Partial Differential Equations. I

MATH/AMSC 673, Fall 2006

Course Information

Lecture4122 CSIC Bldg. #406; TuTh 2-3:15pm
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   Email:
Office Hours By appointment 4119 CSIC Bldg. #406
tel.: x5-0652   Email:
Teaching AssistantWeiran Sun tel.: x5-0664   Email:
TA office hours TuTh 3:30-5pm, 4116 CSIC Bldg. #406
Midterm Tuesday, Oct. 31, 4122 CSIC Bldg. #406
Final Friday, Dec. 15, 10am-1pm, 4122 CSIC Bldg. #406
Grading40% Homework, 40% Final, 20% Midterm

Course Description (Preliminary)

Analysis of boundary value problems for Laplace's equation.  Initial value problems for the heat and wave equations. Fundamental solutions, maximum principles, energy methods. First order nonlinear PDEs. Conservation laws. Characteristics, shock formation, weak solutions. Distributions, Fourier transform.



  • Introduction

    Preliminaries
    The divergence theorem; convolutions (approximate identities); Fourier transform (series); distributions; The Laplacian operator: invariance under rotations and radial representations

    Classifications
    Linear, quasilinear, nonlinear equations
    Cauchy initial data, boundary conditions
    Existence and well-posed problems: Cauchy-Kovalevski: Lewy's counterexample

    Constant coefficients: local solvability; Canonical forms; Classification of 2nd order equations

    Linear and nonlinear examples: The transport equation; The wave, heat and Laplace equations;
    The nonlinear transport and Hamilton-Jacobi equations

  • Elliptic Equations

    Properties of Harmonic functions
    The fundamental solution;  Perron's method; Wyel lemma and regularity
    The Mean-value property (MVP);
    The maximum principle;
    Regularity: Distributions and mollifiers; gain of regularity; analyticity; Harnack's inequalities
    Assignment #1 [ pdf file ] with answers [ pdf file]
    Dirichlet and Neumann problems
    Representation of solutions by Green's function:
    Half space Dirichlet problem -- Poisson kernel;
    Dirichlet problem in ball -- spherical harmonics;
    Potential theory (integral equations)

    Lecture Notes: Dirichlet problem in a ball [ pdf file ]
    Representation of solutions by superposition:
    separation of variables --rectangular and circular geometries
    Lecture Notes: Fourier and Chebyshev expansions [ pdf file ]
    Dirichlet principle -- the weak and variational formulations:

    Elliptic equations of 2nd order (*)
    The maximum principle; Harnack inequality; Gain of regularity

  • Parabolic Equations

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

    The heat equation (Cauchy problem)
    Derivation:
    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 #2 [ pdf file ] with answers [ pdf file ]

  • Hyperbolic Equations

    Where do they come from?
    Transport phenomena; speed of propagation

    The linear transport equation (Cauchy problem)
    Characteristics; The energy method; A maximum principle?

    The wave equation
    Solution by spherical means
    Fourier methods
    The energy method

    Hyperbolic systems (*)
    Dissipation
    Fourier method
    Energy decay (the energy method)

    Midterm [ pdf file ] with Answers [ pdf file ]

  • Nonlinear PDEs

    Conservation laws
    Characteristics; Shocks and entropy condition
    Burgers' equation; existence and uniqueness

    Lecture Notes: Nonlinear conservation laws [ pdf file ]
    Hamilton-Jacobi equations
    Geometrical optics; Characteristics; Legendre transform;

    Assignment #3 [ pdf file ]

  • Extensions

    Weak vs. strong solutions

    Sobolev spaces
    Weak vs. strong derivatives
    Sobolev embeddings and related inequalities
    Compactness and existence of solutions
    Final [ pdf file ] with Answers [ pdf file ]


    Textbooks

    Lawrence C. Evans, Partial Differential Equations (Graduate Studies in Mathematics, V. 19), AMS
    Robert McOwen, Partial Differential Equations, Methods and Applications
    Gerald Folland Introduction to PDEs, Princeton University Press

    TEXTBOOKS -- mostly ELLIPTIC
    D. Gilbarg and N. Trudinger, Elliptic PDEs of 2nd order, Springer-Verlag
    Classical reference for elliptic PDEs with detailed proofs -- more than required for our introductory course

    Eitan Tadmor