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 R
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 )
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
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 f
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 representatio
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
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 con
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 | =
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.
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
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
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
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
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
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/
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
Su
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