lec5v2 - 18.152 - Introduction to PDEs , Fall 2004 Prof....

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Unformatted text preview: 18.152 - Introduction to PDEs , Fall 2004 Prof. Gigliola Staffilani Lecture 5 - Distributions, Continued Convergence of Distributions • Change the definition of D by replacing “differentiable” with “differentiable of any order”. We say that a sequence of distributions f n converges to a distribution f if ( f n , φ ) → in R ( f, φ ) for all φ ∈ D . Fact: If f n f then f f • → n → Proof: • ( f n , φ ) = − ( f n , φ ) ≡ ( f , φ ) ) → − ( f, φ Example: • 1 a − a < x < a Consider the function χ a ( x ) = 2 . 0 x > a | | χ a is a distribution: ( χ a , φ ) = χ a ( x ) φ ( x ) dx R 1 = a φ ( x ) dx. 2 | x | <a What is lim a 0 χ a ? → 1 lim( χ a , φ ) = lim φ ( x ) dx a a 0 2 a x <a → → | | Since φ is differentiable, 1 lim φ ( x ) dx = φ ( a ) . a → 0 2 a | x | <a In fact, 1 1 2 a | x | <a φ ( x ) dx − φ ( a ) = 2 a | x | <a [ φ ( x ) − φ ( a )] dx = 2 1 a | x | <a φ ( x )( x − a ) + . . . dx, a → 0 a −−−→ . → So lim a 0 χ a = δ a . → 1 • Definition: Support of a distribution: Let f ∈ D . Let A = { x | ( ∃ B ( x, r ) |∀ φ ∈ D and supp φ ⊂ B ( x, r ) , ( fφ ) = 0) } . Then supp f = A c . Example: supp δ a = { a } . Fact: If supp f is compact, then we can extend f to C 1 ( R n ) C . The way we extend is this: let g ∈ D → , g ≡ 1 on supp f . Then for φ ∈ C 1 ( R n ) we define ( f, φ ) ≡ ( f, gφ ). This is defined since gφ ∈ D ....
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This note was uploaded on 11/14/2011 for the course MATH 358 taught by Professor Dsd during the Spring '11 term at Middle East Technical University.

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lec5v2 - 18.152 - Introduction to PDEs , Fall 2004 Prof....

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