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The Dirichlet Problem

  • Lester L. HelmsEmail author
Chapter
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Part of the Universitext book series (UTX)

Abstract

In this chapter, approximation of solutions of Laplace’s equation requires the study of sequences of harmonic functions, making use of the integral representations and averaging properties of harmonic functions. The latter property is used to incorporate a larger class of functions called superharmonic functions that are used to approximate solutions. Such approximate solutions culminate in Wiener’s theorem that associates with each continuous boundary function a solution of the Dirichlet problem. Theorems due to Poincaré and Zaremba ensure that a solution will take on the prescribed boundary value at points of the boundary.

Keywords

Superharmonic Function Harmonic Functions Regular Boundary Point Lebesgue Monotone Convergence Theorem Positive Surface Area 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.University of IllinoisUrbanaUSA

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