Math 647: Theory of Partial Differential Equations 1

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