By R. Glowinski, etc.
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Example text
Loo) may be viewed a s an equ a t i o n r e l a t e d t o a multivalued operator (sometimes c a l l e d a setvalued operator). 100) u s i n g t h e concept o f a conjug- ate convex function. 101) @*(u*) = sup [(u*, u ) - * u ) ] . 1) It may be shown ( s e e Moreau /l/) t h a t p i s a proper convex function. 5. e. in Q } . 91) . We i d e n t i f y L'(i2) with i t s dual; t h u s V c L'(Q) c where H-'(i2) i s th,e s p a c e ' o f d i s t r i b u t i o n s o f t h e form Now t h e c o n j u g a t e f u n c t i o n o f @*(u*) = SUP (u*,U) , and hence uEK i s given by V' = f f - ' ( Q ) , (SEC.
22)is an elasto-plasticity problem. 3 admits a unique solution. 4 The problem can be interpreted as follows ( 2 ) : the open domain R consists of two regions and 62, (the elastic region and the plastic region, respectively). g. Lanchon / 2 / , Glowinski-Lanchon /l/). (2) This will be dealt with in more detail in Chapter 3. Examples (SEC. 24) 1 . t o which we add: u = O on and some t r a n s m i s s i o n c o n d i t i o n s which i n c i d e n t a l l y a r e somewhat awkward t o e x p r e s s e x p l i c i t l y ( s e e Br6zis 111) - a t t h e i n t e r f a c e between Sl, and a,,.
78) I (lp I - g)’ dx = IPI,Q ( s i n c e gradu = 0 a t i o n s ) . Hence SUP P n I grad u 1’ dx = 1 5 I grad u 12 dx IPOQ i n t h e r e g i o n e x c l u d e d from t h e i n i t i a l i n t e g r - inf &(u, w , p) 2 U. w 5I - f o r t h e c h o i c e ( 3 . 7 6 ) of p and t h e r e f o r e from ( 3 . 7 8 ) : divp = 1 i n $2 and t h e i n t e g r a l i s e x t e n d e d t o t h e r e g i o n where I p ( x ) 12 g ( I ) The f o l l o w i n g s h o u l d be u n d e r s t o o d : . (SEC. 71),proves the Theorem.