Harmonic Functions

BASIC PROPERTIES OF HARMONIC FUNCTIONS THE LAPLACE EQUATION We reiterate the definition of "harmonic". Let F be a holomorphic function on an open set U ⊆ ℂ. Write F = u + iv, where u and v are realvalued. The real part u satisfies a certain partial differential equation known as Laplace�...

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Bibliographic Details
Published inA Guide to Complex Variables pp. 95 - 110
Format Book Chapter
LanguageEnglish
Published Washington DC The Mathematical Association of America 2011
Subjects
reflection of
subharmonic functions, definition of
as the real part of a holomorphic function
Laplace's equation
harmonic function
solution of for a general domain
convexity
maximum principle
Schwarz reflection principle
Poisson kernel
boundary uniqueness for harmonic functions
for harmonic functions
limit of a sequence of
Dirichlet problem
harmonic conjugate
conditions on the boundary for solving
Harnack's inequality
uniqueness of the solution
smoothness of
subharmonic
mean value property
boundarymaximumandminimumprinciples for harmonic functions
on the disc
real- and complex-valued
for subharmonic functions
on the disc, solution of
properties of
Poisson integral formula
on a general domain
for holomorphic functions
lack of for subharmonic functions
Laplacian
motivation for
and the Dirichlet problem
barriers
ε
Harnack's principle
on a general disc, solution of
sub-mean value property
general versions of
characterizations of
harmonic functions
as real parts of holomorphic functions
mean value property for harmonic functions
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DOI10.5948/UPO9780883859148.008

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Summary:BASIC PROPERTIES OF HARMONIC FUNCTIONS THE LAPLACE EQUATION We reiterate the definition of "harmonic". Let F be a holomorphic function on an open set U ⊆ ℂ. Write F = u + iv, where u and v are realvalued. The real part u satisfies a certain partial differential equation known as Laplace's equation: (The imaginary part v satisfies the same equation.) In this chapter we shall study systematically those C2 functions that satisfy this equation. They are called harmonic functions. (We encountered some of these ideas already in §1.4.) DEFINITION OF HARMONIC FUNCTION Recall the precise definition of harmonic function: A real-valued function u : U → ℝ on an open set U ⊆ ℂ is harmonic if it is C2 on U and where the Laplacian Δu is defined by REAL- AND COMPLEX-VALUED HARMONIC FUNCTIONS The definition of harmonic function just given applies as well to complexvalued functions. A complex-valued function is harmonic if and only if its real and imaginary parts are each harmonic. The first thing that we need to check is that real-valued harmonic functions are just those functions that arise as the real parts of holomorphic functions-at least locally. HARMONIC FUNCTIONS AS THE REAL PARTS OF HOLOMORPHIC FUNCTIONS If u : D(P, r) → ℝ is a harmonic function on a disc D(P, r), then there is a holomorphic function F : D(P, r) → ℂ such that Re F ≡ u on D(P, r).
DOI:10.5948/UPO9780883859148.008