Preliminaries [Fo, pp. 1-33]
The divergence theorem; convolutions (approximate identities); Fourier transform (series); distributions

Classifications
Linear, quasilinear, nonlinear equations
Cauchy initial data, boundary conditions
Existence and well-posed problems: Cauchy-Kovalevski: [Fo, pp. 68-80]; Lewy's counterexample [Fo, pp. 80-83]

Constant coefficients: local solvability [Fo, pp. 83-87]; Canonical forms [We]; Classification of 2nd order equations

Linear examples: the wave, heat and Laplace equations

The Laplacian operator: invariance under rotations [St, pp. 150-154]; 2D polar coordinates; 3D spherical coordinates

Where do they come from?
Diffusion; 2D averaging; random walk; Fourier law

The heat equation (Cauchy problem)
Derivation: [BD pp. 580-584], [Jo, Sec. 7.1]
The 1D Gaussian: Fourier Transform; Detour: mollifiers
The multiD Gaussian: Fourier Transform; Rotation invariance
Irreversibility
The energy method

The heat equation (bounded domains)
Superposition: Separation of variables
The maximum principle
The energy method

Parabolic systems
Dissipation
Fourier method
Energy decay (the energy method)

Assignment #1 [ pdf file ]

Laplace equation
Linearity
Superposition; Separation of variables; Rectangular and circular geometries (BD, pp. 571-579)

Properties of Harmonic functions
The fundamental solution [Fo, pp. 94-102)] Perron's method [Jo, pp. 111-116]; Wyel lemma and regularity [Fo, pp. 103-109]
The Mean-value property (MVP) [Fo, p. 90-92], [GT, p. 14]
The maximum principle [Fo, pp. 93], [GT, pp. 15], [Jo, pp. 103-107]
Regularity: Distributions and mollifiers; gain of regularity; analyticity; Harnack's inequalities

Dirichlet and Neumann problems
Green's function: [Fo, pp. 109-112], [GT, pp. 17-19],[Jo, pp. 106-110]
Half space Dirichlet problem -- Poisson kernel: [Fo, pp. 117-122], [Jo, pp. 106-111],[GT, pp. 19-21]; Dirichlet problem in ball -- spherical harmonics: [Fo, pp. 122-142]; Potential theory (integral equations): [Fo, pp. 150-189]

Dirichlet principle -- the weak and variational formulations: [Fo, pp. 112-117]

Elliptic equations of 2nd order
The maximum principle: [GT, pp. 31-41], [Pr]: Harnack inequality: [GT, pp. 41-45]
Gain of regularity [GT]

Assignment #2 [ pdf file ]

Final [ pdf file ]

[BD] W. Boyce and R. DiPrima ELEMENTARY PDEs and BOUNDARY VALUE PROBLEMS, Wiley
PDEs in one chapter; elementary indeed; popular here...

[Ga] P. Garabedian PDEs
User friendly introductory textbook

[St] W. Strauss PDEs An Introduction Wiley, 1992
illustrates phenomena governed by and solution techniques of PDEs; most suitable for introductory graduate course

[Re] M. Renardy INTRODUCTION TO PDEs 1991

[We] H. Weinberger FIRST COURSE in PDEs, Wiley, 1965
User friendly; the basics

TEXTBOOKS -- Graduate level

[Ca] G. Carrier PDEs 1988

[La51] P. Lax PDEs1951 read it if you find it

[Mc] R. McOwen PDEs. Methods and Applications,  Prentice Hall, 1996

[Ev] C. Evans PDEs Berkeley lecture notes (or the 1998 AMS book)
most suitable for a graduate course

[Fo] G. Folland INTRODUCTION to PDEs
Graduate textbook; the basic trinity in modern treatment

[Ra] J. Rauch PDEs
modern treatment for graduate students

TEXTBOOKS -- advanced

[Mi] S. Mizhota THE THEORY OF PDEs 1973

[Ta] M. Taylor PDEs. Vols I-III Springer, 1996

[Tr] F. Treves  BASIC LINEAR PDEs 1975, Academic Press
The detailed theory; readable

TEXTBOOKS -- the CLASSICS

[CH] R. Courant and D. Hilbert  METHODS OF MATHEMATICAL PHYSICS Wiley
Vols. I and particularly Vol. II -- treatise on 'everything' up to the fifties

[Ho] L. Hormander  THE ANALYSIS OF LINEAR PARTIAL DIFFERENTIAL OPERATORS, Vol. I-IV, Springer-Verlag, 1983-1985
Everything on linear PDEs; not for you -- perhaps the 1963 version...

[Jo] F. John PDEs 1982, Springer-Verlag
The standard graduate textbook; not that easy as the first impression it gives...

TEXTBOOKS -- mostly ELLIPTIC

[Ag] S. Agmon  ELLIPTIC BOUNDARY VALUE PROBLEMS
Lecture notes (from the 60's, right to the point but not in library?

[GT] D. Gilbarg and N. Trudinger  Elliptic PDEs of 2nd ORDER Springer-Verlag
Classical ref. for Elliptic PDEs; detailed Proofs; more than required for introductory course

TEXTBOOKS -- mostly PARABOLIC

[Fr] A. Friedman PDEs of PARABOLIC TYPE, 1964
Classical treatment of Parabolic equations; detailed proofs; consult also Friedman's 1969 PDEs and 1963 generalized functions}

TEXTBOOKS -- mostly HYPERBOLIC

[La60] P. Lax HYPERBOLIC PDEs Stanford Lecture Notes (from the 60's)
Beautiful; too much for one-semester introductory course Special Topics
 

[Pr] M. Protter and H. Weinberger MAXIMUM PRINCIPLE in PDEs Prentice Hall, 1967