New PDF release: Nonlinear Partial Differential Equations: Sequential and

By Elemer E. Rosinger (Eds.)

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2) i2 c r1 - , by the c A~ c A ~ . Indeed, i t i s obvious t h a t whenever i t i s p o s s i b l e t o r e p l a c e m E < A, i n the conditions d e f i n i n g the sequential s o l u t i o n s i n E m E' =G A ' , w i t h E -I E' , A I I A', -+ A, b y general it y and exactness p r o p e r t i e s o f t h e sequenti a1 sol u= A ' , w i l l be improved. t h e s t a b i 1it y tion s i n EI-1 The f o l 1owing two p r o p o s i t i o n s w i l l y i e l d general c o n d i t i o n s under which maximal s t a b i i t y and g e n e r a l i t y , or exactness can be achieved.

G c M(R) o f measurable f u n c t i o n s . ForJ c quences o f measurable f u n c t i o n s , denote b y A& du, t h e l a r g e s t subalgebra 8 c o n l y i f J i s a subalgebra i n Denote b y J uppose g i v e n a subalgebra a v e c t o r subspace o f se= J) such t h a t J . 8 c J . Then o b v i o u s l y J c A G ( J ) i n which case J w i l l be an i d e a l i n A G ( J ) . 4, IG N t h e s e t o f a l l v e c t o r subspaces 7 c G such t h a t 7 n UG = Q, I u UG c AG ( 7 ) . IG i s a subalgebra i n GN, s i n c e 7 c A G ( 7 ) .

Proof T h i s f o l l o w s f r o m Theorem 1 and ( 6 3 ) , ( 6 1 ) , (62) and ( 7 4 ) . E. Rosinger Remark 3 ( a ) F o r t h e method f o r t h e r e g u l a r i z a t i o n o f s i n g u l a r i t i e s presented i n t h i s s e c t i o n , t h e assumption t h a t t h e s i n g u l a r i t i e s a r e concentrated on nowhere-dense subsets i s e s s e n t i a l a t two stages. F i r s t , i n o r d e r t o e s t a b l i s h t h a t lnd i s an i d e a l i n (Co(n))N(see ( 6 6 ) ) , use i s made o f t h e f a c t t h a t t h e union o f two nowhere-dense subsets i s again nowhere-dense.

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