Notes on
Partial Dierential Equations
John K. Hunter
Department of Mathematics, University of California at Davis
Abstract. These are notes from a two-quarter class on PDEs that are heavily
based on the book Partial Dierential Equations by L. C. Evans, to
viii
CHAPTER 7
Hyperbolic Equations
Hyperbolic PDEs arise in physical applications as models of waves, such as
acoustic, elastic, electromagnetic, or gravitational waves. The qualitative properties
of hyperbolic PDEs dier sharply from those of parabolic P
124
CHAPTER 5
The Heat and Schrdinger Equations
o
The heat, or diusion, equation is
(5.1)
ut = u.
Section 4.A derives (5.1) as a model of heat ow.
Steady solutions of the heat equation satisfy Laplaces equation. Using (2.4),
we have for smooth functions t
Appendix
May the Schwartz be with you!4
In this section, we summarize some results about Schwartz functions, tempered
distributions, and the Fourier transform. For complete proofs, see [19, 25].
5.A. The Schwartz space
Since we will study the Fourier tran
viii
6.A. VECTOR-VALUED FUNCTIONS
189
Appendix
In this appendix, we summarize some results about the integration and dierentiation of Banach-space valued functions of a single variable. In a rough sense,
vector-valued integrals of integrable functions hav
CHAPTER 4
Elliptic PDEs
One of the main advantages of extending the class of solutions of a PDE from
classical solutions with continuous derivatives to weak solutions with weak derivatives is that it is easier to prove the existence of weak solutions. Hav
Appendix
4.A. Heat ow
As a simple physical application that leads to second order PDEs, we consider
the problem of nding the temperature distribution inside a body. Similar equations describe the diusion of a solute. Steady temperature distributions satis
Appendix
In this appendix, we describe without proof some results from real analysis
which help to understand weak and distributional derivatives in the simplest context
of functions of a single variable. Proofs are given in [10] or [12], for example.
The
CHAPTER 1
Preliminaries
In this chapter, we collect various denitions and theorems for future use.
Proofs may be found in the references e.g. [3, 10, 19, 27, 32, 33].
1.1. Euclidean space
n
Let R be n-dimensional Euclidean space. We denote the Euclidean n
CHAPTER 2
Laplaces equation
There can be but one option as to the beauty and utility of this
analysis by Laplace; but the manner in which it has hitherto been
presented has seemed repulsive to the ablest mathematicians,
and dicult to ordinary mathematical
Notes on
Partial Dierential Equations
John K. Hunter
Department of Mathematics, University of California at Davis
Abstract. These are notes from a two-quarter class on PDEs that are heavily
based on the book Partial Dierential Equations by L. C. Evans, to