Difference between revisions of "Math 648: Theory of Partial Differential Equations 2"

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=== Prerequisite ===
 
=== Prerequisite ===
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]
+
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]].  Suggestion:  Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.
  
 
=== Description ===
 
=== Description ===
This course develops abstract methods for studying partial differential equations and inclusions.
+
Advanced theory of partial differential equations.  Functional-analytic techniques.
  
 
== Desired Learning Outcomes ==
 
== Desired Learning Outcomes ==
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for
 
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.
 
  
 
=== Prerequisites ===
 
=== Prerequisites ===
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.
+
Students need a thorough understanding of real analysis.
  
 
=== Minimal learning outcomes ===
 
=== Minimal learning outcomes ===
#The Bochner Integral
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Outlined below are topics that all successful Math 648 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.
#*The Pettis theorem
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#*The spaces ''L''<sup>''p''</sup>(&Omega;; ''X'')
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#*Vector measures and Radon Nikodym property in Banach space
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#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(&Omega;;''X'')
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#*Embedding results of Lions and Simon
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#Surjectivity of nonlinear set valued operators.
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#Lion's method of elliptic regularization and evolution equations of mixed type.
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#Weak Derivatives
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#*Morrey's inequality and Rademacher's theorem
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#*Area formula
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#*Integration on manifolds
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#Sobolev spaces
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#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)
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#*Extension theorems for Lipschitz domains
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#*General embedding theorems
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#Korn's Inequality on bounded Lipschitz domains
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#Elliptic regularity and Nirenberg differences
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#The trace spaces of Lions
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#*Traces of Sobolev spaces and fractional order spaces
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#*The half space
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#*A right inverse for the trace for a half space
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#*Intrinsic norms
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#*Fractional order Sobolev spaces
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#*Reflexivity of fractional order Sobolev spaces
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#Sobolev spaces on manifolds
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#*Basic definitions
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#*The trace on the boundary of an open set
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<div style="-moz-column-count:2; column-count:2;">
 
<div style="-moz-column-count:2; column-count:2;">
 
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# Second-order elliptic equations
 +
#* Classification
 +
#* Weak solutions
 +
#** Lax-Milgram theorem
 +
#** Energy estimates
 +
#** Fredholm alternative
 +
#* Regularity
 +
#** Interior
 +
#** Boundary
 +
#* Maximum principles
 +
#** Weak
 +
#** Strong
 +
#** Harnack's inequality
 +
#* Eigenpairs of elliptic operators
 +
#** Symmetric
 +
#** Nonsymmetric
 +
# Linear Evolution Equations
 +
#* Second-order parabolic equations
 +
#** Weak solutions
 +
#** Regularity
 +
#** Maximum principles
 +
#* Second-order hyperbolic equations
 +
#** Weak solutions
 +
#** Regularity
 +
# Calculus of Variations
 +
#* Euler-Lagrange equation
 +
#* Coercivity
 +
#* Convexity
 +
#* Semicontinuity
 +
#* Weak Solutions
 +
#* Regularity
 +
#* Constraints
 +
#* Critical points
 +
#** Mountain pass theorem
 +
# Hamilton-Jacobi equations
 +
#* Viscosity solutions
 
</div>
 
</div>
 +
 
=== Textbooks ===
 
=== Textbooks ===
 
Possible textbooks for this course include (but are not limited to):
 
Possible textbooks for this course include (but are not limited to):
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=== Additional topics ===
 
=== Additional topics ===
The above might be all there is time for, but if there is time for more, it would be nice to consider Mihlin's theorem and the $L^{p}$ theory of elliptic regularity. Other topics could include methods of interpolation in Banach space.
+
If time permits, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.
  
 
=== Courses for which this course is prerequisite ===
 
=== Courses for which this course is prerequisite ===

Latest revision as of 15:45, 3 April 2013

Catalog Information

Title

Theory of Partial Differential Equations 2.

3Credit Hours

(3:3:0)

Offered

F

Prerequisite

Math 641, Math 540, recommended Math 640, Math 647. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.

Description

Advanced theory of partial differential equations. Functional-analytic techniques.

Desired Learning Outcomes

Prerequisites

Students need a thorough understanding of real analysis.

Minimal learning outcomes

Outlined below are topics that all successful Math 648 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. Second-order elliptic equations
    • Classification
    • Weak solutions
      • Lax-Milgram theorem
      • Energy estimates
      • Fredholm alternative
    • Regularity
      • Interior
      • Boundary
    • Maximum principles
      • Weak
      • Strong
      • Harnack's inequality
    • Eigenpairs of elliptic operators
      • Symmetric
      • Nonsymmetric
  2. Linear Evolution Equations
    • Second-order parabolic equations
      • Weak solutions
      • Regularity
      • Maximum principles
    • Second-order hyperbolic equations
      • Weak solutions
      • Regularity
  3. Calculus of Variations
    • Euler-Lagrange equation
    • Coercivity
    • Convexity
    • Semicontinuity
    • Weak Solutions
    • Regularity
    • Constraints
    • Critical points
      • Mountain pass theorem
  4. Hamilton-Jacobi equations
    • Viscosity solutions

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, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.

Courses for which this course is prerequisite

None