chapter 0
preliminaries
a. notations and definitions
b. results from advanced calculus
c. convolutions
d. the fourier transform
e. distributions
f. compact operators
chapter 1
local existence theory
a. basic concepts
b. real first order equations
c. the general cauchy problem
d. the cauchy-kowalevski theorem
e. local solvability: the lewy example
f. constant-coeffcient operators: fundamental solutions
chapter 2
the laplace operator
a. symmetry properties of the laplacian
b. basic properties of harmonic functions
c. the fundamental solution
d. the dirichlet and neumann problems
e. the green''s function
f. dirichlet''s principle
g. the dirichlet problem in a half-space
h. the dirichlet problem in a ball
i. more about harmonic functions
chapter 3
layer potentials
a. the setup
b. integral operators
c. double layer potentials
d. single layer potentials
e. solution of the problems
f. further remarks
chapter 4
the heat operator
a. the gaussian kernel
b. functions of the laplacian
c. the heat equation in bounded domains
chapter 5
the wave operator
a. the cauchy problem
b. solution of the cauchy problem
c. the inhomogeneous equation
d. fourier analysis of the wave operator
e. the wave equation in bounded domains
f. the radon transform
chapter 6
the l2 theory of derivatives
a. sobolev spaces on r
b. further results on sobolev spaces
c. local regularity of elliptic operators
d. constant-coefficient hypoelliptic operators
e. sobolev spaces on bounded domains
chapter 7
elliptic boundary value problems
a. strong ellipticity
b. on integration by parts
c. dirichlet forms and boundary conditions
d. the coercive estimate
e. existence, uniqueness, and eigenvalues
f. regularity at the boundary: the second order case
g. further results and techniques
h. epilogue: the return of the green''s function
chapter 8
pseudodifferential operators
a. basic definitions and properties
b. kernels of pseudodifferential operators
c. asymptotic expansions of symbols
d. amplitudes, adjoints, and products
e. sobolev estimates
f. elliptic operators
g. introduction to microlocal analysis
h. change of coordinates
bibliography
index of symbols
index
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