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Erschienen in:
2019 | OriginalPaper | Buchkapitel
11. Harmonic Functions and Integral Functions
verfasst von : Hemant Kumar Pathak
Erschienen in: Complex Analysis and Applications
Verlag: Springer Singapore
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Abstract
In this chapter, we shall study harmonic function, Harnack’s inequality, and the Dirichlet problem will be solved. The Dirichlet problem consists in determining all regions G such that for any continuous function \(f: \partial G \rightarrow \mathbb {R}\) there is a continuous function \( u :\overline{G} \rightarrow \mathbb {R}\) such that \(u(z) = f(z)\) for \(z \in \partial G\) and u is harmonic in G. Alternatively, the Dirichlet problem consists in determining all regions G such that Laplace’s equation is solvable with arbitrary boundary values.