Basic Properties of Harmonic Functions

  • Sheldon Axler
  • Paul Bourdon
  • Wade Ramey
Part of the Graduate Texts in Mathematics book series (GTM, volume 137)

Abstract

Harmonic functions, for us, live on open subsets of real Euclidean spaces. Throughout this book, n will denote a fixed positive integer greater than 1 and Ω will denote an open, nonempty subset of R n . A twice continuously differentiable, complex-valued function u defined on Ω is harmonic on Ω if
$$\Delta u \equiv 0$$
where Δ = D 1 2 + ⋯ +D n 2 and D j 2 denotes the second partial derivative with respect to the j th coordinate variable. The operator Δ is called the Laplacian, and the equation Δu ≡ 0 is called Laplace’s equation. We say that a function u defined on a (not necessarily open) set ER n is harmonic on E if u can be extended to a function harmonic on an open set containing E.

Keywords

Power Series Harmonic Function Compact Subset Maximum Principle Dirichlet Problem 
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 Science+Business Media New York 2001

Authors and Affiliations

  • Sheldon Axler
    • 1
  • Paul Bourdon
    • 2
  • Wade Ramey
    • 3
  1. 1.Mathematics DepartmentSan Francisco State UniversitySan FranciscoUSA
  2. 2.Mathematics DepartmentWashington and Lee UniversityLexingtonUSA
  3. 3.BerkeleyUSA

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