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Unformatted text preview: 1 Lecture 12 Definition 1 Let be a multiindex. The function D f (which is actually an equivalence class of functions) is called the th weak derivative of f if for any test function C () h D f, i Z D f ( x ) ( x ) dx = ( 1)   h f,D i An equivalent norm on H s is defined by  u k 2 H s = X   s k D u k 2 L 2 Examples: 1) Consider u ( x ) on = (0 , 2) definde by u ( x ) = x 2 < x 1 2 x 2 2 x + 1 1 < x < 2 Note u C 1 () and u ( x ) = 2 x < x 1 4 x 2 1 < x < 2 u 00 does not exist in the classical sense at 1. The weak derivative of u is u 00 ( x ) = 2 < x 1 4 1 < x < 2 u, u , u 00 L 2 (0 , 2). However, the distributional derivative of u 00 is 2 ( x 1) 6 L 2 (0 , 2). Thus, u H 2 (0 , 2) k u k 2 H 2 = Z 2 ( u 2 + ( u ) 2 + ( u 00 ) 2 ) dx = 71 . 37 u H 1 (0 , 2) k u k 2 H 1 = Z 2 (( u ) 2 + ( u 00 ) 2 ) dx = 39 u H (0 , 2) k u k 2 H = Z 2 ( u 00 ) 2 dx = 20 2) Consider u defined on ...
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This note was uploaded on 03/14/2011 for the course MATH 676 taught by Professor Staff during the Fall '08 term at Colorado State.
 Fall '08
 Staff
 Derivative

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