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 ==
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== Catalog Information ==
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<div style="-moz-column-count:2; column-count:2;">
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#  Linear elliptic operators of order ''n''
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=== Title ===
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#* Classification
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Theory of Partial Differential Equations 1.
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#* Strong and weak solutions
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#* Gårding's inequality
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=== Credit Hours ===
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#* Existence of weak solutions for the Dirichlet and Neumann problems
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3
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#* Agmon-Douglis-Nirenberg regularity
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#* Green's formula
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=== Prerequisite ===
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#  Fundamental solutions for general linear differential operators
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[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.
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#  Green's functions for general linear BVPs
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# Dirichlet's Principle for Laplace’s equation in '''R'''<sup>''n''</sup>
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=== Description ===
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#  Poisson's Equation
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Proposed: Classical theory of canonical linear PDEs. Introduction to Sobolev spaces.
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#* Newtonian Potential
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#* Local existence for the Dirichlet Problem with locally Hölder boundary data
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#* Interior Hölder estimates
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#* Kellogg's Theorem
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#  Second-order linear elliptic operators
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#* Weak Maximum Principle
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#* Perron's Method
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#* Uniqueness for the Dirichlet Problem
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#* Hopf's bondary-point lemma
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#* Hopf's Strong Maximum Principle
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#* Alexandroff Maximum Principle
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#* Gidas-Ni-Nirenberg
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#* Uniqueness for the Neumann Problem
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#* Harnack inequality
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#* Finite difference methods
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#* Interior regularity
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#* Schauder estimates
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#* Moser iteration
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#* De Giorgi's theorem
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#* Boundary/Global regularity
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# Second-order quasilinear equations in divergence form
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#* Existence of weak solutions for the Dirichlet problem via the Browder-Minty theorem
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#* Local-in-time existence for reaction-diffusion IBVPs and systems using the contraction mapping principle
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#  Abstract evolution equations
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#* General theory
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#* Existence and reqularity for parabolic IVPs
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#* Existence for hyperbolic IVPs
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#  Viscosity solutions
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</div>
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== Desired Learning Outcomes ==
 
== Desired Learning Outcomes ==
  
 
=== Prerequisites ===
 
=== Prerequisites ===
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Students should understand analysis at the first-year graduate level.
  
 
=== Minimal learning outcomes ===
 
=== Minimal learning outcomes ===
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Outlined below are topics that all successful Math 647 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.
  
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<div style="-moz-column-count:2; column-count:2;">
 
  
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<div style="-moz-column-count:2; column-count:2;">
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# Classical theory for canonical linear PDEs
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#* Transport equation
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#* Laplace's equation
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#** Fundamental solution
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#** Mean-value and maximum principles
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#** Energy methods
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#* Heat equation
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#** Fundamental solution
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#** Mean-value and maximum principles
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#** Energy methods
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#* Wave equation
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#** Spherical means
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#** Energy methods
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# Method of characteristics
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# Sobolev spaces
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#* Traces
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#* Sobolev inequalities
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#* Compactness
 
</div>
 
</div>
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=== Textbooks ===
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Possible textbooks for this course include (but are not limited to):
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* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.
  
 
=== Additional topics ===
 
=== Additional topics ===
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If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.
  
 
=== Courses for which this course is prerequisite ===
 
=== Courses for which this course is prerequisite ===
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[[Math 648]]
  
 
[[Category:Courses|647]]
 
[[Category:Courses|647]]

Latest revision as of 16:45, 3 April 2013

Catalog Information

Title

Theory of Partial Differential Equations 1.

Credit Hours

3

Prerequisite

Math 541, 547. It is proposed that Math 547 be dropped as a prerequisite, as these courses have always operated independently of each other.

Description

Proposed: Classical theory of canonical linear PDEs. Introduction to Sobolev spaces.

Desired Learning Outcomes

Prerequisites

Students should understand analysis at the first-year graduate level.

Minimal learning outcomes

Outlined below are topics that all successful Math 647 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.


  1. Classical theory for canonical linear PDEs
    • Transport equation
    • Laplace's equation
      • Fundamental solution
      • Mean-value and maximum principles
      • Energy methods
    • Heat equation
      • Fundamental solution
      • Mean-value and maximum principles
      • Energy methods
    • Wave equation
      • Spherical means
      • Energy methods
  2. Method of characteristics
  3. Sobolev spaces
    • Traces
    • Sobolev inequalities
    • Compactness

Textbooks

Possible textbooks for this course include (but are not limited to):

  • Lawrence C. Evans, Partial Differential Equations (Second Edition), American Mathematical Society, 2010.

Additional topics

If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.

Courses for which this course is prerequisite

Math 648

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