Lecture 6: Homogeneous distributions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Two-sided Littlewood-Paley partition of unity on Rn
For all k Z, let
k ( ) = (2k ) ,
10
1
k ( ) = (2k ) =
Lecture 10: The wave equation
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Accoustic waves: gas with molecular mass m
Let
u (t , x ) =
n
= particle density at time t at point x .
V
Let ux
Lecture 9: Local solvability for elliptic pdes
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Second-order, real, elliptic pde on open set Rn
For functions aij , bi , c C 1 (), set
n
P (x ,
Lecture 11: Energy conservation and
Finite Propagation Velocity
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Energy
Denition
For a function u (t , x ), we dene the energy of u at time t b
Lecture 12: Fundamental Solution for the
Wave Equation
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Convolution kernel representation of u (t , x )
For f , g S ,
u (t , x ) =
1
(2 )n
sin(
Lecture 15: The Forward Solution for the
Wave Equation
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
If vs (t , x ) satises
t2 vs (t , x ) x vs (t , x ) = 0 ,
v (t , x ) =
t
0
vs (0, x )
Lecture 14: The Inhomogenous Wave
Equation
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Inhomogeneous wave equation with initial conditions
t2 u (t , x ) x u (t , x ) = F (t , x ) ,
u (0,
Lecture 13: The Wave Equation in Higher
Dimensions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Notation
Let Sn1 = cfw_y Rn : |y | = 1 , let d n1 be induced surface
measure on Sn1 , and n
Lecture 8: p.v. kernels and Hlder spaces
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Assume u is homogeneous order n, mean value 0.
Theorem
Suppose that f C s , 0 < s < 1 , and that f ha
Lecture 7: Principal value distributions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Lecture 6
Theorem
Suppose that 0 > s > n , and v ( ) C (Rn \cfw_0) is
homogeneous order s. Let u = v
Lecture 2: Littlewood-Paley, Hlder spaces
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Sobolev spaces of distributions, for k = 0, 1, 2, . . .
Denition
For k Z+ , v H k (Rn ) if x v L2 (R
Lecture 1: Sobolev spaces
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Class logistics
Recommended text:
Michael E. Taylor, Partial Differential Equations I : Basic Theory
Applied Mathema
Lecture 3: Sobolev embedding and Hlder
spaces
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Littlewood-Paley partition of unity on Rn
Let 0 ( ) = ( ), and for k 1
k ( ) = (2k ) ,
0
1
k ( )
Lecture 4: Sobolev embedding and
Hlder/Zygmund spaces
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Last lecture
We proved
Theorem
If f S (Rn ) , then f C s (Rn ) iff
f
Cs
k f
sup 2ks k f
Lecture H: The Heat Equation
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Last lecture
Theorem
Let
Kt (x ) =
|x |2
1
e 4t ,
(4 t )n/2
Then u (t , x ) = (Kt f )(x ) =
t > 0 , x Rn .
Kt (x
Lecture 5: Local Elliptic Regularity
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 557, Autumn 2012
Hart Smith
Math 557
Compactly Supported Elliptic Regularity
Theorem
Suppose that u is compactly supported, and u = v . Then
v
Math 557
Solutions to Homework 1
Autumn 2012
1. Show that, if m Z+ and 0 < s < 1, then C m+s = C m,s . That is, for a
bounded continuous function f , the condition
sup 2k(m+s) k f
L
A
k0
is equivalent to f C m,s , with f
C m,s
A.
Proof. (). Assume k f L