Basic Properties of Harmonic Functions
Part of the Graduate Texts in Mathematics book series (GTM, volume 137)
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
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 E ⊂ R n is harmonic on E if u can be extended to a function harmonic on an open set containing E.
$$\Delta u \equiv 0$$
KeywordsPower Series Harmonic Function Compact Subset Maximum Principle Dirichlet Problem
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© Springer Science+Business Media New York 2001