Difference between revisions of "Math 648: Theory of Partial Differential Equations 2"
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Latest revision as of 15:45, 3 April 2013
Contents
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. Functionalanalytic 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.
 Secondorder elliptic equations
 Classification
 Weak solutions
 LaxMilgram theorem
 Energy estimates
 Fredholm alternative
 Regularity
 Interior
 Boundary
 Maximum principles
 Weak
 Strong
 Harnack's inequality
 Eigenpairs of elliptic operators
 Symmetric
 Nonsymmetric
 Linear Evolution Equations
 Secondorder parabolic equations
 Weak solutions
 Regularity
 Maximum principles
 Secondorder hyperbolic equations
 Weak solutions
 Regularity
 Secondorder parabolic equations
 Calculus of Variations
 EulerLagrange equation
 Coercivity
 Convexity
 Semicontinuity
 Weak Solutions
 Regularity
 Constraints
 Critical points
 Mountain pass theorem
 HamiltonJacobi 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