Vanishing of harmonic functions in planar domains
(2024) In Bachelor's Theses in Mathematical Sciences MATK11 20241Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- We investigate the vanishing conditions for harmonic functions in planar domains. We first prove that if a complex-valued harmonic function $u$ in an open set $\Omega \subseteq \mathbb{C}$ vanishes at a point $a \in \Omega \cap \mathbb{R}$ and satisfies $(\partial+\bar{\partial})^k u(a)=0$ for all $k \geq 0$, then $u$ vanishes identically on the connected component of $\Omega \cap \mathbb{R}$ containing a, where $
\partial=\frac{1}{2}(\frac{\partial}{\partial x}
+\frac{1}{i}\frac{\partial}{\partial y})\quad \text{and}\quad
\bar \partial=\frac{1}{2}(\frac{\partial}{\partial x}
-\frac{1}{i}\frac{\partial}{\partial y}).
$ Additionally, we show that if $u$ vanishes on a circular arc within its domain, then $u \equiv 0$ in $\Omega$.... (More) - We investigate the vanishing conditions for harmonic functions in planar domains. We first prove that if a complex-valued harmonic function $u$ in an open set $\Omega \subseteq \mathbb{C}$ vanishes at a point $a \in \Omega \cap \mathbb{R}$ and satisfies $(\partial+\bar{\partial})^k u(a)=0$ for all $k \geq 0$, then $u$ vanishes identically on the connected component of $\Omega \cap \mathbb{R}$ containing a, where $
\partial=\frac{1}{2}(\frac{\partial}{\partial x}
+\frac{1}{i}\frac{\partial}{\partial y})\quad \text{and}\quad
\bar \partial=\frac{1}{2}(\frac{\partial}{\partial x}
-\frac{1}{i}\frac{\partial}{\partial y}).
$ Additionally, we show that if $u$ vanishes on a circular arc within its domain, then $u \equiv 0$ in $\Omega$. These results extend classical uniqueness theorems for harmonic functions and are validated through analysis in both the open unit disc $D$ and general planar domains. (Less) - Popular Abstract
- Harmonic functions are a special type of functions that appear in many areas of science and engineering, from describing how heat spreads in a material to how fluids flow. These functions are smooth and continuous, meaning that they have no sudden jumps or breaks. They obey a specific equation known as Laplace’s equation. In this paper, we investigate what happens when these harmonic functions become zero, in some certain areas. One of our main findings is that if a harmonic function becomes zero at a single point on the line of real numbers and the rate of change, namely derivatives, are zero at that point, then the function must be zero along the entire segment of the real line that lies within its domain. This is like saying if you mute... (More)
- Harmonic functions are a special type of functions that appear in many areas of science and engineering, from describing how heat spreads in a material to how fluids flow. These functions are smooth and continuous, meaning that they have no sudden jumps or breaks. They obey a specific equation known as Laplace’s equation. In this paper, we investigate what happens when these harmonic functions become zero, in some certain areas. One of our main findings is that if a harmonic function becomes zero at a single point on the line of real numbers and the rate of change, namely derivatives, are zero at that point, then the function must be zero along the entire segment of the real line that lies within its domain. This is like saying if you mute a special note and its harmonics, the whole musical line goes silent. We also discovered that if a harmonic function vanishes along a segment of a circle, then it must be zero everywhere within its domain. Imagine if touching one part of a circular string instrument, then it could silence the entire instrument. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9165405
- author
- Wang, Ruolin LU
- supervisor
- organization
- course
- MATK11 20241
- year
- 2024
- type
- M2 - Bachelor Degree
- subject
- keywords
- Harmonic functions, uniqueness theorem, power series expansion
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUNFMA-4165-2024
- ISSN
- 1654-6229
- other publication id
- 2024:K8
- language
- English
- id
- 9165405
- date added to LUP
- 2025-10-02 16:33:35
- date last changed
- 2025-10-02 16:33:35
@misc{9165405,
abstract = {{We investigate the vanishing conditions for harmonic functions in planar domains. We first prove that if a complex-valued harmonic function $u$ in an open set $\Omega \subseteq \mathbb{C}$ vanishes at a point $a \in \Omega \cap \mathbb{R}$ and satisfies $(\partial+\bar{\partial})^k u(a)=0$ for all $k \geq 0$, then $u$ vanishes identically on the connected component of $\Omega \cap \mathbb{R}$ containing a, where $
\partial=\frac{1}{2}(\frac{\partial}{\partial x}
+\frac{1}{i}\frac{\partial}{\partial y})\quad \text{and}\quad
\bar \partial=\frac{1}{2}(\frac{\partial}{\partial x}
-\frac{1}{i}\frac{\partial}{\partial y}).
$ Additionally, we show that if $u$ vanishes on a circular arc within its domain, then $u \equiv 0$ in $\Omega$. These results extend classical uniqueness theorems for harmonic functions and are validated through analysis in both the open unit disc $D$ and general planar domains.}},
author = {{Wang, Ruolin}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor's Theses in Mathematical Sciences}},
title = {{Vanishing of harmonic functions in planar domains}},
year = {{2024}},
}