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L1 and L∞ Approximation of Vector Fields in the Plane
Lecture Notes in Num. Appl. Anal., 5 , 273-288 (1982) Nonlinear PDE in Applied Science. i7.S.-Japan Seminar, Tokvo, 1982
L1
and
Approximation of Vector F i e l d s i n t h e Plane
L"
Gilbert Strang Massachusetts i n s t i t u t e of Technology
ABSTRACT
We s t u d y f o u r problems, two i n whose analogues i n
L2
L1
and two i n
a r e t h e f a m i l i a r minimum
p r i n c i p l e s which l e a d t o t h e Laplace equation.
One
p o s s i b i l i t y is t o be given t h e boundary v a l u e
6 = Q
and t o minimize point
(x,y)
IIVJrlll
in
Cl
Lmr
or
llV@~lrn;
i s measured by
the gradient a t a lVJrI2
2 = f,
+ Q Y2 '
In t h e o t h e r problems we a r e given a v e c t o r f i e l d v: R
+
R2,
and minimize e i t h e r
IIVw-vlll
or
~IoW-v!lm.
i n each c a s e we use t h e d u a l i t y t h e o r y of convex a n a l y s i s t o g i v e e q u i v a l e n t s t a t e m e n t s of t h e problem, o f t e n with an i n t e r p r e t a t i o n i n mechanics and o f t e n partly solved.
Nevertheless some q u e s t i o n s s t i l l
remain open.
213
274
Gilbert S T R A N G
L2 norm l e a d s t o l i n e a r equations
Approximation i n t h e
(which a r e forbidden a t t h i s c o n f e r e n c e ) .
In
<
LP, 1
P
<
(0,
t h e equations become n o n l i n e a r but much of t h e a n a l y s i s continues t o apply.
In
i s entirely different:
L1
and
Lm, however,
the s i t u a t i o n
it i s not t h e d i f f e r e n t i a l equation
b u t t h e underlying v a r i a t i o n a l p r i n c i p l e t h a t l e a d s t o an e x i s t e n c e theory, and suggests how t o c o n s t r u c t t h e optimal solution. This note s t u d i e s f o u r t y p i c a l problems for s c a l a r valued f u n c t i o n s on a simply connected domain
r.
s u f f i c i e n t l y smooth boundary
fl
C
R2
with
Each of t h e problems has
a dual--a maximization i n s t e a d of a minimization--and a p p l i c a t i o n s t h e dual i s of equal importance.
i n the
Where one
v a r i a t i o n a l statement i s t h e " s t a t i c " formulation of a problem i n mechanics, with s t r e s s e s a s t h e unknown, t h e o t h e r i s t h e "kinematic" form i n terms of v e l o c i t i e s .
We
w i l l study s e v e r a l combinations of boundary c o n d i t i o n s and
inhomogeneous terms, b u t n o t every p o s s i b l e combination, because a l r e a d y we ask t h e r e a d e r ' s consent about one more thing.
In a d d i t i o n t o t h e dual of each problem, t h e r e i s
another p a i r of optimizatlons ( e q u i v a l e n t t o the given one) c r e a t e d by a s p e c i a l s i t u a t i o n i n s o l u t i o n of
div n
= 0
is
T
=
($
R2,
t h a t the general
- $ x ) for some
Y9
4.
Therefore it w i l l happen t h a t each of our problems has four e q u i v a l e n t forms, and t h a t one of them i s simpler t o s o l v e than t h e o t h e r s .
(For t h e o t h e r s we may l e a r n only t h e value
of t h e maximum o r minimum, by d u a l i t y , without f i n d i n g t h e f u n c t i o n which achieves t h a t v a l u e . )
Some q u e s t i o n s w i l l
remain unsolved even w i t h four a l t e r n a t i v e s .
L' and L" Approximation of Vector Fields
275
The problems arise i n t h e s t u d y of p l a s t i c i t y and optimal p l a s t i c design, and elsewhere; we w i l l g i v e r e f e r e n c e s r a t h e r t h a n a complete d e s c r i p t i o n of t h e s e a p p l i c a t i o n s .
And we do
t h e same f o r t h e proofs of d u a l i t y ; i n our problems t h e y come d i r e c t l y from t h e techniques of Ekeland-Temam [ll, who a p p l i e d t h e Moreau-Rockafellar t h e o r y t o a sequence of important examples i n p a r t i a l d i f f e r e n t i a l e q u a t i o n s .
Our c h i e f purpose
i s t o c o n t r i b u t e some a d d i t i o n a l examples, and t h e y have
developed from our j o i n t work w i t h Robert Kohn and Roger Temam. We mention t h a t optimal d e s i g n l e a d s t o more complicated v a r i a t i o n a l problems ([2-31
i s p a r t of a l a r g e l i t e r a t u r e ) ,
and a l s o t h a t p e r f e c t p l a s t i c i t y i n
has r e q u i r e d a new
R3
space of vector-valued displacements and a corresponding analysis
[4-51. The problems i n
t h i s note a r e e a s i e r , b u t
t h e y have n a t u r a l i n t e r e s t and it may be u s e f u l t o organize them more s y s t e m a t i c a l l y . A t t h e end we d i s c u s s some a p p l i c a t i o n s i n optimal d e s i g n
and t h e nonconvex problems t o which t h e y l e a d .
1. The minimum of
l/Vt!!m
w i t h Dirichlet data
It i s t h i s form which can be solved d i r e c t l y , b u t we
begin w i t h t h e f o u r e q u i v a l e n t problems: 1A.
M I N I!n(lm s u b j e c t t o
1B.
MAX
1C.
MIN
1D.
MAX
r
uf
j1VdrIlrn
subject t o subject t o
div o = 0 JslVul 0
rlr = g
=
and
1
on
r
o.n = f
on
r
216
Gilbert S I R A N G
The conversion between 1 A and 1C i s by r o t a t i o n through f = n.n = V $ . t
Therefore
=
of
s/2
It follows t h a t
VQ.
&/&, t h e t a n g e n t i a l d e r i v a t i v e of
6 = g = s f ds
assumed t h a t
a
fl = ( l y J - ? b x ) ,
Jf ds
= 0
on t h e c l o s e d curve
S i m i l a r l y 1B and 1 D a r e connected by i n t e g r a t i o n by p a r t s on
rj r.
up t o a c o n s t a n t on
r.
1B
7
1.
it i s
= (uy,-ux),
and
i s dual t o lA, and 1D t o 1C.
The s o l u t i o n of 1 C comes from an e x t e n s i o n lemma proved independently by McShane and Kirszbraun f o r I d p s c h i t z f u n c t i o n s on a metric space.
Their construction applies
e q u a l l y t o Hblder continuous f u n c t i o n s , and was followed by deeper r e s u l t s of Whitney and Calderon.
In our case t h e
metric i s t h e s h o r t e s t d i s t a n c e w i t h i n
0; i t is Euclidean
distance, i f
Q
i s convex.
Then t h e q u a n t i t y
minimized i s t h e Lipschitz c o n s t a n t f o r g
on
to
l on
Ca
1.
t o be
IIVJIII,
The lemma extends
w i t h no i n c r e a s e i n t h e L i p s c h i t z
c o n s t a n t , by t h e simple c o n s t r u c t i o n
Therefore t h e minimum in 1C i s immediate: Lipschitz c o n s t a n t f o r
it e q u a l s t h e
g, and Problem 1 i s solved.
T h i s provides a n a t u r a l analogue f o r continuous flows of
t h e max flow-min c u t theorem of Ford and Fulkerson.
Instead
of a f i n i t e network with c a p a c i t y c o n s t r a i n t s on t h e edges, t h e flow through i t s c a p a c i t y by
capacity
In1
5
0
i s described by a v e c t o r f i e l d
la1
c(x,y)
fl a r e included i n
0
and
1. The p o s s i b i l i t i e s of varying and nonzero sources
div u
=
F
[6]. I r i c r e a t e d a s i m i l a r t h e o r y
n e a t a p p l i c a t i o n t o t r a f f i c i n Tokyo [ 7 ] .
within
with a
L' and L' Approximation of Vector Fields
211
We g i v e a b r i e f b u t very informal d e r i v a t i o n of t h e o p t i m a l i t y c o n d i t i o n s t h a t connect 1A t o lB, w i t h Lagrange m u l t i p l i e r for t h e c o n s t r a i n t
u(x,y)
as
div n
=
0.
The saddle-
SSIVuI
>
1, and otherwise
p o i n t ( k g r a n g i a n ) form i s
The f i n a l minimum over
it i s
Suf
as i n 1 B .
TT
is
if
-m
The o p t i m a l i t y c o n d i t i o n s are
which gives an i n t e r e s t i n g form f o r
u.
It i s the character-
i s t i c f u n c t on, or more e x a c t l y a m u l t i p l e
of t h e
l//P-Q
c h a r a c t e r i s t i c function, of t h e s e t bounded by t h e l i n e between
and
P
Q
on
I-
--where t h e s e a r e t h e p o i n t s ( n o t
n e c e s s a r i l y unique) a t which Thus
Vu
=
0
g
except a c r o s s t h i s l i n e ; on t h e l i n e ,
s i n g u l a r measure of mass one and magnitude
>
Q = (0,-r).
0
i s t h e normal v e c t o r of
on t h e c i r c l e
The Lipschitz c o n s t a n t i s
x
n
and
1/2r
c1
$,J -trx)
An optimal
u
and
n
r
<
1/2.
P = (0,r)
is zero i n the semicircle
i n t h e complement
= (l/r,O),
of r a d i u s
l/r, between t h e p o i n t s
x
1. 0.
( n o t t h e same as McShane's e x t e n s i o n ) i s =
is a
Vu
Iluli,.
Example 1: g = s i n 0
and
a t t a i n s i t s Lipschitz constant
1
An optimal =
y/r,
t
with
i s normal t o t h e diameter
FQ.
A l e s s t r i v i a l example, with more uniqueness, would s t a r t w i t h an e l l i p t i c a l
n.
278
Gilbert STRANG
2.
The minimum of
with D i r i c h l e t d a t a
\lVllrlll
Again w e g i v e t h e f o u r e q u i v a l e n t extremal p r i n c i p l e s :
SsIflI
PA.
MIN
2B.
MAX s r u f
2C.
MIN J S l V S l
2D.
MAX
n
subject t o div subject t o
and
n = 0
a-n
=
f
1 1 V ~ := ) ~1
subject t o subject t o
1
div
T
r
on
g
=
= 0
and
!\T'la
= 1.
The connections among 2A-2D a r e t h e same a s f o r 1A-1D.
I n t h i s case i t i s a g a i n 2 C ( t h e least g r a d i e n t problem
181) t h a t can be solved most d i r e c t l y , using t h e coarea formula f o r a function
where
yt
length.
4(x,y)
of bounded v a r i a t i o n :
is t h e l e v e l s e t where More p r e c i s e l y , s i n c e
1b
s e t of p o s i t i v e area, we c o n s t r u c t
and then
yt = a E t \ r .
Jr = t
and
is its
IYtl
could be c o n s t a n t over a E~ = ( ( x , y ) E Q: llr(x,y)
This coarea formula and i t s g e n e r a l i z a -
t i o n s a r e a v a l u a b l e t o o l i n geometric measure t h e o r y for a smooth f u n c t i o n
t]
$,
[9-lo];
or a piecewise l i n e a r f u n c t i o n , t h e
proof is s t r a i g h t f o r w a r d . To minimize
of
Yt,
llV@\\l
i s t o minimize f o r each
s u b j e c t t o t h e requirement t h a t
boundary p o i n t s a t which
g
may be required; t h e s e t
Et
g
2
t.
We note t h a t f o r
g
in
L1(r), t h e f u n c t i o n
t
and is i n t e g r a b l e . )
$
=
t.
Yt
t
the length
connects t h e
(Again a more p r e c i s e form
must c o n t a i n t h e s e t on which of bounded v a r i a t i o n , w i t h t r a c e IYtl
i s d e f i n e d for almost a l l
L’ and L L Approximation of Vector Fields
Example 2 .
0. The L2 norm i s s m a l l e s t f o r t h e harmonic f u n c t i o n 6 = r 2 cos 20,
of
g = cos 28
279
on t h e u n i t c i r c l e
z e r o o n l y on t h e f o u r r a y s t o
0 =
norm i s minimized by a f u n c t i o n whole i n s c r i b e d square of s i d e diagonals.
2
\b
J=’
g =
cOS
=
L1
w i t h t h e s e r a y s as
Between t h e s q u a r e and t h e c i r c l e , t h e l e v e l l i n e s
of t h e square, f o r example,
x
But t h e
which v a n i s h e s over t h e
4 = c o n s t a n t a r e s t r a i g h t , t o minimize
lines
2 3s/4.
IT/&,
$ =
To t h e r i g h t
lYt(.
2x2-1:
c o n s t a n t on v e r t i c a l
r
c o n s t a n t , and a g r e e i n g a t t h e boundary
=
1 with
28 = 2 cos2 0 - 1 .
I n t h e d u a l i t y with 2B, t h e o p t i m a l i t y c o n d i t i o n i m i t a t e s
1B above t o g i v e
Therefore
Vu
i s a u n i t v e c t o r i n t h e d i r e c t i o n of
n
# 0, and elsewhere
llr
= 2x2-1
circle. fx u =-
and
~y
< 1.
n = (0,-4x)
Therefore
or
IVul
I n o u r example we had
i n the section
Vu = ( 0 , l )
and
The
L’
x
u = -y.
2
l/n
of t h e
Similarly
i n t h e f o u r q u a r t e r s of t h e c i r c l e - - b u t n o t
uniquely so i n t h e i n s c r i b e d s q u a r e where
3.
where
0
$ = 0
and
CT =
0.
approximation of a v e c t o r f i e l d by a g r a d i e n t
The given v e c t o r f i e l d i s
i s i t s e l f a gradient i f
F
=
v = (v,(x,y),v2(x,y)),
dvl/dy
-
&,/ax
= 0.
and i t
Otherwise
t h e maxima and minima w i l l exceed z e r o i n o u r f o u r e q u i v a l e n t problems :
3A.
280
Gilbert STRANC
3s.
MIN ~ ~ I V w - v I
3C.
MAX JJFb
3D.
MIN
c1
In 3 D ,
subject t o
iSl1-1 s u b j e c t
Cl
-F. through
~ / 2 ;t h e
i s a u t o m a t i c a l l y zero, s o t h a t
Again we use a p r o p e r t y s p e c i a l t o
-F.
=
=
7
Ow-v
= (-wy,wx)
Ow'
-aiv v
7 =
div
i s t h e r o t a t i o n of
7
divergence of div
to
T, lv~lrl 5 1 i n
on
= 0
1C
R2.
In t h i s c a s e it i s 3 C which can be solved a t s i g h t , provided we assume i s t o maximize lvllll
5
F
0.
To make
as l a r g e a s p o s s i b l e
//Fib
among f u n c t i o n s v a n i s h i n g on
$
1. The extremal f u n c t i o n i s
with
r,
Q = distance t o
and
t h i s choice g i v e s t h e optimal values i n 3 A - 3 D . of varying s i g n t h e problem i s much more d i f f i c u l t
F
For
and i n t e r e s t i n g ; we b e l i e v e it t o be unsolved. in
t h e optimal
R1
Q
has
1lr1
On an i n t e r v a l
-
= +1, and t h e "breakpoints"
can be determined. A c o n s t r u c t i o n of t h e optimal
w
(assuming
F
1. 0 )
was
given by Mosolov [U].The computation i s more d e l i c a t e t h a n t h a t of
and i s guided by t h e o p t i m a l i t y c o n d i t i o n
111,
n =
vw-v , F a t every p o i n t
Example 3 .
v
t h i s domain n =
( ey'
-$x)
=
and
F = 2
Vw
# v.
on t h e u n i t c i r c l e .
Ib = I-r, t h e d i s t a n c e t o t h e boundary.
On
Therefore
i s t h e u n i t v e c t o r f i e l d t a n g e n t t o t h e con-
centric circles instance
y,-x)
where
w c 0.
i s s a t i s f i e d by
r = constant.
It happens t h a t i n t h i s
The o p t i m a l i t y c o n d i t i o n d i s p l a y e d above
u
=
-v/IvI,
and t h e r e f o r e
"CaMOt be approximated by a g r a d i e n t . I' mation i s null..
v = (y,-x)
The c l o s e s t approxi-
28 I
L ' and L' Approximation of Vector Fields T h i s choice of
c y l i n d r i c a l rod. -slv.o
The v e c t o r
gives t h e shearing s t r e s s and
d
is t h e a s s o c i a t e d moment r e s i s t i n g an e x t e r n a l f o r c e
t h a t t w i s t s the bar. tion
a r i s e s n a t u r a l l y i n t h e t o r s i o n of a
v
161
5
1
The maximum i n 3A, s u b j e c t t o t h e l i m i t a -
f o r a p l a s t i c m a t e r i a l , i s t h e l i m i t moment of a
b ar w i t h cross-section
61.
T h i s i s t h e l a r g e s t torque the b a r
can resist; t h e angle of t w i s t approaches i n f i n i t y and t h e bar becomes f u l l y p l a s t i c The d u a l v a r i a b l e
( I n 1 = 1 throughout
w(x,y)
Q).
measures t h e "warping" of each
c r o s s - s e c t i o n out of i t s o r i g i n a l plane--and
f o r the circular
b a r of Example 3 each c r o s s - s e c t i o n remains plane and
w = 0.
The minimization 3D has a f u r t h e r mechanical i n t e r p r e t a tion.
It a g a i n r e f e r s t o a b a r w i t h c r o s s - s e c t i o n
subject t o the axial force
F(x,y) .
r e s i s t e d by s h e a r s t r e s s e s
T
expresses e q u i l i b r i u m .
= (7
ss
Then
This body f o r c e i s T
xz' yz
1
17
0 , b u t now
), and
div
7
=
-F
is proportional t o the
minimum weight of a b a r which can withstand t h e load
It
F.
i s a s p e c i a l i z a t i o n of t h e Michell-Prager t h e o r y of optimal design t o t h e c a s e of pure s h e a r [12].
4.
The
Lm
For given
approximation of a v e c t o r f i e l d by a g r a d i e n t
v, w i t h
F = -div v'
and
T
as
= (OW-v)'
before, t h e e q u i v a l e n t problems a r e
4A. MAX - s s d , v 4B.
subject t o
d i v n = 0, n.n
=
0
on
M I N IIVw-v/lrn
4 ~ .MAX
JJF@ s u b j e c t
4D. MIN
/\7/Irn
to
$ =
o
subject t o
div
7
on =
-F.
r, IIv$Il,
= 1
r,
IIulll = 1
Gilbert STRANG
282
Our method i s t o apply t h e coarea formula t o 4 C . show t h a t t h e o p t i m a l function
i s a m u l t i p l e of t h e c h a r a c t e r i s t i c
Q
of some s u b s e t
C
We want t o
E
C
The coarea formula can be
Q.
written as m
where
Et = [ $
is t h e c h a r a c t e r i s t i c f u n c t i o n of
Ct
2. t l .
It i s n o t d i f f i c u l t [ 6 ] t o show t h a t a l s o m
JJF$
n
Suppose
= ,m
(JSFCt) d t .
n
i s t h e maximum i n QC,
M
and suppose t h a t
f o r every c h a r a c t e r i s t i c f u n c t i o n
f o r t h e optimal
t
C.
Then choosing
( o r more p r e c i s e l y
C = Cnt
C = Ct
for a
t n ) w e would c o n t r a d i c t t h e
maximizing sequence of f u n c t i o n s
t.
previous e q u a t i o n s by i n t e g r a t i n g over
Therefore t h e
maximum ( o r supremum) i s a t t a i n e d i n 4C by a normalized characteris tic function
~
= C/\\VC\ll.
I n o t h e r words, Problem 4 C i s e q u i v a l e n t t o t h e s i m p l e r problem
Whenever
F
i s c o n s t a n t , we have an i s o p e r i m e t r i c problem:
n.
maximize area/perirneter w i t h i n weighted i s o p e r i m e t r i c problem.
For
And f o r
F
> F
0
it i s a
of varying s i g n ,
I t i s a g e n e r a l i z e d i s o p e r i m e t r i c problem which i s new t o We have not mentioned t h e boundary c o n d l t i o n seems t o be v i o l a t e d i f
aC
n
r #
0.
JI
US.
= 0, which
Nevertheless t h e a n a l y s i s
L ' and L ' Approximation of Vector Fields
c a n be j u s t i f i e d ;
it i s the condition
t
283
which must be
= 0
r e l a x e d [13], and t h e c o r r e c t form of 4 C i s
S i m i l a r l y Problem 2 C can be r e l a x e d by t h e boundary i n t e g r a l The e f f e c t i n Problem 4 C i s t h a t t h e l e n g t h of
I$-gI.
i s i n c l u d e d i n t h e p e r i m e t e r of
aC
n
r
and we r e a c h t h e i s o p e r i -
C,
m e t r i c problem d e s c r i b e d above. We show by example t h a t t h e o p t i m a l
(at least for
can be computed
~
F = 1 and f o r simple domains).
Since
is
Q
piecewise c o n s t a n t , changing o n l y a t t h e boundary of t h e o p t i m a l set
i n t h e i s o p e r i m e t r i c problem,
C
a C --a
s i n g u l a r measure supported on
= ($
CT
Y' " l i n e of
The o p t i m a l i t y c o n d i t i o n c o n n e c t i n g it t o
T
-$x)
is a
& - f u n c t i o n s .I '
gives only
moderate i n f o r m a t i o n :
We have no method f o r computing Example 4:
v = (y,-x)
and
n.
in
T
n.
on t h e u n i t c i r c l e
F = 2
T h i s i s t h e o l d e s t of a l l i s o p e r i m e t r i c problems, and t h e s u b s e t which maximizes a r e a / p e r i m e t e r i s field
T
is radial, w i t h
gradient t o Example 5 :
v
is
Ow =
v = (y,-x)
with v e r t i c e s a t
div
T
=
-2,
C =
a.
The v e c t o r
and a g a i n t h e n e a r e s t
0.
and
(+ 1/2,+- 1 / 2 ) .
n,
on t h e u n i t s q u a r e
F = 2
The o p t i m a l s u b s e t
n e i t h e r t h e whole square n o r t h e i n s c r i b e d c i r c l e .
C
Instead
it i s a compromise [6,14] whose boundary c o i n c i d e s w i t h except f o r q u a r t e r c i r c l e s of r a d i u s
(2
f
fi)"
is
r
i n the four
284
Gilbert
c o r n e r s of t h e s q u a r e . i s smooth.)
STRANG
(They a r e t a n g e n t t o t h e square, s o
aC
We have s o f a r been u n s u c c e s s f u l i n determining
a corresponding v e c t o r f i e l d
with
T
div
7
= (2
+
fi)!lT\lm.
It e x i s t s , by d u a l i t y t h e o r y , and we have o f f e r e d a modest
p r i z e (10,000 Yen a t t h e U.S.-Japan Seminar) f o r i t s d i s c o v e r y .
L1
It would be i n t e r e s t i n g t o compare t h e s e
and
L”
o p t i m i z a t i o n s with t h e corresponding d i s c r e t e problems i n and
a”.
4’
There t h e o p t i m a l i t y c o n d i t i o n s a r e c l a s s i c a l , and
t h e b e s t approximations must approach o u r s o l u t i o n s ( i n c l u d i n g t h e c h a r a c t e r i s t i c f u n c t i o n s ) i n an i r r e g u l a r b u t c o n s i s t e n t way.
The r a t e of convergence, and t h e p a t t e r n of error i n t h e
d i s c r e t e problems, should be v i s i b l e from numerical experiments-s i n c e this i s a c l a s s of d i s c r e t e problems i n which t h e cont i n u o u s l i m i t can be s o l v e d .
We h e s i t a t e t o propose a more complete l i s t of d u a l v a r i a t i o n a l problems of t h e same t y p e . appear f o r o p t i m i z a t i o n i n
L2,
Harmonic f u n c t i o n s w i l l
and t h e s p e c i a l p r o p e r t y of two
dimensions (which produced a l l t h e second p a i r s of e q u i v a l e n t problems) i n t r o d u c e s t h e conjugate harmonic.
Within t h e l i s t
above t h e r e a r e combinations of c o n d i t i o n s t h a t e a r l i e r e n t e r e d only s e p a r a t e l y , f o r example t h e combination MAX JJF$ s u b j e c t t o
P = g
on
f
and
! l V $ ! l w = 1.
This corresponds t o a p p l y i n g both s h e a r and t o r s i o n t o a c y l i n d r i c a l rod, and it i s solved ( i f t h e c o n s t r a i n t s a r e compatible and
F
>
0) by t h e maximal f u n c t i o n
3i(P) = min [ g ( Q ) + d ( P , Q ) I . A r e l a t e d problem mixes
L2 and
Lm, arid has become a
fundamental example i n t h e t h e o r y of v a r i a t i o n a l i n e q u a l i t i e s :
L’ and L* Approximation of Vector Fields
285
The d u a l minimizes a combined norm
This seems a p p r o p r i a t e a l s o f o r ” r o b u s t s t a t i s t i c s , “ i n which t h e l e a s t squares model (Gauss-Markov l i n e a r r e g r e s s i o n ) i s
natural--except t h a t it a s s i g n s t o o much weight t o o b s e r v a t i o n s t h a t l i e f a y o u t s i d e t h e normal range. 1
significant i n
L
, and
These o u t l i e r s a r e l e s s
a mixed norm i s more r e a l i s t i c .
F i n a l l y we mention optimal design, which i s s u b j e c t t o a l l t h e s e c o n s t r a i n t s and one more:
it begins a s a nonconvex
problem, t o minimize t h e support of
A. t y p i c a l case, f o r
n.
longitudinal shear i n a p l a s t i c cylinder, i s
The i n t e g r a n d jumps from to
at
00
(01
= 1
0
to
1 at
>
(since
u = 0, and from
1 i s inadmissible).
1
The
e q u i v a l e n t “ r e l a x e d problem” r e p l a c e s t h i s i n t e g r a n d by the l a r g e s t convex f u n c t i o n which does n o t exceed it:
10)
for MIN
lul
SJ’IUI
5
1 and
subject t o
m
for div
Id1
CI =
0,
>
1.
n-n
it e q u a l s
I n t h i s new problem = f,
11U11,
2
1
t h e r e e x i s t s an optimal s o l u t i o n (which i s a weak l i m i t
Of
t h e h i g h l y o s c i l l a t o r y minimizing sequences i n t h e o r i g i n a l problem).
The s o l u t i o n can a c t u a l l y be computed by modifying
t h e c o n s t r u c t i o n i n 2 C t o account f o r t h e new c o n s t r a i n t
101
= (V$(
2
1.
It g i v e s t h e admissible s t r u c t u r e of minimum
weight. There i s a l s o a more d e l i c a t e c l a s s of nonconvex problems,
Gilbert STRANC
286
whose relaxed form f a i l s t o be convex:
i n s t e a d i t i s polyconvex.
In a forthcoming paper w i t h Kohn w e s t u d y e l a s t i c design s u b j e c t t o two loads, l e a d i n g t o t h e new minimum p r i n c i p l e
I n t h i s case
n , ~ represents a
loads It would be
2
by
n.
2
by
2
matrix; with
Only t h e c a s e of a s i n g l e l o a d
l e a d s t o an e q u i v a l e n t convex problem; for
n = 2
the
relaxed integrand i s polyconvex--a convex f u n c t i o n of and t h e i r determinant
D =
n
d.7.
a,~,
The underlying theory was
developed a b s t r a c t i y by Morrey [ 1 5 ] , and more r e c e n t l y b y
Ball [16] and Dacorogna [ 1 7 ] .
Perhaps our example i s t h e
f i r s t involving a l l t h r e e arguments i n which t h i s polyconvexif i c a t i o n has been found.
ACKNOWLEDGEMENT W e g r a t e f u l l y acknowledge t h e support of t h e National
Science Foundation (MCS 81-02371 and INT 81-00464) and t h e Army Research Off i c e (DAAG 29-8O-KO033).
L ' and L' Approximation of Vector Fields
287
REFERENCES Ekeland, I . and Temam, R . , Convex Analysis and V a r i a t i o n a l Problems (North-Holland, Amsterdam, 1976) . Kohn, R . and S t r a n g , G., S t r u c t u r a l d e s i g n o p t i m i z a t i o n , homogenization, and r e l a x a t i o n of v a r i a t i o n a l p r o b l e m , i n : Papanicolaou, G . ( e d . ) , Disordered Media, Lecture Notes i n physics 1 5 4 (Springer-Verlag, New York, 1 9 8 2 ) . Rozvany, G . I . N . , Optimal Design of F l e x u r a l Systems (Pergamon, Oxford, 1 9 ' 7 6 ) . Matthies, H . , S t r a n g , G . and C h r i s t i a n s e n , E . , The s a d d l e p o i n t of a d i f f e r e n t i a l program, i n : Glowinski, R . , Rodin, E . , and Zienkiewicz, O . C . ( e d s . ) , Energy Methods i n F i n i t e Element Analysis (John Wiley, New York, 1 9 7 9 ) . Temam, R . and S t r a n g , G . , Functions of bounded deformat i o n , Arch. Rat. Mech. Anal. 75 (1980) 7-21. S t r a n g , G . , Maximal flow through a domain, Mathematical Programming ( t o a p p e a r ) . T r i , M . , Theory of flows i n c o n t i n u a a s approximation t o flows on networks, i n : Prekopa, A . ( e d . ) Survey of Mathematical Programming (North-Holland, Amsterdam, 1978) .
Bombieri, E., DeGiorgi, E . and G i u s t i , E . , Minimal cones and t h e B e r n s t e i n problem, I n v e n t i o n e s Math. 7 (1969) 243-268. Fleming, W . and R i s h e l , R . , An i n t e g r a l formula f o r t o t a l g r a d i e n t v a r i a t i o n , Archiv d e r Mathematik 11 (1960) 218-222. Federer, H., Geometric Measure Theory (Springer-Verlag, New York, 1 9 6 9 ) . M o s O ~ O V J P.P.,
On t h e t o r s i o n of a r i g i d - p l a s t i c c y l i n d e r ,
PMM 4 1 (1977) 344-353.
S t r a n g , G . and Kohn, R . , Optimal d e s i g n of c y l i n d e r s i n s h e a r , i n : Whiteman, J . ( e d . ) , The Hathematics of F i n i t e Elemcnts and n p p l i c a t i o n s TV (Academic Press, Iondon, lU8?). S t r a n g , G . , A f a m i l y of model problems i n p l a s t i c i t y , i n : Glowinski, R . , Lions, J . L. ( e d s . ) , Proceedings of t h e Symposium on Computing Methods i n Applied S c i e n c e s , Lecture Notes i n Mathematics 704 (Springer-Verlag, New York, 1 9 7 9 ) . S t r a n g , G . , A minimax problem i n p l a s t i c i t y theory, i n : Nashed, Z . ( e d . ) , J?unctional Analysis Methods i n Numerical Analysis, Lecture Notes i n Mathematics 701 (Springer-Verlag, New York, 1 9 7 9 ) .
Gilbert STRANC
288
[15] Morrey, C.B., Multiple Integrals in the Calculus of Variations (Springer-Verlag, Berlin, 1966). [16] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Nech. Anal. 63 (1977)
337-403.
[17] Dacoro$na, B., Weak Continuity and Weak b w e r Semi-
continuity of Nonlinear Functionals, Springer Lecture Notes in Mathematics 922 (Springer-Verlag, Berlin, 1982)
L1
and
Approximation of Vector F i e l d s i n t h e Plane
L"
Gilbert Strang Massachusetts i n s t i t u t e of Technology
ABSTRACT
We s t u d y f o u r problems, two i n whose analogues i n
L2
L1
and two i n
a r e t h e f a m i l i a r minimum
p r i n c i p l e s which l e a d t o t h e Laplace equation.
One
p o s s i b i l i t y is t o be given t h e boundary v a l u e
6 = Q
and t o minimize point
(x,y)
IIVJrlll
in
Cl
Lmr
or
llV@~lrn;
i s measured by
the gradient a t a lVJrI2
2 = f,
+ Q Y2 '
In t h e o t h e r problems we a r e given a v e c t o r f i e l d v: R
+
R2,
and minimize e i t h e r
IIVw-vlll
or
~IoW-v!lm.
i n each c a s e we use t h e d u a l i t y t h e o r y of convex a n a l y s i s t o g i v e e q u i v a l e n t s t a t e m e n t s of t h e problem, o f t e n with an i n t e r p r e t a t i o n i n mechanics and o f t e n partly solved.
Nevertheless some q u e s t i o n s s t i l l
remain open.
213
274
Gilbert S T R A N G
L2 norm l e a d s t o l i n e a r equations
Approximation i n t h e
(which a r e forbidden a t t h i s c o n f e r e n c e ) .
In
<
LP, 1
P
<
(0,
t h e equations become n o n l i n e a r but much of t h e a n a l y s i s continues t o apply.
In
i s entirely different:
L1
and
Lm, however,
the s i t u a t i o n
it i s not t h e d i f f e r e n t i a l equation
b u t t h e underlying v a r i a t i o n a l p r i n c i p l e t h a t l e a d s t o an e x i s t e n c e theory, and suggests how t o c o n s t r u c t t h e optimal solution. This note s t u d i e s f o u r t y p i c a l problems for s c a l a r valued f u n c t i o n s on a simply connected domain
r.
s u f f i c i e n t l y smooth boundary
fl
C
R2
with
Each of t h e problems has
a dual--a maximization i n s t e a d of a minimization--and a p p l i c a t i o n s t h e dual i s of equal importance.
i n the
Where one
v a r i a t i o n a l statement i s t h e " s t a t i c " formulation of a problem i n mechanics, with s t r e s s e s a s t h e unknown, t h e o t h e r i s t h e "kinematic" form i n terms of v e l o c i t i e s .
We
w i l l study s e v e r a l combinations of boundary c o n d i t i o n s and
inhomogeneous terms, b u t n o t every p o s s i b l e combination, because a l r e a d y we ask t h e r e a d e r ' s consent about one more thing.
In a d d i t i o n t o t h e dual of each problem, t h e r e i s
another p a i r of optimizatlons ( e q u i v a l e n t t o the given one) c r e a t e d by a s p e c i a l s i t u a t i o n i n s o l u t i o n of
div n
= 0
is
T
=
($
R2,
t h a t the general
- $ x ) for some
Y9
4.
Therefore it w i l l happen t h a t each of our problems has four e q u i v a l e n t forms, and t h a t one of them i s simpler t o s o l v e than t h e o t h e r s .
(For t h e o t h e r s we may l e a r n only t h e value
of t h e maximum o r minimum, by d u a l i t y , without f i n d i n g t h e f u n c t i o n which achieves t h a t v a l u e . )
Some q u e s t i o n s w i l l
remain unsolved even w i t h four a l t e r n a t i v e s .
L' and L" Approximation of Vector Fields
275
The problems arise i n t h e s t u d y of p l a s t i c i t y and optimal p l a s t i c design, and elsewhere; we w i l l g i v e r e f e r e n c e s r a t h e r t h a n a complete d e s c r i p t i o n of t h e s e a p p l i c a t i o n s .
And we do
t h e same f o r t h e proofs of d u a l i t y ; i n our problems t h e y come d i r e c t l y from t h e techniques of Ekeland-Temam [ll, who a p p l i e d t h e Moreau-Rockafellar t h e o r y t o a sequence of important examples i n p a r t i a l d i f f e r e n t i a l e q u a t i o n s .
Our c h i e f purpose
i s t o c o n t r i b u t e some a d d i t i o n a l examples, and t h e y have
developed from our j o i n t work w i t h Robert Kohn and Roger Temam. We mention t h a t optimal d e s i g n l e a d s t o more complicated v a r i a t i o n a l problems ([2-31
i s p a r t of a l a r g e l i t e r a t u r e ) ,
and a l s o t h a t p e r f e c t p l a s t i c i t y i n
has r e q u i r e d a new
R3
space of vector-valued displacements and a corresponding analysis
[4-51. The problems i n
t h i s note a r e e a s i e r , b u t
t h e y have n a t u r a l i n t e r e s t and it may be u s e f u l t o organize them more s y s t e m a t i c a l l y . A t t h e end we d i s c u s s some a p p l i c a t i o n s i n optimal d e s i g n
and t h e nonconvex problems t o which t h e y l e a d .
1. The minimum of
l/Vt!!m
w i t h Dirichlet data
It i s t h i s form which can be solved d i r e c t l y , b u t we
begin w i t h t h e f o u r e q u i v a l e n t problems: 1A.
M I N I!n(lm s u b j e c t t o
1B.
MAX
1C.
MIN
1D.
MAX
r
uf
j1VdrIlrn
subject t o subject t o
div o = 0 JslVul 0
rlr = g
=
and
1
on
r
o.n = f
on
r
216
Gilbert S I R A N G
The conversion between 1 A and 1C i s by r o t a t i o n through f = n.n = V $ . t
Therefore
=
of
s/2
It follows t h a t
VQ.
&/&, t h e t a n g e n t i a l d e r i v a t i v e of
6 = g = s f ds
assumed t h a t
a
fl = ( l y J - ? b x ) ,
Jf ds
= 0
on t h e c l o s e d curve
S i m i l a r l y 1B and 1 D a r e connected by i n t e g r a t i o n by p a r t s on
rj r.
up t o a c o n s t a n t on
r.
1B
7
1.
it i s
= (uy,-ux),
and
i s dual t o lA, and 1D t o 1C.
The s o l u t i o n of 1 C comes from an e x t e n s i o n lemma proved independently by McShane and Kirszbraun f o r I d p s c h i t z f u n c t i o n s on a metric space.
Their construction applies
e q u a l l y t o Hblder continuous f u n c t i o n s , and was followed by deeper r e s u l t s of Whitney and Calderon.
In our case t h e
metric i s t h e s h o r t e s t d i s t a n c e w i t h i n
0; i t is Euclidean
distance, i f
Q
i s convex.
Then t h e q u a n t i t y
minimized i s t h e Lipschitz c o n s t a n t f o r g
on
to
l on
Ca
1.
t o be
IIVJIII,
The lemma extends
w i t h no i n c r e a s e i n t h e L i p s c h i t z
c o n s t a n t , by t h e simple c o n s t r u c t i o n
Therefore t h e minimum in 1C i s immediate: Lipschitz c o n s t a n t f o r
it e q u a l s t h e
g, and Problem 1 i s solved.
T h i s provides a n a t u r a l analogue f o r continuous flows of
t h e max flow-min c u t theorem of Ford and Fulkerson.
Instead
of a f i n i t e network with c a p a c i t y c o n s t r a i n t s on t h e edges, t h e flow through i t s c a p a c i t y by
capacity
In1
5
0
i s described by a v e c t o r f i e l d
la1
c(x,y)
fl a r e included i n
0
and
1. The p o s s i b i l i t i e s of varying and nonzero sources
div u
=
F
[6]. I r i c r e a t e d a s i m i l a r t h e o r y
n e a t a p p l i c a t i o n t o t r a f f i c i n Tokyo [ 7 ] .
within
with a
L' and L' Approximation of Vector Fields
211
We g i v e a b r i e f b u t very informal d e r i v a t i o n of t h e o p t i m a l i t y c o n d i t i o n s t h a t connect 1A t o lB, w i t h Lagrange m u l t i p l i e r for t h e c o n s t r a i n t
u(x,y)
as
div n
=
0.
The saddle-
SSIVuI
>
1, and otherwise
p o i n t ( k g r a n g i a n ) form i s
The f i n a l minimum over
it i s
Suf
as i n 1 B .
TT
is
if
-m
The o p t i m a l i t y c o n d i t i o n s are
which gives an i n t e r e s t i n g form f o r
u.
It i s the character-
i s t i c f u n c t on, or more e x a c t l y a m u l t i p l e
of t h e
l//P-Q
c h a r a c t e r i s t i c function, of t h e s e t bounded by t h e l i n e between
and
P
Q
on
I-
--where t h e s e a r e t h e p o i n t s ( n o t
n e c e s s a r i l y unique) a t which Thus
Vu
=
0
g
except a c r o s s t h i s l i n e ; on t h e l i n e ,
s i n g u l a r measure of mass one and magnitude
>
Q = (0,-r).
0
i s t h e normal v e c t o r of
on t h e c i r c l e
The Lipschitz c o n s t a n t i s
x
n
and
1/2r
c1
$,J -trx)
An optimal
u
and
n
r
<
1/2.
P = (0,r)
is zero i n the semicircle
i n t h e complement
= (l/r,O),
of r a d i u s
l/r, between t h e p o i n t s
x
1. 0.
( n o t t h e same as McShane's e x t e n s i o n ) i s =
is a
Vu
Iluli,.
Example 1: g = s i n 0
and
a t t a i n s i t s Lipschitz constant
1
An optimal =
y/r,
t
with
i s normal t o t h e diameter
FQ.
A l e s s t r i v i a l example, with more uniqueness, would s t a r t w i t h an e l l i p t i c a l
n.
278
Gilbert STRANG
2.
The minimum of
with D i r i c h l e t d a t a
\lVllrlll
Again w e g i v e t h e f o u r e q u i v a l e n t extremal p r i n c i p l e s :
SsIflI
PA.
MIN
2B.
MAX s r u f
2C.
MIN J S l V S l
2D.
MAX
n
subject t o div subject t o
and
n = 0
a-n
=
f
1 1 V ~ := ) ~1
subject t o subject t o
1
div
T
r
on
g
=
= 0
and
!\T'la
= 1.
The connections among 2A-2D a r e t h e same a s f o r 1A-1D.
I n t h i s case i t i s a g a i n 2 C ( t h e least g r a d i e n t problem
181) t h a t can be solved most d i r e c t l y , using t h e coarea formula f o r a function
where
yt
length.
4(x,y)
of bounded v a r i a t i o n :
is t h e l e v e l s e t where More p r e c i s e l y , s i n c e
1b
s e t of p o s i t i v e area, we c o n s t r u c t
and then
yt = a E t \ r .
Jr = t
and
is its
IYtl
could be c o n s t a n t over a E~ = ( ( x , y ) E Q: llr(x,y)
This coarea formula and i t s g e n e r a l i z a -
t i o n s a r e a v a l u a b l e t o o l i n geometric measure t h e o r y for a smooth f u n c t i o n
t]
$,
[9-lo];
or a piecewise l i n e a r f u n c t i o n , t h e
proof is s t r a i g h t f o r w a r d . To minimize
of
Yt,
llV@\\l
i s t o minimize f o r each
s u b j e c t t o t h e requirement t h a t
boundary p o i n t s a t which
g
may be required; t h e s e t
Et
g
2
t.
We note t h a t f o r
g
in
L1(r), t h e f u n c t i o n
t
and is i n t e g r a b l e . )
$
=
t.
Yt
t
the length
connects t h e
(Again a more p r e c i s e form
must c o n t a i n t h e s e t on which of bounded v a r i a t i o n , w i t h t r a c e IYtl
i s d e f i n e d for almost a l l
L’ and L L Approximation of Vector Fields
Example 2 .
0. The L2 norm i s s m a l l e s t f o r t h e harmonic f u n c t i o n 6 = r 2 cos 20,
of
g = cos 28
279
on t h e u n i t c i r c l e
z e r o o n l y on t h e f o u r r a y s t o
0 =
norm i s minimized by a f u n c t i o n whole i n s c r i b e d square of s i d e diagonals.
2
\b
J=’
g =
cOS
=
L1
w i t h t h e s e r a y s as
Between t h e s q u a r e and t h e c i r c l e , t h e l e v e l l i n e s
of t h e square, f o r example,
x
But t h e
which v a n i s h e s over t h e
4 = c o n s t a n t a r e s t r a i g h t , t o minimize
lines
2 3s/4.
IT/&,
$ =
To t h e r i g h t
lYt(.
2x2-1:
c o n s t a n t on v e r t i c a l
r
c o n s t a n t , and a g r e e i n g a t t h e boundary
=
1 with
28 = 2 cos2 0 - 1 .
I n t h e d u a l i t y with 2B, t h e o p t i m a l i t y c o n d i t i o n i m i t a t e s
1B above t o g i v e
Therefore
Vu
i s a u n i t v e c t o r i n t h e d i r e c t i o n of
n
# 0, and elsewhere
llr
= 2x2-1
circle. fx u =-
and
~y
< 1.
n = (0,-4x)
Therefore
or
IVul
I n o u r example we had
i n the section
Vu = ( 0 , l )
and
The
L’
x
u = -y.
2
l/n
of t h e
Similarly
i n t h e f o u r q u a r t e r s of t h e c i r c l e - - b u t n o t
uniquely so i n t h e i n s c r i b e d s q u a r e where
3.
where
0
$ = 0
and
CT =
0.
approximation of a v e c t o r f i e l d by a g r a d i e n t
The given v e c t o r f i e l d i s
i s i t s e l f a gradient i f
F
=
v = (v,(x,y),v2(x,y)),
dvl/dy
-
&,/ax
= 0.
and i t
Otherwise
t h e maxima and minima w i l l exceed z e r o i n o u r f o u r e q u i v a l e n t problems :
3A.
280
Gilbert STRANC
3s.
MIN ~ ~ I V w - v I
3C.
MAX JJFb
3D.
MIN
c1
In 3 D ,
subject t o
iSl1-1 s u b j e c t
Cl
-F. through
~ / 2 ;t h e
i s a u t o m a t i c a l l y zero, s o t h a t
Again we use a p r o p e r t y s p e c i a l t o
-F.
=
=
7
Ow-v
= (-wy,wx)
Ow'
-aiv v
7 =
div
i s t h e r o t a t i o n of
7
divergence of div
to
T, lv~lrl 5 1 i n
on
= 0
1C
R2.
In t h i s c a s e it i s 3 C which can be solved a t s i g h t , provided we assume i s t o maximize lvllll
5
F
0.
To make
as l a r g e a s p o s s i b l e
//Fib
among f u n c t i o n s v a n i s h i n g on
$
1. The extremal f u n c t i o n i s
with
r,
Q = distance t o
and
t h i s choice g i v e s t h e optimal values i n 3 A - 3 D . of varying s i g n t h e problem i s much more d i f f i c u l t
F
For
and i n t e r e s t i n g ; we b e l i e v e it t o be unsolved. in
t h e optimal
R1
Q
has
1lr1
On an i n t e r v a l
-
= +1, and t h e "breakpoints"
can be determined. A c o n s t r u c t i o n of t h e optimal
w
(assuming
F
1. 0 )
was
given by Mosolov [U].The computation i s more d e l i c a t e t h a n t h a t of
and i s guided by t h e o p t i m a l i t y c o n d i t i o n
111,
n =
vw-v , F a t every p o i n t
Example 3 .
v
t h i s domain n =
( ey'
-$x)
=
and
F = 2
Vw
# v.
on t h e u n i t c i r c l e .
Ib = I-r, t h e d i s t a n c e t o t h e boundary.
On
Therefore
i s t h e u n i t v e c t o r f i e l d t a n g e n t t o t h e con-
centric circles instance
y,-x)
where
w c 0.
i s s a t i s f i e d by
r = constant.
It happens t h a t i n t h i s
The o p t i m a l i t y c o n d i t i o n d i s p l a y e d above
u
=
-v/IvI,
and t h e r e f o r e
"CaMOt be approximated by a g r a d i e n t . I' mation i s null..
v = (y,-x)
The c l o s e s t approxi-
28 I
L ' and L' Approximation of Vector Fields T h i s choice of
c y l i n d r i c a l rod. -slv.o
The v e c t o r
gives t h e shearing s t r e s s and
d
is t h e a s s o c i a t e d moment r e s i s t i n g an e x t e r n a l f o r c e
t h a t t w i s t s the bar. tion
a r i s e s n a t u r a l l y i n t h e t o r s i o n of a
v
161
5
1
The maximum i n 3A, s u b j e c t t o t h e l i m i t a -
f o r a p l a s t i c m a t e r i a l , i s t h e l i m i t moment of a
b ar w i t h cross-section
61.
T h i s i s t h e l a r g e s t torque the b a r
can resist; t h e angle of t w i s t approaches i n f i n i t y and t h e bar becomes f u l l y p l a s t i c The d u a l v a r i a b l e
( I n 1 = 1 throughout
w(x,y)
Q).
measures t h e "warping" of each
c r o s s - s e c t i o n out of i t s o r i g i n a l plane--and
f o r the circular
b a r of Example 3 each c r o s s - s e c t i o n remains plane and
w = 0.
The minimization 3D has a f u r t h e r mechanical i n t e r p r e t a tion.
It a g a i n r e f e r s t o a b a r w i t h c r o s s - s e c t i o n
subject t o the axial force
F(x,y) .
r e s i s t e d by s h e a r s t r e s s e s
T
expresses e q u i l i b r i u m .
= (7
ss
Then
This body f o r c e i s T
xz' yz
1
17
0 , b u t now
), and
div
7
=
-F
is proportional t o the
minimum weight of a b a r which can withstand t h e load
It
F.
i s a s p e c i a l i z a t i o n of t h e Michell-Prager t h e o r y of optimal design t o t h e c a s e of pure s h e a r [12].
4.
The
Lm
For given
approximation of a v e c t o r f i e l d by a g r a d i e n t
v, w i t h
F = -div v'
and
T
as
= (OW-v)'
before, t h e e q u i v a l e n t problems a r e
4A. MAX - s s d , v 4B.
subject t o
d i v n = 0, n.n
=
0
on
M I N IIVw-v/lrn
4 ~ .MAX
JJF@ s u b j e c t
4D. MIN
/\7/Irn
to
$ =
o
subject t o
div
7
on =
-F.
r, IIv$Il,
= 1
r,
IIulll = 1
Gilbert STRANG
282
Our method i s t o apply t h e coarea formula t o 4 C . show t h a t t h e o p t i m a l function
i s a m u l t i p l e of t h e c h a r a c t e r i s t i c
Q
of some s u b s e t
C
We want t o
E
C
The coarea formula can be
Q.
written as m
where
Et = [ $
is t h e c h a r a c t e r i s t i c f u n c t i o n of
Ct
2. t l .
It i s n o t d i f f i c u l t [ 6 ] t o show t h a t a l s o m
JJF$
n
Suppose
= ,m
(JSFCt) d t .
n
i s t h e maximum i n QC,
M
and suppose t h a t
f o r every c h a r a c t e r i s t i c f u n c t i o n
f o r t h e optimal
t
C.
Then choosing
( o r more p r e c i s e l y
C = Cnt
C = Ct
for a
t n ) w e would c o n t r a d i c t t h e
maximizing sequence of f u n c t i o n s
t.
previous e q u a t i o n s by i n t e g r a t i n g over
Therefore t h e
maximum ( o r supremum) i s a t t a i n e d i n 4C by a normalized characteris tic function
~
= C/\\VC\ll.
I n o t h e r words, Problem 4 C i s e q u i v a l e n t t o t h e s i m p l e r problem
Whenever
F
i s c o n s t a n t , we have an i s o p e r i m e t r i c problem:
n.
maximize area/perirneter w i t h i n weighted i s o p e r i m e t r i c problem.
For
And f o r
F
> F
0
it i s a
of varying s i g n ,
I t i s a g e n e r a l i z e d i s o p e r i m e t r i c problem which i s new t o We have not mentioned t h e boundary c o n d l t i o n seems t o be v i o l a t e d i f
aC
n
r #
0.
JI
US.
= 0, which
Nevertheless t h e a n a l y s i s
L ' and L ' Approximation of Vector Fields
c a n be j u s t i f i e d ;
it i s the condition
t
283
which must be
= 0
r e l a x e d [13], and t h e c o r r e c t form of 4 C i s
S i m i l a r l y Problem 2 C can be r e l a x e d by t h e boundary i n t e g r a l The e f f e c t i n Problem 4 C i s t h a t t h e l e n g t h of
I$-gI.
i s i n c l u d e d i n t h e p e r i m e t e r of
aC
n
r
and we r e a c h t h e i s o p e r i -
C,
m e t r i c problem d e s c r i b e d above. We show by example t h a t t h e o p t i m a l
(at least for
can be computed
~
F = 1 and f o r simple domains).
Since
is
Q
piecewise c o n s t a n t , changing o n l y a t t h e boundary of t h e o p t i m a l set
i n t h e i s o p e r i m e t r i c problem,
C
a C --a
s i n g u l a r measure supported on
= ($
CT
Y' " l i n e of
The o p t i m a l i t y c o n d i t i o n c o n n e c t i n g it t o
T
-$x)
is a
& - f u n c t i o n s .I '
gives only
moderate i n f o r m a t i o n :
We have no method f o r computing Example 4:
v = (y,-x)
and
n.
in
T
n.
on t h e u n i t c i r c l e
F = 2
T h i s i s t h e o l d e s t of a l l i s o p e r i m e t r i c problems, and t h e s u b s e t which maximizes a r e a / p e r i m e t e r i s field
T
is radial, w i t h
gradient t o Example 5 :
v
is
Ow =
v = (y,-x)
with v e r t i c e s a t
div
T
=
-2,
C =
a.
The v e c t o r
and a g a i n t h e n e a r e s t
0.
and
(+ 1/2,+- 1 / 2 ) .
n,
on t h e u n i t s q u a r e
F = 2
The o p t i m a l s u b s e t
n e i t h e r t h e whole square n o r t h e i n s c r i b e d c i r c l e .
C
Instead
it i s a compromise [6,14] whose boundary c o i n c i d e s w i t h except f o r q u a r t e r c i r c l e s of r a d i u s
(2
f
fi)"
is
r
i n the four
284
Gilbert
c o r n e r s of t h e s q u a r e . i s smooth.)
STRANG
(They a r e t a n g e n t t o t h e square, s o
aC
We have s o f a r been u n s u c c e s s f u l i n determining
a corresponding v e c t o r f i e l d
with
T
div
7
= (2
+
fi)!lT\lm.
It e x i s t s , by d u a l i t y t h e o r y , and we have o f f e r e d a modest
p r i z e (10,000 Yen a t t h e U.S.-Japan Seminar) f o r i t s d i s c o v e r y .
L1
It would be i n t e r e s t i n g t o compare t h e s e
and
L”
o p t i m i z a t i o n s with t h e corresponding d i s c r e t e problems i n and
a”.
4’
There t h e o p t i m a l i t y c o n d i t i o n s a r e c l a s s i c a l , and
t h e b e s t approximations must approach o u r s o l u t i o n s ( i n c l u d i n g t h e c h a r a c t e r i s t i c f u n c t i o n s ) i n an i r r e g u l a r b u t c o n s i s t e n t way.
The r a t e of convergence, and t h e p a t t e r n of error i n t h e
d i s c r e t e problems, should be v i s i b l e from numerical experiments-s i n c e this i s a c l a s s of d i s c r e t e problems i n which t h e cont i n u o u s l i m i t can be s o l v e d .
We h e s i t a t e t o propose a more complete l i s t of d u a l v a r i a t i o n a l problems of t h e same t y p e . appear f o r o p t i m i z a t i o n i n
L2,
Harmonic f u n c t i o n s w i l l
and t h e s p e c i a l p r o p e r t y of two
dimensions (which produced a l l t h e second p a i r s of e q u i v a l e n t problems) i n t r o d u c e s t h e conjugate harmonic.
Within t h e l i s t
above t h e r e a r e combinations of c o n d i t i o n s t h a t e a r l i e r e n t e r e d only s e p a r a t e l y , f o r example t h e combination MAX JJF$ s u b j e c t t o
P = g
on
f
and
! l V $ ! l w = 1.
This corresponds t o a p p l y i n g both s h e a r and t o r s i o n t o a c y l i n d r i c a l rod, and it i s solved ( i f t h e c o n s t r a i n t s a r e compatible and
F
>
0) by t h e maximal f u n c t i o n
3i(P) = min [ g ( Q ) + d ( P , Q ) I . A r e l a t e d problem mixes
L2 and
Lm, arid has become a
fundamental example i n t h e t h e o r y of v a r i a t i o n a l i n e q u a l i t i e s :
L’ and L* Approximation of Vector Fields
285
The d u a l minimizes a combined norm
This seems a p p r o p r i a t e a l s o f o r ” r o b u s t s t a t i s t i c s , “ i n which t h e l e a s t squares model (Gauss-Markov l i n e a r r e g r e s s i o n ) i s
natural--except t h a t it a s s i g n s t o o much weight t o o b s e r v a t i o n s t h a t l i e f a y o u t s i d e t h e normal range. 1
significant i n
L
, and
These o u t l i e r s a r e l e s s
a mixed norm i s more r e a l i s t i c .
F i n a l l y we mention optimal design, which i s s u b j e c t t o a l l t h e s e c o n s t r a i n t s and one more:
it begins a s a nonconvex
problem, t o minimize t h e support of
A. t y p i c a l case, f o r
n.
longitudinal shear i n a p l a s t i c cylinder, i s
The i n t e g r a n d jumps from to
at
00
(01
= 1
0
to
1 at
>
(since
u = 0, and from
1 i s inadmissible).
1
The
e q u i v a l e n t “ r e l a x e d problem” r e p l a c e s t h i s i n t e g r a n d by the l a r g e s t convex f u n c t i o n which does n o t exceed it:
10)
for MIN
lul
SJ’IUI
5
1 and
subject t o
m
for div
Id1
CI =
0,
>
1.
n-n
it e q u a l s
I n t h i s new problem = f,
11U11,
2
1
t h e r e e x i s t s an optimal s o l u t i o n (which i s a weak l i m i t
Of
t h e h i g h l y o s c i l l a t o r y minimizing sequences i n t h e o r i g i n a l problem).
The s o l u t i o n can a c t u a l l y be computed by modifying
t h e c o n s t r u c t i o n i n 2 C t o account f o r t h e new c o n s t r a i n t
101
= (V$(
2
1.
It g i v e s t h e admissible s t r u c t u r e of minimum
weight. There i s a l s o a more d e l i c a t e c l a s s of nonconvex problems,
Gilbert STRANC
286
whose relaxed form f a i l s t o be convex:
i n s t e a d i t i s polyconvex.
In a forthcoming paper w i t h Kohn w e s t u d y e l a s t i c design s u b j e c t t o two loads, l e a d i n g t o t h e new minimum p r i n c i p l e
I n t h i s case
n , ~ represents a
loads It would be
2
by
n.
2
by
2
matrix; with
Only t h e c a s e of a s i n g l e l o a d
l e a d s t o an e q u i v a l e n t convex problem; for
n = 2
the
relaxed integrand i s polyconvex--a convex f u n c t i o n of and t h e i r determinant
D =
n
d.7.
a,~,
The underlying theory was
developed a b s t r a c t i y by Morrey [ 1 5 ] , and more r e c e n t l y b y
Ball [16] and Dacorogna [ 1 7 ] .
Perhaps our example i s t h e
f i r s t involving a l l t h r e e arguments i n which t h i s polyconvexif i c a t i o n has been found.
ACKNOWLEDGEMENT W e g r a t e f u l l y acknowledge t h e support of t h e National
Science Foundation (MCS 81-02371 and INT 81-00464) and t h e Army Research Off i c e (DAAG 29-8O-KO033).
L ' and L' Approximation of Vector Fields
287
REFERENCES Ekeland, I . and Temam, R . , Convex Analysis and V a r i a t i o n a l Problems (North-Holland, Amsterdam, 1976) . Kohn, R . and S t r a n g , G., S t r u c t u r a l d e s i g n o p t i m i z a t i o n , homogenization, and r e l a x a t i o n of v a r i a t i o n a l p r o b l e m , i n : Papanicolaou, G . ( e d . ) , Disordered Media, Lecture Notes i n physics 1 5 4 (Springer-Verlag, New York, 1 9 8 2 ) . Rozvany, G . I . N . , Optimal Design of F l e x u r a l Systems (Pergamon, Oxford, 1 9 ' 7 6 ) . Matthies, H . , S t r a n g , G . and C h r i s t i a n s e n , E . , The s a d d l e p o i n t of a d i f f e r e n t i a l program, i n : Glowinski, R . , Rodin, E . , and Zienkiewicz, O . C . ( e d s . ) , Energy Methods i n F i n i t e Element Analysis (John Wiley, New York, 1 9 7 9 ) . Temam, R . and S t r a n g , G . , Functions of bounded deformat i o n , Arch. Rat. Mech. Anal. 75 (1980) 7-21. S t r a n g , G . , Maximal flow through a domain, Mathematical Programming ( t o a p p e a r ) . T r i , M . , Theory of flows i n c o n t i n u a a s approximation t o flows on networks, i n : Prekopa, A . ( e d . ) Survey of Mathematical Programming (North-Holland, Amsterdam, 1978) .
Bombieri, E., DeGiorgi, E . and G i u s t i , E . , Minimal cones and t h e B e r n s t e i n problem, I n v e n t i o n e s Math. 7 (1969) 243-268. Fleming, W . and R i s h e l , R . , An i n t e g r a l formula f o r t o t a l g r a d i e n t v a r i a t i o n , Archiv d e r Mathematik 11 (1960) 218-222. Federer, H., Geometric Measure Theory (Springer-Verlag, New York, 1 9 6 9 ) . M o s O ~ O V J P.P.,
On t h e t o r s i o n of a r i g i d - p l a s t i c c y l i n d e r ,
PMM 4 1 (1977) 344-353.
S t r a n g , G . and Kohn, R . , Optimal d e s i g n of c y l i n d e r s i n s h e a r , i n : Whiteman, J . ( e d . ) , The Hathematics of F i n i t e Elemcnts and n p p l i c a t i o n s TV (Academic Press, Iondon, lU8?). S t r a n g , G . , A f a m i l y of model problems i n p l a s t i c i t y , i n : Glowinski, R . , Lions, J . L. ( e d s . ) , Proceedings of t h e Symposium on Computing Methods i n Applied S c i e n c e s , Lecture Notes i n Mathematics 704 (Springer-Verlag, New York, 1 9 7 9 ) . S t r a n g , G . , A minimax problem i n p l a s t i c i t y theory, i n : Nashed, Z . ( e d . ) , J?unctional Analysis Methods i n Numerical Analysis, Lecture Notes i n Mathematics 701 (Springer-Verlag, New York, 1 9 7 9 ) .
Gilbert STRANC
288
[15] Morrey, C.B., Multiple Integrals in the Calculus of Variations (Springer-Verlag, Berlin, 1966). [16] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Nech. Anal. 63 (1977)
337-403.
[17] Dacoro$na, B., Weak Continuity and Weak b w e r Semi-
continuity of Nonlinear Functionals, Springer Lecture Notes in Mathematics 922 (Springer-Verlag, Berlin, 1982)