Math 647: Theory of Partial Differential Equations 1

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Catalog Information

Title

Theory of Partial Differential Equations 1.

Credit Hours

3

Prerequisite

Math 541, 547.

Description

Chris Grant's Proposed Core Topics for Math 647/648

  1. Linear elliptic operators of order n
    • Classification
    • Strong and weak solutions
    • Gårding's inequality
    • Existence of weak solutions for the Dirichlet and Neumann problems
    • Agmon-Douglis-Nirenberg regularity
    • Green's formula
  2. Fundamental solutions for general linear differential operators
  3. Green's functions for general linear BVPs
  4. Dirichlet's Principle for Laplace’s equation in Rn
  5. Poisson's Equation
    • Newtonian Potential
    • Local existence for the Dirichlet Problem with locally Hölder boundary data
    • Interior Hölder estimates
    • Kellogg's Theorem
  6. Second-order linear elliptic operators
    • Weak Maximum Principle
    • Perron's Method
    • Uniqueness for the Dirichlet Problem
    • Hopf's bondary-point lemma
    • Hopf's Strong Maximum Principle
    • Alexandroff Maximum Principle
    • Gidas-Ni-Nirenberg
    • Uniqueness for the Neumann Problem
    • Harnack inequality
    • Finite difference methods
    • Interior regularity
    • Schauder estimates
    • Moser iteration
    • De Giorgi's theorem
    • Boundary/Global regularity
  7. Second-order quasilinear equations in divergence form
    • Existence of weak solutions for the Dirichlet problem via the Browder-Minty theorem
    • Local-in-time existence for reaction-diffusion IBVPs and systems using the contraction mapping principle
  8. Abstract evolution equations
    • General theory
    • Existence and reqularity for parabolic IVPs
    • Existence for hyperbolic IVPs
  9. Viscosity solutions

Desired Learning Outcomes

Prerequisites

Minimal learning outcomes

Additional topics

Courses for which this course is prerequisite