aarjaneiro/planetmath huggingface dataset:
proof of maximal modulus principle

proof of maximal modulus principle

f

:

U

→

ℂ

is

holomorphic

and therefore

continuous

, so

|

f

|

will also be continuous on

U

.

K

⊂

U

is

compact

and since

|

f

|

is continuous on

K

it must attain a maximum and a minimum value there.

Suppose the maximum of

|

f

|

is attained at

z

0

in the

interior

of

K

.

By definition there will exist

r

>

0

such that the set

S

r

=

{

z

∈

ℂ

:

|

z

-

z

0

|

2

≤

r

2

}

⊂

K

.

Consider

C

r

the

boundary

of the previous set parameterized counter-clockwise.
Since

f

is holomorphic by

hypothesis

,

Cauchy integral formula

says that

f

⁢

(

z

0

)

=

1

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

z

)

z

-

z

0

⁢

𝑑

z

(1)

A canonical parameterization of

C

r

is

z

=

z

0

+

r

⁢

e

i

⁢

θ

r

, for

θ

∈

[

0

,

2

⁢

π

⁢

r

]

.

f

⁢

(

z

0

)

=

1

2

⁢

π

⁢

r

⁢

∫

0

2

⁢

π

⁢

r

f

⁢

(

z

0

+

r

⁢

e

i

⁢

θ

r

)

⁢

𝑑

θ

(2)

Taking

modulus

on both sides and using the

estimating theorem of contour integral

|

f

⁢

(

z

0

)

|

≤

max

z

∈

C

r

⁡

|

f

⁢

(

z

)

|

Since

|

f

⁢

(

z

0

)

|

is a maximum, the last

inequality

must be verified by having the equality in the

≤

verified.

The proof of the estimating theorem of contour integral (

http://planetmath.org/ProofOfEstimatingTheoremOfContourIntegral

) implies that equality is only verified when

f

⁢

(

z

o

+

r

⁢

e

i

⁢

θ

r

)

r

⁢

e

i

⁢

θ

r

=

λ

⁢

i

⁢

e

i

⁢

θ

r

¯

where

λ

∈

ℂ

is a

constant

.
Therefore,

f

⁢

(

z

o

+

r

⁢

e

i

⁢

θ

r

)

is constant and to verify equation

2

its value must be

f

⁢

(

z

0

)

.

So

f

is holomorphic and constant on a circumference.
It’s a well known result that if 2

holomorphic functions

are equal on a curve, then they are equal on their entire domain, so

f

is constant.

to see this in this particular circumstance is using equation

1

to calculate the value of

f

on a point

ξ

∈

interior

S

r

different than

z

0

. Bearing in mind that

f

⁢

(

z

)

=

f

⁢

(

z

0

)

is constant in

C

r

the

formula

reads

f

⁢

(

ξ

)

=

f

⁢

(

z

0

)

2

⁢

π

⁢

i

⁢

∮

C

r

1

z

-

ξ

⁢

𝑑

z

=

f

⁢

(

z

0

)

. So

f

is really constant in the interior of

S

r

and the only holomorphic function defined in

K

that is constant in the interior of

S

r

is the

constant function

on all

K

.

Thus if the maximum of

|

f

|

is attained in the interior of

K

, then

f

is constant.
If

f

isn’t constant, the maximum must be attained somewhere in

K

, but not in its interior.
Since

K

is compact, by definition it must be attained at

∂

⁡

K

.

Title

proof of maximal modulus principle

Canonical name

ProofOfMaximalModulusPrinciple

Date of creation

2013-03-22 15:46:15

Last modified on

2013-03-22 15:46:15

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

19

Author

cvalente (11260)

Entry type

Proof

Classification

msc 30F15

Classification

msc 31B05

Classification

msc 31A05

Classification

msc 30C80

Generated on Fri Feb 9 20:22:35 2018 by

LaTeXML
proof of weak maximum principle for real domains

proof of weak maximum principle for real domains

First, we show that, if

Δ

⁢

f

>

0

(where

Δ

denotes the

Laplacian

on

ℝ

d

) on

K

, then

f

cannot attain a

maximum

on the interior of

K

. Assume, to the contrary, that

f

did attain a maximum at a point

p

located on the interior of

K

. By the

second derivative test

, the matrix of second

partial derivatives

of

f

at

p

would have to be negative semi-definite. This would imply that the trace of the matrix is negative. But the trace of this matrix is the Laplacian, which was assumed to be strictly positive on

K

, so it is impossible for

f

to attain a maximum on the interior of

K

.

Next, suppose that

Δ

⁢

f

=

0

on

K

but that

f

does not attain its maximum on the boundary of

K

. Since

K

is

compact

,

f

must attain its maximum somewhere, and hence there exists a point

p

located in the interior of

K

at which

f

does attain its maximum. Since

K

is compact, the boundary of

K

is also compact, and hence the image of the boundary of

K

under

f

is also compact. Since every

element of

this image is strictly smaller than

f

⁢

(

p

)

, there must exist a constant

C

such that

f

⁢

(

x

)

<

C

<

f

⁢

(

p

)

whenever

x

lies on the boundary of

K

. Furthermore Since

K

is a compact subset of

ℝ

d

, it is bounded. Hence, there exists a constant

R

>

0

so that

|

x

-

p

|

<

R

for all

x

∈

K

.

Consider the

function

g

defined as

g

⁢

(

x

)

=

f

⁢

(

x

)

+

(

f

⁢

(

p

)

-

C

)

⁢

|

x

-

p

|

2

R

2

At any point

x

∈

K

,

g

⁢

(

x

)

<

f

⁢

(

x

)

+

f

⁢

(

p

)

-

C

In particular, if

x

lies on the boundary of

K

, this implies that

g

⁢

(

x

)

<

f

⁢

(

p

)

Since

g

⁢

(

p

)

=

f

⁢

(

p

)

this

inequality

implies that

g

cannot attain a maximum on the boundary of

K

.

This leads to a contradiction. Note that, since

Δ

⁢

f

=

0

on

K

,

Δ

⁢

g

=

d

⁢

(

f

⁢

(

p

)

-

C

)

R

2

>

0

which implies that

g

cannot attain a maximum on the interior of

K

. However, since

K

is compact,

g

must attain a maximum somewhere on

K

. Since we have ruled out both the possibility that this maximum occurs in the interior and the possibility that it occurs on the boundary, we have a contradiction. The only way out of this contradiction is to conclude that

f

does attain its maximum on the boundary of

K

.

Title

proof of weak maximum principle for real domains

Canonical name

ProofOfWeakMaximumPrincipleForRealDomains

Date of creation

2013-03-22 14:35:21

Last modified on

2013-03-22 14:35:21

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30F15

Classification

msc 31B05

Classification

msc 31A05

Classification

msc 30C80

Generated on Fri Feb 9 20:22:37 2018 by

LaTeXML
Teichmüller space

Teichmüller space

Definition

.

Let

S

0

be a

Riemann surface

. Consider all pairs

(

S

,

f

)

where

S

is a Riemann surface and

f

is a

sense-preserving

quasiconformal
mapping

of

S

0

onto

S

. We say

(

S

1

,

f

1

)

∼

(

S

2

,

f

2

)

if

f

2

∘

f

1

-

1

is

homotopic

to a

conformal mapping

of

S

1

onto

S

2

. In this case we say that

(

S

1

,

f

1

)

and

(

S

2

,

f

2

)

are

Teichmüller

equivalent

. The space of

equivalence classes

under this

relation

is called the

Teichmüller space

T

⁢

(

S

0

)

and

(

S

0

,

I

)

is called the initial
point of

T

⁢

(

S

0

)

. The

equivalence relation

is called

Teichmüller

equivalence

.

Definition

.

There exists a natural

Teichmüller metric

on

T

⁢

(

S

0

)

, where the distance
between

(

S

1

,

f

1

)

and

(

S

2

,

f

2

)

is

log

⁡

K

where

K

is the smallest maximal dilatation of a

mapping

homotopic to

f

2

∘

f

1

-

1

.

There is also a natural isometry between

T

⁢

(

S

0

)

and

T

⁢

(

S

1

)

defined by
a quasiconformal mapping of

S

0

onto

S

1

. The mapping

(

S

,

f

)

↦

(

S

,

f

∘

g

)

induces an isometric mapping of

T

⁢

(

S

1

)

onto

T

⁢

(

S

0

)

. So we could think of

T

⁢

(

⋅

)

as a

contravariant functor

from
the

category

of Riemann surfaces with

quasiconformal maps

to the category of
Teichmüller spaces (as a

subcategory

of metric spaces).

References

1

L. V. Ahlfors.

. Van Nostrand-Reinhold, Princeton, New Jersey, 1966

Title

Teichmüller space

Canonical name

TeichmullerSpace

Date of creation

2013-03-22 14:19:48

Last modified on

2013-03-22 14:19:48

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 30F60

Defines

Teichmüller metric

Defines

Teichmüller equivalence

Defines

Teichmüller equivalent

Generated on Fri Feb 9 20:22:42 2018 by

LaTeXML
monodromy theorem

monodromy theorem

Let

C

⁢

(

t

)

be a one-parameter family of smooth paths in the

complex plane

with common

endpoints

z

0

and

z

1

. (For definiteness, one may suppose that the parameter

t

takes values in the

interval

[

0

,

1

]

.) Suppose that an

analytic function

f

is defined in a

neighborhood

of

z

0

and that it is possible to analytically continue

f

along every path in the family. Then the result of

analytic continuation

does not depend on the choice of path.

Note that it is

crucial

that it be possible to continue

f

along all paths of the family. As the following example shows, the result will no longer hold if it is impossible to analytically continue

f

along even a single path. Let the family of paths be the set of circular arcs (for the present purpose, the

straight line

is to be considered as a degenerate case of a circular arc) with endpoints

+

1

and

-

1

and let

f

⁢

(

z

)

=

z

. It is possible to analytically continue

f

along every arc in the family except the

line segment

passing through

0

. The conclusion of the theorem does not hold in this case because continuing along arcs which lie above

0

leads to

f

⁢

(

z

1

)

=

+

i

whilst continuing along arcs which lie below

0

leads to

f

⁢

(

z

1

)

=

-

i

.

Title

monodromy theorem

Canonical name

MonodromyTheorem

Date of creation

2013-03-22 14:44:35

Last modified on

2013-03-22 14:44:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 30F99

Generated on Fri Feb 9 20:22:47 2018 by

LaTeXML
factorization theorem for H^∞ functions

factorization theorem for

H

∞

functions

Let

H

∞

denote the

bounded

analytic functions

on the

unit disc

.

Theorem

.

Every

f

∈

H

∞

can be written as

f

⁢

(

z

)

=

α

⁢

I

⁢

(

z

)

⁢

F

⁢

(

z

)

where

|

α

|

=

1

,

I

is an

inner function

and

F

is a bounded outer function. Conversely any

function

which can be so written is bounded.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1995.

Title

factorization theorem for

H

∞

functions

Canonical name

FactorizationTheoremForHinftyFunctions

Date of creation

2013-03-22 15:36:23

Last modified on

2013-03-22 15:36:23

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30H05

Related topic

InnerFunction

Generated on Fri Feb 9 20:22:51 2018 by

LaTeXML
inner function

inner function

If

f

:

𝔻

→

ℂ

is an

analytic function

on the

unit disc

, we denote by

f

*

⁢

(

e

i

⁢

θ

)

the radial limit of

f

where it exists, that is

f

*

⁢

(

e

i

⁢

θ

)

:=

lim

r

→

1

,

r

<

1

⁡

f

⁢

(

r

⁢

e

i

⁢

θ

)

.

A

bounded

analytic function on the disc will have radial limits almost everywhere (with respect to the

Lebesgue measure

on the

∂

⁡

𝔻

).

Definition

.

A bounded analytic function

f

is called an

inner function

if

|

f

*

⁢

(

e

i

⁢

θ

)

|

=

1

almost everywhere. If

f

has no zeros on the unit disc, then

f

is called a

singular inner function

.

Theorem

.

Every inner function can be written as

f

⁢

(

z

)

:=

α

⁢

B

⁢

(

z

)

⁢

exp

⁡

(

-

∫

e

i

⁢

θ

+

z

e

i

⁢

θ

-

z

⁢

𝑑

μ

⁢

(

e

i

⁢

θ

)

)

,

where

μ

is a positive

singular measure

on

∂

⁡

D

,

B

⁢

(

z

)

is a

Blaschke product

and

|

α

|

=

1

is a constant.

Note that all the zeros of the

function

come from the Blaschke product.

Definition

.

Let

f

⁢

(

z

)

:=

exp

⁡

(

∫

e

i

⁢

θ

+

z

e

i

⁢

θ

-

z

⁢

h

⁢

(

e

i

⁢

θ

)

⁢

𝑑

m

⁢

(

e

i

⁢

θ

)

)

,

where

h

is a real valued

Lebesgue integrable

function on the unit circle and

m

is the Lebesgue measure. Then

f

is called
an

outer function

.

The significance of these definitions is that every bounded

holomorphic function

can be written as an inner function times an outer function. See the factorization theorem for

H

∞

functions (

http://planetmath.org/FactorizationTheoremForHinftyFunctions

).

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1995.

Title

inner function

Canonical name

InnerFunction

Date of creation

2013-03-22 15:36:20

Last modified on

2013-03-22 15:36:20

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 30H05

Related topic

FactorizationTheoremForHinftyFunctions

Defines

singular inner function

Defines

outer function

Generated on Fri Feb 9 20:22:53 2018 by

LaTeXML
examples of harmonic functions on ℝ^n

examples of harmonic functions on

ℝ

n

Some

real functions

in

ℝ

n

(e.g. any

linear function

, or any affine function) are obviously

harmonic functions

. What are some more interesting harmonic functions?

•

For

n

≥

3

, define (on the punctured space

U

=

ℝ

n

∖

{

0

}

) the

function

f

⁢

(

x

)

=

∥

x

∥

2

-

n

. Then

∂

⁡

f

∂

⁡

x

i

=

(

2

-

n

)

⁢

x

i

∥

x

∥

n

,

and

∂

2

⁡

f

∂

⁡

x

i

2

=

n

⁢

(

n

-

2

)

⁢

x

i

2

∥

x

∥

n

+

2

-

(

n

-

2

)

⁢

1

∥

x

∥

n

Summing over

i

=

1

,

…

,

n

shows

Δ

⁢

f

≡

0

.

•

For

n

=

2

, define (on the punctured plane

U

=

ℝ

2

∖

{

0

}

) the function

f

⁢

(

x

,

y

)

=

log

⁡

(

x

2

+

y

2

)

. Derivation and summing yield

Δ

⁢

f

≡

0

.

•

For

n

=

1

, the condition

(

Δ

⁢

f

)

⁢

(

x

)

=

f

′′

⁢

(

x

)

≡

0

forces

f

to be an affine function on every segment; there are no “interesting” harmonic functions in one dimension.

Title

examples of harmonic functions on

ℝ

n

Canonical name

ExamplesOfHarmonicFunctionsOnmathbbRn

Date of creation

2013-03-22 12:44:23

Last modified on

2013-03-22 12:44:23

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

9

Author

mathwizard (128)

Entry type

Example

Classification

msc 31A05

Classification

msc 31B05

Generated on Fri Feb 9 20:30:03 2018 by

LaTeXML
Gauss’ mean value theorem for harmonic functions

Gauss’ mean value theorem for harmonic functions

If the

function

u

⁢

(

z

)

≡

u

⁢

(

x

,

y

)

is harmonic in a domain of

complex plane

which contains the disc

|

z

-

z

0

|

≦

r

,  then

u

⁢

(

z

0

)

=

1

2

⁢

π

⁢

∫

0

2

⁢

π

u

⁢

(

z

0

+

r

⁢

e

i

⁢

φ

)

⁢

𝑑

φ

.

(1)

Conversely

, if a real function

u

⁢

(

x

,

y

)

is

continuous

in a domain

G

of

ℝ

2

and

satisfies

on all circles of

G

the equation (1), then it is harmonic.

Title

Gauss’ mean value theorem for

harmonic functions

Canonical name

GaussMeanValueTheoremForHarmonicFunctions

Date of creation

2013-03-22 14:57:39

Last modified on

2013-03-22 14:57:39

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 31A05

Classification

msc 30F15

Related topic

GaussMeanValueTheorem

Generated on Fri Feb 9 20:30:05 2018 by

LaTeXML
Rado’s theorem

Rado’s theorem

Theorem

(Rado).

Suppose

Ω

⊂

R

2

is a convex (

http://planetmath.org/ConvexSet

) domain (

http://planetmath.org/Domain2

) with a smooth

boundary

∂

⁡

Ω

and suppose that

D

is the

unit disc

. Then given any

homeomorphism

μ

:

∂

⁡

D

→

∂

⁡

Ω

, there exists a unique

harmonic function

u

:

D

→

Ω

such that

u

=

μ

on

∂

⁡

D

and

u

is a

diffeomorphism

.

References

1

R. Schoen, S. T. Yau.

. International Press, Inc., Boston, Massachusetts, 1997

Title

Rado’s theorem

Canonical name

RadosTheorem

Date of creation

2013-03-22 14:08:07

Last modified on

2013-03-22 14:08:07

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Theorem

Classification

msc 31A05

Related topic

HarmonicFunction

Related topic

PerronFamily

Generated on Fri Feb 9 20:30:06 2018 by

LaTeXML
Schwarz-Christoffel transformation

Schwarz-Christoffel transformation

Let

w

=

f

⁢

(

z

)

=

c

⁢

∫

d

⁢

z

(

z

-

a

1

)

k

1

⁢

(

z

-

a

2

)

k

2

⁢

…

⁢

(

z

-

a

n

)

k

n

+

C

,

where the

a

j

’s are

real numbers

satisfying

a

1

<

a

2

<

…

<

a

n

, the

k

j

’s are real numbers satisfying

|

k

j

|

≦

1

;  the integral

expression

means a

complex

antiderivative,

c

and

C

are complex constants.

The transformation

z

↦

w

maps the

real axis

and the

upper half-plane

conformally (

http://planetmath.org/ConformalMapping

) onto the closed area

bounded

by a broken line.  Some vertices of this line may be in the

infinity

(the corresponding angles are = 0).  When

z

moves on the real axis from

-

∞

to

∞

,

w

moves along the broken line so that the direction turns the amount

k

j

⁢

π

anticlockwise every

z

passes a point

a

j

.  If the broken line closes to a polygon, then

k

1

+

k

2

+

…

+

k

n

=

2

.

This transformation is used in solving two-dimensional potential problems.  The

parameters

a

j

and

k

j

are chosen such that the given polygonal domain in the complex

w

-plane can be obtained.

A half-trivial example of the transformation is

w

=

1

2

⁢

∫

d

⁢

z

(

z

-

0

)

1

2

=

z

,

which maps the upper half-plane onto the first quadrant of the

complex plane

.

Title

Schwarz-Christoffel transformation

Canonical

name

SchwarzChristoffelTransformation

Date of creation

2013-03-22 14:41:02

Last modified on

2013-03-22 14:41:02

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Result

Classification

msc 31A99

Classification

msc 30C20

Synonym

Schwarz’ transformation

Related topic

ConformalMapping

Generated on Fri Feb 9 20:30:17 2018 by

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biharmonic equation

biharmonic equation

Definition. 1

.

A real-valued function

V

:

R

n

→

R

of class (

http://planetmath.org/http://planetmath.org/encyclopedia/Cn.html

)

C

4

, and satisfying the equation

∇

4

⁡

V

=

0

,

(1)

also defines a biharmonic function, and (1) is called the

biharmonic equation

.

Biharmonic operator

is defined as

∇

4

:=

∑

k

=

1

n

∂

4

∂

⁡

x

k

4

+

2

∑

k

=

1

n

-

1

∑

l

=

k

+

1

n

∂

4

∂

⁡

x

k

2

⁢

∂

⁡

x

l

2

⋅

Title

biharmonic equation

Canonical

name

BiharmonicEquation

Date of creation

2013-03-22 16:03:19

Last modified on

2013-03-22 16:03:19

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

9

Author

perucho (2192)

Entry type

Definition

Classification

msc 31B05

Generated on Fri Feb 9 20:30:22 2018 by

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Dirichlet problem

Dirichlet problem

Suppose

Ω

is a domain of

ℝ

n

and

∂

⁡

Ω

is the boundary of

Ω

.
Further suppose

f

is a function

f

:

∂

⁡

Ω

→

ℂ

. Then the

Dirichlet problem

is to find a function

ϕ

:

Ω

∪

∂

⁡

Ω

→

ℂ

such that

ϕ

=

f

,

on

∂

⁡

Ω

,

∇

2

⁡

ϕ

=

0

,

in

Ω

.

Title

Dirichlet problem

Canonical

name

DirichletProblem

Date of creation

2013-03-22 14:57:06

Last modified on

2013-03-22 14:57:06

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Definition

Classification

msc 31B05

Classification

msc 31A05

Classification

msc 31B15

Related topic

HarmonicFunction

Generated on Fri Feb 9 20:30:24 2018 by

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Laplacian

Laplacian

Let

(

x

1

,

…

,

x

n

)

be

Cartesian coordinates

for some open set

Ω

in

ℝ

n

.
Then the

Laplacian

differential operator

Δ

is defined as

Δ

=

∂

2

∂

⁡

x

1

2

+

⋯

+

∂

2

∂

⁡

x

n

2

.

In other words, if

f

is a twice

differentiable function

f

:

Ω

→

ℂ

, then

Δ

⁢

f

=

∂

2

⁡

f

∂

⁡

x

1

2

+

⋯

+

∂

2

⁡

f

∂

⁡

x

n

2

.

A

coordinate

independent

definition

of the Laplacian
is

Δ

=

∇

⋅

∇

, i.e.,

Δ

is the

composition

of

gradient

and codifferential.

A

harmonic function

is one for which the Laplacian

vanishes

.

Notes

An older symbol for the Laplacian is

∇

2

– conceptually the

scalar product

of

∇

with itself. This form is more favoured by physicists.

Derivation

\htmladdnormallink

Click here¡http://planetmath.org/?method=l2h&from=collab&id=76&op=getobj”¿ to see an article that derives the Laplacian in

spherical coordinates

.

Title

Laplacian

Canonical

name

Laplacian

Date of creation

2013-03-22 12:43:48

Last modified on

2013-03-22 12:43:48

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

18

Author

matte (1858)

Entry type

Definition

Classification

msc 31B05

Classification

msc 31B15

Related topic

DAlembertian

Related topic

Codifferential

Defines

Laplace operator

Generated on Fri Feb 9 20:30:26 2018 by

LaTeXML
Perron family

Perron family

Definition

.

Let

G

⊂

ℂ

be a region,

∂

∞

⁡

G

the

extended boundary

of

G

and

S

⁢

(

G

)

the set of

subharmonic functions

on

G

, then
if

f

:

∂

∞

⁡

G

→

ℝ

is a

continuous
function

then the set

𝒫

⁢

(

f

,

G

)

:=

{

φ

:

φ

∈

S

⁢

(

G

)

⁢

and

⁢

lim sup

z

→

a

⁡

φ

⁢

(

z

)

≤

f

⁢

(

a

)

⁢

for all

a

∈

∂

∞

⁡

G

}

,

is called the

Perron family

.

One thing to note is the

𝒫

⁢

(

f

,

G

)

is never empty. This is because

f

is

continuous

on

∂

∞

⁡

G

it attains a

maximum

, say

|

f

|

<

M

,
then the

function

φ

⁢

(

z

)

:=

-

M

is in

𝒫

⁢

(

f

,

G

)

.

Definition

.

Let

G

⊂

ℂ

be a region and

f

:

∂

∞

⁡

G

→

ℝ

be a continuous
function then the function

u

:

G

→

ℝ

u

⁢

(

z

)

:=

sup

⁡

{

ϕ

:

ϕ

∈

𝒫

⁢

(

f

,

G

)

}

,

is called the

Perron function

associated with

f

.

Here is the reason for all these

definitions

.

Theorem

.

Let

G

⊂

C

be a region and suppose

f

:

∂

∞

⁡

G

→

R

is a continuous function.
If

u

:

G

→

R

is the Perron function associated
with

f

, then

u

is a

harmonic function

.

Compare this with Rado’s

theorem

(

http://planetmath.org/RadosTheorem

) which works with
harmonic functions with range in

ℝ

2

, but also gives a much
stronger statement.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

Title

Perron family

Canonical

name

PerronFamily

Date of creation

2013-03-22 14:19:42

Last modified on

2013-03-22 14:19:42

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 31B05

Related topic

RadosTheorem

Defines

Perron function

Generated on Fri Feb 9 20:30:28 2018 by

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