Calculus of Variations (Cambridge Studies in Advanced Mathematics 64)

(t) = 0 if io

0 io

(p (t)=0

if

\t0-t\>6 |*o - * | < 6

fori^io,

t€{l,...,d}.

For this choice of
pto+6

/ h(t)
ft(*Mt)cft^0,

Jto — 6

contradicting our assumption. Thus, necessarily ft (to) = 0 for all to € (a, 6). g.e.d. Lemma 1.1.1 and (1.1.3) imply that a minimizer of I of class C2 has to satisfy the so-called Euler-Lagrange equations, namely: Theorem 1.1.1. Let F € C2([a,b]xRdxRd,R), be a minimizer of

and letu e C 2 ([a,6],I

rb

I(u) = J

F(t,u{t),u(t))dt

Ja

among all functions with prescribed boundary values u(a) andu{b). Then u is a solution of the following system of second order ordinary differential equations, the Euler-Lagrange equations j t (Fp(t,u(t),ii(t)))

- Fu(t,u(t),u(t))

= 0.

(1.1.4)

1.1 The Euler-Lagrange equations. Examples

7

Written out, the Euler-Lagrange equations are Fpp(t, u(t),u{t))u{t) + Fpt{t,u(t),u{t))

+ Fpu(t, u(t), ii(t))u{t)

- Fu(t,u(t),u(t))

= 0,

(1.1.5)

i. e. a system of d ordinary differential equations of second order that are linear in the second derivatives of the unknown function u. Let us compute the Euler-Lagrange equations for our preceding three examples: (1) Here Fu = 0, Fp = d

*
u(t)

_

u{t)2u{t)

il{t)

ii(t)

(yi+w)

3

'

i.e.

u(t) = 0 meaning that u has to be a straight line, a fact that we know of course. (3) For the general example (3), we obtain as Euler-Lagrange equations

u{t)2u(t)

u{t)

7 2 y/l + u{t)2

7t

u{t)

72

hence 0 = il(t) - ^ i i ( t ) (1 + u(tf)

+^

(1 + ii(t)2).

(2) We just need to insert 7 = y/2gu(t) into (1.1.6) to obtain 0 = u(t) + (l+ii(t)2)

* 2u(t)'

(1.1.6)

8

The classical theory

Actually, (2) is an example of an integrand F(t, it, u) that does not depend explicitly on t, i.e. Ft = 0. In this case -(F | ( F - -* FuFP p) =
= 0 by (1.1.4),

and hence every solution of the Euler-Lagrange equation (1.1.4) satisfies F(t,u(t),ii(t))

~u(t)Fp(t,u(t),u(t))

= constant.

(1.1.7)

Conversely, every solution of (1.1.7), with the exception of u = 0, i.e. u = constant, also satisfies (1.1.4). In the case of example (2), we have F = * A j ^ , and (1.1.7) becomes j±-z = u(l + u 2 ), if we denote the constant in (1.1.7) by A. In all examples (l)-(3), we actually had d = 1. If one modifies e.g. (1) and seeks a curve g(t) = (#i(£), ...,&*(£)) C Rd connecting two given points g(a) and #(6), our variational problem becomes

m = £\m\dt = £ r£(jt9i(t)) j dt. The Euler-Lagrange equations in this case are d

,

Q=d dt

/d

d

9iT,(9j)2-9iJ2

. ,,,

9jt}_^

=

_J^

(d

\*

qi

9j9j

(1 . L8)

\*

for i = 1 , . . . , d. We now recall that any smooth curve g(t) C Rd may be parameterized by arc-length, i.e.

U*)\ = 1.

dt"

(1.1.9)

We also know that a reparameterization of a curve g(t) does not change its arc-length 1(g)- Consequently, we may assume (1.1.9) in (1.1.8). The latter then becomes

d f d ,A

, . . ^x~, . dt

0

for

i = 1,... , d

so that we see again that a length minimizing curve in E d is a straight line.

1.1 The Euler-Lagrange equations. Examples

9

Often, one also meets the task of minimizing I(u) = / F(t,u{t),u{t))dt Ja subject to some constraint, for example S(u) = / G(t,u(t),u{t))dt = CQ (a given constant). (1.1.10) Ja As in the case of finite dimensional minimization problems, one then finds a Lagrange multiplier A with 0 = ^ (I(U + ST]) + XS{U + ST))) U Q (1.1.11) as for all 77 G Co([a,6],E d ). This leads to the Euler-Lagrange equations d

Jt

(F + \Gp(t, [lpK p (t, u(t),ii(t))

- {Fu(t,u(t),u(t)) u(t,u(t),u(t))

u(t),u(t)))

+ \Gu(t,u(t),u(t)))

= 0.

(1.1.12)

Example. We wish to miminize

I(u) = / u(tf"2dt Ja

under the constraint rb

/ u(t)2cft=l,

S(u)=

(1.1.13)

Ja

with u(a) = 0 = it(6). (1.1.12) becomes u(t) - Xu(t) = 0.

(1.1.14)

Thus, A is an eigenvalue for the differential operator d2/dt2 under the Dirichlet boundary conditions u(a) = 0 = u(b). Of course, this example can easily be generalized. S u m m a r y . We seek solutions of the variational problem I(u) —» min, with

I(u) = f Ja

b

F(t,u(t),u{t))dt

10

The classical theory

for given F and unknown u : [a, b) —• R d . If F and w are differentiate, one may consider some kind of partial derivative, namely 8I(u,r)) := — I(u + sr})u=0 for 77 G Co ([a, 6],R d ). For a minimizer u then 61 (u, 77) = 0

for all such 77.

If F and u are of class C2, this leads to the Euler-Lagrange equations jtFp(t,

u(t),ii(t))

- Fu(t,u(t),ii(t))

= 0.

The classical strategy for solving the problem I(u) —> min consists in solving the Euler-Lagrange equations and then investigating whether a solution of the equations is a minimum of J or not.

1.2 The idea of the direct methods and some regularity results So far, our formulation of the variational problem I(u) —• min has been rather vague, because we did not specify in which class of functions u we are trying to minimize / . The only things we did prescribe were boundary conditions of Dirichlet type, i.e. we prescribed the values u{a) and u(b) for our functions u : [a, 6] —* Ud. Because of our derivation of the Euler-Lagrange equations in the preceding section, it would be desirable to have a solution u of class C2. So one might want to specify in advance that one minimizes / only among functions of class C2. This, however, directly leads to the question whether / achieves its infimum among functions of class C2 (with prescribed Dirichlet boundary conditions, as always) or not, and if it does, whether the infimum of / in some larger class of functions, say C 1 , could be strictly smaller than the one in C2. In the light of this question, it might be preferable to minimize / in the class of all functions u for which I(u) = / Ja

F(t,u(t),u(t))dt

1.2 Direct methods, regularity results

11

is meaningful. Here, we assume that F(t,u,p) is continuous in u and p and measurable in t. For this purpose, one needs the class of functions for which the derivative u(t) exists almost everywhere and is finite. This is the class AC([a,b}) of absolutely continuous functions. A function u G AC ([a, b]) satisfies for h,t2e [a,6]

u{t2)-u(h)

= /

ii(t)dt.

Jti

Note that F(t, u(t), ii(t)) is a measurable function of t for u € AC by our assumptions on F and the fact that the composition of a measurable and a continuous function is measurablef. The idea of the direct methods in the calculus of variations, as opposed to the classical methods described in the preceding section then consists in minimizing / in a class of functions like AC([a,b]) and then trying to show that a solution u because of its minimizing character actually enjoys better regularity properties, for example to be of class C 2 , provided F satisfies suitable assumptions. This minimizing procedure will be treated later J, since we want to return to the classical theory for a while. Nevertheless, even for the classical theory, one occasionally needs certain regularity results, and therefore we now briefly address the regularity theory. To simplify our notation, we put / := [a,6]§. A class of functions intermediate between C 1 and AC is D 1 ( / , E d ) := {u : / —• Md, u continuous and piecewise continuously differentiable, i.e. there exist a = to < t\ < ... < tm = b with u G C\[tj,tj+1],Rd)tovj = 0,...,m-l}. u G D 1 then has left and right derivatives u~(tj) and u+(tj) even at the points where the derivative is discontinuous, and t—•tj—U

t

j

i"

t Lebesgue integration theory is summarized in Chapter 1 of Part II. The required composition property is stated there as Theorem 1.1.2. Here, this point will not be pursued or used any further. t See Chapter 4 of Part II. § We shall use the same letter J to denote the functional to be minimized and the domain of definition of the functions, inserted into this functional. This conforms to standard notations. The reader should be aware of this and not be confused.

12

The classical theory Examples

Example 1.2.1. [a,6] = [ - l , l ] , d = 1

I(u) = j

(l - (u(t))2f dt,

u(-l) = l =u(l). A minimizer is ^ ) = |t|GD1(/,R) which is not of class C 1 . The minimizer of / is not unique (exercise: determine all minimizers), but none of them is of class C 1 . Example 1.2.2. [a,6] = [-1,1],d = 1 I{u)=

{l-u(t))2u{t)2dt

f

u(-l) = 0

,

ti(l) = l.

Here, the unique minimizer is u u{t)

.

fO = \t

for-l<£<0 forO<*
which again is of class D 1 , but not C 1 . Example 1.2.3. [a, 6] = [-1,1], d = 1

I(u)= f ti(-l) = 0

(2t-ii(t)fu(t)2dt, ,

ti(l) = l.

The unique minimizer is ,x. (0 ® = \l*

u

for-l<*<0 forO<*
which is of class C 1 , but not of class C2. T h e o r e m 1.2.1. Let F(t,u,p) be of class C1 w.r.t. u and p and continuous w.r.t. t {F : I x Rd x Rd - • R), and let u € AC{I,Rd) be a solution of 61{u,rj)=

{Fu{t,u,u)-r] Ja

+ Fp(t,u,u)'r)}dt==0

(1.2.1)

1.2 Direct methods, regularity results

13

for all rj G j4C 0 (/,R d ) («.e. 77 G AC(/,R d )) and we require that if I = [a, 6], f/iere exist a < a\
=

Fuitjiufa^u-itj)),

and analogously for the right derivative. Remark. It actually suffices to assume (1.2.1) for all rj G Co°(/,R d ), because functions in ACQ may be approximated by CQ° functions. If u G C 1 or D 1 , the proof anyway only requires (1.2.1) for rj G Cg or £)Q, respectively (where Dg is defined analogously to CQ). Proof We have, omitting the obvious arguments of F, Fu, etc.,

/ Furjdt = Jj(J

Fudy) r]dt = - f (j

Fudy\ f]dt

(1.2.1) then implies

0 = / [Fp- [

F

udy\ rjdt.

We now make use of: Lemma 1.2.1. Let h G L^JTjR) satisfy rb

/

h{t)(p{t)dt = 0 for all

(1.2.3)

Ja

Then there exists a constant c G M iyi£ft /i(£) = c /or almost all t G / . Remark. It actually suffices to assume (1.2.3) for all

14

The classical theory

and


(h(y)~c)dy.

Ja

Then ip{a) = 0, and
(1.2.4)

./a

Equation (1.2.3) implies 0= /

f (h(y) - c) 2 dy

(h(y)-c)h(y)dy=

Ja

Ja

because of (1.2.4). This implies the claim. q.e.d. We now may complete the proof of Theorem 1.2.1: By Lemma 1.2.1 there exists c G Rrf with Fp(t,u(t),u(t))

= /

Fu(y,u(y),u(y))dy

+c

(1.2.5)

Ja

for almost all t E L Therefore, Fp is of class AC, and differentiating (1.2.5) gives (1.2.3). The claims for u G C1 or D 1 are obvious from the proof. q.e.d. Theorem 1.2.2. Let F : / x Rd x Rd be of class C1, and let Fp be also of class Cl, and let det (Fpipj(t,u(t),u(t))iyj=iy,tt}d) ¥" 0 for all t e I 1 d and a solution u G C (/,]R ) of 61 (u, r])=0

for all r) G Cg(JT, E d ).

T/ien u is of class C2. Proof We define : E x Erf x E d x Rd via (j)(t,u,p,q) := Fp(t,u,p) ~ q. Our assumption d e t F p p ^ 0 makes it possible to apply the implicit function theorem to conclude that
1.2 Direct methods, regularity results

15

F(£o,t£(bPo) f° r an Y ^o G i". Thus, there exists a neighbourhood U of (to,Uo,qo) such that for each (t,u,q) G £/, — 0 has a unique solution p = (p(t, u, q) and that (p : U —• Mrf is of class C 1 . Since we already know a solution of
for t near £0-

1

Since

Theorem 1.2.3. Le£ F satisfy the assumptions of Theorem. 1.2.2, and in addition assume that Fpp is (positive or negative) definite on ft xRd where 0 C E d + 1 contains {(t,u(t)) : t G / } . Let u G AC(I,Rd) satisfy 6I(u, rj) = 0 /or a// 77 G i4C0(JT, Md) (assume that F w (£,u(t),u(£)) and F p (t,tt(t),t6(t)) are integrable). Then ueC2(I,Rd). Proof. Since the uniqueness result of the implicit function theorem is only local, it cannot be applied anymore because u(t) might be discontinuous. We thus need a global argument. Thus, assume that for given (£,11,(7) G ( l x R d , there are two solutions pi,P2 G Rd of 0(£,ii,p, g) = 0, i.e. q = Fp(t,u,pi)

and

q=

Fp(t,u,p2).

Thus / F p p (t,u,p!-f s ( p 2 ~ P i ) ) d s ( P 2 - P i ) = 0 . (1.2.6) Jo By our assumption on F p p , (1.2.6) is invertible, hence p2 = pi, hence uniqueness. Using this global uniqueness together with the existence result of the implicit function theorem, we now see that for any (t,u,q) in a sufficiently small neighbourhood of (to,Uo,qo) (to G / , Uo — u(to), qo = Fp(to,v>o,Po), Po = uo(to)), there is a unique solution
-q = 0

and

16

The classical theory

for almost all t in a neighbourhood of to- Since u(t) and Fp(£, u(t), u(t)) are absolutely continuous w.r.t. t (the latter by Theorem 1.2.1), u(t) coincides for almost all t near to with an absolutely continuous function v(t). We put w(t) := u(t0) + / v(y)dy. Jt0 w then is of class C 1 . Since it is absolutely continuous, by a theorem of Lebesgue u(t) = it(£ 0 )

/

u\y)dy.

Jt 0

Since v = u almost everywhere, we conclude u = w, hence it G C 1 near to, which was arbitrary in I. Theorem 1.2.2 then gives u G C 2 . q.e.d. Corollary 1.2.1. Under the assumptions of Theorem 1.2.3, any ACsolution of 8I(u,r)) = 0 for all rj G ACo(/,Mrf) is a solution of the Euler-Lagrange equations d_ Fp(t,u(t),u(t)) dt or equivalently of Fpp{t, u(t),u(t))u(t)

- Fu(t,u(t)M*))

=0

(L2.7)

4- Fpu(t, u(t), u{t))u{t)

+Fpt(t,u(t),u(t))

- Fu(t,u(t)M*))

= 0.

(1.2.8)

T/ie same ZioMs under the assumptions of Theorem 1.2.2 for a Cl- solution of8I(u,r)) = 0 for all r] G C£(I,Rd). g.e.d. Theorem 1.2.4. Let F : I xRd x Rd ^ R be of class Ck, and let Fp also be of class Ck, k G { 2 , 3 , . . . ,oo}. Suppose u is of class C1 and a solution of 6I(u,r)) = 0 for all 77 G Co(J,R d ), and suppose det (Fpipj{t,u{t),u{t))ij=h_4)

^ 0 for all t G I.

(1.2.9)

Then u G C f c + 1 (J,R d ). {The same result holds if we assume that u G C 1 is a solution of the Euler-Lagrange equations (1.2.8).) Proof. By Theorem 1.2.2, u is of class C 2 , and by Corollary. 1.2.1, it

1.2 Direct methods, regularity results

17

solves (1.2.8). Because of (1.2.9), Fpp(t, u(t), ii(t)) is an invertible matrix, hence u(t) =

Fpp1(t,u(t),u(t)) {-Fpu{t,u{t),u{t))

- Fpt{t,u{t),u{t))

+ Fu{t,u{t),u{t))}

. (1.2.10)

Let now j < fc, and suppose inductively u £ 0 . The right hand side of (1.2.10) then is of class C-7"1. Therefore, u is of class C-7""1, hence u is of class C ' + 1 . g.e.d. The preceding proof most clearly shows the importance of the assumption det(Fptpj(t,u(t),u(t))) 7^ 0 that already occurred in the proof of Theorem 1.2.2. Namely, it implies that the Euler-Lagrange equations (1.2.8) can be solved for u in terms of u and it. Corollary 1.2.2. If under the assumption of Theorem 1.2.3, F and Fp are of class Ck, then a solution u o/<$/(i£, 77) = 0 for all rj G ACo is of class C^1. q.e.d. Summary. If one wants to solve I(u) —> min by a direct minimization procedure, it is preferable to admit a class of comparison functions u that is as large as possible. AC (I, Rd) seems to be a good choice, because this is the largest class for which

I(u) = fF(t,u(t)Mt)) is well defined, assuming F(t,u,p) to be continuous in u and p and measurable in t. However, if one then finds a minimizer w, it might not be a solution of the Euler-Lagrange equations, because it is not regular enough. If the invertibility condition det Fpp ^ 0 is satisfied, however, one may show that a minimizer u is as regular as F allows. Namely, if F and Fp are of class Cfc, k £ {1, 2 , . . . , oo}, then u is of class Ck+l. Examples show that without such an invertibility condition, regularity need not hold. This invertibility condition det Fpp ^ 0 implies that the Euler-Lagrange equations allow the expression of u(i) in terms of u(i) and ii(t).

18

The classical theory 1.3 The second variation. Jacobi fields

We assume that u G Dl(I, Rd) is a critical point of I(u) = / Ja

F(t,u(t),u(t))dt,

i.e.

6I(u,
for all rj G Z ^ ( / , E d ) .

(1.3.1)

We recall that 6I(u,r]) := — I(u + s
+ sri).

If we want to decide if a given solution u minimizes J instead of just being a critical point, we immediately see that a necessary condition would be /"(0) > 0

(1.3.2)

for the above function / and all rj G DQ(I,Rd). theorem, since /'(0) = 0 f(s)-f(0)

= \s2f"(0)+o(s2)

Namely, by Taylor's

fors^O.

More precisely, (1.3.2) is needed for u to minimize J when compared with u -f ST] for sufficiently small s. In other words, we want u to minimize i" in a D 1 -neighbourhood of itself, i.e. among functions veDl{I,Rd) with u(a) = v(a), u(b) = v(b)

and

sup (\u(t) - v(t)\ -f \u-(t) - v-(t)\ 4- \ii+(t) - * + (t)|) < e tei

(1.3.3) (1.3.4)

for some e > 0. (Note: It is not clear that e may be chosen independently of v.) We define the second variation of / at u in the direction rj G DQ as d2 62I(u,f]):= — J( M + 5f7)UsB0.

1.3 The second variation. J'acobi

fields

19

In order that this variation exists, we require for the rest of the section that F is of class C 2 . We then compute 62I(u, n) = ^

J

F(t, u(t) + sr,(t), u(t) + «7(t))d*|.. 0

,5

= /

{Fpipi{t,u{t)Mt))Vi{t)Vj{t)

Ja

+ 2FpiUJ(t,u(t),u(t))fH(t)r1j(t) 4- Fuw(tMt)Mt))m(t)Vj(t)}dt.

(1.3.5)

Here, and in the sequel, we employ the standard summation conventions, e.g. d

i,j = l

We abbreviate (1.3.5) as fb

62I(u, r])=

{Fppriri + 2Fpurjr] + Fuur]r]} dt.

(1.3.6)

Ja

Our preceding considerations imply: Theorem 1.3.1. Suppose F £ C2(IxRdxRdxE) andletu satisfy I(u) < I(v) for all v with {1.3.3), (1.3.4). Then 62I(u,r))>0

forallrieDl(I,Rd).

e

Dl(I,Rd) (1.3.7)

We now put, for given w, min as fb Q(rj) := / cf)(t,r](t),f)(t))dt -> min

among all 77 G Z^(J,R d ). (1.3.8)

If u satisfies the assumptions of Theorem 1.3.1, then Q(r]) > 0

for all 77 e D ^ ,

(1.3.9)

and hence 77 = 0 is a trivial solution of (1.3.8). We are interested in the question whether there are others. The Euler-Lagrange equations for (1.3.8) are

jtM*Mt)M*)) = 4>v(tMt)>W))>

(1-3-10)

20

The classical theory

i.e. - (Fpp(t, u(t), u(t))rj(t) + Fpu(t, u(t), = F^t,

u(t))v(t))

u(t), t*(t))i)(t) + F ^ ( t , ix(t), u(t))rj(t).

(1.3.11)

Since it is considered as given, our first observation is that (1.3.11) is a linear homogeneous system of second order equations for the unknown rj. These equations are called Jacobi equations. Definition 1.3.1. A solution rj £ C2(I,Rd) (1.3.11) is called a Jacobi field along u(t).

of the Jacobi equations

L e m m a 1.3.1. Let F £ C3(I x Rd x K d ,R), detF p p (t,u(t),6(t)) + 0 for all t £ I, u £ C2(I,Rd). Then any solution of rj £ AC0(I,Rd), d 2 6Q(ri,™(t,ri(t),fi(t)) = Fpp(t,u(t),u(t))

for all t and rj

and so the assumption det Fpp(t, u(t), ii(t)) ^ 0, that is seemingly weaker than the one of Theorem 1.2.3, indeed suffices to apply that Theorem. q.e.d. We now derive the so-called necessary Legendre condition: Theorem 1.3.2. Under the assumption of Theorem 1.3.1, i.e. u £ D 1 (/,E r f ) minimizes I in the sense described there, we have that Fpp(t,u(t),u(t))

is positive semidefinite for all t £ I,

i.e. Fpipj (t, u(t), ii(t))?&

> 0

for alH = ( £ \ . . . , £ d ) £ Rd.

(At points where ii(t) is discontinuous, this holds for the left and right derivatives.) Proof. We may assume that to £ I and u is continuous at to. The result at the points where u jumps then follows by taking appropriate limits, and likewise at to — a, b. We then consider 0 < e < min(£o — a, b — to) and define r? £ D^(J,R d ) by

{

0

for a < t < to — e and t0 + c < t < b

e£ linear

for t = t0 for to — e < t < to and for t 0 < t < to -f e

1.3 The second variation. Jacobi fields for given ( e l f

21

Then

{

0 £ -£

for a < t < t0 or t0 + e < t < b for t0 - e < t < t0 for t0 < t < t0 + e.

We apply Theorem 1.3.1 to obtain 0 < 62I(u, rj) = / °

Fpipj (t, u(t), u{t))CZJdt + 0(e 2 )

for c -+ 0,

Jto-e

since all other terms contain a factor e, and we integrate over an interval of length 2e. Hence FPv(toMto)Mto))?Zj

= lim -

/

Fpipj{t,u{t),ii(t))Ctjdt

> 0.

€—0 ^e 7 t 0 - c

g.e.d The Jacobi equations and the notion of Jacobi fields are meaningful for arbitrary solutions of the Euler-Lagrange equations, not only for minimizing ones. In fact, Jacobi fields are solutions of the linearized Euler-Lagrange equations. Namely: Theorem 1.3.3. Let F e C3(I x Rd x R d ,R), and let us(t) be a family of C2-solutions of the Euler-Lagrange equations -Fp(t,us{t),us{t))-Fu(t,us(t),us(t))==0,

(1.3.12)

with us depending differentiably on a parameter s 6 (—e,e). Then

is a Jacobi field along u = uo. Proof. We differentiate (1.3.12) w.r.t. s at s = 0 to obtain ~ (Fpp(t, u(t), u(t))fj(t) + Fpu(t, u(t), ii(t))rj(t)) -Fpu(t,

u(t),u(t))fj(t)

- Fuu{t, u(t), ii(t))rf(t) = 0.

i.e. the Jacobi equation (1.3.11).

q.e.d.

Lemma 1.3.2. Let a < a\ < a2
(1.3.13)

22

The classical theory

Proof. Since is homogeneous of second order in (77,71-), we have 20(t, 77, IT) = (pr,(t, ry, TT)77 4- <M*, V, * > •

Therefore r«2

2/


«/ai

{v(t,V,v)-V + 4>At,il,v)-v}dt.

(1.3.14)

«/ai

Comparing (1.3.10) and (1.3.11), we see that
0(t,r7,7))dt= /

(<M*,77,77)- ^ M ^

since 77 is a Jacobi

7

) ) ) • *7
field.

q.e.d.

As before, let F be of class C 3 , and let u(t) be a solution of class C2 on [a, 6] of the Euler-Lagrange equations d_F (t, u(t),u(t)) - F (t, u(t), u(t)) = 0. ; p u dt Definition 1.3.2. Let a < a\ < 0,2 < b. We call the parameter value a,2 conjugate to a\ and the point (02,1^(02)) conjugate to (ai,i/(ai)) if there exists a not identically vanishing Jacobi field 77 on [01,02] with 77(01) = 0 = 77(02).

We may derive the important result of Jacobi: Theorem 1.3.4. LetF e C 3 ( J x R d x R d , R ) and suppose ue C2{I,Rd). Suppose that Fpp(t,u(t),ii(t)) is positive definite on I. If there exists a* with a < a* < b that is conjugate to a, then u cannot be a local minimum of I. More precisely, for any e > 0, there exists v E Dl(I, Rd) with v(a) = u(a), v(b) = u(b), sup (\u(t) - v(t)\ + \ii(t) - v±(t)\) < e tei

and I(v) < I(u). Proof. Let rj(t) be a nontrivial Jacobi field on [a, a*]. We put *?*(*)

f rj(t) \ 0

fora
1.3 The second variation. Jacobi fields Then 77* e Dl(I,Rd),

23

and by Lemma 1.3.2 Q(V*)=

I
If u were a local minimum, then by Theorem 1.3.1 0 < 62I(u,rj) = Q(r))

for all fj € Dj(7,R d ).

Hence rj* would be a minimizer of Q, hence by Lemma 1.3.1 rj* £ C2(I,md). Since j)*.(a*) = 0, then 77*(a*)=0. Since also rj*(a*) = 0, and since rj* solves the Jacobi equation, a (linear) second order ordinary differential equation, the uniqueness theorem for solutions of such equations implies

a contradiction, because by assumption rj does not vanish identically. Hence u cannot be a local minimizer. q.e.d. In words, Theorem 1.3.4 says that a solution of the Euler-Lagrange equations cannot be minimizing beyond the first conjugate point. Turned the other way round, Theorem 1.3.4 says that if u is a local minimizer, then there cannot be any parameter value a* with a < a* < b that is conjugate to a. It may happen, however, that b is conjugate to a. An example will be given in the next chapter. S u m m a r y . In order to obtain necessary conditions for a solution of the Euler-Lagrange equations d_F (t,u(t),u(t)) ; p dt

=

Fu(t,u(t),u(t))

to minimize I(u)=

I

F(t,u(t),u(t))dt,

Ja

one needs to study the second variation Q{n) := 62I{u,n) = A 2/ ( „ + «,),,_„ for „ € D\. ds

24

The classical theory

If, for fixed it, we consider the variational problem Q(rj) —* 0, we are led to the Jacobi equations - (Fpp(t,u(t),u(t))i)(t) = Fup(tMt)M*))v(t)

+Fpu{tMt)Mt)Mt)) +Fuu(t,u(t),ii(t))i1(t)

for rj. Solutions rj with 77(a) = rj(b) = 0 are called Jacobi fields, a* G (a, 6) for which there exists a nontrivial Jacobi field on [a, a*] is called conjugate to a, and if there exists such a*, u cannot be locally minimizing on [a, 6]. In other words, a solution of the Euler-Lagrange equations cannot be minimizing beyond the first conjugate point.

1.4 Free b o u n d a r y conditions We recall the definition of an n-dimensional embedded differentiable submanifold M of Rd: For every p G M, there have to exist a neighbourhood V = V(p) C Md, an open set t / c l n and an injective differentiable map / : U —> V of everywhere maximal rank n (i.e. for every z G U, the derivative Df(z), a linear map from E n to E d , has rank n) with MHV

= f(U).

An example is the sphere 5 n described in detail in Section 2.1 (Example 2.1.1). The tangent space TPM of M at p then is the vector space Df(z)(Rn). It can be considered as a subspace of the vector space TpMd, the tangent space of Rd at p. As in 1.1, we now consider the variational problem I(u)=

/ F(t,u(t),u(t))dt Ja

—• m i n

with F of class C2. This time, however, we do not impose the Dirichlet boundary condition that the values of u(a) and u(b) were prescribed, but the more general condition that for given submanifolds Mi, M2 (differentiable, embedded) of Md, we require that u(a) eMuu(b)

eM2.

(Dirichlet boundary conditions constitute the special case where M\ and M2 are points.) In this section, we do not consider regularity questions. As an exercise,

1.4 Free boundary conditions

25

the reader should supply the necessary regularity assumptions on F , it, etc. at each step. Let u be a solution. Then, as before, u has to satisfy the EulerLagrange equations, because if u(a) G Mi, 77(a) = 0, then also u(a) + 577(a) G Mi for any s, and likewise at 6, and so we may again consider variations of the form u + srj, rj e DQ. This time, however, also more general variations are admissible. Namely, let us(t) be a family of maps from J into Rd depending differentiably on s G (—e, e), with u(t) = Uo(t) and us(a) G Mi

,

us(b) G M 2 for all s.

Let

Then again

0= ^/(«.)|„ 0 = ^ jf F(t,u(t),u(t))dt^0 = f

{F p (t,u(i),w(t))-J7(i)+i r „(i,u(0,w(i))-ry(<)}di

Jo

= ^{-jtFP =

+ Fuyr1+Fp(t,u(t)Mt))-v(t)\ttZba

Fp(t,u(t),u(t))-v(t)\ttZba,

since it solves the Euler-Lagrange equations. We now observe that 77(a) G TU^M\ (and likewise at 6), since we may find a 'local chart' / as above with Mi f)V(u(a)) = f(U) for a neighbourhood V of u(a) and some open set U C RHl (n\ = dim Mi). By choosing € smaller if necessary, we may assume us(a) G Mi D V = f(U) for s G (~e, e). Since / is injective, there then has to exist a curve 7(5) C U with us(a) = / o 7 ( s ) for all s. Hence 77(a) = £us(a)u=0 = D}'(f1u{a))ir(0) is indeed tangent to Mi at u(a). Moreover, any tangent vector to Mi at u(a) can be realized in this manner. Therefore, since we may choose the values of 77 at a and 6 independently of each other, we conclude F p (a,7i(a),7j(a)) • V = 0 for all V G Tu{a)Mu and likewise F p (6, u(6), u(b)) -W = 0 for all W G Tu{b)M2.

The classical theory

26 We have thus shown:

Theorem 1.4.1. Let u be a critical point of I among curves with u(a) G M\j u(b) G M2 (Mi, M2 given differentiable embedded submanifolds of Rd), i.e. 5^(^s)| s = 0 = 0 for all variations us(t) differentiable in s with us(a) G Mi, us(b) G M 2 for all s G (~e,e) (e > 0). Then u is a solution of the Euler-Lagrange equations for I, and in addition, Fp(a,u(a),u(a)) and Fp(b,u(b),u(b)) are orthogonal to Tu^Mi and TU(5)M2, respectively. In particular, if for example Mi = Rd, then Fp(a,u(a),u(a)) = 0. Summary. If instead of a Dirichlet boundary condition, we more generally impose a free boundary condition that u(a) and u(b) are only required to be contained in given differentiable submanifolds Mi and M2, respectively, of E d , then Fp(a,u(a),u(a)) and F p (6, u(b),u(b)) are orthogonal to these submanifolds for a critical point of / under those boundary conditions.

1.5 Symmetries and the theorem of E. Noether In the variational problems of classical mechanics, one often encounters conserved quantities, like energy, momentum, or angular momentum. It was realized by E. Noether that all those conservation laws result from a general theorem stating that invariance properties of the variational integral / lead to corresponding conserved quantities. We first treat a special case. Theorem 1.5.1. We consider the variational integral I(u) = /

F(t,u(t),u(t))dt,

Ja

with F G C2([a, b] x Rd x Md, E). We suppose that there exists a smooth one-parameter family of differentiable maps hs : Rd -> Rd (the precise smoothness requirement is that h(s,z) := hs(z) is of class C 2 ((-e 0 ,e 0 ) x Md,M) for some e0 > 0), with h0(z) = z

for all

zeRd

1.5 Symmetries and the theorem of E. Noether

27

and satisfying J

FUhs(u(t)),fths(u(t))\dt

=J

FUu{t),jtu{t)\dt

(1.5.1)

for all s G (-e,e) and all u G C 2 ([a,6],R d ). Then, for any solution u(t) of the Euler-Lagrange equations (1.1.4) fori, Fp(t,u(t),u(t))

^-hs(u(t))\s=0

(1.5.2)

is constant in t G [a, 6]. Definition 1.5.1. A quantity C(£,ii(£),it(£)) that is constant in t for each solution of the Euler-Lagrange equations of a variational integral I(u) is called a (first) integral of motion. Proof of Theorem 1.5.1: Equation (1.5.1) yields for any to G [a, 6], using h0(z) = z, 0 = ^ j ° F Uhs(u(t)), = J°

jtha(u(t))\

dt\s=0

{Fu (t, u(t), u(t)) ^h,(u(t))

(1.5.3)

+F p (*,ti(*)^(*))^^aK*))}*U=o. We recall the Euler-Lagrange equations (1.1.4) for u: 0 = ~FP (t,u(t),ii(t)) - Fu (t,u(t),ii(t)).

(1.5.4)

Using (1.5.4) in (1.5.3) to replace F u , we obtain 0=

/ ° {jtFp (*'"(*)'«('))

zh,{u{t))

+F p (t,u(t),«(*)) ^^/i»(«(0)}
(1-5-5)

to

r d_ Fp(t,u{t),u(t)) — hs{u(t))\s=0 dt V "'"'""-"~'~"ds Ja dt

) dt.

Therefore Fp(to,u(to),u(to))~h,-rh s{u(t 0))\s=o s(u(t 0))\s=o

(a,u{a),u(a)) = Fp(a,u(a),u(a))

—

hs(u(a))\s=o (1.5.6) for any to G [a,6]. This means that (1.5.2) is constant on [a, 6]. q.e.d.

28

The classical theory Examples

Example 1.5.1. We consider for u : E —> E 3 n , u = ( i / i , . . . , it n ) with

i.e. a mechanical system in M3 with point masses m^, and a potential V(u) that is independent of the third coordinates of the Ui. Then ha(z) = z + se$, where es is the third unit vector in M3, leaves F invariant in the sense of Theorem 1.5.1. Since d

u I

— ns \s=0 — e3>

as we conclude that n

i=l

i.e. the third component of the momentum vector of the system is conserved. Example 1.5.2. Similarly, if a system as in Example 1.5.1 is invariant under rotations about the e3-axis, and if h8 now denotes such rotations, then (up to a constant factor) d

L

,

— ha\s=oUi

= e3

AUi.

as Hence, the conserved quantity is the angular momentum w.r.t. the e$axis, n

E i=l

F e3 A Ui =

P

E (m*^) " (e3 A Ui) = E (Wi A m^«) * e3t

i

We now come to the general form of E. Noether's theorem Theorem 1.5.2 (Theorem of E. Noether). We consider the variational integral I(u) = / Ja

F(t,u(t),ii(t))dt

1.5 Symmetries and the theorem of E. Noether

29

with F G C 2 ([a,6] x Rd x E d , E ) . We suppose that there exists a smooth one-parameter family of differentiable maps hs = (h°s,hs) : [a,6] x Rd -> E x Rd as

(s G (—eo,€o)

before) with

h0(t, z) = (t, z)

for all (t, z) G [a, 6] x Rd

and satisfying

rh°*{b) /

(

d

F(ts,hs(u(ts)),—

JhO(a)

\

rb

\ hs(u(ts))

«t s

)dts= /

/

F{t,u(t),u(t))dt

Ja

(1.5.7) forts — h?s(t), all s G (-e 0 ,e 0 ) and all u G C 2 ([a,6],R d ). Then, for any solution u(t) of the Euler-Lagrange equations {1.1.4) for I,

F p (*,ti(*),tiW)x fc -N*))l-=o + (F(*,u(*),
^2WI-=o

(1.5.8)

w constant in t G [a, 6]. Proo/. We reduce the statement to the one of Theorem 1.5.1 by artificially considering t as a dependent variable on the same footing with u. Thus, we consider the integrand F(*(T),«(<(T)),^,^«(*(T))

/

,

4-u(t(T))\

:=F\tMt),dT

*

dt

)&

( L5 - 9 )

eft

F(t,«(«),u(*))- d T . Then

= I F{t,u(t),u{t))dt,

if t(T0) = a, t(n) = b

(1.5.10)

•/a

/(«). By our assumption, F remains invariant under replacing (t,u) by hs(t,u). Consequently, Theorem 1.5.1 applied to I yields that d (u{t))\s=0 s=0 Fp{t,u{t)yu{t)) ,— — hss{u(t))\ as

d Fpo{t,u(t),u(t))~-. + Fpo{t,uW as

30

The classical theory

with p° standing for the place of the argument ^ of F (while p stands as before for the arguments u), is invariant. Since, by (1.5.9), F —F Fpo = F - Fpti at s = 0 (note ^ = 1 for s = 0 since /i[}(£) = £), this implies the invariance of (1.5.8). q.e.d. Example 1.5.3. Suppose F = F(i/,t/), i.e. F does not depend explicitly on t. Then ha(t,z) = ( t - f s , z ) leaves I invariant as required in Theorem 1.5.2. Therefore, the 'energy' F{t, u(t),ii(t))

- Fp(t, u{t), it{t))u{t)

is conserved. We shall see another proof of this fact in Section 4.1. S u m m a r y . The theorem of E. Noether identifies a quantity that is preserved along any solution u(t) of the Euler-Lagrange equations of a variational integral, a so-called first integral of motion, with any differentiable symmetry of the integrand. For example, in classical mechanics, conservation of momentum and angular momentum correspond to translational and rotational invariance of the integral, respectively, while time invariance leads to the conservation of energy.

Exercises 1.1

For mappings u : [a, 6] —• E^, consider

E(u)~\Ja \u(t)\2dt (| • | is the Euclidean norm of Md, i.e. for z = (zl,... ,z d ), \z\2 = J2i=\(z%)2)' Compute the Euler-Lagrange equations and the second variation. Also, let L(u) := / \it(t)\dt. Ja Show that

Exercises

1.2

31

with equality if \ii(t)\ = constant almost everywhere. (What is an appropriate regularity class for the mappings u that are considered here?) Determine all minimizers of the variational integral

I(u)= f

(l-u{t))2dt

withu(-l) = 0 = ii(l). 1.3

1.4

Develop a theory of Jacobi fields for variational problems with free boundary conditions. In particular, you should obtain an analogue of Jacobi's theorem. For mappings u : [a, 6] —• E d , consider

Compute the first and second variation of / and the Jacobi equation. Can you find Jacobi fields?

2 A geometric example: geodesic curves

2.1 The length and energy of curves We let M be an n-dirnensional embedded submanifold of Rd. In this section, we assume that / is of class C 3 , i.e. that all local charts are thrice differentiable. We let c € AC([0,T],M) be a curve on M. This means that c is an absolutely continuous map from the interval [0, T] into Rd with the property that c(t) € M for every t e [0,T]. The derivative of c w.r.t. t will be denoted by a dot ',

e«) :=§«,. The length of c is given by L(c):=£\c(t)\dt = £

Hr(caA

dt,

(2.1.1)

where ( c 1 , . . . , cd) are the coordinates of c = c(t). We also define the energy of c as

E(c) := \ £ |c(t)|2 dt=\£j2

(caf dt.

(2.1.2)

We let now f:U->V

,

f(U) =

MDV

be a local chart for M as defined in Section 1.4. We assume for a moment that c([0, T]) is contained in f(U). Since / maps U bijectively onto f(U), there exists a curve l(t) C U 32

2.1 The length and energy of curves

33

with c(0 = / ( 7 ( * ) ) .

(2.1-3)

Since the derivative Df(z) has maximal rank everywhere (by definition of a chart, cf. 1.4), 7 is absolutely continuous, since c is, and we have the chain rule c(t) = (Df) h(t)) 0 7 ( 0 , or

*°(«) = §7(7(<))7iW, where the index i is summed from 1 to n. Thus

*>-jf( ^(l(f))7''«)f£(7(f))V(<) I

*

and 1 fT dfa E{c) =

dfa

2 Jo ^r(^))^W^j^W)^W*-

In these formulae, and in sequel, the index is summed from 1 to d. For zeU, we put 9f a dfa ftjM - ^ W ^ j W (2-1-4) With this notation, the preceding formulae become

and

£ ( c ) = / (sy(7(*))7<(«)y(«))i* Jo

(2.1-5)

E(c)=l-j

(2.1.6)

ffyfrtoftWW*-

Definition 2.1.1.

is called £/ie metric tensor of M w.r.L the chart f U —>V. We note that (9ij(z))ij=1^n

is symmetric, i.e.

gij(z) =9ji(z)

for all i,j

and positive definite, i.e. 9ij{z)rfrf

> 0 whenever rj = (77 1 ,..., rf1) ^ 0 € W1.

34

Geodesic curves

Remark 2.1.1. The use of local charts for M seems to have the obvious disadvantage that the expressions for length and energy of curves become more complicated. The advantage of this approach, namely not to consider curves on M as curves in Rd satisfying a constraint, is that this constraint now is automatically fulfilled. All curves represented in local charts lie on M. This more than compensates for the complication in the formulae for L and E. Our aim will be to find curves of shortest length or of smallest energy on M, i.e. to minimize the functional L and E among curves on M. For this purpose it will be useful to observe certain invariance properties of L and E. First of all, whenever i : Rd —> Rd is a Euclidean isometry, i.e. i(y) = Ay + b with A 6 0(d), the orthogonal group, and b 6 R d , then L(i(c)) = L(c)

(2.1.7)

E(i(c)) = E(c)

(2.1.8)

for any curve c : [0, T] - • Rd. Secondly, L is parameterization invariant in the sense that whenever r:[0,S]-[0,T] is a diffeomorphism (i.e. r is bijective, and both r and its inverse r _ 1 are everywhere differentiable), then L(c) = I ( c o r ) ,

for any curve c : [0, T] — Rd.

Namely C1A

L(cor)

J h^

1

C0T s s

^ T

0

/l(5c)H T

f\c(t)\dt.

\dTi

\\

ds

(2.1.9)

2.1 The length and energy of curves

35

E, however, is not parameterization invariant. By the Schwarz inequality, we have instead \

T

2

/

f dt\ -If

T1

\ 2

\c(t)\2dt)

^V^^E(cj, (2.1.10)

with equality iff \c(t)\ = constant

for almost all t.

(2.1.11)

We have shown: Lemma 2.1.1. For every c e AC([0,T],R d ) L(c)

<

V^>/E{C),

with strict inequality, unless \c(t)\ = constant

almost everywhere.

\c(t)\ = constant

almost everywhere ,

If

we say that the curve c is parameterized proportionally to arc-length, and if |c(*)| = 1, we say that it is parameterized by arc-length. We recall that a Jordan curve, i.e. an injective curve c : [0, T] —> Mrf, is rectifiable if it is absolutely continuous (which we always assume), and this implies that it may be parameterized by arc-length, i.e. there exists a diffeomorphism r:[0,L(c)]-+[0,T] with — (c o r)(s) = 1 for almost all s, I ®s I i.e. the reparameterized curve c = cor is parameterized by arc-length. From Lemma 2.1.1, we obtain: Corollary 2.1.1. Let c : [0, L(c)] —• Rd be a curve parameterized on [0,L(c)]. Among all reparameterizations r:[0,L(c)]-[0,L(c)]

36

Geodesic curves

(i.e. we keep the interval of definition fixed, namely [0, L(c)]), the parameterization by arc-length leads to the smallest energy. Namely, if c : [0, L(c)] —• E d is parameterized by arc-length L(c) = 2JE?(c),

(2.1.12)

whereas for any other parameterization of c on the same interval, L(c) < 2E(c).

(2.1.13)

We now return to those curves c that are confined to lie on M, in order to discover a third invariance. Namely,we compare the two expressions (2.1.1) and (2.1.5) for the length of c, and similarly (2.1.2) and (2.1.6) for its energy. (2.1.1) is obviously independent of the chart / : U —• V and its metric tensor, and therefore (2.1.5) has to be independent of them, too. In order to study this more closely, let f:U-+V be another chart with c([0,T}) C f(U). Then there exists a curve 7 in U with c(t) = /(7(t)) for all t. Putting

dfa

dfa

^):=^(*)^r(2)

brzeu,

we then also have 1

T

L{C) =

L (^W^'W)'*-

(2.1.14)

In order to study this invariance property more closely, we define V := I'1

of:f-1

(f(U) n f(U))

- r

1

(/([/) n /(£/))

(see Figure 2.1). if is called a coordinate transformation, (p is a diffeomorphism, i.e. a bijective map between open subsets of E n whose derivative D(7«),

hence

^(t)

= |£(7W)V W

and from / > ( z ) ) = /(z)

(2.1.15)

2.1 The length and energy of curves

37

Figure 2.1. we get 9iA*)=hiM*))^(z)^(*)-

(2-1-16)

From (2.1.15) and(2.1.16), we see 9ii ( 7 ( t ) ) 7 W « =0« (7(*))7*(*)7 (t),

(2.1.17)

and this shows again the equivalence of (2.1.5) and (2.1.1), and likewise for the corresponding expressions of the energy. The important transformation formula (2.1.16) shows how the metric tensor transforms under coordinate transformations. This invariance property of L and E makes it possible to express the length and energy of an arbitrary curve c on M that is not necessarily contained in the image of a single chart as follows: One finds a subdivision t0 = 0 < U < ... < t m _ i < tm = T of [0, T] with the property that c ([t„_i,t„]) is contained in the image of a single chart

for each v = 1 , . . . , m. Let (<7^(z)). . = 1

be the metric tensor of M

Geodesic curves

38 w.r.t. the chart / „ . Then m

m

**

x

= E / WMt)H(tH(t)V dt where c(t) = }v°lv{i) for t £ [£„_i,£„]. By the preceding considerations, this does not depend on the choice of charts fu. For this reason, one usually just says that for a curve c on M L(c)=

f (ff«(7(*))7 i (t)y(t)) i dt, Jo

(2.1.18)

where 7 is the representation for c w.r.t. a local chart, and (flfy)»,i=i,.-,n is the metric tensor of M w.r.t. this chart. Similarly E{C) =

\1 ^(7(*)M<)7W.

(2.1.19)

We now assume that the charts for M are twice differentiate and return to the question of finding shortest curves on M, for example between two given points. By Corollary. 2.1.1, it is preferable to minimize E instead of L, because a minimizer for E contains more information than one for L; namely, minimizers for E are precisely those minimizers for L that are parameterized proportionally to arc-length. Thus, minimizing E not only selects shortest curves but also convenient parameterizations of such curves. We now compute the Euler-Lagrange equations for E as given by (2.1.19): d 0 = ~v:Eji — £ y

for i = 1 , . . . , m

<=» 0 = ! (2s«(7(t))V(t)) - ( ^ j f l y )

(i(t))ik(tW(t)

(the factor 2 in the first term results from the symmetry gij = gji) «• 0 = 29ii'f 4- 2 ^ 5 i j 7 V - £i9kjikijWe now introduce some further notation: (9ij) • , w / t , j = l,...,n

(2-1.20)

2.1 The length and energy of curves

39

is the matrix inverse to ( i-eI 1

f o r ? —~~ jk*

9ij,k :— Qzk9iJi

and finally the Christoffel symbols r

j*

:=

2gtl(gjl*~*~gklJ

"

9jk

^'

Equation (2.1.20) then becomes

o = f + Wl $9i3,kikij - 9kj,nk¥) = 7* + §0*' (&',* + 0w»i ~ 9jk,i) 7*7* by using symmetries. Thus: L e m m a 2.1.2. T/ie Euler-Lagrange curves on M are 0 = f ( i ) + r;. fe ( 7 (t))7 J '(t)7 fe (<)

equations for the energy E for

/ort = l,...,n.

(2.1.21)

The theorem of Picard-Lindelof about solutions of ordinary differential equations implies: L e m m a 2.1.3. For any z G U, v G M n , the system (2.1.21) has a unique solution y(t) with 7(0) = z , 7(0) = v for t G [—€, e] and some e > 0. Moreover, ^(t) depends differentiably on the initial values z, v. Definition 2.1.2. The solutions of (2.1.21) are called geodesies on M.

Examples Example 2.1.1. The sphere

{

n+l

^

(x 1 ,..., x n+1 ) G M n + \ Y^ (xif = l \ C Mn+1

is a differentiable manifold of dimension n. In order to construct local charts, we put fix := 5 " \ { ( 0 , 0 , . . . ,0,1)} ,n2:=Sn\{(0,0,--,0,-1)}

40

Geodesic curves

and define 01 : fix -+ E n , g2 : fi2 -+ Mn as

and 92(lV x

- ° + , ) = (r^T

i^«)

(<7I and 02 are the stereographic projections from the south and north pole, respectively). We then obtain charts /i=Sf1:Rn-Sn\{(0,...,0,l)} /2=52-1:M"^5n\{(0,...)0,-l)}. More explicitly, f\ can be computed as follows: With

[Z ,...,Z

1=

)

Q a x

x

- ^1_xn+l'---'1_xn+l

>

/'

= * V ( 1 - x n + 1 ) 2 4- x n + 1 x n + 1 ,

hence z*V 4-1 and then CJ =

_

(j — 1

n).

Thus

For the metric tensor, we compute df{ _ 26jk dzk ~ 1 4 ^z* df^1 dzk

_

4z*zk (1 4 ^ ^ ) 2

4zk

~(14^^)2'

(j,k = l , . . . , n )

2.1 The length and energy of curves

41

Hence 9ij(z)

QfocQfa = Tr-Tr- =

4 o—&ij'

(2.1.22)

Actually, the metric tensor w.r.t. the chart fi is given by the same formula. In order to compute the expression for geodesies, we also need to compute the Christoffel symbols. It turns out that adding a little generality will actually facilitate the computations. We consider a metric of the form 9ij = ^ « y ,

(2.1.23)

where : K n —+ M+ is positive and differentiable. Then 9ij = ( / > % .

(2.1.24)

We also put

* - dzk ~

ij

0 3 dz* "

ij

0 2 dz* '

Next ^k

_ 1 fcf

r ^ = 2^

( # U + » M - #i,«)

a
a^

(2-1.25)

a
Thus, r f • vanishes if all three indices i, j , k are distinct, and for all z, j r

J< = r « = - ^ .

andr

Ji = ^

******

In the present case, ^ = log(l + | z | 2 ) - l o g 2 hence dtp _ 2z* dz^~

1 + Id2'

(2-1"26)

Geodesic curves

42

Therefore, the equations for geodesies become n

0

n

= ? + 2 E r « W ^ - r«(7)ff + £ r ii(7)W (using the symmetry T^ = T^) 7

2V

f

27

J

2

V +V

^tti + W

7

2yV.

2

(2.1.27)

.tti + W

We now claim that the geodesic *y(t) through the origin, i.e. 7(0) = 0, with 7(0) = a £ E n is given by -y(t) = aa(t),

(2.1.28)

where a : R —• R then satisfies a(0) = 0, d(0) = 1. Making the ansatz (2.1.28) in (2.1.27) leads to

fr{l

2\a\2a

&

~Z

v

+ a2\a\2

fril

\,2

, ,2 o d

l-f|a|^a2

+ a2\a\2

i = l,...,n. 7

Since we may assume o / O (otherwise the solution with 7(0) = a is a point curve, hence uninteresting), this equation holds, if a(t) satisfies the ordinary differential equation (ODE) 2 \a\ a .9 /n ., rt^N L« — a 2 . (2.1.29) 1 4- |a| a2 The theorem of Picard-Lindelof implies that (2.1.29) has a unique solution in a neighbourhood of t = 0. We then have found a solution *y(t) of (2.1.27) of the desired form (2.1.28). The image of j(t) is a straight line through 0. By Lemma 2.1.3, we have thus found all solutions through 0. The images of the straight lines under the chart /1 are the great circles on 5 n through the south pole. We can now use a symmetry argument to conclude that all the geodesic lines on Sn are given by the great circles on Sn. Namely, the south pole does not play any distinguished role, and we could have constructed a local chart by stereographic projection from any other point on Sn as well, and the metric tensor would have assumed the same form (2.1.22). More generally, one may also argue as follows: We want to find the geodesic arc j(t) on Sn with 7(0) = po, 7(0) = VQ for some p0 G Sn, V0 £ TPoSn. Let c0(t) be the great circle on n

0= a

2.2 Fields of geodesic curves

43

Sn parameterized such that Co(0) = po, Co(0) = Vo- c$ is contained in a unique two-dimensional plane through the origin in E n + 1 . Let i denote the reflection across this plane. This is an isometry of R n + 1 mapping Sn onto itself. It therefore maps geodesies on Sn onto geodesies, because we have observed that the length and energy functionals are invariant under isometries, and so isometries have to map critical points to critical points. Now i maps po a n d Vo to themselves. If 7 were not invariant under i, i o 7 would be another geodesic with initial values p0, Vo, contradicting the uniqueness result of Lemma 2.1.3. Therefore, 2 0 7 = 7, and therefore 7 = CQ. We draw some conclusions: The geodesic arc through two given points need not be unique. Namely, let p, q be antipodal points on 5 n , e.g. north and south pole. Then there exist infinitely many great circles that pass through both p and q. We shall later on see that the first conjugate point of a point p £ Sn along a great circle is the antipodal point q of p. One also sees by explicit comparison that a geodesic arc on Sn ceases to be minimizing beyond the first conjugate point, in accordance with Theorem 1.3.4.

2.2 Fields of geodesic curves Let M be an embedded, differentiate submanifold of E d , or, more generally, a Riemannian manifold of dimension nf, again of class C 3 . Let MQ be a submanifold of M; this means that Mo itself is a differentiate submanifold of E d , respectively a Riemannian manifold, and that the inclusion i : M 0 c-> M is a differentiate embedding. We assume that MQ has dimension n — 1, and that it is also of class C 3 . Theorem 2.2.1. For any x0 in M0 , there exist a neighbourhood V of XQ in M, and a chart f : U —• V with the following properties: (i) U contains the origin o / E n 7 /(0) = #o(ii) M0nV = f{Un{xn=0}) % (iii) The curves x = Q , C% = constant, i = l , . . . , n — 1, are geodesies parameterized by arc-length. The arcs £1 < xn < £2 on any such f We do not introduce the concept of an abstract Riemannian manifold here, but some readers may know that concept already, and in fact it provides the natural setting for the theory of geodesies. On the other hand, the embedding theorem of J.Nash says that any Riemannian manifold can be isometrically embedded into some Euclidean space ]Rd, hence considered as a submanifold of Rd. Therefore, from that point of view, no generality is gained by considering Riemannian manifolds instead of submanifolds of R d .

Geodesic curves

44

curve between the hypersurfaces xn = £1 and xn = £2 we all of the same length £1 — £2 • (iv) The metric tensor on U satisfies 9nn = 1,

9%n = 0 for all i = 1 , . . . , n - 1

(2.2.1)

(T/ie second relation means that the curves xl = Q , i = 1 , . . . , n — 1, intersect the hypersurfaces xn = constant orthogonally.) Proof. Since Mo is a hypersurface, for every p G Mo, there exist two unit normal vectors n±(p) to Mo at p, i.e. n±(p) G TpM, \\n±(P)\\ = l (n ± (p), v) = 0 for all v G TPM0 C TPM. In a sufficiently small neighbourhood V of #o, we may assume that such a normal vector n(p) may be chosen so that it depends smoothly on p G Mo fl V =: Vo. We assume that there is a local chart (p0 : UQ —* Vo for Mo (Uo C M 71-1 ), possibly choosing V smaller, if necessary. For every p G Mo fl V, we then consider the geodesic arc 7P(£) with = P, 7P(0) = n(p). 7P(0)

(2.2.2)

This geodesic exists for |£| < e = e(p) by Lemma 2.1.3. By choosing V smaller if necessary, we may assume that e > 0 is independent of p. Instead of 7P(£), we write ~/(p,t). Since the solution of (2.2.2) depends differentiably on its initial values (see Lemma 2.1.3), hence on p, the map / : tfo x ( - € , € ) - M (x,t) ->7(
^ " ' • • •' S J ^ T ' n ( ^ ( x ) ) ( n o t e

tnat

7(^(^)>0) = <^(x) a n d

n

(^(x))

2.2 Fields of geodesic curves

45

are orthogonal to all the vectors -^ G T ^ ^ M o , j = 1 , . . . , n — 1). Therefore, by the inverse function theorem, / yields a chart in some neighbourhood U of (0,0) G Uo x (-e,e). / obviously satisfies (i), (ii) (after redefining V). (iii) also holds by construction (putting xn = t). Next, gnn = 1, since the curves x% = Q , namely / ( c i , . . . , cn__i, £), £ G (—c, e), are geodesies parameterized by arc-length, hence gnn = (-gf, -gf) = 1. Finally, the system of equations for these curves to be geodesic is d2xk (dxn)2

^k dxl dxj + I « 7 rtJ- r inr - rn dx dx

(*"=<)

forfc=l,..-,n.

Hence in particular r *nn = 0

forfc=l,

Now =

^nn

9^

(2gnlyTl

-

<7nn,/) = <7 <7n/,n,

since # n n = 1. Therefore gnkyU = 0 for all k = 1 , . . . , n. Since furthermore gnk{x1 ^... , x n _ 1 , 0 ) = 0, because the geodesic arc xn = £, xl = Q = constant, is orthogonal to the surface (p(xx,..., x n _ 1 ) = / ( a : 1 , . . . , xn~l, 0), we obtain £nfc = 0. q.e.d. Definition 2.2.1. The coordinates whose existence is affirmed by Theorem 2.2.1 are called geodesic parallel coordinates based on the hypersurface M0. Theorem 2.2.2. Let f : U —• V be a chart with the properties described in Theorem 2.2.1. In particular, the curves xl = c«, c« = const ant, for i = l , . . . , n — 1 are geodesic arcs. Then any such curve is the shortest connection of its endpoints when compared with all curves contained entirely in U and having the same endpoints. Proof We consider the geodesic 7 (t)

= {x* = Ci,a:n = * , - € < * < € } ,

where U = U0x (-e, e). Let 7(f), h < t < t2 be another curve in U with 7(^i) = 7 ( - e ) , 7(t 2 ) = 7(e). We have to prove L(7) > L(7),

(2.2.3)

46

Geodesic curves

with strict inequality, unless 7 is a reparameterization of 7. Now

£(7) - f 2 '/tl

( E

9a (7 to) 4 W to + ( V ' t o )

\i,i=l

j

A,

(2.2.4)

/

since # n n = 1, ^ n = 0 for i = 1 , . . . , n - 1 by Theorem 2.2.1(iv), rt<2 1 ^

(t)| * > 7 n (t 2 ) - 7 n (*i) = 7"(e) - 7 n (~e)

= L( 7 ). The first inequality is strict, unless 7* is constant for i = 1 , . . . , n — 1, and the second one is strict, unless jn(t) is monotonic. q.e.d. Following Weierstrafi, we say that the geodesies 7 (£)

= {x{ = a,xn

=

t,-c
constitute a field of geodesies. Theorem 2.2.2 essentially says that any geodesic arc in this field is shorter than any other curve with the same endpoints in the region covered by the field. Both properties are essential. Namely geodesic arcs on Sn that are longer than a great semicircle show that geodesies not embedded in a field need not minimize the length between their endpoints. And geodesic arcs on a cylinder, contained in meridians, but longer than a semicircle show that there may be shorter curves not contained in the field. We observe that if 7(2) solves (2.1.21), so does 7(A£) for A = constant. We fix ZQ G U and denote the geodesic arc 7 of Lemma 2.1.2 with 7 (0)

= z 0 ,7(0) = ^

by 7„. Then by the above observation 7
forA^O.

(2.2.5)

Thus 7AV is defined on [ ^ , j], if 7 is defined on [—c, c]. Since 7^ depends differentiably on v, and since v G R n , \v\ = 1, is compact, there exists eo > 0 with the property that for all v with \v\ = 1, j v is defined on [—eo, eo]. Prom (2.2.5), we then conclude that for any w G E n with M < eo> yw is defined on [—1,1]. For later purposes, we also note that by Lemma 2.1.3, eo may be chosen to depend continuously on ZQ.

2.2 Fields of geodesic curves

47

We now define a map e = eZ0 : {w G Rn : M < e 0 } -+ J7 WH+7^,(1). Then e(0) = z0. We compute the derivative of e at 0 as De(0){v) = |7t»(l)|,=o = ^7,(%=„

by (2.2.5)

= 7«(0) =

V.

Hence, the derivative of e at 0 G l n is the identity, and the inverse mapping theorem implies: Theorem 2.2.3. e maps a neighbourhood of 0 G E n diffeomorphically (i.e. e is bijective, and both e and e~l are differentiate) onto a neighbourhood of ZQ G U. q.e.d. We want to normalize our chart / : U —• V for M. First of all, we may assume zQ = 0

(2.2.6)

for the point Zo G U under consideration. Secondly, the transformation formula (2.1.16) implies that we may perform a linear change of coordinates (i.e. replace f by f o A, where A G GL(n,R)) in order to achieve fti(0) = * « .

(2.2.7)

We assume that / : U —> V satisfies these normalizations. We then replace / by / o e defined on {w G E n : \w\ < e0}. Theorem 2.2A. In this new chart, the metric tensor satisfies gijM^Sij r$fc(O) = O = 0y,fc(O)

(2.2.8) for alii , j , k.

(2.2.9)

Proof. By (2.1.16), gij = 6ij holds, since the metric tensor w.r.t. the chart / satisfies this property and De(0) is the identity by the proof of Theorem 2.2.3. In order to verify (2.2.9), we observe that in our new chart, the straight lines tv (v G E n , t \v\ < e) are geodesies. Namely, tv is mapped to 7t v (l) = 7v(£) (see (2.2.5)), where *yv(t) is the geodesic with

48

Geodesic curves

initial direction v. We thus insert 7(2) = tv into the geodesic equation (2.1.21). Then 7 = 0, hence rjk(tv)vjvk

= 0

for t = l , . . . , n .

In particular, inserting t = 0, we get r)k(0)vjvk

= 0 for all v G R n , i = 1 , . . . , n.

We use t; = e*, where (ei)l=1

n

is an orthonormal basis of R n . Then

r{i(0) = 0 for a l i i and/. We next insert v = ^(ej 4- e m ), £ ^ m. The symmetry Tljk — T^. (which directly follows from the definition of H fc and the symmetry gjk = #fcj) then yields rj m (0) = 0 for a l l i , / , m . The vanishing of gij^ for all i,j, k then is an easy exercise in linear algebra. q.e.d. Definition 2.2.2. The local coordinates xl,...,xn constructed before Theorem 2.2.4 are called Riemannian normal coordinates. We let x 1 , . . . , x n be Riemannian normal coordinates. We transform them into polar coordinates r, (p1,..., ipn~~l in the standard manner (e.g. if n = 2, x1 = r c o s ^ 1 , x2 = r s i n ^ 1 ) . This coordinate transformation is of course singular at 0. We now express the metric tensor w.r.t. these polar coordinates. We write grr instead of # n , and we write gr(p instead of g\i, I = 2 , . . . ,n, and g^ instead of (9ki)k,i=2,...,d' * n Particular, by Theorem 2.2.4 and the transformation rule (2.1.16) ffrr(0) = l,ffr V (0)=0-

(2.2.10)

The lines through the origin are geodesies by the construction of Riemannian normal coordinates, and in polar coordinates, they now become the curves

with fixed
Therefore, the geodesic equation (2.1.21) gives Trr = 0

for all i

(where of course Tlrr stands for T ^ ) , i.e. -gil (2gri,r ~ 9rr,i) = 0 for all i,

2.2 Fields of geodesic curves

49

hence 2#w,r ~ 9rr,i = 0

for all/.

(2.2.11)

Putting r — I gives grrr = u, and with (2.2.10) then grr = 1.

(2.2.12)

Using this in (2.2.11) gives 9rip,r

==:

U,

hence with (2.2.10) again gry = 0.

(2.2.13)

We have thus shown: Theorem 2.2.5. In the preceding coordinates, so called Riemannian polar coordinates, that are obtained by transforming Riemannian normal coordinates into polar coordinates, the metric tensor has the form /l

I° I :

0

... r

9 w ( ,
\0

0\

1 I ' /

where g^ stands for the (n — 1) x (n — 1)-matrix of the components of the metric tensor w.r.t. the angular variables y? 1 ,..., (pn~l. Note that this generalizes the situation for Euclidean polar coordinates. The Euclidean metric on M2, written in polar coordinates, e.g. takes the form

(J r")Note that Theorem 2.2.5, in contrast to Theorem 2.2.4, is valid on the whole chart, not only at the origin. Corollary 2.2.1. Riemannian polar coordinates are geodesic parallel coordinates based on the hypersurfaces r = constant (r ^ 0, since r = 0 corresponds to a single point, and not a hypersurface). Proof. By Theorem 2.2.5, all properties stated in Theorem 2.2.1 hold. q.e.d.

Geodesic curves

50

By Corollary 2.2.1 and Theorem 2.2.1, the curves (p = constant, r\ < r < r 2 5 are shortest connections between their end points among all curves lying in the chart. We are now going to observe that this holds even globally, i.e. also in comparison with curves that may leave the chart: Theorem 2.2.6. For each p E M, there exists e0 > 0 with the property that Riemannian polar coordinates centered at p may be introduced with domain {(r,^):0 0 : c(r) 6 B(p, e) for 0 < r < t}. Since w.l.o.g. e > 0 and c is continuous, t0 is positive. We are going to show that VCl[o.'o]J

> 6.

(2.2.14)

Since the curve (t,(po), 0 < t < e, has length e as easily follows from Theorem 2.2.5, this will imply the claim. In order to verify (2.2.14), we proceed as follows: L c

( !Io.^) =

/t0(ftj(c(*))ci(t)^(*))i* J 0

(identifying Cj

with its coordinate representation) [to

> / Jo

J

(9rrrf)2 dt

2.3 The existence of geodesies by Theorem 2.2.5 and since gw

51

is positive definite (writing c(t) —

(r{t)Mt))) rto

— \ Jo

\r\ dt,

again by Theorem 2.2.5

>

rdt = r(t0) = e. Jo Here, equality only holds if gw 0, i.e. if q . is a straight line through the origin. The second claim now easily follows. q.e.d. Corollary 2.2.2. If M is compact, there exists e0 > 0 with the property that for every p G M, there exist Riemannian polar coordinates with domain {(r,^):0
2.3 The existence of geodesies Definition 2.3.1. Let M be a connected differentiate submanifold of Euclidean space R.d, or, more generally^, a connected Riemannian manifold. The distance between p,q G M is d(p,q) := inf{L(c)| c : [a, 6] —• M rectifiable curve with

c(a) = p, c(b) = q}.

Theorem 2.3.1. Let M (as in Definition 2.3.1) be compact. There exists eo > 0 with the property that any two points p,q G M with d(p, q) < e0 can be connected by a unique shortest geodesic arc (i.e. of length d(p,q)). This geodesic arc depends continuously on p and q. Proof. We take e0 as described in Corollary 2.2.2. This gives a unique shortest geodesic arc from p to q which furthermore depends continuously on q. Exchanging the roles of p and q then yields continuous dependence on p, too. q.e.d. f See footnote on p. 43.

Geodesic curves

52

We now proceed to establish a global result: Theorem 2.3.2. Let M be a compact connected differentiable submanifold ofW*, or, more generally, a compact connected Riemannian manifold. Then any two points p,q £ M can be connected by a shortest geodesic arc (i.e. of length d(p,q)). Proof. Let (c n ) n€ N be a minimizing sequence. We may assume w.l.o.g. that all cn are parameterized on the interval [0,1] and proportionally to arc-length. Thus Cn(0) = p , C n ( l ) = g ,

L(cn) —> d(p, q)

for n —> oo.

For each n, we may find t(),n = 0 < tiiTl < . . . < t m , n = 1 with L Cn

(

i[«,-..».«i,»i)^ c °'

with eo given by Theorem 2.3.1. By Theorem 2.3.1, there exists a unique shortest geodesic arc between c n (fy_i jn ) =: Pj-i, n and c n (tj,n) =: Pj, n . We replace cn\ , by this shortest geodesic arc and obtain a t new minimizing sequence, again denoted by c n , that now is piecewise geodesic. Since the length of the cn are bounded because of the minimizing property, we may actually assume that m is independent of n. Since M is compact, after selecting a subsequence of c n , the points p^n converge to limit points pj, (j = 0 , . . . , m) as n —• oo. cn\ , the t unique shortest geodesic arc between Pj_i, n and p J > n , then converges to the unique shortest geodesic arc between Pj~\ and pj (for this point, one verifies that limits of geodesic arcs are again geodesic arcs, that limits of shortest arcs are again shortest arcs, that d(pj~i,pj) < eo, and one uses Theorem 2.3.1). We thus obtain a piecewise geodesic limit curve c, with c(0) = p, c(l) = g, and L(c) = lim

L(cn),

n—+oo

since we have for the geodesic pieces L

(%-».',l)=„^oI'(C»l^-i.-.«i.-l)

for all j (tj = lim t, n). Since the c n constitute a minimizing sequence, n—+oo

L(c) = d(p,q),

2.3 The existence of geodesies

53

and c thus is of shortest possible length. This implies that c is geodesic. Namely, otherwise we could find 0 < S\ < S2 < 1 with L Icj[s s J < e0, but with C|(s s j not being geodesic. Replacing c\{ ] by the shortest geodesic arc between c(s\) and c(«2) would yield a shorter curve (cf. Theorem 2.2.6.), contradicting the minimizing property of c. q.e.d. Thus, any two points on a compact M may be connected by a shortest geodesic. We now pose the question whether they can be connected by more than one geodesic, not necessarily the shortest. On 5 n , for example, this is clearly the case. Actually, the answer is that it is the case on any compact M. That result needs a topological result that is not available to us here, however. Therefore, we will restrict ourselves to a special case which, however, already displays the crucial geometric idea of the construction for the general case, too. Theorem 2.3.3. Let M be a differentiable submanifold of Euclidean space Rd, (or more generally], a Riemannian manifold), diffeomorphic to the sphere S2. The latter condition means that there exists a bijective map h:S2

^M

that is differentiable in both directions. Then any two points p, q £ M can be connected by at least two geodesies. Proof. M is compact and connected since diffeomorphic to S2 which is compact and connected. Let us assume p ^ q. We leave it to the reader to modify our constructions in order that they also apply to the case p = q. (In that case, Thm 2.3.3 asserts the existence of a nonconstant geodesic c : [0,1] —• M with c(0) = p = c(l).) One may then construct a diffeomorphism h0 : S2 -+ M with the following properties: Let S2 = { ( x 1 , ^ 2 , ^ 3 ) e M3 : |x| = l } . Then P = M0,0,1),

g = MO, 0,-1)

and a shortest geodesic arc c : [0,1] —• M with c(0) = p, c(l) = q is given by c(i) — M0,sin7r£,cos7r£). f See footnote on p. 43.

54

Geodesic curves

Let us point out that these normalizations are not at all essential, but only convenient for our constructions. We look at the family of curves 7(^,5) = /i0(sin27rssin7r£, cos27rssin7r£,cos7r£),

0 < s}t < 1. (2.3.1)

Then 7 (£,

0) = 7(£, 1) = c(i)

for all t

and 7(0, s) = c(0),

7(1, s) = c(l)

for all s.

We find some number K with Hn(;8))
for all s.

(2.3.2)

Redefining the parameter t, we may also assume that all curves 7(-,s) are parameterized proportionally to arc-length. By Theorem 2.3.1, there exists 60 > 0 such that the shortest geodesic between any p, q £ M, with d(p, q) < e0 *s u r n c l u e - Let 0 = £0 < h < . . . < tm = 1 be a partition of [0,1] with 'i-«;-!<]£

forj = l , . . . , m .

(2.3.3)

Let another partition (TI, . . . , r m ) satisfy To = ^O < T\ < h < T2 < . . . < Tm < tm = T m + i

and T

J~TJ~i<^

forj

= l , . . . , r o + l.

(2.3.4)

If 7 : [0,1] —• M is any curve parameterized proportionally to arc-length with L(l) < K, we then have for j = 1 , . . . , m d(1(tj^),1(tj))
2.3 The existence of geodesies Lemma 2.3.1. Suppose e0 for all j .

55

< e0 and d(7( r j)>7( r j-i)) <

rf(7(^),7(^-1))

r(7) : = r 2 o n ( 7 )

L(r(7))
(2-3.5)

wn£/i equality iff 7 is geodesic. Proof. By uniqueness of the shortest geodesic between 7(fy_i) and 7(£j), we have L(n(7))
Likewise, for every curve 7', L (7,'

) < €Q for all j ,

MMV))<W) with equality only in case

r2(y) =7 ; Therefore L(r(7))n(7) (= ro...or(7)) converges uniformly to a geodesic with the same endpoints as 7.

56

Geodesic curves

Proof. Each curve r n (7), n E N, is a piecewise geodesic with corners r n 7 ( r i ) , . . . , r n 7 ( r m ) and endpoints r n 7(r 0 ) = 7(0), r n 7 ( r m + i ) = 7(1). The individual segments are the unique shortest connections between these points. Therefore, each such curve is uniquely determined by the m-tupel A" := (r n 7(Ti),...,r"7(T r o )) e M x ... x M . m times Since M is compact, a subsequence of An converges to some limit (Pi,...,Pm) 6 M x ... x M. r n (7) then converges uniformly towards the piecewise geodesic 70 with endpoints 7 0 (0) = 7(0),7o(l) = 7(1) and nodes 70(r,) = Pi (i = 1 , . . . , m) with segments 70j, being the shortest geodesic arcs ber tween their endpoints. This follows from the continuous dependence of the occurring geodesic arcs on their endpoints (Theorem 2.3.1). We denote the convergent subsequence of ( r n ( 7 ) ) n € N by (7z,)„€N. For all v € N then 7l/+1

= r n ( l / ) 7^

with n(y) € N.

By the minimizing property of the subsegments of the 7 „ , L

= d

(l» ( r i - i ) >7" fo)),

('Mr,-.!,^)

hence ra+1 L

d

(>) = X)

(?" ( r i - i ) ' > ( r J'))'

j=l

Since 7I/(T 7 ) converges to p^ = 70(T^), L(7I/) converges to m+l L

(7o) = X ] d (7o to-i), 70 (r,-))

for ^ —• 00. Then also L( 7 o) = lim L( 7 „+i) = lim L(rn<">7„) 1/—>oo

1/—>oo

< lim L(7i/)

by Lemma 2.3.1

= i(7o), and equality has to hold throughout. Moreover, r(7„) converges to

r(f0),

2.3 The existence of geodesies

57

and L ( r ( 7 o ) ) = lim L(r( 7 „)) > lim L (rn^7„ =

)

by Lemma 2.3.1 again

L{10).

Lemma 2.3.1 then implies that 70 is geodesic. q.e.d. We now return to the proof of Theorem 2.3.3: We apply the preceding curve shortening process to all curves 7(-, s), $£ [0,1], simultaneously. For each 5, a subsequence of r n 7(-,$) then converges to a geodesic from p to q. We want to exclude the situation that all those limit geodesies coincide with c. Let K0 : = £ ( c ) , and tti := sup n lim L(r n 7(-,s)). o<s K0. We distinguish two cases: (1) Ki > K0

Since 7(-, 5) is continuous in s, so is r n 7(-, s) for every n G N. We now claim: Whenever s u p i ( r n 7 ( - , s ) ) < Kx + e

(2.3.6)

s

there exists s n € [0,1] with L(r" 7 (-, *»)) " ^ ( r n + 1 7 ( - , »»)) < 2e

(2.3.7)

i(rn7(-,*n))>«i-c-

(2-3-8)

and

Indeed, otherwise supL(rn+17(-,s)) < « ! - € , S

contradicting the definition of K\ (note that sup 5 L(r n " f l 7(-,5))

58

Geodesic curves is monotonically decreasing in n by Lemma 2.3.1). By definition of «i, there exists a subsequence (e n ) n € ^ —• 0 with supL(r n 7(-,5)) < Ki + e n . s

A subsequence of (r n 7(-,s n )) n(E N has to converge to some limit curve c as above, and because of (2.3.7) with e = e n , we conclude as in the proof of Lemma 2.3.2 that L(r(c)) = L(c), and c is hence geodesic by Lemma 2.3.1. Because of (2.3.8) and continuity of L in the limit as in the proof of Lemma 2.3.2, we get L(c) = K\. Since c and c are both defined on [0,1] and have different lengths, they have to be different curves. Thus, c is the desired second geodesic. (2) Ki = ft0

We are going to show that in this case, there even exist infinitely many geodesies from p to q. For that purpose, we consider the curve 700 :=7(2> 5 )This is a closed curve with 7(0) = 7(1) = c{\) (see Figure 2.2). Since ho is a diffeomorphism and r n 7(t, 5) is obtained through a process that can easily be made continuous from 7(^,5) = /io(sin27T5sin7rt,cos27r5sin7rt,cos7rt), r n 7(£,s) has to map [0,1] x [0,1] surjectively onto M. Therefore, for every n £ N and every 5 G [0,1], there exists &n(s) with 7(5) €r n 7(-,<7 n (5))=: 7 n , 5 (-) (in other words, r n 7(-,<7 n (s)) is a curve passing through 7(5)). 7n,5(*) then is a curve with 7n,s(0) =c(0) =p,7n,«(l) = c(l) =q, and because of K\ = K0J we obtain lim L(7 n , a (-)) < sup

lim L(r n 7 (-, 5 )) = ^o.

(2.3.9)

2.3 The existence of geodesies

59

q Figure 2.2. After selection of a subsequence, (7n,s(*))n€N again converges to some limit curve cs(-) with ca(0) = p , c 5 ( l ) =q and 7(s)eca(-).

By (2.3.5), L(ca(-)) < K 0 , and since Ko is the infimum of the energies of all curves from p to q («o = £(c), a n d c is minimizing), cs(-) is a minimizing curve itself, hence geodesic. Therefore, we have shown that for every 5, there exists a geodesic from p to q that passes through 7(5). Hence there exist infinitely many geodesies from p to g, as claimed. q.e.d. Remarks: (1) Lemmas 2.3.1 and 2.3.2 do not need that M is diffeomorphic to S2. Compactness suffices.

60

Geodesic curves (2) We may construct the curves 7n,s(*) at the end of the proof also in case K\ > K0. In that case, however, limits of such curves need not be geodesic anymore. (3) See Section 3.1 for an abstract version of the argument at the end of the preceding proof.

Exercises 2.1

2.2

1

For curves
6 R 2 | x 2 > 0}, con-

Compute the Euler-Lagrange equations and determine all solutions. For curves d

7(t) = ( 7 1 , . . t 7 d ) : ^ { ( x 1 , . , x d ) 6 l d | 5 ] ( x l ) 2 < l } ) t=i

consider

2.3

Compute the Euler-Lagrange equations and determine all solutions. Determine all geodesies between two given points on a cylinder { ( x , y , z ) e R 3 : x 2 - f y2 = l } .

2.4

Let £ be a surface of revolution in R 3 , i.e. £ = { ( x , y , * ) e R 3 : x 2 + y2 = / ( * ) }

2.5 2.6

for a smooth, positive / : R —*• R. What can you say about geodesies on £? For example, are the curves (x, y) = constant geodesies? When are the curves z = constant geodesies? Determine Riemannian polar coordinates on the sphere Sn with a domain of definition that is as large as possible. Let p be the center of Riemannian polar coordinates on M, with domain of definition {v e Rrf : \\v\\ < g}. Let c : [0,e] -+ M be a geodesic with c(0) — p that is parameterized by arc-length, 0 < e < Q. Show that c([0, e]) does not contain a point that is conjugate to p.

Exercises 2.7

2.8

61

Let M be a differentiable submanifold of Rd that is diffeomorphic to S 2 . Show that for any p G M, there exists a nonconstant geodesic c : [0,1] —• M with c(0) = c(l) = p. Try to find other topological classes of manifolds with the property that there always exists more than one geodesic connection between any two points.

3 Saddle point constructions

3.1 A finite dimensional example d

Let F : R —• R be a function of class C1 which is bounded from below and which is 'proper' in the following sense: F(x) -+ oc

for |x| -* oc.

(3.1.1)

Since F is bounded from below, (3.1.1) is equivalent to: For every s £ R, {x e R d : F(x) < $}

is compact.

(3.1.2)

Therefore, F assumes its infimum. Namely, we take any s0 > inf

F(x).

x€Rd

Then {x e Rd : F(x) < so} is compact and nonempty, and since F is continuous, it has to assume its infimum on that set. We now assume that F even has two relative minima, #i, #2 in R d , and that they are strict in the following sense: For x = £i,#2, w e have 360Vy

with

0 < \y-x\

< 60 : F(y) > F(x).

(3.1.3)

Theorem 3.1.1. Under the above assumptions, F has a third critical point £3 (i.e. VF(xs) = 0) with F(x3) > max(F(xi),F(x 2 )) =: «o Proof. We consider curves 7 : [0,1] —• Rd with 7(0) = xi ,7(1) =x2. 62

(3.1.4)

3.1 A finite dimensional example

63

We first observe that there exists a > 0 with the property that for any such curve, there exists to € (0,1) with F(7(t0))>Ko

+ <*.

(3.1.5)

In order to verity this, we may assume w.l.o.g. F(xi) <

F(x2).

We then choose 6 with 0 < 6 < min(<50, - \xi - x2\).

(3.1.6)

For every y with \y — x2\ = 5 then by (3.1.3) F(y) >

F(x2),

and since {\y — x2\ — 6} is compact, F assumes its minimum on this set, hence for some a > 0 min

F(y) > F(x2) + a = n0 + a.

(3.1.7)

Since for every curve 7 with (3.1.4) we have |7(1) - x2\ = 0, |7(0) - x2\ = \xi -

x2\,

there has to exist some t0 € [0,1] with h(t0)-x2\

=6

(recall (3.1.6)) .

By (3.1.7) then i W o ) ) > «o 4- a, and (3.1.5) follows indeed. We now define K\ := inf sup F(7(£)), 7

*€[o,i]

d

where 7 again is a curve in R with 7(0) = #i, 7(1) = #2- By (3.1.5) «i > « 0 . Our intention now is to find a critical point £3 of F with F(x 3 ) = «i. Since F(xi),F(x2)<«0,

(3.1.8)

64

Saddle point constructions

Xs will then be necessarily be different from X\ and #2- As a step towards the existence of such a point x 3 , we claim Ve > 0

3<5 > 0 V curves 7 with

7(0) = #1,7(1) = x2

with sup F ( 7 ( * ) ) < « i + *

(3.1.9)

*€[0,1]

3t0£

[0,1] with: F{i{t0))

> «i - 6

(3.1.10)

|(VF)( 7 (*o))|<e.

(3.1.H)

Suppose this is not the case. Then 3e0>0

VneN

3

curve

7„

between

X\

and

x^

with

suPF(7n(«))
with

(3.1.12)

F(7„(«o)) > «i - eo

(3-1-13)

|(VF)(7B(«O))|>CO.

(3-1-14)

For s > 0, we define a new curve 7 n>s by 7n,,(0:=7n(0-5(VF)(7n(0). Since x\ and #2 are minima, V F ( x i ) = 0 = VF(x 2 )> and so 7n,s(0) =Xi,7„, 5 (l) = X2, so that the curves 7 n>s are valid comparison curves. By our properness assumption (3.1.2) and (3.1.12), 7 n (£) stays in a bounded subset of E d , and V F will then be bounded on that bounded set, and hence for any s0 > 0 and all 0 < s < s0, the curves 7n,s(£) stay in some bounded set, too. This set is independent of n (as long as 0 < 5 < 5 0 , for fixed s0 > 0). By Taylor's formula F(7n,.(0) = F(yn(t))

- sVF(ln(t))

• VF( 7 „(<)) + o(s).

Since F is continuously differentiate and 7n,5(£) is contained in a bounded set, o(s) can be estimated independently of n and t (as long as 0 < s < s0). In particular, after possibly choosing s0 > 0 smaller, F(7n,.W) < F ( 7 „ ( 0 ) - | |VF( 7 „(t))| 2

(3.1.15)

3.1 A finite dimensional example

65

for all n, s with 0 < s < so, and t with |VF( 7 nW)l > co.

(3.1.16)

Thus, in particular, F(ln,S0(t))

< F(7„(0) - yeg

(3.1.17)

for all such t and all n. We now simply choose n so large that

i < ?* Then by our assumption, all £Q with F(yn(to)) and hence for all such to F(jn,so(to))

(M.«» > «i ~~ eo satisfy (3.1.14),

< F( 7 „(«o)) - y e g
+

< «i

I _ f 2 e 2 by (3.1.12) n z by (3.1.18).

(3.1.19)

Having proved (3.1.19), there are now various ways to construct a path 7 from X\ to #2 with F ( 7 ( 0 ) < «i

for all t € [0,1].

(3.1.20)

One way is to refine the above construction by letting s depend on t as follows: we choose a smooth function a(t):[0,l]-[0,a0] with a(t) = 0

whenever

F(yn(t))

< K\ — e0

a(t) = so

whenever

F(yn(t))

> K\ — —.

and

We then look at the path j(t) = 7n,yn(t)) < «x -

F(7(*)) < F(7B(t)) - ^

~

<M- ? -

^

66

Saddle point constructions

(cf. (3.1.15), (3.1.16), (3.1.14)), and finally for all t with F(7 n (*)) > * i -

T

F(7(*)) = i ! '(7n,.o(0)<«i

(cf-(3.1.19)).

Thus, (3.1.20) holds indeed. This, however, contradicts the definition of K\. Therefore, the assumption that our claim was not correct led to a contradiction, and the claim holds. It is now simple to prove the theorem. Namely, we let e n —• 0 for n —» oo, and for e = e n , we find <5 = 6n as in the claim. We than choose a curve yn from x\ to x
< «i +min(c„,«„).

(3.1.21)

t€[0,l]

According to the claim, there exists tn £ [0,1] with F(7n(*n))>Kl-€n

(3.1.22)

|(VF)( 7 n(*n))|
(3.1.23)

After selection of a subsequence, (7n(^n))n€N then converges to some point £3, because of (3.1.2) and (3.1.21). £3 then satisfies by continuity of F and V F F(x 3 ) = «i

(3.1.24)

VF(x 3 ) = 0.

(3.1.25)

Thus, £3 is the desired critical point. q.e.d. Theorem 3.1.1 may be refined as follows: Theorem 3.1.2. Let F as above again have two relative minima, not necessarily strict anymore. Then either F has a critical point £3 with F(x 3 ) > max(F(xi),F(x 2 )) = «o> or it has infinitely many critical points. Proof. For the argument of the proof of Theorem 3.1.1, we only need inf sup F(7(*)) > « 0 , 7

(3.1.26)

t€[0,lj

where the infimum again is taken over curves 7 : [0,1] —• Rd with 7(0) = x\, 7(1) = X2. So, suppose that (3.1.26) does not hold. We then want to

3.2 The construction of Lyusternik-Schnirelman

67

show the existence of infinitely many critical points. As in the proof of Theorem 3.1.1, we may assume F(xx) < F{x2). The argument at the beginning of the proof of Theorem 3.1.1 then shows that (3.1.26) holds if x2 is a strict relative minimum. If x2 is a relative minimum, which is not strict, for all sufficiently small 8 > 0, say 8 < <5Q, we have F(x2) < F(x)

for all x

with

\x - x2\ < 80

(3.1.27)

and there always exists some x$ with 0 < \xs — x2\ < 8 and F(x6) = F(x2).

(3.1.28)

We then put 8\ = 80/2. Then x&x is a relative minimum of F by (3.1.27), (3.1.28), hence a critical point. Having found a critical point xsn with 0 < \xsn — x2\ < \xsn_1 - x2\, we put 8n+1 = -\x6n

-x2\

and find a critical point xsn+1 with 0

< | ^ n + 1 -x2\

< (5 n+ i.

Thus, xsn+1 is a critical point of F different from all preceding ones. q.e.d. Remark. It is not very hard to sharpen the statement of Theorem 3.1.2 from 'infinitely many' to 'uncountably many'.

3.2 The construction of Lyusternik-Schnirelman In this section, we want to prove the following theorem, in order to exhibit some important global construction in the calculus of variations, introduced by Lyusternik-Schnirelman. The result presented is much more elementary than the theorem of Lyusternik-Schnirelman, which says that on any surface with a Riemannian metric, e.g. a surface embedded in some Euclidean space, diffeomorphic to the two-dimensional sphere, there exist at least three closed geodesies without self-intersections. The more elementary character of our setting allows us to bypass essential geometric difficulties encountered in a detailed proof of the LyusternikSchnirelman Theorem.

68

Saddle point constructions

Figure 3.1. T h e o r e m 3.2.1. Let 7 be a closed convex Jordan curved of class C1 in the plane M2. (7 then divides the plane into a bounded region A, and an unbounded one, by the Jordan curve Theorem. That 7 is convex means that the straight line between any two points of 7 is contained in the closure A of A.) Then there exist at least two such straight lines between points on 7 meeting 7 orthogonally at both end points (see Figure 3.1). Proof. We start by finding one such line. Let £ be the set of all straight lines / in A with dl C 7. We say that a sequence (ln)nen C C converges to / E £, if the end points of the ln converge to those of /. In order to have a closed space, we allow lines to be trivial i.e. to consist of a single point on 7 only. We denote the space of these point curves on 7 by Co. We let / := [0,1] be the unit interval. We consider continuous maps v:I-+C with the following two properties: (i) v(0) = v(l). (ii) To any such family, we may assign two subregions A\(t) and A2(t) of A in a certain manner. Namely, we let A\(t) and A2(t) be the two regions into which v(t) divides A. Having chosen A\(0) and A 2 (0), A\(t) and A2(t) then are determined by the continuity t A closed Jordan curve is a curve 7 : [0, T) -> Rd with 7(0) : : 7(T) that is injective on [0, T). Cf. the definition of a Jordan curve on p. 35.

3.2 The construction of Lyusternik-Schnirelman

69

Figure 3.2. requirement. We then require A1(1) = A2(Q). We let Vi be the class of all such families v. The construction is visualized in Figure 3.2. (0 corresponds to 0 E J, /to±,//to§,///to£, ltol) Actually, in order to simplify the visualization, if v(0) is a point curve (on 7), i) may be relaxed to just requiring that v(l) also is a point curve (on 7), not necessarily coinciding with v(0) (see Figure 3.3). Namely, any point curves can be connected through point curves, i.e. with vanishing length. We denote by L(l) the length of / G £ and define K\ :— inf

s\ipL(v(t)).

veVi tei

Figure 3.3.

Saddle point constructions

70 We want to show that

«i > 0.

For this purpose, let p > 0 be the inner radius of 7, i.e. the largest p for which there exists a disc B(x0jp)

C A

for some XQ G A (B(xo,p) := {x G E 2 : \x - x 0 | < p})« Then «i > «i •= inf supL(v(t)nB(xo,/o)). We let A'i(t) := ^ ( ^ ) n B(xo,p), i = 1,2. Because of (ii) and the continuous dependence of Ai(t) and hence also of Af{(t) on t, there exists some to £ I with Area (A[(t0)) = Area(A ; 2 (t 0 )). Thus v(to) divides B(xo,p) into two subregions of equal area. v(to) then has to be a diameter of J5(xo,p), i.e. L(v(t0)nB(x0,p))

= 2p.

Therefore «i > «i = 2p > 0 and «i is positive indeed. We are now going to show by a line of reasoning already familiar from Sections 2.3 and 3.1 that K\ is realized by a critical point / of L among all lines with end points in 7, i.e. by / meeting 7 orthogonally (see Theorem 1.4.1). For that purpose we shall assume for the moment that 7 is of class C 3 . Later on, we shall reduce the case where 7 is only C 1 to the present one by an approximation argument. We now claim Ve>0

36 >0: sup L(v(t))

Vv € Vi with < K1+6

tei

3t0 e I and where a\(l)

with

L (y(to)) > K\ - e

|cos(ai(v(t 0 )))| , |cos (a 2 (v (t0)))\ < e,

and 0*2(1) are the angles of / at its endpoints with 7.

3.2 The construction of Lyusternik-Schnirelman

71

Otherwise 3e 0 > 0 : Vn G N

3vn G Vi with

supL(i; n 0O) < «i + ^ Vt 0 with L (vn(*o)) > tfi - co |cosai(v n (t 0 ))| > e0 or

|cosa 2 (vn(*o))| > eo-

The idea to reach a contradiction from that assumption is simple, once the following Lemma is proved: L e m m a 3.2.1. For every planar closed Jordan curve 7 of class C 3 , there exists (3 > 0 with the following property: Whenever # G E 2 satisfies dist(x,7) := inf \x - y\ < (3 ye-y

there exists a unique y G 7 with dist(x,7) = \x — y\. Proof We consider 7 as an embedded submanifold of the Euclidean plane E 2 . 7 is then covered by the images of charts / : U —• V of the type constructed in Theorem 2.2.1. Here, U and V are open in E 2 , and 7

n v = f (u n {x2 = 0}).

Furthermore, the curves x1 = constant in U correspond to geodesies, i.e. straight lines in V perpendicular to 7, and they form shortest connections to 7 fl V. By shrinking U, if necessary, we may assume that it is of the form (—£, £) x (—77,77), with £ > 0, 77 > 0. Since 7 is compact, it can be covered by finitely many such charts fi

:

( - 6 , 6 ) x (-WiVi) -* Vi

, i = l,...,ra.

If we then restrict fi to (—6,6) x ("i 21 , ^)» the lines x1 = constant, :z ± ^ < x2 < ^ , then correspond to shortest geodesies to 7, since the part of 7 not contained in Vi is not contained in the image of fi, and hence has distance at least %*• from the image of the smaller set (—&, &) x (-^2i, ?f). This is indicated in Figure 3.4 where the broken lines correspond to x2 = Jk, and this is depicted for two different indices i. Therefore, (3 := min ( ^ ) satisfies the claim. t=l,...,n

q.e.d.

72

Saddle point constructions

Figure 3.4.

Vn(to)

Figure 3.5. We now return to the proof of Theorem 3.2.1: Without loss of generality eo < f3 < ^ . Assume e.\ cosai (vn(t0))

> e0.

The following construction is depicted in Figure 3.5. Choose

SI(£Q)

€

3.2 The construction of Lyustemik-Schnirelman

73

vn(to) with l*i(to)-Pi(to)|=£ where p\(to) is the endpoint of vn(to) where it forms the angle a\(to) with 7. We replace the subarc v^tto) of vn(to) between p\(to) and s\(to) by the shortest line segment vfn(to) from s\(to) to 7. By the theorem of Pythagoras and the convexity of 7 L (v'n (t0)) < L {yl (t0)) sinai (vn (t0))


+

L(vl(t0))

< l K ( t „ ) ) ^ + I(^((o)) = /3y/l^4

+

L(v2n(t0)).

Since L (y\ {to)) = j3, we have L(i£(to)) 0. Hence L«(*o)) <*!->?. We now continuously select points s\(t), S2(t) on vn(t) for every £ G J with \Pi(t) - Si(t)\ = /?,

whenever

L (v n (t)) > Ki - (3

and Pi(t) = s^(t),

whenever

L (i>n (£)) < K\ — 2/3 i = 1,2

and lftW-*iWI
for all*.

74

Saddle point constructions

We then choose again the shortest lines from Si(t) to 7 and replace vn(t) by the straight line v^{t) between those points, where these two shortest lines meet 7. By our geometric argument above L (v* (t)) < K\ — 77 for some rj > 0, whenever L(vn(t))

> fti - 6 0 .

Since also always L(v'n(t))
contradicting the definition of K\. Consequently, our claim is correct. We then find a sequence (tn)neN C J and (vn)neN C V\ with sup L(vn(t)) tei L(vn(tn))

KI n

|cos (ai (vn (tn)))\, |cos (a 2 (vn (tn)))\ < - . A subsequence of (vn (tn))ne^ then converges to a straight line l\ in A of length K\ meeting 7 orthogonally at its endpoints. In order to construct a second line I2 meeting 7 orthogonally at its endpoints, we proceed as follows: We denote by V2 the class of all continuous maps v : / x / —• C with v({0} x /) and v({l} x I) C C0 and with the following property: For all continuous maps T:I-*IXI

T(8) =

(h(8)Ms))

(3.2.1)

3.2 The construction of Lyusternik-Schnirelman

r2 = 0

f2=l/4

r2 = 1/2

f2«3/4

75

>2=l

Figure 3.6. with *i(l) = 1 ~ *i(0),t 2 (0) = 0,t 2 (l) = l,

(3.2.2)

we have v o r € Vi. Let us exhibit an example of such a v € V2 (see Figure 3.6). We consider the vi 6 Vi of Figure 3.5 where i>i(0) and i>i(l) were point curves on 7, and we rotate v\ via the parameter t2 so that at t2 = 1 we have the same picture as at t2 = 0, but with t\ interchanged with 1 — t\. Equation (3.2.2) then holds. We note that I x I becomes a Mobius strip, when we identify the parameter ti on the line t2 = 1 with the parameter 1 — t\ on the line £2=0. We define K2 := inf sup2 v€V2

L(v(t)).

tei

Then K2 >

tti,

and ^2 again is realized by some straight line l2 in A meeting 7 orthogonally at its endpoints. We consider two cases: (1) K2 > * i . Then L(l2) =

K2

>

K,\

=

L(/I),

and l2 hence is different from l\.

(2) K2 = Ki.

We claim that in this case, we even get infinitely many solutions of our problem, i.e. lines in A meeting 7 orthogonally. Namely, we let VQ € V2 be any critical family, i.e. satisfying tel2

76

Saddle point constructions (It is not hard to see that in the present case such a VQ € V2 indeed exists.)

We then have for any r : J -+ I2 with (3.2.2) sup L (vo (r (s)))
(3.2.3)

*i <supL{v0{T{s))),

(3.2.4)

and since K\ = K2, we have equality in (3.2.3) and (3.2.4). This means that VOOT is a critical family for «i, and it then has to contain a solution lr of our problem. Let S C {(s,t) € I x I\ L(vo(s,t)) = K2)} denote the set in J x J corresponding to all solutions induced by vo. After carrying out the identification prescribed by (3.2.2), which makes I x I into a Mobius strip, we see that the complement of S in this Mobius strip is not path connected. Namely, otherwise we could find r satisfying (3.2.2) for which T(I) avoids 5, and for such a r, VQ o r would then not contain a solution, as S is the set of all solutions in the family vo. This, however, contradicts what has just been said (see Figure 3.7). In fact, S has to carry a one dimensional cyclef on the Mobius strip. Otherwise, S would be contractible (in the Mobius strip) and one could reparameterize VQ on I2 so that the set of solutions corresponds to a finite number of points. But this is incompatible with K2 = «i as we have just seen. Since for each path r as in (3.2.2) with T(I) C S, VO O T G V\ is nonconstant by (3.2.1) and (3.2.2), we obtain an uncountable number of solutions. We thus have shown our result if 7 is of class C 3 . If 7 is only of class 1 C , we choose a sequence of curves 7 n of class C3 approximating 7. This means that there are parameterizations 7nW» l(T) by arc-length with lim sup ( |7 n (r) - 7 ( r ) | + ^ 7 n ( r ) - ^ 7 ( r ) •+00

0.

We then let l\,n and l2yn be the corresponding solutions for 7 n . After selection of subsequences, l\,n and l2yU then converge to solutions Zi, l2 for 7, and those l\ and l2 realize the critical values K\ and K2, respectively. Since the argument to produce infinitely many solutions in case K\ = K2 f We have to employ here some constructions from algebraic topology. A reference is any good book on that subject, e.g. M.Greenberg, Lectures on Algebraic Topology, Benjamin, Reading, Mass., 1967, pp. 33-45, 186. While this is somewhat technical we strongly urge the reader to try to understand the essential geometric idea of the preceding construction.

3.2 The construction of Lyusternik-Schnirelman

/

77

r(I)

h Figure 3.7.

did not depend on a higher differentiability assumption on 7, it is still applicable here, and we thus can complete the proof as before. q.e.d. The variational content of Theorem 3.2.1 is that we produce two geodesies in E 2 that meet a given convex Jordan curve orthogonally. In fact, this statement generalizes to any closed convex Jordan curve on some surface, enclosing a domain homeomorphic to the unit disk. In Sections 2.3, 3.2, we could only treat variational problems that could be reduced to finite dimensional problems, because we did not yet develop tools to show the existence of critical points of functionals defined on infinite dimensional spaces. We shall develop such tools in Part II, and consequently in Chaper 9 of Part II, we shall be able to present general results about the existence of unstable critical points in the spirit of the preceding results. The crucial notion will be the PalaisSmale condition that guarantees that the type of reasoning presented in Section 3.1 extends to certain functionals defined on infinite dimensional spaces. Also, the reasoning employed in Section 3.2 that infinitely many critical points can be found if two suitable critical values coincide will be given an axiomatic treatment in Section 9.3 of Part II.

Saddle point constructions

78

Exercises 3.1

X

Let F € C (M, R) (M an embedded, connected, differentiable submanifold of Rd) be bounded from below and proper (i.e. for all s € R, {x € M : F(#) < 5} is compact), and suppose F has two relative minima £i,#2- Let tt0 := max(F(xi), F(x2)).

3.2

3.3

Show that F either possesses a critical point £3 with F(#3) > KQ, or that it has uncountably many critical points. Let F € C ^ R ^ R ) be bounded from below and proper, and suppose it has three strict relative minima xi,X2,x 3 . Try to identify conditions under which F then has to possess more than two additional critical points, e.g. three or four. Let A be a compact convex subset of the unit sphere S2 C R 3 , and suppose dA is a smooth curve 7; the convexity condition here means that for any two points in A, one can find precisely one geodesic arc inside A that connects them. Show the existence of at least two geodesic arcs in A that meet 7 orthogonally at both endpoints.

4 The theory of Hamilton and Jacobi

4.1 The canonical equations We let t be a real parameter varying between t\ and ^ . We consider the variational integral 1=

f2 L

fax1®,...,xn(t),x\t),...

,xn{t)) dt

(4.1.1)

Jti

for the unknown functions x(t) = ( x 1 ( t ) , . . . , xn(t)) with fixed endpoints x(t\) and x(t2). Here, dt' We assume that L is of class C 2 . The Euler-Lagrange equations for i" are ±Lit-Lxi=0

(i = l,...,n).

(4.1.2)

We assume the invertibility condition det L±i±i ^0.

(4.1.3)

As shown in 1.2, this implies that solutions of (4.1.2) are of class C 2 . (4.1.3) also implies that we may perform a Legendre transformation. Namely, by the implicit function theorem, we may then locally solve Pi = L±i

(4.1.4)

w.r.t. x\ i.e. xi =xi(t,x,p)

(p= ( p i , . . . , p „ ) ) . 79

(4.1.5)

80

The theory of Hamilton and Jacobi

The expressions pi are called momenta. The Hamiltonian H is defined as H(t,x,p)

:=i*pt-£(*,s,£).

(4.1.6)

We obtain dx^ dx^ ^ 'd^~Lii'd^~Lxi'

Hxi Pj

and with (4.1.4) then Hxi =

~Lxi.

and with (4.1.2) and (4.1.4) then Hxi = -n.

(4.1.7)

Also dxj dpi

ti

dxj dpi

and thus again with (4.1.4) HPi=x\

(4.1.8)

(4.1.7) and (4.1.8) constitute a so-called canonical system. We are going to see that (4.1.7) and (4.1.8) also arise as Euler-Lagrange equations of the variational problem obtained by expressing L in (4.1.1) through H via (4.1.6). Namely, 1=

f2 (xjPj

-H(t,x,p))

dt,

(4.1.9)

Jti

where the unknown functions are x(t) and p(t), has Euler-Lagrange equations (4.1.7) and (4.1.8), and so does / = - f2(xjpj

+ H{t,x,p))dt.

(4.1.10)

Jti

Before proceeding, we observe that if H does not depend explicitely on t, i.e. H = H(x,p), then if is a constant of motion, i.e. constant along any solution x(t) of the equations, Namely, ~H (x(t),p(t)) by (4.1.7) and (4.1.8).

= Hxix* + HPipi = 0

(4.1.11)

4-2 The Hamilton-Jacobi equation

81

Example. For L = | | i | — V(x), we have H=\\p\2

+ V(x),

and the canonical equations become x =p P=-Vx. This example, which describes the Newtonian motion of a particle of unit mass subject to a potential F , is helpful for remembering the signs in the canonical equations.

4.2 The Hamilton-Jacobi equation Assumption. There is given a set fi C M n + 1 = { ( ^ x 1 , . . . , x n )} with the property that for any points A, B € fi, A = (a, K,1, . . . , ttn), B = (s,*? 1 ,.. . ,
(4.2.1)

and also Pi =gi(t;s,q1,...,

qn; a,*1,...

,Kn) = L±i.

(4.2.2)

In particular, Ki = fi(a;s,q\...,qn;a,K\...)Kn) qi =

(4.2.3)

fi(s;s,q\...,qn;a,K\...,Kn).

We also define
(4.2.4)

v< := gi(s; 5, g 1 , . . . , g n ; a, ft1,..., Kn) = L4i (s, q, q). In the sequel, fl etc. will mean a derivative w.r.t. the first independent variable, fl etc. a derivative w.r.t. the second one. Inserting (4.2.1), (4.2.2) into i", we obtain J = /(*,
(4.2.5)

82

The theory of Hamilton and Jacobi

and call this expression the geodesic distance betweeen A and B. In this connection, / is called eiconal. Recalling (4.1.9), we may write /=

/

( Pi i< -H(t,x,p))dt.

(4.2.6)

We want to compute the derivatives of I(s, q, a, K).

Is = Viq1 - H(s, q, v) + J' (g'J1 + gj1' - HXif

- HPigty dt.

Equations (4.1.7) and (4.1.8) yield Hx< = —&, HPi = /*, and thus la = Vi? ~ H(s, q,v)+

(dif)

'dt

.,\t=s

+ gifx

= fi
for t = s

f

for * = a,

= 0

and thus altogether / s = Viq% - H(s, q, v) - Vitf =

-H{s,q,v)

= L(s,q,q)-qiL^

(4.2.7)

Next ^

= /

7 0 f +9iTh dqi1 +9idqi

Ja

~ Hxi^j -H ,^\ "* dq* *pp' dqi

dt

r\ £
= 9i^-\

again by (4.1.7), (4.1.8)

°1j \t=
by (4.2.3) dK a nAd ' = n

d<

?

x

^ ° ' ^ J = ^-

Thus Iqj =Vj=

Lg,{s,q,q).

(4.2.8)

4-2 The Hamilton-Jacobi

equation

83

Analogously, Ia = H(a, K, ip) = -L(a, K, k) + A*L*«.

(4.2.9)

4 ; = -¥>j = - i « i ( ^ *,«).

(4.2.10)

Inserting (4.2.8) into(4.2.7), we obtain I8 + H{s,qjq)=0.

(4.2.11)

Thus, the geodesic distance as function of the endpoint satisfies (4.2.11), a Hamilton-Jacobi equation. In the present context that equation then is also called eiconal equation. We observed at the end of Section 4.1 that H is constant along solutions if it does not depend on t explicitly. In that case, (4.2.11) implies that / then depends linearly on 5. It may be useful for understanding the preceding formulae if we derive them without the use of the Legendre transformation. Thus

/ L{t,x{t),x{t))dt = J

L{t,fJ)dt

and

Is = L(s, q, q) + f (Lxif' + W * ' ) dt. The Euler-Lagrange equations give d_ dt

I>xx ~ 37-^irM

and so

= L(8,q,q) + J'±(L±
L±ifl \t=(T

As before, we obtain from (4.2.3) /*' = - / * /*' = 0

fort = 5 for t = a,

hence Is = L(s,q,q)~

Lqiq\

84

The theory of Hamilton and Jacobi

i.e. (4.2.7). Likewise,

i¥ = f

LTt

L±t dp U

TTT

+

LT' TT-;

dt

t= S

w

i.e. (4.2.8). Thus, the Hamilton-Jacobi equation (4.2.11) is J s - L ( 5 , ? , ( j ) + Jqi
(4.2.12)

We have seen in the preceding how solutions of the canonical equations yield solutions of the Hamilton-Jacobi equations. We now want to establish a converse result. Let
(4.2.13)

with Po =
Definition 4.2.1. / /

with GeC2

(4.2.14)

and det(GxiXj)ij=1

n^0

(4.2.15)

is a family of solutions of (4-2.13) depending on n parameters A i , . . . , An, we call if = G(t, x 1 , . . . , xn, A i , . . . , An) + A

(4.2.16)

(where A is a free real parameter) a complete integral of (4-2.13). We have the following theorem of Jacobi: Theorem 4.2.1. Let (p = G(t, x 1 , . . . , x n , A i , . . . , An) -f A be a complete integral of (4.2.13). Then one may obtain a family of solutions of the

4-2 The Hamilton-Jacobi equatiIon

85

canonical equations = xl

(4.2 .17)

Hxl = ~Pi

(4.2 .18)

Hpi

depending on 2n parameters A i , . . . , A n ,

/i , . . . , / x "

by solving

Gx< = S

(4.2 .19)

Gxi

(4.2 .20)

= Pi-

Proof. Because of (4.2.15), (4.2.19) may be solved w.r.t. #*, xl = z t ( £ , A i , . . . , A n , / / \ . . . , / / n ) . Inserting this into (4.2.20) then yields Pi = Pi(t, A i , . . . , An, [i , . . . , fin). We have to show that xl and pi satisfy the canonical equations. For this purpose, we differentiate (4.2.13) w.r.t. xl and obtain: GtXi + HPkGxkxi

+ Hxi = 0.

(4.2.21)

Differentiating (4.2.13) w.r.t. A*, we obtain GtXi+HPkGxkXi=0,

(4.2.22)

since the terms containing ^- cancel by (4.2.21). Differentiating (4.2.19) w.r.t. t, we obtain dxk GXit + GXixk—=0. (4.2.23) Comparing (4.2.22) and (4.2.23) and recalling (4.2.15) yields (4.2.17). Differentiating (4.2.20) w.r.t. t, we obtain ^ i = GxH + Gxixk^-.

(4.2.24)

Comparing (4.2.24) and (4.2.21) and using the relation (4.2.17) just derived, we then obtain (4.2.18). q.e.d. The canonical equations are a system of ODE whereas the HamiltonJacobi equation is a 1 s t order partial differential equation (PDE). The preceding considerations show the equivalence of these equations. While in general, one may consider a PDE as being more difficult than a system of ODE, in applications, one may often find a solution of the canonical

86

The theory of Hamilton and Jacobi

equations by solving the Hamilton-Jacobi equation. Here, it is typically of great help that the Hamilton-Jacobi equation does not depend on the unknown function itself, but only on its derivatives. Let us consider the following example of geometric optics: ip{t,x)y/\1 + x2dt

= /

{if (t, x) > 0),

already explained in Example (3) of Section 1.1 in a slightly different notation. The physical meaning is that x(t) is considered as the graph of a light ray travelling in a medium with light velocity 'ux\, where c is the velocity of light in vacuum. In this example, putting L(t, x, x) =
(4.2.25)

we have p = Li =

H = px-L

y/TTx1

= - vV~P2-

(4.2.26)

I(s, q, (7, K) here is the time that a light ray needs to travel from A = (a, K) to B = (5, q). The Hamilton-Jacobi equation Is -f H (5, g, Iq) = 0 becomes the eiconal equation I2s+I2q=V2-

(4-2.27)

The surfaces I(s, q) = constant are called wave fronts. Another simple example comes from a quadratic L(t, x, x) = -(x2 + ax2)

(a = constant).

(4.2.28)

Then p = L± = x,H

= px-L=

-(p2 - ax2),

(4.2.29)

and the Hamilton-Jacobi equation becomes Jt + i ( / x 2 - a r r 2 ) = 0 .

(4.2.30)

If we substitute J = p(t)x2, we are led to the Riccati equation p + 2p2-%=0.

(4.2.31)

4-3 Geodesies

87

If we substitute / = — Xt 4- ip(x) with a parameter A, we obtain from (4.2.30) -A + - (V»'(rr)2 - ocx2) = 0, i.e.


>/a£ 2 + 2Adf.

(4.2.32)

The equation I\(t,X,\) means

t+

- f-r

= [I

di 2

V ^ 4- 2A Jo \/a< This can be solved for x; let us assume for example a < 0; then the solution is / 2A x = \ sin ( \ / ^ a (t 4- /^)) . V —ex x of course solves the Euler-Lagrange equation for (4.2.28) x = ax. A physical realization is the harmonic oscillator, where x(t) is the displacement of an oscillating spring, with a = - ^ (m = mass, k = spring constant). Since p = Ix , It + H(x, Ix) = 0, we obtain from (4.2.32) A=

H{x,p),

i.e. A is the energy of the spring.

4.3 Geodesies We consider the case where L is homogeneous of degree 1, i.e. L = L&xi.

(4.3.1)

det L±iXJ = 0,

(4.3.2)

Then

88

The theory of Hamilton and Jacobi

and we cannot perform a Legendre transformation as in Section 4.1. We have H^-L

+ tfLv = 0,

(4.3.3)

and the computations of Section 4.2 yield (writing L& instead of pi etc.) J, = L ( * , g , g ) - g % « = 0

(4.3.4)

An example are the geodesic lines considered in Chapter 2. Here, L=y/Q with Q = gij(x\...,xn)xixj.

(4.3.5)

The Euler-Lagrange equations are

(43 6)

Z{7QQ")-JQQ*-°-

-

Since t does not occur explicitely in (4.3.5) and since / is invariant under transformations of t, we may choose t such that Q = 1,

(4.3.7)

i.e. that solutions are parameterized by arc-length. Equation (4.3.6) then becomes jtQ±* - Qx* = 0.

(4.3.8)

Conversely, along a solution of (4.3.8), we have Q = constant, justifying our choice of t. Namely, Q is homogeneous of degree 2 w.r.t. the variables £*, hence Qiii* = 2Q.

(4.3.9)

Differentiating (4.3.9) w.r.t. t along a solution, jtQ±i J i* 4- Q±iS? = 2jtQ

= 2Qxiii

and (4.3.8) indeed yields —-Q — 0 along a solution, at

-f 2Q^x*,

4-4 Fields of extremals

89

As already demonstrated in 2.1, (4.3.8) are the Euler-Lagrange equations for E=

Q(x(t)^W)dt=\J\ij(x(t))x\t)x'(t)dt

\j

(4.3.10)

We recall (Lemma 2.1.1) that the Schwarz inequality implies /

y/Qdt <(s-
J Qdt)

2

with equality precisely if Q = constant, and the extremals of E are precisely those extremals of i" parameterized proportionally to arc-length. In contrast to I, E is no longer invariant under transformations of t. Therefore, for solutions of the Euler-Lagrange equations corresponding to E, the parameterization is determined up to a constant factor. The Hamiltonian for E is H = Q±ix{ -Q

=Q

because of (4.3.9) .

(4.3.11)

Moreover, Pi = Qx*=29ijij-

(4.3.12)

Thus (with g* = (gij)-1).

H = -/hiPj

(4.3.13)

The Hamilton-Jacobi equation becomes Et + ^gijEx<EXJ

= 0 cf. (4.3.13), (4.2.11), (4.3.10)

(4.3.14)

and the canonical equations are x'^l^Pj

cf. (4.1.8), (4.3.13)

ldakj Pi = - 4 - ^ r P f c P i

cf

(4.3.15)

- t 4 - 1 - 7 )- ( 4 - 3 - n )> ( 4 - 3 - 5 )-

As observed at the end of Section 4.2, E depends linearly on t.

4.4 Fields of extremals Let ft C M tion

n+1

satisfy the assumptions of 4.2, T G ( ^ ( f y R ) . The equaT((T,^1,...,^n)=0

(4.4.1)

90

The theory of Hamilton and Jacobi

then defines a possibly degenerate hypersurface E (assume E ^ 0). Given B = (s, q1,..., qn) G fi, we seek A = (a, K1, . . . , Kn) G E that minimizes I(s,q\...,qn,a,K\...,Kn) as a function of (a, ) satisfying (4.4.1). At such a minimizing A, we have with some Lagrange multiplier A Ia + A i ; = 0 7^-fAT^=0

(4.4.2) (j = l , . . . , n ) .

Unless the situation is degenerate (A = 0 or Ta = TKi = 0 for all z), this means that the vector ( 1 ^ , i ^ i , . . . , i ^ ) is proportional to the gradient of T, hence orthogonal to E. From (4.2.9), (4.2.10), we then obtain -H(
(4.4.3)

These are equations for the tangent vector (K 1 , . . . , £ n ) of the solution from A to £ . A solution satisfying (4.4.3) is called orthogonal to E. We want to use the following: Assumption. Through each point of CI, there is precisely one solution orthogonal to E. For each B = ( s , ^ 1 , . . . ,
K

a

unique

(5, g))

the geodesic distance from the hypersurface E. Theorem 4.4.1. Given such a field of solutions orthogonal to E, the geodesic distance satisfies J9 = -H(s,q,L4)

(4.4.4)

Jqj =L#,

(4.4.5)

and

hence also the eiconal equation Js + H(sy g, Jq) = 0.

(4.4.6)

4.4 Fields of extremals

91

Js = Is + Iaas + IKiKis

(4.4.7)

Proof.

T(a(s, g), tt(s, #)) = 0 implies a a T„ + n\TKi = 0 and likewise

If we then use (4.4.2), we obtain in (4.4.7)

Jgi — -fgi >

and the result follows from (4.2.7), (4.2.8), (4.2.11). q.e.d. Conversely Theorem 4.4.2. If J(s,q) is a solution of (4*4-G) °f c^ass C2> there exists a field of solutions orthogonal to the hypersurf aces J(s, q) = constant, and J is the geodesic distance from the hypersurface J = 0. Proof Let J satisfy (4.4.6). We put JH:=Jq*(s,q).

(4.4.8)

The following system of ODE (f = # p , ( S , ^ , J g , )

(4.4.9)

then defines an n-parameter family of curves. By (4.4.8), we have along any such curve Pi = Jqia "J" JqiqiQ > and (4.4.6) gives JSqi "+" Hq* "+" HpjJq3qx

— 0.

Recalling (4.4.9), we obtain Pi = ~Hqi.

(4.4.10)

92

The theory of Hamilton and Jacobi

Equations (4.4.9) and (4.4.10) state that the curves q(s) constitute a field of solutions. (4.4.6) and (4.4.8) yield -H = Js

This means that (4.4.3) is satisfied for T = J with A = 1, and the solutions are orthogonal to the hypersurfaces J = constant. q.e.d. Theorem 4.4.1 gives solutions of the Hamilton-Jacobi equation (4.4.6) depending on an arbitrarily given function T £ C 1 (R n + 1 ) (namely, we obtain those solutions that start on T = 0), whereas Theorem 4.4.2 implies that all solutions are obtained in that way. The surfaces J = constant are called parallel surfaces of the field. In the special case where the hypersurface T — 0 degenerates into a point, we recover the considerations of Section 4.2.

4.5 Hilbert's invariant integral and Jacobi's theorem For a solution J(£, re 1 ,..., xn) of the Hamilton-Jacobi equation, we put again Pi '-— JxiV"ix

> • • • » # )•

If A = (a, K1, . . . , Kn) and B = (5, q1,..., qn) are connected by an arbitrary differentiable path x*(r), the integral J(B)-J(A) =

£jtJ(r,x(r))dr + JT )dr

does not depend on this particular path, but only on the end points A and B. We rewrite this integral as

fA

p~-H(T,x(T),p(r)))dT

(4.5.1)

and call it Hilbert's invariant integral Conversely now let functions Pi{T,xx,... , # n ) be given in a region O C E n + 1 for which the integral (4.5.1) does not depend on the path x{r) connecting A = (a, x (a)) and

4-5 Hilbert's invariant integral and Jacobi's theorem

93

B = (s,x (s)). Thus, we may define J : Q —> R by J(B) - J(A) = j ' (p~

- H ( r , x ( T ) , p ( r ) ) ) dr.

(4.5.2)

Since this integral does not depend on the path connecting A and JB, we must have Jx* = Pi Jt =

(4-5.3)

-H(t,x,p).

J then solves the Hamilton-Jacobi equation. By Theorem 4.4.2, any solution of the Hamilton-Jacobi equation is the geodesic distance function for a field of solutions of the canonical equations. Thus, any invariant integral of the form (4.5.1) yields a field of solutions. Let us now reconsider Jacobi's Theorem 4.2.1. Let ip = G(t,x\...,xn,\u...,\n)

+\

(4.5.4)

p + H{t, x\ . . . , x n , P l , . . . ,p n ) = 0

(4.5.5)

be a complete integral of

(with p =
m

particular det(GxtXj)?0.

(4.5.6)

Jacobi's theorem says that we obtain a 2n-parameter family of solutions of the canonical equations by solving

Gxi

=Pi,

where the parameters are A i , . . . , An, /i 1 ,. • • > / / \ For fixed values of A i , . . . , An, A, G determines a field of solutions of the canonical equations, and by the preceding consideration, it is given by the corresponding invariant integral G(B) - G(A) = j = £

S

(W ^ - H \ d r { L (r, x*(r), X\T)) + (^

(4.5.7) - i r ^ r ) ) L» j dr,

where xl(r) now denotes the derivative in the direction of the solution and not in the direction of the arbitrary curve x*(r) connecting A and B. We now vary A i , . . . , An, but keep the curve x*(r) fixed. Then the field

The theory of Hamilton and Jacobi

94

of solutions varies, and so then does xl{r). We also determine A so that G(A) = 0. Differentiating (4.5.7) then yields

G

i, L

dT

(458)

"-f((^- ) ^) '

'-

In the same way as G{B), this expression only depends on B {A is kept fixed for the moment) but not on the particular x J (r). For each 2?, we find Bo on the surface G(*,x\...,£n,Ai,...,An) = 0 that can be connected with B by a solution of the canonical equations. Along such a solution, we have dx>

.7

and the integrand in (4.5.8) thus vanishes along this curve. Instead of integrating from A to B, it therefore suffices to integrate from A to JBo, and we obtain GXi = IM\

(4.5.9)

1

l

with fi being the value of the integral from A to Bo. Thus, fi can be considered as a constant for the solution passing through Bo. If, conversely, (4.5.9) defined a family of curves xl(t, X3r, //•?) (the family is locally unique because of (4.5.6)), then, since G\3 is constant, the integrand in (4.5.8) has to vanish along any curve of the family. Thus dr

-it

)L±JXi=Q

(i = l , . . . , n ) .

(4.5.10)

In our field we have (cf. (4.2.8)) L±i — Gxj,

hence by assumption (4.5.6) det L±JXi

= detGXJXl

^ 0.

Equation (4.5.10) then implies

this means that the curves defined by (4.5.9) are solutions of the canonical equations contained in the field defined by G(t, rr 1 ,..., rrn, A i , . . . , A n ). We also observe that the parameter A is only used for specifying the surface G = 0 and has no geometric meaning beside that.

4-6 Canonical

transformations

95

4.6 Canonical transformations We want to find transformations, i.e. diffeornorphismsf

(x,p)

h-> (£,TT),

that preserve the canonical equations x — Hp P=-Hx.

(4.6.1)

This means that £ = £(#,p), n = 7r(x,p) satisfy

* = -#£.

(4.6.2)

with #*(t,£(x,p),7r(x,p)) = H(t,x,p). Equation (4.6.1) constitutes a system of ODE and if the assumptions of the Picard-Lindelof theorem are satisfied, a solution exists for given initial values x(t0) = x0, p(to) — po on some interval [t 0 ,ti]. For any i € [^o^i]5 we then obtain such a transformation by letting £(x,p) = x(f), 7r(x,p) = p(t) where (#(£),p(£)) is the solution of (4.6.1) with x(to) = x,p(to) = p. Thus, the evolution of (4.6.1) in time t, the so-called Hamiltonian flow, yields 'canonical transformations'. However, the concept of canonical transformations is more general as we now shall see. Since

dx1

dpi

_ d&

*> = M

a& x +

e£p<

f A diffeomorphism is a bijective map that together with its inverse is everywhere differentiable.

96

The theory of Hamilton

and Jacobi

and dx%

dpi l

dlT4

dpi

Hx .8X\H x d&

jsa

P

we obtain the conditions dpj _ dnj

dx1 _ die* ~~ dpi dx1

d& dx1'

d& dpi' dnj dnj

(4.6.3)

or in matrix notation

rSi

SL-i

\ dx dn

dp dn

lite

aFJ

-l

\(%YT .-(S)

-(g)

(4.6.4)

(H)

where ^4T denotes the transpose of a matrix A. Obviously, this is a condition that does not depend anymore on the particular Hamiltonian H. Definition 4.6.1. A diffeomorphism ip : R 2 n - • R 2 n , (x,p) *-> fan), satisfying (^.£.#) (or,equivalently (4-6-4)) i>s called canonical transformation. Canonical transformations can often be used to simplify the canonical equations. Before we return to that topic, however, we interrupt the discussion of the Hamilton-Jacobi theory in order to describe some basic points of symplectic geometry (for more information on that subject, we refer to D.Mc Duff, D.Salamon, Introduction to Symplectic Topology, Oxford University Press, Oxford, 1995). We denote the (n x n) unit matrix by I n and put J:=

0 Jn

-V 0

Then obviously J 2 = - I2n-

(4.6.5)

4-6 Canonical transformations

97

Equation (4.6.4) may then be written as (Dil>)-1 = -J(DiP)TJ,

(4.6.6)

or equivalently (Di/jfjDi/j

= J.

(4.6.7)

In this connection, a ty satisfying (4.6.7), i.e. a canonical transformation, is also called symplectomorphism. Prom these relations, one also easily sees that ^ is a canonical transformation iff ip~l is. In terms of J, the canonical equations (4.6.1) can also be written as z = -JV°H(t,z)

(4.6.8)

where z = (x,p), V°H(t,z) = (HX,HP). For a reader who knows the calculus of exterior differential forms, the following explanation should be useful. We consider the two-form uj = dxi A dpi

on E 2 n

(here, as always, we use a summation convention: dxl A dpi means S r = i dxlAdpi). According to the transformation rules for exterior differential forms (i.e. d^ — ^*jdxl etc.), we have, for £ = £(#,p), n = 7r(x,p), d£J A dnjJ =

^

T^TTT-^ - TT^-Tr-^

\dx*dpk

dpkdx')

dx% A ofe. yk

Thus, a; remains invariant under the transformation ?/>, i.e. dtf ArfTr^= dxl A dp<

(4.6.9)

precisely if ip is a canonical transformation. In fact, this is often used as the definition of a canonical transformation. If UJ is left invariant under i/>, so is ujn := UJA---A<JJ = nli-l)****^

dx1 A- • -AdxnAdpi A- • -Adpn. (4.6.10)

n times

Since G^1 A• • • Ad£ n Ad7Ti A• • • Ad7rn = (det Dip)dxl A • • • Acten Adpi A • • • Adp n , we conclude Liouville's:

98

The theory of Hamilton and Jacobi

Theorem 4.6.1. Every canonical transformation i/>: R 2 n —> R 2 n satisfies det£>V> = l.

(4.6.11) q.e.d.

One also expresses this result by saying that a canonical transformation is volume preserving in phase space as dx1 A • • • A dxn A dpi A • • • A dpn can be interpreted as the volume form of R 2 n . By what was observed in the beginning of this section, this applies in particular to the Hamiltonian flow which constitutes Liouville's original statement. After this excursion and interruption, we return to our canonical equations (4.6.1) and try to simplify them by suitable canonical transformations. Canonical transformations may be easily obtained from the variational integral

/ = /

L(t,x,x)dt

Jti

with L(t,x,x)

= x-p-

H(t,x,p)

(p = L±).

If W is any differentiate function, then

has the same critical points as / , because

r = i + w(t2)-w(t!), so that I* and J differ only by a constant independent of the particular path x(t). Thus, we may for example take any function W(t,x,£) and require that for all choices of x, £, x, £ dW x-pH(t,x,p) = £ • n - #*(*,£, TT) + — . (4.6.12) Then, with

/*= f'A(t,s,£)dt, differs from J only by a constant. Thus, if x(t) is a critical path for J,

4-6 Canonical transformations

99

£ (#(£), p(£)) then becomes a critical path for /*. Since dW -— = Wt + Wx.x

+

WrS,

(4.6.12) becomes * • (P ~ Wx) - £ • (n + Wz) - H + H* - Wi = 0

(4.6.13)

Since (4.6.13) is required to hold for all choices of x, £, x, £, we obtain: Theorem 4.6.2. Given an arbitrary (differentiable) function W(t, x, £), a canonical transformation (transforming (4-0.1) into (4-6.2)) is obtained through the equations V^WX 7T = -Wf:

(4.6.14)

H* = # , i.e.

Wt=0.

Wt = 0 of course means that W = W(x, £). In the same manner, we may also take a function W(t,p, £), W(t, x, TT) or W(t,p,7r). In the first case, we obtain for example the equations x = Wp H* =H , i.e. Wt = 0. Here and above, of course H* — if*(£,£,7r). We may now easily explain Jacobi's method for solving the canonical equations. We try to find W(x,£) satisfying H(t,x,Wx(x,0)

= H*(0,

(4.6.15)

i.e. reduce the Hamiltonian to a function of the variable £ alone. We have to require that detWxiV

T^O.

(4.6.16)

This ensures that the equation n = —W^ determines x, and p then is determined from p = Wx. If (4.6.15) holds, (4.6.2) becomes

7r = - # f

(4.6.17)

This implies that ^ 1 , . . . , ^ n are constants of motion (i.e. independent of t), or so-called integrals of the Hamiltonian flow. A system for which

100

The theory of Hamilton and Jacobi

n independent integrals can be found is called completely integrable. Thus, if we can find a so-called generating function W(x, £) of the above type reducing the Hamiltonian to a function of £ alone, the canonical system is completely integrable. Clearly, since in this case £*,..., £ n are constant in £, the relation n — —H^(^) can then be used to determine 7Ti,..., 7rn. In other words, a completely integrable canonical system may be solved explicitly through quadratures. Actually, one may show in this case that the sets Tc = {£* = c 1 , . . . , £ n = cn} for a constant vector c = ( c 1 , . . . , c n ) are n-dimensional tori, if compact and connected. Thus, the so-called phase space {(x,p) G R2n} is foliated by tori that are invariant under the motion, and on each such torus, the motion is given by straight lines. It should be pointed out, however, that completely integrable dynamical systems are quite rare, in the sense that the complete integrability usually depends on particular symmetries, and their dynamical behaviour is quite exceptional in the class of all Hamiltonian systems. The invariant tori may disappear under arbitrarily small perturbations. By way of contrast, the Kolmogorov-Arnold-Moser theory asserts that these invariant tori persist under sufficiently small and smooth perturbations if the coordinates of HZ are rationally independent and satisfy certain Diophantine inequalities, and if the matrix Hit of second derivatives is invertible. In the older literature, the notion of 'canonical transformation' is usually applied to any transformation ip : R2n —> R2n that preserves the form of the canonical equations, i.e. (4.6.1) is transformed into (4.6.2), but without requiring that

H'(t,Z,ir)=H(t,x,p). An example of a canonical transformation in this wider sense is £ = 2X , 7T = p with

H* = 2H.

If we now take a generating function W (t, x, £) as above, the Hamiltonian is transformed into H* =:H + Wt

(4.6.18)

while the first two relations of (4.6.14), i.e. p = Wx ,7r = - W e

(4.6.19)

4-6 Canonical transformations

101

still hold. This may be used to explain Jacobi's theorem once more, as we now shall see. Let I(t , x , . . . , x n , A i , . . . , An) be a solution of the Hamilton—Jacobi equation It + H(t,x,Ix)

= 0,

(4.6.20)

depending on parameters A i , . . . , An and satisfying as usually det/ x i A i ^ 0 .

(4.6.21)

We now choose W(t,x1,...,xn,€i,...,4n) = /(t,a:1,...,xn^i,...,4n). The corresponding transformation then is

7T = - J e

(4.6.22)

j r ( t , £ , 7 r ) = ff(t,x,p) + I t . Because of (4.6.20), JTsO. Thus, the new canonical equations are just £= 0 7T = 0 .

Solutions are of course £ = A = constant 7T = —I\ = —fi = constant. We have thus obtained the statement of Jacobi's Theorem 4.2.1, namely that from a solution of (4.6.20) with (4.6.21), we may obtain solutions of the canonical equations by solving h =V IX=P with parameters A = ( A i , . . . , A n ), \i = (/i 1 ,..., \in).

102

The theory of Hamilton and Jacobi

Classical references for this chapter include: C.G.J. Jacobi, Vorlesungen iiber Analytische Mechanik (ed. H. Pulte), Vieweg, Braunschweig, Wiesbaden 1996, C. Caratheodory, Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Teubner, Leipzig 1935, R. Courant, D. Hilbert, Methoden der Mathematischen Physik II, Springer, Berlin, 2nd edition, 1968. The global aspects are developed in V.I. Arnold, Mathematical Methods of Classical Mechanics, GTM60, Springer, New York, 1978. A recent advanced monograph is H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhauser, Basel, 1994. That text will give readers a good perspective on the present research directions in the field.

Exercises 4.1

4.2

Discuss the relation between the canonical equations for the energy functional E and the equations for geodesies derived in Chapter 2. (Kepler problem) Consider the Lagrangian L{x,x) = I \x\2 + A 2 |x|

4.3

for x € M3.

Compute the corresponding Hamiltonian and write down the canonical equations. Show that the three components of the angular momentum x Ax are integrals of the Hamiltonian flow. For smooth functions F, G : R 2n —• E, define their Poisson bracket as

where z = (x,p) = (xl,..., xn,p\,... nates of E 2 n . Let z(t) = (x(t),p(t)) system

,pn) are Euclidean coordibe a solution of a canonical

x = Hp P=

-Hx

Exercises

103

for some Hamiltonian H(x,p) that is independent of t. Show that for any (smooth) F : E 2 n - • E

±F{z{t)) = {F,H}. Show that the Poisson bracket is antisymmetric, i.e. {F,G} =

-{G,F}

and satisfies the Jacobi identity {{F, G}, L} + {{G, L}, F} + {{L, F}, G) = 0 4.4

for all smooth F, G, L. Show that a diffeomorphism i/> : E 2 n ~> E 2 n is a canonical transformation if

for all smooth F, G.

5 Dynamic optimization

Optimal control theory is concerned with time dependent processes that can be influenced or controlled via the tuning of certain parameters. The aim is to choose these parameters in such a manner that a desired result is achieved and the cost resulting from the intermediate states of the process and from the application or change of the parameters is minimized. In some problems, the control parameter can be applied only at discrete time steps, while other problems can be continuously controlled. As we shall see, however, the discrete and the continuous case can be treated by the same principles. Since the end result may be prescribed, and the value of a parameter at some given time influences the state of the system at subsequent times and therefore typically will also contribute through this influence to the cost of the process at those later times, the determination of the optimal control parameters is best performed in a backward manner. This means in the discrete case that one first selects the best value of the control parameter at the last stage, whatever state the system is in at that time, then the value at the second-to-last stage, so that at this step the contribution of the value of the control parameter at the last stage to the total cost function is already determined and one only needs to optimize the cost function w.r.t. the second-to-last parameter value, and so on.

5.1 Discrete control problems We consider a process with n states # i , . . . , xn £ Md. At each state #i, we may choose a control parameter KeAi, 104

(5.1.1)

5.1 Discrete control problems

105

where A* is a given control restriction (A* C Rc) to determine Xi+i =(pi(xi,\i)

(5.1.2)

with cost

The total cost of the process starting at the initial state xv is n

j M x „ , A „ , . . . , A n ) := ^ f c t ( x i , A i ) ,

with

xi+i = y><(x<,A<). (5.1.3)

We wish to minimize the total cost of the process and define the Bellman function J„(x„) :=

inf

Ku(xu,\u,...,\n)

(i/ = 1 , . . . ,n).

(5.1.4)

A i €A i

t = f,.. .,n

T h e o r e m 5.1.1. The Bellman function satisfies the Bellman equation J„(x„) =

inf (fc„ (x„, A„) + J„+i (yv (x„, A„)))

/or i/ = 1 , . . . , n

A„€A„

(5.1.5) (ftene, we pttf 7 n + i = 0). Furthermore, (A„,...,A n ) G A^ x • • • x A n , (x^,... ,x n ) mf/i (5.1.2) are solutions of (5.1.4) iff Ij(xj) = kj(xj, Xj) + Ij+xixj+i)

forj = v,..., n.

(5.1.6)

Proof. Since Kv\xv\

A „ , . . . , An) = ku(Xi,, \v) -h Kv+i {(PvyXi") Ai/); Aj,_|_i,..., A n )),

we get J„(x„) =

inf

jFf„(x„;A„,...,A n )

t = t/,. , . , n

inf

=

inf

inf

M^><M +

A„€A„ \ \

=

Kv(xv\ \v,...,

inf x

j£Aj

An)

jFf„+i (y>„(x„, A„); A„+i,..., An) I

j = v + l,...,n

/

inf (fc„(x„, A„) + I„+i (
which is (5.1.5). For (A„,..., An) G A^ x • • • x A n , Xj+i =


for

106

Dynamic

optimization

j = i/,..., n,

< kvyXy, Xu) + • • • -h rC n (x n , A n j = Ky\Xyi

An, . . . , A n j .

If the infimum w.r.t. Xj G Aj (j = z/,..., n) is realized, we must have equality, and (5.1.6) follows. q.e.d. Corollary 5.1.1. (Ai,... ,A„) G Ai x - - - x A n , ( x i , . . . , x „ ) with (5.1.2) is a solution of (5.1.4), iff for all v = I,... ,n, (A„,..., An) G A„ x • • • x A n , ( a v , . . . , x n ) with (5.1.2) is a solution of (5.1.4)Corollary 5.1.2. (Bellman's method) An optimal solution of the process can be calculated as follows: For any value of xn, compute Xn(xn) minimizing (5.1.5) for v = n. Having computed XJ(XJ) for j = i / + l , . . . , n , compute Xu(xu) for any value of xv as to minimize (5.1.5) and put x„+i = (pu(xu1Xu(xu)). For an arbitrary initial value x\, an optimal process thus is given by: Ai := Ai(xi) , x2 := Vi(xi,Ai) , A2 = A 2 (£ 2 ),... •

5.2 Continuous control problems We want to minimize K(ti,x(h))

for a path x : [t0,h] -+ Rd

under the following conditions: We have the initial condition x(t0) = x0 and the final condition x(h) G Bx with a given set B\ eRd.

We have the control equation

x(t) = f(t, x(t), X(t))

for almost all t G (t0l h)

for a piecewise continuous control function X(t) satisfying X(t) G A

5.2 Continuous control problems

107

for some given A C E c . Pairs (\(t),x(t)) satisfying all these restrictions are called admissible, and the set of admissible pairs is called P(£o,#o). We put I(t0,x0):=

inf

K(tux(ti))

(A(t),x(t))€P(to,x 0 )

(Bellman function). Lemma 5.2.1. (i) I(t\,xi) = K(ti,xi) for all x\ G B\ (ii) For any path (A(t), x(t)) G P(to, XQ), I(t, x(t)) is a monotonically increasing function oft£ [to,t\]. Proof (i) is obvious. For (ii), if to < T\ < r2 < ti, the set of all admissible paths from (T2,X(T2)) to (t\,B\) can be considered as a subset of those ones from ( T I , X ( T I ) ) to (t\,x(ti)). Namely, if we have any path from (T2,£(T2)) to (ti,x\) for some x\ G fii, we may compose it with x(t)\. r j to obtain a path from ( T I , X ( T I ) ) to (ti,#i). Thus, every endpoint in B\ that can be reached from (T2,X(T 2 )) by an admissible path can also be reached from ( T I , X ( T I ) ) by an admissible path. This implies monotonicity. q.e.d. Theorem 5.2.1. (A(t),ir(t)) is a solution of the problem, if I(t,x(t)) is constant in t. Moreover, if there exist a function J(t,x) that satisfies J ( t i , # i ) = K(ti,xi) for all x\ G B\ and is monotonically increasing along any admissible path, and an admissible path (\(t),x(t)), along which J is constant, then that path is a solution of the problem. Proof. For a solution, /(to, *o) = K{tux{h))

= J(ti,x(ti)

(x 0 = x(t0)),

(5.2.1)

I(t,x(t)) then is constant by Lemma 5.2.1 (ii). If 7(t,x(t)) is constant, then (5.2.1) holds, and by Lemma 5.2.1, we have a solution. Given J as described, by the monotonicity of J , for any admissible path J{to,x0) < K(ti,x(t\)) and for the path (A(£),x(t)), J(t0,x0) = J(ti,5(ti)) = K(tux(h)), and optimality follows. q.e.d.

108

Dynamic

optimization

Lemma 5.2.1 implies that for those t for which I(t,x(t)) tiable ((\(t),x(t)) G P(t0,x0)) It(t, x(t)) + /*(*, x(t))f(t,

is differen-

x(t), X(t)) > 0.

For an optimal (A(t),x(t)), we have by Theorem 5.2.1 then hit, x{t)) + h{t, xit))fit,

xit), A(t)) = 0.

Corollary 5.2.1. iBellman equation) Let t G [to,£i]> £ £ ^ d - Assume that for every X G A, t/iere exists on admissible pair (A(t),x(t)) mt/i A(r) = A, ar(r) = 6 TTien ini:(/t(r,0 + / « ( r , 0 / ( r , € , A ) ) = 0 . A6A

Proof. This follows from the proof of Lemma 5.2.1. Namely, the assumption implies that we may select A such that the path is optimal at the point (r, £) under consideration. q.e.d. Example. We want to minimize the integral

f1 {u2it)+X2it))dt Jt0 with the initial condition uit0) = u0 and the control equation u(t) = auit) + p\it)

with given a, p G R.

(5.2.2)

In order to express this problem as a control problem, we introduce a new dependent variable v{i) as solution of the equation v(t) = u2it) + A2(t) , vit0) = 0.

(5.2.3)

We then want to minimize

Given /9 : [to, *i] —>• R with p(*i) = 0 and satisfying the Riccati equation p(t) = - 2 o p ( t ) + / 3 a p a ( t ) - l ,

(5.2.4)

5.3 The Pontryagin maximum principle

109

we put J{t,u,v)

p(t)u2(t)+v(t).

=

Then J(h,u(t1),v(t1))=v(t1) and from (5.2.2), (5.2.3), (5.2.4) ^-J(t,u(t),v(t)) at

= (32p2u2 + 2p(3u\ + A2 = (/Jpu + A)2 > 0,

and this expression vanishes precisely if \(t) = -0p(t)u(t). By Theorem 5.2.1, x(t) = (u(t),v(t)) optimal solution.

(5.2.5)

and \(t) = ~/3p(t)u(t) yield an

If we substitute X(t) through the control equation (5.2.2) in the variational integral, we obtain the integral

£ (w^2+(i+$)u{t)2 ~ 2-^u{t)ii{t))dt' which is essentially the same as the one considered at the end of 4.2 with integrand given by (4.2.28). We recall that the latter one had also been reduced to a Riccati equation. Equation (5.2.5) expresses the control parameter as a function of the state of the system. We just have a feedback control: knowing the state at a given time determines the control needed to reach an optimal state at the next time.

5.3 The Pontryagin maximum principle We consider the control problem / F(£,x(£), X(t))dt —• min Jt0 (5.3.1) with the control conditions x(t0) = x 0 x(t) =

f(t,x(t),X(t))

(5.3.2)

Dynamic

110

optimization

with controls X(t) G A C R c and the end condition (5.3.3)

0(
Here, X(t) is required to be piecewise continuous, and x(t) to be continuous. (Equation (5.3.2) then has to be interpreted as an integral equation x(t) = x0 + Jt / ( r , # ( T ) , A(r))dr.) F, / , and g are required to be of class C 1 . Also, to is fixed, whereas t\ > to is variable subject to the restriction (5.3.3). We define the Pontryagin function H(x,\,p,t,no)

:=P-/(*,S,A)

-nQF(t,x,\).

We now state the Pontryagin maximum principle Theorem 5.3.1. If (x(t),\(t)) is a solution of the control problem, there exist A£o > 0, a = ( a i , . . . , a^) € Md (a ^ 0 if HQ = 0) and a continuous p = ( p i , . . . ,Pd) on [£o, h] such that at all points where X(t) is continuous, we have H(x(t), \(t),p(t)91,

fio) = max H(x(t)9 A,p(*), *, W>)

(5.3.4)

p = -Hx ,x = Hp

(5.3.5)

and

and at the end point t\, we have the transversality condition daj p(t1) = ~-±-(t1,x(t1))-aj.

(5.3.6)

There also exists a continuous function rj : [to,t\] —> K s^ch that at all points where X(t) is continuous v(t)=H(x(t),\(t),p(t),t,iio)

(5.3.7)

and f){t) = Ht V(ti)^^(tux(ti))^ Also, one may always achieve fiQ = 0 or I.

(5.3.8) (5.3.9)

5.3 The Pontryagin maximum principle

111

Remarks: (1) The equation x = Hp is just the control equation

x=

f(t,x(t),\(t)).

(2) If A = Mc, then (5.3.4) becomes «A(x(t),A(t),p(t),t,/io)=0. (3) If we want to guarantee a fixed end time i\, we simply introduce an additional variable xd+1 = t with control conditions xd+1 = 1 xd+l{to)

= t0

xd+1{h)

= h.

and end condition

We now want to exhibit the Hamilton-Jacobi theory as a special case of optimal control theory. Concretely, we want to derive the EulerLagrange equations which are equivalent to the canonical equations of Chapter 4 from the Pontryagin maximum principle. We thus consider the variational problem L(t, #(£), x(t))dt —• min J to

with x(t0) = #o, #(£i) = #i, x : [^o>^i] ~• R d a n d where x(t) is required to have piece wise continuous first derivatives. We introduce the control variable through the control equation

\{t) = x(t) with A = Md, i.e. no constraint imposed. We have g(ti,x(t\)) x(t\). The Pontryagin function of this problem is

= x\ —

H(x, A,p, t, fi0) = p • A - noL(t, x, A). By Theorem 5.3.1 there exists fi0 = 0 or 1, a G Rd (a ^ 0

for fi0 = 0)

112

Dynamic

optimization

and p e C°([t0, *i], Rd) with V=~HX p(ti) = a H{t,x(t),\(t),p(t),ii0)

= maxW(*,x(t),A,p(t),/x 0 )

andr/€C°([t 0 ,ti],]R) with t7(t)=W(t,a:(t),A(t),p(t), W ,) *) = Wt > t?(*i)=0. We now want to exclude that fio = 0. In that case, we would have f] = Ht = 0

, hence

77 = 0

since

77(^1) = 0

p= a

since

p(ti) = a.

and p = — Hx = 0

, hence

Thus W = a • A, and since Ht = 0, W(x(t), A(t),p(t), t, 0) = 0, and thus a = 0, contradicting the statement of the theorem that a ^ 0 in case /L^O = 0. We may thus assume fio = 1. The Pontryagin maximum principle then gives the Weierstrafi condition L(t, x(t), A) - L(t, x(t), i(t)) > p • (A - £(*))

for all XeRd

(5.3.10)

and «A(t,x(t),A(t),p(t),l)=0

(5.3.11)

and the Legendre condition £±z (£,#(£),#(£))

is positive semidefinite.

Equation (5.3.11) implies P = L±, and together with P =

—

ria; =

LOJ»

(5.3.12)

5.3 The Pontryagin maximum principle

113

we obtain the Euler-Lagrange equations ~L± = Lx. (5.3.13) at A basic reference for the variational aspects of optimization and control theory where also a detailed proof of the Pontryagin maximum principle together with many applications is given is E. Zeidler, Nonlinear Functional Analysis and its Applications, III, Springer, New York, 1984, pp. 93-6, 422-40.

Part two Multiple integrals in the calculus of variations

1 Lebesgue measure and integration theory

1.1 The Lebesgue measure and the Lebesgue integral In this section, we recall the basic notions and results about the Lebesgue measure and the Lebesgue integral that will be used in the sequel. Most proofs are omitted as they can be readily found in standard textbooks, e.g. J. Jost, Postmodern Analysis, Springer, Berlin, 1998, pp. 151-97 and 209-15. Definition 1.1.1. A collection S of subsets ofRd (on Rd) if

is called a a-algebra

(i) E d 6 E (ii) IfAeT,, then also Rd \ A e E (iii) / / An e E, n = 1,2,3..., then also U^Li An e E. The Borel a-algebra is the smallest a-algebra containing all open subsets of Rd. The elements of the Borel o-algebra are called Borel sets.

Easy consequences of (i)-(iii) are (iv) 0 e E (v) If An e E, n = 1,2,3..., then also (X°=i € E. (vi) UA,Be

E, then also A - B := A \ (A n B) e E.

Definition 1.1.2. Let E be a o-algebra. A measure \i on E is a countably additive function \i : E -+ E + fl {oc}. 117

118

Lebesgue measure and integration theory

'Countably additive' here means that oo

(U

\

An

n=l

/

oo

I = 5Z ^ A «) 7n=l

/or any collection of mutually disjoint (AmCiAn = 0 /or ra ^ n) elements of E. A measure defined on the Borel a-algebra is called a Borel measure. A Borel measure [i is called a Radon measure if n(K) < oo for every compact K C Rd and fi(B) — sup{fi(K) \ K C B, K compact} for every Borel set B. A measure / i o n E enjoys the following properties: (vii) /x(0) = 0 (viii) If A, B G E, A c B, then fi(A) < fi(B) (ix) If An e E, n = 1,2,3,... and An C An+i for all n, then M | I J A n J = lim \n=l

fi(An).

/

Theorem 1.1.1. There exist a {unique) a-algebra E on Rd and a (unique) measure \i in E satisfying (x) ;4ra/ open subset ofRd is contained in E (i.e. E contains the Borel a-algebra) (xi) For Q : = { x = ( x 1 , . . . ,xd) G Rd | a, < xj < bj , j = 1 , . . . ,d} , /or numbers a\,...,

a^, 6 i , . . . , bd, we have d

i=i

(xii) (translation invariance) For x G Rd, A G E we /love x + A := {x + 2/1 y G A} G E and fi(x + A) = n(A) (xiii) If A C B, B £ T,, l*(B) — 0, f/ien A G E (and, consequently, H(A) = 0). This [i is called Lebesgue measure, and the elements of E are called (Lebesgue) measurable. In later chapters, we shall however write meas in place of fi for Lebesgue measure.

1.1 The Lebesgue measure and the Lebesgue integral

119

One should note that the a-algebra of (Lebesgue) measurable sets is larger than the Borel a-algebra. We say that a property holds almost everywhere in A C E d if it holds on A \ B for some B C A with n(B) = 0. We say that two functions f,g : A —> EU {±00} are equivalent if f(x) = g(x) for almost all x E A. A set contained in a set of measure 0 is called a null set. We usually write measA instead of n(A) for a measurable set A. Definition 1.1.3. Let A C Rd be measurable. A function f :

A-+RU{±oc}

is called measurable if {xeA\f(x)<\} is measurable for every A £ E. If/ n , n e N, are measurable, c e E, then / i + / 2 , c/i, /1/2, max(/i, / 2 ) , min(/i, / 2 ) , limsup n _> 00 / n , liminfn_>oo / n are likewise measurable. Any continuous function / is measurable, because in that case {f(x) < A} is open in its domain of definition. We have the following important composition property: Theorem 1.1.2. Let g : A —• E c be measurable (i.e. g = (g1,... ,gc), and each component gi is measurable), y : E c —• E continuous. Then y o g is measurable. The characteristic function \A oi A C Rd is defined as if xe A otherwise. Thus, A is measurable if and only if its characteristic function \A is measurable. More generally, s : A —• E is called a simple function or a step function if it assumes only finitely many values, say s(A) = { A i , . . . , AjJ, and if all the sets {s(x) = A*} are measurable. Thus

XA(*):={J0

k

S=

22*iX{aix)=\i}i=l

Theorem 1.1.3. / : A —• E is measurable if and only if it is the pointwise limit of a sequence of simple functions. If f : A —• E is measurable and bounded, then it is the uniform limit of a sequence of simple functions.

120

Lebesgue measure and integration theory

Definition 1.1.4. (1) Let A cRd

be measurable with [i(A) < oo, k

a simple function on A. The Lebesgue integral of s is

L

k

s(x)dx := ] P A^({s(x) = A*}).

(2) Let A be as in (1), f : A —» R measurable and bounded. Let sn : A —» E 6e a sequence of simple functions converging uniformly to f according to Theorem 1.1.3. The Lebesgue integral of f then is I f(x)dx JA

:= n lim / sn(x)da ~*°° J A

(this integral is independent of the choice of the sequence (sn)n€N)(3) A as in (i), / : i - ^ R U {±00} measurable. Put

{

m n

if }{x) < m if f(x) > n

f(x) We say that f is integrable if lim

ifm
fmin(x)da

exists. That limit then is called the Lebesgue integral fA

f(x)dx

off(4) A c K d measurable, f : A —• R U {±00} measurable, f is called integrable if for any increasing sequence A\ C A
/ f{x)XAn{x)dx

n-^ooJA

exists. That limit then is independent of the choice of (An) and called the Lebesgue integral fA f(x)dx of f. Theorem 1.1.4. The Lebesgue integral is a linear nonnegative functional on Cl(A), the vector space of Lebesgue integrable functions on a measurable set A, and it satisfies:

1.1 The Lebesgue measure and the Lebesgue integral

121

(1) If f € CX(A), and if f = g almost everywhere on A, i.e. li{xeA\f(x)^g(x)}=0, then g € Cl(A), and I f(x)dx=

/

g(x)dx.

In particular, f f(x)dx = Oifti(A) = 0. (2) Iff£

Cl(A), then \f\ € C\A), and \f \JA

f(x)dx\< I

[ \f(x)\dx. JA

(3) If f e CX(A), h: A-+ RU {±00} measurable with \h\ < / , then h€Cl{A) and I h(x)dx\ < I f(x)dx. I JA

\JA

(4) If ix(A) < 00, / : A —• K measurable with m < f < M, then f£C\A), and mfji(A) < J f(x)dx

< Mfi{A).

(5) / / (-An)neN is a sequence of mutually disjoint (Am O An = 0 for m^n) measurable sets, A := U^=i An, f £ £l{An) for every n, and if 00

V; / then f eCl(A),

.

\f(x)\dx<<x>,

and [ f(x)dx

Conversely, if f £ Cl(A), sequence {An)neN.

= T

I

f(x)dx.

then this equation holds for any such

122

Lebesgue measure and integration theory

(6) / / / G CX(A), then for every e > 0, there exists (f G Co(Md) := {g : R d -> R continuous\ {x \ g(x) ^ 0} bounded} with

/ 1/0*0 ~" ^0*01 ^ < 6Theorem 1.1.5 (Fubini). Let A C Rc, B C Rd be measurable, and write x = (£,77) G Ax B. If f : A x JE? —• E U {±oc} zs integrable, then f

f(x)dx=

f ( [

JAXB

m,v)dv)dt

J A \JB

/

{Here, for example fB /(£, n)dn exists for almost all £ G A.) For f e Cl(A), we put

We then have Jensen's inequality: Theorem 1.1.6. Let A C R d be bounded and measurable, f a convex function. Then for all ip G Cl(A)

'

(

/

/

)

*

/

/

•

*

•

1.2 Convergence t h e o r e m s In this section, again no proofs are given, and the reader is referred to J. Jost, loc. cit., pp. 199-208. Theorem 1.2.1 ( B . Levi). Let A C R d be measurable, and let fn : A —• R U {±00} be a monotonically increasing sequence (i.e. fn(%) ^ / n + i ( x ) for all x G A, n € N) of integrable functions. If lim / fn(x)dx

< 00,

n-^ooJA

then f := l i m n - ^ fn (pointwise limit) is integrable, and / f(x)dx = n lim / JA ^°° J A

fn(x)dx.

1.2 Convergence theorems

123

Corollary 1.2.1. Let A C Rd be measurable, fn : A -* R+ U {±00} (nonnegative and) integrable. If 00

r

v)dx £ / U(X)

< oc

then YlnLi fn is integrable, and

/ y]fn(x)dx==y] / fn(x)dxTheorem 1.2.2 (Fatou). Let A C Rd be measurable, fn : A - • l U {ztoc} integrable for n G N. Assume £/ia£ £/iere exists some integrable F : A-+RU{±oo} with fn>F

L

fn(x)dx

for all n G N,

< K < oc /or some if independent of n.

1A

Then lim infn^oo / n is integrable, and / liminf fn(x)dx J A n-*°°

< liminf / n -*°° J A

fn(x)dx.

Theorem 1.2.3 (Lebesgue). Let A C Rd be measurable, fn : A —* RU{±oo} a sequence of integrable functions converging pointwise almost everywhere on A to some function f : A —» R U {±00}. Suppose there exists some integrable F : A —• R U {±00} with \fn\
for all n.

Then f is integrable, and / f(x)dx JA

= n lim / ^°° J A

fn(x)dx.

Thoerem 1.2.3 is called the theorem on dominated convergence. Let us consider an example that shows the necessity of the hypotheses in the previous results: fn : [0,1] —• R is defined as

/»(*) := { n0

for 1/n < x < 2/n otherwise.

,

>

~

^\

124

Lebesgue measure and integration theory

Then lim fn = 0, n—+oo

and lim / fn(x)dx = 1 ^ 0 = / f(x)dx. -+°° Jo Jo The fn do not form a monotonically increasing sequence so that B.Levi's theorem does not apply, and they are not bounded by some integrable function that is independent of n so that Lebesgue's theorem does not apply either. Considering — fn instead of / n , we finally obtain a sequence for which Fatou's theorem does not hold. As a corollary of Theorem 1.2.3 one has (approximate the derivative by difference quotients): n

Corollary 1.2.2 (Differentiation under the integral). Let I C R be an open interval, A C R d measurable, and suppose f : A x I —* R U {±00} satisfies (i) for any t £ I, /(•, t) is integrable on A (ii) for almost all x £ A, / ( # , •) is differentiable on I (Hi) there exists an integrable cj>: A —• RU {±00} with the property that for all t € I and almost all x G A

^ / ( M ) | <*(*)• Then

ip(t):= J is a differentiable function oftEl,

W

f(x,t)dx with d

f(x,t)dx.

A Ot

q.e.d.

2 Banach spaces

In this chapter, we present some results from functional analysis that will be needed in the sequel, in particular in the next chapter. All proofs are supplied. As a reference, one may use any good book on functional analysis, e.g. K. Yosida, Functional Analysis, Springer, Berlin, 5th edition, 1978, pp. 52-5, 81-3, 90-92, 102-28, 139-45 or F. Hirzebruch, W. Scharlau, Einfuhrung in die Funktionalanalysis, Bibliograph. Inst., Mannheim, 1971, pp. 60-88, 107-12. (These were also our main sources when compiling this chapter.)

2.1 Definition and basic properties of Banach and Hilbert spaces Definition 2.1.1. A vector space V overR is called a normed space if there exists a map ||-|| : V —• E, called norm, satisfying (i) | H | >0 for allv e V, v^O (ii) ||Ai;|| = |A| |M| for all\eR,veV (iii) \\v -f w\\ < \\v\\ -f \\w\\ for all v,w £ V (triangle inequality) A sequence (fn)n£N C V is said to converge to v € V if lim \\vn — v\\ = 0. n—+00

(In order to distinguish the notion of convergence just defined from the notion of weak convergence to be defined in the next section, we sometimes call it norm convergence or strong convergence.) 125

126

Banach spaces

A sequence (vn)neN C V is called a Cauchy sequence if for every e > 0 we may find N G N such that for all n,m> N \\Vn-Vm\\

< £•

A normed space (V, ||-||) is called a Banach space if it is complete w.r.t the notion of convergence just defined, i.e. if every Cauchy sequence converges to some v £ V.

Examples (1) Every finite dimensional normed vector space is a Banach space, for example Rd with its Euclidean norm |-|. (2) Let K C Rd be compact. C°(K) := {/ : K ~+ R continuous}, ll/lloo : = suPa:GiC l/( x )l f° r f € C°(K), defines a Banach space. If we equip Cm(K) := {/ : K —• R m-times continuously differentiable}, m G N, with the norm HH^, it is not a Banach space, because it is not complete. Namely the convergence w.r.t. 11-1loo is uniform convergence, and while the uniform limit of continuous functions is continuous, in general the uniform limit of differentiable functions is not necessarily differentiable. (3) Let (V, 11-||) be a Banach space, W dV & linear subspace that is closed w.r.t. ||-|| i.e. if (u>n)n€N C W converges to v G V^limn^oo \\wn - v\\ = 0), then v G W. Then (W, ||-||) is a Banach space itself. Definition 2.1.2. A Hilbert space is a vector space H overR equipped with a scalar product, i.e. a map (•,•): JJ x if —• R satisfying (i) (v, w) = (w, v) for all v,w G H (ii) (Aifi + A2t>2,w) = \\{vi,w) + A2(t>2,w) for all Ai,A2 G R, vi,v2)w G H (iii) (v, v) > 0 for alive H\ {0}. In addition, we require (iv) H is complete w.r.t. the norm \\v\\ := (v, v)*, i.e. a Banach space. In order to justify the preceding definition, we need to verify that \\v\\ = (v,v)* defines indeed a norm in the sense of Definition 2.1.1. Since the properties (i), (ii) of Definition 2.1.1 are clearly satisfied, we only need to check the triangle inequality:

2.1 Basic properties of Banach and Hilbert spaces

127

Lemma 2.1.1. Let ( v ) : ffxfl-+R satisfy (i)~(iii) of Definition 2.1.2. Then we have the Schwarz inequality: \(v,w)\ < \\v\\ • \\w\\ for all v,w G H, with equality if and only if v and w are linearly dependent. Proof. We have for v, w G H, X G R (v + Xw, v + Xw) > 0 by (iii) . Inserting A = — ,y™( and expanding with the help of (i), (ii) yields the Schwarz inequality. Since \\v + w\\2 = (t; -f tc, v + tc) = ||f||2 + I M | 2 + 2(v, w), the Schwarz inequality in turn implies the triangle inequality. q.e.d. Definition 2.1.3. Let V be a vector space (overR, as always). M CV is called convex if whenever x, y G M, then also tx + (1 - t)y GM

for all 0
Example 2.1.1. Let (F, ||-||) be a normed space. Then for every fi < 0, JE?M := {# G V | ||x|| < fi} is convex. Namely if x,y G JBM, i.e. |x| < //, |y| < /x, then for 0 < t < 1 |te + ( l - t ) y | < t | x | + ( l - t ) | y | < j x , hence to + (1 - £)y € B^. The following definition contains a sharpening of the convexity of the balls B^. It will be formulated only for fi = 1, but by homogeneity ((ii) of Definition 2.1.1), it implies an analogous condition for any [i > 0. Definition 2.1.4. A normed space (V, ||-||) is called uniformly convex if for all e > 0 there exists <5 > 0 with the property that for all x, y G V with \\x\\ = \\y\\ = 1, we have 2^ + V) > 1 - £ =H|x - 2/|| < c.

(2.1.1)

Remark 2.1.1. An equivalent form of the implication (2.1.1) is x-V\\ (again for ||x|

> 1 =»

= 1).

11 x +

K

2/)

< 1

(2.1.2)

Banach spaces

128

Example 2.1.2. In a Hilbert space (if, (•, •)), we have the parallelogram identity •2

^IMI' +^IMI2-^-*/!!2

jffr + tf) 1

(2.1.3)

which follows by expanding the norms in terms of the scalar product. Therefore, any Hilbert space is uniformly convex. Lemma 2.1.2. In Definition 2.1.3, the condition \\x\\ = \\y\\ = 1 may be replaced by

<

Ml < i ,

I.

Proof. In the situation of Definition 2.1.3, for €o > 0, we may find So > 0 such that for all z,w with \\z\\ = \\w\\ = 1, we have

j(* + H > i~<50 =» I k - H I < e o-

(2.1.4)

Let now c > 0, ||x|| < 1, \\y\\ < 1. If for 6 < \ l-«<

2^

+ y)

then ||x||>l-2«,

||»||>l-2&

In particular, x, y ^ 0, and by the triangle inequality,

1(JL + JL) 2viwi

2^x+y)

>

iiwiiy

y

y

> 1 - 3<5.

We apply (2.1.4) with z = j^fjj, w = J ^ J , e0 = f. If 36 < 60, we then get

_f

y_

W|

||2/!|

<2-

Now

k-2/ll <

+

x

y

\\y\\ by the triangle inequality

<4*

+

i.

+

Nil

y \\y\\

2.1 Basic properties of Banach and Hilbert spaces

129

Choosing 6 = min(3<5o,e/8), we have shown the implication |1

\(* + v) > l-6=>\\x-y\\

<e

for ||*|| < 1, \\y\\ < 1. q.e.d. Lemma 2.1.3. Let (V, ||-||) be a uniformly convex Banach space. Let (#n)n€N C V be a sequence with limsup ||x n || < 1 for all n G

(2.1.5)

and lim

~\Xn

= I.

-h Xm)

n,m—+c»

(2.1.6)

Then (xn) converges to some x G V with \\x\\ = 1. Proof. Let e > 0. (2.1.5) and (2.1.6) imply lim||x n || = 1. Therefore, by replacing xn by nf^rr, we may assume w.l.o.g ||x n || = 1. Because of (2.1.5), we may apply Lemma 2.1.2. By (2.1.6), we may find N G N such that for n,m> N ly^n

i %m)

>

1-6,

with 6 determined by Lemma 2.1.2. We obtain \\xn

xm 11
i.e. (xn)nGN is a Cauchy sequence. Since (V, ||-| is a Banach space, it has a limit x, and ||x|| = lim ||x n || = 1. n—*oo

q.e.d. In order to formulate the Hahn-Banach theorem, a fundamental extension result for linear functionals from a linear space to the whole space, we need: Definition 2.1.5. Let V be a (real) vector space. p : V -+ M+

(E+:={tEl|t>0})

130

Banach spaces

is called convex if (i) p(x + y) < p{x) + p{y) for all x, y e V (ii) p(Xx) = Xp(x) for all x eV, A > 0 Example 2.1.3. The norm on a normed vector space. Let VQ be a linear subspace of the vector space V, /o : VQ —• R linear. A linear / : V —* R is called an extension of /o if /Ivo = /oTheorem 2.1.1 (Hahn-Banach). Let VQ 6e a Zineor subspace of the vector space V, p : V —• R + convex. Suppose that fo:Vo~+Ris linear and satisfies fo{x)
forallxeVo.

(2.1.7)

Then there exists an extension f : V —• R o/ /o t^it/i f(x) < p(x)

for all xeV.

(2.1.8)

Remark 2.1.2. We shall need the Hahn-Banach theorem only in the case where V possesses a countable basis, i.e. is separable (see p. 130). Proof We may assume VQ 7^ V. Let v £ V \ Vb, V\ be the linear subspace of V spanned by VQ and v, i.e. Vi:={x

+ tv\xeVo,

,te

R}.

We shall now investigate how /o can be extended to f\ : V\ —>• R with /i(a?) < p(x)

for all a; € Vi.

(2.1.9)

We put fi{v) =: a. Then as an extension of /o, /i satisfies fi(x + tv) = /o(x) + ta. Equation (2.1.9) requires /o(a;) -h t a < p(x -h tv).

(2.1.10)

For t > 0, this is equivalent to «^p(? + ")-/o(f).

(2-1-H)

»>-p(-f-»)-/„(f).

(2.U2)

and for t < 0 to

2.1 Basic properties of Banach and Hilbert spaces

131

For #1, x2 £ V, we have /0O2) - fo(xi) < p{x2 - xi) = p((x2 + v) - {xi + v)) < p(x2 4- v) + p(-Xi - v), hence ~fo{x2) +p{x2 + v) > -fo{xi)

- p ( - x i - v).

(2.1.13)

Thus a 2 := inf ( - / 0 (x 2 ) + p(x2 + tv)) > ax : = s u p ( - / 0 ( x i ) - p ( - x i

-v)).

Therefore, any a with ot\ < a < a2 satisfies (2.1.11) and (2.1.12), hence (2.1.10). Thus, the desired extension /1 exists. If V possesses a countable basis, we may use the preceding construction to extend /o inductively to all of V. If V does not possess a countable basis, we need to use Zorn's lemma to complete the proof. For that purpose, let :={ R, i = 1,2, we have 2|wi = 0, /o : VQ -> E linear with \fo(x)\ < A ||x||

for all x £ V0.

Banach spaces

132

Then there exists an extension / : F - + l o / / o with \f(x)\ < X\\x\\

forallxeV.

Theorem 2.1.2 (Helly). Let (V, ||-||) be a Banach space, / i , . . . , / n linear functional V -* R that are continuous w.r.t. the norm convergence, /i, a\,...,an E R . Suppose that for any Ai,. » A n G

1n

n

1

(2.1.14)

<M TTien for each e > 0, there exists xe G V with fi(x€) = a;

for t = 1,2,..., n

(2.1.15)

and ||x € || < / * + €. Proof Let m < n be the maximal number of linearly independent /*, z = l , . . . , n . It suffices to consider ra linearly independent /*, w.l.o.g. fi, • • •»/m, since the remaining ones are easily seen to be taken care of by (2.1.14). F(x) := ( / i ( x ) , . . . , fm(x)) may then be considered as a linear map onto Rm. We equip Mm with its Euclidean structure. Let B H+e

•-

{xeV\

||x||
Then F(B M+C ) is a convex set containing 0 as an interior point. Also, F(£?M+C) is balanced in the sense that with p G R m it also contains -p. We now assume that a i , . . . , a m is not contained in F(B M + C ). Because of the properties of F(B M+C ) just noted, we may then find A = (Ai,. , Am) with / J AiCki >

sup

X>/<(*) =1

t=l

771

(*X + €) t= l

contradicting (2.1.14). Thus ( a i , . . . , a m ) G claim.

F(JBM+C),

2.2 Dual spaces and weak convergence Let V be a vector space. The linear functionals f:V-+R

implying the g.e.d.

2.2 Dual spaces and weak convergence

133

then also form a vector space. If (V, ||-||) is a normed vector space, we define the norm of a linear functional / : V —• R as #

:= sup ! ^ i G R+ U {oo}.

(2.2.1)

Lemma 2.2.1. A linear functional f :V -+R is continuous if and only z/||/|U < oo. The easy proof is left to the reader. (See also Lemma 2.3.1 below.) q.e.d. Definition 2.2.1. V* := {/ : V —• R linear with \\f\\^ < oo} equipped with the norm (2.2.1) is called the dual space of (V, ||-||). (It is easy to verify that (2.2.1) defines a norm on V* in the sense of Definition 2.1.1.) L e m m a 2.2.2. (V*, ||-||J is a Banach space. Proof Let (fn)neN C V* be a Cauchy sequence. For every e > 0 we may then find N G N such that for n, m G N \\fn-fm\l<e. By (2.2.1), this implies that for every x G V \fn(x)

- fm(x)\

< C.

Therefore, since R is complete, (fn(x))neN converges for every x G X. We denote the limit by f(x). f : V —• R then is a linear functional. It is an easy consequence of the triangle inequality that \\f\\^ < oo and that lim n _oo | | / n - / | | + = 0. This implies that (fn)nen converges to / G V*, and (V*, | H U therefore is complete, hence a Banach space. q.e.d. Remark 2.2.1. We did not assume that V itself is a Banach space. We now consider (V*)* =: V**, the dual space of V*, with norm denoted by ||-|| ## . Any x £ V defines a linear functional i(x) : V* -* R i(x)(f):=(flX):=f(x).

134

Banach spaces

Lemma 2.2.3. ||i(#)|| ++ = ||x||. Thus, the linear functional i(x) : V* —> E is contained in V**, i.e. we have a linear isometric map i : V —» V**. Proof. We have

l(/,*)l
IM|> s u p i j M = ||t(x)|L.

(2.2.2)

Conversely, let x G V. Let /o(te) :=*||a;||

for t e R.

By the Hahn-Banach theorem (Corollary 2.1.1), we may extend / 0 from {tx\ t G l } to V as a linear functional / with

and

l(/,»)l = lkllTherefore ll^)IL=supKMi>|N|. fev* II/IU

(2.2.3)

Equations (2.2.2) and (2.2.3) imply the result. q.e.d. Definition 2.2.2. A normed linear space (V, ||-||) is called reflexive if i:V-*V** is a bijective isometry {i.e. \\x\\ = ||i(x)|| ++ for all x G V). Remark 2.2.2. (1) Since (V**, |||| + + ) is a Banach space by Lemma 2.2.2, any reflexive space is complete, i.e. a Banach space. (2) By the remark before Definition 2.2.2, the crucial condition in that definition is the surjectivity of i.

2.2 Dual spaces and weak convergence

135

Definition 2.2.3. (i) Let (V,||-||)&e a normed linear space. (#n)nGN C V is said to be weakly convergent to x G V if f(xn) converges to f(x) for all f G V*, in symbols: xn

v

x.

(ii) Let (V*, ||-||J be the dual of a normed linear space. (fn)nen C V* is said to be weak* convergent to f G V* if fn(x) converges to f(x) for all x G V. Theorem 2.2.1. Let V be a separable} normed linear space. Let {fn)n€N C V* be bounded, i.e. ||/ n ||* < constant (independent of n). Then (fn) contains a weak* convergent subsequence. Proof. Let (y„)„£?$ by a dense subset of V. Since {fn{yi))neN is bounded, a subsequence (fn(yi)) of (fn(yi)) converges. Having iteratively found a subsequence (/™) of (/ n ) for which (/™(2/z,))n€N converges for 1 < v < m, we may find a subsequence (/™ +1 ) of (/™) for which also (/^T+1(2/m+i))nGN converges. The diagonal sequence (/™)n€N then converges at every yv, v G N, and since (yv)vm is dense in V, (/™(x))nGN has to converge for every x eV. Thus, we have found a weak* convergent subsequence of (/n)n€Nq.e.d. Remark 2.2.3. (1) The argument employed in the preceding proof is called Cantor diagonalization. (2) Theorem 2.2.1 remains true without the assumption that V is separable, and so does the following: Corollary 2.2.1. Let (V,||-||)6e a separable reflexive Banach space. Then every bounded sequence (xn)nen contains a weakly convergent subsequence. Proof. By (2.2.2) or reflexivity, (i(xn))nen is a bounded sequence in V** and therefore contains a weak* convergent subsequence. Since V is f

Separable means that V contains a countable subset (yv)u£N that is dense w.r.t. 11-||, i.e. for every y € V, € > 0 there exists yu with \\y — yu\\ < e.

136

Banach spaces

reflexive, the limit is of the form i(x) for some x eV. f(xn)

- (/,a?n) -* (/,») = f{x)

Thus

for every / G T

so that (x n ) n€ N converges weakly to x. q.e.d. T h e o r e m 2.2.2. Any weakly convergent sequence (xn)n^ space is bounded. Proof. We shall show that i(xn) { / e r | | | / | L < 1 } . Then also

is

uniformly

in a Banach bounded

ll*n|| = \\i(*n)\\„ = sup J ^ i

on

(2.2.4)

is bounded (see Lemma 2.2.3 for the first equality). Since i(xn) is linear, it suffices to show uniform boundedness on some ball in V*. Otherwise, we find a sequence Bj of closed balls, Bj = {/ € V* | | | / - £11 <

Qi}

for some fj e V* , 6j > 0

with Bj+\ C

JBJ

and

lim Qj = 0 3-

and a subsequence (x^) of (xn) with |(/,x;)|>j

for all f€Bj.

(2.2.5)

By construction, (fj)jen forms a Cauchy sequence and therefore converges to some /o G T , with oo

/oef|^. Because of (2.2.5), we have

\(fo,x'n)\>j

foralljeN.

This is impossible since ( / o , ^ ) converges because (x^) n€ m converges weakly. q.e.d. Example 2.2.1. (1) In a finite dimensional normed vector space (which automatically is complete, i.e. a Banach space), weak convergence is just componentwise convergence and therefore equivalent to the usual convergence w.r.t. the norm.

2.2 Dual spaces and weak convergence

137

(2) In an infinite dimensional reflexive Banach space (V, ||-||), this is no longer so, because one may always find a sequence (e n ) nG N C V with ||ei|| < 1 for all i and ||e* — ej\\ > 1 for i ^ j . Such a sequence cannot converge w.r.t. | | | | , because it is not a Cauchy sequence, but it always contains a weakly convergent subsequence according to Corollary 2.2.1 (we have shown Corollary 2.2.1 only under the assumption of separability, but it holds true in general). Lemma 2.2.4. Let (V, ||-||)6e a separable normed space. Then V* satisfies the first axiom of countability w.r.t. the weak* topology, i.e. for each f £ V*f there exists a sequence (U^ueN of subsets of V* that are open in the weak* topology such that every U that is open in this topology and contains x is contained in some Un. Consequently, if (V, \\-\\)is also reflexive, then V* satisfies the first axiom of countability w.r.t. the weak topology. Proof. Let / £ V*. Every neighbourhood of / w.r.t. the weak* topology contains a neighbourhood of the form

UttVu...,Vk(f):={geV*\

\g(vi) - m)\

< e for i = 1,... , * } .

Since V is separable, there exists a sequence (wn)ne^ C V that is dense w.r.t the 11-|| topology. We claim that the neighbourhoods of the form U±tWilt...tWik(f) form a basis of the neighbourhood system of / of the required type, i.e. every U€;Vli.„iVk(f) contains some such U±.w. fW, (/). For that purpose, we choose n with | < e and Wix,..., Wik with \VJ - Wi3 \ < ^ for j = 1 , . . . , k. For g e [A (/), we then have

\9(Vj) ~ f(vj)\ < k K ) - / K ) | + \(9 - f)(vj - ti;y)l < \ + I < €, i.e. g e Ve-Vl,...,vk(f) as required. Finally, if V is reflexive, then the weak* and the weak topology of V* coincide. q.e.d. We now present some further applications of the Hahn-Banach theorem that will be used in Chapter 3. Lemma 2.2.5. Let (V, ||-||) be a normed space, Vo a closed linear subspace. Then VQ is also closed w.r.t. weak convergence.

138

Banach spaces

Proof. By the Hahn-Banach theorem (Corollary 2.1.1), for every XQ G V \ Vb, we may find a continuous linear functional /o : V —• R with /o(x 0 ) = 1

/ok=0. Thus, #o cannot be a weak limit of a sequence in Voq.e.d. Lemma 2.2.6. Let (V,||-||)6e a reflexive Banach space, Vo a closed linear subspace. Then Vo is reflexive. Proof. We may identify VQ* with a subspace of V**, by putting v(f) = v(f\Vo) for / G V*, v G V0**. Let v e V£*. Since V is reflexive, there exists x G V with v(/) = / ( x ) for all / G T . We claim x G Vo- Otherwise, by the Hahn-Banach theorem (Corollary 2.1.1), there exists f eV* with

m ±o f\v0 = 0. Since f(x) = f (/|vb) by the above, this is impossible. Since every /o G VQ* can be extended to / G V*, again by Hahn-Banach, we conclude v(fo) = fo(x) Thus, t; = i(x). This implies

VQ**

for a l l / eV0*.

= i(Vfo), i.e. reflexivity of Vog.e.d.

Corollary 2.2.2. -4 Banach space (V, ||-||)is reflexive if and only if its dual (V*, 11-|U is reflexive. Proof If V = V**, then also F* = F***. Thus, if V is reflexive, so is V*. Consequently, if conversely V* is reflexive, so then is V**. Since V can be identified with a closed subspace of V** by Lemma 2.2.2, Lemma 2.2.6 then yields reflexivity of V. q.e.d. Lemma 2.2.7. Let (V, ||-||)6e a normed space, and suppose that (x n ) nG N C V converges weakly to x eV. Then ||x||
2.2 Dual spaces and weak convergence

139

Proof. After selection of a subsequence, we may assume that ||# n || converges (see Theorem 2.2.2). Assume ||x|| > lim | | x n | | . n—+oo

As in the proof of Lemma 2.2.3, we may find f EV* with

11/11. = 1 l/(x)| = IW|. But then | / ( x ) | > lim ||x n || > l i m s u p | / ( x n ) | , n-+oo

n~*oo

while the weak convergence of (# n ) n eN to x implies f(x) = lim

f(xn).

n-+oo

This contradiction establishes the claim. q.e.d. Theorem 2.2.3 (Milman). Any uniformly convex Banach space is reflexive. Proof (Kakutani). Let (V, ||*||)be a uniformly convex Banach space, and let XQ* € V**. We need to show that there exists some xo € V with i(x0) = x£*

(2.2.6)

(see Remark 2 after Definition 2.2.2). We may assume w.l.o.g. that 11*0*11 = 1-

(2-2-7)

For every n 6 N, we may then find fn € V* with | | / n | | = 1 and 1 - - < x*0*(fn) < 1. n We now claim that for every n G N, we may find xn E V with fi(xn)=x*0*(fi)

for i = l , . . . , n

(2.2.8)

(2.2.9)

and

I M | < 11*5*11 + ^ = l + i .

(2.2.10)

For any A i , . . . , An £ R, we have

X>*o*(/o U=i

r° \^ /i = l' ***

< WxoW

X>/J |t=l

140

Banach spaces

and so the claim follows from Helly's Theorem 2.1.2. Since in addition to (2.2.10) also I K H = | | / n | | l k n | | > fn(xn)

= XQ*(fn)

> 1 ~

~,

we must have lim ||x n || = 1. n—>oo

For ra > n, we have 2 2 - -

2 < fn(xn)

+ / n ^ m ) < | | x n + Xm\\

< \\xm\\

+ \\xm\\

< 2 +

-.

n n By Lemma 2.1.3, (xn)n€N is a Cauchy sequence and converges to some xo eV, satisfying Hxoll = 1 (2.2.11) and fi(xo)=x*0*(fi)

for i = 1,2,3,...

(2.2.12)

The solution #o of (2.2.11), (2.2.12) is unique. Namely, if there were another solution £Q, on one hand, we would have

IN+ 411 < 2

(2.2.13)

by uniform convexity. On the other hand fi{x0 + x'0) = 2XQ*(/<)

for all z,

hence

2 - - < 2x*0*(fi) = /<(x0 + 4) < IN + *oll , hence

IN + 4ll>2. This contradicts (2.2.13), and so Xo is unique. We now claim that /o(*o) = x*0*(f0)

for any / 0 G V*,

(2.2.14)

so that XQ* = i(xo), proving the theorem. Let this /o G V* be given. In the above reasoning, we replace the sequence / i , / 2 , / 3 , - - - by /o> /l> /2, /3, — We then obtain X Q G F with

IKII = i and fi(x'0) = x*0*(fi)

fori = 0 , 1 , 2 , 3 , . . .

(2.2.15)

2.2 Dual spaces and weak convergence

141

Since the solution XQ of (2.2.11), (2.2.12) was shown to be unique, however, we must have xr0 = #o- Equation (2.2.15) for i = 0 then is (2.2.14). q.e.d. Corollary 2.2.3 (Riesz). Any Hilbert space (H, (•, •)) can be identified with its dual H*. Proof. Since a Hilbert space is uniformly convex, Therem 2.2.3 implies H = H**. On the other hand, any x £ H induces an fx G H* by fx{y) '= (x,y)

for y G H.

We have 11/0-11= sup (x,y) < \\x\\ IMI=i and fx(x) = (x,x) = ||x|| 2 , hence

ll/xll = IN|. Thus, H is isometrically embedded into H*. For the same reason, H* is isometrically embedded into H**, and since H = if**, one readily verifies that these embeddings must be surjective, hence H = H* = H**. q.e.d. Let M be a linear subspace of a Hilbert space H. The orthogonal complement ML of M is defined as M1 := {x e H : (z, y) = 0

for all y G M} .

It is clear that M1 is a closed linear subspace of H. M need not be closed here, but the orthogonal complement of M is the same as the one of its closure M in H. Corollary 2.2.4. Let M be a closed linear subspace of the Hilbert space H. Then every x G H can be uniquely decomposed as x = xi + x2 with xi € M,x2 € ML. Proof. By the proof of Corollary 2.2.3, x G H corresponds to fx G H* with fx(y) = (x,y) for all y G H. We let f^

be the restriction of /^ to M . M, since closed, is a Hilbert

142

Banach spaces

space itself, and f*f is an element of the dual M*. By Corollary 2.2.3, it corresponds to some X\ € M, i.e. f™(y) = (xi,y) for all

y€M.

We put X2 := x — x\. Then for all y € M, (x - xu y) = / , ( » ) - / f (y) = 0 since fx = / f on M. Therefore, #2 € M- 1 . Thus, we have constructed the required decomposition. Concerning uniqueness, if x = X\ + #2 = x'i + #2 w ^ n #i> #i

G

M, X2, ^2 £ M x ,

then for all y € M (x,t/) = (xx,y) =

(x'vy),

and by Corollary 2.2.3 applied to M, X\ = x ; j, and therefore also #2 = #2q.e.d. Of course, the reader knows the preceding result in the case where H is finite dimensional, i.e. a Euclidean space. x\ is interpreted as the orthogonal projection of x onto the subspace M, and therefore Corollary 2.2.4 is called the projection theorem. The next result will be needed for Sections 4.2 and 4.3 when we establish the existence of minimizers for lower semicontinuous, convex functionals. Theorem 2.2.4 (Mazur). Suppose (xn)n€N converges weakly to x in some Banach space V. For every e > 0, we may then find a convex combination N

N

] P An Xn (An > 0, ] P An = 1) 71=1

71=1

with

IN I / ^ Anxn

< e.

(2.2.16)

|n=l

Proof. We consider the set Co of all convex combinations of the x n , i.e.

{

N

2 J A„xn 71=1

N

}

with An > 0, 2 ^ An = 1 > . n=l

J

2.2 Dual spaces and weak convergence

143

Replacing all xn by xn — x\ and x by x - #i, we may assume 0 E Co. If (2.2.16) is not true, then there exists e > 0 with ||x-y|| >£forallt/eC0.

(2.2.17)

C\ := {z £ V : ||* - y\\ < - for some y € C0} is convex and contains the ball with radius | and center 0. We consider the Minkowski functional p of C\ defined by p(z) : = i n f { A > 0 ;

X^zed}.

p is convex in the sense of Definition 2.1.5 since C\ is convex, and continuous since C\ contains the ball of radius | > 0 about 0. Since, because of (2.2.17), ||x - z\\ > - for every z e C\, we have p(x) > 1. More precisely, there exists t/o with x = A_1i/o,

0 < A< 1

p(Vo) = 1. We consider the linear subspace V0 =

{fiyo^eR}cV

and the linear functional /o(») =

fionVo.

Then fo <pon

Vo,

and by the Hahn-Banach Theorem 2.1.1, there exists an extension / of /o to all V with /
<\-1=f(\-1yo)

= f(x).

Banach spaces

144

This, however, contradicts the fact that (x„) n€ iv C Co converges weakly to x. Thus, (2.2.17) cannot hold, and (2.2.16) is established. q.e.d.

2.3 Linear operators between Banach spaces The results of this section will be used in Chapter 8. In Section 2.2, we considered linear functional f:V-+R; in the beginning, V was a normed linear space, with norm denoted by 11-||, and later, we also assumed that V was complete, i.e. a Banach space. In the present section, we replace the target E by a general Banach space W, with norm also denoted by ||*||. We thus consider linear operators

T-.V-+W, and we put

||r||:=supi3GR+U{cx)}. L e m m a 2.3.1. The linear operator T : V only if\\T\\ < oc.

(2.3.1)

W is continuous if and

Proof. If ||T|| < oo, then the inequality (2.3.2)

\\TX\\<\\T\\\\X\\

implies that T is continuous. (Of course, this uses the linearity of T.) Conversely, if T is continuous, we recall the usual e — 6 criterion for continuity, and so for e = 1, we find some 6 > 0 with the property that \\Ty\\ < 1 if |M| < 6. For x G V \ {0}, we then have with y = <5pn (||j/|| < 6) \\Tx\\ =

[

Ty

<-sM

Thus

imi< ^
2.3 Linear operators between Banach spaces

145

The space of continuous linear operators T : V —• W between the normed spaces (V, ||-||) and (W, ||-||) is denoted by L(V, W). It becomes a normed space with norm ||T||. Lemma 2.3.2. //(W, ||-||) is a Banach spacef then so is (L(V, W), ||-||). The proofIs the same as the one of Lemma 2.2.2, simply replacing (M, | • |)

by WH-II). Remark 2.3.1. Again, (V, ||-||) need not be a Banach space here. Lemma 2.3.3. Let T € L(V, W). Then kerT:={x€V

: Tx = 0}

is a closed linear sub space of V. Proof, ker T = T - 1 (0) is the pre-image of a closed set under a continuous map, hence closed. q.e.d. In the sequel, we shall encounter bijective continuous linear operators T: K - > W between Banach spaces. It is a general theorem in functional analysis, the inverse operator theorem, that the inverse of T, denoted by T - 1 , is then continuous as well. Here, however, we do not want to prove that result, and we shall therefore frequently assume that T " 1 is continuous although that assumption is automatically fulfilled in the light of that theorem. Lemma 2.3.4. Let T.V-+W be a bijective continuous linear map between Banach spaces, with a continuous inverse T - 1 . If S £ L(V,W) satisfies

l|T-S||
S=

T{Id-T~1{T-S)).

(2.3.3)

146

Banach spaces

As with the geometric series, the inverse of S then is given by

Br'tr-sirV1,

(2.3.4)

provided that series converges. However,

2(r- 1 (r-5))" < E HCr- i ( r _ 5))1| 1 i/=m

<E(ir Mllir -5|ir, and since |JT* -111 \\T — S\\ < 1 by assumption, the series satisfies the Cauchy property and hence converges to a linear operator with finite norm. q.e.d. If V is a vector space, we say that V is the direct sum of the subspaces Vi, V2, V = Vi 0 V2 if for every x G V, we can find unique elements #i G Vi, #2 € V2, with x = # i 4- #2-

We then also call V\ and V2 complementary subspaces of V. Easy linear algebra also shows that if V\ possesses a complementary subspace of finite dimension, then the dimension of that space is uniquely determined, i.e. if Vi 0 V2 = Vi 0 V2', then dim V2 = dim V2'. We now consider a normed vector space (V, ||-||). Then every finite dimensional subspace Vo is complete, hence closed. We also have: Lemma 2.3.5. Let V0 C V be a finite dimensional subspace of the normed vector space (V, ||*||). Then VQ possesses a closed complementary subspace V\, i.e. V = Vo 0 V\. Proof. Let ei,..., en be a basis of V0, /Q : Vo —> R be the linear functiona l with /o( e ») = 6v

( M = l»»-,n).

By Corollary 2.1.1, we may find extensions /«? : V —• R with fL = /<J.

2.3 Linear operators between Banach spaces

147

We define 7r: V —> V as n

7ra?:= J ^ / ^ a : ) ^ . 7r is continuous, with TT(V) = Vo. V"i := ker7r then is closed as the kernel of a continuous linear operator (Lemma 2.3.3), and every x € V admits the unique decomposition x = 7r(ar) -f (x — 7r(#)) with 7r(a:) € Vo, a: — 7r(a:) € Vi, because w on = n. q.e.d. Definition 2.3.1. Let T : V —+ W be a continuous linear operator between Banach spaces (V, ||-||) and (W, ||-||). T is called a Fredholm operator if the following conditions hold: (i) Vo = kerT is finite dimensional Consequently, according to Lemma 2.3.5, there exists a closed subspace V\ ofV with V = Vb0Vi.

(2.3.5)

(ii) There exists a finite dimensional subspace WQ of W, called the cokernel ofT (cokerT) giving rise to a decomposition ofW into closed subspaces W = W0 0 Wi

(2.3.6)

with Wx = T(V) =: R(T)

(range

ofT).

Thus, T yields bijective continuous linear operator T\ : V\ —>W\. We finally require (iii) T~l : W\ —> V\ is continuous. For a Fredholm operator T, we call ind T := dim V0 - dim Wo

(= dim ker T - dim coker T)

the index ofT. The set of all Fredholm operators T : V —> W is denoted byF{V,W). Remark 2.3.2. Question to the reader: Why is F(V, W) not a vector space?

148

Banach spaces

Remark 2.3.3. As mentioned, condition (iii) is automatically satisfied as a consequence of the inverse operator theorem. Remark 2.3.4- I*1 o u r conventions, the cokernel of T is only determined up to isomorphism, i.e. any Wo satisfying (2.3.6) with W\ = T(V) is a cokernel. Usually, one defines the cokernel as the quotient space W/W\, but here we do not want to introduce quotient spaces of Banach spaces. Theorem 2.3.1. Let V,W L(V,W), and

be Banach spaces. F(V,W)

is open in

ind : F(V, W) -> Z is continuous, hence constant on each connected component of F(V, W). Proof. Let T : V —> W be a Fredholm operator. We use the decompositions V^VbeVi

with V0 = kerT

W = W0®W1

with W0 = cokerT

of Definition 2.3.1. For S £ L(V, W), we define a continuous linear operator S':VixW0->W (x,z) *-+ Sx + z, and we obtain a continuous linear operator L(V, W) -> L(Vi x W 0 ,W) Since T\ : V\ —> Wi is bijective with a continuous inverse, T' is also bijective with a continuous inverse, and by Lemma 2.3.4 this then also holds for all S in some neighbourhood of T. For such 5, S'(Vi) is closed as V\ is closed and S" is continuous, and we have the decomposition

w = 5 , (Vi)e5 , (w 0 ), and since S'(Vi) = S(Vi) also iy = 5(Vi)e5,(W0),

(2.3.7)

and since Wo is finite dimensional, so is Sf(Wo). Then 5 ( F ) D S(V\) is also closed since S(V\) is closed and possesses a complementary subspace of finite dimension. Finally, the dimension of the kernel of S is upper semicontinuous.

2.3 Linear operators between Banach spaces

149

Namely, if S is in our above neighbourhood of T, then since S is bijective, S is injective on Vi, and hence the kernel of S is contained in some complementary subspace of V\, and as observed above, the dimension of such a subspace equals the one of Vo- Thus dim ker S < dim ker T

(2.3.8)

if 5 is in a suitable neighbourhood of T in L(V, W). Altogether, we have verified that S is a Predholm operator if it is sufficiently close to T. Prom the preceding, we see that there exist finite dimensional subspaces VQ = ker S and VQ of V with

v = v£®v£'®vu and thus dim VJf + dim Vg = dim V0

(V0 = ker T).

(2.3.9)

S thus is injective on VQ ® V\, and since S coincides with Sf on V\, we get a decomposition W = 5(Vi)®5(V r o / / )eWS, with H^ = cokerS' and from (2.3.7) dim S{V£) + dim WJ = dim S'(W0) = dim W0

(2.3.10)

since 5 ; is bijective. Consequently ind S = dim ker S — dim coker S = dimV r 0 / -dimW'o = (dim V0 - dimF 0 ") - (dim W0 - dimS(Vf)) = dim VQ - dim W0

by (2.3.9), (2.3.10)

since 5 is injective on VQ

= indT. for S in some neigborhood of T. g.e.d. The following result motivates the definition of a Predholm operator: Theorem 2.3.2 (Fredholm alternative). Let V be a Banach space, T : V —> V a Fredholm operator of index 0. We consider the equation Tx = y.

(2.3.11)

150

Banach spaces

Either (i) Either Tx = y is solvable for all y, and thus T is surjective, hence also injective as indT = 0, and so the solution x is uniquely determined by y, or (ii) Tx = y is only solvable if y is contained in some proper subspace ofV (with a finite dimensional complementary subspace), and for each such y, the solutions x constitute a finite dimensional affine subspace. Proof A direct consequence of the definition. q.e.d.

2.4 Calculus in Banach spaces In this section, we collect some material that will only be used in Chapters 8 and 9. Definition 2.4.1. Let (V, \\-\\v)> (W, \\-\\w) be Banach spaces, F : V —• W a map. F is called differentiable (in the sense of Prechet) at u € V if there exists a bounded linear map DF(u)

:V-*W

with Hm

^o°,

\\F(u + v)-

F(u) - DF(u)(v)\\w

= Q

IMIv

/ is called differentiable in U C V if it is differentiable at every u G U. f is said to be of class C1 if DF(u) depends continuously on u. f is said to be of class C2 if DF(u) is differentiable in u and the derivative D2F(u) := D(DF)(u) depends continuously on u. It is easy to show that a differentiable map is continuous. We now wish to derive the implicit and inverse function theorems in Banach spaces that will be used in Chapter 8. We shall need a technical tool, the Banach fixed point theorem: Lemma 2.4.1. Let A be a closed subset of some Banach space (V, ||-||). Let 0 < q < 1, and suppose G : A —• A satisfies \\Gyi - Gy2\\ < q\\yi - y2\\

for all yuy2

€ A.

(2.4.2)

2.4 Calculus in Banach spaces

151

Then there exists a unique y £ A with Gy = y.

(2.4.3)

/ / we have a continuous family G(x) where all the G(x) satisfy (2.4-2) (with q not depending on x), then the solution y = y(x) of (2.4-3) depends continuously on X. Proof. We choose y0 £ A and put iteratively yn :=

Gyn-\.

We have n

n

yn = J2 (th - vi-i)+2/0 = 53 ( G ' _1 ^ t=l

Gi l

~ y°)+y°-

(2-4-4)

i=l

We obtain from (2.4.2)

Y,\\Gl-1vi-Gi-1yo\\

<E ^

1

llyi -2/0II < ^— llyi -itoll-

Consequently, the series yn in (2.4.4) converges absolutely and uniformly to some y £ A1 noting that A is assumed to be closed and the limit function y = y(x) is continuous. We have y = lim Gyn = G ( lim yn) = Gy, n—+00

\n—+00

/

hence (2.4.3). The uniqueness of a solution of (2.4.3) follows from (2.4.2), since q < 1. q.e.d. Theorem 2.4.1 (Implicit Function Theorem). Let Vi,V2,W be Banach spaces with all norms denoted by \\-\\, U C V\ x V2 open, (xo^yo) £ U', F £ CX{U^ W), i.e. F is continuously differentiable. For purposes of normalization solely, we assume F(x0lyo)=0.

(2.4.5)

We also suppose that D2F(x0,yo)'-V2-+W, the derivative of F(x$, -) :V2 ^ W aty = yo, is invertible. By our differentiability assumption, D2F(xo1yo) is continuous, and we assume that

152

Battach spaces

its inverse is likewise continuous. Then there exist open neighbourhoods U\ of XQ, U2 ofyo with U\ x U2 € U, and a differentiate map

with F(xyip(x))

=0

(2.4.6)

and D
(fix)))-1

o DiF(x, (p(x))

for all

xeUi (2.4.7)

(D\F(-,y) :V\—>W is the derivative of F(-,y) : V\ —• W). In fact, for every x € U\9
=0

can be solved locally uniquely for y as a function of #, if the derivative of F w.r.t. y is continuously invertible. Proof. The idea is to transform the problem into a fixed point problem for which the Banach fixed point theorem is applicable. We put l:=D2F(X(hyo)With this notation, our fixed point equation is * ( * , y) ••= y - rlF{x,

y)

=y

(2.4.8)

which clearly is equivalent to our orginal equation F(x, y) = 0. For every x, we thus want to find a fixed point of yt-+$(x,y). x

Using l~ ol = id (note that / is invertible by assumption), we get *(ar, yi) - *(x, y2) = r1(D2

F(x0j y0)(yi - y2) - (F(ar, t/i) - F(x,

y2)).

In Lemma 2.4.1, we take q = ~, and by the differentiability of F at (#o>2/o) and the continuity of Z"1, we may find 6' > 0,£ > 0 with the property that for Wx-xoW^S' and \\yi - Vo\\ < ^, I life - yo\\ < e

( hence also \\yi - y2\\ < 2e ),

2.4 Calculus in Banach spaces

153

we have ||*(rc,yi) - *(a?,i&)|| < - \\yi - !fe||. Furthermore, we may find 6" > 0 with the property that for \\x — sco11 < <$"> we have ||*(a?,»b)-*(a?o,»d)ll < 2* Since $(#o,2/o) = 2/o by assumption, we then have for \\y — yo\\ < e \\Q(x,y) - voll < ||*(»,») " *(aMfo)ll + 11*0*. 0o) " *(*o, tt>)|| <^\\y-yo\\

+

^<e

whenever \\x — XQ\\ < 6 := min(<$',<$"). This means that if \\x — XQ\\ < <5, $ (#,?/) maps the closed ball A:={y£V2:\\y-y0\\<e} onto itself. By Lemma 2.4.1, for every x with \\x — XQ\\ < <5, there exists a unique y =: (£>(#) with \\y — y0\\ < e and y = $(x,t/), i.e. F(x1y) = 0. Moreover, t/ depends continuously on x. We consider the open balls Ui := {x : \\x - x0\\ < (5}, t/2 := {2/: ||v - Vo\\ < £>• ($(#,•) also maps the open ball U2 onto itself.) By choosing <$,£ > 0 smaller, if necessary, we may assume f / i x t / 2 C U. It remains to show that
U2,

and abbreviate y\ := iF(a?i,0i),J 2 :=

D2F(xuyi).

Since F is differentiable, we may write

F(x,y) =h(x-xi)

+ l2{x-x2)

+r(x,y)

where the remainder term satisfies

lim

x-+xi y - • 2/i

^ 4

-=0.

\\x - xi\\ + \\y - yi\\

(2.4.9)

154

Banach spaces

Since F(x,
+2/i -l2Xr{x,
(2.4.10)

By (2.4.9), we may find r),p> 0 such that for I k - a : i | |
\\r(x,y)\\


< 2||ri||(Ha -

Xl

H

+

H» - WID-

Thus ||r(x, (*i)ID- (2-4.11) By (2.4.10), (2.4.11), ||v>(x) - v»(x,)t| < WtfhW \\x - and + ^ ||x - xi|| + i | M x ) - ^(xi)||, hence \\(a?) - ^(a?i) = -l^hix

- a?i) -f r 0 (x),

(2.4.12)

with lim „ x-+xi

r

°^ „ = 0

from (2.4.9).

(2.4.13)

\\x — Xi\\

(2.4.12) and (2.4.13) yields the differentiability of

(2.4.14)

Proof. We shall apply Theorem 2.4.1 to F(x1y) := f(y) - x, and find an open neighbourhood U\ of XQ and a differentiable function
2.4 Calculus in Banach spaces

155

with ^>(U\) C C/2 for a neighbourhood U2 of yo1 with
= x for x G C/"i.

As y>(£/i) = f~1(U\) is open, we may redefine [/2 as ^(C/i), and tp then yields a bijection between U\ and C/2. As f(
• ity(*o)) = id, i.e. (2.4.14). g.e.d.

The next topic concerns ordinary differential equations in Banach spaces. In Chapter 9, we shall use the Picard-Lindelof theorem in a Banach space that we shall now derive. We need the integral of a continuous function x.I^V from some interval / = [a, 6] C R into some Banach space V,

1

x(t)dt.

Ja

This can be defined as a Riemann integral as in the case of real-valued functions through approximation by step functions. Given a continouous

we say that x(t) solves the ODE (ordinary differential equation) on / ,

if for

--x{t) = x(t) = $(t, x(t)) at dlltel x(t)=x0+ /

with x(a) = x0

${T,x(r))dT.

(2.4.15)

(2.4.16)

Ja

Theorem 2.4.2 (Picard-Lindelof). Suppose that $ is uniformly Lipschitz continuous, i.e. suppose there exists some L < oc with | | * ( t i , * i ) - * ( t 2 , x 2 ) | | < L (|*i - t2\ 4- ||xi - x 2 ||) for all t e I,xux2 e V. Then for any XQ € V, there exists a unique solution of (2.4-15).

(2.4.17)

156

Banach spaces

Proof. We shall solve (2.4.16) with the help of Lemma 2.4.1. For a continuous y : / —• V1 we define Gy G C ° ( / , V), (Gy)(t):=x0+

f *(r,i/(r))dr. Ja We note that C°(/, V), the space of continuous functions from / with values in V, is a Banach space w.r.t. the norm \\y\\co-SMv\\y{t)\\. tet (To verify this, one just needs to observe that any sequence (yn)neN C C°(I,V) with lim

\\yn - ym\\Co [ =

n,m—• oo

\

lim sup \yn{t) - ym{t)\) n,m—*oo ^ j

=0 J

converges uniformly to some continuous function y : / —• V.) We have \\Gyi -Gy2\\Co

= sup / ($(r,*/i(r)) *€/ |«/a <\t — a\ L\\y\ — 2/211 c70

$(T,2/2(T)))GM

I because of (2.4.17).

We choose e > 0 so small that

Lemma 2.4.1 with V replaced by C°([a, a -f e], V) and with q = \ then implies that there exists a unique t/GC°([a,a + e],F) with fb

Gy(t) = xo + / $ ( r , y(r))dr iov a
x(0) = a?0.

(2.4.18)

Exercises

157

(:r(0), the value at 'time' 0, is called initial value). We denote the solution by x(xo,t). Then for s, t > 0, X(XQ, t + s) = x(x(t), s)

(semigroup property).

Thus, the solution with initial value XQ at 'time11 + 5 is the same as the solution with initial value x(t) computed at 'time1 s. Proof. This follows from the uniqueness statement in Theorem 2.4.2, as both sides of (2.4.18) are solutions. q.e.d.

Exercises 2.1

Let (V, \\'\\v) (W, \\'\\w) be normed linear spaces. For a linear functional

f:V-+W, put sup

ll/(*)ll W

z€V\{0}

2.2

iFllV

Show that / is continuous iff | | / | | < 00. Let L(V,W) := {/ : V -> W linear with | | / | | < 00}. Show that if (W^IHI^) is a Banach space then so is (L(V, W), ||-||). Show that a normed space (V, ||-||) is uniformly convex if the following condition holds: Whenever (xn)neN, (yn)neN C V satisfy limsup||ar n || < 1 , limsup ||y n || < 1 and lim ||arn + y n || = 2, then lim (xn - yn) = 0. n—+oo

2.3

A normed space (V, ||-||) is called strictly normed if the following condition holds: Whenever x,y £V, x,y ^ 0 satisfy

ll* + vll = IMI + llvlU then there exists a > 0 with x = ay.

158 2.4

Banach spaces Show that any uniformly convex normed space is strictly normed. Does the Banach fixed point theorem (Lemma 2.4.1) continue to hold if we replace (2.4.2) by the condition \\Gyi - Gy2\\ < \\yi - y2\\ for all Vl,y2

e A?

3 LP and Sobolev spaces

3.1 Lp spaces In the sequel, instead of functions f : A -> RU {±00} (A measurable), we shall consider equivalence classes of functions, where / and g are equivalent if f(x) = g(x) for almost all x € A. We shall be lax with the notation, however, not distinguishing between a function and its equivalence class. The equivalence class of the zero function is called the null class, and a function in that class is called a null function. Definition 3.1.1. Let A C Rd be measurable, LP(A)

peR\{0}.

= {(equivalence classes of) measurable functions f : A —• R U {±oc} with

\f(x)\"eCHA)}. For f G LP (A), we put

\\f\\p--=\\f\\l,(A)--=(JA\m\Pdxy

•

(3.1.1)

The notation suggests that ||-|| is a norm, and we now proceed to verify this for p > 1. First of all, ||/|| p = 0 & f is a null function.

(3.1.2)

Thus, ||-|| is positive definite (on the set of equivalence classes). Next, for c € R, l|c/|| p = | c | | | / | | p .

(3.1.3)

It remains to verify the triangle inequality. This is obvious for p — 1:

||/i + MLHA)

^

WMLHA) 159

+ WMILHA) •

(3-1-4)

Lp and Sobolev spaces

160 For p > 1, we need

Lemma 3.1.1 (Holder's inequality). Letp,q > 1 satisfy ~ 4- ~ = 1, fr € If(A), f2 € L«(4). Then fuf2 € L 1 ^ ) , and ||/i/2|li<||/i||p||/2||fl.

(3.1.5)

Proof, By homogeneity, we may assume w.l.o.g. Il/i|l p = l , ll/allg = l.

(3.1.6)

Recalling Young's inequality, namely ap bq ab < — 4- — p g

1 1 for a, b > 0 , p, q > 1 , - + - = 1, p g

(3.1.7)

we have for x £ A /1W/2W < — - — + —-—> p q hence by our normalization (3.1.6) / l/i(^)/a(a:)|€te < 1 + 1 = 1 = H/ill^ll/alUq.e.d. We now obtain the triangle inequality: Lemma 3.1.2 (Minkowski's inequality). Let / 1 , f2 € £ P 0A), p > 1. T/ien ll/i + / 2 | | p < | | / i | | p + | | / 2 | | p .

(3.1.8)

Proof The case p = 1 is given by (3.1.4). We now consider p > 1 and put q := ^ (so that \ + ± = 1). For (*) := l/i(*) + / 2 ( z ) r \ we have

^ = i/i+/ 2 r, q

i.e. V € L {A). Since l/iCx) + / 2 (x)| p < \h(x)*{x)\

+

\MxMx)\,

we get ||/i + / 2 || P '<||/iV<|| 1 + ll/2

= (ll/l|| p + ll/ 2 |l P )ll/l+/2||lp

3.1 LP spaces

161

Since p - 2 = 1, (3.1.8) follows. q.e.d. We have thus verified that ||-|| is a norm on LP(A). In fact, we have: Theorem 3.1.1 (Riesz-Fischer). Let A be measurable, p > 1. Then LP(A) is a Banach space. Proof. Let (/n)n€N C LP(A) be a Cauchy sequence. For every v € N, we may then find nv € N with ll/n - / n j | p < 2^

for

a11 U

^ n^*

This implies that the series oo

<31-9)

\\u\P+J2\\f^-^\\p converges. We claim that the series CO

then converges in LP(A). Since all elements of the series are nonnegative, (<7m)m€N converges to some g : A —• M+ U {oo} pointwise in A, and Corollary 1.2.1 implies that (gm) also converges to g in L P (A). In particular, g(x) < oo for almost all x e A. Thus, our original sequence (3.1.10) is absolutely convergent for almost all x € A, towards some / with | / | < g -f |/m|; in particular / G L p (0). We interrupt the proof to record: Lemma 3.1.3. Let (/ n )n€N converge to f in LP(A). quence converges pointwise almost everywhere to f.

Then some subse-

In order to complete the proofs of Lemma 3.1.3 and Theorem 3.1.1, it remains to show that the series (3.1.10) converges to / in LP(A). (Then a subsequence of (/ n ) converges to / in LP(A). Since (fn) was assumed to be a Cauchy sequence in LP(A), the whole sequence has to converge in LP(A). It is in general not true, however, that the whole sequence also converges pointwise almost everywhere to /.) This is easy: oo /n,W + £

(/n„+1(x) " /»„(*)) "

fix)

Lp and Sobolev spaces

162

converges to 0 almost everywhere in A, and since

fni (*) + Yl C^+i ^) ~ ^ ^) " - ^ <2| 5 (x)| + 2|/ n i (x)|, t/=i

we may apply Lebesgue's Theorem 1.2.3 on dominated convergence to conclude that we get convergence also w.r.t. ||-|L. q.e.d. Corollary 3.1.1. L2(A) is a Hilberi space with scalar product

(/i,/ 2 ):= [ h(x)f2(x)dx. JA

Proof. It follows from Holder's inequality (Lemma 3.1.1) that

i(/i,/ 2 )i R U {±00} measurable. ess sup/(x) := inf {A G R | f(x) < A for almost all x G A) xeA (essential supremum), and L°°(A) := {(equivalence classes of) measurable functions f : A —• R U {±00} with ll/lloo ••= I I / I I L ~ ( A ) : = e s s s u p | / ( x ) | < 00} x£A

Theorem 3.1.2. L°°(A) is a Banach space. Proof If is straightforward to verify that IHI^ is a norm. It remains to show completeness. Thus, let (fn)neN be a Cauchy sequence in L°°. For v G N, we find n„ G N such that for m,n>_nv Win

/ m | l o o < cyv •

Thus

\x*A\

|/n(rr)-/m(rr)|>lj

3.1 L* spaces

163

is a null set for ra, n > nu, and so then is

N:=

(J

|xGi4|

\fn(x)-fm(x)\>^

m,n>nl

as the countable union of null sets. Since

\fn(x) ~ fm(x)\ < ± for Vfi^n>mnv and x G A \ iV, / n converges uniformly on A \ N towards some / . We simply put f(x) = 0 for x G N. Then ess sup |/„(x) - / ( x ) | = ess sup \fn(x) - / ( a ) | , since the essential supremum is not affected by null sets,

< jp and fn converges to / in L°°(A). q.e.d. We also note that Holder's inequality admits the following extension to the case p = 1, q = oo: Lemma 3.1.4. Let fx G L1(A), f2 G L°°(A).

Then fxf2 G L1(A), and

ll/i/alli < IIAIIi ll/alloo -

(3.1.H)

Proof.

/ |/i(x)/ 2 (a:))|dx<esssup|/2(x)| / JA

x£A

\fi(x)\dx

JA

= ll/ 2 IUI/illiq.e.d. Theorem 3.1.3. Let A C Rd be measurable. Let 1 < p < oo, q = -^y, i.e. -P + jQ = 1. Then Lq(A) is the dual space of LP(A). In particular, LP(A) is reflexive. Remark 3.1.1. The dual space of LX(A) is given by L°°(A) while the dual space of L°°(A) is larger than Ll(A). Therefore, neither Ll(A) nor L°°(A) is reflexive. In order to prepare the proof of Theorem 3.1.3, we first derive:

164

LP and Sobolev spaces

Theorem 3.1.4 (Clarkson). Let A C Rd be measurable, 2 < q < oo. Then Lq(A) is uniformly convex. Remark 3.1.2. Clarkson's theorem holds more generally for 1 < q < oo. The proof for 1 < q < 2 is a little more complicated than the one for 2 < q < oo. The proof of Theorem 3.1.4 is based on: L e m m a 3.1.5. Let 2 < q < oo, f,g G Lq(A).

Then

11/+g\\qq +11/ - *n; < 2*- 1 (H/H; + M I ; ) .

(3.1.12)

Proof. For x, y > 0, we have (xq + yq)* < ( x 2 + i / 2 ) 5 < 2 ^ ( x 9 + i / 9 ) i

(3.1.13)

(In order to verify the left inequality in (3.1.13), we may assume w.l.o.g. x2 -f y2 = 1. Then xq < x2, yq < y2 since q < 2, and the desired inequality easily follows. The right inequality follows for example from Holder's inequality (Lemma 3.1.1) applied to the following functions /i,/2:(-l,l)-»R / i = l, 2

x ut) = {% b 2

for - 1 < t < 0 for 0 < t < 1. )

The left hand side of (3.1.13) implies (|o + b\q + \a- fe|9)' < (|o + bf + \a- 6| 2 ) *
(3.1.14)

for a, 6 € R, and by the right-hand-side of (3.1.13), we have V2(a2 + b2)^ < 2 ^ (\a\q + |6| 9 )' .

(3.1.15)

Equations (3.1.14) and (3.1.15) imply |/0r) + g(x)\" + \f(x) - g{x)\" < 2"~l (|/(x)| 9 + |ff(x)|'),

(3.1.16)

and (3.1.12) follows by integrating (3.1.16). q.e.d. Proof (Theorem 3.14)- Let f,g e Lq(A) with

3.1 LP spaces

165

By (3.1.12),

H/+ffll2 + li/-0li2<2'. Therefore, for e > 0, we may find 6 > 0 such that

ll/-3ll,<< whenever | | | ( / + #)|| > 1 — 5. This shows uniform convexity. g.e.d. Proof (Theorem 3.1.3). We consider the map i: Lp(A) -> Lq(A) with <(/)(») := /

f(x)g(x)dx.

JA

By Holder's inequality (Lemma 3.1.1) ||i(/)||=

sup

|i(/)(s)|<||/||p.

(3.1.17)

g€Ll(A) \\g\\q
Thus i(f) is indeed an element of Lq(A)*. We claim that we have equality in (3.1.17). This means that there exists some g £ Lq with /

f(x)g(x)dx

\JA

We put g(x) := sign/(x) \f(x)\p-\ and / f(x)g(x)dx\^ I.M

HI/IUML.

(3.1.18)

Then \g\q = | / | p , hence g e

Lq(A),

/ |/(x)p(ar)|da? I

JA

[\f(x)\pdx

=

=

JA

(A

l/(X)|PrfX P l/(X)|PrfX 9

) (^

)

= ll/ltllffllgThis verifies (3.1.18), hence equality in (3.1.17). Equality in (3.1.17) implies that i is an isometry, in particular injective. In order to complete the proof we need to show that i is surjective. Suppose on the contrary that L"{A)*\i{U{A))^%.

Lp and Sobolev spaces

166

Since LP{A) is complete and i is continuous, i(Lp(A)) is complete, hence closed. By the Hahn-Banach theorem (Corollary 2.1.1), there then exists veL«(A)**,v ^ 0 , with v\i(LP(A))

= 0.

We now suppose for a moment that 1 < p < 2. Then 2 < g < oo, and Lq{A) is reflexive by Theorems 3.1.4 and 2.2.3. We may therefore find a g in Lq{A) with F(g) = v(F)

for all F E

Lq{A)\

We then have for any y> E L P (A) 0 = v(i(
3.2 Approximation of Lp functions by smooth functions (mollification) In this section, we shall smooth out LP functions by integrating them against smooth kernels. As these kernels approach the Dirac distribution, these regularizations will tend towards the original function. For that purpose, we need some g E C^{Rd)\ with

/ Jud

g(x)dx(= \

f JB{O,I)

g{x) > 0 for all x E Rd

(3.2.1)

g(x) = 0

(3.2.2)

g(x)dx)=l.

for |x| > 1

(3.2.3)

J

Such a g is called a Friedrichs mollifier. In this §, fi will always denote an open subset of Rd. Let / E L x (0). We extend / to all of Rd by putting t For O C K d open, C*Q°(0) is the space of all C°° functions

3.2 Approximation of LP functions by smooth functions

167

f{x) = 0 for x e Rd \ ft. Let h > 0.

/fc( ):=

* £/,X^) / ( y ) d *-

(324)

--

fh is called the mollification of / with parameter h. In order to appreciate this definition, we first observe suppQ \j^~)

C B(Vlh)

:= {z£Rd\

\z-y\<

h},

(3.2.5)

where Q ( ^ ^ ) is considered as a function of x, and

hL'icir)*-*-

<326)

--

For these reasons, one expects that fh tends towards / as h tends to 0. It remains to clarify the type of convergence, however. The advantage of approximating / by fh comes from: Lemma 3.2.1. Let Q,' CC fif, h < dist(fy,dfi). Then fh € C°°(fi'). Proof By Corollary 1.2.2, we may differentiate w.r.t. x under the integral sign in (3.2.4), and since g £ C°° so then is fhq.e.d. We now start investigating the convergence of fh towards / . Lemma 3.2.2. If f £ C°(Q), then for each ftf CC ft, fh converges uniformly to f on fi' as h —• 0. In symbols: fh—^f on ftf as h —• 0. Proof We have f{x) = / g(w)f{x)dw J\w\
by (3.2.3)

(3.2.7)

and fh(x) = [ g(w)f(x - hw)dw J\w\
(3.2.8)

in (3.2.4). For Qf CC ft and h <

f tQ* C C fi' means that the closure of fi' is compact and contained in fi. We say that Q' is relatively compact in Q.

Lp and Sobolev spaces

168

\ dist(n',dfi), we then have sup |/(x) - h{x)\

< sup /

x£Q'

Q(W) \f(x) - f(x -

hw)\dw

xeQ' l\w\ J\w\
< sup |/(x) -

f(x-hw)\

\W\<1

using (3.2.3) once more. Since fi' is bounded, {x G fi1 dist(x, fi') < h) is compact (recall the choice of h). Therefore, / is uniformly continuous on that set, and we conclude that sup |/(x) - fh(x)\ -> 0 as h -+ 0, x£Q'

i.e. uniform convergence. q.e.d. Theorem 3.2.1. Let f G LP(Q), 1 < p < oo. T/ien /^ converges to f in Lp(Q) as h -+ 0. Froo/. We have for p G 17 (f2) / Jn

\gh(x)\pdx Q(w)g(x — hw)dwdx

= JQJ\W\
< / I /

g(w)dw J

JQ \J\W\<1

J

I /

g(w) \g(x - /ut;)|pcfe; J .

\J\u)\
J

by Holder's inequality = / J\w\
g(w) / \g(x -

hw)\pdxdw,

JQ

using (3.2.3) and Fubini's theorem, = / J\w\
\g{y)\pdydw

Q(w) / d

JR

= j \g(y)\pdy, JQ

using (3.2.3) again. Thus \\9h\\LPiQ)<\\9\\LP{Qy

(3.2.9)

3.2 Approximation of LP functions by smooth functions

169

Let e > 0. By Theorem 1.1.4, (6), we may find ip G C#(R d ) with II/-VIILP(R-)<|-

( 3 - 2 - 10 )

Since (p has compact support, we may apply Lemma 3.2.2 to conclude that for sufficiently small h > 0, ^ 3'

(3.2.11)


( 3 - 2 - 12 )

II¥>-WIIILP(R*)

Applying (3.2.9) to / -
(3.2.10)-(3.2.12) yield

11/ - A l l ^ n ) < 11/ - /nlli>(R-) < «•

(3-2.13) g.e.d.

Corollary 3.2.1. For 1 < p < oo, C£°(n) is dense in L p (fi). Proo/. Let / G L p (0), e > 0 . We may then find O ' C C f i with H/llLp(n\n')

<

2*

We put / ' := fxLp{Q')- Then H/-//llLp(n)<|-

( 3 - 2 - 14 )

By Theorem 3.2.1, for sufficiently small h, ll//-/fcllLp(n)<5-

(3-2-15)

By (3.2.13), (3.2.14) 11/ ~~ //IIILP(Q)

<

2*

Since f'h G C^(Q) for h < dist(fi',dfi), the claim follows. q.e.d. Corollary 3.2.2. Lp(fl) is separable for 1 < p < oc. jBvcry / G L P (Q) con 6e approximated by piecewise constant functions. Proof By Corollary 3.2.1, it suffices to find a countable subset BQ of LP(Q) with the property that for every (p G CQ°(Q) and every e > 0, there exists some a G BQ with

IIP-allien) <«•

(3-2.16)

Lp and Sobolev spaces

170

Let B the set of all functions a on Md of the following form: There exist some fc, N £ N and rational numbers c*i,..., a* and cubes Q i , . . . , Qfc G Md with corners having all their coordinates in jjZ and of edge length •^ such that for x E Qi otherwise. Clearly, B is countable. Since a continuous function (p with compact support is uniformly continuous, we may easily find some a £ B with

"<*> = { ?

11° -

^ILP(Q)

^ ll a ~


< e-

(3.2.17)

We put BQ := {axn I « £ # } . # n is likewise countable, and from (3.2.15), (3.2.16), we conclude that BQ is dense in LP(Q). q.e.d. Remark 3.2.1. The separability of LP(Q) can also be seen by using Corollary 3.2.1 and the Weierstrass approximation theorem that allows the approximation of continuous function with compact support by polynomials with rational coefficients. The preceding results do not hold for L°°(fi). Namely, if a sequence of continuous functions converges w.r.t. IHI^oom), then it converges uniformly, and therefore, the limit is again continuous. Therefore, noncontinuous elements of L°°(Q) cannot be approximated by continuous functions in the L°°-norm. Also, L°°(Q) is not separable. To see this, let (a>n)neN be any subsequence of {0,1}, i.e. an € {0,1} for all n. To (a n ), we associate the function /( a n ) on (0,1) defined by , - J 1 ' ^ - " t o ff

for^<x<^rrifafc = l for^<x<^Tifa, = 0

torkeN

-

Then for any two different sequences (a n ), (6 n ), ||/(a n ) -/(&n)|lr,~((o,i)) = = 1 Since the set of subsequences of {0,1} is uncountable, this implies that L°°((0,1)) is not separable. Of course, a similar construction is possible for L°°(fi), fi any open subset of Rd. We finally note: L e m m a 3.2.3. Let f € L2(Vt), and suppose that for all tp e

[ f(x)
Then / = 0.

CQ°(Q)

3.3 Sobolev spaces

111

Proof. Since Cg°(fi) is dense in L 2 (fi), and since gy-> I Jn

f(x)g(x)dx

is a continuous linear functional on L 2 (fi), we obtain that / f(x)g(x)dx /n

= 0 for all g G L 2 (0).

Putting # = / yields the result. q.e.d.

3.3 Sobolev spaces In this section, we wish to introduce certain extensions of the LP spaces, the so-called Sobolev spaces. They will play a fundamental role in subsequent chapters because they constitute function spaces that are complete w.r.t. norms naturally occurring in variational problems. In this section, ft will always denote an open subset of Rd. We shall use the following notation: For a d-tuple a := ( a i , . . . ,a
N-t«..«.=-(sr)"'-(s'rDefinition 3.3.1. Let u,v G L 1 (fi). Then v is said to be the ot-th weak derivative of u, v := Dau, if I\pvdx = ( - 1 ) N J uDaipdx

(3.3.1)

for every (p G Cg . We can now define, for k G N and 1 < p < oo, the Sobolev space Wk*(Q) := {u G Lp(Q) | Dau exists and lies in LP(Q)forall\a\
liv*.p(fl) : =

] C / lD«ulP

Fino/fe fe* #*' P (Q) andH^p(Q) be the closures o / C ° ° ( l ] ) n ^ ( ( l ) fc and C$° n W 'P(fi), respectively in Wk*(Q).

Lp and Sobolev spaces

172

We shall use the following abbreviations for u G W 1 ' 1 (fi), 1 < i < d. D{U is the weak derivative for the multiindex ( 0 , . . . , 0 , 1 , 0 , . . . , 0), 1 at the i t h position, and Du is the vector ( D i t t , . . . , Ddu) of all first weak derivatives. The following result is obvious. L e m m a 3.3.1. Letu G Ck(Q), and suppose all derivatives ofu of order < k are in LP(Q). Then u G Wk,p(Q), and the weak derivatives are given by the ordinary derivatives. q.e.d. Thus, the WkyP spaces constitute a generalization of the spaces of k times differentiate functions. The Wk)P norm is considerably weaker than the Cfc-norm, and so the Wk*p spaces are larger than the Ck spaces. Before investigating the properties of these spaces, it should be useful to consider an example: Let ft = (—1,1) C M, u(x) := \x\. We claim that u G Wl,p(Q) for 1 < p < oc. In order to see this it suffices that the first weak derivative of u is given by

Indeed, we have for (p G /

CQ((-1,

1))

(p(x)v(x)dx = — /


We claim, however, that u is not contained in W2,p(ft). Namely if w(x) were the second weak derivative of it, it would have to be the first weak derivative of v) and consequently, we would have w(x) = 0 for x ^ 0. The rule for integration by parts (3.3.1) would then require that for all y?e Q 1 ( ( - i , i ) ) 0= /

(f(x)w(x)dx (pf(x)v(x)dx

= — = /

(p'(x)dx — /

ip'{x)di

= 2(p(0) which is not the case. Thus, v does not have a first weak derivative.

3.3 Sobolev spaces

173

Remark 3.3.1. Some readers may have encountered the notion of a distributional derivative. It is important to distinguish between weak and distributional derivatives. Any L 1 (fi) function possesses distributional derivatives of any order, but as the preceding example shows, not necessarily weak derivatives. In the example, of course, the second distributional derivative of u is 2<5o, where <5o is the Dirac delta distribution at 0. u does not possess a second weak derivative because the delta distribution cannot be represented by an L 1 function. Theorem 3.3.1. The Sobolev spaces Wk,p(fl) spaces w.r.t. ||-|liy*,P(n)-

are separable Banach

Proof. That ||-||^fc,p(n) 1S a norm follows from the fact that IHI^pm) is a norm (see section 3.1). Similarly, we shall now derive completeness of Wk*{Q) from the completeness of the LP(Q) spaces (Theorem 3.1.1). Thus, let (vn)n€N C WkiP(Q) be a Cauchy sequence w.r.t. ||-||w*.p(n)* This implies that (Daun)n^ is a Cauchy sequence w.r.t. IHI^pm) for all \OL\ < k. By Theorem 3.1.1, (Daun) therefore converges in Lp(fl) towards some va. [ Daun

• ip = (-l)l a > / unDa
JQ

(3.3.2)

JQ

Therefore, va is the a-th weak derivative of t>o, the L p -limit of (un)new, and consequently VQ € Wk,p(ft). The separability again follows from the corresponding property for L p (fi) (Corollary 3.2.2). q.e.d. Theorem 3.3.2. Wk*(Q) =

Hk*(n).

This result says that elements of WkyP(Q) can be approximated by C°°(fi) functions w.r.t. |Hlwfc,P(m. I*1 general, however, for k > 1 one has ifo' p (0) ^ Wfc'p(fi) so that Wk*(fl) functions cannot be approximated by Cg°(fi) functions, in contrast to LP(Q) functions where this is possible (Corollary 3.2.1). This is seen from the following simple example: 11 = (—1,1) C K, u(x) = 1. If ((pn)neN C CQ° (ft) converges to u in L 1 (fi), then after selection of a subsequence, it converges pointwise almost everywhere (Lemma 3.1.3), and therefore, for sufficiently large n, there exists xn G (-1,1) with (pn(xn) > \. Since n(l)> this implies that

j _ \ 1 .

Lp and Sobolev spaces

174

Therefore, p(Q) can be approximated by C°°(fi) functions. As in §3.2, we extend u to be 0 outside fi and consider the mollifications Uh € C°°(fi). We compute A*(u/i(a?)) = yi J

Da,xQ ( —j£- J • w(y)dy

(using Corollary 1.2.2)

where Da,x is the derivative w.r.t. x,

= (-l)°jf - i?«,,e(^)-«(y)d» = j ^ Q ( ^ ) A,u(y)d» by definition of Dau = (Dau)h(x).

(3.3.3)

Thus, the derivative of the mollification is the mollification of the derivative. Since Dau € L p (n), by Theorem 3.2.1, (Dau)h converges to Dau in L P (Q) for h —• 0. By (3.3.4), we conclude that Da(uh) converges to Dau in 1^(0), for all | a | < fc, and this means that UH converges to u in

wk*{n).

q.e.d. Theorem 3.3.3. Wk*p(Sl) is reflexive for k e N, 1 < p < oo. Proof. It follows from Theorem 3.1.3 that the dual space of WkiP(fl) is given by Wk,q(Ql)) with £ -f ^ = 1. This implies reflexivity. g.e.d. Theorem 3.3.4. iJo'^fi) is closed under weak convergence in WkyP(Q). Proof This follows from Lemma 2.2.5, since HQ,P(Q) by its definition is a closed subspace (w.r.t. strong convergence) of Wk'p(U). q.e.d. Theorem 3.3.5. For 1 < p < oo, k G N, any sequence in Wk,p(Q) that is bounded w.r.t. ||*||v^fc,p(n) contains a weakly convergent subsequence.

3.4 Rellich's theorem, Poincare and Sobolev inequalities

175

Proof. By Theorems 3.3.1 and 3.3.3, Wk'p(Q) is separable and reflexive. Therefore, the result follows from Corollary 2.2.1. q.e.d.

3.4 Rellich's theorem and the Poincare and Sobolev inequalities The compactness theorem of Rellich is: Theorem 3.4.1. Let ft C Rd be open and bounded. Let (un)ne^ C H0yP(ft) be bounded, i.e. \\un\\Wi,p,Q\ < c (independent of n). Then a subsequence of (un)n£^ converges in Lp(fl). Remark 3.4.1- Rellich originally proved the theorem for p = 2. Kondrachev proved the stronger result that some subsequence converges in Z/*(n) for 1 < q < ^ if p < d and for 1 < q < oo if p > d. Of course, these exponents come from the Sobolev Embedding Theorem (see (3.4.12)). See Corollary 3.4.1 below. Proof. Since un G H0,P(Q), for every n G N and e > 0, there exists some vneC^{Q) with | P n ~ Vn||jyi,p(n) < «•

(3.4.1)

Therefore ||t>n||||ri,p(n) < C '

(=C+f)'

(3A2)

We consider the mollification v

nAx) = yt J e \h^)

V

^y)dy

of vn and estimate \vn(x)

-Vnyh(x)\

/ g(w)(vn(x) |^NI
< / Q{w) / J\w\
- vn(x - hw))dw |

Q

\°r

\^-vn(x-r'd) drdw

by (3.2.7), (3.2.8) w with ^ = 7 - 7 . \w\

(3.4.3)

LP and Sobolev spaces

176 This implies

/ \vn(x) - vUyh(x)\p dx JQ

fhM I d

f ( f < / I /

g(w) /

JQ \J\W\
f ( f

= / I / JQ \J\W\
JO

/

\

g(w)p / Jo

\—vn(x r

\°

dx

J

ft

rh\w\ I

\ '

Y

— vd) \drdw\

\or

(g(w)1~p)

V

I

\—vn(x

I

\

— rfi)\drdw I

J dx /

I [ g(w)hp\w\p f \Dvn(x)\pdxdw) , vM
< ( / g(w)dw) VM
using Holder's inequality, Fubini's theorem and the notation n

(

d

d

Since fiwi<1 g(w)dw = 1 (by (3.2.3)), we obtain K - Vn,h\\LP{n) ^ h WDvn\\LP(Q) < he' by (3.4.2) < -

if h is sufficiently small.

(3.4.4)

Next, \VnMx)\ ^

yiCo\\Vn\\Li{U) with Co :— sup^ g(z) by definition of vn,h

< -jcoimeasny-p \\vn\\LP(n) by Holder's inequality,

(3.4.5)

and similarly O

-j

(3.4.6) with Ci := sup 2 | ^ r ^ ( ^ ) | - From (3.4.2), (3.4.5), (3.4.6), we see that for fixed h > 0, |t>n,h|| C i (n) < constant

(3.4.7)

(where the constant depends on h). Therefore, (vn,/i)n€N contains a uniformly convergent subsequence by the Arzela-Ascoli theorem. Since uni-

3.4 Rdlich's theorem, Poincare and Sobolev inequalities

177

form convergence implies Lp-convergence (e.g. by Theorem 1.2.3), the closure of t>n,h, is compact in L P (Q). Since a compact subset of a metric space (e.g. a Banach space) is totally bounded, there exist finitely many u?i,..., WN £ LP(Q) such that for every n € N there exists 1 < j < N with e IKft-WjllLp(n) < g-

(3-4-8)

By (3.4.1), (3.4.4), (3.4.8), for every n <E N we find 1 < j < N with \un

w

"^ €'

j\\LP{Q)

Thus, (wn)n€N is totally bounded in Lp(ft). Therefore, the closure of (un)n€f$ in L p (fi) is compact (again, a general result for metric spaces), and it thus contains a convergent subsequence in L p (ft). q.e.d. We now come to the Poincare inequality: Theorem 3.4.2. Let ft C Rd be open and bounded. For any u E

HQ,P(Q)

i

||u|

d

W) * ( ™ r ) H ^ ' W

(3-4-9)

lu/iere u;^ is the Lebesgue measure of the unit ball in Rd. Proof. Since CQ(Q) is dense in H Q ' ^ Q ) , we may assume u e Co (ft). We put u(x) = 0 for all x e Kd \ ft. For d e Rd with |tf| = 1, we have £

/•DO

w(x) = — / Jo

Trw(x + or

r

$)dr.

Integration w.r.t. $ yields

I

1

f°° f

d

\u(x)\ = \ — -,— / / —uix + ntydfidr I dud Jo J\0\=i dr

<xr/l—^rrl^(»)l*. <^d JQ

\X

- vf

178

LP and Sobolev spaces

Therefore p

(jf|u(*)f<'dx

P

-l

by Holder's inequality

^(/w»»'*)f(/„^)

<3-4i(,»

using Fubini's theorem to exchange the order of integration in the first factor. In order to control JQ

\x-y\

we choose R with meas ft = meas B(y, R) = uJdR (B(y,R):={zeRd\ Since

\z\ < R}). 1 1 —-j—r for lla? —*«i/l -> ^ ^ d - i -< R-^J—r d-i 1

1 JZT > 1 ^ n

Ix-yl"" -^-

for \x-y\<

R,

we have / jzrdx JQ \X - y\

< I I^i^x JB(y,R) \x - y\ = dwdR = dwd

a

(3.4.11)

3

(rneas 0 ) .

Equations (3.4.10) and (3.4.11) yield (3.4.9). q.e.d.

3.4 Rellich's theorem, Poincare and Sobolev inequalities

179

We now come to somewhat stronger results that will however only be needed in Chapter 9. Namely, we have the Sobolev inequalities. T h e o r e m 3.4.3. Let u G H^P(Q). (i) Ifp
then u G L ^ F ( f i ) , and IHI^L

(ii) Ifp>d,


(3.4.12)

then u G C°(fi), and sup\u\ < c(measfi)^~p \\Du\L v n

(3.4.13)

with constants c depending only onp and d. (Actually, by a Theorem of Morrey, forp > d, u G H0,p(ft) is even Holder continuous with exponent 1 — ^.) We only prove (i) as (ii) will not be used in the present book: Proof. We first assume u G CQ(Q). have

Since u has compact support, we

oo

|Diu(y)|dy* fori = l , 2 , . . . , d .

/

-oo

Multiplying these inequalities for i = 1 , . . . , d yields

/

7=T

W(y)\

d

r°°

\ ^

<\JlJ_ IA"(2/)l
•

Using Holder's inequalityf, we compute °°

d

/

\u(x)\*^

dx1

-co

IDMy^dy1) -co

a

/ /

M]/

dx1

IAti(»)|d»*

J — co \i=2^"°°

/ 1

oo

\ JZT / < *

JDmMlVJ

/*oo

\

3^r

(n/_ lAtid/JId^dx1) .

f More precisely, one uses Exercise (2) below with p\ = • • • = Pd-i ~

d—\.

Lp and Sobolev spaces

180

Iteratively also integrating w.r.t dx2,...,

dxd finally yields

iHiLA(n)^(n/ni^wi^) < 3 f

Y\Diu(x)\dx

This is (3.4.12) for p = 1. The case of general p may now be obtained by applying (3.4.14) to \u\^ for suitable \i > 1 and using Holder's inequality. Namely, from (3.4.14) for |u|M in place of u

lun^K^Jjuixr-'iDuwidx <^\\\ur1

\\Du\\LP Li d by Holder's inequality.

forj + i p

q

For p < d, we may take \i = *•d—p 7 'p and obtain

H

M

^

ft

IIM—1 L.llM-1

II

iin-.M

11 r-k

LP

'

ud_ < -i \wr ud \\Du which yields (3.4.12), since £ = ^ .

q.e.d.

As a consequence, we obtain the theorem of Kondrachev: Corollary 3.4.1. Let ft E Rd be open and bounded. Let (un)ne^ C H0,p(ft) be bounded for some 1 < p < d. Then a subsequence converges inLq(fl) for any I p as otherwise the result is an easy consequence of Holder's inequality since ft is bounded. We denote this converging subsequence again by (un). Prom Holder's inequality, we obtain \\un — um\\La(Q) < \\Un — Um\\Li(Q) \\un ~ ^m||

d£_

if [i satisfies - = /i -f (1 — /i) ( q \p d
(3.4.15)

Exercises

181

Since Dun is bounded in LP(Q) by assumption, and (un) is a Cauchy sequence in LP(Q), hence also in I/ 1 (fi), (3.4.15) then implies the Cauchy property in Lq(ft). q.e.d.

Exercises 3.1

Let Ax := {x G Rd

> 1} , A2 := {x G

<1>.

and consider f(x) = \\x\\x

for A G E .

For which values of d,p, A is / G L*(Ai), or / G U)(A2)cl 3.2

Let A C R be measurable. Let p i , . . . ,Pfc > 1» ]Ci=i p = *> /< G LP'(A) for i = 1 , . . . , k. Show / i •... • fk G L X (A), with

II*

^n«/iiip. i

3.3

<=i

Let A C R d be measurable, meas A < oo, 1

(meas A) p

TII/III (meas A) «

(Hint: Apply Holder's inequality with fi = I, f2 = f) 3.4

Let A C i d be measurable, 1 < p < tf < r, ^ = f + / G LP(A) n Lr(A). Then / G L*(A), and L«(A)

3.5

<

ii

7£i,

l-Q

LP(A) UJ \\Lr(A)

Let A C R d be measurable, meas A < 00, / : A -> R U {±00} measurable. Then lim

1

T 11/11LP{A)

P-*°° (meas A) p

L°°(A)

(where we allow these quantities to be infinite).

Lp and Sobolev spaces Let A C Md be measurable, (fn)neN C LP(A) with 11fn||p < constant. Suppose fn converges pointwise almost everywhere on A to some / . Is / e LP(A), and do we necessarily get ll/n-/||p-0

asn~+oo?

Let A\,A2,f be as in exercise 1). For which d, fc,p, A is / in Wk*(Ax) or in Wk*(A2)? Consider the sequence (sin(nx)) ne N in L 2 ((0,1)). Does it converge in the L 2 -norm? Does it converge weakly? If so, what is the limit?

4

The direct methods in the calculus of variations

4.1 Description of the problem and its solution The typical problem of the calculus of variations is to minimize an integral of the form F(u) := /

f(x,u(x),Du(x))dx

Jn

where fi is some open subset of Rd (in most cases, Q is bounded), among functions u: n - > R belonging to some suitable class of functions and satisfying a boundary condition, for example a Dirichlet boundary condition u(v) = 9(y)

for

ytdft

for some given g : dQ —• R. Thus, the problem is F(u) —> min

for u EC,

where C is some space of functions. The strategy of the direct method is very simple: Take a minimizing sequence (wn)neN C C, i.e. lim F(un) = inf F(u), n—>oo

u€C

and show that some subsequence of (un) converges to a minimizer u G C. To make this strategy be successful, several conditions should be met: (1) Some compactness condition has to hold so that a minimizing sequence contains a convergent subsequence. This requires the careful selection of a suitable topology on C. 183

184

Direct methods The limit u of such a subsequence should be contained in C. This is a closedness condition on C. In particular, for (1) and (2) to hold, C should not be too restrictive. In other words, one should not specify too many properties for a solution u in advance. Some lower semicontinuity condition of the form F(u) < liminf F(un)

if un converges to u

n—>-oo

has to hold, in order to ensure that the limit of a minimizing sequence is indeed a minimizer for F. The lower semicontinuity condition becomes easier if the topology of C is more restrictive, because the stronger the convergence of un to u is, the easier that condition is satisfied. That is at variance, however, with the requirement of (1) since for too strong a topology, sequences do not always contain convergent subsequences. Therefore, we expect that the topology for C has to be carefully chosen so as to balance these various requirements. In order to gain some insights into this aspect, it is useful to approach the problem from an abstract point of view. Thus, we shall return to the concrete integral variational problem raised in the beginning only later.

4.2 Lower semicontinuity We say that a topological space X satisfies the first axiom of countability, if the neighbourhood system of each point x G l has a countable base, i.e. there exists a sequence (t/I/)1/eN of open subsets of X with x €.UV with the property that for every open set U C X with x G U there exists n e N with

VncV. X satisfies the second axiom of countability if its topology has a countable base, i.e. there exists a family {Uu)y^n of open subsets of X with the property that for every open subset V of X, there exists n <£ N with

UnCV. We note that separable metric spaces X satisfy the second axiom of countability. In fact, let (X^)^ € N be a dense subset of X, and let ( r ^ ) ^ ^ be dense in 1R+. Then {7(x„,rM) := {x e X : d(x,x„) < r^}

4-2 Lower semicontinuity

185

(d(-, •) the distance function of X) forms a countable base for the topology. If the first countability axiom is satisfied, topological notions usually admit sequential characterizations. For example, if (#n)n£N C X is a sequence in a topological space X satisfying the first axiom of countability, then any accumulation point of (xn) (i.e. any x G X with the property that for every neighbourhood U of x and any m G N, there exists n > m with xn G U) can be obtained as the limit of some subsequence of (xn). Although we shall often employ weak topologies which typically do not satisfy the first axiom of countability, for our purposes it will usually be sufficient to use sequential versions of topological properties. For that reason, we shall define our topological notions in sequential terms, without adding the word 'sequentially'. Definition 4.2.1. Let X be a topological space. A function F : X —• 1 := R U { i o o } is called lower semicontinuous (Isc) at x if F(x) < liminfF(x n ) n—+oo

for any sequence (xn)nen C X converging to x. F is called lower semicontinuous if it is Isc at every x G X. The following properties are immediate: Lemma 4.2.1. (i) IfF:X-+Ris Isc, A > 0, then \F is Isc. (ii) If F,G : X ~+ R are Isc, and if their sum F + G is well defined {i.e. there is no x G X for which one of the values F(x),G(x) is +oo and the other one is —oo), then F + G is also Isc. (iii) For F, G : X -+ R Isc, inf (F, G) is also Isc. (iv) If (Fi)i£i is a family of Isc functions, then s\xpieI Fj is also Isc. Examples. (1) Any continuous function is lower semicontinuous. (2) If X satisfies the first axiom of countability, then A C X is open if and only if its characteristic function \A is Isc. Definition 4.2.2. (i) Let X be a normed space, with norm ||-||. F : X —• R is weakly proper, if for every sequence (x n ) n € N C X with \\xn\\ —• oo we have F(xn) —• oo for n —• oo.

186

Direct methods

(ii) Let X be a topological space. F : X —• E is coercive if every sequence (xn) C X with F(xn) < constant (independent of n) has an accumulation point We now formulate the following general existence theorem for minirnizers: Theorem 4.2.1. Let X be a separable reflexive Banach space, F : X —• R weakly proper and lower semicontinuous w.r.t. weak convergence. Then there exists a minimizer XQ for F, i.e. F(x0) = inf F(x) Proof. Let (xn)n^

(> -oo).

be a minimizing sequence for F , i.e. lim F(xn) = inf F(x).

Since F is weakly proper, ||x n || is bounded. Since X is reflexive, after selection of a subsequence, xn converges weakly to some xo G X by Corollary 2.2.1. By lower semicontinuity of F , F(x 0 ) < lim F(xn) = inf F(x), n—>oo

x£X

and since XQ G X, we must have in fact equality. Also, since F assumes only finite values by assumption, this implies that inf F(x) > —oo. xex q.e.d. Remark 4-2.1. The argument of the preceding proof also shows that in a separable reflexive Banach space, a weakly proper functional is coercive w.r.t. the weak topology. Lower semicontinuity w.r.t. weak convergence is a rather strong property, in fact much stronger than lower semicontinuity w.r.t. to the Banach space topology of X. Fortunately, there exists a general class of functionals, namely the convex ones for which the latter property implies the former. Definition 4.2.3. Let V be a convex subset of a vector space; F : V —• R is called convex if for any x,y £V, 0
t)F(y)

(convexity of V means that tx + (1 — i)y G V whenever x,y G V, 0 < * < 1).

4-3 Existence of minimizers

187

Lemma 4.2.2. Let V be a convex subset of a separable reflexive Banach space, F : V —» R convex and lower semicontinuous. Then F is also lower semicontinuous w.r.t. weak convergence. Proof Let (xn)n£N C V converge weakly to x G V. We may assume that F(xn) converges to some K G R. By Theorem 2.2.4, for every m G N and every e > 0, we may find a convex combination N

N

nXn

Vm *•= Yl ^

n

(^ > °» y ^ n = 1)

n—m

n=m

with \\Vm -x\\

< e.

Since F is convex, N

F(ym) < J2 XnF(xn).

(4.2.1)

n=ra

Given e > 0, we choose m = m(e) G N so large that for all n > m, F(xn)
Since F is lower semicontinuous F(x) < liminf F(ym) < limsupF(y m ) < K = lim F ( x n ) . m->oo

m->oo

This shows weak lower semicontinuity of F. q.e.d.

4.3 The existence of minimizers for convex variational problems We return to the concrete variational problem discussed in Section 4.1 and begin with: Lemma 4.3.1. Let Q c Rd be open, f : ft x Rd -+ R, with /(.,v) measurable for all v G R d , / ( # , •) continuous for all x G fi, and /(a,v) > - a ( x ) + & M P

188

Direct methods

for almost all x G ft, and all v G Rd, with a G ^(ft), 77ien $(t;) := /

Jn

b G R, p > 1.

f(x,v(x))dx

is a lower semicontinuous functional on Lp(ft),

$ : Lp(ft) —> E U {oo}.

Proof. Since / is continuous in v, f(x,v(x)) is a measurable function, and so $ is well-defined on Lp(ft), by Theorem 1.1.2. Suppose (vn)neN converges to v in Lp(ft). Then a subsequence converges pointwise almost everywhere to v by Lemma 3.1.3. We shall denote this subsequence again by (^n)> noting that the subsequent arguments may also be applied to any remaining subsequence. Since / is continuous in v (actually, it would suffice to have / lower semicontinuous in v), we have / ( x , v(x)) - b \v(x)\p < lim inf (/(x, vn(x)) - b

\vn(x)\p).

n—•»oo

Because of the lower bound / ( x , vn(x)) - b \vn(x)\p > -a(x) with a G I/ 1 (fi), we may apply the Theorem 1.2.2 of Fatou to conclude / (f(x,v(x))

Jn

- b\v(x)\p)dx

(f(x,vn(x))-b\vn(x)\p)dx.

< liminf / n

^°° Jn

Since vn converges to v in L p (fi), / b\v(x)\pdx

= lim / 6|v n (x)| p dx,

and we conclude lower semicontinuity, namely / f(x,v(x))dx Jn

< liminf / n ^ ° ° Jn

f(x,vn(x))dx. q.e.d.

L e m m a 4.3.2. Under the assumptions of Lemma 4-3-1, assume that /(#,-) is a convex function on R d for every x G ft. Then $(v) := fQf(x,v(x))dx defines a convex functional on Lp(ft).

4.3 Existence of minimizers

189

Proof. Let v,w € LP{Q), 0 < t < 1. Then $(tt; + (1 - t)tu) = / f(x,tv{x)

+ (1 - t)u;(a:))da:

< / {tf{x,v{x)) +

{l-t)f(x,w(x))}dx

JQ

by the convexity of / = t$(v) + (1 -

t)$(w). q.e.d.

We may now obtain a general existence result for the rninimizer of a convex variational problem. Theorem 4.3.1. Let Q C Rd be open, and suppose f : Q x Rd —• R (i) /(•,!>) w measurable for all v G Md. (ii) / ( # , •) 25 convex for all x Eft. (iii) f(x,v) > —a(x) + 6|t;|p /or almost all x E ft, all v £ Rd, with ae ^(ft), b>0,p> l. Let g G H1*^),

and let A := g + H^P(Q). F(u) := /

Then

f(x,Du(x))dx

JQ

assumes its infimum on A, i.e. there exists uo G A with F(u0) = inf F(u). ueA Proof. By Lemma 4.3.1, F is lower semicontinuous w.r.t. H1,p(ft) convergencef, and by Lemma 4.2.2, F then is also lower semicontinuous w.r.t. weak H1,p(ft) convergence, since H 1>P(Q) is separable and reflexive for p > 1 (see Theorems 3.3.1 and 3.3.3). Let (un)neN be a minimizing sequence in A, i.e. lim F(un) = inf F(u). n—>oo

u£A

Since / \Dun\p < \F(un) o

JQ

+ lf a(x)dx, b JQ

(Dun)n£N is bounded in Lp(ft), hence (wn)n£N C g+H0,p(ft) is bounded in Hl*(Sl) by the Poincare inequality (see Theorem 3.4.2). Since H^p(ft) t Note that convex functions on Rd are continuous.

Direct methods

190

is a separable reflexive Banach space, by Theorem 3.3.5, after selection of a subsequence, (t/n)n€N converges weakly to some uo E A (A is closed under weak convergence, Theorem 3.3.4). Since F is convex by Lemma 4.3.2 and lower semicontinuous by Lemma 4.3.1, it is also lower semicontinuous w.r.t. weak HliP(ft) convergence by Lemma 4.4.2. Therefore F(u0) < lim F(un) = inf F(u), and since uo € A, we must have equality. q.e.d. Remark 4.3.1. The condition u 6 g+HQ,P(Q), i.e. u-g € HQ,P(Q), is a (generalized) Dirichlet boundary condition. It means that u = g on dft in the sense of Sobolev spaces.

4.4 Convex functionate on Hilbert spaces and Moreau-Yosida approximation In this section, we develop a more abstract method for showing the existence of minimizers of variational problems. It has the advantages that it does not need the concept of weak convergence and that it provides a constructive approach for finding the minimizer. In order to concentrate on the essential aspects, we shall only treat a special situation. Definition 4.4.1. Let X be a metric space with metric d(-,-), and let F : X —• E U {oo} be a functional For X > 0, we define the MoreauYosida approximation Fx of F as F\x)

:= mf(\F(y)

+ d2(x, y))

(4.4.1)

y€X

for x € X. Remark 4-4-1' This is different from the definition in Section 5.1 where we shall take d(x,y) instead of d2(x,y). Here, one might take da(x,y) for any exponent a > 1. For our present purposes, it is most convenient to work with a = 2. We now let if be a Hilbert space with scalar product (•, •) and norm ||-|| and induced metric d(x,y) = ||ar —y||. Let D(F) C if, and let F : D(F) -> E be a functional. We say that F is densely defined if D(F)

4-4 Convex Junctionals

191

is dense in H. For x ^ D(F), we put F(x) = oo. We say that F is convex if whenever 7 : [0,1] —> H is a straight line segment, then for 0 < t < 1 F( 7 (t)) < tF( 7 (0)) + (1 - t ) F ( 7 ( l ) ) .

(4.4.2)

In particular, if 7(0), 7(1) G £>(F), then also 7(f) G D(F) for 0 < t < 1. Lemma 4.4.1. Let F : i f —• E U {00} 6e convex, bounded from below, and lower semicontinuous. Then for every x G H and X > 0, there exists a unique yx =: J\x) with Fx{x) = XF{yx) + d 2 (x, yA)

(4.4.3)

Proof. We have to show that the infimum in (4.4.1) is realized by a unique yx. Uniqueness: Let y^y^ be solutions of (4.4.3), and let 2/o = ^(2/iA + 2/2) be their mean value. By convexity of F F(ti)<\{F{v?)+F{ti)),

(4.4.4)

and by Euclidean geometry, if yx ^ y$, we have

l k - % X H 2 < ^ ( l k - ^ i r + ||x-^|| 2 ),

(4.4.5)

hence \F(yx)

+ \\x-yx\\2<\F(yx) = ^F(yx)

+ +

\\x-yx\\2 \\x-yx\\2,

contradicting the minimizing property of yx and y£. Thus, we must have Vi = 2/2» P r o v m S uniqueness. Existence: (4.4.5) may be refined as follows: For 2/1,2/2 € # and 2/o := 2(2/1+2/2) we have for any x £ H \\* - J/oll2 = 5 (||x - j/xll2 + ||x - y 2 || 2 ) - \ l b ! - y2\\2 •

(4.4.6)

192

Direct methods

We now let (yn)neN be a minimizing sequence, i.e. XF(yn) + \\x - 2/„||2 -

inf (xF(y) + ||x - y|| 2 ) = : « A .

(4.4.7)

We claim that (yn) is a Cauchy sequence. For /, k G N, we put Vi,k '•= 9 ^ f c

+

^)'

Using the convexity of F as in (4.4.4) and (4.4.6), we obtain \F(yktl) < \ (\F(yk)

+ \\x - yk\\2)

+

\\x-yktl\\2 + \\x - » | | 2 ) - \ \\Vk

+ \ (\F(yi)

l|2

(4.4.8) By definition of K\ (see (4.4.7)), the left hand side of (4.4.8) cannot be smaller than «A, and so we conclude that

llifc-wll 2 -o as fc, I —> oo, establishing the Cauchy property. Since the norm is continuous and F is assumed to be lower semicontinuous, the limit yx of (2/n)n€N then solves (4.4.3). q.e.d. Lemma 4.4.2. Let F and yx = Jx(x) in the closure of D(F). Then

be as in Lemma 4.4.I. Let x be

x = lim Jx(x).

(4.4.9)

A—0

Proof. Since x is in the closure of D(F), for every 6 > 0, we may find x6 £ B(x, 6):={yeH:\\x~

y\\ < 6}

with F(xs) < oc. Then lim (\F(x6)

4- ||x - x^|| 2 ) < &2

and therefore lim sup K\ < 0

(4.4.10)

A-+0

(see (4.4.7) for the definition of K,\). Let us now assume that there exists a sequence An —> 0 for n —• 00 with \\x-yXn\\2

>a>0

for all n.

(4.4.11)

4-4 Convex junctionals

193

Then from (4.4.10) limsup(A n F(i/ A n ) 4 - | | x - i / A n | | 2 ) < 0,

(4.4.12)

hence F (yXn) —•-oo

asn^oo.

(4.4.13)

(4.4.12) and (4.4.13) imply F(yx) + \\x - y1^2
+ \\x-yXn\\2

-+-oo

as n -+ oo

which is impossible. Thus, (4.4.11) cannot hold, and (4.4.9) follows. q.e.d. T h e o r e m 4.4.1. Let F : H —• MU{oo} 6e convex, bounded from below, and lower semicontinuous, and F ^ oo. For x € M, we let yx = Jx(x) as in Lemma 4-4-1- If (yXn)nen is bounded for some sequence Xn —> oo, then (yx)\>o converges to a minimizer of F as X —• oo. Proof Since (yXn)neN is bounded and since yXn minimizes ~\\x-y\\2,

F(y) + we obtain F(yx")^

inf F(y) y€H

so that (yXn)neN is a minimizing sequence for F. We now claim that \\*-yx\\2 is monotonically increasing in A. Indeed, let 0 < ii\ < fa- Then by definition of t/Ml W

2

||ar — y»\\2 > F(y^)

) H Mi

+ — ||x - y^\\2 , Mi

hence W )

H

1

,,

-i.2 ^

D/.un

,

1

ii

..uiii2

||ar — y " a | f > F ( ^ ) + — ||x - i/"1

M2

M2

+

(^-^)(^-^lH2-||a:-^ll2)

This is compatible with the minimizing property of y^2 only if \x-y^\\2>\\x-y^\\2

194

Direct methods

and monotonicity follows. This monotonicity then implies that All2

II

II

^ II

is bounded independently of A since it is assumed to be bounded for the sequence An —> oo. We next claim that F(yx) monotonically decreases towards

miF(y). Indeed, from the definition of t/A, F(yx) =

inf

{y.\\x-y\\<\\x-yx\\}

F(y),

and therefore yx has to decrease since | \x — yx 11 increases. The limit has to be inf y € # F(y) since this is so for the subsequence (yAn)n€N- We now claim that (yx)\>o satisfies the Cauchy property, i.e. for every e > 0, there exists Ao > 0 such that for all A,/u > A0 \\yx-y»\\2<e. For that purpose, we choose Ao so large that for A, /x > Ao \x - yx\\2 - \\x - y»

< \

(4-4.14)

which is possible by the preceding monotonicity and boundedness results. We may also assume F(yx) > F(y").

(4.4.15)

We let

y^:=l(y^+y"). Then from the convexity of F , (4.4.15), and (4.4.6),

*V'") + ^ | | * - 2 / A ' 1 | 2

)+1-(\\X-y>\\2

+

by (4.4.14).

l-\\\y>-y»\

4-5 Euler-Lagrange equations

195

This, however, is compatible with the minimizing property of yx only if \\yx

-y»\\2

< <•

x

Thus {y )\>o satisfies the Cauchy property for A —• oo, and it therefore converges to some y £ H. y then minimizes F , because F(yx) decreases towards inf y€ /f F(y) for A —• oo, and F is lower semicontinuous. q.e.d. The preceding reasoning is adapted from J. Jost, Convex functionals and generalized harmonic maps between metric spaces. Comment. Math. Helv. 70 (1995), 659-673. For a more general construction, see J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Birkhauser, Basel, 1997, pp. 61-4. In particular, the method also works in uniformly convex Banach spaces. General references for Moreau-Yosida approximation are the books of Attouch and dal Maso quoted in Chapter 6. Theorem 4.4.1 yields an alternative proof of Theorem 4.3.1 in case p = 2. Namely, Lemma 4.3.1 implies the lower semicontinuity, Lemma 4.3.2 the convexity of the functional, and the Poincare inequality the boundedness of any minimizing sequence, as described in the proof of Theorem 4.3.1. The present proof, however, does not need the concept of weak convergence. As mentioned, the method extends to uniformly convex Banach spaces, and thus can handle also arbitrary values of p > 1 (see Remark 3.1.2).

4.5 The Euler-Lagrange equations and regularity questions In this section, we return to the variational problems considered in Sections 4.1 and 4.3; we consider variational integrals of the form *( M ) : = f / ( x , u(x), Du(x))dx, Jn

for u £

Hx'p(n)

on a bounded, open subset ft of R d , and we make the following assumptions o n / : f t x R x R ( i - > R = RU {±oo}: (i) /(•, u, v) is measurable for all u £ R, v £ Rd. (ii) / ( # , •, •) is differentiable for almost all x £ ft. (iii) \f(x,u,v)\ < Co -f ci \u\p -f c2 \v\p, c 0 ,ci,c 2 constants, for almost all x G f i , and all u £ R, v £ R d . Condition (iii) implies that $(u) is finite for u £ Hl,p(fl), since O is bounded. (If fi is unbounded, this still holds provided c 0 = 0.) In the

196

Direct methods

preceding section, we have obtained some results on the existence of a minimizer for # in the class g + ifQ ,p (fi), for given g E HlyP(fl). In the present section, we wish to characterize such minimizers by necessary conditions. These conditions will assume the form of differential equations. In fact, these differential equations will hold for arbitrary critical points of # (as specified in the assumptions of our subsequent results), and not only for minimizers. Theorem 4.5.1. Let f satisfy (in addition to (i)-(iii)) (iv)

d£ (x,u,v) du

d

f(

^ < c3 -f c4 |u| p + c5 |t>|p ,

C3, C4, C5 constants, for almost all x E fi, and all u GR, V E M.d. Let u be a minimizer for # in the class g + H0,p(Q) (g E H1,p(ft) We then have for all (p E CQ°(Q)

given).

= 0.

(4.5.1)

Proof. Since u is a minimizer for $ in g + if 0 ' p (O), *(u) < *(u 4- *
(4.5.2)

We have $(w + t ^ ) = / Jn

f(x,u(x)+tip(x),Du(x)+tDip(x))dx.

By (ii), (iii), (iv), we may apply Corollary 1.2.2 to conclude that $(u+tip) is differentiate w.r.t.
dt

$(u + tip)

— \

l—(x,u(x)+tip(x),Du(x)+tDip(x))ip(x) dip(x)"

+ E dii & <x)+^(x)' ^ ^ ) + " W ) -f§-}dxt=i

(4-5-3)

4.5 Euler-Lagrange equations

197

Furthermore, (4.5.2) implies

Equations (4.5.3) and (4.5.4) imply (4.5.1). q.e.d. Remark 4-5.1. From the preceding proof, it is clear that we do not need to assume that u is a minimizer for 3>. If suffices that u is a critical point for # in the sense that ~ $ ( u + tip)\t=0 for all if € C5°(fi). (4.5.5) at Corollary 4.5.1. Suppose that f satisfies (i)-(iv), and in addition, f € C2. If u E C2(Q) minimizes $ in the class g + HQ}P(SI) (or, more generally, satisfies (4-5.5)), then \r*

d2

f

(t

< x r. .xx

S 5^ *'*

(x)

'^

2

( x ) )d u

&^

Js

d2f

,

,

x

^ ,

XN

3u

+E ^ ( ^ " W . ^ ) ) - |(^,«w,^(x))=o. t=i

(4.5.6) Definition 4.5.1. Equation (4-5.6) is called the Euler-Lagrange equation for $ . Proof (Corollary 4-5.1). By the differentiability assumptions made, we may integrate (4.5.1) by parts to obtain

t= l

t=l

From Lemma 3.2.3 (applied to supp
Direct methods

198

Equations (4.5.6) constitutes a quasilinear partial differential equation of second order for u. Many such partial differential equations arise as Euler-Lagrange equations of variational problems. Therefore, if one wants to solve such an equation, one might try to find a minimizer of the associated variational problem. However, the existence theory for minimizers as described in Section 4.3 naturally yields an element u of the Sobolev space HQ'P(Q), whereas in Corollary 4.5.1 it is required that u be of class C 2 (Q). Thus, there exists a gap, since in general elements of H0,P(Q) are not of class C2. It is the task of regularity theory to bridge this gap, i.e. to show that under suitable assumptions on / , any minimizer of # is smooth, and specifically here of class C2. The theory of partial differential equations indicates that such a result does not hold without additional assumptions on / , like an ellipticity assumption, meaning that the matrix (at,7'(£))i,j=i,...,d with coefficients atJ(x) = dvi$vj (x, u(x), Du(x)) is positive definite. Indeed, examples show that without such an assumption, in general one does not get smoothness of minimizers. On the positive side, however, we do have de Giorgi's and Nash's: Theorem 4.5.2. Let Q, be open and bounded inRd, of class C°°, with

f : Q, x Rd —» R be

0) AH 2 )
for all x G fi, u E l , t>, £ G Mrf, with constants A > 0, A < oc,

oil &<*«> Let u / n

< M ( l + \v\)

G g + HQ' (ft)

f(x,Du(x))dx

for a constant M < oo.

be a bounded minimizer

of F(u)

:=

(g G H1,P(Q) given). Then u is smooth in fi (u G

C°°(ft)).

The proof of the theorem of de Giorgi and Nash is too long to be presented here. We refer to M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhauser, Basel, 1993, pp. 76-99 and

4-5 Euler-Lagrange equations

199

J. Jost, Partielle Differentialgleichungen, Springer, Berlin, 1998 where a detailed proof is given. Of course, there also exist extensions of this result to more general integrands of the form / ( # , u, v). We refer the interested reader to O. Ladyzhenskaya, N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968 (translated from the Russian), Chapters IV-VI. One remark is in order here: Since Sobolev functions are only equivalence classes of functions (in the sense specified at the beginning of Section 3.1), a more precise version of Theorem 4.5.2 is: Under the stated assumptions, the equivalence class of u contains a function of class C°°. This point, however, usually is assumed to be implicitly understood in statements of regularity theorems. In order to display at least one regularity result, however, we consider a particular example: For a bounded, open fi C Md, g £ H1,2(Q), we wish to minimize Dirichlet's integral D(u) := f \Du(x)\2dx

(4.5.8)

Jn in the class g + H0,P(Q). By Theorem 4.3.1, a minimizer u exists, and by Theorem 4.5.1, it satisfies / Du(x) • Dip(x)dx = 0

foran>eC£°(n)

(4.5.9)

JQ

(here Du(x) • Dip(x) := Yli=i Diu(x)Di
( A : = | : ^

(A is called Laplace operator.) by Corollary 4.5.1, i.e. it is harmonic. This is the famous Dirichlet principle: obtain a harmonic function u in fi with boundary values g by minimizing the Dirichlet integral among all functions with those boundary values. In order to justify Dirichlet's principle it thus remains to show that any solution of (4.5.9) is of class C2. Actually, one can show more, namely, u e C°° (in fact, u is even real analytic in Q but this will not be demonstrated here), and at the same time weaken the assumption. Namely, we have:

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Direct methods

Theorem 4.5.3 (Weyl's lemma). Let u € L 1 (fi) satisfy /

u(x)A
for all v € C§°(fi).

(4.5.10)

JQ /n

ThenueC°°{n). Remark 4-5.2. (1) Clearly, (4.5.9) implies (4.5.10) by definition of Du. (2) The remark made after Theorem 4.5.2 again applies. Proof {Theorem 4-5.3). We consider the mollifications with a rotationally symmetric p (and we express this by writing p as a function of |x|)

Mx)

JaQ{^ir)u{v)dv

"h

as in Section 3.2. Given tp € Co°(f2), we restrict h to be smaller than dist(supp
=

J2

e[—r—J

= / u(x)A
u(y)dyA(p(x)dx (4.5.11)

JQ

using Fubini's theorem. q.e.d. Remark 4-5.3. We have also used the fact that A commutes with mollification, i.e. (A
(4.5.12)

For this, one needs that Q is a function of |x| only, i.e. rotationally symmetric. Also, this point needs the rotational invariance of the Laplace operator A. Therefore, the present proof does not generalize to other variational problems. After this interruption, we return to (4.5.11) and conclude that /

uh(x)A(p(x)dx

=0

(4.5.13)

JQ

by applying (4.5.10) to
=0

4-5 Euler-Lagrange equations

201

in Qh := {% £ fi | dist(x,dft) > /i}. Also /

\uh{y)\dy

JQh

< J \u(x)\dx

(4.5.14)

JQ

I f f\x — y\\ by Fubini's theorem, using - j / f -—-—- 1 dy = 1 by (3.2.3) < oc

since u £ L 1 ^ ) .

Therefore, the functions it^ are uniformly bounded in L 1 . We now need Lemma 4.5.1. Let f € C 2 (ft) 6e harmonic, i.e. Af(x)

= 0

m n.

TTien / satisfies the mean value property, i.e. for every ball B(xo, r) C 0,, /(so) = ~^1 ^d

f(*)dx

I

r

= TTT^T

I

dwdrd

JB(x0,r)

f{x)da{x) JdB(x0,r)

where Wd is the volume of the unit ball in E d . Proof. For 0 < g < r 0= /

Af{x)dx

JB(xn,p) 'B(xo,Q)

I

dv

JdB(x0,Q)

-(x)da(x),

where v denotes the exterior normal of B(xo7 Q)

J

-J-(y +

d l

Qu)Q

- dw

i) OQ

in polar coordinates UJ = ^z^ ^

Q

j

Qd~l%- / ^Q

4-1

d

dg „d-id

JdB(0,l) JdB(0,l)

gl-d

f(v+

f JdB(0,l)

&»)*» f(x)d
d 1 i dg \\cLj (kjddeQ-d 1 JddB(xo,Q)

I

(4.5.15)

Direct methods

202 Thus, -

TTY /

f(x)da(x)

is constant in Q, and since its limit for g —• 0 is f{xo) as / is continuous, it has to coincide with f(xo) for all 0 < g < r. Since (-T-VT / f{x)da(x) I gd~ldg, d 1 \^dQ JdB(X0,6) J

-i-r / f(x)dx = ~ r rd W JB(X0,e) h

UdTad

the first inequality in (4.5.15) also follows. q.e.d. We return to the proof of Theorem 4.5.3: Since Uh is harmonic, it satisfies the mean value properties of Lemma 4.5.1. Since the family Uh is bounded in L 1 , Uh{xo) =

J /

UdTa

uh(x)dx

JB{xv,r)

is bounded for fixed r with B(xo,r) C fi^. Therefore, the Uh are uniformly bounded in flhQ for 0 < /i < 4&. Furthermore, from (4.5.15) \uh(xi) - uh(x2)\ < j — ( - ) / dWd

XT X

I

J

'

B(x1,r)\B(x2,r) UB(x2,r)\B(x1,r)

< c(r) |xi - X2I

\uh{x)\dx (4.5.16)

for some constant depending on r, if B(a;i, r), B(^2, r) C O^0. Therefore, the gradient of Uh is also uniformly bounded on Qh0 • Likewise, derivatives of Uh of all orders can be uniformly bounded on H^0 (0 < h < ^ - ) , either by repeating the same procedure, or by observing that together with Uh, also all derivatives of Uh are harmonic so that (4.5.16) can be iteratively applied to all derivatives in order to convert a bound on some derivative into a bound for a higher one. Therefore, a subsequence of Uh converges towards some smooth function 1;, together with all its derivatives, as h —• 0. Since all the Uh satisfy Auh = 0 so then does v: Av = 0 in ft. Since on the other hand Uh converges to u in Ll(ft) by Theorem 3.2.1, the two limits have to coincide (e.g. by Lemma 3.1.3). Therefore u = v, and consequently u is smooth and harmonic. q.e.d. As an application, we consider the following

Exercises

203

Example 4-5.1. Let a : R —• R be Lipschitz continuous with 0 < A < a(y) < A < oo

for all y e R.

d

Let ft C R be open. We want to minimize

r F(u):=

d

/ y2a(u(x))Diu(x)Diu(x)dx

in the class A := # + # d ' P ( n ) , w i t n S i v e n 9 e H1*^). Lindelof theorem, the ordinary differential equation dU

dv

(4.5.17) By the Picard-

X

y/a(u)

(4.5.18)

admits a solution u(v) of class C 1 ' 1 . We then have , sdu du

,

Since ^ > A" 2 > 0, the inverse function v(u) exists and is of class C ' as well, and we have by (4.5.19) and a chain rule for Sobolev functions that easily follows from the chain rule for differentiable functions by an approximation argument that d

^

d

a(u)DiuDiU = ] P DivDiV.

Therefore, (4.5.17) is transformed into Dirichlet's integral F{u) = D{v). Since the latter admits a smooth minimizer, the original problem (4.5.17) then admits a minimizer that is of class C 1 ' 1 in ft.

Exercises 4.1 4.2

in (iv) of TheWeaken the growth assumption required for orem 4.5.1. Hint: Use the Sobolev Embedding Theorem. Compute the Euler-Lagrange equations for the variational integral A(u) := / y/l + \Du{x)2\dx. Jn {A(u) represents the volume of the graph of u over ft. Critical points are minimal hypersurfaces that can be represented as graphs over ft.)

Direct methods

204 4.3

Compute the Euler-Lagrange equations for E(u) := / gij(x)Diu(x)Dju(x)

Jn

(detgij(x))^

dx,

where (gl-i(x))i,j=ii...id is the inverse matrix of ((fo(£))t,j=i,...,d. Assume that (<7ij(x))i,j=i,...,d is positive definite for all x G ft. Show that for given g G iif1 , 2 (0), there exists a unique minimizer of E among all u G Hla(9) with u- g G if 1 ' 2 (f2). (Minimizers for E are harmonic functions w.r.t. the metric gij(x).)

5 Nonconvex functionals. Relaxation

5.1 Nonlower semicontinuous functionals and relaxation From Section 4.3, we recall the following Theorem 5.1.1. Let Q C Rd be open, I < p < oo, / : fi x i d ^ measurable and suppose:

1

(i) For almost all x £ Q, f(x, •) is convex on Rd (ii) There exist a G L1^), b G R with f(x,v)

> -a{x) + b\v\p

for almost all x € Q and all v G Md. Then F(u):=

[ Jn

f{x,Du(x))dx

is Isc and convex on H1,p(i}) equipped with its weak topology and assumes its infimum in the class of all f G HltP(£l) with f — g G H0,P(Q) for some given g G H1,P(Q). Here, (ii) is just a coercivity condition ensuring that a minimizing sequence stays bounded w.r.t. the HliP -norm (w.l.o.g. F ^ oo) (i) implies that F is lsc, w.r.t. the norm topology of Hx'p, and the convexity then implies that F is also lsc w.r.t. the weak H1,p topology. Since bounded sequences in H1,p have weakly convergent subsequences, any minimizing sequence has a convergent subsequence, and a limit of such a subsequence then minimizes F by lower semicontinuity. Not all functionals that one wishes to consider in the calculus of variations are convex, however. As a motivation for what follows, we consider 205

206

Nonconvex functionals.

Relaxation

the following example of Bolza: n = (0,l)cR, F(u) = /

u:(0,l)->R,

u(0)=0

= u(l)

(u2(x) + (u'(x)2 - l ) 2 ) dx.

We claim that : u G H*A((0,1))}

M{F(u)

= 0.

(5.1.1)

For the proof, we consider 'sawtooth'-functions: Let n G N, i un{x) := <

2*

r

£

f 0r

n , t+ 1 -x H n

(i = 0 , l , . . . , n - l ) . u n is contained in Hhoo((0,1))

2z + 1 < X <

2n 2n 2z 4-1 ^ 2i 4- 2 for — — < x < —-— 2n 2n

f

C # M ( ( 0 , 1 ) ) and satisfies:

For all x e (0,1)

0 < un{x) < —, zn

t i n ( 0 ) = 0 = ti„(l), for almost all x G (0,1)

\u'n(x)\ = 1.

(5.1.2) (5.1.3) (5.1.4)

Consequently lim F(un)

= 0.

n—+CXD

Since F(u) is nonnegative for every w, (5.1.1) follows. The inflmum of F therefore cannot be realized by any i ^ ' function, because if we had F(u) = 0, then u(x) = 0 for almost all x G (0,1) and |w;(x)| = 1 for almost all x G (0,1), and these two conditions are not compatible. (In fact, since d = 1 here, any u G # Q ' 4 ( ( 0 , 1)) is absolutely continuous, and so u = 0 if u{x) = 0 a.e., hence u is differentiable and u' = 0. (More generally, any Sobolev function that is constant on some set A has a representative u whose derivative Du vanishes on A.) We have thus shown that the problem F(u) -> min in # 0 M (fi) does not have a solution.

5.1 Nonlower semicontinuous Junctionals and relaxation

207

We observe that our minimizing sequence (un) converges to zero rl,4 weakly in H0y , by (5.1.2) and / u'n(x) 0 Jo Jo However, F(0) = 1 > 0 = lim

for all

F(un).

n—»oo

Therefore, F is not lsc w.r.t. weak H1 ^-convergence although the integrand is continuous in u'. As we shall see this results from the lack of convexity of the integrand. We also observe that any sequence of sawtooth functions un, i.e. satisfying

1^1 = 1 a.e. 2

that converges to 0 in L is a minimizing sequence for F. Remark 5.1.1. Functional of the type of our example often arise in optimal control theory as described in Section 5.2 of Part I. For example, one considers problems of the following type rT

f(t, u(t),a(t))dt / /o Jo

-+ min

(5.1.5)

under the side conditions u(0) = u0,

u(T) = uT

u'(t)=g(tMt),°(t))

(5.1.6) (5.L7)

with given functions / and g. u is called a state variable, a a control variable. This means that one assumes that u describes the state of some system evolving in time t whose derivative or rate of change can be controlled through a parameter a. The aim then is to choose a in such a manner that the functional, often considered as 'cost function', is minimized. Thus, one needs to find some equation a(t) =
208

Nonconvex functionals.

Relaxation

and this leads to minimizing functionals of the type / Jo

f{t,u{t),u\t))dt.

Expressions of the type (u'(i)2 — l ) 2 can occur in many technical examples, like boats sailing against the wind. Faced with a problem that one cannot solve, one may contemplate several options: One could try to modify the problem, or one might generalize the concept of a solution, or both. We shall discuss several such strategies. We first modify the problem via relaxation. This is an important method in the calculus of variations, and we therefore discuss it in some generality. Definition 5.1.1. Let X be a topological space, F : X —• E. We define the lower semicontinuous envelope or relaxed function sc~F of F as follows: (sc~F)(x) := sup {$(#) : $ : X —• R is lower semicontinuous with $(y) < F(y) for allyeX} Lemma 5.1.1. sc~F is the largest Isc function on X that is < F everywhere. In particular, F is lower semicontinuous if and only if F = sc~F. Proof. sc~F is lsc as a supremum of lsc functions, see Lemma 4.2.1 (iv). Obviously, sc~F < F , and for all lsc $ with # < F , we have $ < sc~F by definition of sc~ F. q.e.d. Theorem 5.1.2. Let X be a topological space, F : X —> R a function. Then every accumulation point of a minimizing sequence for F is a minimum point for sc~F. Consequently, if F is coercive, then sc~F assumes its minimum, and min sc~ F = inf F. x x

5.1 Nonlower semicontinuous junctionals and relaxation

209

Proof. Let (#n)neN C X be a minimizing sequence for F with accumulation point xo- Then (sc~F)(x0)

< liminf(sc~F)(x n )

< liminf F(xn)

by lower semicontinuity of sc F (see Lemma 5.1.1)

since sc~F < F

n—>oo

= inf F(y)

since (x n ) isaminlmizing sequence tor F.

(5.1.8)

On the other hand, the constant function $ ( x ) = inf F(y) is lsc and < F , hence by Lemma 5.1.1 for every x £ X inf F(t/) < (sc-F)(x).

(5.1.9)

From (5.1.8) and (5.1.9) we conclude (sc~F)(xQ) = inf F(y) = mm(sc~F)(x). y£X

(5.1.10)

x£X

This implies the first claim. If F is coercive, then every minimizing sequence has an accumulation point, and the second claim also follows. q.e.d. What does Theorem 5.1.2 tell us for our example? It simply says that if we cannot minimize our original functional F due to its lack of lower semicontinuity, we then minimize another functional instead, one that is lower semicontinuous and as close as possible to F. Theorem 5.1.2 then says that limits (or more generally, accumulation points) of minimizing sequences for F do not minimize F , but the relaxed functional sc~F. Since sc~F is the largest lsc functional < F by Lemma 5.1.1 that is the best one can hope for. It then remains the task to determine the relaxed functional of some given F. Before proceeding to do so for our example, let us relax ourselves a little and derive some easy consequences of the definition of the relaxed functional and consider some easier examples first. Lemma 5.1.2. Let X satisfy the first axiom of countability. Then sc~F is the relaxed function for F : X —> ft iff the following two conditions are satisfied:

210

Nonconvex junctionals. (i) whenever XJI

Relaxation

•X (sc"F)(x)

< liminf F ( x n ) n—»oo

(ii) for every x £ X, there exists a sequence xn —• x with (sc"F)(x)

> lim F(xn) n—>oo

Proof. We claim that, since X satisfies the first axiom of countability, (sc~F)(x) = inf{liminf F(xn) : xn -» x in X } .

(5.1.11)

We denote the right hand side of (5.1.11) by F~(x). Then F~~ is lsc. In order to verify this, we have to check liminf (inf {liminf F (y^n) : yvn —• yv}) > inf {liminf F(xn) : xn -> x} (5.1.12) whenever yv —> x. Indeed, otherwise, for some <5 > 0, we would find some diagonal sequence yv,nv ~¥ % as v —» oo with ^ (jfi/.u,,) < inf {lim inf F ( £ n ) : xn -» a:} - 5 which is impossible. Thus, F~~ is sequentially lsc, hence lsc, because X is assumed to satisfy the first axiom of countability. Also, F ~ < F , and for every lsc $ < F , we have for xn —> £ $(#) < liminf # ( x n ) < liminf F ( x n ) , n—••oo

n—»oo

and hence *(z) <

F~{x).

Thus, F ~ is the largest lsc functional < F , and (5.1.11) follows from Lemma 5.1.1. It is then easy to see (and left as an exercise) that F~(x) satisfies and is characterized by the properties (i) and (ii). q.e.d. Example 5.1.1. Let X be a topological space, A c l a subset. The indicator function %A is defined by

t \ := <J°

IA{X)

if:

'xe A ' x £ A. { oo if:

We then have sc iA -

where A is the closure of A in X.

iA,

5.1 Nonlower semicontinuous Junctionals and relaxation

211

The characteristic function \A is defined by

^

, , f 1 if x €A (x):= iA.

i o if*.

Then sc XA = XA where A is the complement of X \ A. Example 5.1,2. Let fl C E d be open, 1 < p < oo, J : L p (fi) -» R denned by

/(„) : = I /n 1^1" d * + In M" <** * « € C 1 ^) 1 oo

otherwise.

(Note that J(it) may also be infinite for some u £ Cl(ti).) We claim (sc-I)(u)

= I In \Du\P I oo

dx

+ In l«lP d *

if u

€ if ^ ( f l ) otherwise.

In order to show this, we shall verify the conditions of Lemma 5.1.2: (i) (sc~I) is lower semicontinuous on Lp which yields condition (i). The lower semicontinuity is seen as follows: Suppose un —» u in L p (fi). For the purpose of lower semicontinuity, we may select a subsequence (wu)v^n C (un)ne^ with lim (sc~"J)(w„) = liminf(sc~J)(ii n ), v—+00

n—•KX)

and we may also assume that this limit is finite. (yjv)v^ then is bounded in H1,p(fl). A subsequence of (wu) then converges weakly in H 1 ' p (f2) (Theorem 3.3.5), and by the Rellich-Kondrachev compactness Theorem 3.4.1, it also converges strongly in Lp(fl). The limit has to be u, because the original sequence (un) was assumed to converge to this limit. Since the Jcf1'p-norm is lsc w.r.t. weak Hl,p convergence (Lemma 2.2.7), we have (sc~I)(u) < lim

(sc~I)(wl/)

= liminf(5C~/)(ii n ).

212

Nonconvex junctionals.

Relaxation

(ii) Let u e H^p(ft). Since C^fl) 0 Hl*(tt) is dense in Hhp(fl), may find a sequence (un)ne^ C C1(f2) D HllP(fl) with lim ( f \Dun\p + / \un\p)

we

= f \Du\p + \u\p ,

i.e. lim/(ii n ) = If ugH1*^),

(sc~I)(u).

then I(u) = (sc~I)(u) = oo.

This verifies condition (ii). Example 5.1.3. Similarly, for °

W

'

if u G L p (fi) \ CQ1 (fi),

\oo

the relaxed functional is I cx)

otherwise.

Remark 5.1.2. We may also define the above functionals / , I0 on Lfoc(f2) instead of Lp(fl). The relaxed functionals will be given by the same formulae. Remark 5.1.3. For p = 1, the relaxations of / and 70 a r e not given anymore by the H 1 ' 1 -norm, but by the BV-norm which is defined in Chapter 7. In metric spaces, there is an alternative useful characterization of the relaxation of a given functional which we now want to describe. Definition 5.1.2. Let X be a metric space with distance function d(-, -), F : X —• R U {oo} be bounded from below, F ^ oo. For \ > 0, we define the Moreau-Yosida transform of F as Fx(x) := inf (F(y) + \d(x,y)). yex Theorem 5.1.3. The functionals F\ satisfy |F A (*i) - Fx(x2)\

< Ad(xi,a?2)

(5.1.13)

(5.1.14)

for every X > 0, xi, x2 € X. In particular, they are Lipschitz continuous. For any x € X (sc~F)(x) = lim Fx(x). A—+oo

(5.1.15)

5.2 Representation of relaxed Junctionals via convex envelopes 213 Proof. For xi,X2,t/ G X, A > 0, we obtain from the triangle inequality F(y) + Xd(xuy)

< F(y) + \d(x2,y)

+

Xd(xux2).

The definition of F\{x2) implies then inf (F(y) + Xd(xuy)) yex

< Fx(x2) +

\d(xux2),

hence Fx{xi) < Fx(x2) + Ad(xi,x 2 ). Interchanging the roles of X\ and x2, we conclude \Fx(xi) - Fx(x2)\ <

Xd(xux2).

Since we have now shown that F\ is Lipschitz continuous, and since ^A < F, we obtain Fx < sc-F, hence for all x £ X supF A (x) < (sc'F)(x).

(5.1.16)

A>0

For any A > 0, we find x\ £ X with F(xx) + \d(x,xx)
+j .

Therefore lim x\ = x A—+oo

and (s
(5.1.17)

A—*oo

Equations (5.1.16) and (5.1.17) imply (5.1.15). q.e.d.

5.2 Representation of relaxed functionals via convex envelopes Theorem 5.2.1. Let Q, C Rd be open, I