The Dirichlet Problem
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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.
KeywordsSuperharmonic Function Harmonic Functions Regular Boundary Point Lebesgue Monotone Convergence Theorem Positive Surface Area
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