Math 648: Theory of Partial Differential Equations 2

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Catalog Information


Theory of Partial Differential Equations 2.

3Credit Hours





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.


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

Desired Learning Outcomes


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


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