valued function on I whose real coefficients at the points t E I are the real numbers Lq, 97 (t, q(t), q(t)), i, j = 1 , ... , n. Lg4 and L94 are defined accordingly.

Note that the set M"(IR) of all real it x it matrices is in a canonical way a Banach space and can be regarded as the Banach space C(IR", IR") of all (continuous) linear mappings from the real Hilbert space IR" into the Hilbert space R. We denote the scalar product in IR" by (-, .); thus

(x,y)xjyi for x,yEIR". 1=r

The Frechet derivatives of the action function can then be rewritten

DW(q)(h) =

D2W(q)(h,h)=

J

(Lq(t) -

d L4(t), h(t))dt,

(5.2.16 a)

f{(h(t),Lgq(t)h(t))+2(h(t),Lg4(t)h(t)) r (5.2. 16 b) + (h(t), Ls4(t)h(t))} dt -

Taking into account the fact that h E E°(I)" and assuming that Lg4 E C'(I,Ma), we can bring (5.2.16b) into the form we used earlier. First of all we note that O -.lab dt (h(t), L94(t)h(t)) dt

_

I

dt{(h(t), atL94(t)h(t)) + (h(t), L94(t)h(t)) + (h(t), L94(t)h(t))}

.

If Lg4(t) is a symmetric matrix, i.e. (x,Lg4y) = (Lg4x,y), Vx,y E IR", then we have as before

D2W(q)(h, h) _ f{(h(t),A(t)h(t)) + (h(t), B(t)h(t))} dt i

(5.2.17a)

where

A(t) = Al9I(t) = Lqq(t) - dtLg4(t),

(5.2.17 b)

B(t) = B191(t) = L94(t).

(5.2.17 c)

5.2 Hamilton's Principle in Classical Mechanics

97

These equations serve to illustrate the formal analogy between this. case and a system with one degree of freedom. If D2W(q) # 0, then

DW(q)(h) = 0, D2W(q)(h, h) > 0

(5.2.18 a) (5.2.18 b)

necessarily hold for all h E E°(I)" for a trajectory q E El (1)' which minimises the action W. If we now substitute

h(t)=X(t)ei, 7=1,...,n, into (5.2.16a) for DW(q)(h), where {e;};=1 ... n is the canonical basis of IR" and X E E°(1), then, with the aid of Lemma 5.2.1, the condition (5.2.18 a) gives

Lq(t) - dtL4(t) = 0, which means dtL4,(t,q(t),9(t))=Lv,(t,9(t),4(t)),

7=1,...,n.

(5.2.19)

These are the Euler equations (Euler-Lagrange equations) for the action function W.

The analysis of the condition (5.2.18b) is somewhat more complicated than in the one-dimensional case. We shall start with the simple part which leads to Legendre's necessary condition. To this end we need the following (trivial) generalisation of Lemma 5.2.2. Lemma 5.2.11. Let A,, B E C(I, M"). Let the functional

J(h) =

{(h(t), A(t)h(t)) + (h(t), B(t)h(t))} dt JI

be nonnegative on E° (I)", or, in other words, J(h) > 0, Vh E E°(1)". Then

B(t) is a positive definite matrix for all t E I, i.e. (y, B(t)y) > 0, Vy E IR" and Vt E I.

If we apply Lemma 5.2.11 to the functional (5.2.17) then we obtain Legendre's necessary condition for a minimising trajectory q E E1(I)". In other words, the matrix B(t) = L94(t) must be positive definite for all t E I:

(q,L44(t)q) > 0,

Vt E I, Vy E IR".

(5.2.20)

In our investigations of the necessary and sufficient conditions we shall further assume, for the same reason as in the one-dimensional case, the natural strengthening of the Legendre condition (5.2.20), that is, we let B(t) = L44(t) be strictly positive definite Vt E I, so that

(y, B(t)y) = (y,L44(t)y) > 0,

Vt E I, Vy E 1R" \ {0}.

(5.2.21)

5. Classical Variational Problems

98

We investigate on E°(1)" general quadratic functionals of the form

J(h) =

r

11{(h(t),A(t)h(t)) + (h(t), B(t)h(t))} dt

(5.2.22 a)

where

AEC(I,M"), A(t)'=A(t), VtEI,

(5.2.22b)

and

B E C(I, M.),

B(t) > bin

,

b > 0.

(5.2.22 c)

A clarification of the question as to when a functional such as J on E°(1)" is positive definite or strictly positive definite will produce not only necessary but also sufficient conditions (Jacobi's conditions) for trajectories which minimise the action W.

The condition B(t) > bi,,, b > 0, implies that we can define B'1(t) _ (B(t))-1 and B+1/2(t) = (B(t))1/2. Then B-1(t)B(t) = B(t)B-1(t) =

B-112(t)B'12(t)

B112(t)B'12 (1) = B(t).

For an arbitrary function w E C1(I, M") and all h E E°(I)", the identity

O=f d (h(t), w(t)h(t)) dt = Jdt{(h,wh)+(h,wh)+(h,wh)} holds. Thus, here too, we can rewrite the functional (5.2.22 a) in the following way:

J(h)

J dt{(B'/2h+B-'"2wh,B'"2h+B-'/2wh) + (h, [A+th - w`B-lwlh)}

(5.2.23)

Here w' denotes the adjoint matrix of w. The positivity properties of the functional J(h) can again be easily ex-

plained if we are able to write J as an integral over the square of a real function, i.e. if the differential equation

A + L = w `B-' w

(5.2.24)

for the matrix w has a solution in C'(I, M"). Since w'B-'w is symmetric, every solution of (5.2.24) for the symmetric initial condition w(a) = w'(a) is itself symmetric, w(t) = w*(t), Vt E I, since a solution w of the differential equation (5.2.24) satisfies the integral equation t w(t)=w(a)-JA(r)dr+ !(w B `w)(r)dr. °

5.2 Hamilton's Principle in Classical Mechanics

99

As we might expect from our consideration of the one-dimensional case, and as we can indeed show, the existence or nonexistence of a solution w E C' (I, Mn) of the Ricatti-type differential equation (5.2.24) can be characterised with the

aid of the concept of the conjugate point for the Euler equation associated with the functional J(h):

-dt (Bu) + Au = 0,

u E C2(I,1R") .

(5.2.25)

As a first step, let us discuss a characterisation of conjugate points. Let A E C(I, M") and B E C1(I, M") be given, matrix-valued functions with the following properties:

(i) A(t)' = A(t), Vt E I, (ii) B(t)>bIl", b>0, VtEI. The Euler equation (5.2.25) associated with the functional J is often known as the Jacobi equation of the functions A and B. One then refers to a point

c E (a, b) as a conjugate point of a with respect to A and B, as long as a nontrivial solution of the Jacobi equation exists which vanishes at the points a and c. Making these assumptions, the Jacobi equation (5.2.25) has 2n linearly independent solutions on I. As in the one-dimensional case, we distinguish the following solutions. Let vo 1 be the solution of (5.2.25) with

vo(a)=0, vo,r(a)=ej, j=1,...,n,

(5.2.26a)

and let via be the solution of (5.2.25) with

v13,0(a)=ei,

v1,0(a)=0,

j=1,...,n.

(5.2.26b)

Here (e1 , ... , en) denotes the canonical basis of 1R". We now interpret the solutions vo 1 , ... , vo 1 as the columns of an n x n matrix V01 and, analogously, the solutions vi,0 , ... , v1 0 as the columns of an n x n matrix V10. The functions

Vo1 E C2 (I, M") and Vio E C2(I, Mn) are then solutions of the differential equation

-dt(BV)+AV =0, V EC2(I,M") for the initial conditions

Va,(a)=0, Vio(a)=1l

Vm(a)=I",

Vio(a)=0. The choice of the initial conditions guarantees that the solutions (vo 1(t), ,

vi 0(t)}, j = l , ... , n, generate the solution space of the Jacobi equation. For any solution u of the differential equation (5.2.25), we thus know that

100

5. Classical Variational Problems n

u(t) _

[vi,o(t)uj(a) + vo,l(t)uj(a)]

(5.2.27)

j=l

For 2n solutions u1,...,u2n, this relation can be conveniently written as a

matrix equation: vl (o,)...u2n(a)

ul(t)...u2n(t)

uln(a)... vlo(a)

vol(a)... v01(a)

ulo(t)...v10(t)

v01(t)...D01(t))

_ (Vio(a) Vio(t)

ul(a)... U2 n(a) 1(a)...u2n(a)

Vol(a)'\(ul(a)...u2n(a)

Vol(t)

ul(a)...u2n(a)

One now refers to AU1...u z..(t) = det (

ul(t)... u2u(t)

as the Mayer determinant of the solutions ul I... , u2n and

Wu,...ua.(t) =det

(ul(t)...

2n(t)

is known as the Wronskian (or Wronski's determinant) of these solutions. By virtue of the initial conditions, equation (5.2.27) gives the important relation (5.2.28) au' .2-(t)= Dol(t)Wu,..., urn (t) where

Dol(t) = det Vol(t)

This relation is the basis of the characterisation of conjugate points. The following lemma holds:

Lemma 5.2.12. Subject to the assumption above, c E (a, b] is a point conjugate to a with respect to A and B if, and only if, one of the following equivalent conditions is true: (a) Dol(c) = 0.

(b) The Mayer determinant 4n, ... u:.. vanishes at the point c for a system of solutions u1,...,u2n with Wn, ... uzn # 0; in other words, 4n, 0.

Proof. The equivalence of the statements (a) and (b) is justified by equation (5.2.28). If c E [a, b] is a point conjugate to a, then a nontrivial solution u of the Jacobi equation (5.2.25) exists, such that u(a) = u(c) = 0. Also, the nontriviality of u (i.e. it # 0) requires that u(a) # 0. Equation (5.2.27) implies that n u=

uj(a)vol = Vol it(a)

j=l

5.2 Hamilton's Principle in Classical Mechanics

101

and accordingly, since u(a) # 0, one has

0 = u(c) = Vol (c) - d(a)

,

0 = Doj (c) = det Vo l (c).

Thus, (a) follows.

Conversely, if (a) holds, then, because det Voi(c) = 0, the equation V0i(c)A = 0 possesses a nontrivial solution A E IR", A # 0. Put

u(') = VolA=EA1vo, (') i=1

It follows that

u(a)=Vm(a)-A=0-a=0, u(c) = Vai(c) . A = 0,

'h(a)=Vo,(a).A=A#0. is a nontrivial solution of Jacobi's differential equation (5.2.25) which vanishes at a and c. It follows that c is a point conjugate to a. 0 Thus

The following theorem characterises the positivity properties of the functional (5.2.22 a) in terms of conjugate points, assuming that (5.2.22b) and (5.2.22 c) hold.

Theorem 5.2.13.

(a) The functional J defined by (5.2.22 a) on E°(I)" is strictly positive definite if, and only if, no point conjugate to a with respect to A and B exists in (a, b].

(b) If the functional J(h) = ftdt{(h,Ah)+(h,Bh)} is positive definite on E° (I)", then the open interval (a, b) contains no point c conjugate to a with respect to A and B. Proof. Let the functional J be (strictly) positive definite on E°(1)". Then, by contradiction, we want to rule out the existence of a conjugate point. Suppose

the assertion to be false. Then there would exist a point cl conjugate to a with respect to the functions A and B. If c, = b, then there would exist a ul E E°(I)", ul # 0, which was a solution to the Jacobi equation, which implies

J(ui) _ f dt(ui,Au' r

- dt(Bid)) = 0.

If J(h) > 0, Vh E E°(I)", h # 0, then c, = b is not possible as a conjugate point. From this we can assume that c, E (a, b). We are led to a contradiction,

as in the one-dimensional case, by deforming the given functional J = J, continuously into the strictly positive functional Jo(h) = if dt(h,h). if

102

5. Classical Variational Problems

To this end, consider the family of functionals Ja, A E [0,1], on Eo(I)^: r

J,,(h)= 1 {(h,Bah)+(h,AAh)}dt

I

where

Ba=AB+(1-A)ll,,, Ax=AA, and the corresponding family of Jacobi equations

-dt(B,U)+A,U=O,

U

By Lemma 5.2,12, the conjugate points with respect to the functions Aa and Ba can be characterised by means of the solution Ua of these equations for the initial conditions U,(a) = 0,

U(a) = Il.

These solutions depend continuously on A. Thus, the assumption that a conjugate point c1 E (a, b) exists for A = +1 allows us to conclude, as in the proof of Theorem 5.2.8, that a conjugate point co E (a, b) for the functions Ao = 0 and Bo = Iln must also exist for A = 0. But this is a contradiction, since the solution Uo of dt

(BoU) + AoU = -U = 0

for the initial conditions Uo(a) = 0 and Uo(a) = 1n is

Uo(t)=(t-a)l

.

Thus (b) follows, as does the first part of assertion (a). Conversely, let there be no point conjugate to a with respect to A and B in (a, b]. Giving our heuristic observations in connection with equation (5.2.23) and (5.2.24) a more precise form, it is possible to show that the functional J is strictly positive definite. If there is no conjugate point in (a, b], then the solution V = Vor can be inverted on (a, b], i.e.

t -+ V-1(t) = V(t)-' It follows that

is a function in C' ((a, b],

t -. w(t) = -(BVV-')(t) is also an element in C' ((a, b],

(5.2.29)

with derivative

rv = -A-wVV-' = -A+wB-'w; thus

t.+A=wB-'w.

(5.2.24')

5.2 Hamilton's Principle in Classical Mechanics

103

Now we show the symmetry of the solution defined by (5.2.29) on (a, b] of

the differential equation (5.2.24'). Note that for arbitrary solutions U, W of the Jacobi equation (d/dt)(BU) = AU the following relation holds:

U'BW - U*BW = cont.

(5.2.30)

To prove this, one shows that because of the Jacobi equation and the symmetry of A and B, the derivative of

(Ux, BWy) - (Ox, BWy) vanishes for all x, y E ]R". If we now apply (5.2.30) to W = V = Vol = U, it follows that V*BV

- V*BV = (V*BV - V BV)(a) = 0,

so that

V' BV = V' BV , and thus

(-w)" = (BVV-')` = -w. It follows that (w(t)) ' = w(t) ,

Vt E (a, b] .

Thus the function w defined by (5.2.29) is also a solution of the differential equation (5.2.24). The representation (5.2.23) of the functional yields b

J(h) = lim

etie+Ja+e

dt{(h, Ah) + (h, Bh)}

to

= lim

dt{(BI/2h+B-'12wh,B'f2h+B-l/2wh) e_0+ J + (h, [A + w - w*B-'w]h)} Ib

=

dt(B1t2h + B-112wh, B112h + B-'t2wh) > 0,

because, since h E E°(I)u, w(t)h(t) has a limit value for t - a, namely -B(a)h(a). The absence of conjugate points in (a, b] implies therefore that the functional J is positive definite on E°(I)". Now let h E E°(1)" and let J(h) = 0. the formula above, i.e.

J(h) = with 11

J dtlIB1/2h + B-'t2wh112

shows that h must satisfy the differential equation B'"2h +

B-''2wh = 0. Moreover, w(t) is well defined at the point t = b. Since h E E°(I)", it follows that h(b) = h(b) = 0. Since h is a solution of the linear

104

5. Classical Variational Problems

differential equation h + B-'wh = 0, h is the trivial solution, i.e. h = 0. Thus the functional J is also strictly positive on E°(I)". To summarise, we can formulate the following characterisation of a minimising trajectory q E El(I)" of a system with n degrees of freedom.

Theorem 5.2.14 (Principle of Least Action).

(a) Necessary condi-

tions. If q E E1(I)" is a trajectory for which the action W(q) = f1L(t,q(t), 4(t))dt attains a local minimum, and if the second Frechet derivative of W does not vanish at the point q, then the following hold:

(i) Euler's equations d 49L

dt aq (t, q(t), 4(t)) =

a(t, q(t), 4(t)) ,

j =1,

,n, VtEI.

(ii) Legendre's condition. The matrix z

L44(t)= W. (t, q(t), 4 (t)))

a

-l

is positive definite for all t E I. (iii) Jacobi's condition. There is no point conjugate to a in (a, b) with respect to the functions A)9) and B(Q) defined by A(9)(t) = Lqq(t) -

B(9)(t)

d LQ9(t)

= L44(t)

Here we assume that A(9) E C(I, M") and B(q) E C'(1, M").

(b) Sufficient conditions. A trajectory q E El (I)" may satisfy the following conditions.

(i) q is a solution of Euler's equations, i.e.

dt 8q (t,q(t),4(t)) = 8g (t,q(t),4(t)),

Nt E I,

j = 1,...,n.

(ii) For the functions At4) and B(q) defined in terms of q in (a) (iii), AM C

C(I,M") and B() E C'(I,M") hold. (iii) B(9)(t) is strictly positive definite for all t E I. (iv) There is no point conjugate to a in the interval (a, b] with respect to the functions Al9) and B)9).

Then the action W(q) = f1L(t,q(t),4(t))dt has a strict local minimum for the trajectory q.

5.2 Hamilton's Principle in Classical Mechanics

105

5.2.3 An Example from Classical Mechanics For many systems in classical mechanics with n degrees of freedom, the Lagrangian L has the form L(t, q, 4) = 2 (4, M4) - V (q) ,

where the matrix M = M" is symmetric, i.e. Mil = MM;, i, j = 1 , ... , n, and V is a potential energy which we take to be twice continuously differentiable. It follows then that

Lq= -Vq, L4=M4, Lg4 =0,

L44 =M,

L49 = -Vqq.

The corresponding Euler equation is dt (M4) _ -Vq,

which is just Newton's equation. Now we want to discuss exactly when a solution q of this equation minimises the action function

W(q) =

J L(t, q(t), 4(t))dt

for a time interval I = [a, b]. Legendre's necessary condition requires that L44 = M be positive definite, and the strengthened form of Legendre's condition requires in addtition that M be strictly positive definite, or in other words, M > m]l , m > 0. Since M defines the quadratic form of the kinetic energy T of the system, 1

T = 2 (4, MO

these are quite natural requirements for the Lagrangian given above. It is a lot more difficult to satisfy Jacobi's conditions. The functions A = A( and B = B(q) associated with a solution q of Euler's equation are given by A(t) = Lqq = -Vgq(q(t)) and

B(t) = L44 = M in this case. The Jacobi equation associated with A and B, i.e. the Euler equation of the rrfunctional J(h) on E°(I)" where

J(h)=J {(h,Ah)+(h,Bh)}dt=Jl{(hMii)-(h,Vggh)}dt, I

is accordingly,

106

5. Classical Variational Problems

-dt(MU) - Vgq(q(t))U = 0. If the matrix M is time-independent and strictly positive, then this equation can be written U + M-'Vgq(q(t))U = 0. For a general potential, it is not easy to decide whether a nontrivial solution U E C2(IIR") with U(a) = 0 and U(t) # 0, Vt E (a, b], exists for this equation,

i.e. whether no conjugate point with respect to the functions A = M and B = -Vqq exists in (a, b]. We shall thus discuss an example. Let

V(q)=2(q,Kq), K=K`, KEM . It follows that Vgq(q(t)) = K, Vt E I, and one thus has the Jacobi equation

U+M-'KU=0. Let the matrix M-'K E M. be diagonalised by the matrix C E .M,,, i.e.

GM-' KG-' = D -

al

0

...

0

A

0

0

... a0

0

where A5 E IR. We then have for it = GU

h+Dh=0 or h5+a5 hj = 0, j = 1, ... , n. The solutions for the initial conditions h5 (a) = 0 are ht(t) = d5sin(wt(t - a)), or

hf(t) = di sinh(wi(t - a)).

wl = +/, Aj > 0

wt = +/, A5 < 0

where d5 E IR.

Thus, if A5 < 0 for j = 1, ... , n, no point conjugate to a exists in (a, b] and the trajectory q minimises the action W. On the other hand, if A5 > 0 for some j E { 1, 2 , ... , n}, then we have to distinguish two cases.

If (b - a) max(X j E {1,...,n}, A5 > 0) < x, then the corresponding solutions hi have no further zeros in (a, b]. Then (a, b] contains no point which is conjugate to a and in this case the action is a minimum.

However, if (b- a) max(aJ j E {1,...,n), A5 > 0) > qr, then the corresponding solutions uj have a least one zero in (a, b). It follows that solutions it = (hl , ... , exist with

h(a)=h(c)=0, a
5.3 Symmetries and Conservation Laws in Classical Mechanics

107

Remark 5.2.5. Legendre's conditions, i.e.

L44 >0 or L44>0 express a certain partial convexity of the Lagrangian L(t,q(t),4(t)) in the variables 4(t).

Remark 5.2.6. We have formulated all results only for a local minimum of the

action functional W. Similar results also hold mutatis mutandis for a local maximum.

5.3 Symmetries and Conservation Laws in Classical Mechanics 5.3.1 Hamiltonian Formulation of Classical Mechanics In the previous section we discussed in detail the variational approach to the Lagrangian formulation of classical mechanics. However, in physics the socalled Hamiltonian formulation of classical mechanics is also very important, since traditional formulations of quantum mechanics usually start from it. Thus we shall now describe briefly the variational approach to this formulation, which is just as important as the Lagrangian formulation. We start with the Lagrangian L E CZ(I,IR2°) of a system with n > 1 degrees of freedom. The canonical momenta of the system are then well defined by

(t,4,4),

k-1,...,n.

If we assume in addition that

detL44=det

8zL a4j a4k

(t,q,q))

0,

then the equations pk = (8L/04k)(t,q,4) can, with the aid of the implicit

function theorem, be solved at least locally for 4j, j = 1 , ... , ra, 4j = 4j (t, 4, P)

In this case the Hamiltionian H(t, q, p), H(t, p, q)

= {EPk4k - L(t, q, 4)1 k=1

4=4h'9,P)

is obtained from the Lagrangian by means of a Legendre transformation (or contact transformation). The designation Hamiltonian is not quite legitimate,

108

5. Classical Variational Problems

because Lagrange was the first to investigate this function - using the same notation H, moreover! The Legendre transformation takes the Euler-Lagrange equations aL _ d aL

ON

dt apk

into the so-called Ham.iltonian equations of motion

aH

OH

qk=apk, Pk=-aPk, -

aL

OH

_

,

k=1, ..,n,

since, on the one hand, the total differential of H is

dH =

dt+{.OHdgk+rdPk} ll

k=1l 9

P

and, on the other hand, from our definition, n

dH = E {gkdpk +pkdgk} -

aL

aL

dt -

k=1

k=1

l

((

q

aL

dgk + aqk dgk l aqk

{(

dgkJdt, 49L

I

The Hamiltionian equations of motion follow easily, using

aL

d aL

apk

dt aqk =

qk

Instead of n second-order Euler-Lagrange equations in configuration space we now have 2n first-order Hamilton equations in phase space. In the variational approach to these equations we have to investigate the action as a function of trajectories in the phase space P = {(q, p)Iq G IRn,p E }. Let I = [a, b] be a time interval and 1I £1(I) = {(q,P) E Cl(I;IR2n)Iq(a) = gs,q(b) = qE

Eo (I) = {(q, P) E C1(Ii 1R2n)Iq(a) = 0 = q(b)} .

By virtue of L dt = ELI pkdgk - H dt, the action W = fI L dt can also be written as a line integral in phase space ((q, P) E El (1)): W(q,p) =

Jr {EPk@)k@) k=1

- H(t,q(t),P(t)) 1 dt. 111

The Frechet derivative of the function W : £1(I) -* IR is the linear mapping £o (I) -* ]R defined by

5.3 Symmetries and Conservation Laws in Classical Mechanics

DW(q,P)(h 9) = f

/ +Pk(t)hk(t) a9 hk(t)} dt 1H, - hk(t) (MO + a4 11 dt aP

{gk(t) 1 4k(t)

aP

I k_1

E Jr k=1

109

{ gk(I) (4k@)

.

_k

Accordingly, by our previous arguments, the action W being stationary for the trajectory (q,p) E £1(I) is equivalent to this trajectory being a solution of the Hamilton equations.

5.3.2 Coordinate Transformations and Integrals of Motion We start with a short characterisation of the integrals of motion of a system in terms of the Hamiltonian. A function f : lR x P - IR is called an integral of motion if f is constant on all trajectories in phase space. Hamilton's equations hold along a trajectory (q,p) E C1(IR, lR2n). Thus a C'-function f(t, q, p) is an integral of motion if, and only if, 0 = (d/df) f (t, q(t), p(t)) for every trajectory t -. (q(t), p(t)) in phase space. Now

+{ag4i(t)+afmt)}

dtf(t,q(t),p(t))=

=1

Of+E{af aH

of OHl

at

api aqi

1=1

aqi api

f +{H,f},

'

if the Poisson bracket (f, g} of two functions f, g : P -. IR is defined by

i=1

of ag (api

aqi

of ag aqi api

It follows that a C1-function f : IR x P -+ 1R is an integral of motion if, and only if,

+{H,f}=0. The variational formulation of the equations of motion as "points" of sta-

tionary action can be used if we want to investigate the invariance of the equations of motion under certain transformation and thus discover the integrals of motion. The fundamental result here is E. Noether's Theorem [5.1] which we shall now discuss. First of all, let us recall that two Lagrangians L and L' which differ only in a summand of the form dt

G(t, q)

110

5. Classical Variational Problems

lead to action functions W(q) = ft L dt and W'(q) = f L' dt which differ on E'(I)" in an additive constant: from

L'=L+dtG we have, for q E E'(I)" = E'(I; gs,gE)",

W'(q)=W(9)+I dtd G(t,q(t))= W(q)+G(b,gE)-G(a,qs). As we learned earlier, the following relationship exists between the equations of motion

(L]i(q)(t) =

OL

d IL dt 84,

8qi

(t,q(t),4(t)) = 0,

7 = 1,...,n

associated with a Lagrangian L, and the Frechet derivative DW(q) of the action function defined by L: DW(q)(h) = r Y[L]t(q)(t)hj(t)dt, Jtj_1

for all h E Eo (I)". Thus, the Lagrangians L and L' lead to the same equations of motion. We begin our investigations of the invariance of Lagrangians under coordinate transformations by a few remarks about one-parameter groups of transformations in IR". Let (h,),E]. be a one-parameter group of diffeomorphisms h, : 1R" -, IR" of IR", where ho = idpt., = I". Let (s, x) -+ h,(x) be a function in C2(IR x 1R"; 1R"). A mapping

H, : E'(I; qs, qE)" -, E'(I; h,(qs), he(gE))" is induced by h, by virtue of

q-*H,(q)=hsoq. We shall now determine the derivative of this mapping. Let q E E'(I; qs, qE)" and f E E,1(1)n. The differentiability of h, yields for all t E I H=(q + f)(t) = he(q(t) + f(t))

0 h,

= H3(q)(t) + ax (q(t))f(t) + o(f(t)).

It follows that DH,(q) is the linear mapping DH,(q) : E0(1)" - Eo(I)" defined by

(DH3(q)(f))(t) = ax (q(i))f(t)

5.3 Symmetries and Conservation Laws in Classical Mechanics

111

Since h, is a diffeomorphism, it has at all points x E 1R" an invertible Jacobi matrix

A,

Oh,

\invertible

It follows that DH,(q) is an

mapping E,1(1)'

E01(I)' for all

q E E'(I) Let Q, E El(I;h,(4s),h,(gE))" be defined by

Q,(t) = H,(q)(t) = h,(q(t)) for q E El (I;gs,gE)". It follows that

Q,(t) = jt Q,(t) = DHa(4)(4)(t), if we use the notation DH,(q) for the natural extension of DH,(q) to C(I, IR"). Finally, let L be a Lagrangian with standard properties that is "essentially invariant" under (h,),E]R; in other words, let there be a C2-function IR x IR, x 1R" , (s, t, x) -+ G,(t, x) E IR

such that for all q E E'(I; qs, qE)" and all s E 1R

L(t,Q,(t),Q(t))=L(t,4(t),4(t))+dtG,(t,Q,(t)),

tEI.

It then follows that

W(Q,) = W(H,(q)) = f L(t, Q,(t), Q,(t))dt t = W(q) + G,(b, Q,(b)) - G,(a, G,(a)) and thus, for all s E 1R, DW(4) = DW(H,(q))DH,(q).

Since we saw above that DH,(q) is invertible, the invariance of the equations of motion under h,, s E 1R, follows, viz. DW(q) = 0 if, and only if, DW(H,)(q)) = 0 for all s E IR, or, in more detail,

[L]j(4)=0,

j=1,...,n,

if, and only if,

[L]j(Q,)=0,

j=1,...,n, forallsElR.

We shall now show that the one-parameter group (h,),E R allows us, in the case above, to determine an integral of motion for the Lagrangian L. It follows from

dtG,(t,Q2)

112

5. Classical Variational Problems

that 0

=

d_L(t,q,4) aL

dQ, ds

aL

d

+ a4(t,Q., Q,)dQ, -

dd ds

dtG,(t,Q,)

Since G is a C2-function, we can change the order of the derivatives in the last summand. Similarly, from the assumption for )h

d Q,=dtld3 Finally, let us consider the Euler-Lagrange equations for the trajectory q and thus, as we showed above, for the trajectories Q, = H,(q). This gives

o= a4(t,Q,(t),Q,(t))dQ,(t)+ a4(t,Q,(t),Q3(t))dt

dQ,(t)

- dt d G.(t,Q,(t)) -G.(t,Q,(t))}

= t {ag(t,Q,(t),Q,(t))asQ,(t) Thus,

HL(t,q,4)=

aq(t,4,4)dH,(q)-d G.(t,H.(4))} ,=o

is an integral of motion, taking into account the fact that Ho(q) = q. We have thus proved the following theorem.

Theorem 5.3.1 (E.Noether). Let h,

:

lR" -. 1R" s E lit, be a one-

parameter group of diffeomorphisms of IR", where ho = idRn. Let (s, x) h,(x) be a function in C2(IR X IR."; IR."). Let L be a Lagrangian for which we make the standard assumptions. Furthermore, let a C2-function

IRx Rxli"3(s,t,x)-.G3(t,x)EIR exist such that for all q E E'(I; qS, qE)" and for all s E lR,

L(t,Q,(t),Q,(t))=L(t,q(t),4(t))+dtG,(t,Q,(t)),

tEI,

(let Q, =h, o q and Q. be as defined above). Then

H(t,4,9)=

" aL i=r

d

d

a47(t,q,4)dsh,,i(q)-dsG,(t,h,(4))

is an integral of motion for the Lagrangian L.

,=0

5.4 The Brachystochrone Problem

113

Remark 5.3.1. This theorem, which is due to E. Noether, has found many important applications particularly in physics [5.2]. For example, it can be used to show easily [5.3] that 10 integrals of motion exist for the class of Lagrangians of the form

1
t=1

V(Igi-qil)

where V is a smooth two-particle interaction. These are the 10 classical integrals of motion (conservation of energy, centre of gravity, total linear momentum, total angular momentum) which follow from the invariance of this Lagrangian under the ten-parameter Galilei group.

5.4 The Brachystochrone Problem As we mentioned in the introduction the brachystochrone problem is one of the oldest problems in variational calculus. The first solution was given by Johann Bernoulli in 1696; we shall look at his solution briefly at the end of this section. There are other methods of solution put forward by, for example, Jacob Bernoulli, Leibniz and Newton.

The problem is to determine a curve C passing through two points A and B in a plane perpendicular to the earth's surface such that a point mass in = 1 moving along C under the influence of gravity travels from A to B in the shortest possible time. For our variational treatment of the problem we choose a coordinate system in this plane, with the positive x-axis pointing in the direction of the gravitational force. Furthermore, we can take the coordinates of the points A and B in this system to be

A=(0,0), B=(b,b'), b>0, b'>0. The motion of the point mass can be described by the laws of classical mechanics. As we have already seen, the time functional T which is to be minimised is the function T(q) of the "trajectory q E E1([0, b}; 0, b' )" given by /

b (1 + q1(x)2 )

1/2

dx.

The "Lagrangian" for this problem is

L(x,9, 4)=

C1+

2\ 1/2 /11

It does not satisfy our standard requirements. However, since the singularity of L at x = 0 is integrable, it can be shown that the results obtained earlier can be extended to this case. We thus have

114

5. Classical Variational Problems

DT(q)(h)

_

D2T(a)(h, h) =

[2gx(l + gx(x)2)]'/2 h(x)dx, I

h (x)2

J, [2gx(l+q'(x)2)1112 1+q'(x)2

dx,

for all h E Eo(I) and all q E E'(I;0,b'), I = [0,b]. D2T(q) is thus strictly positive definite. Thus every stationary point q of T is a local minimising point (Theorem 3.3.1). A simple calculation shows that the function T is convex on E'(I; 0, b'). A local minimum is therefore exactly that absolute minimum of T(q) which we wanted to determine (Theorem 1.1.3). It is thus sufficient to determine a stationary point of T. The condition

O=DT(q)(h) =

[L

[29x(1+4l(x)2)]112h(x)dx

for all h e Eo(I) requires that d dx

4'(x)

[2gx(l+q'(x)2)11/2

0

for all xE7

and thus that q '(x)

[2gx(1 + q'(x)2)]1/2

It follows that

= k = cont. for all x E I.

g2 +q

q2(x)=2gk2x,

1

0<x
and thus 2gk2b =

4 (b)2 < 1, 1 + q'(b)2 -

so that k2 < 1/2gb. If we put f = 1/4gk2, then from 42(l - 2gk2x) = 2gk2x we get

g2 rr\1

l 2x/

where

2n <1.

27{, Since we are looking for solutions q E E1(I;0,b) we can rule out the case b/2q{ = 1. When b/2 7t < 1, exactly one to E (0,1r) exists such that b = 9{(1 - costo). The points x E [0, b] can then the parametrised:

x=x(t)=N(1-cost), 0
5.4 The Brachystochrort' a r,

lLl

It follows that Y(t)2 = q'(x(t))2712 sine t = 7{21-cost

1}cost

27{

1 - x(t)/27{

7{2 sinz

t

sin2 t = H2(1 -cos t) 2

and thus

y(t) = ±71(1 - cost). Using the initial condition y(O) = 0 we obtain the solution

y(t) = ±7t(t - sint). We have thus obtained the solution to the Euler equation for the brachystochrone problem in parametric form: x(t) = 7{(1 - cost),

y(t) =±71(t-sint), 0
(x)

We can thus determine to from the ratio b' : 6. V

_ to - sin to

1,

1 - costo

0 < to Cr.

(A)

7{ can then be found very easily. The function

sint t- f(t)= t1 -- cos t

is monotone increasing in t E (0, r) and its maximum value is f(a) = a/2. Thus there is exactly one solution of (A) for b'/b < a/2 and thus exactly one solution to the brachystochrone problem. This solution is given by the equations above as part of a cycloid through the points A = (0, 0) and B = (6, 6'). When b'/b > a/2 there is no solution to this problem. In conclusion, we would like to look briefly at the solution to the brachystochrone problem suggested by Johann Bernoulli. We start with a property which is one of the first things we learn in mechanics, namely, that a point mass falling from an initial position A along an arbitrary curve C has at every point P a velocity which is proportional to vT, where h is the vertical distance between A and P. We now divide space into many thin horizontal

116

5. Classical Variational Problems

layers of thickness d. Then we assume that the velocity changes from layer to layer in small steps. The velocity in the first layer is cv, in the second layer it is c 2d and in the nth layer it is cVn-d. The trajectory is supposed to be a straight line within each layer. In order to minimise the total time for crossing all layers, Snell's law of refraction from geometrical optics is applied. This states that sin al sin as sin a sin a2 where a; is the angle between the trajectory and the normal to the boundary between layer i and layer i - 1. Now let us imagine, together with Johann Bernoulli, that the thickness d of the layers gets smaller and smaller. Taking this limit, the conditions stipulated by the law of diffraction still hold. It thus follows that the solution is a curve C with the property that, if a is the angle between the tangent and the vertical at an arbitrary point P of C and if h is the vertical distance between A and P, then (sin a)/vfh- is constant for all points P on C. One can convince oneself very easily that this geometrical property characterises the cycloid. Johan Bernoulli's considerations are in no way a rigorous proof but they are very ingenious and illustrative.

Remark 5.4.1. The problem of the brachystochrone with Coulomb friction has been investigated by N. Ashby et at [5.4]. In this case too, the solution of the Euler-Lagrange equations can be expressed with the aid of elementary functions.

5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory In many cases, the equations of motion of classical mechanics follow from a variational principle, namely Hamilton's principle, which is also known as the principle of (least) stationary action. The Lagrangian L(q, 4) for a system of n > 1 point masses, where q = (qt , ... , qi E ad, i = 1 , ... , n, depends on the position qi and the velocity 4i of each point mass. In this formulation of classical mechanics, the equations of motion are the Euler (-Lagrange) equations for the Lagrangian. These equations make evident the fact that the action functional

W(q)=I L(q,4)dt, I=[a,hl, takes on a stationary value for a solution q of these equations. In hydrodynamics, electrodynamics or in the theory of gravitation we are dealing with systems which possess an infinite number of degrees of freedom. The physical situation is then described by means of a field ¢(x, t), x E IRd, t E IR, or several fields. A field theory is a generalisation of classical mechanics

5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory

117

in which the field variables ¢(x, t) play the role of the dynamic variables q;(t),

1,... , n. The discrete index i, 1 < i < n, now becomes the continuous variable x E lad and F,1 is replaced by fRs dx. In making this analogy we would expect for a classical field theory that the action functional W should be given as an integral over space and time, or, in the language of differential i

forms, as the integral of the (d+l)-form L (the Lagrangian form, here identified with its density with respect to the volume form):

W=J ga+1 G. One of the advantages of the Lagrangian formulation is that it yields a conservation law for every one-parameter group which leaves the Lagrangian density L invariant. This is E. Noether's theorem. If the Lagrangian density C at the point (t, x) depends only on the value of ¢ and a finite number of partial

derivatives D°¢ IaI < p, at that same point, then the Lagrangian density is said to be local. One also speaks of a local, classical field theory. From now on we shall assume that L depends only on 0 and the first derivative DO. As we shall see, this implies that the field equations are partial differential equations of at most second order. We saw earlier (Sect. 5.3) that in obtaining the equations of motion as Euler-Lagrange equations for C, we had a great deal of freedom in choosing the Lagrangian C depending on the underlying space of potential trajectories. We shall have a corresponding freedom in classical local field theory as far as the underlying space of potential "field configurations" is concerned: the Lagrangian densities £(', 0(.), DOO) and

L'(', 0('), DO(.)) = £(, 0O, DO(.)) + div f(-, 0(')) will lead to the same field equations, which will be Euler equations (for C or

v). 5.5.1 Hamilton's Principle in Local Field Theory In this section we shall generalise Hamilton's principle from classical mechanics to classical, local, scalar field theory. Instead of a time interval I = [a, b] C IR

we now take a nonempty open subset .(l C IR", n > 1, with compact closure

Ti and smooth boundary 00 = fl \ £1. Our starting point is to take the "submanifold"

E'(0) = {f E C'(Q;IR) nC°(Q;IR) If [an= g} = E'(0;g) of the Banach space C' (0, IR) as the space of all potential field configurations

whose values on the boundary Ofl are given by a function g : 80 -, IR. As standard hypothesis for the Lagrangian densities, we shall assume

118

5. Classical Variational Problems

E E CZ(dl x IR. x 1R'; lit) .

The corresponding fundamental action W = WC is then well defined on E1(fl) as

W(O) = jn

E(x,

O(x), DO(x))dx,

O E E1(S2) .

(*)

If fo is some fixed element of E'(fl) and if Eo(fl) denotes the subspace

E01(f)={fEC1(fl;lR.)nC°(2;l)If ran= 0) of C1 (fl; lR), then the representation

E1(fl) = fo + Eo (fl) for E1(fl) results.

Thus, we can easily calculate the Frechet derivative of W : E'(fl) IR in a fashion analogous to previous calculations. Let us take G to be a function of the variables x, d and pj = 80/8xj, with LO and Cps denoting the corresponding partial derivatives. The Frechet derivative DW(6) at a point d E E'(fl) is then the continuous linear mapping from Eo(fl) into lR, given by

DW(O)(h)=fn{4h+

t

fn(Emh+V

j=1

Now, it is true that divs(hVpC) = V 1h OpE + h divz vpx , and therefore

V1h-VpCdx= jndiv(hVpE)dxr-

hdivVpCdx 12

=Ian hVpCdo -

)hdivV,Cdx.

n

The condition h tan= 0 implies that DW(&)(h) = 1 [EO - divVpC]hdx n for all ¢ E E'(fl) and for all h E Eo(fl). The vanishing of the first Frechet derivative of W for a function d E El (fl) is thus equivalent to this function satisfying the Euler equation

EO(x, #(x),V (x)) - div V,C(x, O(x),V (x)) = 0,

Vx E S? .

The second Frechet derivative of W(O) follows from Taylor's formula for functions of several variables, i.e.

5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory

119

G(x,¢+h,V¢+Vh)=,C(x,¢,0¢)+G,6h+VpLVh [DOG(h, h) + 2hD#DpEVh + (DpC)(Vh, Vh)] + o(h, Oh).

+ We thus have

DZW(¢)(h, h) =

Jn

dx [(D,G)(h, h) + 2(DODp C)(h, V h) + (DnG)(V h, V h)]

where the argument of D2 C, D,DpG and D2L is (x, ¢(x),V (x)). Now we can again rearrange the second term:

2 J (D,DpG)(h,Vh)=2EJ Gg,hOihdx=EJ dxG,6p,B;(h2) ;=1 n

j=1

Jn dx ;=1 [a'

(h2rmp1)

r

- h2a;Gbp1 ]

= J dxdiv(h2DpD,G) n

f dx h2 divDD#G n

_ - J dx h2 div Dpr, since h E Eo (fl)) . n

This yields r

D2W(O)(h, h) =

Jrn

[(D2G)(Oh, Oh) + (GOO - div D,LO)h2] dx,

D2W(&)(h, h) =

Jn

{(Vh,GppVh) + h(1HH - div Vprp)h} dx

forall0EE'(fl) and all h E Eo'(fl). The second Frechet derivative therefore has the same form as it does for systems with finitely many degrees of freedom, i.e.

J(h)=D2W(¢)(h,h)=I {Ah2+(Oh,BVh)}dx where n

2

A(x) = a2z (x, m(x), V (x)) - E a, aP 'C(x, OW" VOW) =1

B(x)=(Bii(x), 1
.

, (x,fi(x),0O(x))

Using well-known arguments, for positive definite J on Eo (fl), B has to be a positive definite matrix for all x E Si. This is Legendre's condition.

120

5. Classical Variational Problems

Assuming appropriate differentiability properties of the functions involved, and taking into account

(Vh, BVh) = div(hBVh) - h div(BVh) and then GauB-Stokes' theorem, i.e.

Jn

div(hBVh)dx =

r

Jan

hBVh do = 0,

we obtain the following representation for J(h):

J(h) =) h[Ah - div(BV h)]dx . n

The Euler equation associated with the quadratic functional J(h) is therefore

Ah - div(BVh) = 0. This is Jacobi's equation for the problem. We shall assume Legendre's sufficient condition in our further investigations of the necessary and sufficient conditions for a local minimum of the functional

W(M)=foG(x,0,V )dx. This means that the matrix 5z G

B»(x)= ap, ap, (x,d(x),V (x)) constructed for a solution 0 of the Euler equation associated with G is strictly positive definite (for almost all x E 2). For every function w E C'(2,IR") and all h E E0'(.2), the identity

0 =- fan Ow da = f div(h2w)dx = an

n

fn

(2(hwh, w) + h2 div w)dx ,

holds. We can rearrange J(h) using this identity, yielding

J(h)=f (Ah2+(Oh,BVh)+2(Vh,hw)+h2divw)dx n

= fn [IIBrI2vh+B-i"2hw112+(A+divw-(w,B-lw))h2] dx. As soon as a solution Oo E C2(2) of Jacobi's equation

Aq5o - div(BV o) = 0 which has no zeros in 12 exists, the equation

5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory

121

w = -Boo 1Ooo yields a solution for the partial differential equation

A + divw - (w, B-'w) = 0 in C1 (fl, IR"). Indeed, for such a 00,

-A00 = -div(BD¢o) = div(¢ow) = (Voo,w) +0a divw. We thus have

A+divw=It then follows for all h E E01(2) that

J(h) = f9 IIB'"2Vh+B-'/2hw112dx > 0. Now if J(h) = 0 for one h E Eo (2), then this function h satisfies the equations

Vh+B-'hw=0, Vh - h¢o'DOo = 0, or

Vh - hOlogOo =0 in U. The solution of this equation is h = coo. If ¢o is not an element of EE(fl), then J(h) is strictly positive. Remark 5.5.1. Our previous considerations have shown that under appropriate assumptions classical field equations have the form of the Euler-Lagrange equations of a suitable Lagrangian density L, i.e., by virtue of Hamilton's principle, these equations express precisely the fact that the associated local action functional Wc,n (see page 118, equation (*)) is stationary. These considerations also showed exactly how to do this. Globally, that is for ,2 = IR", difficulties generally arise because G(x, ¢(x), Do(x)) is not necessarily integrable over IR". In this case, we talk about the principle of stationary action in the following sense. Let the field equations be given on IR" in the form of the Euler-Lagrange equations C0(x, O(x), DO(x)) - divV C(x, O(x), DO(x)) = 0

of a suitable Lagrangian density G. These equations are then equivalent to the condition that all local action functionals Wc,12(0) =

G(x, O(x), DO(x))dx

fl C lR" (fl as above) be stationary in 0.

,

122

5. Classical Variational Problems

In special cases, one can obtain the field equations as conditions for the globally defined action functional

W() = f C(x,O(x),DO(x))dx nz

to take on stationary values. In this case, the choice of the space of possible field configurations depends sensitively on the Lagrangian density L (see Chap. 9).

5.5.2 Examples of Local Classical Field Theories (a) Nonlinear Elliptic Field Equations

In general, it is not easy to decide when a function ¢ E E'(2) minimises the functional

u

f G(x,0,VO)dx. n

For example, let AX, O(x), V d(x)) = 2 (V O(x), M(x)V $(x)) +VW

where V E C2(IR), and let M E C'(,fl, Bbl"(IR")) be symmetric so that M = M*. It follows that

L,=V'(¢), DpG=Mp,

DpG = M, Giq = V"(0), DpLO = 0. Eider's equation, i.e. the field equation of the local field theory we are considering, is, in this case,

V'(O(x)) - div(M(x)V (x)) = 0. Legendre's condition requires that

M(x)>0, VxE,fl, and Jacobi's equation reads here V"(4,(x))h(x) - div(M(x)h(x)) = 0.

In many applications, the term 2 (VO, MVO) can be interpreted as the kinetic

energy of the field 0. Then M(x) > 0, Vx E l2, seems to be a plausible requirement. But again, in general it is practically impossible to decide when Jacobi's equation for a solution 0 of Euler's equation possesses a solution h which does not vanish in fl, i.e. when conjugate surfaces do not exist. Consider now a special case:

M(x) = ll" ,

Vx E S?,

5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory 2

V(O) _ 2 02,

123

mEIR.

Then Euler's equation is

m20+4¢=0 and Jacobi's equation is

m2h+dh=0.

our general results therefore assert in this case that if Jacobi's equation has a nontrivial solution which does not vanish in ,f2, then the solution of Euler's equation minimises the functional

- 22 02) J R dx 1211v0112 \ (b) Nonlinear Klein-Gordon Equation

Let 12 C W. Consider the Lagrangian density

[(aµ0)2 - rn'20]

in ElR,

,

where )2 = (at0)2 - 1,7.

(aµ0)2 = (at0)2 - (a=,0)2 - (3x20)2 - (O

012

This Lagrangian density gives the following Euler equation (the so-called Klein-Gordon equation): (U + m2)0(t, x) = 0 where

=8,2 -4.

If we want to have a source term j(t,x) in the Klein-Gordon equation we must add a term j4' to the Lagrangian density. For in = 0 therefore, the wave equation results. If the scalar field ¢ interacts with itself (as it does, for example, in anharmonic vibrations of a crystal), then the simplest terms which can be added to G° are proportional to 0' and 04. For example, for the Lagrangian density

G=G°+4¢4+jO we have the Euler equation ( + m2)O(t,x) = A

3

+

j.

In applications to physics the parameter in is interpreted as the mass of a particle.

5. Classical Variational Problems

124

In the case of the Klein-Gordon equation, i.e.

we can integrate the equation of motion immediately with the aid of a Fourier transform. Let 0(t, x)e-'P'dx . 0(t,P) = (2x)-3I2 IR.

Then

-(m2 +P2) , and for given initial values of 0(t, x) and at¢(t,x) at t = 0, the solution is sin (t

0(t,P) = cos (t

O(t, P) =

-

m2 +PZ) 0(O,P) +

.m2 + P2 sin (t

Due to the factor

m2 + p2)

m2 +p2

0(0'P)

tn2 + P2) RO, P) + cos (t m2 + P2) 0(0, P)

m.2 +p2 in the formula for

.

these relations define a

mapping

t -, Mt, .), at(O(t, .)) from IR+ = [0, co) into H1(1R3)$L2(IR3), in other words, a "flow" in H1(IR3)e

L2(IR3). For a definition of the Sobolev space H'(IR3) see Chapter 9 or the Appendices.

5.6 Noether's Theorem in Classical Field Theory We would like to use the variational formulation of the equation of motion to derive conservation laws from invariance properties of the Lagrangian density, just as we did in classical (Lagrangian) mechanics. Noether's theorem is concerned with just this. First of all, we want to explain that the role of the integrals of motion of classical mechanics is taken, in classical local field theory, by conserved charges. If, say, 0 _ (¢r , ... , ¢N) E E1(1R4; IRN) are fields over space-time 1R4 1R x 1R3, then for every C1-function I : IR4 x RN x MY' -> IR4, the C1-function 10 : li4 -* 1R.°, Ii(x) = I(x,4)(x), 0O(x)),

which is associated with ¢, is called a current of 0. We interpret the points x = (xo, x1, x2, x3) of IR4 as time (xo) and space (x = (x1, x2, x3 )) coordinates, with corresponding notation for the components of 10 = (10 1, I1).

Subject to well-known integrability conditions, the fact that the "charge" Qr°(x0) =

IO(x0,x)d3IIi'

5.6 Noether's Theorem in Classical Field Theory

125

is constant with respect to time is equivalent to the fact that the divergence of the current 0 vanishes:

,a

0

=o.

s=a

Thus, associated with every current rm satisfying div 10 = 0, there is, in this sense, a conservation law, namely, the conservation of the "charge" (so E lit arbitrary)

(r#) = Q10 (xo) .

W

In a field theory, these conserved charges play the same role as the integrals of motion in classical mechanics. In this context we speak about a conservation law if a divergence-free current exists (with suitable integrability conditions).

Let us consider again, for a subset fl c It" as above and a function g : 8fl --, IR.N, the submanifold E' (fl, g) = {¢ E C' (fl; IR'v) n CO (fl; IRN) ¢ ( OP = g} of the Banach space C' (.fl; IRN). As before, we associate a local action functional

W=W' :E'(Q,g)-'1R, W(4)=J C(x,`5(x),V (x))d"x n

with a Lagrangian density C E CZ(,2 x IRN x IR'N; IR). The equation of motion for the fields ¢, i.e.

[,C,],(s) = ag (_, O(x), VO(x))-divap. 49 (, O(x), V (s)) = 0, j =1, ... , N, in the variational formulation are just the Euler condition DW() = 0 for W to take on stationary values in ys E E'(fl, g). We shall use the notation

and

aG 8yj (X,y,P) and

= V C(x,y,P).

8L aP. (X, y, P) = Oj. L(x, y, P)

Now let an r-parameter Lie group of diffeomorphisms he of IR" X IRN be given: let e = (El , . . . , e,) be the parameters of this Lie group. It is known that the finite transformations for a Lie group can be constructed from the infinitesimal transformations (in the connected component of the neutral element). It is thus sufficient if we study the effects of the infinitesimal transformations. We shall restrict our considerations to those cases which are very important in physics. We thus assume that the diffeomorphisms hE have the form:

126

5. Classical Variational Problems he(x,y)1e=o =(x,y),

he(x,y) = (0e(x),1Pe(x,y)),

where Vie is a diffeomorphism of IR". Put pp

Sv(x) '=

awe uEy

(x)

which results in the following representation for e --, 0: r 'P e(x) = x + E e (x)e + oi(e;x), v=1

I

y)E,. + 02(E; x, y)

Ye(x, Y) = Y + Y=1

It follows that

Tx- (x)

= Iln + E 5x (x)E, + Vol(E;x) v=1

and thus, after a simple calculation, det

80

e (x) = 1 + div

0i (E; x)

ax-

.

Define for every transformation he a mapping He : El (fl, g) -, E1(121,ye) by means of the following equations:

fle = ye : OHe = Y'e(0Q) -,

IRN,,ye(xe)

= 'Qe(we 1(xe),y(Ye 1(x E))), EI/Ji' 1('nc,9e),

He(0)

- e(Y'e 1(xe),

&(xe) after

It follows

calculation that

aTE(, )aYE' (xe)+aTe(,

nI'

and

a0,(xe)_

axe

elementary

l (xe)))'

ax

ax

=ao(x)+ ax -

1

ay

ax

ax

Ey0

aoe(x,

ax

(x) C` aea(x)Es + o(E;x).

(xe)

=1

ay

ax(

5.6 Noether's Theorem in Classical Field Theory

127

Now we would like to investigate the consequences of the following invariance properties of the Lagrangian density G under the diffeomorphisms he:

a` (x)

L(V,<(x),0.(

= £(x, `5(x), V (x)) + div f(e, x, O(x))

(*)

where f : IR' xaR" x]RN -. IR" is aC2 function such that f(0; ) = 0. In other words, C is invariant with respect to the diffeomorphism we are considering apart from the divergence of some vector field f. We shall discuss the meaning of the invariance property (*) later. Using the formulas above, we calculate the derivatives of the left-hand side of (*)

ae;

[G (Ye(x),#

axe( e(x)))Idet a

=L (x,O(x), aX(x))' -

(det ax (x)'\Ie=o

+- -C(&e(x), We( e(x)), a = &(x) aiv e1(x) +

ax

e=0

I x, m(x), L, (X)) °

+, ay (x, O(x), ax(x))a+E aP (X, Ow, a-x(=)) ae I

j=1

e=0

&'e(x))le=0

(5 (x))le=°

(x) + E

= G"(x)div e, (x) +

a=°

;_,

L9Yj

a©la0

N acs aoj ai; + apj { axe (x, OW) + y` (x, OW) ax (x) - ax (x) ax (x) i-1 N

E

=

O(x)) j=1

+div{ C'(x)6.(x)+E a-(x)ej(x,m(x))-x fill

+N{dive j=1

j=1

-J

j=1

e -j

(x)+c 1(x)gt,(x)diva'(x)}.

(x)}

5. Classical Variational Problems

128

Differentiating the right-hand side of (*) gives where

yf(E,x,O(x))le=o

Thus we have shown that

E[Gd]j(x)&j,(x,O(x)) = div { j=1

E

l 5=1

N

o

-J

g

o f,4x) + MX, OW)

m

O(x)) - G'(x)C,(x) 7

j=1aP ((,

1

(diver ; =1

aP

(x)

"O1i;,,(x)+Ly

90'(x)

ll

))1111

If we now put N

=F j=1

1(x)e,.(x)+f,.(x,O(x))

O' CO

-j

aN

- d(x)&(x) j=1

-aLO --

1

-)

for v = 1,...,r, then the equation (*) asserts that r linearly independent

equations N

ax (x)

j=1

,,(x)]=divIt(x),

v=1,...,r,

result, where Bi is a matrix of maximal rank. In this way, we obtain, for solutions 0 of the field equations [G0];(x) = 0, v = 1 , ... , r, and j = 1 , ... , N, r linearly independent conserved currents thus, subject to the known integrability conditions, r independent conserved charges Q(Im), v = 1 , ... , r. We shall start from a variational form of the equations of motion in order to interpret the invariance properties (*) of the Lagrangian density in terms of the transformation properties of the associated action functional. We define an action functional on El(.fQE,g6) by WE(0)

£(x"#(xe), aO(xe))d"xe,

E E'(J2E,ge).

Then We o H, is an action functional on El (S?, g). The invariance condition (*) for the Lagrangian density L implies for W. o HE that

5.6 Noether's Theorem in Classical Field Theory W. o H£ (O) = W (O) + L div f (e, x,

O(x))da

x

R

= W(O) +

JaR

129

f(e,x,g(x))da,

O E ES(J?,g),

and thus

DW(¢) = D(W£ o H£)(0) , and this is, according to Hamilton's principle, just the transformation property of the equation of motion. - Then D(W oH£) can be determined by means of an elementary but tedious calculation. The result is a tranformation formula for the equations of motion: [GO]; _ E [Gh`]j(x) j=1

agi

(x, O(x)) + E div

ap;(x,O(x))

8y;

P-i

j=1

- J£(x) j=1 =1 { a-pl (x) µ=1 X

a

ay (x) + Y

8xv

(x, O(x))

( aim. (x))

> as

µ=1

1`

` pj

I (x)

1

9Lm

aJe(x)

µ=1

ayi

£

ap

j=1 =1

t

ayi

where we have used the abbreviations

[Ed.

1j(X) = (DI - div aP ) (0=(x),

JC(x)= det ac(x)l,

(x)), aI£ (0 (x))) ax(0£(x))-

I£(x)

When J£(x) = 1 and Ir(x) = a (e)6µ the equations of motion are invariant, since then

[rd]i(x) =

a

` (x, O(x))

j=1

and the matrix (&¢/&y)(x, y) is invertible for all x, y. Thus, it follows that

[C ]i(x)=0, i=1,..., N, if, and only if, [D"]&) = 0, j = 1 , ... , N. Let us summarise the above.

Theorem 5.6.1 (E. Noether). Let C E C2(IR" IRN IR^N) be a Lagrangian density which possesses the transformation property (*) under an r-parameter Lie group of diffeornorphisms h£. Then there are r linearly independent combinations of the Baler expressions [GO]j, j = 1, ... , N, every one of which is the divergence of a current It of ¢:

5. Classical Variational Problems

130

.

003

N

ax

i=1

1

div It(x),

v = 1,...,r.

N

_

E L,, (x) a' j=1 a8i N

- > j=1

(x)

00' ap

(x)9j (x, fi(x)).

-]

The transformation of the equations of motion of the field 0 under the difeomorphisms h, is determined by the formula (A).

Remark 5.6.1. As we have emphasised several times, it is possible to obtain from every conserved current J a quantity QJ(xo) = f Jo(xo,x)d3x which is constant in time. This corollary of Noether's theorem is used in many physical applications. See [5.3, 5-8] for explicit applications of Noether's theorem to field theory.

5.7 The Principle of Symmetric Criticality In many areas, including mathematics and physics, it has proved extremely useful to look for symmetries and to exploit them, if they exist, in problem solving. The success of this procedure is based on the "principle of symmetric criticality". In order to illustrate the idea of this principle, let us describe a typical situation where it is used (usually implicitly). Suppose a classical field theory, on ]R3 let us say, is described in terms of a rotationally invariant action functional W(O) of the type considered in Sect. 5.5 and one looks for rotationally symmetric critical points of W, e.g. one looks for field configurations 0 which are invariant under the action of the rotation group SO(3) and which are critical points of W. We take as an "ansatz" that the field components are functions of the Euclidean norm r = Ix I of the points x E IR3 and then we compute the variation of W with respect to all fields that also satisfy this ansatz, for instance along the lines of Sect. 5.2. Setting the first variation of W, calculated in this way, equal to zero yields a system of ordinary differential equations for the field components, which can be solved explicitly in some cases. Now this system is used as a necessary and sufficient condition for a rotationally symmetric field configuration to be a critical point of W. Obviously this system represents a necessary condition for a field configuration to be a critical point of W under these circumstances. But the converse is a claim that needs proof. Actually, it can be proved that the above conditions are also sufficient; this then means that a critical rotationally symmetric field configuration is a rotationally symmetric critical field configuration for W. This last statement is just the statement of the principle

5.7 The Principle of Symmetric Criticality

131

of symmetric criticality, in the context of this example. A suggestive formulation of this principle now is that "any critical symmetric point is a symmetric critical point". Unfortunately, the principle is not valid in this general form, as we shall see later. The principle has been used in many applications of the calculus of variations, in particular in theoretical physics, without being particularly noticed. An early, typical example of the implicit use of this principle can be found in Weyl's derivation of the Schwarzschild solution of the Einstein field equations [Ref. 5.9, p. 252 or Ref. 5.10, p.165]. About fifteen years ago this principle was made reference to explicitely by Coleman [Ref. 5.11., App. 4]. However, it was not until 1979 that R. S. Palais gave a precise formulation of it and a proof of its validity in fairly general circumstances [5.12]. We discuss here a simplified version of his presentation.

The example given leads by abstraction to the following mathematical framework in which the above heuristic discussion will attain a precise meaning (the notions used here are made more precise later on). Let M be a smooth manifold on which a group G acts by diffeomorphisms, and suppose F : M -, IR is a G-invariant smooth function on M, e.g. F(g-p) _ F(p), Vg E G, Vp E M. The set of symmetric points is, by definition,

E=E(G,M)={pEMI g-p=p,Vg EG}.

(5.7.1)

The restriction of F to E is denoted by FE = F I E. The set of critical points of F (or FE) is K(F) = {p E MI DpF = 0} (5.7.2) (or K(FE) = {p E EI DpFE = 0}). The principle of symmetric criticality then claims that

K(FE) C E fl K(F),

(P)

at least under appropriate hypotheses on the triple (M, G, F). Naturally, in order to be able to speak about critical symmetric points, i.e. critical points of FE, E has to be a differentiable manifold, and this is assured by ensuring that E is a smooth submanifold of M. Thus a first hypothesis

on (M, G) has to ensure that the set E of symmetric points is a smooth submanifold of M. If this is the case, the tangent space TTE of E at p E E is welldefined and

the tangent space TTM of M at p E E naturally splits into TpS and a part TP E transversal to TpS:

TM = TS + TS.

(5.7.3)

Furthermore, it follows that the differential DpFE of FE at a point p E E is given by

DPFE = DpF [ TpS.

(5.7.4)

132

5. Classical Variational Problems

Thus we see immediately:

(i) Any symmetric critical point of F, e.g. any point of EfnK(F), is a critical symmetric point; (ii) In order for (P) to hold, the implication

DrF (TpE=O=DPF [Tp E=0

(5.7.5)

has to be realised, e.g. if DyF(v) vanishes for all directions v parallel to E then it also vanishes for all directions v transverse to E. Because of (i) we could also formulate (P) as

K(FE)=I'nK(F).

(P')

Before we derive some general results, let us discuss two classes of examples. Within the first very simple class there are examples for which (P) does not hold. The second class is a particular case of that discussed at the beginning of this section. It represents an illustration of a typical and quite effective application of (P). Example 5.7.1. A Class of Counterexamples. Let the additive group 1R act on

M=IRZby &r(x, y) = (x + ykt, y)

for all p = (x, y) E M and all t E IR for some fixed k E IN. Obviously this is a smooth action of IR on 1R2 which is linear in the case k = 1. The set of symmetric points of G = 10, It E IR} is the subspace

E=E(G,M)=(p=(x,0)IxcJR) and a smooth function F : 1R2 - IR is G-invariant if, and only if, there is a smooth function f : IR -, 1R such that F(x,y) = f(y)

V(x,y) E 1112.

It follows that FE(x,O) = f(0), and thus DpFE = 0 for all p = (x,0) E S. e.g. K(FE) = E. On the other hand we have K(F) = {p = (x, y) E IR2 I DyF = 0} = {(x, y) E 1112 I f'(y) = 0}

and therefore

E if f'(0) = 0. The cases of functions f with f'(0) # 0 thus provide counterexamples to (P). Some less trivial counterexamples are discussed in [5.12].

5.7 The Principle of Symmetric Criticality

133

Example 5.7.2. A Class of Examples for (P) to Hold. For 0 < rt < r2 < 03 write R = {x E IR" I rl < IxI < r2) where IxI is the Euclidean norm of x E IR". The boundaries of the set R are the spheres Sl = {x E IR" I Ix I = r;}, i = 1, 2. The rotation group G = SO(n) acts linearly on the Banach space

C2(R) _ {u : R -, IR I hull = sup{ID"u(x)i Ix E R, IaI < 2} < oo} by

(9-u)(x)=u(g-rx)for all xER,9EGand uEC2(R).

The subset of functions in C2(R) having prescribed constant values Cr, C2 E IR on the boundary of R is a smooth (Banach) manifold, denoted by M = M(C1):

M={uEC2(R)Iu rS;=C;, i=1,2}.

The tangent space of M at it E M is

rSi=0, i

1, 21.

Now we want to give a convenient description of the set E = E(G, M) of symmetric points of M under the above action of the rotation group G. To this end consider the Banach space C2([ri,r2]) of twice continuously differentiable real functions on the interval [rr, r2] with norm

IIno1Io=sup{Iupjl(r)l Irr
The Banach space C2([rj,r2]) is mapped onto the subspace of G-invariant elements of C2(R) by the map

r: C2([ri,r27) ti C2(R),

(ruo)(x)= uo(Ixl),

x E R,

no E

C2([rl,r2])

Thus we obviously have

r5o9E={uEM19-u=u, VgEG}. Conversely it can be shown easily that for every G-invariant it E C2(R) there exists a no E C2([rr,r2]) such that it = ruo [5.7, p.5]. Define uo(r), r E [rr,r2], simply by uo(r) = u(re) where e is any fixed unit vector in IR". By invariance we then have ruo = it. It follows that

E=TEO. The tangent space of Eo in no E Eo is T"QEo = {vo E C2([ri, r2]) 1 vo(rr) = vo(r2) = 0)

Now it is not hard to show that E is a smooth submanifold of M and that the following relation for the tangent spaces holds. If it = ruo, uo E Eo, then

5. Classical Variational Problems

134

r(T,.0Eo)

= {v = rvo I vo E C2(Qri,r2)), vo(ri) = vo(r2) = 0}

.

As G-invariant function F on M we consider the Dirichlet functional

F(u)=2 J

(x)I2dx, uEM.

1

R

TM -. IR, is given by

Its differential in it E M,

D,.F(v) =

JR

Du(x) . Ov(x) dx, v E TTM

and therefore a critical point u E Mrof F is characterised by

0= for all v E

R

Du - Vvdx=J Y -(v2u)dx-J vaudx R

R

By GauB' theorem we have

V.(vVu)dx=J IR v

v

for all v E C2(R), v ( OR = 0, implies Au(x) = 0 for

0

xER. A harmonic function it on R is by definition an element it E C2(R) such that An = 0. Therefore

K(F) = {u E H(R) I u r Si = Ci, i = 1, 2} if H(R) denotes the set of harmonic functions on R. Now for it E E we have it = ruo with no E Eo and thus

Vu(x)=uo(IXDI1 X This allows us to give a simplified expression for

FE=F(E by using polar coordinates:

r FE(u) = 2 JR

/'r2

IM(ruo)(x)12 dx = W. J

vo(r)2rn-1 dr = Fo(uo)

where w denotes the volume of the unit sphere in lR°, e.g. as E = rEo, it follows that FE or = FO : Eo -, R. A simple calculation implies for it = ruo, no E ED, and all v = rvo, vo E that

5.7 The Principle of Symmetric Criticality

135

(D.FE)(v) = (D..Fo)(vo) = wn J r2 uo(r)vo(r)r"-' dr. r,

Therefore,

K(FE) = {u E E I D"FE = 0) = r{uo E Eo I D"aFo = 0} = rK(Fo) if K(Fo) denotes the set of critical points of Fo : Eo -t IR. But K(Fo) is easily determined: ry

no E K(Fo) G 0 =

uo(r)vo(r)ru-r dr

J

_- J

a {r"-huo(r)})vo(r)dr

for all vo E C2 ([rl,r2]), vo(rl) = vo(r2) = 0 i.e., uo E K(Fo)

d .

dr

{r"-'uo(r)) = 0,

and this ordinary differential equation is easily solved:

n = 2:uo(r)=a+blogr, n>3:uo(r)=a+br2-n, where the constants a and b are fixed by the boundary values of uo : uo(r;) _ Obviously we have in both cases "

Qruoairuo=0, =r

i.e. the function nuo determined by the ordinary differential equation above and the boundary values is harmonic. Therefore,

K(FE) = rK(Fo) C E n K(F) and (P) holds in this case. Suppose conversely that (P) holds (for instance by general arguments such as Corollary 7.5.3). Then we know

K(FE) = E n K(F) = E n H(R). As we have seen above K(FE) is easily determined by solving a simple ordinary differential equation with prescribed boundary values. Therefore (P) determines all rotationally invariant harmonic functions in R in a simple way! (Let the boundary values C; vary.) We now present some general results. These results rely on simplifying assumptions on the pair (M, G). Suppose E is a real Banach space on which

136

5. Classical Variational Problems

a group G is represented by continuous linear maps T. : E - E; i.e. T : G .C(E,E) satisfies Ta. Tg2 = Tg, 92 for all g; E G. Then the set of symmetric points

E= {xCEI T9x=xVgEG) is a closed subspace of E. Now if F E C1(E, IR) is a G-invariant function; the equation F(T9x) = F(x)

Vg E G Vx E E

implies by the chain rule and the linearity of the action of G that

oT9 = DxF Vg E G Vx E E and in particular, for x E E,

DxF is a G-invariant continuous linear functional on E:

I-T9=1, VgEG}. As usual we denote by E° the annihilator of the subspace E in E':

E° = (I E E'l I C E = 0). Thus ge get

K(FE)={xEEIDzF11=(D,F) f E=0} and a first simple but quite effective result follows.

Theorem 5.7.1. Let a group G be represented by continuous linear maps on a Banach space E and use the notation introduced above. Then

z' fl E° = {0} implies (P) for all G-invariant functions F E C' (E, IR).

Proof. E n K(F) = {x E E I

0}. By the above expression for K(FE)

the statement follows.

Corollary 5.7.2. If E = N is a Hilbert space and g H T9 a unitary representation of a group G on N-i then (P) holds for all G-invariant functions F E C' (9-t, IR).

5.7 The Principle of Symmetric Criticality

137

proof. If (- , -) denotes the scalar product of the Hilbert space 71, an anti-

isomorphism J of N onto its topological dual space 7{' is defined by y H (y) = IV E *H': 1y(x) :_ (y, x)

b'x E 7{.

Under this isomorphism J the closed subspace E of symmetric points is mapped onto the subspace E' of G-invariant functionals:

E' _ .r(E) C 't' as Ty is unitary, and the orthogonal complement

El= {y E7{I(y,x)=0HxEE} of E is mapped onto the annihilator of E in 7{':

E° =

J(EJ-).

But in a Hilbert space we always have z n E1 = {0},

and thus E' n E° = {0} follows and the corollary is proved by Theorem 5.7.1.

Corollary 5.7.3. If a compact group G is represented on a Banach space E by continuous linear maps T. : E -. E and if this representation is strongly continuous (i.e. if g Tyx is a continuous map G -. E for each x E E) then (P) holds for all G-invariant functions F E C'(E,IR).

Proof. Introduce the spaces E, E' and E° as above and suppose that there

is an I E E' n E°, I # 0. Then there is an xo E E with 1(xo) = 1. The compact group G has a normalized Haar measure p which is invariant [5.131. By assumption, g - Tyx° is a continuous function on the compact space G; thus this function is integrable: fG Tyxo dp(g) = To E E.

Also, I E E' C E' implies

1(yo)=Il(T9x°)dp(g)=f cI(xo)dp(g)=1, c

and the invariance of p implies for all g' E G: Ty,To = JGTV,TVxo dp(g) =

IcTy'ogxo

dp(g)

_ o,

5. Classical Variational Problems

138

i.e. To E E. But for ! E E°, E C Ker 1. This contradiction proves E'nE° = {0} and again Theorem 5.7.1 proves this corollary. Remark 5.7.3. There is another approach to Corollaries 5.7.2 and 5.7.3 which is a little less direct but which shows their common basis. This basis is the following observation: If the Banach space E and the group G satisfy the hypothesis of Theorem

5.7.1 and if every closed G-invariant hyperplane H in E, 0 E H, has a Ginvariant complemented subspace, then

E'n2°={o} follows.

The proof of this relies on the fact that each such hyperplane H is the kernel of an element I E E', H = Ker 1, and conversely. The corollaries then follow by proving the hypotheses of the above observation. In actual application one quite often has to deal with a G-invariant smooth function F on the smooth manifold M modelled on a Banach space E instead of a function on the Banach space itself. Without explaining all necessary details about Banach-manifolds [5.14] we indicate the basic ideas about how to reduce this more general case to the simpler situation discussed above. The basic fact about a smooth manifold M modelled on a Banach space E which we have to use here is the following. For every p E M there is an open set U of M containing p and a diffeomorphism W from U onto an open neighborhood y(U) C E such that w(p) = 0 E E. The pair (U, gyp) is called a coordinate system or a chart at p E M. It follows that the differential Dpco of W at p is an isomorphism from the tangent space TM of M in p onto the Banach space E. Suppose now that a (Lie) group G acts smoothly on M, i.e. for each g E G

PF gop is a smooth map M -. M of class C', let us say. Then the differential Dpg of this map in p E M is a continuous linear map of the tangent space TTM of M in p into the tangent space T9opM of M in g op. The chain rule implies Dp(gr92) = (D920p9r) o Dp92

Thus for a fixed symmetric point

pEE={p'EM19op =P, VgEG} we get that g - Dpg is a representation of G by continuous linear maps TyM -, TyM and thus g1

; Dpp o Dpg o Dpp-1

is a representation of G by continuous linear maps E --, E.

5.7 The Principle of Symmetric Criticality

139

The basic definition now is the following. The action of G on M is called linearisable at p E M if, and only if, there is a chart (U, (p) at p such that the action looks linear:

A

V

r'P(U)=wo9ow-1.

9

9EG §EL(E,E)

As usual, L(E, E) denotes the space of linear maps E -> E. A linear map on E is uniquely determined by its values on an open neighborhood of 0 E E. Thus it follows that

(i) there is at most one such g E L(E, E), (ii) g is continuous, (iii) g 1--1 g' is a representation of G by elements of G(E, E), the space of

continuous linear maps E - E. Differentiation at 0 E E implies

Do (wogow-')=D9opwo DgoDow-1 = Ds.pw o Dpg o

(Dpw)-'

and thus, for p E E,

DpwoDp9o(Dpw)-' _:T9 and g T9 E G(E, E) is a representation of G. For p E E we also introduce the G-invariant closed subspace E. of E by

Ep=Ep(G,E)={xEEI Tgx=x, VgEG}. For p' E U n Z it follows that

T"'P(P)=wogow-'(w(P))=w(P) and conversely, for x E Ep n w(U), we get for p' = w-'(x):

9oP =w-1 owogow-'(x)=w-1o7'yx=w '(x) =P This calculation shows that

w:UnE-1 p(U)nEp. Thus, in a suitable neighborhood U of a symmetric point p E Z, the symmetric points are mapped by yo onto the symmetric points in yy(U) C E with respect to the action T9 of G on E. As Ep is a closed subspace of E it follows that E is a smooth submanifold of M and that the tangent space of E in p is given by TpE = (Dpw) r Ep

5. Classical Variational Problems

140

Therefore, locally we are back to the situation where a group G acts on a Banach space E, i.e. in the situation of Theorem 5.7.1. We proceed accordingly and introduce

Ep={l EE'l IoT9 =1VgEG}, Ep°={lEE'l EpCKerl}.

Suppose that a function F E C' (M, IR) is G-invariant. Then F(g o p) = F(p) implies

DyopF o Dpg = DpF Vg E G. Thus for p E E it follows that

DpF o (Dp p)-' oT9 p = DpF o (Dp p)-' Vg E G, i.e.

lp = DyF o (Dpy)-' E E'p

.

As, again, DpFE = DpF [ T,E, the set of critical symmetric points is given by

cEpnE°°} and therefore the assumption

EpfEP={0) VpEE implies K(FE) C E n K(F), i.e. (P).

Theorem 5.7.4. Suppose M is a smooth manifold modelled on a Banach space E with a C' action of a Lie group G on it. (a) If the action of G on M is linearisable at every

pEE=E(G,M)={pEMlgop=p, VgEG}, then the set E of symmetric points is a smooth submanifold of M.

(b) For P E E define Ep, E,, and E°° as above, If Ep n Ep = {0) holds for all p E E then (P) holds for all G-invariant functions F E C' (M, IR). We want to apply Theorem 5.7.4 to prove the corresponding results of Corollaries 5.7.2 and 5.7.3 for the case of manifolds. Thus, corresponding to Corollary 5.7.2, we assume M to be a Riemann manifold, i.e. M is supposed to be locally diffeomorphic to a Hilbert space N and on each tangent space TyM there is a scalar product (., .) that varies smoothly with p. Furthermore, we assume that a Lie group G acts on M by isometries, i.e., for each p E M, Dpg is supposed to be an isometry, TyM -. TyopM. Then for a symmetric point p E E,

g'-' Dpg is a unitary representation on the Hilbert space TpM. In order to see that the action of G is linearisable at every symmetric point p, we use the crucial

5.7 The Principle of Symmetric Criticality

141

.fact that in the present case of a Riemann manifold the exponential map 0 exp exists, which maps a neighborhood V of zero of the Hilbert space TTM diffeomorphically onto a neighborhood b(V) of p in M with 0(0) = p. Thus MV), r-I) can be regarded as a chart in p (geodesic normal coordinates at p). For p E .V the crucial property of the exponential map

r/'oDpg(v)=goO(v) VvEV implies

Dpg(v) = 0' ° g o O(v) for v E V, i.e. the action of G on M is indeed linearisable. Therefore Theorem 5.7.4 implies the following corollary.

Corollary 5.7.5. Let M be a Riemann manifold and suppose that a group G acts on M by isometrics. Then the following holds: (a) The set S _ {p E M I g ° p = p, Vg E G) of symmetric points is a smooth submanifold of M; (b) (P) holds for every G-invariant function F E C'(M,1R). Furthermore, corresponding to Corollary 5.7.3, Theorem 5.7.4 also implies Corollary 5.7.6.

Corollary 5.7.6. If M is a smooth Banach manifold on which a compact Lie group acts smoothly, then (a) the set E of symmetric points is a smooth submanifold of M, (b) (P) holds for all G-invariant functions F E C' (M, IR).

The proof of this consists in showing first that compactness of G and continuity of G x M -, M, (g, p) i--* g o p, imply the existence of arbitrarily small G-invariant neighborhoods of symmetric points and then using such a neighborhood to construct a chart which linearises the action of G. Then Theorem 5.7.4 is applied.

6. The Variational Approach to Linear Boundary and Eigenvalue Problems

One of the first areas in which the basic variational concepts and methods are applied is the solution of large, important classes of linear boundary and eigenvalue problems. The results discussed in this chapter follow in part from the results on nonlinear boundary and eigenvalue problems which we shall discuss in Chap. 8; nonetheless we present proofs in this chapter which emphasise the simplicity and elementary character of the variational approach.

We start with a variational proof of the spectral theorem for compact selfadjoint operators, followed by a general version of the projection theorem (for convex closed sets). We need to prepare for tackling the problem of determining the spectrum of an elliptic second-order operator by first of all reformulating the problem

so that variational concepts and methods (quadratic forms) are applicable. Then we must prove a few results specific to this problem. The solutions of particular elliptic boundary and eigenvalue problems are then fairly simple special cases of these results.

6.1 The Spectral Theorem for Compact Self-Adjoint Operators. Courant's Classical Minimax Principle. Projection Theorem We shall show first of all how we can quite easily obtain a well-known result from the spectral theory of linear operators in Hilbert spaces (see [6.1], for example) with the aid of the general variational-calculus results we have already obtained. As a preparatory exercise, we suggest you prove the spectral theorem for Hermitian matrices A # 0 by determining the critical points of the function f(x) = (x, Ax)

subject to the constraint O(x) = (x, x) - 1 = 0. The same idea also yields a proof of the spectral theorem for compact self-adjoint operators. Recall that a compact operator A in a separable Hilbert space 1{ is characterised by in N, the the property that, for every weakly convergent sequence

6.1 The Spectral Theorem for Compact Self-Adjoint Operators

143

sequence of images (Ax ),En converges strongly. This definition simply states that the functions X-, Il Axll and x--, (x,Ax) are weakly continuous on N. This is immediately obvious for the first function. The weak continuity of the function x -, (x, Ax) can be proved as follows. If

xo = w -

x,,, then it follows that Ax0 = s -

Ax and thus

(x0, Axo) - (xn, Ax,,) = (x0 - x,,, Ax0) + (xu, Axo - Ax,,) converges towards zero because (xss),EIN is strongly bounded.

Theorem 6.1.1. Spectral Theorem for Compact Self-Adjoint Operators (Hilbert-Schmidt, Riesz-Schauder). Let N be a separable Hilbert space and A $ 0 a compact self-adjoint operator on W. Then A possesses an orthonormal system {ej}jEl4 of eigenvectors with real eigenvalues (Aj)jErj, Aei = Ajej, which has the following properties. (i) The eigenvalues are arranged in descending order of magnitude: Iar1 >

... > 1A11> Iai+rI _ ...

(ii) Either finitely many of the eigenvalue are nonzero, or all eigenvalues are nonzero, and limj-. Aj = 0 (0 is the only clusterpoint of the aj, j E IN). (iii) The multiplicity of each nonzero eigenvalue is finite:

dimKer(A - AjI) < oo for all j E IN. (iv) The orthonormal system of eigenvectors {ej, j c IN} is complete if, and only if, A is injective. (a) We start off by determining the eigenvalue of greatest magnitude. Proof. The following holds: IIAII =

sup

IlAull =

ss EBy(R)

sup

Ucsi(h)

I(n, Au)l = IIAIIB(x),

where

Br(x)={xExlllxll<1) and S,(R)={xENIIIxII=1}. The closed unit ball Bl(W) of a Hilbert space is weakly (sequentially) compact and u -, IlAull and it -, I(u,An)I are weakly continuous. Theorem 1.1.2 then gives the existence of a maximising point er E B, (R), i.e. IIAII = IIAer11=

sup

uEB,(H)

IlAull

A $ 0 implies el $ 0, and so Ile,11 = 1, since if 0 < Iler11 < 1, then el II

e

el e S1(fl) II

144

6. Variational Approach to Linear Boundary and Eigenvalue Problems

and it would follow that IIAei II < IIAer II =

IIAei 11, II

II

which is a contradiction. (b) The function x -, Q(x) = (x, Ax) is differentiable and has the Frechet derivative Q'(x)(h) = 2(Ax, h) , Vh E 7l . The following holds: sup xESi(7f)

Q(x) = ±IIAII.

The constraint ¢(x) = (x, x)-1 = 0 is defined by a mapping ¢ which is regular at all points of the level surface 0-1(0) = S1(7{), since 0'(x)(h) = 2(x, h) for

hE7{. We established in (a) that el E Sl(7{) is an extremal point of the function Q. Theorem 4.2.3 (Ljusternik) then gives the existence of a Al E IR such that

Q'(el) = A,O'(el),

i.e.

Act = act

,

and IA1I = IIAII.

(c) Let 7{1 = {e1}' C 7{ be the subspace of 7{ orthogonal to el. Since A is self-adjoint, it maps the subspace 7{r into itself: for x E Ni, (Ax, el) = (x, Ael) = (x, Ael) = 0,

i.e.

Ax E {el } 1 .

Thus the restriction Al = A { 7{1 of A to the subspace H1 is a well defined operator on the Hilbert space 7H1. This operator is likewise self-adjoint and compact, and its norm is not larger than that of A: IIA1IIs(7i,) : IJA1l5(71)

If Al # 0, then our considerations from (a) and (b) can be applied to the compact self-adjoint operator Al in the Hilbert space 711. They give the existence of a normed eigenvector e2 E 7{r of Al with eigenvalue A2, where IX2I = IIA1IIs(x,) <_ IIAIIs(x) = IAI I

and

(e2, el) = 0.

(d) The reduction step described in (c) can be repeated successively. It is necessary to differentiate between two cases, depending on whether the procedure stops after finitely many steps or not. However, no further variational arguments are needed to conclude the proof, and we thus refer the reader to the literature [6.1]. Theorem 6.1.1 establishes the existence and some of the properties of the eigenvalues of a compact operator. Furthermore, a method for calculating these eigenvalues is hinted at in the proof. Working out this method in detail leads to the classical rainim.ax principle of Courant-Weyl-Fischer-Poincare.

6.1 The Spectral Theorem for Compact Self-Adjoint Operators

145

Theorem 6.1.2. Minimax Principle. Let 7{ be a real, separable Hilbert space and A> 0 a self-adjoint operator on 7{ with spectrum ,(A) = {Am in E ]N},, Am+1 >_ Am,. Let E. denote the family of all m-dimensional subspaces E,,,, of 7L. Then the eigenvalues (Am)mEIN can be calculated as follows:

Am = min max

(v,Av)

E-EE,,, VEE,,, (v,v)

proof We determine the value distribution of the Rayleigh quotient

R(v) = (v, Av) (v, v)

by expanding the vectors v E H in terms of the eigenvectors (ej)jEpq of A as determined in Theorem 6.1.1. For 00

00

v = E aiei,

(v, v) = E as < co,

j=1

i=1

we get

(aiei,Aajej) _

(v,Av) i,j=1

Ajal j=1

and so Lid a,a, R(v) -_

Now let V,,, = lin{el eigenvectors. It follows that

max R(v) =

ai2 denote the subspace generated by the first m

max

vEV.,.

E.

Em =1 Ej=1 ai

= A,, = R(em),

and thus it remains for us to show that maxvEEr R(v) > A. for every other subspace E. E £m. Let Em # Vm be such a subspace; then E. n V # {0} and it follows that max

.EE- R(v) >

vEmo

R(v)

Ej>m+1 ',jal J Am+1 max Vj>-+l ajejEEm Ej>m+l a7

Am

which completes the proof.

Remark 6.1.1. Later on, when we discuss the Ljusternik-Schnirelman theory, we shall take the following simple reformulation of Theorem 6.1.2 as our starting point for a very important generalisation of the classical minimax principle.

146

6. Variational Approach to Linear Boundary and Eigenvalue Problems

Let S{m} denote the set of unit spheres S. in ra-dimensional subspaces E. of 7f,

S,n=EmnS,(f), Si(x)={vEW,IIvII=1), Em EE,n. Then according to Theorem 6.1.2, min

max (v, Av) .

Sr,ES,v vESm

The following version of the projection theorem for closed convex sets of a Hilbert space represents a generalisation of the well-known projection theorem

for subspaces and it shows that the essence of the proof is a "variational" argument. Although a direct proof is known (see, for example [6.2]), we want to present the variational proof here.

Theorem 6.1.3. Projection Theorem for Convex Closed Sets (Beppo Levi). Let 7{ be a real Hilbert space and K C 7{ a nonempty convex closed subset. Then there is a function PK :H -, K which has the following proper. ties:

(a) IIx - PK(X)II = inf:EK IIx - zll, Vx E 7t. (b) pK(x) is characterised for x E 71 by

(x - PK(x), Z - PK(X)) < 0,

VzEK.

(c) PK is (Lipschitz) continuous:

IIPK(xl)-PK(x2)II <-Ilxl-x2II,

Vx1,x2 E 7C.

Proof. We intend to define the function PK by property (a). For this we have to show that for arbitrary, but fixed, x E 7{, the function

V.(z)=llx-zll for zEK has exactly one minimising point in K, which we shall call PK(x). The first step is to show the existence of minimising points: to this end, let x E 7l be arbitrary, but fixed. Since W. is obviously a convex continuous function on K, the set

M = {x E K I co.(z) < W.(zo)) is, for arbitrary, but fixed, zo E K, a nonempty convex closed set. If K is unbounded, then cz(z) -+ +oo, IzII - co, and it follows that M is also bounded and thus, by Theorem 1.2.2, weakly compact. By Lemma 1.2.4, Theorem 1.2.5 gives the existence of at least one minimising point uy E K: W(vz) = inf Mz)

.EK

6.1 The Spectral Theorem for Compact Self-Adjoint Operators

147

The uniqueness of uz follows from the fact that Hilbert spaces are strictly normed (i.e. whenever x, y E 7i satisfy lIx + till = llxll + Ilyll, there exists a A > 0 such that y = Ax. Let us E K be another minimising point; then

(2(uz+u,))

2Wz(uz)+2(ur)

due to the convexity of Wz. Thus

which means that

II2(uz-x)+2(u'x -x)II = II2(u--x)II+II2(u,r - x)II 1

2(uz-x)=A2(t

-x)

for some A > 0. llu. - xll = Iluz - xil implies A = 1, and thus u'z = uz. This shows that PK : 7{ - K is well defined by

uz,

PK(x) = = Ix

for xEN, xOK, for x E K.

Remark 6.1.2. This proof works in every strictly normed reflexive Banach space and thus also yields the existence of best approximation element for these spaces.

In the next step we show that all minimising points of cot are characterised by the inequality (b). If uz is a minimising point of cot in K, then it follows

for all z K and all I C [0,1] that 0 < IIx - uz + t(uz - z)112 - IIx - unll2 = 2t(x - ux, uz - z) + t2lluz - zMI2

and thus (x - ui, uz - z) > 0, which is (b). Conversely, if uz is a point in K for which this inequality holds for all z E K, then it follows that l1

(x - Z,Z - U) _ (x - ux, z - uz) + (ux - Z, Z - ux)

<0, i.e.llx - uzll
148

6. Variational Approach to Linear Boundary and Eigenvalue Problems

If X1, x2 G R, then inequality (b) states that (x1 - PK(x1), PK(x2) - Prc(x1)) < 0 and

(x2 -PK(X2),PK(x1) -PK(x2)) co. Therefore

(x1 - x2 +PK(X2) - PK(xr ), PK(x2) - PK(x1)) < 0

and thus IIPK(x2) -PK(x1)II2

(x2 - x1,PK(x2) -PK(x1)) IIx2 -X111 IIPK(x2) -PK(x1)II,

which proves (c).

6.2 Differential Operators and Forms Generally speaking, the possibility of solving linear partial differential equa-

tions using the direct methods of the variational calculus is based on the simple idea of representing the particular differential operator as the Frechet derivative of a quadratic form on a suitable function space; this form is then investigated with the aid of general theorems and, in particular, its critical points are determined. The boundary conditions for potential solutions are already taken into account through the choice of function space on which the quadratic form is being investigated. This simple formulation leads to our goal for both linear boundary value and eigenvalue problems. If the quadratic form is replaced by a general function, then this treatment can also be successfully applied to nonlinear boundary and eigenvalue problems (Chaps. 7 and 8). We shall demonstrate these methods explicitly for linear and quasi-linear secondorder differential operators (which are the most important in applications).

It should then become sufficiently clear what is to be done in the case of higher-order differential operators. The approach is based upon suitable classes of function spaces and their embedding relations. Specifically, the function spaces used here are the socalled Sobolev spaces; the most important properties and embedding relations of these spaces are discussed in Appendix D. The theory of Sobolev spaces was developed essentially to be used in this approach. A newer, more complete exposition of this theory can be found in [6.3]. Obviously, not all (linear) differential operators can be treated following the approch described above: only those which have "divergence form" can be treated thus. The reason for this restriction will very soon become clear. A further restriction consists in the fact that, up to now, it has only been possible to give the most interesting results for differential operators on bounded

6.2 Differential Operators and Forms

149

sets. This restriction essentially stems from the fact that embedding relations between Sobolev spaces are used. We shall later discuss successful attempts of extending this approach to differential operators on IR".

To get down to details, let 1l C 1R' be an open nonempty bounded set and let A be a linear differential operator in divergence form on fl, i.e. A acts

"follows on functions u : fl - lR:

i-r

aii=aii.

j-r

(6.2.1)

The eigenvalue problem for the differential operator A then consists in finding numbers A and functions ua # 0 on fl such that Aua = An,

(6.2.2)

Appropriate choice of the space in which we want to look for the eigenfunctions ua is naturally a decisive factor for the solubility of this problem. In deciding

which space of potential solutions to choose, an important viewpoint results from establishing the behaviour of the potential solutions on the boundary i' = 8fl of fl, i.e. from the boundary conditions for the differential operator A.

Let the coefficients of the differential operator A be essentially bounded on 0, that is ao, a,i E L°°(2). Then A is certainly defined on D(Q) = Co (.2) and for gyp, 0 E D(Q), we have

(W,AO)2 = (5,,ao'lk)2 - ( 5',Eai \T j=1

ai'a'o/

=(W,no0)2+E\aiW,Eaiiai0/ j=1

i=1

2

-J

d"nEai\pEaiiai?k 12

j=1

where ( , ) is the scalar product of L2(Q). If .Q has a boundary sufficiently smooth for Gauf' integral theorem to be applicable, we have

J

n

d'xEai{WEaiiait}= f EdajWo i_j

l i=i

ani-1

aiiaio=0. i=i

In this case, therefore (W, AO)2 = (W, aob)2 + (VW, aVV))2

(6.2.3)

for all W, 0 E D(Q). The right-hand side of this equation defines a quadratic form Q = QA on D(Q). This quadratic form possesses many extensions. A natural upper bound for these extensions in L2(fl) has the domain

{u c L2(Q) IOju c L2(fl), j = 1,..., n} = H'(fl) ,

(6.2.4)

150

6. Variational Approach to Linear Boundary and Eigenvalue Problems

where the derivative is meant in the sense of distributions. H1(Q) is a Hilbert space with the scalar product (u,v)2 + (Vu, Vv)2. We shall show in Appendix D that for Q C lR° with sufficiently smooth boundary Ho (Sl) = closure of V(S2) in H'(92)

={vEHi(.2)1 v180=0}.

(6.2.5)

We thus expect from the derivation of (6.2.3) that

Ana=Aua,

UACHo(S2),

(6.2.6 a)

if, and only if,

Q(v,uA)- A(v,ua)=0

(6.2.6b)

for all v E Ho'(SQ), that is if uA is a critical point of the function u -4 Q.\ (u) = Q(u,u) - A(u,u)2

(6.2.6c)

on Ho (.fl).

Remark 6.2.1. Equation (6.2.6 a) expresses the fact that ua is an eigenfunction of the differential operator A in the Hilbert space L2(f2) with Dirichlet boundary conditions (ua r 8S2 = 0). We shall go into the formulation of other boundary conditions in more detail later on, when we come to discuss boundary value problems. Equation (6.2.6b) is also referred to as the variational form of the egenvalue problem (6.2.6 a). The proof of the equivalence assertion (6.2.6 a-c) is simple. First of all Qa(u)(h) = 2QA(h, u)

,

u, h E Ho (Sl) ,

is the Frechet derivative of the function Qa on Hp (Q) (all functions are realvalued). Thus ua is a critical point of QA if, and only if, (6.2.6b) holds, i.e. if 0 = (v, (ao - A)ua)2 + (Vv,aVu)1)2 (*)

for all v E Hd(Q), ua E Ho(S2) implies that aVua E L2(Q)" and it is thus differentiable in the sense of distributions. It follows that ua is a critical point of Qa if, and only if,

(ao - A)ua - V (aVua) = 0 in D'(Q)

(x)

since D(?) is dense in Ho(SQ). (ao - A)ua E L2(Sl) implies that V V. (aVua) also belongs to L2(Sl). Therefore, (x) also holds in L2(Q), i.e.

Aua = aua in L2(Q)

,

6.2 Differential Operators and Forms

151

and ua is an eigenfunction of the differential operator A in L2(fl) with Dirichlet boundary conditions. The converse statement results in just the same way, taking into consideration in the derivation of (6.2.3) that D(Q) = Ho (Q). We have not been able to say very much above the solubility of the eigenvalue problem (6.2.6 a-c) in the general situations we have treated up to now. The so-called ellipticity conditions prove to be sufficiently general and, as far as the solubility is concerned, very effective hypotheses; we shall present them here in their simplest form: The differential operator A is (strongly) elliptic if positive numbers 0 < m < M < oo exist such that for almost all x E ,f2 C IR"

and all t'EIR" (6.2.7)

j=1

j,i=1

j=1

Moreover, let

0Cao(x)
m

d"x 1` 9ju(x)aji(x)a:u(x)

d"x L.(aju)2(x) < Jn

Jn

j=1

j,=1

<MJ d"x Y(7jn)2(x) R

j=1

results, and thus m(Vu, Du)2 < Q(u, U) < M(Vu, Vu)2 + K(u, u)2 .

(6.2.8)

The second inequality in (6.2.8) implies that the quadratic form it -r Q(u, u) is continuous on the Hilbert space Ho (Q). Let us consider the Poincare inequality (Appendix D) IIv112

C A Vv112,

Vv E Ho(fl) ,

(6.2.9)

where A = A(.Q) < oo. Then the first inequality in (6.2.8) implies that the quadratic form Q on Ho (Q) is also coercive: 2 Q(u,u)> 1+a2(IIu1122 +IIou112).

(6.2.10)

Now H,'(52) is continuously and densely embedded in L2(f2). Moreover, Rellich's theorem for open bounded sets J2 C lR" with smooth boundaries states that the identical embedding of Ho (fl) into L2(Q) is compact. We are then able to recognise the eigenvalue problem (6.2.6 a) in the variational formulation (6.2.6 b) as a special case of the following general eigenvalue problem:

152

6. Variational Approach to Linear Boundary and Eigenvalue Problems

Two real Hilbert spaces 7{r and ?l2 with scalar products ( , .)1 and are given. It is assumed that (i) 711 is densely contained in 7{2;

(E)

(ii) the identical embedding i : 7{r - 112 is compact. We have to determine, for a continuous symmetric bilinear form Q on 7{r, all real numbers A for which there is a uA E 74, uA # 0, such

that Q(u, uA) = A(u, u4 2

for all u E 7{r.

We shall solve the eigenvalue problem (6.2.1) for the differential operator A in L2(2) by solving the general eigenvalue problem. (E). This solution is based on several abstract results which will be formulated and proved in the next section.

6.3 The Theorem of Lax-Milgram and Some Generalisations The original theorem of Lax-Milgram is a simple result for linear operators in Hilbert spaces. We shall use it to prove a result which, with the aid of the spectral theorem for compact operators, fairly easily yields the solution of the general eigenvalue problem.

Lemma 6.3.1. Let 7i be a (complex) Hilbert space and a : 7{ x 71 -,

a

continuous sesquilinear form on 7{ with a(u,u) > cllull2 ,

Vu E 71,

c > 0.

Then there is exactly one bounded linear operator A on 71 such that

a(u,v) = (Au,v),

Vu,v E H ,

(6.3.1)

where (.,.) denotes the scalar product of H. A is bnjective and has a continuous

inverse operators A-r. Proof. By virtue of the theorem of Riesz-Frechet, every continuous sesquilinear

form a has the representation a(u, v) = (Au, v),

where A : 7{ - 7{ is a uniquely determined bounded linear operator [6.1]. In the case of a coercive sesquilinear form this operator is injective, as is its adjoint: cIIulI2 < a(u,u),

(Au,u) < hull IlAull,

and (u,A`u) < pull jA'uhl

6.3 The Theorem of Lax-Milgram and Some Generalisations

153

therefore

(+)

dIIn1I s IIAuII' IIA*uII-

The well-known relat ion

(Ran A)1 = Ker A' yields

Ran A = (Ran A)11 = {0}1 = 1-l .

it follows easily from cllull <_ IlAull that RanA is closed. Thus RanA = 7i and A : 71 -, N is bijective. The estimate cllull $ IlAull implies also that A-1 is bounded and thus continuous Vv ER. IIA'vIi<_1IIvII, c

Theorem 6.3.2. Let 1-l be a real Hilbert space and E a Hausdorff locally convex topological vector space over IR. Suppose the following:

(i) N C E is dense in E. (ii) The identical embedding i : H -, E is continuous. (iii) a : 7{ x H -* IR is a coercive continuous bilinear form. Then for every continuous linear functional f on E there is exactly one u f E H such that

a(uf,v)= f oi(v)

(6.3.2)

holds for all v E N.

(a) Let f E E' be given. Because of (ii), f of : 7{ - IR is a continuous linear functional on 1{ such that the theorem of Riesz-Frechet guarantees the existence of exactly one = £f E 7{ with Proof.

foi(v)=(ef,v), VvEN. We have in this way a well-defined mapping J : E' -* 7{, J(f) = t f. It is evident that it is linear; but it is also injective since according to (i) H is dense in E. (b) Define

D(A) = {u E H I v - a(u, v) is an E-continuous linear form on 7-1}

.

The fact that it E D(A) therefore means that 7i D v - a(u, v) is a linear form on 7-l which is continuous with respect to the topology induced by E on R. Let it E D(A); since 7{ is dense in E, the linear form a(u, v) has a unique extension to a continuous linear form Au on E: we have a(u,v) = (Au,i(v)),

Vv E H.

(6.3.3)

154

6. Variational Approach to Linear Boundary and Eigenvalue Problems

where ( , ) denotes the duality between E and the topological dual space E'. This defines a mapping A : D(A) - E'. A is obviously linear. An = 0 implies a(u,v) = 0, Vv E 7i, and thus (v = u) it = 0. Thus, A is injective. (c) It remains for us to show that A is also surjective. Let f E E' be given.

Put u f := A-'J(f) where A : W -, 7i denotes the bijective operator from Lemma 6.3.1. It follows that u f E D(A) since because

a(uf,v)=(Auf,v)=(J(f),v)=foi(v),

VvE7-l,

v -+ a(u f, v) is an E-continuous linear form on N. Since H is dense in E, it follows from

(Au f, i(v)) = a(u f, v) = (f, i(v))

,

Vv E H ,

that

Auf= f. A is thus surjective. Theorem 6.3.2 is mostly applied in the more specialised version in which E is a Banach space or a Hilbert space. We therefore note the following.

Corollary 6.3.3. Let N1 and R2 be two real Hilbert spaces with scalar products (.I.), and (., -)2 where (i) Hi is dense in 7{2; (ii) the identical embedding of N1 in H2 is continuous.

If a : N1 x W, - IR is now a coercive continuous bilinear form on N1, then for every l; E R2 there is exactly one tie E H1 such that a(uf,v) = (6,i(v))2

for all v E N1.

The analogy to Browder's results for more general, not necessarily quadratic, coercive functionals becomes clear in the more general version of this corollary in the form of Theorem 7.2.3 (cf. Chap. 7). We note only that the coercivity of the bilinear form a : 7i x 7{ -. IR implies the monotonicity of the linear operators A : 7i -> R and A : D(A) C 7i -* E' defined by Lemma 6.3.1 and Theorem 6.3.2, respectively. For we have

(Au-Av,u-v)=a(u-v,u-v)>cllu-vll2>0, Vu,vEH, and for all u, v E D(A) (A(u) - A(v),i(u - v)) = a(u - v,u - v) > cllu - vjj2 > 0.

In the case of not necessarily quadratic functionals, a weakened coercivity

condition and a monotonicity condition of the above type are used.

6.3 The Theorem of Lax-Milgram and Some Generalisations

155

The complete solution of the general eigenvalue problem will now be deejibed in the following theorem.

Theorem 6.3.4. Let 7li and R2 be two real Hilbert spaces with scalar prodnets (I -)j and .)2, dim7{1 = oo, and assume the following: (i) 711 is dense in 7{2; (ii) the identical embedding i : 7{1 -, 7{2 is (continuous and) compact.

Then for every symmetric bilinear form Q on R, which satisfies

O

Q is continuous, (iv) Q is coercive, i.e.

Q(u,u)?c1lull2,

c>0, Vu E7{l,

there is a monotone increasing sequence (Am) of eigenvalues,

0 < Al < A2 < A,v m-.w +oo,

and an orthonormal basis {e,,,},,,EN C 7t1 of 7{2 such that

Q(em,v)=Am(em,v)2,

VvE7-11,

VrnEIN.

{v,,,} = {A7112em} is an orthonormal basis of 71k with respect to the scalar product on 'H1. Proof.

(a) By Lemma 6.3.1, Q(u,v) = (u,Au)1 where A is a self-adjoint

operator: A : 7{1 - * 'H],

c < A, A-1 < c 1

.

It follows that CIIUI12 < Q(u, u) < 1IAII Ju1li

for all it E 7{1

Thus

(u,v) -. (u,v)

'Q(u,v) defines a scalar product on 7{1 equivalent to -)1(b) If we identify 712 with its dual space 7{2, then Corollary 6.3.3 states that for every E 7{2 there is exactly one ut E 7{1 such that v) _ (C,i(v))2 for all v E R j. A mapping B : H2 -* 7{1

is therefore well defined by

BC:=uE,

f E7{2.

The relations (Be, v) = Q(Be, v) _ (f, i(v))2

156

6. Variational Approach to Linear Boundary and Eigenvalue Problems

and

(Be, v)1 = (6, A-1 v)2

,

Vv G Hi

,

imply that B is an injective continuous linear operator from 7-12 into N1 (c) Since the identical embedding i : 7{i --1 H2 is compact,

C = Boi:fl -17{1 is a continuous compact operator on (7-ti, (. )). C is also self-adjoint and ,

positive since

(Cu,v) = (i(u),i(v))2 = (u,Cv). Since B is injective, C is also injective since Ni is dense in H2. Thus, by Theorem 6.1.1, there is an orthonormal basis {Vm}mEIN of (H1, ( , )) and a sequence of numbers (pm)mEIN where Pm

pm m-.co

0, and Co. = pmvm

,

Vm E N.

It follows that the sequence of numbers A. = 1/pm satisfies

Ammoo -1 +oo and Am < Am+1 For the vectors e,,, :=

A1/2

,

Vm E IN.

vm E 7-t1 we have

Q(em,v) = (em,v) = Am(Cem,v) = Am(i(em),i(v))2

and thus (en, em)2 = (2(eu), i(em))2 = (vn, vm) = 8nm

{em },NN C 7{1 is therefore an orthonormal system in H2 consisting of "eigenit follows vectors" of Q. Since {vm}mEIN is an orthonormal basis of (7{1,

from f EH,nfe.jnElN)' that 0 = (i(en),i(f))2 = An1/2(vn,f),

Vn E IN

and so f = 0. Since N1 is dense in H2, the system {em}, m E IN, is also complete in 7{2.

6.4 The Spectrum of Elliptic Differential Operators in a Bounded Domain. Some Problems from Classical Potential Theory The solution of the eigenvalue problem (6.2.2) for the differential operator (6.2.1) in the variational form (6.2.6 a-c) becomes rather simple if we use the abstract results of the last section. The solution of our original problem proves to be a special case of the solution in Theorem 6.3.4 of the general eigenvalue problem (E).

6.4 The Spectrum of Elliptic Differential Operators in a Bounded Domain

157

-Theorem 6.4.1. Let J? C IR" be a nonempty open bounded set with a suffciently smooth boundary P = aft. Then assuming the ellipticity condition (6.2.7) holds, the eigenvalue problem

Au=aou -

a;(a;;(9;u)=Au,

it

lP=0

h4s a monotone increasing sequence of eigenvalues Ai:

0<ar
a7->+oo i*00

and a sequence {eJ}jEIN C Ho(fl) of eigenvectors ej in Ho(fl) which form an orthonormal basis of L2(fl). f,\112 et 3 E IN} is an orthonormal basis of HH(fl) with respect to the scalar product (,) = Q(-,.). The eigenvalues can be calculated with the aid of the m.inimax principle:

An, = min

max Q(v,v) I2

E,,, E£,,, vEE,,,

V 2

n

Q(u, v) = (u, aon)2 +E (a, u, aijajv)2

,

(6.4.1)

where £m is the family of all rn-dimensional subspaces of Ho (fl).

Proof. We show that the hypotheses of Theorem 6.3.4 are satisfied in the situation described above. According to Appendix D, 711 = Ho (fl) is dense in 7t2 = L2(fl) and the identical embedding i : 7{r -, 712 is compact. Theorem

6.3.4 guarantees that a continuous symmetric coercive bilinear form Q on 7-ti = Ho(fl) is defined by (6.4.1). Finally, we show that the eigenvalues can be calculated by means of the minimax principle. This is obvious firstly for the eigenvalues µ,,, of the compact self-adjoint operator C > 0 on 71r, defined by C = B o i, Q(Bu, v) = (u, i(v))2, Vu E 712 and Vv E 7t1, by virtue of the proof of Theorem 6.3.4. The eigenvalues A. of A that we require are characterised

by (6.2.6b) and can be calculated from A. = 1/p,,; the assertion follows. Remark 6.4.1. Theorem 6.4.1 determines, among other things, the spectrum of the Laplace operator (ao = 0, aid = bid) with Dirichlet boundary conditions on open bounded sets with sufficiently smooth boundaries.

In conclusion let us discuss the variational solution of a few simple linear boundary value problems which, in their simplest forms, have played an important role in classical potential theory. The abstract method of solution of these problems is indicated by Theorem 6.3.2 and Corollary 6.3.3. Let us consider a differential operator A in the form (6.2.1) over an open set fl C 1R"

158

6. Variational Approach to Linear Boundary and Eigenvalue Problems

with smooth boundary, subject to the assumptions of Sect. 6.2. The following problem has to be solved for a given function f : fl -, IR. Determine a function u : ,(l -* IR such that

(i) Au=aou-1: ai(Eaiiaiu)=f j=1

in fl

(6.4.2 a)

i=1

and (

u ra,fl=0.

(6.4.2 b)

A slight generalisation of our treatment in Sect. 6.2 results in the variational formulation and precise specification of this problem in the following form. For given T E Ho (fl)', determine UT E Hp (.fl) such that Q(UT, V) = T(v)

for all vEHo'(Q),

(6.4.2 a')

where Q is defined by n

Q(u,v) _ (u,aov)2 +

(8,u,aiiOiv)2i

u,v E Hd((l).

(6.4.2 b')

i,i=r

The boundary condition u r 8S2 = 0 is again satisfied here simply by the choice of the space of potential solutions (see Appendix D). Problem (6.4.2) always has a solution in the version (6.4.2').

Theorem 6.4.2. Let R C it be an open bounded set with smooth boundary and let A be a linear differential operator in divergence form which satisfies the ellipticity condition (6.2.7). Then the following holds. For every continuous linear form T on Ho (fl) there is exactly one UT E Ho (dl) such that Q(uT, V) = T(v)

,

by E Ho (fl)

.

The mapping T - UT from (Ha(ft))' into Hd(Sl) is linear, continuous and bijective. UT is the minimising point of the functional

ET(v) = ZQ(v, v) - T(v)

onHo(H), i.e. ET(UT) = inf{ET(v) I v E Ho(fl)}.

Proof. Due to the boundedness of fl, the ellipticity condition (6.2.7) implies, as above, that the symmetric bilinear form Q on the Hilbert space 7{ = Ho ((l) is well defined, continuous and (strictly) coercive. Applying Theorem 6.3.2 for 7{ = E = Ho (0), the first part of the assertion follows immediately. Following

the method of proof of Theorem 6.3.2, we have UT = A-'JT, which shows the second part of the assertion, since J here is the canonical isomorphism of the dual space 7{' on 7i. The identity

6.5 Variational Solution of Parabolic Differential Equations

159

Ef(uT + h) = ET(UT) + Q(UT, h) - T(h) + Q(h, h) = ET(UT) + Q(h, h) shows immediately that the critical point UT is indeed a minimising point of !ET.

gemark 6.4.2. This theorem can also be extended to unbounded domains if the coercivity of Q on Ho (fl) can be established using stronger hypotheses for A, and without using the Poincare inequality. It is also possible to do without assuming the boundary of fl is "smooth", even if an interpretation of the boundary condition is then not so obvious. When n = 3 and An = -Au, Theorem 6.4.2 asserts the unique solubility of the classical Poisson problem. It is well known [6.4] that the unique solubility of this problem is equivalent to the unique solubility of the classical Dirichlet

problem: "determine a function u : -l -> lR such that -Au = 0 in 0 and u j8,fl = a for a given `surface charge' a : OS? -> IR". Finally, the unique solubility of this problem is also equivalent to the existence of exactly one

Green's function = G0a :.0 x 0 -. IR for the Laplace operator A and the domain fl, that is for arbitrary but fixed y E G, the function G must satisfy the conditions

(i) -A.G(x,y) = b(x - y) and (ii) G(x, y) = 0, Vx E 8.f1. Thus Theorem 6.4.2 (for more general elliptic differential operators as well) can be used to show the unique solubility of the corresponding Dirichlet problem and the existence of a Green's function.

Remark 6.4.3. If n = 1, this is a boundary value problem for ordinary secondorder differential equations. J. Sturm and his friend J. Liouville were the first to systematically investigate such problems, in 1836; these problems are known nowadays as Sturm-Liouville boundary value problems. For more details we refer to [6.5, 6].

6.5 Variational Solution of Parabolic Differential Equations. The Heat Conduction Equation. The Stokes Equations As we saw in the previous section, elliptic boundary and eigenvalue problems can be solved relatively easily using the methods of variational calculus. We shall show in this section that one can also use the methods to solve parabolic differential equations. To this end, we shall discuss two specific problems,

6. Variational Approach to Linear Boundary and Eigenvalue Problems

160

the heat conduction equation and the Stokes equations of hydrodynamics, in more detail. These will illustrate and provide us with a reason for seeking a general framework for variational solutions of parabolic differential equations. A treatment which goes somewhat further than ours can be found in [6.7, g]. Parabolic differential equations are mainly used in physics to describe the behaviour of systems over time. The heat conduction equation is a simple and not untypical example which, in convenient units, takes on the form

(t, x)-Qsu(t,x) = f(t,x), (W)

1

(t, x) E (0, T) x 1? C ]R4

(6.5.1 a)

n(0,x)=uo(x), xEazCR

(6.5.1b)

u(t,x) = 0,

(6.5.1c)

3 ,

x E 80, t E [0, T],

for isotropic homogeneous media.

The problem (W) then consists in determining the time behaviour t --> u(t, ) of the temperature distribution in 12, for a given heat source f and initial

temperature distribution u0, such that the temperature at the boundary 8!? of 12 stays constant (and equal to zero). Thus, as far as the time behaviour is concerned, this is an initial value problem according to (6.5.1 b), and for the spatial coordinates it is a boundary value problem according to (6.5.1 c) for the solution of a linear inhomogeneous partial differential equation. We have already seen how to treat boundary value problems like (6.5.1 c): we satisfy (6.5.1 c) "automatically" if we search for solutions of (6.5.1 a) which

are "functions" [0, T] 3 t - ut with values in the Sobolev space Ho ((2). (We obviously make the implicit assumption that the initial conditions also belong to the Sobolev space Ho (fl).) We can express the differential equation (6.5.1 a), as before, by means of an equation of the form dt(vt,v)2+Q(ut,v)=(ft,v)2,

VvEHO(Q), 0
(6.5.2)

if we put

ft,v2=fn f(t,x)v(x)d'x, Q(ut, V) = J V u(t, x) - V v(x)d3x = (Vut, Vv)2 . n

(6.5.2')

Conversely, if t - ut is a solution of (6.5.2) with ut=s = no, then (t,x) -, ut(x) is also a weak solution of (6.5.1 a) with the correct initial and boundary conditions (6.5.1 b) and (5.5.1 c).

The strategy which leads to the solution of (6.5.2) can be generalised without too much effort. Let us first of all discuss a general framework for the treatment of parabolic differential equations. We shall obtain as a special case the solution of (6.5.2) and thus the weak solution of (6.5.1).

6.5 Variational Solution of Parabolic Differential Equations

161

ga A General Framework for the Variational Solution of Parabolic Problems The framework introduced here proves to be a natural generalisation of the Mess which lead to a solution of the Dirichlet boundary value problem. The following notation and facts serve as preparation for a formulation of this ismework. We denote by C([O, T], X), for a Hilbert space X with scalar product ( , -)x and norm II' II x and a positive number T, the space of all continuous functions -tbe interval [0, T] with values in X. The space C([0, T], X) with norm

IIvIIc([o,Tl,x) = sup IIv(t)IIx o
is a Banach space. Correspondingly, L2([0,T],X) denotes the space of all (equivalence classes of) Lebesgue-measurable square-integrable functions on [0,T] with value in X, that is functions [0,T] -, X for which t .-a IIv(t)IIx

is Lebesgue-measurable and IIvIIL=((0,T[,x)

_

f

T

1/2

IIv(t)IIX dt)

0

finite. It follows by polarisation for two such functions u and v that

t -* (u(0, v(t))x is also Lebesgue-measurable and integrable over [0, T]. Thus T

(u,v)L=([o,TI,x) =

J0

(u(t),v(t)fx dt

defines a scalar product on L2([O,T], X) and makes this space a Hilbert space. We now come to a formulation of a general framework for the variational solution of parabolic problems. We assume we have the following:

(a) Two real Hilbert spaces 711 and fl2 with the following properties: (i) Ni is dense in f12; (ii) the identical embedding of Nl into f12 is continuous.

The scalar products and norms on 7t1, i = 1, 2, are denoted by ( , ), and II' IIi.

(b) A continuous bilinear form Q on W1. (c) An "initial condition" u0 E 712. (d) An "external force" f E L2([0,T],f12).

6. Variational Approach to Linear Boundary and Eigenvalue Problems

162

We are looking for all functions u which have the following properties.

(a) u E C([O,T],H2)nL2([0,T],7i1). (p) u(O) = to.

(y) For all vE7{1,

dt(u(t),V)2 -Q(u(t),v) = (f(t),v)2

(6.5.3)

holds in the sense of distributions on (0, T). Remark 6.5.1. The restriction of solutions to those in C([O,T],7-t2) allows us to formulate the initial condition (p). The assumptions that u should belong to L2([O,T],7-11) and that 7{1 is continuously embedded in 1-12 imply that the equation (y) makes sense in D'((O,T)).

The heat conduction problem (6.5.1) fits into this framework in the reformulated form (6.5.2) and (6.5.2'). One need only put 7{1 = Ho(Q) and 7{2 = L2(Q). The condition f E L2([0,T],H2) should then be regarded as making our assumptions specific. In the version above, with its weak assumptions, one is able to say very little about the solubility of the general parabolic problem given by the conditions (a)-(y). However, if we make the same assumptions as in our treatment

of general elliptic boundary value problems, then the following useful and quite general theorem results.

Theorem 6.5.1. The general parabolic problem given by conditions (a) to (-y), subject to the following hypotheses, always possesses exactly one solution.

(E) The bilinear form Q is strictly coercive on 7{l, i.e. a positive number a exists such that Q(v,v) > alIvIIl, Vv E 7{l. (K) The identical embedding i : R1 -+ H2 is compact. (S) The bilinear form Q is symmetric. (a) According to Theorem 6.3.4, we know that under the above assumptions, there is an increasing sequence {Ai}iE¢N of eigenvalues Proof.

0
Aia-. -,}00,

of Q and an orthonormal basis {wi}iEIN of H2 whose elements wi are all in N1 such that

Q(v,w1)=Ai(v,wi)2,

VvE7{r,

ViEIN.

Moreover, vi = A112wi is an orthonormal basis of Ni with respect to the scalar

product ( , ) = Q(, ).

6.5 Variational Solution of Parabolic Differential Equations

163

(b) Instead of determining the one function we are looking for, u E 'j2([O,T1,7{1) fl C([O,T],7-12) which satisfies (/9) and (-y), let us first of all .determine all components

i = 1, 2, ... ,

ui(t) = (wi,u(t))2,

,of this function with respect to the orthonormal basis {wi} of 7{2. According to (a)-(y), these components have to satisfy the conditions fT Iui(t)I2 dt < oo, (a') u; E C([0, T], R), u° (f) ui(O) = = (wi,uo)2, (. ) dtui(t) + A1ui(t) = fi(t) = (wi, f(t))2,

where (y') can be obtained from Q(wi, u(t)) = Ai (wi, u(t))2

This problem is easily and uniquely solved by /e

ui(t) = exp(-Ait)u° + J fi(r) exp[-Ai(t - r)] dr 0

xi(t)+yi(t),

0
xi(t) = exp(-Ait)uo is a C°°-function and yi(t) is absolutely continuous; thus ui is certainly a solution of the differential equation (7') in the sense of L' functions. The initial conditions (/3') and u1 E C([O,T], IR) are obvious. Furthermore, the following estimates hold for 0 < t < T: xi(t)I < Iu°I

Iyi(t)I =

,

Ai

J0T

Ixi(t)I2 dt < Z Iu?I2 If;(r)12dr1-exp(

f tfi(r)exp[-Ai(t-r)]dr <{ fT l

a

2A1T)}h/2

2Aj

/or

T

Al T Iyi(t)I2 dt < T 0

J0

I fi(t)I2 dt .

It follows that 1 °°

rT Ai i=1

J

Ixi(t)I2 dt <

1

In112 = ,=1

0

z IIuoII1

and

0

rT

=1

m (T If;(t)IZdt=TIIfIIL'([0,T[,H,),

lyi(t)12dt

A1 0

,=1

0

and thus by virtue of (a) it is also certain that

17

6. Variational Approach to Linear Boundary and Eigenvalue Problems

164

1UT

lu;(t)12 dt < oo,

1=1

so that

t - u(t)

u;(t)w; E L2([O,T],7{z)

follows. We then also have t

a;u;(t)w; E L2([O,T],7{2). ;=1

(c) We have already shown in (b) that

u = n-, lim S. oo where

in

L2([O,T],7-12)

n

S.(.) = Eu;Ow; E C([0,T],7t1) C C([0,T],7{2). 1=1

The estimates given in (b) for xi(t) and yi(t) easily yield the fact that (S.)IEIN is a Cauchy sequence in C([O,T],7{2). It follows that there is a uniquely determined element u2 E C([O,T],7{2) where

u2 = lim Sn in C([O,T],7{2) Now, for all v E C([O,T], H2), II VIIL2([O,Tl,x,)
holds. It therefore follows for all n E IN that IIu2 - UIIL'([o,TI,H,) s Ilu2 - Sn11L2((D,21,H2) + Ilsn - U11L2([o,Tl,H2) < T1/2 11u2 - Sn11c([D,TI,Hz) +

HIS. - u11L2([e,2j,H=)

and thus that u2 = u in L2([0,T],7{2), which means that u E C([O, T], 7t2) .

(d) We shall show in a similar fashion that u E L2( [0, T],7{I) also holds. We shall use the coerciveness of Q and the fact that (w;/ai 2);Ew is an orthonormal basis of 7{I with respect to Q(y ) in order to show that the sequence (Sn)nEIN in L2([0, T], 7t1) is a Cauchy sequence. Thus it follows for n > rn that

6.5 Variational Solution of Parabolic Differential Equations n

/T

IISn

2

> ui(t)wi

SmIIL2([O,Tl,7Lt) = J

dt

H,

i=m+1

<

rT

J

/

\

aQl E u;(t)w;, ,=m+1 n

u;(t)w;J dt i=m+1

T

Ai

i

j Iui(t)I2 dt

i=m+1

<

165

Z

U

Ai

a i=m+1

J

T{Ixi(t)IZ + Iyi(t)I2}dt.

The estimates proved in (b) therefore show that (Sn)nElN is also a Cauchy sequence in L2([O,T],7{1). There is thus a uniquely determined element ul E L2([0, 11, N1) satisfying

u1 = lim Sn in L2([O,T],7{1). n-. Since 711 is continuously embedded in 712, a constant c > 0 exists such that IIvIIH, < cIIvIIH Vv E N1. It follows that IItIIL-((O,TI,H,) <- ciIvIIL'((Q,Tl,H1)

for all v E L2([O,T],7{1) and thus, for all n E IN, IIu' -UIIL2([0,71,H,) < IIu1- STIIL-Q0,T[,H2) +

< cllul -

IISn - UIIL2(0,Tl,H2)

Sf hhL,([o,zn,H,) + IISn - uIIL1([0,7I,H,)

and consequently u1 = it in L2([0,TI,N,), which means it also belongs to L2([0,T],7{1). This yields, with (c),

u E C([O,T],7{2) fl L2([O,T],7{,) C L2([0, T], 7{2).

(e) Since y(O) = 0 and x;(0) = u°, Vi E IN, it follows immediately that u(t) = E°°1 u;(t)w; satisfies the initial condition. It thus remains for us to show the validity of the differential equation (y). The ui are solutions of equation (6.8') in D'((O,T)), which means that E D((O,T)),

0

T[-w (t)ni(t) + aiui(t)w(t)] dt = T w(t)fi(t)dt . 0

For arbitrary v E 7{i and for all N E IN, it follows that

6. Variational Approach to Linear Boundary and Eigenvalue Problems

166

T

J

EN

(v, wi)2fi(t) dt

'P(t) e

i=11jI(

=JT J0

_ 0

l V'(t)E(v,wi)2ui(t) + E Ai(v, wi)2u;(t)P(t)}dt

l

i=1

i=1

111J

N

T

l

wi)2ui(t) +Qv - (t) (v, ;=1

N i=1

JJ1J

Since {w;};EpN is an orthonormal basis of 9-12, Ei=1(v,w;)2fi(t) converges in

L2([O,T]) to (v, f(t))2. Since iNlui(.)w; converges to u(.) in L2([O,T],7-i1) and L2([O,T],7-12), then EN1(v,wi)2ui(-) converges to (v,u(-))2 in L2([O,T]) and Q(v, ENl

converges to Q(v,u(-)) in L2([O,T]), since Q is contin-

uous on ftl. We can thus take the limit N -1 oo in the above equation and we obtain

f 0o

[- (t)(v,u(t))2 + Q(v, u(t))] = T w(t)(v,f(t))2 dt, 0

which is the differential equation (-y). The uniqueness of it follows from the uniqueness of the components ui with respect to the orthonormal basis {w;}iENN of 7{2. This completes the proof of Theorem 6.5.1.

6.5.2 The Heat Conduction Equation The heat conduction equation (6.5.1) can now be solved quite simply in its variational form (6.5.2) and (6.5.2'). Theorem 6.5.2. Let 92 C R' be an open, bounded set with a smooth boundary. Then the heat conduction problem (6.5.1) has exactly one solution it = u f,,,p E C([O, T], L2 (.2)) n L2 ([0, T], Ho (j?))

for all initial conditions no E H0(fl) and for all "heat sources" f E L2([O,T], L2(Q)). In the spectral representation of the Laplace operator in ,2 with Dirichlet boundary conditions, this solution is given by 00

u(t) =

E{exP(-A;t)(wi,uo)L2(n) i=1

t

exP[-Ai(t-r)](wi,f(t))L2(.rt)dr}wi,

+ 0

where (-Ai,w;);Epv are the eigenvalues and eigenfunctions of the Laplace operator AD with Dirichlet boundary conditions.

6.5 Variational Solution of Parabolic Differential Equations

167

proof. We establish in Appendix D the fact that 1.4 = Ho '(J?) is, subject to certain assumptions, compact and densely embedded in 9{2 = L2(dl). Q(u, v) = (Vu, Vu)L2(12) is obviously a continuous symmetric bilinear form on Ha (Q), since we are considering real-valued functions, and by virtue of our previous considerations (Sects. 4.2 and 4.4). Q is also coercive on 7t1. Thus, Theorem 6.5.1 is applicable. In this theorem, Ai and wi are eigenvalues and eigenfunctions of the bilinear form Q: Q(v, wi) = Ai (v, wi)x,

,

Vv E Ho (f2) i

here this means that (Vv, Vwi)L2(n) = A1(v,wi)L2(n),

Vv E HO(ST)

where wi E Ho (S2). Theorem 6.5.2 follows.

Remark 6.5.2. Formally speaking, equations (6.5.1 a) and (6.5.1 b) can be integrated using the function

rt

u(t, x) = exp(t4D)uo(x)+J eXp[(t - T)DD]f(T, x) dT , 0

where dD is the Laplace operator on Sl with Dirichlet boundary conditions. A more exact investigation of the semigroup of operators in L2(S2), (exp(tdD))t>o, which is generated by 1D shows that some meaning can be attached to the formal solution above. However, this "operator solution" of (6.5.1 a) and (6.5.1 b) will not necessarily satisfy the boundary condition (6.5.1 c) for arbitrary f E L2([0,T]),L2(Sl)). Substituting the spectral representation for exp(t4D), also proved in Theorem 6.5.2,

exp(t4D)=L.eXp(-Ait)Iwi)(wil,

wiEHd(0),

i=r lwi)(wiI being the orthogonal projector onto the one dimensional subspace

spanned by wi, the operator solution yields the solution given in Theorem 6.5.2.

6.5.3 The Stokes Equations in Hydrodynamics We now want to briefly discuss another application of the general results of Sect. 6.5.1, namely the solubility of the so-called Stokes equations. A bounded domain Sl C IR" with a "smooth" boundary P = 851, an n-tuple uo = of functions not Sl - IR, an n-tuple (f, ,...,f") of :

functions

fi:flT-.IR,

SQT=(0,T)XSl,

6. Variational Approach to Linear Boundary and Eigenvalue Problems

168

and a positive constant p are given. We are looking for an n-tuple it = (u, , ... , un) of functions ui

:

,2T -+ IR and a function p : 2T -+ ]R

which are solutions of the system of equations (S):

(S)

81u-µdu+gradp=f in

,2T,

div u = 0 u=0

.2T ,

in on in

[0,T) x r, P.

A solution of this system of equations is interpreted as the velocity and pressure field (u,p) of an incompressible viscous fluid with kinematic viscosity

µ, in a domain 5? during the time interval [0, T], and on which an external force with density f is acting. It is assumed here that neither the velocity u nor the initial velocity no is too large; the motion of the fluid is then sufficiently slow so that the system of Stokes equations (S) can actually be regarded as a linear approximation of the Navier-Stokes equations n

a,u-µdu+

u;8;u}grade= f in 2T

with the same constraint div it = 0 in 2T and the same boundary and initial conditions. The Stokes and Navier-Stokes equations are treated in detail in [6.8].

So that we shall be able to discuss the problem of the solubility of the Stokes equations within the general framework given above, we define first of all suitable Hilbert spaces 7{1 and 7-12 and a continuous symmetric bilinear form Q and 7-11, and then we check that the assumptions of Theorem 6.5.1 are satisfied. An obvious choice for 7{1 seems to be

hi ={vEHd(Q)"I divv=0}. If we choose 7{1 thus, the boundary and divergence conditions for the Stokes equations will be automatically taken into account in a weak sense. The Hilbert space 7{2 is essentially fixed by virtue of the fact that the hypotheses of Theorem 6.5.1 must be satisfied: 7{2 must be chosen to be 7{2 = closure of 7{r in L2 (.Q)n .

Then 7{1 is continuous and densely embedded in 7{2 We need a more exact characterisation of 7{2 for further analysis, and we take this from the following (see [6.8]).

Lemma 6.5.3. If ,2 C IR° is a bounded domain with a sufficiently smooth boundary r = 8.2, then

7{2=fu EL2(Q)nI

tr=0}

6.5 Variational Solution of Parabolic Differential Equations

169

where n is the unit normal vector on I. If E denotes the orthogonal projector from L2 (Q)" onto the subspace H2, it follows that H2 = EL2(.Q)"

and

7{r = EHo (Q)" .

Hd(Q) is compactly embedded in L2(Q) for bounded domains 1? (Rellich's theorem); it follows that Ho (Q)" is also compactly embedded in L2(fl)". This representation of 7{r and 7{2 shows easily that, since the product of a compact operator and a bounded operator is also compact, N1 is also compactly embedded in 7{2. A suitable bilinear form Q on H, is Q(n,v) = I2(Du, Vv)L2(n)n

which is obviously symmetric and continuous on H1. By virtue of earlier considerations, Q is also coercive on 7{1. Thus Theorem 6.5.1 can be applied and we obtain the following.

Theorem 6.5.4. For all f E L2([0,T],L2(Q)") and for all no E N2, there is exactly one

u=uf,"o E Ni, D,(v,u(t))x2

+ p(Vv,Vu(t))x, = (v,f(t))L2(n) in D'((O,T))

and u(O) = no.

Proof. Ev = v holds for v E 7t1 and thus so does

(v,f(t))L'(n)" = (v,Ef(t))L'(n)" = (v,g(t))'H2 where g = Ef E L2([O, T], N2); we are then indeed dealing with the situation of Theorem 6.5.1. It remains for us to clarify in which sense Theorem 6.5.4 describes the

solution of the Stokes equations (S). Since D(Q)" is not contained in 7{1, a distributional interpretation of the solution does not immediately follow canonically. According to Theorem 6.5.4,

S=D=u-µ4u-f is an n-tuple of distributions on Si = (0, T) x f2. It follows from Theorem 6.5.4 for all v E D(Q)", where div v = 0, i.e. v E 7-11, and for all 0 E D((O,T)) that T

S(V) (D v)_ J

'(t){D,(v,u@)L'(n)- + I1(w, VU)L2(12)..

0

- (v, f (t)) Le (n)n } dt

=0.

170

6. Variational Approach to Linear Boundary and Eigenvalue Problems

Making use of a fundamental result due to G. de Rham [6.8], it follows from

S(b ®v) = 0, Vv E D((2)"

,

div v = 0,

VO E D((O, T)) ,

that a distribution p E D'(.2T) exists such that

S= - grad, p. Thus, the solution u of Theorem 6.5.4 is given by

Btu - µd.u + grade p = f in D'(,flT)", i.e. this theorem yields the existence and uniqueness of a solution u E L2([O,T],7{r) fl C([0,T],7-12) which satisfies the Stokes differential equation

and the condition that there should be no sources, in the sense of distributions on (0, T) x .(l. The fact that the boundary condition u = 0 on [0, T] x Q is satisfied in the weak sense is contained in the statement u E L2([0,T],7-1) C L2([0,T],Ho((1)"). It follows from it E C([O,T],7{2) that the initial condition u(0) = us is satisfied in L2(Q)" in the sense of limt-,o u(t) = uo. Remark 6.5.3. If one makes additional assumptions about the regularity of 0, uo and f, it is possible to obtain corresponding assertions about the regularity for the solution (u,p) [6.8].

T.

Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

7.1 Forms and Operators - Boundary Value Problems In the previous chapter we saw how to solve linear elliptic boundary value problems using variational methods. In this chapter we shall discuss some of F. E. Browder's generalisations to nonlinear, or, more specifically, quasi-linear elliptic boundary value problems [7.1]. Our attempts at solving boundary value problems for quasi-linear differential equations in "divergence form",

A(u)(x)=Ao(x,u(x),Vn(x))->ja1Ai(x,u(x),Vu(x))=f(x)

(7.1.1)

i=

using the methods of variational calculus start off in a very similar fashion to our attempts at solving linear differential equations. Here too, we start by considering a suitable "generalised Dirichlet form" a associated with A, on a distribution space which takes into account the boundary conditions for the potential solutions of (7.1.1). We shall limit our investigations to secondorder differential equations (see [7.1] for a natural generalisation to differential equations of order 2m, m > 1) and thus we consider the following generalised Dirichlet form on the Sobolev space W1 LP(G), where G C 1R° is open: a(u,v) = (Ao(-,Y(u)),v)2 + E(Aie,Y(u)),ajv)2

(7.1.1')

i=1

where Y(u)

_ (Y0(u),Y1(u),"',Yn(u)) _ (U 91n,...,ann)

and f(x)9(x)dnx,

(f, 9) 2 =

c

assuming all values to be in R. A first group of hypotheses for the coefficient functions Ai : Gx IRn}1 _ 1R will establish the following property of a: Ia(u,v)I C h(aIuuul,P)IIII1,P,

u,v E W1,P(G)

(7.1.2)

172

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators /P

IIuII1,P = ( > II"II<_1

J

ID"u(x)IP dx) G

where the function h : IR+ -i IR+ is bounded on bounded sets. If (7.1.2) is satisfied, then for arbitrary, but fixed, it E W',P(G), v -> a(u, v) is a continuous linear form T(u) on X - W1,P(G) for which

(T(u), v) = a(u,v),

Vu,v E X ,

(7.1.3)

holds, where ( , ) denotes the duality of X and X'. Thus (7.1.3) defines a mapping T from X into X', assuming that (7.1.2) holds. We shall introduce boundary conditions in the following, initially abstract, way. We choose a closed subspace V of WI,P(G) which satisfies

Wo'P(G) C V C W1'P(G).

(7.1.4)

We can then also take a to be a generalised Dirichlet form on V which, due to (7.1.2), defines a mapping T : V -. V' by means of the duality (7.1.3). (V` is the topological dual space of the Banach space V with norm 11 II'.) We can now formulate the following problem with these conditions. Variational Boundary Value Problem. For given V, a and f E V', determine all it E V which satisfy

a(u,v) = (f,v), Vv E V .

(7.1.5)

Equation (7.1.5) and the requirement that it be in V not only mean that it must satisfy (at least in a weak sense, since D(G) e V) the differential equation A(u)(x) = Ao(x,y(u)(x)) -

8jAj(x,y(u)(x)) = f(x),

(7.1.6)

j=1

but also that it satisfy certain boundary conditions. The origin of these boundary conditions is: (i) The choice of V; this boundary condition can be expressed as "u E V". It becomes effective particularly when V "is very much smaller than W1'P(G)". This is the only boundary condition for V = Wo'P(G), i.e. for the Dirichlet problem. (ii) On the other hand, if V "is considerably larger than Wo'P(G)", then (7.1.5) gives the so-called natural, boundary conditions for it, since the equa-

tions A(u) = f and a(u,v) = (f,v), Vv E V, assert that the boundary terms occuring in partial integration of the equation must vanish. These natural boundary conditions can be regarded as being nonlinear analogues of Neumann's boundary conditions for nonlinear differential operators in divergence form, v(x)n(x).A(x,y(u)(x)) = v(x)Y:nj(x)A5(x,y(u)(x)) = 0 j=1

7.2 Surjectivity of Coercive Monotonic Operators

173

for all x E 8G and all v E V. Here n(x) denotes the outer unit normal to 8G at

the point x. Obviously, we shall not be able to express these natural boundary iconditions in the explicit form given above without making some assumptions about the smoothness of the boundary of G. The variational formulation then seems to be a suitable generalisation of this case. Further hypotheses on the coefficient functions Aj imply additional properties of the mapping T : V -. V': the monotonicity

(T(v) - T(u), v - u) >0, Vv,uE V,

(7.1.7)

and the coerciveness

(T(u),u) > CQjujj)jjuIj,

Vu E V,

(7.1.8)

where the function C : [0, oo) -. 1R has the property C(s) -. +oo for s -. +oo.

The surjectivity of the mapping T then follows by means of a result due to Browder (and Minty), and hence the solubility of the variational boundary value problem.

These abstract results will be proved in the following section together with a few generalisations. Finally, in Sect. 7.3, the abstract properties of the monotonicity and coerciveness given above will be transformed into concrete hypotheses for the coefficient functions Aj. To this end, we shall first of all recall the properties of Niemytski operators. The remaining part of the proof of the solubility of the variational boundary value problem then consists more or less of concrete estimates, using well-known inequalities. Remark 7.1.1. A complete discussion of the boundary conditions with specific examples can be found in [7.2-4].

7.2 Surjectivity of Coercive Monotonic Operators. Theorems of Browder and Minty Let E be a real Banach space and T a mapping from E into the topological dual space E' of E. We emphasise that the "operator" T is not assumed to be linear. However, in the sense of the definitions (7.1.7) and (7.1.8) we assume T be monotone and coercive. The underlying Banach space E will always be separable in our applications and we shall therefore prove results about the surjectivity of coercive monotone operators for the separable case only. Generalisations to nonseparable Banach spaces have been shown (by Browder and Carrol).

The basic theorem on the surjectivity of continuous coercive monotone operators was discovered independently by F. E. Browder and G. Minty in 1963. This theorem is the nonlinear analogue of the Lax-Milgram theorem.

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

174

Since we consider this chapter to be simply an illustration of the effectiveness of variational methods, we shall be content just to discuss the simplest versions of Browder's results [7.5]. First of all we consider the following two lemmas. Lemma 7.2.1. Let E be a Banach space and T : E -* E' a monotone map-

ping.

(a) Let T be continuous. Then for a given f E E', every solution of the equation T(u) = f is characterised by

(T(v)-f,v-u)>0, VvEE. The solution set T-1(f) for the equation T(u) = f is closed and convex. (b) Let T be coercive, i.e. a function C : [0,00) -* IR exists, where C(r) -->

+co for r - +oo, such that (T(u),u) > Qlull)llull, Vu E E. Then the solution set T-1(f) is bounded for every f E E':

T-'(f) S {u E El h ull S o(Ilfll')}, where

C(s) := sup{r.I C(r) < s). Proof.

(a) The monotonicity of T implies that the inequality above must

hold for a solution u. Conversely, if this inequality holds, then for arbitrary, but

fixed, z E E we can put v = u+tz, t > 0. It follows that (T(u+tz)- f, tz) > 0 and thus (T(u + tz) - f, z) > 0. Since T is continuous, the limit t \, 0 gives (T(u)-f, z) > 0. Since z is an arbitrary point in E it follows that T(u)-f = 0. Thus T-1(f) can be written as

T-1(f) = n {u E E (T(v) - f, v - u) > 0}

.

vEE

T-1(f) is thus the intersection of closed half-spaces

{u -E E I(T(v) - f,v - u) > 0} and therefore a closed convex set.

(b) For u E T-'(f), JJuJJ

(IIuII) < (T(u),u) = (f, u) s 11111' lull

It follows that C(lluIU <-11fll'

or

lull <- C(Ilfll').

Roughly speaking, the surjectivity of continuous coercive monotone operators is implied by the combination of the following two facts:

7.2 Surjectivity of Coercive Monotonic Operators

175

(i) continuous coercive operators on finite-dimensional Banach spaces are surjective; (ii) a generalised Galerkin approximation converges as a result of the monotonicity of the operators.

This indicates the central role played by the following lemma.

Lemma 7.2.2. Let T : F - F' be a continuous coercive mapping on the finite-dimensional Banach space F. Then T is surjective: T(F) = F'.

(a) To show that T(F) = F' it is sufficient to show that 0 E T(F), proof. since T(u) = f if, and only if, Tf(u) = T(u) - f = 0. Tf is continuous and coercive if, and only if, T has these properties.

Note, furthermore, that the hypotheses of this lemma remain true if we pass to an equivalent Banach space (possibly with another function C : [0, co) -* IR from the same class). Since we have assumed that dim F < oo, we can also assume that F is a Hilbert space; F and F can thus be identified. (b) For sufficiently large R > 0 we know C(R) > 0. For such a number R

we thus have, for s(u) = u - T(u) (s(u),u) < IIuoP

,

Vu E F,

]lull = R.

Let BR = {u E F I IIu1I < R}. The radial retraction r from F to BR is given by

v,

r(v)

for v E BR,

1v, for v O BR.

It follows that r(B j) C 8BR and, furthermore, that

f(u) = r o s(u) is a continuous mapping from BR into BR. Brouwer's fixed point theorem states that f has a fixed point uo in BR. If IuhIo < R, then the definition of r and the fact that uo = f(uo) show that f(uo) = s(uo) = uo,

and thus T(uo) = 0.

The case when IIuoII = R can be excluded, since it follows first of all from

IIuoII = R and no = f(uo) that s(up) E BR U MR, and thus a = Ils(uo)II >- R.

176

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

Then, f(uo) = r(s(uo)) =

s(uo),

P

which, however, gives a contradiction: R2 = IIuoII2 = (f(uo),uo)

Ps(uo),uo\ = P (s(uo),uo) <

R2

The following theorem, due to Browder and Minty, can now be proved quite easily. First of all, we introduce a suitable generalised Galerkin approximation. Then, Lemma 7.2.2 can be used for the approximating mappings.

Finally, we can pass to the limit, using the monotonicity property and the characterisation of the solution set T-1(f) by Lema 7.2.1. Theorem 7.2.3. (Browder-Minty). Let E be a separable, reflexive Banach space with dual space E' and let T : E -' E' be a continuous, coercive, mono. tone mapping. Then T is surjective: T(E) = E'. For fixed f E E', T-1(f) is a bounded, closed, convex subset of E. Proof. To show that T(E) = E' it is sufficient, as in Lemma 7.2.2, to show that 0 E T(E). 1st step: Definition of a suitable Galerkin approximation. Since E is separable, there is an increasing sequence (E of finite-dimensional subspaces E of E whose union is dense in E. Let On : E -. E denote the identical embedding and let 0', : E' -i E;, denote the adjoint projection. For given T : E -> E', put din,

nEIN.

(7.2.1)

It follows that all T. have the properties of T. This is obvious as far as the continuity is concerned. The monotonicity of the T on E. can be shown as follows. For arbitrary u, v E E. we have (Tn(u) - T,(v), v. - v)E., = (On,(,((,,T o Y',,(u) - To'^(v)), a -//v) E, = (T(Y'n(u)) - T(11'Yn(v)), 0.(u) - Wn(v))

>0. The "uniform" coerciveness of the T. can be shown similarly. For all it E E,,, the coerciveness of T on E implies (Tn(V'),u)E., = (T(W4)), W,(u)) 1 C'(Il4n(v)II)II0n(u)II = C(IIuII)IIuII,

where (,)E. denotes the duality of E and E;,.

7.2 Surjectivity of Coercive Monotonic Operators

177

$nd step: Proof of the surjectivity of the T.. Since T. : En --i E; is a continuous coercive mapping of the finite-dimensional Banach space En, Lemmas 7.2.2 and 7.2.1 give the existence of a un E E. such that T0(un) = 0 and

Ilunll < M = 15(0) < oo.

3rd step: Passage to the limit. Since E is a reflexive Banach space we see that the bounded sequence of the solutions u., has a weakly convergent subsequence (unj )jEN, Unj

' *w no

.

Now we show that T(u0) = 0. Let v E Em, m E IN, be arbitrary, but fixed. We have Em C En for all n > m and thus, by virtue of the monotonicity of Tn on En, 0 < (T,(v) - Tn(vn),v - vn)E.. = (T(v),v - un)E = (T(v),v - un).

passing to the limit gives

0 < (T(v), v - uo) and thus

0 < (T(v), v - uo) ,

Vv E U Em . mEN

Since T is continuous and UmEN Em is dense in E, it follows that

0 < (T(v), v - no)

,

Vv E E,

and thus, by Lemma 7.2.1 (a), T(uo) = 0. This shows that 0 E T(E), and thus

T(E)=E'. The monotonicity of T was not used above until the third step of the proof, to conclude from no = to - limj. uni that T(uo) = limj.. ,T(unj ).

From this it should be clear that every other property of T which allows to reach this conclusion also implies, using coerciveness, the surjectivity of T. A property of T like this which has proved itself useful in applications is given by the Smale condition: ( For every sequence (un),EN C E, the conditions (S)

u=w - limun and (T(un)-T(u),un-u) 0 mply u = s - lim nn .

The condition (S) can thus be considered as a generalised monotonicity condition.

The following theorem shows that the surjectivity of coercive mappings is still valid if, in the Browder-Minty theorem, the monotonicity condition is replaced by the Smale condition with T bounded. The boundedness of T means that T maps bounded sets onto bounded sets.

178

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

Theorem 7.2.4. Let E be a separable reflexive Banach space and let T : E -. E' be a continuous coercive mapping which is bounded and which satisfies the Smale condition. Then T is surjective: T(E) = E'.

Proof. Define the Galerkin approximation (En, Tf)SEN as in the proof of The-

orem 7.2.3. Now let f E E' be given. By Lemmas 7.2.2 and 7.2.1 there is a sequence (un)nEIN which has the following properties:

un EEn, T(un)=#;,(f),

IIuj <M
Since E is reflexive, a weakly convergent subsequence (unj )jEIN exists with the limit no. Since T is bounded, {T(un) In E IN} is also bounded. Now let m E IN and v E Em; E,,, C En for all it > in, and thus (T(un),v) = (Tn(un),v)E.. _ (b (f),v)E = (f,v) Therefore

lim (T(un), v) = (f, v)

.

n-m Since (T(un))-En is bounded in E' and U. Em is dense in E, it follows that

f = w - lim T(un) n-.oo and thus also that

Jim (T(unj) - T(uo),unj - u0)

lim (T(unj),unj) - lim (T(uo),unj -u0) - lim (T(unj),uo) = j-.oo j-too j-.oo

= lim (f,unj) - (f,uo) = 0. j-.oo and by virtue of the The Smale condition implies that uo = s - limj.oo continuity of T we also have T(uo) = s - limj.oo T(unj ). We have already

shown that f = w - limj.0T(unj) and so it follows that f = T(uo). Thus, T is surjective.

7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution The general strategy for the solution of the variational boundary value problem

has been indicated in Sects. 7.1 and 7.2 by Browder's abstract theorems, Theorems 7.2.3 and 7.2.4. We must now satisfy the hypotheses of these general theorems in terms of specific conditions on the coefficient functions A0 , ... , An

7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution

179

of the generalised Dirichlet form. In a first step we ensure, under the most general hypotheses possible, that a generalised Dirichlet form a is well defined on W''P(G) according to (7.1.1') and that this then has the continuity property (7.1.2). The crux of the proof that the hypotheses formulated in (Hl) for Ai

are indeed sufficient for this purpose can be found in [Ref. 7.6, paragraphs 18-201. For convenience we have summarised the relevant results below. For a given measurable subset B C R' we shall refer to a function g : B x ]g." -+ IR as a C function (Caratheodory function if the following conditions hold:

(i) y g(x, y) is a continuous function on IR" for almost all x E B. (ii) x F-4 g(x, y) is a measurable function on B for all y E 111". If g is a C function on B x IR" and if vi , ... , v" are measurable functions on B which are finite almost everywhere, then gv, given by 9v(x) = g(x,vi(x),...,v"(x))

(7.3.1)

for almost all x E B, is also a measurable function of B. If, moreover, B has gv is also continuous in measure. a finite Lebesgue measure, then v If B C Hi"` is a Lebesgue-measurable set and if g = (gl , ... , gN) are C functions on B x IR" which define via (7.3.1) mappings = (gr , ... , gN) from LP'n(B) = LP(B) x . . x LP(B) (n times) into LP; (B), j = 1, ... , N, then the mappings gi : LP,"(B) -. LP+(B) are continuous and bounded. g is thus a continuous bounded mapping from LP'"(B) into LP'(B) x . . x LPN(B), .

.

1
The operator g defined in this way is called the Niemytski operator, associated with g.

Remark 7.3.1. The statement above asserts in particular that the mappings gi associated with the C functions gi are already continuous and bounded (as mappings from LP'"(B) into LP' (B)) if they are defined only on the whole of LP'"(B) and take values in LPj (B). The following criteria express when this is the case:

(a) gi maps LP'"(B) into LPI (B), 1 < pi < oo, if, and only if, a function ai E LP+(B) and a constant b > 0 exist such that

jgi(x,y)I <_ ai(x) +bE

yiIPIP'

(7.3.2)

=1

for almost allxEBandally=lR". (b) gi maps LP'"(B) into L°°(B) if, and only if, a constant c > 0 exists such that Igi(x, y)I < c

for all yelit" and almost all xCB.

(7.3.2')

180

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

The hypotheses (H1) below for the coefficient functions Ao, Al , ... , An are

thus seen to be quite natural. (H1) (i) Let Ao, AI , ... , An be C functions on G x IRn+1 (ii) The following estimates for j = 0,1, ... , n hold almost everywhere

inxEGandforallyElRn+1: n

IAi(x,y)I

n

aj(x)+)3i(x)IyoIP'+E9ji(x)IyiI'''+)7fji(x)IyoIr'' Iyils'; i=1

i=1

(7.3.3)

subject to the following constraints: the exponents pj, qji, rji and sji are all nonnegative and satisfy

Pi,4ii:5 P,=P-

rii+sii SP-

and the nonnegative functions aj, /3j, fji and gji satisfy aji E LP'(G), Qj E LP/(P-1-Pi)(G),

P=P 9ji E LPAp-1-v,')(G),

fji E LPAP-1-';:-rii)(G). Vainberg's results, which we mentioned above, now make it fairly easy to prove the next theorem.

Theorem 7.3.1. If the functions Ao, Al , ... , An satisfy the hypotheses (H1), then (7.1.1') gives a well-defined generalised Dirichlet form a on W1'P(G) which has the continuity property (7.1.2). The mapping thus defined by (7.1.3), (W1'P(G))', is continuous and bounded. T : W1,P(G)

(a) Since it -, y(u) = (u, 81u,..., 8nu) is a continuous linear mapProof. ping from W'-P(G) into LP,n+l(G), we first of all show that the Aj are welldefined mappings from LP'n+1(G) into LP '(G). Vainberg's results then ensure the continuity of these mappings. It follows that u --' Aju(u),

Ajy(u)(x) = Aj(x,y(u)(x)),

is a well-defined continuous mapping from W1'P(G) into LP '(G). (b) F o r g = (go, 91, ... , 9n) E LP.n+l(G), the inequality (7.3.3) gives the estimate n

IIAigllp' < Ilajllp'+Ilpjlgolp'IIp'+YfIlgjilgjl'''IIP'+Ilfjilgolr"19i1g"llp i=1

i=1

The rest of the proof consists in repeated application of Holder's inequality, taking into account the restrictions on the exponents formulated in (H1).

7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution

181

Finally, we get the following estimate which shows that Aj is a mapping from the whole of LP,n+1(G) into LP(G):

jAigil"

IIajIIP'+II9iIIPI(P-1-Pi)IIyojIp' n

+E{II9iiIIP1(p_i_g1,)IIgjIIP" +

Ifill)PAP-1- ,i-ai,)Il9ollPLvillp"}

i=1

(7.3.4)

where the hj are functions which are bounded on bounded sets.

(c) The estimate (7.3.4) proves that it - Ajy(u) is well defined and a continuous mapping from W1=P(G) into LP (G). It is now easy to show for arbitrary it, v e W1"P(G) that n

Ia(u,v)I = (Aoy(u), v)2 + Y(`4iy(U), Ojv)2 j=1 n

1/p'

j=D

1/P

j=1

Thus, with h(s)P

hj(s)P,

j=o

and taking

(7.3.5)

Iy(u)IILP.'+1 = IvII1,P

into account, we get the estimate (7.3.6)

Ia(1L,v)I < h(IIuIi1,P)IIvil1,p'

(d) For arbitrary u, uo E W 1'P(G), we have, by (7.1.3),

JIT(u)-T(uo)lli,p=

sup

la(u,v) - a(uo,v) I

vE IIvIIi,P<1

<

sup

{I(Aay(u) - Aoy(uo),v)2I

IIvII1.P <_1

(Ajy(u) - Aiy(uo),Div)2l}

+ i=1

1/p

n

{

IIAjy(u) - Ajy(uo)II;

}

j_o

Since it r-, Ajy(u) is continuous, the continuity of the mapping T also follows, and the proof of Theorem 7.3.1 is thus complete. It follows from (7.3.6) that

182

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators IIT(u)IRi,p 5 h(IIuIll,n)

where the function h is bounded on bounded sets.

11

To prove the solubility of the variational boundary value problem with the aid of Theorem 7.2.3, we need additional hypotheses which imply the monotonicity and coerciveness of the mapping T. The following hypotheses provide a simple basis for this. (H2) Monotonicity. For almost all x E G and all y,y' E IR"+' we have ,[Ai(x,y)

- A1(x, y')][yi - 4) ? 0.

(7.3.7)

(H3) Coerciveness. Suppose there are a constant a > 0 and nonnegative func-

tions g, E LpAp-'A' 0 < rj < p, such that for almost all x E G and all y E 1Rn+' we have n

n

Ai(x, y)yi > a i=o

gi(x)Iyi fir'

Iyi Ip i=o

(7.3.8)

i=o

Remark 7.3.2. The coerciveness condition (H3) can, in this general situation, be regarded as a suitable generalisation of the ellipticity condition assumed in the linear problem. Later on, we shall discuss a weakening of the coerciveness condition which we shall show to be sufficient in situations in which the Sobolev embedding theorem can be used. This weakening will consist in omitting the summands Ao()yo and lyolp in the first two sums in the inequality (7.3.8), just as we did in the linear problem. The coerciveness hypotheses (H'3) is, in this version, a considerable weakening of the ellipticity condition used in linear problems (cf. Chap. 6).

We now come to the proof of the solubility of the variational boundary value problem. Theorem 7.3.2. For every open, nonempty set G C IR', the variational

boundary value problem (7.1.5),

A(u) = f with respect to V, has a solution for every closed subspace V satisfying

W0'p(G) C V C W''p(G),

and for every f E V' if the coefficient functions Ao, Al , ... , An satisfy the hypotheses (HI), (H2) and (H3). For fixed f E V', the family of all solutions is a closed, convex, bounded subset of V.

7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution

183

(a) Since V is a subspace of W'-P(G) it has the topology induced proof. by W1"P(G). Thus we can substitute V for W1'P(G)in Theorem 7.3.1. It follows that (7.1.1') defines a generalised Dirichlet form on V which satisfies the estimate (7.1.2) for u, v E V. In addition, (7.3.9) (T(u),v):= a(u,v), W E V defines a continuous mapping from V into W. (b) The monotonicity of T follows simply from Hypothesis (H2). If u, v E V, then it follows form (7.3.7) and (7.1.1') that (T(u) - T(v),u - v) = a/(u,u - v) - a(v,u - v)

= J dnx E{Aj(x, y(u)(x)) - Aj(x, y(v)(x))) c j-o x {yj(u)(x) - yj(v)(x)}

>0. (c) The coerciveness of the mapping T follows from (H3) applying Holder's inequality. For arbitrary it E V, we have (T(u),u) = a(u,u) = Jc

dnx

to

lyi

> JGdnx{

l

Aj(x, y(u)(x))yj(u)(x) j=o (u)(x)lp

-

j=O

j=o

9j(x)lyj(u)(x)1 '

J

n

_ a1Ialh,p -

III

j=o n

aIIuII1,,-

II9jIIP1(P-rj)llyj(u)llp' j=0

Since Ilyj(u)IIp <_ IIuIIi,p and 0 < rj < p, this inequality implies, by virtue of

the fact that p - 1 > 0, that (T(u),u) ? C(IIuIII,P)IInIII,P

with a function C : [0, oo) -. IR which has the property that C(s) -. +oo for s +oo. (d) Since V is a reflexive Banach space, part (a)-(c) of the proof show that T satisfies the hypotheses of Theorem 7.2.3. Finally, applying this theorem proves Theorem 7.3.2.

Remark 7.3.3. We would like to draw particular attention to the fact that, in Theorem 7.3.2, no special assumptions were used about G. It would, for example, be possible for G to be unbounded and not have a smooth boundary (meaning that Sobolev's embedding theorems could not be used).

184

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

If we assume that G is an open, bounded subset of IR", then for arbitrary r,q > 1 we have IGI(1-1/4)/rIfllr9

Ilfllr

where

IGI =

JC

d"x,

which means that L' (G) is continuously embedded in Lr(G). If, in addition, we assume that G also has a smooth boundary, then we may use Sobolev's embedding theorems, i.e. for

01p

1

1

n

r

W1"P(G) is continuously embedded in Lr(G), and when

---<-
1

1

this embedding is compact and the Sobolev inequality Ill hr < K(IGI,p,r) lIVfl1P holds. We can, of course, use this additional information to weaken Hypothesis (H1) somewhat, one possibility being given by Browder [7.7[. However, it is much more important to take the opportunity to weaken Hypotheses (H2) and (H3) considerably and still be able to guarantee the solubility of the variational boundary value problem with the aid of Theorem 7.2.4.

We shall discuss the following version in some detail, since, firstly, it uses the direct generalisation (H3) of the ellipticity condition assumed for linear problems and also because the proof of this version solves a problem explicitly which is typical in this variational calculus, i.e. it derives strong convergence of a sequence from its weak convergence and suitable additional hypotheses.

In order to be able to conclude from hypothesis (H3) that the mapping T is coercive, we need a strengthening of Hypothesis (H1). We thus make the following assumptions.

(Hi): Hypothesis (H1) with the restrictions 0

rot+so:
(H`2): For all almost x E G and all yo E IR. we have {A3(x,yo,y) - At(x,yo,Y)}{yi - y,} > 0 i=1

for y# y'. (H3): A constant a > 0 and a nonnegative function

go E LP/(P-ro)(G)

0
7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution

185

exist such that for almost all x E G and all y C IR+' we have

EAi(x,y)yj > a> IyjIP j=1

-9o(x)lyoI'

j=1

We can thus prove the following theorem.

Theorem 7.3.3. For every open nonempty set G C lRn which is bounded and which has a "smooth boundary", the variational boundary value problem (7.1.5), A(u) = f with respect to V, has a solution for every closed subspace

V where Wo'p(G) C V C W' P(G) and for every f C V' if the coefficient functions satisfy the hypotheses (Hi), (H2) and (H3). (a) Hypothesis (H'1) implies, as above, that (7.1.1') and (7.1.3) define a continuous mapping T from V into V' which is bounded on bounded subsets of V. This time, however, it is rather more difficult to prove that T is coercive. proof.

(H3) gives E(Aiy(u),yi(u))2 = J dnx E AI(x,y(u)(x))yi(u)(x) j=1

j=1

JG

d°x{ aE lyi(u)(x)IP - 9o(x)lu(x)Ir° }

l

j=1

allonllp - Ilsolul'lh > allonllp -

Ilgollp/(P-r0)Ilullp

It follows that n

(T(u),u) _ (Aoy(u),u)2+E(Ajy(u),yj(u))2 j=1

allonllp-II90llp/(P-ro)IIUIIP -IIAoy(u)Ilp'llnllp Estimate (7.3.4) shows that under Hypothesis (Hi) we have IlAoy(u)llp, < b(Ilulll,p)llull7 P

where b is a bounded function b : IR+ -, 1R+ and 0 < s < p - 1. The Sobolev inequality states that IlVfllp and Ifll1,p are equivalent norms, i.e.

IIVflip 5 IIfII1,p
(

(u),u) >

Ivullp

allonllp-1-

allonllp

llgollpl(P-ro) Illlull

- C2IIVUIIp

-1-

+oo for IlVullP -'oo,

IIP -

b(llulll.p) IlloullP b(IIuII1,p)Cs+11IVullp

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

186

since p - 1 > ro - 1 and p - 1 > s. It also follows that (T(u),u) _ = +00,

lim hull=,,

IIkIIi,p

and thus that T is coercive. (b) Since V is a separable reflexive Banach space, Theorem 7.3.3 follows from Theorem 7.2.4 if we can prove that T also satisfies the condition (S). Let {ui}iEIN C V be a weakly convergent sequence with the limit u, for which

(T(ui) - T(u), ui - u) -* 0.

(7.3.10)

For 1 < p < n, W''P(G) is compactly embedded in LP(G) (for p = n in L'(G),

1/r > 1/p - 1/n = 0) and so V is also compactly embedded in LP(G). The weak convergence in V implies that {ui I i E IN} is strongly bounded in V. Then jut I i E IN} is relatively compact in LP(G), i.e. a subsequence {uik}kEM

and a set N C G of measure zero exist such that

(i) uik -+ u strongly in LP(G) k-+oo

(ii) uik(x) ;u(x)pointwiseinG\N. }

(*)

We shall simplify the notation by again denoting this subsequence by {ui}iEN.

Its image under the mapping y : W',P(G) - LP'n+r(G) has the following properties:

<M < oo, Vi E IN, (') (ii) {yo(ui)}iEt satisfies (*), (iii) {yj(ui))iEJ converges weakly in LP(G) to yo(u), j = 1,...,n.

(7.3.11)

We still have to show that the sequences {yi (ui) , .... y (ui)}iEIN converge strongly in LP(G), and to this end it is helpful to recall the following facts.

(a) A weakly convergent sequence in a normed space converges strongly if, and only if, it contains a strongly convergent subsequence. (0) Since G has a finite Lebesgue measure, Vitali's convergence theorem [Ref. 7.8, Theorem 111.6.15] states that {y3(ui)}iEW converges strongly in LP(G) if the following conditions are satisfied: (i) {y3(ui)}iEN converges pointwise almost everywhere.

(ii) The sequence {yj(ui)}iEn is uniformly absolutely continuous, i.e. for every e > 0 there is a S. > 0 such that for all measurable subsets A C G with measure JAI < 6, and all i E IN, Iyi(ui)(x)Ipd"x < E. The strong convergence of the sequence {yj(ui)}jEN in LP(G) follows if

we can prove the latter two conditions for a suitable subsequence.

7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution

187

(c).Our first step is to reduce the condition (7.3.10). Since {Aoy(ui)}iE¢.i is bounded in LP (G) (inequality (7.3.4)), the strong convergence of {yo(ui)}iEw L7(G) gives

lim (Aoy(ui) - Aoy(u), ui - u)2 = 0. It thus follows that Urn Y(Aiy(ui) - A1y(u),Yj(u;) - Yj(U))2 = 0. i=1

Since

Aj : L""'+1(G) -, LP (G) is continuous, we also have lim Ai(-,yo(ui),y(u)) = A5(-,yo(u),yo(u))

in Lo (G), so that (7.3.10) finally reduces to the following assertion of convergence: Jim d"x '-'°° Jo

E{Ai(x,ui(x),y(ut)) - Ai(x,ui(x),y(u))} j=1

x {yi(ui) - yi(u)} = 0.

(7.3.12)

By Hypothesis (H2), the integrand Ii(x) in (7.3.12) is nonnegative, Ii(x) > 0, so that (7.3.12) expresses exactly the strong convergence of {Ii};EE in L'(G):

Ii -. 0 in L'(G),

II (x) > 0.

(7.3.13)

It follows that a subsequence {Ii, }kEiN and a set N' C G of measure zero exist such that (7.3.13') Ii,(x)k 0, VxEG\N'.

(d) The next step is to show that Condition (i) of (3) above holds for a suitable subsequence of {yi(ui)}iE]N. A simple rearrangement gives n

E Ai (x, y(u'i ))yi (u i) i=1

n

_ E Ai (x, y(ui))y1(u ) i=1 n

+

E Ai(x, yo(ui), y(u))Iyi(ui) - yi(u)I

j=1

+11(x). Thus, by virtue of (H2), we can make the following estimate:

(7.3.14)

188

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

E y1(ui)jP < 1a EAi(x,y(ui))yi(ui)+ a9o(x)jyo(ui)jr0 j=1

1=1

9o(x)lyo(ui)1r0 +

LA1(x,y(ui))yi(u)

j=1

(7.3.15) r=1

Estimate (7.3.3) for the Al can also be represented for lied < 1, in the following way. We have, for suitable 0 < o, p, a + p < p - 1,

'+7i(x)lyl' '+6J(x)lyol

lAi(x,y)l Cai(x)+3 (x)lyoI" lyl' where the functions ai, ... , 61 are all finite outside a common set No C G of measure zero. If we substitute this estimate in (7.3.15), then it follows for

xEG\Nothat

E lyi(ui)(x)IP < i=1

19o(x)lyo(ui)(x)Ir0

a

+1 a Ii(x)

+ F(x,yo(ui)(x),b(u)(x))Iy(ui)(x)IP-1

(7.3.16)

where F is a function which is bounded on bounded sets in G x 1R"+1 However,

since Iik and yo(uik) converge at all points it E G \ No U N U N', estimate (7.3.16) shows that the sequence {y(uik)}kEP4 is bounded at all these points. For an arbitrary, but fixed, point it E G \ No U N U N', let y(uikj(x)) be a subsequence which converges to a point y E IR'. Since we also have uik(x) - u(x), it follows from (7.3.13') and the fact that the Al are continuous in y E 1R"+1 that n

E)A1(x, u(x), y) - Ai (x, u(x), y(u)(x)))(yi - yi (u)(x)) = 0. i=1 If we had y

5,1

y(u)(x), then there would be a contradiction to (HZ). WVe

therefore have

lim y(uikt(x) = y(u)(x) . This shows that every convergent subsequence {y(uiki)(x)}1EV of the bounded subsequence y(uik)(x) has the same limit, i.e. y(u)(x). Then {y(uik)(x)}kEN itself converges to y(u)(x). Since we chose it E G \ No U NUN' to be arbitrary, then the pointwise converges in C \ No U N U N' follows, and thus so does the condition (i) in (/3) above for {y(uik))kENN.

(e) In the final step we shall show the uniform absolute continuity of the sequences {y(uik)}kE]N in LP(G). We integrate (7.3.15) over an arbitrary measurable subset H C G and then estimate termwise:

7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution

J

H

d"x E Iyj(ui)(x)Ip<«J d-xli(x)+I fH H

189

dnx9o(x)lui(x)Iro

j=1

+

-

H

ex j=1 Aj(x,y(ui))yi(u)(x)

ui, y(u))[y](ui) - yj(u)ld°x . + a H A1(x, r-1 (7.3.17)

If E > 0 is now given, then an ie E IN exists such that for all i > i, we have

f dnxli(x) <

,

-

c 4 since 0 < Ii converges to zero in L1(G). This estimate then holds for all measurable subsets H C G. Let us therefore choose a bo > 0 such that for all measurable H C G, where IHI < 60, we have

JH

d"xIi(x) <4 ,

i=1,...,8E-1,

which implies J H dnxIi(x)

<4 for all i E IN and all such H C G. By virtue of

fx

d^x9o(x)lui(x)ll S

<M1119o1Ir.^(H),

where P

and go E L°(G) ,

p - ro a 61 > 0 exists such that for measurable subsets H C G, with IHI < Si, it follows that

a 1 x d"x9o(x)lui(x)I- <

if

Vi E N.

4

The estimate of the third term in (7.3.17) follows in a similar fashion: Ai(x,y(ui))yi(u) < h(IIui111,p)Ilull

d"x

H

j=1

and the boundedness of {h(IIuilll,p)}iEN shows that a 62 > 0 exists such that for measurable H C G, IHI < 62, it follows that

L

a J

C

j=1

,

biEIN.

190

7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators

Since an estimate for the last term in (7.3.17) can also be done similarly, i.e.. J dnx E Ai(x,ui,y(u))(yi(ui) - yi(u)]

a H

i.l

n

1 I}Ai(-, ui, y(u))1ILn' (H){IlYi(ui)IIp + Iyi(u)IIp} j=1 n

M3 E II A3(', ui, y(u)II Ln# (H), j=1

the previous explicit estimates of IlAj( ...)11

and the convergence of {ui)iEP7

in LP(G) show that a 63 > 0 exists such that this term is also less than or equal to e/4 for all measurable subsets H C G, JHI < 63, and all i c N. If we choose S = min{bo, 61, 62, 63}, then by (7.3.17) it does indeed follow for all

i E IN and all measurable H C G, IHI < 6, that

JH

dnx1: lyj(ui)(x)lp<_e, j_1

which is Condition (ii) of ()3) for {y(ui)}iEpy, and thus also for the subsequence {y(uik)}kEu.

(f) It follows that the remarks (a) and ()3) above imply that {yj(ui)}iEN converges strongly in LP(G) to yj(u) and thus that the sequence {ui}IEN converges strongly in W1 "p(G) to u. The mapping T thus has the property (S) and Theorem 7.2.4 completes the proof. Remark 7.3.4. Theorems 7.3.2 and 7.3.3 solve a large class of boundary value problems for the quasi-linear differential equation (7.1.1) in the sense of distributions of G. However, these theorems say nothing about the important problem of the regularity of these weak solutions, namely whether the solu-

tions it in V, WJ'p(G) C V C W',P(G), are also solutions in the classical sense.

Example 7.3.5. Let G C W' be an open, nonempty, bounded subset with a smooth boundary, and let V be a closed subspace of W1"2(G) = H'(G), where Ho (G) C V. Then, with the aid of Theorem 7.3.3, it is very easy to solve the variational boundary value problem for n > 2,

for arbitrary f E V' with boundary conditions defined via V. We only have to put p = 2 in Theorem 7.3.3. All hypotheses (Hi'), i = 1, 2, 3, are then trivially satisfied and the solubility follows.

7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution

191

Obviously, solution by means of the linear theory of Chap. 6 is even simpler. This is, however, not the case if a simple nonlinear modification is made to this equation, say for n = 2. Let the Caratheodory functions p;j, i, j = 1, 2, be given on G x IR, with the following properties:

(i) There is an a > 0 such that for all (x, yo) E G x ]R and all z E 1R2 we have

2

2

z.P.j(x,yo)zj > a Ez2

(z,P(x,yo)z) _

i,j=1

j=1

(ii) Functions fij E L°°(G) exist such that for all yo E lit and almost all x E G we have

Paj(x,yo)J
Aj(x, u(x), Vu(x)) = f(x) - EO j=1 with boundary conditions defined by V and for given f E V' for the case when 2

Aj(x,yo,y) _

Pji(x,yo)yi =1

To this end, we show that the hypotheses (H;), i = 1, 2, 3 are satisfied. Condition (i) above easily implies the monotonicity hypothesis (H'2) and the

coerciveness hypothesis (H3) for p =2 = n. It follows form (ii) that (Hl) is also satisfied for p = 2. Theorem 7.3.3 thus guarantees the solubility of this boundary value problem, which reduces to the linear boundary value problem

-du = f when p,j(x, yo) = 6;j.

8. Nonlinear Elliptic Eigenvalue Problems

8.1 Introduction As a further application of the direct methods of the calculus of variations let us discuss a special class of nonlinear eigenvalue problems. As far as the technical framework is concerned, we proceed here as in our treatment of nonlinear boundary value problems in Chap. 7, which means, that if we are seeking solutions in a region G C 1R", we work in an appropriate Sobolev space Wm'p(G) = E. The starting point of this method of solution is the following simple application of Theorem 4.2.3 on Lagrange multipliers. If f and h are two C1 functions on E with the derivatives Df = f and Dh = h', then we can solve the nonlinear eigenvalue equation

f'(u) = Ah'(u),

u E E, A E IR,

(8.1.1)

in a simple way by determining the critical points of the function h on suitable level surfaces f-1(c) of f or, conversely, by determining the critical points of f on sutiable level surfaces h-1(c) of h. The eigenvalue A appears thereby as a Lagrange multiplier. To locate the critical points precisely, it is now necessary to impose certain restrictions for the level surfaces and the functions whose critical points on the level surfaces are to be determined. This introduces a certain asymmetry in the roles of f and h, which is also manifest in the hypotheses for f and h.

For our particular application in this direction, we start with an elliptic differential operator A of second order in divergence form and an operator B of zero order. This means that A and B can be expressed as follows. For a function it : G -* 1R we put Y(u) = (Y0(u),Y1(u),. . .,Y"(u)) = (n, 81u ,.. .,8"u).

(8.1.2)

For the functions Fi : G x IRn+1 -. 1R we have, therefore, Vx E G,

F1(Y(u))(x) =Fi(x,Y(u)(x)),

(8.1.3a)

and similarly, for a function Ho : G x 1R, we have Ho(u)(x) = Ho(x,u(x)),

Vx E G

.

(8.1.3 b)

8.1 Introduction Now let

193

n

A(u) = Fo(y(u)) - E aiF'i(y(u))

(8.1.4 a)

i=r

B(u) = Ho(u).

(8.1.4 b)

Here we assume that the functions Fj are the partial derivatives of a function

F:GxIR"+r-1R,i.e. Fi (x, y) = aF (x, y)

(8.1.5 a)

and, similarly, Ho(x,yo) =

(8.1.5 b)

ay (x, yo)

where H is a function H : G x IR , IR. Now it is necessary to state precisely what we understand by a solution of the quasi-linear elliptic eigenvalue equation

A(u) = AB(u)

(8.1.6)

on G. This is given in the following definition.

Definition 8.1.1. Consider an open set G C IR" and two quasi-linear partial differential operators A and B on G, as given by the (8.1.2-8.1.5) in divergence

form. Furthermore, let V be a closed subspace of the Sobolev space E _ Then u is said to be a solution of the eigenvalue problem (8.1.6) for the variational boundary conditions defined by V if there exists a real number A = A(u) such that a(u,v) = Ab(u,v), Vv E V. (8.1.7)

is the generalised Dirichlet form for the operator A on V and similarly b is the generalised Dirichlet form for the operator B on V; in other Here

words, for all u, v E V we have a(u,

yi(v))2=1: i=o

(

i=o c

d"xFi(x, y(u)(x))yj(v)(x), (8.1.8 a)

b(u,v) = (Ho(u),v)2 = J d"xHo(x,u(x))v(x).

(8.1.8 b)

G The connection with the eigenvalue problem given by (8.1.1) can be es-

tablished if we can show under suitable hypotheses that

a(u,v) = (f'(u),v), b(u, v) = (h'(u), v) Vu, v E V

,

(8.1.9 a) (8.1.9 b)

194

8. Nonlinear Elliptic Eigenvalue Problems

where we set

f(u)=fd"xF(x,(y)(u)(x))

(8.1.10a)

h(u)=J d"xH(x,u(x)).

(8.1.10b)

c

and

c In (8.1.9), ( , ) denotes the duality between V` and V where V' is the topological dual of V.

Remark 8.1.1. There is no difficulty in principle in extending the theoretical framework to eigenvalue problems of the order 2m, m > 1, and this extension has been explicitly carried out by Browder [8.1, 2]. We shall, however, restrict ourselves to the simpler case of m = 1, because (i) its treatment is much more transparent and brings out clearly the general strategy; (ii) in most of the applications, particularly in physics, this is the case needed. An overview will make the orientation easier. Section 8.2 deals with the determination of a solution to the eigenvalue problem (8.1.6). In analogy to linear eigenvalue theory, this can be regarded as equivalent to the problem of determining the ground state. The approach takes two steps. First we develop an abstract theory for the boundary value problem. Then we state the hypotheses for the coefficient functions for the differential operators A and B in the form (8.1.4), from which we deduce the hypotheses for the abstract theory. We shall then give explicit examples to

illustrate the different hypotheses. In this way the solution for the ground state of (8.1.6) will be obtained by determining the minimum of h on the level

surface f-1(c) off (or the maximum of h, = h r f-1(c)). The determination of eigenfunctions other than the solution of the ground state is already indicated by linear eigenvalue theory. The classical minimax principle of Poincare-Fischer-Weyl-Courant et al. gives the most convenient machinery known for the derivation and investigation of eigenvalues of higher order in the linear case, by utilising variational methods. For example, the minimax principle enables one to find the eigenvalues and the beginning of the essential spectrum of a self-adjoint operator A bounded from below in a Hilbert space ?f [Ref. 8.3, Theorem XIII.1]. An essentially topological version of the minimax principle which covers in particular nonlinear problems was first proposed by Ljusternik [8.4] and then further developed by him and others. This version of the minimax principle shows the relation between the theory of critical points and the LjusternikSchnirelman theory of "category" of subsets of manifolds on which the critical points are sought [8.5, 6]. Very important steps for the further development of Ljusternik-Schnirelman theory, were the extension to smooth Hilbert manifolds by J. T. Schwartz

8.2 Determination of the Ground State

195

and the extensions to smooth Banach manifolds (Finsler manifolds) by R. S. Palais [8.7]. F. E. Browder [8.1, 2] proposed a further extension of the LStheory on Finsler manifolds including investigations of concrete eigenvalue problems. We refer the reader to Palais [8.7], Browder (8.1] and Rabinowitz [8,8] for a detailed list of sources on the development of LS-theory.

If we want to realise the level surfaces f-1(c), c E IR, of a C' function f : W1,'(G) IR as Finsler manifolds and then apply the fairly demanding LS-theory for Finsler manifold, then we must assume p > 2 as shown in [8.2]. This means that additional conditions are needed for the coefficients F and F. We therefore prefer F. E. Browder's method of Galerkin approximation [8.9], which

(i) only requires the LS-theory for C' manifolds and hence is much simpler; (ii) allows finite-dimensional approximations for an infinite-dimensional eigenvalue problem, a property which is very useful not only from a practical point of view;

(iii) is also capable of treating the cases of C' functions f on W"(G) where

1
The Ljusternik-Schnirelman theory for compact manifold in 1R" is discussed in Sect. 8.3. In Sect. 8.4 we begin with the investigation of special level surfaces f-' (c), for which, finally, a "Galerkin approximation" is developed. This approxima-

tion then enables us to implement the results of the LS-theory for compact manifolds in lR" to suitable level surfaces in a separable Banach space. In this

way we obtain an abstract existence theorem which gives the lower bound for the number of solutions of equation (8.1.6). Finally, a class of concrete quasi-linear elliptic eigenvalue problems is treated. A few explicit examples are given to illustrate this quite general result.

8.2 Determination of the Ground State in Nonlinear Elliptic Eigenvalue Problems 8.2.1 Abstract Versions of Some Existence Theorems We present here two abstract existence theorems for the eigenvalue problem (8.1.1). Roughly speaking, the two theorems differ in the fact that the two functions f and It exchange their roles in the variational approach (maximising or minimising, respectively, under constraints). In the second result, the solution is characterised by a minimality property and can therefore be directly interpreted as the "ground state solution". It is characteristic for the proof of both theorems that one again applies a generalised Galerkin or, more precisely, a Rayleigh-Ritz approximation.

196

8. Nonlinear Elliptic Eigenvalue Problems

Theorem 8.2.1. Let E be a real, separable, reflexive Banach space whose dual space is denoted by E'. Let the functions f, h E C1(E,IR) have the following properties:

(a) Hypothesis for the constraint f. For c E IR let the following be true: (ai) Mc(f) = f-1(c) = {u E E I f(u) = c} is bounded. (a2) (f'(u),u) $ 0, Vu E MM(f). (a3) f' : E -* E' is bounded. (a4) f' satisfies Condition (S) (Sect. 7.2).

(b) Hypothesis for the function h which is to be maximised on M0(f). (b1) h' : E -+ E' is compact. (b2) h is weakly continuous on bounded sets of E.

(c) Hypothesis of "relative boundedness" of h' and f'. For every sequence {un}fEIN C MM(f) there exists a co > 0 such that (h'(un),un)I ? c0I(f'(un),un)I,

Vn E IN

is true.

It follows then that there exists an u E M0(f) and a A E IR satisfying (1) h(u) = maxvEM,(f) h(v),

(ii) f'(u) = Ah'(u). Proof. Step 1. Rayleigh-Ritz Approximation. Since E is separable there exists a total sequence {xn}nEN in E which contains a total subsequence in MM(f). Then we have En = lin{x1,...,xn},

a sequence of finite-dimensional subspaces of E, with En C E;,+1

and E= U E .

(8.2.1)

nEN

Now, setting

Mqn= MM(f) n En,

it E IN,

(8.2.2)

we get similarly

M n 9 M.n+r

and MM(f) = U nEIN

Here the relation

U Mc,n c Mc(f) nEIN

is evident, and the assumption

(8.2.3)

8.2 Determination of the Ground State

197

Mc(f) \ U Mc,n # 0 nEIN

can be easily shown to lead to contradiction.

Let Wn : En -* E denote the identical embedding and Wn : E' -, E;, the

dual projection of Wn. Then, setting

fn=foWn=f tEn, nEIN it follows

that fn=Wn

(8.2.4 a)

of o W.

(8.2.4 b)

because f E C1(E,IR) means

f(x + y) - f(x) = (f'(x),y) + o(y), Vx,y E E

.

In particular, for x = wn(u), y = tpn(h), u, h E En, fn(u + h) - fn(u) = f(pn(u) + Wn(h)) - f(Wn(u))

_

(f'(Wn(u)),Wn(h)) + o(Wn(h))

_ (Wn o f' o Wn(u), h) + O(Wn(h))

and hence (8.2.4 b) holds. As a consequence of (a2), Mc(f) is a C' manifold in E of codimension 1.

According to (8.2.2) and (8.2.4 a), we have Mc,n = MM(fn), so that by (8.2.4) and (a2), Mc,n is again a C1 submanifold of codimension 1 in En since

(f. (u), u) # 0,

Vu E Mc,n .

(8.2.5)

Setting also hn = h t En, we get

h' = pn o h' o Yn.

(8.2.6)

Step 2. Solution of the Finite-Dimensional Eigenvalue Problem. According to (a,), MM(f) is bounded; therefore, all Mc,n in the finite-dimensional Banach space E. are bounded. Further, as they are all closed, it follows that all M,,n are compact C' submanifolds in En. Weierstrass' theorem gives therefore that for all n E IN there exists un E Mc,n

such that hn(un) = sup h(u).

(8.2.7)

-EM...

According to (8.1.5) the theorem of Lagrange multipliers is applicable. It follows therefore that for all n E IN there exists An E IR such that h.(un) = A.Wn(un)

is true.

(8.2.8)

8. Nonlinear Elliptic Eigenvalue Problems

198

Step S. The Limit. This part of the proof is the real core of the theorem. While in the two foregoing steps we have utilised only the easier part of the hypothesis, we shall see now that the hypotheses of Theorem 8.2.1 are strong enough in their entirety to control the limiting process. (a) From (8.1.8) we get (hn(un), un) = An(fn(uv), un) ,

do E IN.

The hypothesis (c) gives us that ( = 1/Aw)fEUJ is bounded. Since the sequence is bounded in norm and E is reflexive, there exist subsequences {u,y

and { f,,, } jEiN so that j-+oo

it in E and fn, -* f E IR.

(8.2.9)

The hypothesis (b2) gives

h(u) = jlim h(u,,;). According to (8.2.6) and (8.2.7), we have for all j E IN sup

h(un,) = hn,(un,) =

h(v) <

h(v) =

sup

vEMt .

,

and therefore h(u) = sup jEIN

sup

h(v) =

sup

h(v).

(8.2.10)

vEM.(f)

This shows that the function It on the level surface Mjf) attains its supremum at a point u E E. This point can be obtained as the weak limit of the eigenfunction }jEIN of the finite-dimensional approximation of the eigenvalue problem.

(fl) It remains for us to show that this point u really lies on the level surface. As soon as we can prove u E MM(f), it follows easily that u is also an eigenfunction of the pair f', h'; because from (8.2.10) we then have

h(u) =vEMAf) max h(v), so that by assumption (a2) the theorem of Lagrange multipliers, guarantees the existence of a number p E IR such that pf'(u) = h'(u). We show that it E MM(f) by proving that the sequence u,,., j E IN converges not only weakly but also strongly towards it. Condition (S), which f satisfies according to (al), suffices for this.

(y) Verification of the hypothesis of Condition (S) for the sequence Let M E IN and v E E be arbitrary, but fixed. The hypothesis (b1) guarantees the strong convergence of the sequence {h'(u,y)}jEN in E', say towards w E E'. For nj > m it follows from (8.2.4b), (8.2.6) and (8.2.18)

that

8.2 Determination of the Ground State

199

(f'(unj),uni -v) = (Wn. of oWnj(unj),unj -v) , _ (fnj Wnj o h' otpnj (unj), unj -v)

= Snj (h'(u,1), unj - v) ,

and therefore by (8.1.9) and the strong convergence h'(unj)

lim (f'(unj),unj - v) = jam lim {£nj(h'(unj),unj -v)} = £(w,u - v).

Thus, for all v E Un En, jly (f'(uj ), unj - v) = e(w, u - v)

(8.2.11)

We now show that (8.2.11) is valid for all v E E. We have by (a3) that C1 = SuPjEIN IIf'(unj )II' is finite. Now let v E E and e > 0 be given. According to (8.2.1) there exists a vo E (_f nEIN En such that

IIv - voll Ce :=

1 IIwII')e.

2(G1+m By (8.2.11) there exists a jo = jo(vo,e') corresponding to vo such that for all

j?jo I(f'(unj ),unj - vo) - e(w, u - vo)I <- 2 . Hence it follows for all j > jo that (f'(u nj ), unj - v) - S (W, u - v) <- I(f'(unj ), unj - vo) - C(w, u - vo)I + I(f'(unj) - ew, vo - v)I

< Z +{IIf'(unj)II'+Iei IIwII'}Ilvo -vII <-

+ 2 =E.

Therefore (8.2.11) is valid for all v E E. For v = u, in particular, we get

jllm (f'(unj ), unj

- u) = 0

and thus by (8.2.9)

hm (f'(unj) -f'(u), unj jo0

- u) = 0.

This means that the weakly convergent sequence {unj }jr- EC E satisfies the hypothesis of Condition (S) for the function f'. This condition therefore implies the strong convergence of this sequence

u=s - lim unj j400 (6) Since f is continuous we have

f(u) = jlm f(unj) = C,

8. Nonlinear Elliptic Eigenvalue Problems

200

which means that it E Mc(f). It follows in addition that

(h'(u),u) = lim (h'(unj),u,,) _ (w, u) (f'(u),n) _ .1iMW (u nj), nnj Hypothesis (c) with co = co({unj}FEIN) and (a2) allow us to conclude that I(h'(n),u)I

I(f'(n),u)I

=

lim I(h'(unj),un,)I > CO>(),

j-j-- (f'(n'nj),unj)I

and therefore that (h'(u),u)

P =

0.

(f#(u),u)

Thus we obtain the assertion (ii) where A = 1/p. Finally, it follows that p = f = limi.c Sn and hence that A = The following theorem proves the existence of a solution of a nonlinear eigenvalue problem of the form f'(u) = Ah'(u) by determing the "ground state" of f on the level surface of h in accordance with variational theory. The hypothesis and the methods of proof are therefore similar to those of Theorem 8.2.1.

Theorem 8.2.2. Let E be a real, separable, reflexive Banach space whose dual space is E'. Let the functions f, h E C,(E,1R) have the following properties:

(a) Hypothesis for "constraint". For c E lit let the following hold: (ar) (h'(u),u) > 0, Vu E Mp(h) = In E E Ih(u) = c}.

(a2) VR > 0, 3c(R) > 0 : (h'(u),u) > c(R), Vu E Mc(h) fl BR(0). (as) h' : E -, E' is compact. (a4) h is weakly continuous on bounded sets. (b) Hypothesis for the function to be minimised:

(br) f(u) --, oo for (lull -++00. (b2) f' : E -, E' is bounded. (b3)

f' : E -. E'

satisfies the condition (S).

Then it follows that there exists a it E M,(h) and a A E lR such that (i)

(ii)

f(u) =

Ui

f(v),

f'(u) = Ah'(u).

8.2 Determination of the Ground State

201

Proof Step 1. Rayleigh-Ritz Approzirnation. Let us define the finite-dimensional subspaces E. as in Step 1 for the proof of Theorem 8.2.1. Then it follows similarly that Mc,n = MM(h) fl En C Mc,n+i .

M0(h) = U Mc,n,

(8.2.12)

nEIN

MM(h) is a C' submanifold of codimension 1 in E and so is Me,n in En for sufficiently large n E IN. Further, (8.2.4) and (8.2.6) are valid. Step 2. Solution of the Finite-Dimensional Eigenvalue Problem. The formula

f(u)-f(v)=10

1

dT (f '(v +'r (u - v)), u - v)

a

and the hypothesis (b2) imply that f is bounded on all bounded sets of E. By (b1) there exists an R, > 0 such that inf{f(u) I u E MM(h)} = inf{f(u) I u E MM(h) fl BR--,(O)} - m > -oo. (8.2.13) For all n E IN it follows that

mn_1 > mn := inf{f(u) I u E Mc,n} > m.

(8.2.14)

Let v1 E Mc,1. According to (b1) there exists an R > 0 such that 11VII > R implies f(v) > f(vl ). It follows then for all n E IN that mn = inf{ fn(u) u E Mc,n} = inf{fn(u) In E Mc,n fl BR(0)}

(8.2.15)

But again, Mc,n n BR(0) is a compact subset of En. Therefore it follows in the usual way that for all n E IN there exists a un E Mc,n fl BR(0) and a A. E IR satisfying fn(un) = min fn(v),

fn(un) = Anh'n(un).

(8.2.16 a)

(8.2.16b)

Step S. The Limit. By (8.2.4c) and (8.2.6) we have f(un) = fn(un) =

v

in.

f(v)

> vEm R+tf(v) = f(un+1)

Since

MM(h) = U Me,n nEIN

holds, it follows that f(un) \n_.c m. For R > 0 fixed as in (8.2.15) choose c(R) > 0 by (a2). It follows then that (hn(un),un) > c(R),

Vn E IN,

8. Nonlinear Elliptic Eigenvalue Problems

202

and therefore by (8.2./16b) and (b2) that Iani

(fn(un),un)

_ (h

R

sup W(vA' <00

c(R) VEB,(O)

I

for all nEIN. In the usual manner we then obtain the subsequences {unj }jEN

and

{Anj }jEN with

un. -°1 uEE and an' . -i AEIR.

(8.2.17)

The strong convergence of h'(unj) follows from (a3), namely,

to = lim h'(un.) E E'. ' jEIN (a4) implies h(ung)

h(u) and thus u E Mp(h). As in (ry) in the proof of

Theorem 8.2.1, the strong convergence of the sequence {unj }nEN follows from (b3). We therefore obtain

f(u) = jhrn f (unj) = M =

i vE inf(T)

f (v)

Thus u E Mc(h) is a minimising point of f on Mp(h). It follows further that to

lim

j-too f'(unj)

= 7(u)

For V E UnEN En, say v E En, we have from (8.2.4), (8.2.6) and (8.2.16 b)

(f'(u),v) = jlim (f'(uoj),v) _ jli m (f'(p'nj(unj)),rpnj(u))

jw

= lim (fn (un'), v) = lim (A..h'(un,),v) J-. ' ' ' = lim Anj (h'(unj ), v) = A(h'(u), v) . j-.oo

Since U,EN En is dense in En, it follows that f'(u) = Ah'(u), which is the assertion (ii).

Addendum to Theorem. 8.2.1 and Theorem 8.2.2. The eigenfunction u and the eigenvalue A whose existence is assured by Theorems 5.2.1 and 5.2.2, can be obtained as strong limits of subsequences (unj)jEN and (anj)JEW of the sequences of eigenfunctions (un)nEIN and the eigenvalues (anj )nEN of the approximating finite-dimensional eigenvalue problem

fn(un) = Anhn(un),

un E En ,

An E I t, n IN,

so that

u = s - lim un. , A = lim An. .

8.2 Determination of the Ground State

203

proof. u and a are defined in this manner. Before applying these existence theorems to concrete nonlinear eigenvalue problems, let us briefly discuss the hypothesis of relative boundedness of f' and h', i.e. (c) in Theorem 8.2.1. This hypothesis has an important role in the proof: it guarantees the boundedness of the sequence of eigenvalues for the approximating finite-dimensional problems, and consequently contributes in a decisive way to the proof of the hypothesis of Condition (S). Naturally, the hypothesis of relative boundedness of f and h' can be

realised in different ways with the other hypotheses. In Theorem 14 in [8.2], Theorem 8.2.1 (c) has been replaced by the following hypothesis (c'):

(c) There exists a no E M,(f) and an r > 0 such that for all u E MM(f) with h(u) > h(uo), the inequality (h'(u),u) > r holds. Hypothesis (c') gives the desired result for the sequence of eigenfunctions {un}nEN of the approximating finite-dimensional eigenvalue problems as can

be easily verified, since we may assume, without restriction, that no in (c') belongs to El; consequently, it follows for all n E IN that hn(un) = h(un) > h(uo) and therefore

(hn(un),u.) = (h'(un),un) >_ r > 0 for all n E IN. Since MM(f) is bounded and f' is bounded on bounded sets, it follows for the sequence un E M, ,n C MM(f) that lanl =

(f'(u,), un)

1

r

r

s r sup IIf (un)II IInnII < 00. (h'(nn),un) Similarly, it should be clear that Theorem 8.2.2 remains valid when we replace

(a2) in that theorem by (c) or (c') for the relative boundedness of f and h'. Another version of the hypothesis of relative boundedness of f' and h' will be used in Theorem 8.4.5, (B1). Theorem 8.2.1 remains valid with this hypothesis instead of (c). 8.2.2 Determining the Ground State Solution

for Nonlinear Elliptic Eigenvalue Problems If the hypotheses of Theorems 8.2.1 and 8.2.2 for the coefficient functions of nonlinear elliptic eigenvalue problems are used, one obtains the existence and a method for calculation of the solutions. The starting point of this problem is again the quasi-linear differential operators A(u) and B(u), determined by the coefficient functions Fo, Fl , ... , F, and Ho, respectively.

8. Nonlinear Elliptic Eigenvalue Problems

204

Theorem 8.2.3. Let G C 1R° be an open, bounded subset with smooth boundary. Let then + 1 C functions Fo, F,.... F : G x lRn}1 -, IR be the partial derivatives BF/aye of the C function F : G x ]Rn+1 -* R. Let the functions F and Fj satisfy Hypotheses (D1), (D2), (K) and (M) of Sect. 8.4 for a p E (1, n). Further, let H : G x IR -+ IR be a C function whose derivative

Ho(x, Y) =

OH(x,

y)

5Jo

is again a C function. The functions H and Ho satisfy Hypotheses (D3) and (D4) of Sect. 8.4 for the same p E (1,n). Finally, let V be a closed subspace of the Sobolev space W1"p(G) with Wo''(G) C V. Now, and

f(u) = J F(x,y(u)(x))dnx

h(u) _ f H(x,u(x))dnx,

c

G

(8.2.18)

are continuously Frechet-differentiable functions on V and possess the following derivatives (u,v E V):

(.f'(u),v) _

c

dnxFF1(x,y(u)(x))yi(v)(x)

(8.2.19 a)

i=o

and

(h'(u),v)=JGdnxHo(x,u(x))v(x).

(8.2.19b)

The following two existence results are then valid for the eigenvalue problem A(u) = AB(u): (A) For c E lR let

(f'(u),u)

0,

Vu E Wf) = f-1(c).

Further, let a co > 0 and a u1 E MM(f) exist such that for all u E Mjf) where h(u) > h(uo) one has always (h'(u), u) > co

Then the function h attains its maximum on the level surface M,(f) at a point u e M,(f) which solves the eigenvalue problem A(u) = AB(u), A E a, under the boundary conditions defined through V which means (f'(u), v) = A(h'(u), v)

,

Vv E V .

(B) For a c E lR let the following be true for the level surface Mp(h) _ h-1(c):

(i) (h'(u),u) > 0, Vu E Mp(h), (ii) VR > 0, 3c(R) > 0 : (h'(u), u) > c(R), Vu E M,(h) n BR(0).

8.3 Ljusternik-Schnirelman Theory for Compact Manifolds

205

under these conditions the function f attains its minimum on M,(h) at a point u E Mp(h). This point is a solution of the eigenvalue problem A(u) = AB(u), A E IR, with the variational boundary condition defined by V. proof.

(a) The hypotheses for the coefficient functions F, Fj, H and Ho

enable us to apply Lemmas 8.4.6 and 8.4.6'. From Lemma 8.4.6 we find that f : v _+ lR is bounded on bounded sets and is continuously Frechet-differentiable with derivative (8.2.19 a). f' : V -, V' is continuous, coercive and bounded

bounded sets. Condition (S) is satisfied by f'. For h, Lemma 8.4.6' gives h : V -* lR is well defined, bounded on bounded sets and continuously Frechet-differentiable with derivative (8.2.19b). h' : V -+ V' is completely continuous, bounded on bounded sets and weakly conon

tinuous.

It follows from Theorem 8.4.2 that f(u) -> co for ]lull -# co that MM(f) is sphere-like for sufficiently large c.

(f) The coerciveness of f' implies (f'(u),u) > 0 for all u E MM(f), if c is sufficiently large. The rest of the hypothesis in (A) is exactly the hypothesis of relative boundedness of f' and h' in the fom (c'). We have proved all the hypotheses of Theorem 8.2.1 in (a) and therefore the assertion (A) follows. (y) The additional hypothesis in (B) together with the properties of f, f', h and h' proved in (a) enable us to apply Theorem 8.2.2 so that the assertion follows.

Remark 8.2.1. The statements formulated in the addendum to Theorems 8.2.1 and 8.2.2 are obviously also valid in the context Theorem 8.2.3.

8.3 Ljusternik-Schnirelman Theory for Compact Manifolds 8.3.1 The Topological Basis of the Generalised Minimax Principle Let X be a topological space and K C X a closed subset. Further, let it X - IR be a continuous real function on X and F a family of subsets of X

:

on which h is bounded. For such a family we set

m(h,.F) = inf sup h(x) = inf sup h(A) . AEF xEA

AEF

(8.3.1)

Then m(h,F) is called the minimax value of h with respect to F. Our aim is to isolate effective conditions for which the minimax value is a value of the function, and in fact the value of h at a point in K. This is achieved in the following way. Let 0 be a deformation of X, which means that (x,t) -'V)t(x)

206

8. Nonlinear Elliptic Eigenvalue Problems

is a continuous mapping X x lR.+ -* X with Y'o = id,, i.e.

fi(x)=x, HxEX. Let v& effectively decrease the values of h in X \ K, i.e. let Eli satisfy Hypothesis (A), following.

For all -oo0 such that the following is true: Ot, (hb) C h° := h-i ([-oo, a])

(A)

Under this assumption we have the following theorem.

Theorem 8.3.1 (Generalised Minimax Principle). Let {X, K, h, F, ti,} be as defined above; let ,F be O,-invariant, i.e. A E F, t > 0 = E '-; let h(K) C IR be closed and m(h,.F) finite. Then there exists an xo E K, so that m(h,.F) = h(xa).

(8.3.2)

Proof. The very simple proof of this theorem will be given indirectly. Assume m(h,.F) E IR \ h(K); then a, b E IR exist such that

(i) a < m(h,.F) < b, (ii) K n h- '([a, b]) = 0. According to (i) there exists an A E .F, such that m(h,.F) < sup h(A) < b, and thus A C V. According to (ii) there exists a t > 0 such that 0,(h') C h° holds. It follows then that 0,(A) C h°, a contradiction, since ali,(A) E F implies

m(h, )) < sup h(efi,(A)) < sup h(h°) = a. Therefore it follows that

E h(K), which means exactly (8.3.2).

Remark 8.3.1. (a) The assumption that h(K) is closed will be realised in applications by showing that

Knh-r([a, b]) is compact in X for all - oo < a < b < oo. (b) The proof of Theorem 8.3.1 shows that Hypothesis (A) on the decrease of the values of the function h in X \K by 0 can be weakened to the hypothesis (A'), below.

For all -oo < a < b < oo with h-'([a, b]) n K = 0 and all B E .F let there be a to = to(B, a, b) > 0 such that lp,,(B n h&) C ha Example 8.3.2. (a) For .F = {X } one has m(h,.F) = sup.ex h(x).

(b) For .F = {{x} I x E X} one has m(h,.F) = infseY h(x).

(A')

8.3 Ljusternik-Schnirelman Theory for Compact Manifolds

207

(c) Let Sk be an k-dimensional unit sphere and i; a homotopy class of spy Sk -. X. For such a class f we set

men 1(e) is invariant for all deformations in X. An especially important application of Theorem 8.3.1 is the determination of critical points. Let X be a C' manifold and h E C1 (X, IR.) a C' function on

X. Then the set

K=K(h)={xEXI h'(x)=0} of critical points of h is closed in X. Now if the pair (.F,,') satisfies the hypotheses of Theorem 8.3.1, then the minimax value of h with respect to F is a critical point, and h possesses at least one critical point. It is clear, then, that in order to determine as many critical points as possible we construct

(i) deformations 0 of X with the property (A) with respect to K = K(h); (ii) 4-invariant families.F of subsets of X which give a measure for the number of ciritical points.

The next two sections deal with this construction.

8.3.2 The Deformation Theorem The deformation theorem shows how to construct deformations with the property (A). The main technique here is that of "gradient" (or gradient-like) flows on X, i.e. the "method of steepest (or sufficiently steep) descent". We shall discuss this technique here.

In order to keep the framework introductory and elementary, we shall prove the deformation theorem only for the simple case of a compact C' manifold X. The more difficult extension of this theorem to the case of a "Finsler

manifold" X can be found in [8.7] and [8.1]. Thanks to the very effective reduction technique ("Galerkin approximation") of Browder [8.1] our elemen-

tary version of the deformation theorem is sufficient for treating the most far-reaching applications in nonlinear eigenvalue theory.

Let X be a compact C' manifold and h : X - IR a C' function on X. Then the gradient Vh of h is the vector field on X defined by x -* Vh(x) where

(Vh(x),v,)z=D,h(v"), Vv"ETT(X). Here T. (X) is the tangent space of X at the point x and (,) the scalar product on T"(X). The Schwarz inequality shows that h decreases fastest at the point x E X in the direction -Vh(x):

D"h(-Vh(x)) = -(Vh(x), Vh(x))"

.

8. Nonlinear Elliptic Eigenvalue Problems

208

Let t - f t be the one-parameter group of diffeomorphisms of X generated by the vector field -Oh, which means ¢t is the flow of the vector field _ph and as such is characterised by the following properties [Ref. 8.10, paragraph 8; 8.11]

dtOt (x) = -Vh(¢t(x))

,

V(x,t) E X x a.

Oo(x) = x,

(8.3.3)

Consequently,

dth(Ot(x)) = D#.(r)h(dt0t(x)) = -Iloh(0t(x))II2,(=)

(8.3.4)

and we have the following alternatives:

(a) Oh(x) = 0, i.e. x is a critical point of h and therefore 0t(x) = x, Vt E 1R.

(b) Oh(x) # 0, and then (i) (ii)

h(¢t)(x)) < h(x),

Vt > 0.

h(@t(x)) > h(x),

Vt < 0.

(8.3.5)

In particular, (8.3.5 a) contains the following statement: The critical points K(h) of h E C1(X,1R) are exactly the fixed points of the flow qti associated with -Oh, 0t : X -, X for t i4 0. Our aim is to show now that the flow ¢t can play the role of V)t in Theorem 8.3.1 and thus possesses the property (A). The following lemma is a preparation for the proof.

Lemma 8.3.2. Let xo E X be a regular point of h (i.e. Oh(xo) # 0), and let c = h(xo) be the corresponding regular value. Then there exists an e > 0 and an open neighborhood U = U(xo) of xo in X, so that ¢1(U) C h`-` holds. Proof. According to 8.3.4 we have

dt(h(mt(xo)) = -Iloh(,At(xo))II2,(=o) < 0 and Thus

t-.h(&t(xo)),

d h(0t(xo))It=o < 0.

t>0,

decreases strictly monotonically in a neighborhood of t = 0, i.e.

h(#r(xo)) < h(xo) = c,

or h(#1(xo)) < c - 2e for a suitable e > 0. The continuity or x - h(¢1 (x)) now guarantees the existence of an open neighborhood U = U(xo) of xo in X such that h(01 (x)) < c - e, Vx E U, i.e. 01(U) C h`-` Thus, the central point of the so-called deformation theorem, namely, the effective decrease of the values of the function in the neighborhood of a regular

8.3 Ljusternik-Schnirelman Theory for Compact Manifolds

209

point, has already been proved by applying the method of trajectories of steepest descent.

Theorem 8.3.3 (Deformation Theorem). Let X be a compact C' manifold and h : X -* IR a C1 function. Let 0 be the flow defined on X by _Oh. Then there exists for given c E lit. and a given open neighborhood U of

Sc = h-1(x) n K(h) in X an e > 0 so that one has 0,(hc+e \ U) C hc-`.

(8.3.6)

In particular, for every regular value c of h there is an e > 0 for which

01(hc+') C hc-`

(8.3.6')

holds.

proof. Every point of Y = h-1(c) \ U is a regular point of h, so that Lemma 8.3.2 guarantees the existence of an open neighborhood U. of x in X and the existence of an e > 0 with the property ¢,(U,) C he"-. Now Y is closed in X and is therefore compact, and UyEY U1 covers Y. Thus there exist points

Setting co = min{er, ..... e,.}, we

21 , ... X. E Y, such that Y C U j., get

N

h".

U

j=1

Now W = U U UN I is an open covering of h`1(c). The continuity of h ensures the existence of an e E (0,eo) with h-1([c-e,c+e]) C W. Observing

that h`+` C hc-`Uh-'([c-e,c+e]) we get N

he+e \ U C hc_e U W\ U C hc-` U U U1, j=1

and therefore N ,,//,,

Y'

1(hc+E \ U) C 01(h")

UU

// w1(UZ,)

/` hc-EO U hC e,

j=1

since according to (8.3.5) one has ¢1(hc-`) C h`-`

Under suitable assumptions the deformation theorem can also be proved for noncompact manifolds, and even for certain infinite-dimensional manifolds, the so-called Finsler manifolds [8.1, 7 and references therein]. There the main

problem is that the gradient field -Vh does not necessarily possess a global flow on noncompact manifolds X. Fortunately, we only need a realisation of a deformation of X with the property (A), and not necessarily trajectories

8. Nonlinear Elliptic Eigenvalue Problems

210

of steepest descent such as those associated with -Oh, but only those with "sufficiently steep" descent. This enables us to change the gradient field -Vh to a vector field having a global flow 0 on X and satisfying the condition (A) (pseudo-gradient field in [8.7], quasi-gradient field in [8.1]).

For the case when X is not a compact manifold one needs some further restriction for h, including condition (C) of Palais and Smale [8.12]. This states: If the sequence of function values {h(xn)}nEN is bounded for a sequence and also (PS)

IIvh(xn)II=. -' 0, then there exists a convergent subsequence {xnlil}IEr,,

.

As a consequence of the continuity of Oh, the limiting point x of the subsequence {xn(])}JEar is a critical point of it. With reference to Remark 8.3.1, we therefore state in this context:

If h satisfies the PS-condition, then h(K) is closed.

(8.3.7)

This is because the PS-condition implies, as can easily be seen, that for any a < b, K(h) n h-1 ([a, b]) is compact. The PS-condition can therefore be regarded as a generalised compactness criterion.

8.3.3 The Ljusternik-Schnirelman Category and the Genus of a Set In this section we shall study different deformation-invariant classes of subsets of a C' manifold X in connection with the minimax principle. These different classes of sets will be characterised by a quantity which does not decrease under arbitrary (odd) deformations. This finally enables us to derive lower bounds on the number of critical points by applying Theorem 8.3.1. Originally, and in much important later work, the concept of Ljusternik-

Schnirelman category, cat(A; X), of a set A in a topological space X was introduced. We define cat(¢,X) = 0 and cat(A; X) = 1 if A is closed and

contractible in X to a point. In other words, cat(A;X) = 1 if, and only if, the embedding i : A - X is homotopic to a constant mapping. For an arbitrary set A C X which can be covered by finitely many closed sets which are contractible in X to a point, we define cat(A; X) as the minimal number of such sets required for the covering. If a subset does not have any such covering,

then we put cat(A;X) = +oo. The most important properties of this function on the power set P(X ),

cat(; X) : P(X) - IN U {too} , are its monotony, subadditivity and its behaviour under deformations:

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211

a) If G : X x [0,1] -t X is a homotopy of id x, then cat(gi (A); X) > cat(A; X) ,

VA E P(X) ,

where y1(x) = G(x, 1). (b) If g is a homeomorphism from X onto X, then

cat(g(A);X)=cat(A;X), VAEP(X). Now defining for k = 1, 2 , ... ,

.rk(x):={ACXIcat(A;X)>k}, we obtain deformation-invariant families of subsets of X which can be used in Theorem 8.3.1. This is done in the work of Ljusternik, Schnirelman, Palais and Browder, and leads to results such as: an even function h E C1(X; ]R) on X possesses at least cat(X;X) critical points.

The determination of the category of a subset A of X is, in general, a rather complicated matter which is based on nontrivial results of cohomology and homotopy theory. We therefore prefer an elementary approach based on the concept of "genus" of a set and at the end of this section comment briefly on the relation between genus and category of a set. For a real Banach space E, let f(E) denote the family of all closed subsets A C E \ {0} which have reflection symmetry (x E A -x E A). If A and B are reflection-symmetric sets in the same (different) Banach space (s), then denotes the collection of all odd continuous mappings of A into B. Obviously, one can have also C. (A, B) = 0.

Definition 8.3.4. Let E denote a real Banach space and E(E) the family of all closed subsets of E \ {0} which are symmetric under reflection. Define a function y on E(E) as follows: (i) y(0) = 0, (ii) y(A) _ +oo, if C. (A, IR" \ {0}) = 0, Vn E IN, (iii) y(A) = inf{n E IN I IR" \ {0}) 54 0}. Then 7(A) is called the genus of the set A E E(E).

Thus the genus y(A) of a set A E E(E) is the smallest integer it > 1 such that there exists a continuous odd mapping of A in IR" which does not have any zeros. The concept of the genus of a set was introduced by Krasnoselskij [8.13].

Example 8.3.9. shows that

(a) Let A E E(E) be finite: a simple explicit construction IR \ {0}) # 0; thus, y(A) = 1. Let A = {±x1 , ... , ± x"}.

We set f(±xi) = ±1; f : A -+ IR \ {0} is continuous and odd. (b) For x E E\{0}, let B,(x) = {y E E l lly-xll < r}; if 0 < r < lIx1j, then

Br(x) C E \ {0} holds. Setting Cx = B,(x) U B,(-x), it follows that Cz E

212

8. Nonlinear Elliptic Eigenvalue Problems

E(E). Again, a simple explicit construction shows that and hence 7(Cz) = 1.

IR' \ {0}) #

(c) Let A E E(E) where A fl (-A) = 0 is assumed. Then the genus of A U(-A) is exactly 1. For the proof it suffices to define a nonzero constant function on A and to set fi(x)

fa, -a,

for all x E A,

for all x E -A, a #0.

Then 0 is an odd function from A U(-A) into IR\ {O} which is also continuous

due to A fl (-A) = 0. Thus 7(A U(-A)) = 1. The following lemma gives the most important properties of the genus of sets. One can find further information in [8.14].

Lemma 8.3.5. The following properties hold for A, B E E(E): (i) Cn(A, B) # 0 7(A) < 7(B), (ii) A C B 7(A) < 7(B), (iii)

C

00

7(A) = 7(B).

In particular, one also has 7(A) = 7(B) if there exists an odd homeomorphism of A onto B.

(iv) 7(A UB) < 7(A) + 7(B). (v) 7(B) < oo, 7(A \ B) ? 7(A) - 7(B)(vi) For compact A E E(E), 7(A) < co and 38 > 0 such that y(Ab) = 7(A) (Ab = {y E E I dist(y, A) < 8}). (vii) Let V C E be a finite-dimensional subspace and p the linear continuous projection onto V. Then from A E E(E) and -y(A) > k = dim V, it follows that A fl (I - p)E - A fl V1 # 0. Proof. (i)-(iii) are trivial and (v) follows immediately from (iii) and (iv). For (iv), when 7(A) = +oo or 7(B) = +oo the statement is trivial. Therefore, consider 7(A) = n < +oo, 7(B) = rn < +oo. Consequently, there exists

FGAEC.(A,IR"\{0)) and 42BEC"(B,IRm\{0}). Now set 'PA(x) V,A(x) _ 1 + IWA(x)I

It follows that ',A is a bounded function in C. (A, IR." \ {0}). The extension theorem of Tietze-Urysohn [8.15] gives an extension OA of t/iA to a continuous

function on all of E. Denoting by A the odd part,

&(x) =

1 {?GA(x)

- OA(-X)},

8.3 Ljusternik-Schnirelman Theory for Compact Manifolds

213

we have

'PA EC"(E, IR") Similarly, let Y'B E

and WA(x)=O1(x) forx EA.

IR') be constructed. Setting

o =(?A [AUB,''GB I AUB), we have

<' E C"(A U B, R- x am), and co does not have any zeros. For, if x E A U B, say x E A; then

1G(x) = (''A(x),B(x)) and OA(x) E IR" \ {0} gives the assertion. Therefore, Cu(A U B, lR"++" \ {0}) #

0, and hence 7(A U B) < n + in.

For (vi), for x E A let us choose 0 < r. < 11x1j. The sets C. _ Thus Br, (x) U Bri (-x), x E A, are an open covering of A : A C UREA Cz...... C..,, cover the compact set A. It follows that A C Uj=1 C C. From (ii) and Example 8.3.3 (b) we obtain n

7(A) < E7(C:;) = n < oo. i=1

Let, for example 7(A) = m < n. Then there exists a SPA E C.(A,IR' \ {0}) which can be extended, in analogy to (iv), to a continuous odd mapping ip E C.(E,1Rt). 3 A = SPA implies as a consequence of the continuity of cp and the compactness of A the existence of a 6 > 0, such that