By Robert A. Adams, John J. F. Fournier
ISBN-10: 0120441438
ISBN-13: 9780120441433
ISBN-10: 1435608100
ISBN-13: 9781435608108
Sobolev areas offers an creation to the speculation of Sobolev areas and different similar areas of functionality, additionally to the imbedding features of those areas. This conception is wide-spread in natural and utilized arithmetic and within the actual Sciences. This moment variation of Adam's 'classic' reference textual content comprises many additions and lots more and plenty modernizing and refining of fabric. the fundamental premise of the e-book continues to be unchanged: Sobolev areas is meant to supply an exceptional origin in those areas for graduate scholars and researchers alike. * Self-contained and available for readers in different disciplines. * Written at simple point making it obtainable to graduate scholars.
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Extra info for Sobolev Spaces. Pure and applied Mathematics
Sample text
First assume that L 6 [LP(~)] ' is given and IlL; [LP(~2)]'II = 1. Then there exists a sequence {w,} 6 L P ( ~ ) satisfying Ilwnllp = 1 and such that lim,__,~ IL(w,)l = 1. We m a y assume that IL(w,)l > 1/2 for each n, and, replacing w, by a suitable multiple of w,, by a complex n u m b e r of unit modulus, that L(llOn) is real and positive. Let e > 0. By the definition of uniform convexity, there exists a positive n u m b e r 6 > 0 such that if u and v belong to the unit ball of L P ( ~ ) and if II(u + v)/2llp > 1 - 6 , then Ilu - Vllp < ~.
2) Proof. If p - 1, then (2) is an obvious equality. For p > 1, the function t p is convex on [0, cx~); that is, its graph lies below the chord line joining the points The Lebesgue Spaces L p (~2) 24 (a, a p) and (b, bP). Thus (a-+-b) p a p - I - b p < 2 2 ' from which (2) follows at once. 1 If u, v ~ L p (~), then integrating lu(x) + v(x)l p < (lu(x)l + Iv(x)l) p _< 2p-l(lu(x)l p --i-Iv(x)l p) over f2 confirms that u + v E LP (f2). 3 by We shall verify presently that the functional [l" lip defined (The L , N o r m ) Ilullp - (f )l/p [u(x)Pdx is a norm on L p ( ~ ) provided 1 < p < oe.
P r o o f . First assume that L 6 [LP(~)] ' is given and IlL; [LP(~2)]'II = 1. Then there exists a sequence {w,} 6 L P ( ~ ) satisfying Ilwnllp = 1 and such that lim,__,~ IL(w,)l = 1. We m a y assume that IL(w,)l > 1/2 for each n, and, replacing w, by a suitable multiple of w,, by a complex n u m b e r of unit modulus, that L(llOn) is real and positive. Let e > 0. By the definition of uniform convexity, there exists a positive n u m b e r 6 > 0 such that if u and v belong to the unit ball of L P ( ~ ) and if II(u + v)/2llp > 1 - 6 , then Ilu - Vllp < ~.