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By Wiktor Eckhaus (Eds.)

ISBN-10: 0444111077

ISBN-13: 9780444111074

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3. AUXILIARY RESULTS L e t A be an a r b i t r a r y nonempty subdomain o f R N , N t 2 , and l e t k be a nonnegative i n t e g e r . Suppose t h a t o < s < 1 , and l e t p > 1 . (~) such t h a t f o r every multiindex K w i t h nach space w i t h t h e norm IKI = k . ,M , i n R N , covering r , and C3+a-diffeomorphi ms $ j o f U . onto the open u n ' t b a l l B" o f R N p c h t h a t $ j ( U . (U. ,M , Moreover, w i hout loss i f i e n e r a l i t y , we can (and w i l l ! ) assume t h a t $j E C3+u(Uj) and +sf E C3+a($) a hN .

1968) P e r t u r b a t i o n s s i n g u l i h r e s d'In6q. V a r . 932-934. Rat. Analysis. [lo] Bensoussan, A . L. (1975) Diffusion Processes . i n Probabilistic Methods i n I Z i f f e r e n t i a l Equations , Lecture Notes i n Mathematics, ' S p r i n g e r , 451 , pp. E 2 5 . [ l l ] Brbzis, H. P e r s o n a l communication. LIONS 111. VARIATIONAL INEQUALITIES OF EVOLUTION ORIENTATION [2], t h e s o l u t i o n of a s t o c h a s t i c optimal stopping time problem i n terms of V . I .

1). Denote by n1 Then t h e hypotheses imply t h a t an a r b i t r a r y component o f { x E R u,(x) < u,(x)l t h e f u n c t i o n u := u1 - u2 s a t i s f i e s the i n e q u a l i t i e s . 9) . 8). Hence, t h e g e n e r a l i z e d maximum p r i n c i p l e 122, pp. 0. . 9. 1). F o r example, i t i s now p o s s i b l e t o g i v e a d e t a i l e d s t u d y o f t h e n o n l i n e a r e i g e n v a l u e problem . Au = x f(x,u) in Bu = x on g(x,u) , r , R i n p a r t i c u l a r w i t h respect t o t h e question o f t h e existence o f m u l t i p l e solutions.

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New Developments in Differential Equations, Proceedings of the Conference on Analytical and Numerical Approachesto Asymptotic Problems, Universityof Nijmegen by Wiktor Eckhaus (Eds.)


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