Math 547: Partial Differential Equations 1

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Chris Grant's Proposed Core Topics for Math 547/548

  1. General Cauchy problem
    • Cauchy-Kovalevskaya Theorem
    • Lewy Example
  2. Method of characteristics for first-order equations
    • Semilinear case
    • Quasilinear case
    • General case
  3. Quasilinear systems of conservation laws on a line
    • Riemann problem
    • Rankine-Hugoniot jump condition
    • Entropy condition
    • Shocks
    • Rarefaction waves
  4. Classification of general second-order equations
  5. Canonical forms for semilinear second-order equations
  6. Classical theory for the canonical second-order linear equations on Rn
    • Laplace's equation
      • Green's first and second identities
      • Mean Value Principle and its converse
      • Weak and strong maximum principles
      • Uniqueness for the Dirichlet problem
      • Poisson integral formula
      • Existence for the Dirichlet Problem on a ball
      • Fundamental solutions
      • Green's functions
      • Harnack inequality
      • Liouville's Theorem
      • Harnack's Convergence Theorem
      • Existence for the Dirichlet Problem on domains with regular boundaries and for continuous boundary data
      • Interior and exterior sphere conditions
    • Wave equation
      • Method of spherical means
      • Hadamard’s method of descent
      • Huygen’s Principle
      • Conservation of Energy
      • Domain of Dependence
    • Heat equation
      • Fourier transforms
      • The heat kernel
      • Existence for the IVP
      • Weak and strong maximum principles
      • Uniqueness for the IBVP

Desired Learning Outcomes


Minimal learning outcomes

Additional topics

Courses for which this course is prerequisite