Potential Theory and Fundamental Solutions of Elliptic Equations

  • Alexander I. Saichev
  • Wojbor A. Woyczyński
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


This chapter is devoted to the theory of linear elliptic partial differential equations and the related problems of potential theory. The basic concept of the Green’s function and the source solution are introduced and explored. This is followed by a detailed analysis of the Helmholtz equation in one, two, and three dimensions with applications to the diffraction problem for monochromatic waves. The inhomogeneous media case sets the stage for the Helmholtz equation with a variable coefficient and an analysis of waves in waveguides. The latter can be reduced to the celebrated Sturm–Liouville problem, and we study properties of its eigenvalues and eigenfunctions.


Fundamental Solution Poisson Equation Radiation Condition Helmholtz Equation Complex Amplitude 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander I. Saichev
    • 1
    • 2
  • Wojbor A. Woyczyński
    • 3
  1. 1.Department of Management, Technology, and EconomicsETH ZürichZürichSwitzerland
  2. 2.Department of Radio PhysicsUniversity of Nizhniy NovgorodNizhniy NovgorodRussia
  3. 3.Department of Mathematics, Applied Mathematics and Statistics, and Center for Stochastic and Chaotic Processes in Science and TechnologyCase Western Reserve UniversityClevelandUSA

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