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140

and for each bounded set B e B(H) there is a finite time Tg > 0 satisfying

S(t)

(D(ip°°)

nfljcBo for all

t>T

By the above lemma, we can construct a global attractor for {S(t)}. In fact, the global

attractor Ao of {£>(*)} is given by

Ax. = Us{B

For a detail

proof,

see [9]. Here, by a global attractor Ao of {£(*)} we mean that

(i) Ao is

non-empty and compact subset of H;

(ii) for each bounded set B e B(H),

dist

(S(t)z,

Aoo)

—• 0 uniformly inze

D{ip°°)

n B

as t

—•

oo;

(iii) Aoo = S(t)Ao for all t > 0.

Now, let us mention our main result in this paper.

Theorem 3.1. Let {</>'} e

$({a

}) and assume that (A1)-(A6) and (3.1)

hold.

Then the set

A:= U w

(B) (3.3)

BeB(H)

satisfies that

(i) A

C Aoo

d At is non-empty and compact in H;

(ii) for each bounded set B G B(H),

distn{E{t, s)z, A

) —> 0 uniformly in s > 0 and z € D(ip

) n B

as t ^r +co;

(iii) A C S(i)A /or any t e R+.

In our proof of Theorem 3.1 one of the important steps is to show the following lemma

on the relationship between w-limit sets under E(t,s) and an absorbing set B

for S(t).

Lemma 3.3. Let B

be the compact absorbing set for {S(t);t e R

} obtained in Lemma

3.2. Then,

{B) C B

for each B € B(H).

In the proof of Lemma 3.3 the time dependence condition (*) on {</?'} is essentially

used, and the proof is a modification of that of [3, Lemma 4.1] to the multivalued case.

Proof of Theorem 3.1: We give a sketch of the

proof.

By Lemma 3.3, we observe that

At C Bo- From this relation together with assertion (iii) proved below we have (i). Also,

(ii) of Theorem 3.1 follows from the definition (3.3) of A and Remark 3.1 in a way similar

to that of the proof of [3, Theorem 5.1].

141

Next, let us show (iii) of Theorem 3.1. By Lemma 3.1, we easily see that for each

XCB

S(t)X C

S(t)X,

R+. (3.4)

Now, let x be any element of A*. By the definition of A,, there exist sequences

} C B(H) and {x

} C H with x

U>E{B„)

such that

x„

—>•

in H as n

—>

+oo. (3.5)

It follows from Remark 3.1 that for each n, there exist sequences {t

,j} C R+ with

,j -> +oo, {s

} C R

, {z

j} C B„ with z

j G D(ip

"J) and {«J

} C if with

u„j G E(t

j + s

j, s

j)z

n<j

such that

»„j —• a;

in H as j

—>

+oo.

Let t be any time in R

. We note that

for j with t

- > t + 1. Hence there is a w

j G E(t

j + s„j

—

i, s

j)z

j such that

Here by the global estimates obtained in Theorem 2.1, {w

j e ff ; j 6 iV with

^nj >

+ 1} is relatively compact in if, so that we may assume that the sequence w

converges to some element w

G H as j

—>

+oo. Clearly, tu

LOE(B

Therefore from

(ii) of Lemma 3.1 we observe that

S{t)w

c S{t)u

which implies that

(J S(t)u)

), Vn > 1.

n>l

On account of (3.4) and (3.5), we see that

x G [J S(t)cu

) = S(t) |J uj

) C S(t) (J u

) C S{t)A,.

n>l

n>l n>l

Hence A, is semi-invariant under S(t), namely A, C S(t)A, for all £ G R

. Q

Theorem 3.1 says that the attracting set A

is semi-invariant under S(t) associated

with the limiting autonomous system (Eoo), in general. We now give a sufficient condition

in order that A, is invariant under S(t).

Definition 3.2. Let z G D((f°°). Then, we say that S(t)z is regularly approximated

by E(t + s, s) as s

—>

+00, if for each finite T > 0 there are sequences {s

} C R

with

—>

+00 and {z

} C H with z

G D((p

) and z

—» z in ii satisfying the following

property: for any function u

l,2

(0, T; ii) satisfying u(t) G 5(i)z for all t G

[0,

T] there

is a sequence {«„} C W

1,2

(0, T; ii) such that u

(t) e,E(t + s

, s

for all t G

[0,

T] and

—>

u in C([0,

T];

i/) asn-> +00.

142

Now, let us mention a result on the invariance of A, under S(t).

Theorem 3.2 Assume that for any z of

A*,

S(t)z is regularly approximated by E(t

s, s)

as s

—>

+00. Then,

= S(t)A* C Aoo for all T 6 R

. (3.6)

Moreover, if S(t)z is regularly approximated by E(t+s, s) as s

—•

+00 for all z e A^, then

Aoo.

(3.7)

Proof.

By Theorem 3.1, the attracting set A

is semi-invariant under S(t). So we have

only to show that S(t)A* C A* for any t 6 R

Now let y e S(t)A». Then, by (iii) of Theorem 3.1 we see that

y e S{t)A. C S{t + T)A» for all r > 0,

which shows that there are sequences

{T„}

with

T„

—>

+00 and {x

} c A, such that

y€S(t +

Now by our assumption we observe that for each n there are sequences {s

j} C R

,j) C H and {y

j} C H such that

j->+00, x

- e

D(<p

->),

j-> x

in H

and

,j, Vn,j —• 2/ in -ff

as j

—>

+00. Therefore, by the usual diagonal argument, we can find a subsequence {j

}

of {j} such that x

:= aJ

j„>

Vn '•— yn,j„

d s

:= s

satisfy

< -,

€ £(i +

T„

+ s„, s

)x„, \y

- y\

< -

n n

for every n = 1,

2,....

Since {£„} is bounded in #, say {£„} C i? for some B £ B(H), the

above fact implies (cf. Remark 3.1) that y e u

{B) c -A». Thus 5(f).4„ C A*.

Now here, we have A

= S(t)A* for all t € i?+. Therefore it follows from the invariance

of A* and ./"loo under S(t) that

= S(t)A, C S(t)Aoo = Ax> /or aH t e

JR+.

Thus (3.6) holds. Assertion (3.7) is proved by a slight modification of that of (3.6). <C>

We say in this paper that the set A*, having the properties in Theorem 3.1, is a global

attractor for {E(t, s)}.

4 Examples

In order to illustrate our theorems we give two simple examples of non-autonomous flows

in the one dimensional space H = R, which show that .4, ^ Aoo in general. For a given

interval

[a,b]

we denote by

,b]{-)

the indicator function on [a,b].

143

Example 1. We consider a family {ip

} of proper l.s.c. convex functions on R and a

function g(t, u) on R defined as follows:

f /[!,!](•) for 0 < t < 2,

<?'(•) := | /[!_!](•) for 2 < t < +oo

[ /[o,i](•) for t = +oo

and

g(t, u) := —\/u for 0 < u < 1.

It is easily checked that they satisfy all the conditions in section 2. Now, consider the

non-autonomous and autonomous evolution equations (E

) with / = 0 and (Eco) with

f°° = 0, and the corresponding non-autonomous and autonomous systems E(t, s) and

S(t),

respectively. Also, let A* and A*, be the global attractors associated with them.

Then, observing that

E(t + s, s)u

= min{l, -At + c)

} with c = 2^j% e [-, 1] for s > 2,

S(t)v

= min{l, -(t + c)

} with c = 20^

(0,1]

and S(i)0 is the set [0,1] for all sufficient large t > 0, we have A

= {1} and Aao =

[0,1].

By the way, in this example, S(t)0 is not regularly approximated by E(t+s, s) as s

—>

+oo.

Example 2. We next consider {</?'} and g(t,u) as follows:

/[_!,!)(•) for 0 < t < 2,

/[_!,!](•) for 2 < t < +oo,

7[o,i](-) fort = +co

and

g(t,u)

-y/u for 0 < t < +oo, 0 < u < 1,

u 1

for 0 < t < 2, - - < u < 0,

2 - - ' 2 ~

u 1

for 2 < t < oo, < u < 0.

Just as in Example 1, they satisfy all the conditions in section 2, and we can generate the

non-autonomous and autonomous systems E(t, s) and S(t) associated with the evolution

equations (E

) with / = 0 and (Soo) with /°° = 0, respectively. In this case it is easy

to see that A, = A^, = [0,1] and for any z 6

[0,1],

S(t)z is regularly approximated by

E(t + s, s) as s

—>

+oo.

It seems difficult to give any example showing that A* / S(t)A* in such simple

situations as above.

References

[1] H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans

les Espaces de Hilbert, North-Holland, Amsterdam, 1973.

144

[2] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and

Monographs 25, Amer. Math. Soc, Providence, R. I., 1988.

[3] A. Ito, N. Yamazaki and N. Kenmochi, Attractors of nonlinear evolution systems

generated by time-dependent subdifferentials in Hilbert spaces, pp. 327-349, Dynam-

ical Systems and Differential Equations, Missouri 1996 Volume 1, ed. W. Chen and

S. Hu, Southwest Missouri State University, 1998.

[4] A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by

dif-

ferential inclusions and their approximations, Abstract and Applied Anal., 5(2000),

33-46.

[5] V. S. Mel'nik and J. Valero, On global attractors of multivalued semiprocesses and

nonautonomous evolution inclusions, Set-Valued Anal., 8(2000), 375-403.

[6] U. Mosco, Convergence of convex sets and of solutions variational inequalities, Ad-

vances Math. 3(1969), 510-585.

[7] M. Otani, Nonmonotone perturbations for nonlinear parabolic equations associated

with subdifferential operators, Cauchy problems, J. Differential Equations, 46(1982),

268-299.

[8] K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolu-

tion equations governed by subdifferentials, pp. 287-310, in Recent Developments in

Domain Decomposition Methods and Flow Problems, GAKUTO Intern. Ser. Math.

Appl., vol 11, Gakkotosho, Tokyo, 1998.

[9] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,

Springer-Verlag, Berlin, 1988.

[10] N. Yamazaki, A. Ito and N. Kenmochi, Global attractor of time-dependent dou-

ble obstacle problems, pp.

288-301,

Functional Analysis and Global Analysis, ed. T.

Sunada and P. W. Sy, Springer-Verlag, Singapore, 1997.

Quasiconvex extreme points of convex sets

Martin Kruzik

Center of Advanced European Studies and Research,

Friedensplatz 16, 53 111 Bonn, Germany*

Email: kruzik@caesar.de

and

Institute of Information Theory and Automation,

Academy of Sciences of the Czech Republic,

Pod vodarenskou vezi 4,

182 08 Praha 8, Czech Republic.

Email: kruzik@utia.cas.cz

Abstract

Zhang [17] showed that if the quasiconvex hull of a compact set in K

" is con-

vex then also its rank-1 convex hull is convex. In this note we show that a reason

for that is in a special structure of quasiconvex extreme points of compact convex

sets.

In particular, we show that compact convex sets are lamination convex hulls

of their quasiconvex extreme points. We also give a clear geometric characteriza-

tion of quasiconvex extreme points of compact convex sets. Finally, we show that

quasiconvex exposed points of convex compact sets coincide with their lamination

convex extreme.

1 Introduction

It was proved in [8, 16] that generalized Krein-Milman type theorems ([7]) hold for com-

pact quasiconvex and rank-1 convex sets in TR

mxn

(the Euclidean space of real matrices

m x n). In particular there exist minimal generators of those sets called quasiconvex and

rank-1 convex extreme points, respectively. An example in [8] shows that quasiconvex

and rank-1 convex extreme points differ in a general situation. In this short paper we

give a simple geometric description of quasiconvex extreme points on compact convex

sets.

More precisely, we show that quasiconvex extreme points on compact convex sets

are those points which cannot be written as a convex combination of two mutually

dif-

ferent rank-1 connected matrices. In other words, for convex compact sets the notion

of lamination convex extreme points coincides with that of rank-1 and quasiconvex ex-

treme points. This gives a concrete description of this, generally a very abstract, notion.

* address for correspondence

145

146

Further, we get that convex compact sets are generated by the sets of their quasiconvex

extreme points via sequential lamination. An example in [8] shows that this does not hold

for nonconvex sets, however. Moreover, our result implies that if the quasiconvex hull is

convex then also the rank-1 convex hull must be convex, the fact which was proved in

[17].

Finally, we show that recently defined quasiconvex exposed points ([18]) coincide

with quasiconvex extreme points in case of compact convex sets. Proofs of our results

heavily rely on separation theorems for convex sets (see e.g. [14]) and on Krein-Milman

type theorems for quasiconvex and rank-1 convex sets.

We say that a function / : IR

" ->• IR is quasiconvex if for any ip G W

-°°(]R

; IR

(0,

l)"-periodic, and any A

f(A)<[ f(A +

(x))dx

. (1)

Quasiconvexity plays a crucial role in the calculus of variations. Namely, the sequential

weak* lower semicontinuity of / :

IR") -> IR, I(u) = /

/(Vu(i)) dx, Q C IR" a

bounded domain is equivalent to the quasiconvexity of / : M

mxn

-» H, see [1, 10, 11].

Further we say that / : R

" -> IR is rank-1 convex if for any A, B G H

rank(A - B) < 1 and any 0 <

< 1

f(XA + (1 - X)B) < Xf(A) + (1 - X)f(B) . (2)

Rank-1 convexity is a necessary condition for quasiconvexity but it is not a sufficient

condition if m > 3 and n > 2 as shown in [15]. If min(m, n) = 1 then both the notions

are equivalent to each other and the case m

—

2, n > 2 still remains open.

Let if be a compact set in ]R

mxn

. Besides the convex hull C(K) we define the

quasiconvex hull and rank-1 convex hull Q{K) and R{K) of K, respectively, as follows

(see e.g. [12])

Q{K) := {A

mxn

; f{A) < sup/ V/ : R.

" -> IR quasiconvex } , (3)

R(K) := {A

mxn

; f(A) < sup/ V/ : H

" ->• E. rank-1 convex } . (4)

Quasiconvex hulls play an important role in the theory of martensitic transformations.

Namely, the quasiconvex hull of the zero set of a material stored energy density is the set

of stress free configurations. The rank-1 convex hull is an inner approximation of this set.

Finally, we define the so-called lamination convex hull L{K) of K as L(K) := [J

{

(K),

ielNu{0}

where L

(K) := K and

Li{K) := {XA + (1 - X)B; 0 <

< 1 , rank(A -B) = l,A,Be L^K)}

i e IN .

Clearly, we have L(K) C R(K) C Q(K) and contrary to quasiconvex and rank-1

convex hulls the lamination convex hull is relatively easy to construct.

The notion of quasiconvexity is closely related to gradient Young measures. Let Q C

IR" be a bounded domain and K C IR

" compact. It is known ([3, 6]) that for any

147

sequence

}

C W

'°°(ft;Irl

) such that, for almost all x e Q, Vu

(x) 6 K there

exists its subsequence (here denoted by the same way) and a family of probability measures

}x&i, supported on K such that for any continuous function v : K —> ]R and any

geL

(Q)

lim / v(Vu

(x))g(x) dx = / / v(A)i>JdA)g(x) dx . (5)

k-HxJn

JSIJK

The family of probability measures {v

}xm f°

which the above limit passage holds

is called a gradient Young measure generated by {Vw^jtejN If {v

}

m is independent

of x we call such a measure a homogeneous gradient Young measure. It follows from

the analysis by Kinderlehrer and Pedregal [6] who found an explicit characterization of

gradient Young measures that a probability measure v supported on a compact set K is a

homogeneous gradient Young measure if and only if for any quasiconvex / : M

—>

/ (J

Au(dA)j < j

}{A)u{dA) . (6)

A homogeneous gradient Young measures which satisfies (6) even for all rank-one

convex functions will be called a homogeneous laminate ; cf. [13].

It is well known that the quasiconvex (rank-1 convex) hull of a compact set K c M

m><n

coincides with the set of first moments of all homogeneous gradient Young measures

(homogeneous laminates) supported on K.

We say that A in K is a quasiconvex extreme point of K if the only homogeneous

gradient Young measure supported on K with the first moment A is

(Dirac's mass at

A).

Following Zhang [16] we denote the set of all quasiconvex extreme points of K by

q<e

Analogously, we denote by K

T%e

the set of rank-1 convex extreme points of K, i.e.

the set of points which can only be represented by a Dirac mass amongst all homogeneous

laminates. As homogeneous laminates are a strict subset of homogeneous gradient Young

measures ([15]) we have

qfi

Tfi

We say that A e K is a lamination extreme point if the following holds:

A = \B + (1 - X)C , 0 <

< 1 , B, C e K , rank(B - C) < 1 => B = C = A

and we denote their set by Ki

The definition above is a particular case of D-convex extreme points defined in [9].

Finally, we denote by K

the set of convex (usual) extreme points of K, by Aff K

the affine hull of K, Aff K := A+ span(ii' - K) for any fixed A e K and the dimension

of K, dim K := dim span(if

—

K). We say that A,B e JR

mxn

are rank-1 connected if

rank(A

—

B) = 1. Aff K has rank-1 connections if it contains rank-1 connected matrices.

2 Properties of convex compact sets

From now on we suppose that K has more than one point. It holds for a compact set

K c IR

" such that Q(K) = K that K = Q{K

qfi

), cf. [16], and, in general, L{K

qfi

) C K,

strictly. Surprisingly, if K is convex we have the following result.

148

Proposition 2.1. Let

K c

" be a compact convex set. Then

K = L ^

j^(K

qye

Proof.

We proceed by induction. If dim

K =

1 and Aff

has rank-1 connections then

the assertion clearly holds due

to the

Krein-Milman theorem because

qfi

;

cf. [16].

If Aff

has no rank-1 connections then

K = if

by [4, Th. 4.1] and

K =

Li(K). Suppose

that

the

assertion holds

for

sets

the dimension less than

and that dim

K = d.

Take

£ dK

and

let H

supporting hyperplane

to K

through p.

KnH is

compact, convex

and its dimension is less than d. By the induction hypothesis

KnH = L ^^

jj((Kf]

if),,

But if A

{K n

q>e

c K n H

then

if,,

because

K is in

one

the half

spaces determined

by H.

Thus

p E K n H C

(

^-(if,,

)

_i(if,,

If Aff

has rank-1 connections then L

(if)

Li(dK). Therefore

K C

Li(I/<i_i(if

gie

)). Hence,

qfi

On the other hand, as

if,,

C K

we have L

qtf

C if,

i.e.,

if =

qfi

If Aff

has no rank-1 connections then

K = if

9)e

by [4, Th. 4.1] and

K = L

dim

(if

„_,,).

•

The proposition above shows how convex sets are generated

means

their quasi-

convex extreme points and

answers

question raised

[16]

this special case.

Corollary

2.2. Let K C R

" be

compact,

dim K = d and

such that

Q(K) is

convex. Then L

)

= Q{K).

Proof.

We have

Q(if) =

{Q(K)

qfi

) because <2(if)

convex

On the

other hand,

Q(if),,

c if,,

;

see [16].

' •

The following result has been first obtained by Zhang [17] and

new proof has recently

been found

[5]. We get

here

as a

simple consequence

Corollary 2.2.

Corollary 2.3.

Let K c

compact,

dim K = d and

such that

Q{K) is

convex. Then

(K) = Q{K).

Proof.

have L

(if,,

)

(if)

c Q(K). But due to

Corollary

2.2

(if„,

)

Q(if) and therefore L

(if)

Q(if).

' •

can

give

simple geometric description

quasiconvex extreme points

convex

sets.

Proposition 2.4.

Let K c

mxn

compact convex set. Then

qfi

= K

ifi

Therefore,

A £ if is a

quasiconvex extreme point

of K if

and only

there

is no

interval

with distinct rank-1 connected endpoints and with

A as an

inner point.

Proof.

As in

general

C if

C if|,

must only show that

C if,

. We

proceed

induction.

dim

if =

1 and

Aff if

has rank-1 connections then

the

assertion

clearly holds.

If Aff if

has

rank-1 connections then

if = if,

[4,

Th.

4.1]

and the

assertion holds again. Suppose that the assertion holds for sets of the dimension less than

d and that dim

K =

Aff

has no rank-1 connections then the argument is the same

for the

dimension one,

suppose that

Aff if

has rank-1 connections. Take

ifj,

Then

must

be in dK

and

let H be a

supporting hyperplane

to if

through

A. KnH is

149

convex, compact

and its

dimension

less than

Moreover,

by the

induction hypothesis

and lamination extremality

A € (K n H)i,

C (K n

qfi

is a

homogeneous gradient

Young measure supported

on K

with

the

first moment

A it

must

supported

KCiH

entirely stays

in one of the

half spaces defined

by H and we

have that

v =

5A,

i.e.,

e K

qfi

This yields

if,

•

There

is a

geometric description

quasiconvex extreme points

means

qua-

siconvex exposed points given

in [18].

Recall that argmax

:= {A e M; f(A) =

maxBgM

f(B)} for a

compact

set M and a

continuous function

defined

on M.

Definition

2.5. (see [18]) Let K c

nxn

quasiconvex.

A e K is

called

level

quasiconvex exposed point

of K if

there

is a

sequence

compact subsets

{lfj}*_

such

that

:= K and K^ := {A} and

quasiconvex functions {/i}*

such that

i+l

:= {Ae K; Ae

argmax

i+1

}

,i = 0,...,k-l .

The

set of

all

quasiconvex exposed points of finite levels

denoted

by K

q>ex

Zhang showed that

for a

compact

set K the set of

level

quasiconvex exposed points

is dense

in K

q<e

. It is not

clear whether

general

qje

= K

q<ex

. If

we restrict ourselves

functions which

are

quasiconvex only

on K, i.e.,

which satisfy

the

Jensen inequality

(6)

for

all

gradient Young measures supported

on K

then

it is

really true that

qfi

= K

qfix

Namely,

as K

qfi

is the

Silov boundary

of K, cf. [2, 8], we

easily

see

that each point from

even

level

quasiconvex exposed point.

In general,

for

convex sets

we can

prove

the

following.

Proposition

2.6. Let K c

mxn

compact

and

convex. Then

q<e

= K

qex

Proof.

suffices

prove

the

first equality.

again proceed

induction.

If dim

= 1

then

if K has

rank-1 connections

it has

only

two

extreme points

and the

assertion

follows from

the

theorem

[18, Th. 1.1] by

Zhang.

If K has no

rank-1 connections then

can

translate

such that

Aff K is a

subspace without rank-1 matrices

and the

result

again follows

Zhang

[18, Th. 1.3].

Suppose that

dim K = d and

that

the

assertion holds

for

sets

up to the

dimension

—

1. If K has no

rank-1 connections

we are

done again;

cf. [18, Th. 1.3], as all

point

are

quasiconvex exposed points. Otherwise

if A 6 K

q<e

must

be at the

boundary

Take

supporting hyperplane

through

A to K.

Then

A e (H fl

q<e

and by the

induction hypothesis

A € (H

q>ex

By the

definition above

means that there

is a

sequence

sets

:= (H n

K),...

M^ := {A} and a

sequence

quasiconvex functions

/i,..../*

such that

i+1

= {P

/i+i(P)

maxf

i+l

{B),B

e #;}.

But

H n K =

argmax^/, where

/ is an

affine function such that

H = {x; f(x) = c}

and

K is in the

region where

/ < c.

Therefore,

we define

the

sequence

sets

:= K,

= M;_i, i =

1,...,

k +

and a

sequence

functions

v{ := f, v

= /

i, we get

that

is quasiconvex exposed point because

the

definition above

satisfied

for {iV,} and {u;}.