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A characterization of pseudo-monotone differential operators in divergence form. a
L. Boccardo & B. Dacorogna a
b
Dipartimento di Matematica, Università di Roma II,
b
Département de Mathématiques, Ecole Polytechnique Féderale de Lausanne, Lausanne, 1015 Published online: 08 May 2007.
To cite this article: L. Boccardo & B. Dacorogna (1984): A characterization of pseudo-monotone differential operators in divergence form. , Communications in Partial Differential Equations, 9:11, 1107-1117 To link to this article: http://dx.doi.org/10.1080/03605308408820358
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I n t h i s a r t i c l e w e w i l l b e c o n c e r n e d w i t h pseudo-monotone
cpe-
r a t o r s , w h i c h h a v e b e e n s t u d i e d i n t s n s i v e l y by B r e z i s [ ~ r l l , ~ r o w d e r [ ~ o l ] ,H a r t m a n n - S t s m p a c c h i a
[ I I S ~ ] ,'riass r ~ e l ] ,L e r a y - L i o n s
.
[LL~;..
We w i l l g i v e a c h a r a c t e r i z a t i o n o f .;llch q p e r a t n r s when t h e y are lifferential operators,
i n d i v e r g e n c e f o r m . Our r e s u l t w i l l u s e t h e
methods of t h e c a l c u l u s o f v a r i a t i o n s t h a t a r e c o m o n l y u s e d i n or-
. .l t .l s n s f s r weak ? = y - r s c - i c = n t i n u i : y ; d e r to o b r e F n n e c e s s a r y c o n d ~
( s e e , among o t h e r s x o r e c l o s e l y [MS:]
all, [ ~ o l l , a ill,
-
all
and we w i l l f o l l o w
and [ E T ~ ] ) .
W e now g i v e t h e d e f i n i t i o n ( s e e [ ~ r l l )o f p s e u d o - m o n o t o n i c i t y .
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1107 Copyright @ 1984 by Marcel Dekker, inc.
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resuir.
I n o r d e r t o p r o v e t h e a b o v e t h e o r e m we w i l l n e e d two lemmas
L e m a 1: L z t I
c lRa
b o u n d a r y .and l e t u-,*l b e
b e a bounded Dpen s e t wit!, 2
Lipschitz
s - c h that
LLL
*
where 0 > o i s f i x e d and - d e n o t e s
e x i s t a sequence v such t h a t m
weak
*
c o n v e r g e n c e . Then t 5 e r e
-
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-, t i o n s and r e l i e s o n a lerm.2 o r -. Mae snafie ( s s e
8
~
-".
7
~
p,Z/h). i i Let ~
Eef i n e u (x) m
Observe t h a t since if x
E
x
if
x E 31
E
I
i s l o c a l l y L i p s c h i t z w i t h c o n s t a n t 25 on m ii 21, I_ and y E 21, choose {El = [ x , ~ !n 31 , t h e n 11,
-
.
-m
Using Hac S h a n e ' s fying (2.3)
fm
if
(2.5);
T i
a n e x t e n s i o n of v t o I s a t i s m m i t t h e r e f o r e remains t o show ( 2 . 6 ) . F o r s i m -
lemma we f i n d v
9 l i c i t y we d e n o t e t h e e x p r e s s i o n on t h e r i g h t hand s i d e of ( 2 . 6 ) as r
m
.
Using t h e d e f i n i t i o n of v
II?
we g e t
T 7.
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a\),
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, t h e n g = A ( u ) ) and i s in d i v e r g e n c e form t h e n a ( x , s , .) i s a n o p e r a r o r oi L y p e i:
(i.e.,
e i t h e r l i n e a r o r monotone. Note, i n p a r t i c u l a r , c h a t a weakly cont i n u o u s o p e r a t o r i s of t y p e M and i t h a s b e e n shown by G i a c h e t t i [ ~ i l :t h a t i f A i s weakly c o n t i n u o u s and i n d i v e r g e n c e f o r m ( c . f . ( 1 . 3 ) ) t h e n a ( x , s , E ) must b e l i n e a r w i t h respecrr co r h e v a r i a b l e
