MATH 673/AMSC 673 - Partial Differential equations I

Fall 2011


Instructor: Antoine Mellet

Office: Math Building 4104

Lectures: MWF 2:00pm- 2:50pm (MTH 0104)

Office Hours: W 3-4pm and by appointment

Prerequisites:
MATH 411 (Advanced Calculus II) or equivalent.

Textbook:


The syllabus of Math 673/AMSC 673 consists of the core material in Chapters 1-3 and of selected topics from Chapters 4 and 6:

Further reading

Grading:

Homework will be assigned typically a week or two before their due dates. Late homework will not be accepted.
Your solutions are required to be legible and clear. Illegible problems will not be graded.

The midterm will take place during discussion session on Wednesday October 19th

The final exam will take place on Monday December 19th, 1:30pm-3:30pm

Homework

Tentative list of topics: (download the pdf file )

  1. Introduction

    • Definition;
      Examples: The transport equation; The wave, heat and Laplace equations; Conservation laws; Hamilton-Jacobi equations. [Chap. 1]
    • Cauchy initial data, boundary conditions: Notion of well posedness [Chap. 1]
    • Example: Transport equation with constant coefficient (IVP, method of characteristics, weak solution, non-homogeneous problem) [Sec. 2.1]
      Reading: Appendix A3 (Notation for functions; Notations for derivatives; Function spaces)

  2. Elliptic equations [Sec. 2.2, 6.1, 6.4]

    • Properties of harmonic functions [Sec. 2.2.1-2.2.3]
      • Laplace and Poisson equation; Dirichlet and Neumann BVP.
        Reading: Appendix C1 (Boundaries)
      • The fundamental solution: Derivation; Solving Poisson's equation [Sec. 2.2.1]
        Reading: C2 (Gauss-Green Theorems)
      • Mean Value formulas [Sec. 2.2.2]
      • Weak harmonic functions; Weyl's lemma.
      • Properties of harmonic functions: Maximum principle; Regularity; Liouville theorems; Harnack's inequality. [Sec. 2.2.3]
        Reading: Appendix C5 (Convolution and smoothing)
    • Dirichlet and Neumann problems [Sec. 2.2.4]
      • Green's functions; Representation formula.
      • Dirichlet problem in a half-space and in a Ball: Poisson's kernel;
    • Energy methods [Sec. 2.2.5]
      Dirichlet principle; Weak and variational formulations.
    • Solutions by superposition: Separation of variables (rectangular and circular geometries)
    • General Second order Elliptic equations [Sec. 6.1, 6.4]
      • Definition [Sec. 6.1]
      • Weak maximum principle [Sec. 6.4.1]
      • Strong maximum principle [Sec. 6.4.2]
      • Harnack's inequality. [Sec. 6.4.3]

  3. Parabolic equations (The heat equation)

    • Initial value problem in the whole space (Cauchy problem).
      • Fundamental solution: Derivation; Solution of IVP; Non-homogeneous problem [Sec 2.3.1]
      • Fourier transform [Sec. 4.3.1]
    • Heat equation in bounded domain
      • Initial Boundary Value Problems.
      • Mean value formula; Maximum principle; Uniqueness; Regularity [Sec. 2.3.2, 2.3.3]
      • Energy methods. [Sec. 2.3.4]
      • Separation of variables [Sec. 4.1]

  4. The Wave equation [Sec. 2.4]

    • Derivations (physical interpretation), IVP
    • D'Alembert formula (n=1) [Sec. 2.4.1]
    • Solution by spherical means (n=2) [Sec. 2.4.1]
    • Non-homogeneous problem [Sec. 2.4.2]
    • Energy methods [Sec. 2.4.3]

  5. First-order (hyperbolic) PDEs [Chap 3]

    • Linear transport equation
      • Method of characterstics
    • Nonlinear first order PDE
      • Method of Characteristics
    • Calculus of variations and Hamilton-Jacobi equations
      • Calculus of variation
      • Legendre transform
      • Hopf-Lax formula
      • Weak (viscosity) solutions
    • Conservations laws (Burger's equation)
      • Shock formation and entropy condition
      • Weak solutions
      • The Riemann problem.
      • Applications in gas dynamics, materials science and fluid mechanics.