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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## 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.

### Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online