weak-derivatives

weak-derivatives - 3 68 BRUCE K. DRIVER 19. Weak and Strong...

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BRUCE K. DRIVER 19. Weak and Strong Derivatives For this section, let be an open subset of R d , p,q,r [1 , ] ,L p ( )= L p ( , B ,m ) and L p loc ( )= L p loc ( , B ,m ) , where m is Lebesgue measure on B R d and B is the Borel σ —a lgebraon . If = R d , we will simply write L p and L p loc for L p ( R d ) and L p loc ( R d ) respectively. Also let h f,g i := Z fgdm for any pair of measurable functions f,g : C such that fg L 1 ( ) . For example, by Hölder’s inequality, if h f,g i is de f ned for f L p ( ) and g L q ( ) when q = p p 1 . De f nition 19.1. A sequence { u n } n =1 L p loc ( ) is said to converge to u L p loc ( ) if lim n →∞ k u u n k L q ( K ) =0 for all compact subsets K . The following simple but useful remark will be used (typically without further comment) in the sequel. Remark 19.2 . Suppose r, p, q [1 , ] are such that r 1 = p 1 + q 1 and f t f in L p ( ) and g t g in L q ( ) as t 0 , then f t g t fg in L r ( ) . Indeed, k f t g t fg k r = k ( f t f ) g t + f ( g t g ) k r k f t f k p k g t k q + k f k p k g t g k q 0 as t 0 19.1. Basic De f nitions and Properties. De f nition 19.3 (Weak Di f erentiability) . Let v R d and u L p ( )( u L p loc ( )) then v u is said to exist weakly in L p ( )( L p loc ( )) if there exists a function g L p ( )( g L p loc ( )) such that (19.1) h u, ∂ v φ i = h g,φ i for all φ C c ( ) . The function g if it exists will be denoted by ( w ) v u. Similarly if α N d 0 and α is as in Notation 11.10, we say α u exists weakly in L p ( )( L p loc ( )) i f there exists g L p ( )( L p loc ( )) such that h u, ∂ α φ i =( 1) | α | h g,φ i for all φ C c ( ) . More generally if p ( ξ )= P | α | N a α ξ α is a polynomial in ξ R n , then p ( ) u exists weakly in L p ( )( L p loc ( )) i f there exists g L p ( )( L p loc ( )) such that (19.2) h u, p ( ) φ i = h g,φ i for all φ C c ( ) and we denote g by w p ( ) u. By Corollary 11.28, there is at most one g L 1 loc ( ) such that Eq. (19.2) holds, so w p ( ) u is well de f ned. Lemma 19.4. Let p ( ξ ) be a polynomial on R d ,k =deg( p ) N , and u L 1 loc ( ) such that p ( ) u exists weakly in L 1 loc ( ) . Then (1) supp m (w p ( ) u ) supp m ( u ) , where supp m ( u ) is the essential support of u relative to Lebesgue measure, see De f nition 11.14.
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weak-derivatives - 3 68 BRUCE K. DRIVER 19. Weak and Strong...

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