Difference between revisions of "Math 647: Theory of Partial Differential Equations 1"
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| + | == Chris Grant's Proposed Core Topics for Math 647/648 == | ||
| + | <div style="-moz-column-count:2; column-count:2;"> | ||
| + | # 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 | ||
| + | # Fundamental solutions for general linear differential operators | ||
| + | # Green's functions for general linear BVPs | ||
| + | # Dirichlet's Principle for Laplace’s equation in '''R'''<sup>''n''</sup> | ||
| + | # Poisson's Equation | ||
| + | #* Newtonian Potential | ||
| + | #* Local existence for the Dirichlet Problem with locally Hölder boundary data | ||
| + | #* Interior Hölder estimates | ||
| + | #* Kellogg's Theorem | ||
| + | # 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 | ||
| + | # 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 | ||
| + | # Abstract evolution equations | ||
| + | #* General theory | ||
| + | #* Existence and reqularity for parabolic IVPs | ||
| + | #* Existence for hyperbolic IVPs | ||
| + | # Viscosity solutions | ||
| + | </div> | ||
| + | |||
== Desired Learning Outcomes == | == Desired Learning Outcomes == | ||
Revision as of 14:36, 8 May 2008
Contents
Chris Grant's Proposed Core Topics for Math 647/648
- 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
- Fundamental solutions for general linear differential operators
- Green's functions for general linear BVPs
- Dirichlet's Principle for Laplace’s equation in Rn
- Poisson's Equation
- Newtonian Potential
- Local existence for the Dirichlet Problem with locally Hölder boundary data
- Interior Hölder estimates
- Kellogg's Theorem
- 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
- 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
- Abstract evolution equations
- General theory
- Existence and reqularity for parabolic IVPs
- Existence for hyperbolic IVPs
- Viscosity solutions
