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 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.


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

  1. The Bochner Integral
    • The Pettis theorem
    • The spaces Lp(Ω; X)
    • Vector measures and Radon Nikodym property in Banach space
    • Riesz representation theorem for the duals of Lp(Ω;X)
    • Embedding results of Lions and Simon
  2. Surjectivity of nonlinear set valued operators.
  3. Lion's method of elliptic regularization and evolution equations of mixed type.
  4. Weak Derivatives
    • Morrey's inequality and Rademacher's theorem
    • Area formula
    • Integration on manifolds
  5. Sobolev spaces
    • Embedding theorems for Wm, p(Rn)
    • Extension theorems for Lipschitz domains
    • General embedding theorems
  6. Korn's Inequality on bounded Lipschitz domains
  7. Elliptic regularity and Nirenberg differences
  8. The trace spaces of Lions
    • Traces of Sobolev spaces and fractional order spaces
    • The half space
    • A right inverse for the trace for a half space
    • Intrinsic norms
    • Fractional order Sobolev spaces
    • Reflexivity of fractional order Sobolev spaces
  9. Sobolev spaces on manifolds
    • Basic definitions
    • The trace on the boundary of an open set


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

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.

Courses for which this course is prerequisite