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A. S. Fokas : Differential Forms, Spectral Theory and Boundary Value Problems

A new method will be reviewed for analyzing boundary value problems for linear and integrable nonlinear PDEs. This method involves the following:

  1. Given a PDE for q(x), x in Rn, construct a closed (n-1)-differential form W(x,k), k in Cn-1.
  2. Given a convex domain D contained in Rn, perform the spectral analysis of W
  3. Given appropriate boundary conditions, analyze the global relation
    \int_{\partial D} W

For linear PDEs, the relation of this method with the Ehrenpreis-Palamodov principle, as well as the relations with applied techniques such as the Weiner-Hopf technique will be discussed.

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