Math 648: Theory of Partial Differential Equations 2

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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

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

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 Lp spaces and the space C0. 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.

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