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inverse_lecture12

# inverse_lecture12 - 1 Lecture 12 Denition 1 Let be a...

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1 Lecture 12 Definition 1 Let α be a multi-index. The function D α f (which is actually an equivalence class of functions) is called the α th weak derivative of f if for any test function φ C 0 (Ω) D α f, φ Ω D α f ( x ) φ ( x ) dx = ( - 1) | α | f, D α φ An equivalent norm on H s is defined by || u 2 H s = | α |≤ s D α u 2 L 2 Examples: 1) Consider u ( x ) on Ω = (0 , 2) definde by u ( x ) = x 2 0 < x 1 2 x 2 - 2 x + 1 1 < x < 2 Note u C 1 (Ω) and u ( x ) = 2 x 0 < x 1 4 x - 2 1 < x < 2 u does not exist in the classical sense at 1. The weak derivative of u is u ( x ) = 2 0 < x 1 4 1 < x < 2 u, u , u L 2 (0 , 2). However, the distributional derivative of u is 2 δ ( x - 1) L 2 (0 , 2). Thus, u H 2 (0 , 2) u 2 H 2 = 2 0 ( u 2 + ( u ) 2 + ( u ) 2 ) dx = 71 . 37 u H 1 (0 , 2) u 2 H 1 = 2 0 (( u ) 2 + ( u ) 2 ) dx = 39 u H 0 (0 , 2) u 2 H 0 = 2 0 ( u ) 2 dx = 20 2) Consider u defined on Ω - ( - 1 , 1) × ( - 1 , 1) defined by u ( x, y ) = x x > 0 0 x 0 Let φ C 0 (Ω). Then Ω ∂u ∂y φ ( x, y ) dxdy = - Ω u ∂φ ∂y dxdy = - 1 0 1 - 1 x ∂φ ∂y dydx = - 1 0 x ( φ ( x, 1) - φ ( x, - 1)) dx = 0 since φ C 0 (Ω) 1

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Thus, ∂u ∂y = 0.
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inverse_lecture12 - 1 Lecture 12 Denition 1 Let be a...

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