A Comparison Of Harmonic And Holomorphic Functions

School of Education, Culture and Communication

Division of Applied Mathematics

BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS

A Comparison Of Harmonic And Holomorphic Functions

by

Adrian Daniel Renz

Kandidatarbete i matematik / till¨

ampad matematik

DIVISION OF APPLIED MATHEMATICS

M¨ALARDALEN UNIVERSITY SE-721 23 V¨ASTER˚AS, SWEDEN

School of Education, Culture and Communication

Division of Applied Mathematics

Bachelor thesis in mathematics / applied mathematics

Date:

2020-14-05

Project name:

A Comparison Of Holomorphic And Harmonic Functions

Author :

Adrian Daniel Renz

Supervisor : Linus Carlsson Reviewer : Anatoliy Malyarenko Examiner : Masood Aryapoor Comprising : 15 ECTS credits

Abstract

Many results in real and complex analysis are the consequence of mean value properties and theo-rems. This is the case for harmonic and holomorphic functions as well. The mean value property builds the foundation for several properties of each set of functions. Using this property one can de-rive more properties like the maximum principle for harmonic functions and the maximum modulus principle for holomorphic functions. These results are then used to show other properties. The goal is to compare the theorems and proofs for harmonic and holomorphic functions and to understand why the results seem to be similar.

Acknowledgments

Contents

1 Introduction 1

2 Holomorphic and Harmonic functions 3

3 Mean Value Property 5

4 Maximum Principle 9

4.1 Maximum Principle for harmonic functions . . . 9 4.2 Maximum Modulus Principle . . . 11

5 Uniqueness 15

5.1 Poisson’s equation (Harmonic functions) . . . 15 5.2 Holomorphic functions . . . 17 6 Liouville’s Theorem 17 6.1 Holomorphic functions . . . 17 6.2 Harmonic functions . . . 19 7 C∞ _{functions 23} 7.1 Holomorphic functions . . . 23 7.2 Harmonic functions . . . 25 8 Conclusion 27 9 Appendix 30

1

Introduction

The general task is to study harmonic functions in the real plane and the holomorphic functions in the complex plane. Here we will study similarities and differences between the properties of the two classes of functions. Furthermore, we analyse the similarities in theorems and proofs that we show during the course of this thesis and where those originate from.

We restrict ourselves to R2

and C.

Harmonic functions are functions that satisfy Laplace’s equation. This equation, which is with-out a doubt one of the most important equations in mathematics, is a partial differential equation (PDE) that is widely used in several areas, for example in physics and engineering. The physical meaning of the Laplace equation is that the potential satisfies the equation in a domain without a source. The equation was found by the French mathematician, physicist and astronomer Pierre-Simon Laplace (1749–1827). There is a whole field called Potential Theory, which is dedicated to the study of this equation.

Holomorphic functions, which are denoted as analytic functions in several sources are complex-valued functions, which can be locally represented by the power series. This theory was a product of several parallel researches conducted by the French mathematician Augustin-Louis Cauchy (1789-1857) who published his work in [1] and the German mathematicians Karl Weierstraß (1815-1897) and Bernhardt Riemann (1826-1866), whose work is published in [5]. Holomorphic functions are the main focus of study in complex analysis in present days.

Literature Review

As this is a bachelor thesis in mathematics, the research is already conducted and stated in a lot of books and papers. Those we used are part of our literature review. However, many books cover exactly the same material.

One of the most important equations are partial differential equations. Evans book [2] studies several of those partial differential equations, from linear, to non-linear PDE’s. The focus in partic-ular are the Laplace equation, the Heat equation, the Wave equation and the transport equation. The author goes deep into the properties of those equations as well as solving those PDE’s in several ways. One of the properties for the Laplace equation the author states in his book is the Poisson equation, which equation plays an important part in the uniqueness of harmonic functions. There

are several different boundary conditions that can be used, but Jomaa and Macaskill’s [3] conduct their research with Dirichlet boundary conditions, because those are the easiest to analyse. We will also focus on Dirichlet boundary conditions in this thesis. In this paper [3], Jomaa and Macaskill analyse the Poisson equation in one and two dimensions separately and discuss the error expres-sions, which goes way further than the scope of this thesis. More properties as well as different perspectives on proofs are stated in Walter Rudin’s book [6], which has wide range of topics in the field of real and complex analysis covered. One is of course the properties of harmonic func-tions. However, there are several more topics ranging from vector spaces and calculation in those spaces over transformations (such as the M¨obius transformation) to holomorphic functions. This book goes way deeper into holomorphic functions, due to the fact that those are very important for several chapters in this book because of the focus on complex analysis. The main focus is on Cauchy’s theorem, which can be found in [1]. Walter Rudin introduces this theorem early on and refers to it in several chapters. Another more basic written book which summarizes topics in Real and Complex analysis is [7]. The focus in this book is on holomorphic functions and therefore of course on Cauchy’s work [1] as well. In comparison to Rudin, this book has a different focus, as harmonic functions are only stated in a subsection of the book. Properties of harmonic functions are not stated at all. Properties of holomorphic functions (analytic functions in this book) are stated and shown across all chapters.

All theorems and most of the proofs are stated in those books, which makes the research seem to be complete in this field. The focus of this thesis is to use the above stated literature to show how properties of holomorphic and harmonic functions relate to each other. We adapted several of the proofs, so that they fit in the context of this thesis. In the beginning of each subsection we state the reference of which book gave us the direction for the theorem and proof.

Because there are several ways to denote mathematical objects, we unified them. The table below shows the notation that are used in this thesis.

notation description

z A point in the complex plane

x A point in the real plane of the form (x0, x1)

d(x,r) An open disk centered at x with radius r

d(x,r) The closure of the disk centered at x with radius r

Ω An open set

Ω The closure of the open set Ω f (z) A complex-valued function u(x) A function with real variables α = (α1, α2) An index vector, where αi∈ {0, 1, 2, ...}

k = |α| = α1+ α2 The length of the vector α

uxi =

∂u

∂xi Partial derivative of u with respect to the variable xi

2

Holomorphic and Harmonic functions

The following definitions are stated in Saff and Sniders book [7] in the case of holomorphic functions and of course Lawrence C. Evans book [2] for harmonic functions.

The derivative of a complex-valued function in a neighborhood of a point z0 is defined in the

following way:

Definition 1. Provided that the limit exists, the derivative of a complex-valued function f , that is defined in a neighborhood of z0 is given by

df

dz(z0) := lim∂z→0

f (z0+ ∂z) − f (z0)

∂z .

With this knowledge, we can define what it means for a function to be holomorphic.

Definition 2. A complex-valued function is said to be holomorphic on an open set Ω if it has a derivative at every point of Ω.

If the function f is holomorphic, i.e. the derivative exists at every point, then the function is continuous. That is lim

z−>z0

f (z) = f (z0), ∀z0, z ∈ Ω.

This Definition, however, is restricted on an open set Ω. This begs the question what happens if a function is holomorphic on the whole of C.

Definition 3. A function that is holomorphic on the whole of C is called an entire function. The real counterpart for holomorphic functions are the functions, that satisfy the elliptic partial differential equation of second degree called Laplace equation. The Laplace equation is given by

∆u = 0 (1)

Definition 4. Let u(x0, x1) be a real-valued function. This function is said to be harmonic if its

second partial derivatives are continuous and it satisfies (1).

Another two important partial differential equations are the Cauchy Riemann equations.

∂u ∂x = ∂v ∂y (2) ∂u ∂y = − ∂v ∂x (3)

Using equations (2) and (3), we can show that any holomorphic function f (z) can be written in the form f (z) = u(x0, x1) + iv(x0, x1), where both u(x0, x1) and v(x0, x1) are harmonic functions

and therefore functions with respect to the real variables x0 and x1. So, the real and imaginary

parts of a holomorphic functions are harmonic. This can be easily shown by using the fact that partial derivatives can be taken in any way and that the imaginary as well as the real part of holomorphic functions are infinitely differentiable.

We can create ∂2_u ∂x0∂x1 = ∂ 2_u ∂x1∂x0

where we can use the Cauchy Riemann equations ∂2_v ∂x2 1 = −∂ 2_v ∂x2 0

which is obviously equivalent to the Laplace equation (1). Thus, the real and imaginary part of holomorphic functions are harmonic functions.

An example of a holomorphic function is f (z) = ux0(x0, x1)−iux1(x0, x1), where u is a harmonic

function. This obviously satisfies the Cauchy Riemann equations (2) and (3). Thus, ux0 and ux1

Now we know that a harmonic function is a locally real part of a holomorphic function. Let’s assume that there exists a known harmonic function u(x0, x1) that is harmonic in a neighborhood

d(z0,r). Then we can find a v(x0, x1), such that we can create a holomorphic function of the form

f (z) = u(x0, x1) + iv(x0, x1). This is of course a local property. Let u(x0, x1) be a harmonic

function in a convex domain. A convex set is a set, where the line segment between any two points in this set is also in this set.

Let u(x_e0,xe1) be a point in the domain. Then

U (x0, x1) = Z x0 e x0 u(t, x1) dt We define f (z) = f (x0, x1) = Ux0(x0, x1) − iUx1(x0, x1) = u(x0, x1) − i Z x0 e x0 u(t, x1) dt

which shows that we can find a v(x0, x1) if there exists a known harmonic function u(x0, x1).

We could have used this property to show local phenomenas of harmonic functions when we have knowledge of holomorphic functions. However, we will not proceed in that manner and instead show several of the more classic proofs for properties of holomorphic and harmonic functions.

3

Mean Value Property

The following part is about the mean value property for harmonic functions. Assume we have a harmonic function u within an open set Ω. Furthermore, there exists a disk d(x,r)⊂ Ω, where r is

the radius. The mean value property states that value of the function u in the center is the same as the average value of the function u over the disk d(x,r) or the average value of the function u

over the boundary circle ∂d(x,r) for any radius r.

Theorem 1 is stated extensively in Evans book [2], the proof, however, is different in several parts.

Theorem 1. Assume that u is an a harmonic function inside an open set inside Ω, which lies in R2 and let p = (a, b) ∈ Ω. Then

u(p) := 1 πr2 Z Z d(p,r) u(x0, x1) dx1dx0= 1 2πr I ∂d(p,r) u(x0, x1) ds

which holds for all r > 0 such that d(p,r) ⊂ Ω

Proof.

This proof consists out of two parts as we have to show two equalities.

1. In the following part of the proof we use Greens divergence Theorem. Let F be a continuously differentiable vector field defined as F =



 M N



and let • define the dot product. Then, for any radius 0 ≤ R ≤ r Z Z d_(p,R) ∇ • F dx0dx1= I ∂d_(p,R) M dx1− N dx0

Let F = ∇u. Then M = _∂x∂u

0 and N = ∂u ∂x1. We now have, Z Z d(p,R) ∇ • ∇u dx0dx1= I ∂d(p,R) ∂u ∂x0 dx1− ∂u ∂x1 dx0

Since u is harmonic, we have

0 = I ∂d(p,R) ∂u ∂x0 dx1− ∂u ∂x1 dx0 Parametizing ∂d(p,R) gives us α(t) = (x0(t), x1(t)) = (a + R cos t, b + R sin t), 0 ≤ t ≤ 2π α0(t) = (x0₀(t), x0₁(t)) = (−R sin t, R cos t), 0 ≤ t ≤ 2π |α0(t)| = R

Now we need to plug this into our equation. 0 = I ∂d(p,R) ∂u ∂x0 dx1− ∂u ∂x1 dx0 = Z 2π 0  ∂u ∂x0

(a + R cos t, b + R sin t)R cos t + ∂u ∂x1

(a + R cos t, b + R sin t)R sin t 

dt Using the Chain rule backwards gives us

0 = Z 2π

0

∂

which, by Leibniz we can write as 0 = ∂

∂R Z 2π

0

u(a + R cos t, b + R sin t)dt Integrating both sides from 0 to r 0 = Z r 0 ∂ ∂R Z 2π 0

u(a + R cos t, b + R sin t)dtdR

Fubini’s theorem lets us interchange the order of integration, hence 0 =

Z 2π

0

u(a + r cos t, b + r sin t)dt − 2πu(a, b)

=1 r

Z 2π

0

u(a + r cos t, b + r sin t)rdt − 2πu(a, b)

=1 r Z 2π 0 u(x0, x1)ds − 2πu(a, b) Thus, we get u(p) = 1 2πr I ∂d(p,r) u(x0, x1) ds 2. Show that _πr12 R R d(p,r)u(x0, x1) dx1dx0 = 1 2πr H

∂d(p,r)u(x0, x1) ds = u(p). We show that by

using the co-area formula.

We start off by showing what the co-area formula tells us. Let f (x) be a function which is continuous on the disk d(x,R). Then

Z Z d(x,R) f (x) dA = Z R 0 Z ∂d(x,t) f (x) dtds We can now parametrize ∂d(x,t)as x0= t cos θ and x1= t sin θ. Then we get

s(θ) = (x0(θ), x1(θ))

s0(θ) = (−x1(θ), x0(θ))

|s0(θ)| = t which means that

Z R 0 Z ∂d(x,R) f (x) dA = Z R 0 Z 2π 0 f (s(θ))|r0(θ)| dθdt = Z R 0 Z 2π 0 f (t cos θ, t sin θ)t dθdt

Using this knowledge, we continue with our proof. Z Z d(p,r) u(x0, x1)dx0dx1= Z r 0 Z ∂d(p,r) u(x0, x1)ds ! dR = Z r 0 2πR 1 2πR Z ∂d(p,r) u(x0, x1)ds ! dR

Using part 1 of the proof Z Z d(p,r) u(x0, x1)dx0dx1= u(p)π Z r 0 2R dR = u(p)πr2 Thus, u(p) = 1 πr2 Z Z d(p,r) u(x0, x1)dx0dx1  Before we go on we introduce the following Lemma.

Lemma 1. Assume that f : R → R is a non-negative continuous function such that f (x0) = δ > 0.

Then there exists an interval I = (x0− , x0+ ) such that f (x) > δ₂ when x ∈ I. In particular,

R

If (x) dx > 0.

The proof of this Lemma is not considered in this thesis. However, Figure 1 provides an illus-tration of this lemma.

x0 _x0+

x0

-f(x0)=δ

4

Maximum Principle

4.1

Maximum Principle for harmonic functions

The maximum principle essentially states that a non-constant harmonic function in a domain can only attain its maximum on the boundary and not in the interior. So, the maximum and minimum bounds the values of a harmonic function on the boundary if the domain is bounded.

The following theorem is stated in [2], where Lawrence C. Evans writes about properties for harmonic functions. The proof is inspired by this book but rewritten so it fits the context of this thesis.

Theorem 2. Maximum Principle

Let u be a harmonic function within the domain U . Furthermore, suppose that u is twice differentiable on U and differentiable once on U , i.e. u ∈ C2_{(U ) ∩ C}1_{(U ).}

(i) Then

max

∂U u = max_U u.

(ii) Let x0∈ U with the condition that U is connected1 such that

u(x0) = max U

u, then u is constant within U.

Note that (i) is commonly denoted as the weak maximum principle, where (ii) is usually said to be the strong maximum principle. It is also important to note that Theorem 5 can be modified by using −u instead of u. The result would then be the minimum principle.

Proof.

We start our proof of the maximum principle by showing (ii) because (ii) implies (i).

This proof is conducted similarly to the proof of the Maximum Modulus Principle for holomor-phic functions, which we state in the next subsection.

Let u be harmonic on the disk d(x,r) such that u(x) ≥ u(x1) ∀ x1∈ d(x,r). We will prove that

this forces u to be constant on d(x,r).

In order to show that we assume that u is not constant in d(x,r). Then we know that there

exists an x1 ∈ d(x,r) with the property u(x1) < u(x). Now we can use Lemma 1, as illustrated in

Figure 2, on a circle with radius R = r1= |x1|.

r R

(2 )

x

Figure 2: Illustration of Lemma 1 used on a circle

u(x) = 1 2πr I ∂d(x,r) u(x1) ds < 1 2πr I ∂d(x,r) u(x) ds = u(x)

This is obviously a contradiction, thus u has to be constant on the disk d(x,r). Now all that’s

left is to show that u is constant in the domain U .

Denote x as the point where u achieves its maximum in the domain U . Then there exists a disk d(x,r)⊂ U centered at x. We select any point x16= x inside the disk d(x,r). We know already that

u is constant inside this disk and therefore we know that u(x1) = u(x).

We can create another disk d(x1,r1) ⊂ U around x1 and select another point x2 6= x1 inside

d(x1,r1). Then because u has to be constant in that disk, we know that u(x2) = u(x1).

Using the same pattern over and over again until we reach any point xn in the domain U , tells

In order to understand why (i) holds as well we need to look at the extreme value theorem for compact sets, which states that every continuous function attains a maximum and minimum at some point in a non-empty compact space. We know that the closure of the domain U is a compact set since it is the subset of a closed and bounded set U ⊂ R2. Therefore, we know that the continuous function u on the compact set U attains its maximum and minimum on U .

Thus there are two possibilities. Either u attains its maximum in the interior of U or not. If u does not attain its maximum in the interior of U , it does on the boundary ∂U . If u does attain its maximum in the interior of U , then (ii) holds and we know that u is a constant function.

Therefore, (ii) holds and of course (ii) implies that (i) holds as well.



4.2

Maximum Modulus Principle

In order to get an understanding for the maximum modulus principle, we first need to get the mean value property for holomorphic functions. The mean value property for holomorphic functions is an immediate consequence from Cauchy’s formula.

Recall Cauchy’s formula stated in [1]. Let f be holomorphic on an open set Ω that contains the disk d(z0,r) with the boundary cr, then

f (z0) = 1 2πi Z cr f (z) z − z0 dz (4)

We can parametrize the circle cr using z = z0+ reiθ, 0 ≤ θ ≤ 2π. This leads us to the mean

value property for holomorphic functions. f (z0) = 1 2πi Z 2π 0 f (z0+ reiθ) z0+ reiθ− z0 ireiθdθ = 1 2π Z 2π 0 f (z0+ reiθ) reiθ re iθ_dθ = 1 2π Z 2π 0 f (z0+ reiθ) dθ

What this property tells us, is that if we have a function f on a disk and f is holomorphic within the disk, then the value of f in the center z0is equal to the average value on the boundary.

Theorem 3. Let f be holomorphic on the open disk d(z0,R) and continuous on its closure d(z0,R),

with z0∈ d(z0,R)

If |f (z0)| ≥ |f (z)| ∀ z ∈ d(z0,R)

then f (z) = f (z0) is a constant function on d(z0,R)

Proof.

Assume that f (z) is not constant in d(z0,R). Then there exists a w0in d(z0,R)such that |f (z0)| >

|f (w0)|. Now we can use Lemma 1 on a circle with radius r = |z0− w0|.

Since _2π1 R2π 0 1dθ = 1, we get |f (z0)| = 1 2π Z 2π 0 |f (z0)| dθ

where we can use lemma 1 to get > 1

2π Z 2π

0

|f (z0+ reiθ)| dθ.

Applying the triangle inequality and the mean value property leads us to 1 2π Z 2π 0 |f (z0+ reiθ)| dθ ≥     1 2πr Z 2π 0 f (z0+ reiθ) dθ     = |f (z0)|

This is obviously a contradiction. Thus, |f (z)| has to be constant. We will see in the proof of Theorem 4 that this also implies, that f (z) has to be constant.

 This is a local Theorem. However, we need to use Theorem 3 in order to get a more global version of the maximum modulus theorem.

The next theorem is stated extensively in [7]. The proof in the way we conducted the research, however, is not stated in any book.

Theorem 4. Let Ω be an open connected set and let f be holomorphic on Ω. If |f (z)| achieves its maximum value at a point z0 in Ω, then f is constant in Ω.

Proof.

Assume that f (z) is holomorphic on an open connected set Ω. Denote z0 as the point where

z1in the interior of the disk d(z0,R), then we know by Theorem 3, that |f (z1)| = |f (z0)|. This also

means that f (z) is constant on d(z0,R).

Now we can create a second disk d(z0,R) ⊂ Ω centered at z1 and select a point z2 in the

interior of said disk. Using Theorem 3 again tells us that f (z) is constant on DR1 and therefore

|f (z2)| = |f (z1)| holds.

Continuing this pattern up to any zn ∈ Ω. on the set Ω tells us that |f (z)| has to be constant

in Ω. In Figure 3 we have n = 6, so we also have z6.

Figure 3: Illustration of the proof for Theorem 3 What is left to show is that |f (z)| = f (z).

Let f (z) = u(x0, x1) + iv(x0, x1). We know that |f (z)| is constant, so |f (z)|2 =

√

u2_{+ v}22 ₌

u2_{+ v}2 _{is constant as well. Furthermore, we know that} ∂|f |2 ∂x0 =

∂|f |2

∂x1 = 0 again due to the fact

that |f (z)| is constant. Using the Cauchy-Riemann equations as well as these two relations we get f0(z) = 0.

∂|f |2 ∂x0 = 2u∂u ∂x0 + 2v ∂v ∂x0 = 0 ∂|f |2 ∂x1 = 2u∂u ∂x1 + 2v ∂v ∂x1 = −2u ∂v ∂x0 + 2v∂u ∂x0 = 0 multiplying those with v and u respectively gives us

2vu∂u ∂x0 + 2v2 ∂v ∂x0 = 0 2vu∂u ∂x0 − 2u2 ∂v ∂x0 = 0 subtracting those equation from each other gives

2 ∂v ∂x0

v2+ u2
= 0 ⇒ ∂v ∂x0

= 0 unless u = v = 0 Multiplying with u and v respectively, we get

2u2 ∂u ∂x0 + 2vu∂v ∂x0 = 0 2v2∂u ∂x0 − 2vu∂v ∂x0 = 0 Adding the two equations

(u2+ v2)2 ∂u ∂x0

= 0 ⇒ ∂u ∂x0

= 0 unless u = v = 0

The points where u = v = 0 holds, have to be (1) of lower dimension than 2, because otherwise there would be an open subset where the derivatives would be zero, or (2) all points must satisfy u = v = 0 everywhere.

This means we have to show that both cases hold. Case (1):

We know that u and v are harmonic functions. The maximum principle tells us that if a harmonic function is constant in an open set, then it has to be constant everywhere. Now, since

∂u ∂x0 and

∂v

∂x0 are continuous functions, they have to be zero, even on this lower dimensional set.

Therefore, f0(z) = 0. Case (2):

u = v = 0 everywhere implies f0(z) = 0.

This of course means that f (z) has to be constant in Ω.

The Theorem above considers an open connected set. If we take a look at a bounded open connected set, we can modify Theorem 3.

Theorem 5. Let Ω be a set that is open, and bounded. Let f (z) be a holomorphic function that is continuous on Ω. Then the maximum modulus of f (z) is on the boundary ∂Ω.

Proof.

The closure of Ω, Ω is a compact set. Furthermore, let v ∈ Ω be a point, where |f (z)| attains its maximum. Namely, |f (v)| ≥ |f (z)| ∀z ∈ Ω. We know that |f (v)| < ∞ because we have a continuous function on a closed and bounded set.

If we assume that v is on the boundary of Ω, we get |f (v)| = max

z∈∂Ω|f (z)| = max_z∈Ω|f (z)|. By

Theorem 4, if v ∈ Ω, we get that f (z) must be constant. Thus we know that max

z∈∂Ω|f (z)| = maxz∈Ω

|f (z)| holds as well.



5

Uniqueness

5.1

Poisson’s equation (Harmonic functions)

This subsection’s style of writing the Poisson equation is stated in the paper [3], however the theo-rem and proof are stated in Lawrence C. Evans book [2] and W. Rudin’s book [6]. The proof below differs in several parts from those in these books.

Definition 5. On an open set Ω, the Poisson equation is given by

∆u = f (5)

as well as some boundary conditions, where ∆ is the Laplacian operator, f is a function and u is our desired solution.

Note that if f ≡ 0, then we get the Laplace equation (1), above.

There are several possible boundary conditions. For simplicity, we just focus on the most important one, namely the Dirichlet boundary conditions.

Because the Poisson equation is a linear equation, we have that the uniqueness of the solution to the Laplace equation is equivalent to the uniqueness of the solution to the Poisson equation. This is the case because the Laplace equation is a special case of the Poisson equation. Namely let f = 0 in equation (5) such that we get the Laplace equation ∆u = 0 (1). As a result we have that if a function vanishes on the boundary, it vanishes everywhere. In order to get a solution for our Poisson equation, we have to solve the Dirichlet problem for the Poisson equation. However, this is not the goal of this thesis, as we examine the uniqueness of those solutions and not any specific solution.

Theorem 6. Let Ω be a bounded, open set. If a solution exists to the Poisson equation with Dirich-let boundary conditions, that is u solves

     ∆u = f in Ω u = g on ∂Ω (6)

then the solution u is unique. Proof.

Suppose that u1is one solution to (5) and u2 is another one.

We define h = u1− u2and therefore get:

     ∆h = 0 in Ω h = 0 on ∂Ω Therefore, it follows that h is a harmonic function.

Because we have a harmonic function in h, we can use the maximum principle. So, h must have its maximum and minimum on the boundary. That is,

h(x) ≤ max y∈Ω h(y) = max y∈∂Ωh(y) = 0 and conversely, 0 = min

y∈∂Ωh(y) = miny∈Ω

So obviously h ≡ 0 and 0 ≡ u1− u2, which gives us u1≡ u2. Therefore, the solution to Poisson’s

equation is unique.



5.2

Holomorphic functions

Now, that we showed the uniqueness of harmonic functions with Dirichlet

boundary conditions, the next step is to show that two holomorphic functions with the same boundary values are in fact the same function.

Theorem 7. Let f (z) and g(z) be holomorphic functions inside the domain Ω. Let C be a contour2

in Ω. If f (z) = g(z) on C, then f (z) = g(z) holds everywhere inside the contour C. Proof.

Assume that f (z) = g(z) on the contour C. Then we have h(z) = f (z) − g(z) = 0.

We can now use the Maximum Modulus Principle, which states that the maximum modulus value inside Ω is the same as the maximum modulus value on the closed contour C, i.e. its boundary. This then tells us that if we pick any x0inside C, then we know that that is equivalent

to the boundary value. As the function h(z) = 0 on the boundary, we have that this condition holds everywhere, which means that f (z) = g(z) at any point inside the contour C.



6

Liouville’s Theorem

The next topic we are investigating is the so-called Liouville’s theorem.

6.1

Holomorphic functions

The following subsection uses theorems and notations from [7]. As mentioned in the literature review, the proofs are rewritten and adjusted so they fit in the context of this thesis.

Before starting this subsection we need to introduce another lemma, the ML-Inequality which is also commonly denoted as the estimation lemma. The Lemma is stated in S. Ponnusamy and H. Silverman’s book [4] as well as the proof which will be part of the Appendix.

Lemma 2 (ML-Inequality). Let C be a Contour and with length L. Let f (z) be continuous on C. Then     Z C f (z) dz     ≤ Z C |f (z)| |dz| ≤ M Z C |dz| = M L where |f (z)| ≤ M < ∞

Now we can state a theorem for the estimates of every derivative at a certain point, because we use it in order to show Liouville’s Theorem for holomorphic functions.

Theorem 8. Let f (z) be a holomorphic function on the disk d(z0,R) centered at z0. Furthermore,

let max |f (z)| = M < ∞ ∀z ∈ d(z0,R). Then,

|f(n)_(z 0)| ≤

n!M

Rn ∀n ∈ N

holds for all derivatives of f (z) at z0.

Proof.

We start at the general Cauchy’s Integral formula, which gives us f(n)(z0) = n! 2πi Z d_(z0,R) f (z) (z − z0)n+1 dz

Note that if n = 0, we get the specific equation (4) as stated in section 4.2. Now taking the absolute value, leads to

|f(n)_(z 0)| =     n! 2πi          Z d_(z0,R) f (z) (z − z0)n+1 dz     

where we can use the estimation lemma, which gives us |f(n)_(z 0)| ≤     n! 2πi     Z d_(z0,R)     f (z) (z − z0)n+1     |dz| = n! 2π Z d_(z0,R) M Rn+1 |dz|

Because we know that the length of ∂d(z0,R) is 2πR, we get |f(n)_(z 0)| ≤ n! 2π M Rn+12πR = n!M Rn

which is exactly what we wanted to prove.

 Now, let f (z) be holomorphic and bounded by some M on the plane C. Take n=1 and let R → ∞. Then, because the theorem above holds for all z0, we see that the first derivative of f (z)

vanishes everywhere. A more rigorous proof is found after the theorem. So, we get the following. Theorem 9 (Liouville’s Theorem). Let f (z) be a bounded, entire function. Then f (z) is constant. Proof.

Since f (z) is entire, we know that it is differentiable everywhere and that |f (z)| ≤ M ∀z ∈ C. Let z0 be any point in C. We know that f (z) is entire, so it is also holomorphic in any disk

with radius R > 0 around z0 by definition. By the Cauchy Estimates we have

|f0(z0)| ≤

M R Letting the radius R → ∞ gives

0 ≤ |f0(z0)| ≤ 0

Thus,|f0(z0)| = 0, And since our z0 is arbitrary, we know that this holds for all z ∈ C. Thus, f (z)

has to be constant.



6.2

Harmonic functions

The following theorems and properties are adjusted from the theorems and properties in Lawrence C. Evans book [2].

We just stated Liouville’s theorem for holomorphic functions. Of course this theorem exists for harmonic functions as well. Again, we start off by showing the estimates on derivatives of harmonic

functions and use that to show Liouville’s theorem for harmonic functions. Using the mean value formula again, we can derive estimates for derivatives.

Definition 6. For any α = (α1, α2) with k = |α|, we define

Dαu(x) = ∂ k ∂xα1 1 ∂x α2 2 u We also introduce the Gauss-Green Theorem on a disk d(x,r).

Theorem 10. Z Z d(x,r) uxidx = Z ∂d(x,r) νiu(x) ds (i = 1, 2)

where νi _{is the ith component of the outer unit normal vector}3_.

We will use this theorem later in a proof.

Theorem 11. Let u(x) be a harmonic function in the disk d(x,s)∈ R2. Then, for any 0 < r < s

it holds that

|Dαu(x)| ≤ C

ry in ∂dsup(x,r)

|u (y) |

where Dα _{denotes any partial derivatives of order k = 1, C is a constant and sup is the supremum.}

Proof.

We can set our x to 0 by translation of coordinates without losing our generality. Then, by the man value property as well as the Gauss-Green Theorem, we get

Dαu(0) = 1 πr2 Z Z d(0,r) Dαu(x)dx = 1 πr2 Z ∂d(0,r) νiu(x) ds

Applying this to the absolute value gives us |Dα_{u(0)| =}      1 πr2 Z Z d_(0,r) uxi dx      =      1 πr2 Z ∂d(0,r) νiu ds      ≤ 1 |πr2_| Z ∂d(0,r) |u| ds ≤ C r∂dsup(0,r) |u|

3_{On a surface, there are two possible unit normal vectors. One pointing inside and one pointing outside. This is}

This concludes the proof.

 Now we know that our estimate on derivative in the center of the disk d(x,r) is the same as on

the supremum of u(x) on the boundary ∂d(x,r).

The next Theorem is going to show the general formula that is used to estimate derivatives of harmonic functions.

Theorem 12. Let u(x) be a harmonic function in R2_{. Moreover, let d}

(x,r)⊂ Ω be a disk of radius

r > 0 around x and let d(x,r)⊂ R2 for every x ∈ R2. Then,

|Dα_{u(x)| ≤} Ck

r2+k

Z Z

d(x,r)

|u(x)| dx where Dα _{denotes any partial derivatives of order k.}

Proof.

For simplicity, we prove this Theorem for the cases k = 0 and k = 1. Any higher case k can be proven by induction. The proof for k = 0 is an immediate consequence of the mean value theorem.

|u(x)| ≤ 1 πr2 Z Z d(x,r) |u(x)| dx = C r2 Z Z d(x,r) |u(x)| dx

This means all that’s left is to show the proof for the case k = 1. Let x0 be the center of a disk d(_x0,r

2). Furthermore, let u be continuous on the closed and

bounded disk d(x0,r2). Then we know that the maximum of u is taken on the boundary ∂d(x0,r2).

We do not know exactly where that point is but we know it exists by the maximum principle. Using Theorem 2 on said disk, we get obtain some point x1on the boundary ∂d(x0,r2). Then we can create

another disk centered at x1 of radius r₂.

Namely, d(x1,r2)⊂ d(x0,r)⊂ R

2

|u(x1)| ≤ 4 πr2 Z Z d_(x1,r 2) |u(x)| dx

Because we obviously have Z Z d_(x1,r 2) |u(x)| dx ≤ Z Z d_(x0,r) |u(x)| dx we get |u(x1)| ≤ 4 πr2 Z Z d_(x0,r) |u(x)| dx for any |x1=r₂|

Now we can use Theorem 11 with the above obtained inequality.

|Dαu(x0)| ≤ C r |u(x1)| ≤ C r 4 πr2 Z Z d_(x0,r) |u(x)| dx = C r3 Z Z d_(x0,r) |u(x)| dx

which is exactly what we needed.

 Using the estimates on derivatives, we can again show Liouville’s Theorem similarly to the holomorphic case.

Theorem 13. Let u(x) be a harmonic, bounded function in R2_{. Then u is a constant function.}

Proof.

In Theorem 15, we will see that u(x) is smooth. Choose any point x0in R2and let d(x0,r) be a

disk with radius r centered at x0. By Theorem 11, we get

|uxi(x)| ≤

C r∂dsup_(x0,r)

Now, let r → ∞, then we know because the supremum of u is bounded 0 ≤ |uxi(x0)| ≤ 0

Since |uxi(x0)| = 0 and we chose an arbitrary x0, we know that this holds for all x0∈ R

2_{. So}

our function u(x) has to be constant.



7

C

functions

7.1

Holomorphic functions

This subsection is about the differentiability of holomorphic functions. We touched this topic briefly while defining what it means for a function to be holomorphic. Now we want to show, that a holomorphic function that is by Definition 2 differentiable at every point, is in fact infinitely differentiable. Another common description is that if a function f is infinitely differentiable, it is denoted as a C∞function.

Definition 7 and 8 are inspired by the definitions in Saff and Sniders book [7]. We start off by introducing the power series.

Definition 7. A power series is a series of the formP∞

n=0cn(z −z0)n, where cnare the coefficients.

We also know that the power series converges absolutely and uniformly on a closed disk D(z0,r)

for any r < R, where R ∈ [0, ∞].

Definition 8. A function f (z) is holomorphic at a point z0, if there exists a power series expansion

that is defined on a disk D(z0,r) and converges in a neighborhood around the point z0.

A series expansion expands a function into an infinite sum of terms. So, Definition 8 tells us that any analytic function has a power series expansion, which leads us to the main theorem of our subsection.

The following theorem is used stated in [6]. The proof of this Theorem is taken from Rudins book, with slight adjustments for clarification.

Theorem 14. Let f (z) be a holomorphic function in the domain Ω which is representable by a power series in Ω. Then the first derivative f0(z) is also representable by a power series in the domain Ω. Furthermore, if f (z) = ∞ X n=0 cn(z − z0)n on a disk D(z0,r), then f0(z) = ∞ X n=0 ncn(z − z0)n−1 Proof.

We start off by setting z0= 0. That is f (z) =P ∞

n=0cnzn. That makes the calculations easier

while not losing any generality in our proof. Let f (z) converge on a disk D(z0,r). Now we can use

the Root Test to show that the radius of convergence of f (z) is the same as for its derivative f0(z). So, because p|n| = 1, we have Rn −1= lim sup

n→∞

n

p|cn| = lim sup n→∞

n

p|ncn|, where R ≥ r denotes the

radius of convergence. Thus, we know that the power series of f0(z) has to converge in the same radius of convergence r as f (z). We now set g(z) = f0(z) and take a point φ and a radius R such that φ ∈ D(z0,r) and 0 < |φ| < s < r < R. Let z 6= φ, then

f (z) − f (φ) z − φ − g(w) = ∞ X n=1 cn  zn_{− φ}n z − φ − nφ n−1 

Here, we get two cases. If n = 1, we get the first term on the right hand side to be z − φ z − φ− 1φ 0_{= 0} But if n > 1, we have zn−1φ0+ zn−2φ + zn−3φ2+ ... + z0φn−1− nφn−1 = zn−1− φn−1
_{+ z}n−2_{φ − φ}n−1
_{+ z}n−3_φ2_{− φ}n−1
_{+ ... + zφ}n−2_{− φ}n−1
_{+ φ}n−1_{− φ}n−1
= (z − φ)(zn−2_{+ ... + φ}n−2_{) + φ(z}n−3_{+ ... + φ}n−3_{) + ... + φ}n−3_{(z + φ) + φ}n−2 = (z − φ) n−1 X k=1 kφk−1zn−k−1

Let |z| and |φ| be smaller than min(1₂,s₂), such that      n−1 X k=1 kφk−1zn−k−1      <     n(n − 1) 2 s 2 n−2   Then     f (z) − f (φ) z − φ − g(φ)     ≤ |z − φ| ∞ X n=2 |cn| n2 2n−2s n−2

We know that the right side converges absolutely because s < R. As z → φ, the left side goes to 0. So we have g(φ) = f0(φ).



7.2

Harmonic functions

This whole sub section is inpired by the way Lawrence C. Evans [2] shows it in his book. Lawrence C. Evans [2] writes this Theorem as follows.

Like for holomorphic functions, we want to show that if all second partial derivatives of harmonic functions exist, i.e. u ∈ C2_{, then all partial derivatives exist. This means we want to show u ∈ C}∞_.

As usual, we will work in R2 _{in this subsection as well.}

Theorem 15. Let u be a harmonic function. Then u ∈ C∞ in Ω

There are several definitions and properties we need to understand before starting with this proof. The first one is Convolution.

Definition 9. We define Convolution of two functions g and f as [f ∗ g](t) =

Z

R2

f (p)g(t − p)dp

Now we can introduce the standard mollifier. Let Ω ⊂ R2 _{be an open and bounded set and Ω}

• φ has to be compactly supported

• R

R2φ= 1

We now can define the function

η(x) =      Ce 1 |x|2 −1 _{if |x| < 1} 0 if |x| ≥ 1 where C is a constant such that the second property of mollifiersR

R2η = 1 is satisfied. It can be

proven by induction that η is infinitely differentiable, i.e. η ∈ C∞_(R2). The standard mollifier is a function η: R2→ R defined as

η(x) = 1 2η  |x|  

with the same properties as the mollifier, in addition to it being smooth.

Furthermore, one can modify one function by using convolution with our standard mollifier. Let f be an ordinary function. Then the mollification of f is denoted as

f(x) ≡ [f ∗ η](x) =

Z

R2

f (p)η(x − p)dp for any x ∈ Ω

where the distance between x and the boundary ∂Ω is  > 0, i.e. Ω:= {x ∈ Ω : dist(x, ∂Ω) > }.

This function is infinitely differentiable, so all derivatives of f _{are continuous on Ω} .

With these definitions and properties, we can now proof Theorem 15. In Evans book [2], the proof is written as follows.

Proof.

Let u _{:= η}

∗ u in Ω. Then we know that u ∈ C∞(Ω). This means if we can show that

u_{(x) ≡ u(x) for all x ∈ Ω, we know that u(x is infinitely differentiable. Given x ∈ Ω, we have}

dist(x, ∂Ω) = ˜ > 0. We choose a  =₃˜. Then

u(x) = Z Z Ω u(p)η(x − p) dA = 1 2 Z Z d(x,) u(p)η  |x − p|   dA

by the co-area formula we get the following, where η does not depend on the angle u(x) = 1 2 Z  0 η r   Z ∂d(x,r) u(p) ds ! dr

The mean-value property (Theorem 1) on a circle then gives us u(x) = u(x)1 2 Z  0 η r   nα(n)rn−1 dr

where n = 2 and α(n) is the area of the unit disk, so in our case we have α(2) = π u(x) = u(x)2π 2 Z  0 η r   r dr = u(x)1 2 Z 2π 0 dθ Z  0 η r   r dr = u(x)1 2 Z 2π 0 Z  0 η r   r drdθ

We can now make a change of variables using t = r_, which of course also means that dt = dr_. Thus, u(x) = u(x)1 2 Z 2π 0 Z  0 η(t) t2 dtdθ

Using the co-area formula again we get u(x) = u(x)

Z Z

d_(x,1)

η(p) dA

= u(x)

So, we get u_{(x) = u(x), which of course means that u(x) ∈ C}∞_(Ω ).

 Thus, we know that harmonic functions are infinitely differentiable.

8

Conclusion

The main theorem that many proofs build on is the mean value property. Essentially, all theorems build up on this simple property. That is the reason why a lot of proofs are similar for holomorphic and for harmonic functions. There are several other theorems that could be compared to each other and would give us similar results for each respective function. A good illustration for this are

estimates of derivatives. In both cases we have a constant multiplied by the maximum value of the function divided by the radius. Just by looking at the proofs one can see how similar both sets of functions behave.

There are several ways of taking this thesis even further. One of them is by studying subharmonic functions, which are semi-continuous. Here one could compare the properties and proofs between those and our two functions used in this thesis. Furthermore, one could work with weak derivatives and see how this affects several proofs. Another extension would be examining different properties in different spaces such as Hilbert and Banach spaces. In general, there are theorems in this thesis where one can adjust a small part and investigate the difference it makes. For example we can replace the compact set U in the maximum principle and replace it with a pseudocompact set.

References

[1] A-L. Cauchy. Œuvres compl`etesd’Augustin Cauchy. Academie de Sciences, 1885.

[2] L.C. Evans. Partial Differential Equations: Second Edition. Graduate Studies in Mathematics. AMS, 2 edition, 2010.

[3] Z. Jomaa and C. Macaskill. The embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions. Journal of Computational Physics, 202:488–506, 01 2005.

[4] S. Ponnusamy and H. Silverman. Complex Variables with Applications. Birkh¨auser Boston, 2007.

[5] B. Riemann. Grundlagen f¨ur eine allgemeine Theorie der Functionen einer ver¨anderlichen com-plexen Gr¨osse. In H. Weber, editor, Gessamelte Mathematische Werke, pages 3–47. Teubner, Leipzig, 1876. Inaugural Dissertation, G¨ottingen, 1851.

[6] W. Rudin. Real and Complex Analysis, 3rd Ed. McGraw-Hill, Inc., USA, 1987.

[7] E.B. Saff and A.D. Snider. Fundamentals of Complex Analysis with Applications to Engineering and Science. Prentice Hall, 2003.

9

Appendix

Proof of the ML-Inequality (Lemma 2) from Subsection 6.1. Proof.

We start off by parametrizing C into z(t) on an interval [a, b]. This gives     Z C f (z) dz     =      Z b a f (z(t))z0(t) dt     

Using the triangle inequality gives us     Z C f (z) dz     ≤ Z b a |f (z(t))| |z0_{(t)| dt}

Now because we have |f (z)| ≤ M we have     Z C f (z) dz     ≤ M Z b a |z0_{(t)| dt = M L}

which concludes the proof.