Abstract
Let Ω be a domain in ℝn and u a C 2(Ω) function. The Laplacian of u, denoted Δu, is defined by
. The function u is called harmonic (subharmonic, superharmonic) in Ω if it satisfies there
. In this chapter we develop some basic properties of harmonic, subharmonic and superharmonic functions which we use to study the solvability of the classical Dirichlet problem for Laplace’s equation, Δu = 0. As mentioned in Chapter 1, Laplace’s equation and its inhomogeneous form, Poisson’s equation, are basic models of linear elliptic equations.
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© 2001 Springer-Verlag Berlin, Heidelberg
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Gilbarg, D., Trudinger, N.S. (2001). Laplace’s Equation. In: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, vol 224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61798-0_2
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DOI: https://doi.org/10.1007/978-3-642-61798-0_2
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41160-4
Online ISBN: 978-3-642-61798-0
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