For questions regarding harmonic functions.

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72 views

Differentiating this integral,

I want to show that $u_{xx} + u_{yy} = 0$ for the integral given below, so I think I want to differentiate under the integral with respect to both $x$ and $y$. The goal is to show that $u$ is ...
3
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1answer
35 views

How to construct such a harmonic function on the upper half plane of $\mathbb{C}$ satisfying the following condition?

(1)Let u be a bounded harmonic function on the upper half plane of $\mathbb{C}$. Show that $\forall y$ we have $u(x+iy)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y\cdot u(t)}{(t-x)^2+y^2}dt$ for $x,y\...
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1answer
51 views

Some basic questions regarding the Maximum Principle for Harmonic Functions,

I've seen a uniqueness argument come up a few times but I don't really understand it. The argument is that if two harmonic functions $f$ and $g$ agree on the boundary of some domain, then $f-g$ or $g-...
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2answers
38 views

Find all possible functions, $F(r)$, harmonic in $2$ and $3$ dimensions,

Sources: this is an old advanced calculus exam question, which I think is asking for harmonic functions. The problem statement is: Suppose $F(r)$ is a smooth function of $r$ for $r>0$ . Define ...
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1answer
48 views

Holomorphic versus harmonic functions

Is it true that any holomorphic function on domain $D$ is of the form $u_x-iu_y$ for some harmonic function $u$? Motivation for this question is the problem of existence of harmonic conjugates.
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1answer
84 views

Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
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1answer
28 views

Why does $v(z)=\text{Im}\left[\left(\frac{1+z}{1-z}\right)^2\right]$ not contradict maximum principle?

Since $\left(\frac{1+z}{1-z}\right)^2$ is holomorphic in $\mathbb{D}$, its imaginary part is harmonic, and we have $$\underset{r \uparrow 1}{\lim}v(re^{i\theta})=0 \quad \forall ~ \theta \in [0, 2\pi)$...
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2answers
32 views

What is an example of a nonconstant subharmonic function that attains a minimum?

Let $D$ be a domain in $\mathbb{C}$. What would be an example of a nonconstant subharmonic function that attains its minimum in the domain?
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0answers
33 views

2D Laplace Equation with Sine-squared BC

I am having a bit of trouble solving the 2D Laplace Equation $$\nabla^2u(y,z) = 0$$ with 2 BCs being $\left.\dfrac{\partial u}{\partial y}\right|_{y=0} = 0$ $u\left(y=\frac{h}{2},z\right) = c\sin^...
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3answers
84 views

Looking for pathologic counterexample: Nonzero harmonic function which is zero on the unit circle except 1

From my Complex Variables class: Let $C_1$ be the unit circle, $B_1$ the open unit disc and $\Gamma = C_1 \backslash \{1\} $. I am looking for a nonzero function $u \in C(B_1 \cup \Gamma)$ which is ...
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1answer
34 views

Prove $v$ is harmonic and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$

Prove that if $v(z) = \mathrm{Im}[(\frac{1+z}{1-z})^2]$, then $v$ is harmonic on the unit disc and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$ for all $\theta \in [0,2\pi)$. Explain why this does not ...
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1answer
100 views

Laplace equation in two dimensions and complex analysis

I was playing around with Laplace's equation: $$\frac{\partial^2 u}{\partial x ^2}+\frac{\partial^2 u}{\partial y ^2}=0$$ It occured to me that it can be written as: $$\bigg(\frac{\partial}{\partial ...
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1answer
106 views

Dirichlet problem on upper half disk

In our current Complex Variables homework assignment, we are given the following problem: Let $\Omega = \lbrace z \in \Bbb C : |z|<1, Im(z)>0\rbrace $ and $f \in C(\partial\Omega,\Bbb R)$ ...
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0answers
38 views

Real zeros of a function holomorphic in the upper half plane

Suppose that $f(z)$ is holomorphic in the upper half-plane and $\lim_{\, |z| \to \infty} f(z) = 1$ along any ray in the open upper half-plane. Suppose also that $f(z)$ has a continuous extension to ...
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2answers
41 views

Prove ln(n) diverges as quickly as the harmonic series.

Question: Find a (simple) $f(n)$ so that $\lim\limits_{n \rightarrow \infty} \frac{n \sum\limits_{k=1}^{n} \frac{1}{k}}{f(n)} = 1$ My Attempt: I know the answer, by using Mathematica, is $f(n) = n \...
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43 views

Energy method for harmonic functions

I have two questions about the informations bellow that can be found in the book elliptic partial differential equations -QING HAN and FANGHUA LIN - chapter 1-pg 19 If $a_{ij} \in C(B_{1}(0))$ ...
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1answer
20 views

Comparison principle for functions $u,v$ such that $\Delta u-f(u)\ge \Delta v-f(v)$

Let $\Omega \in {{R}^{n}}$ and $u,v\in {{C}^{2}}\left( \Omega \right)\cap C\left( {\bar{\Omega }} \right),\text{ }f\in {{C}^{1}}\left( R \right)$ such that ${f}'\left( t \right)\ge 0$, for all $t\in ...
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44 views

How is $\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$?

I'm looking at some material on harmonic functions and it says $$\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$$ where $n$ is the number of dimensions and $\omega_n$ ...
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65 views

Show that the “Hartogs Regularity Radius” $R(z)$ is subharmonic

Exercise I'm a little stuck on an Exercise in Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range. The Exercise (E.II.5.1) is as follows (here $\...
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1answer
76 views

The proof of Hopf lemma for harmonic functions

I would like to understand a passage from the proof of Hopf lemma. . In the second image above the author says: Therefore by theorem 1.29 (Maximum Principle for Subharmonic Functions) $h_{\...
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0answers
46 views

Harmonic non-negative function is constant

I'm having some trouble with the following: Let $u:\mathbb{R}^2\setminus\{0\}\rightarrow[0,\infty)$ be a harmonic function. Show that $u$ is constant. I have seen different proves for this. However, ...
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16 views

Steklov eigenvalue for the upper half plane

I have been thinking on this problem. Maybe someone know some previous work on this problem. The question is quite neat, consider the nonnegative solution $u\geq 0$ of the following equation $$\begin{...
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1answer
35 views

Given $u$ subharmonic, show that $u^p$ subharmonic for $p\geq 1$

Exercise I'm currently reading through Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range, and I've been tasked with the following (Exercise II.4.4): ...
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65 views

Poissons formula from unit disk to upper half plane

I am trying to derive the Poisson's formula for the upper half plane from the formula on the unit disk using a conformal map. A conformal map from the unit disk to the upper half plane is (I think) $z ...
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2answers
62 views

Laplace $2$-D Heat Conduction

Consider the following steady state problem $$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$ $$ T(0,y) = 300, \space \space T(4,...
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1answer
28 views

Proving harmonic function is zero

I'm having trouble with a homework assignment. This is the question: Suppose that $\Omega \subset \!R^3$ is a path connected bounded region and that $f : \Omega \rightarrow \!R$ satisfies $\Delta f(\...
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1answer
82 views

Sign of Laplacian Green's function in 3D

I am trying to prove that on a "nice" domain $\Omega$ in $\mathbb{R}^{3}$, the Green's function $G$ of $\bigtriangleup$ (the Laplacian) on $\Omega$ is always negative. I would like to use the Maximum ...
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1answer
43 views

Hardy-Littlewood theorem about the Poisson integral for $p=1$

(Hardy-Littlewood Theorem) : Let ‎$ u(r,‎\theta)‎$‎ be the Poisson integral of ‎$ ‎\varphi ‎\in L‎^{p}‎‎$‎, ‎$ 1<P ‎\leqslant‎ ‎\infty‎$‎ , and let $ U(‎\theta)=\sup‎_{r<1}‎|u(r,‎\theta)|‎ $‎. ...
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$G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$. My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?
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1answer
48 views

Alternate proof of Poisson Formula

I know how to prove this in general (I have a proof for Poisson formula I did), however, I have not been able to prove this way. Please help. Assume the theorem for $R=1$ deduce statement for any $R&...
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23 views

Neumann and Dirichlet Conditions for Schwarz-Christoffel Map

I'm looking to solve Laplace's equation on a polygon with Dirichlet and homogenous Neumann conditions using Schwarz-Christoffel (CS) mapping. I'm able to map the polygon to the upper-half plane using ...
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1answer
33 views

Proving $v$ is harmonic

Let $u$ be a harmonic function in $\mathbb{R}^3$ and let $a > 0$. Show that the function $v$ defined in spherical coordinates by $v(r,\theta,\psi )=\frac{a}{r}u(\frac{a^2}{r},\theta,\psi)$ is ...
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1answer
68 views

All second partial derivatives of harmonic function are $0$

I am given this question as a homework assignment. Assume that $f$ is from $\mathbb R^2$ to $\mathbb R$ and has a strict local maximum at $(x_0, y_0)$. prove that all second partial derivatives of ...
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2answers
203 views

Is there a conformal mapping that sends the upper semi-circle to the positive (or negative) real line,

and the real interval [-1,1] to the other half of the real line? I am considering the upper semi-circle ${|z|=1, 0<arg(z)<\pi}$, with the line [-1,1] that closes the loop. I want to map the ...
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1answer
34 views

Showing an equation satisfies laplace equation

I'm completely lost with the laplace equation I've searched different explanations of it on google and on this website and nothing is helping explain it. The question I was given is: Show that the ...
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1answer
112 views

Using Green's identity to show that a harmonic function with zero boundary values is identically zero

I am confused how to do this question. I need to use Green's first identity and if $\nabla(f)=0$ then $f$ is constant on $\Omega$ since $\Omega$ is path connected. I have subbed in the information ...
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0answers
45 views

How to find a family of analytic functions with a modulus of boundary value = 1, and no zeros inside the unit disk?

a) Find all $f(z)$ satisfying i) is analytic in $|z|\le 1$ ii) has no zeros in $|z|<1$ iii) $|f(e^{i\theta})|=1$ for $0<\theta<2\pi$ b) What are the possible $f(z)$ if (ii) is replaced by ...
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1answer
56 views

Finding a harmonic function on half-disk that is equal to $1$ on the semi-circle and $0$ on the diameter

I first showed that the mapping $$z + \frac{1}{z}$$ sends the upper semi-disk, $\{|z|<1, \operatorname{Im} z >0\}$, along with the real line from $-1$ to $1$, to the whole of the real line in ...
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0answers
19 views

Laplace's equation in cylindrical coordinates for a function that only depends on the angle

I need to solve Laplace's equation: $$\nabla^2\Phi = 0$$ with the boundary conditions: $$\Phi(\theta=0)=0$$ $$\Phi(\theta=\pi)=a_1$$ In cylindrical coordinates ($r,\theta,z$), for $\Phi =\Phi(\theta)$...
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1answer
15 views

Harmonic measure function and the surjectivity of the diagonal map

Let $\Omega$ be a finitely connected bounded domain in the complex plane bounded by $n+1$ analytic jordan curves. Letting $\partial \Omega$ denote the boundary of $\Omega$, we write $\partial \Omega = ...
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1answer
56 views

Solving Laplace's equation for semi-infinite bar using conformal mapping

I am trying to solve the 2d version of Laplace's equation ($\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}=0$) for the semi-infinite bar shown below in the shaded region. That ...
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2answers
37 views

How to prove this function is bounded?

I've got the following function: $v(r,\theta) = \frac{1}{2\pi} \int_{0}^{2\pi} \frac{r^{2} - 1}{r^{2} + 1 - 2 r \cos\left(\psi - \theta)\right)}f(\psi)d\psi$ The function $f$ is NOT specified, but ...
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0answers
45 views

valid kernel and invertibility

Perhaps it is an easy question and I should be able to do it. However I have no idea how to work with it. I am trying to show that following kernal is valid: K(x,y) = 1/(1-xy) where x and y are (-...
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0answers
36 views

Helmoltz equation on the torus

I am looking for the solution of the Helmoltz equation (or even Laplace, if not available) on the torus, that is, the manifold of line element \begin{equation} ds^2 = (c + a \sin(\theta))^2 d\varphi^...
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53 views

Convex function on $\mathbb{R}^n$ composed with $n$ subharmonic functions

Currently, I am working on the following question (which appears in Hörmander's Notions of Convexity, Exercise 3.2.6) Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is a convex function which is increasing ...
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2answers
65 views

Harmonic function and Neumann Compatibility Condition

______________________________________________________________ Conversely, in a different approach using Green's 1st identity, I showed that by choosing v ≡ 1, that the compatibility condition is ...
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2answers
52 views

Show that $u(x,y)$ is constant.

Let $u(x,y)$ be a harmonic function on domain s.t all the partial derivatives of $u(x,y)$ vanish at the same point of , then $u(x,y)$ is constant. Now the thing is if the harmonic conjugate of $u(x,...
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0answers
35 views

Prove Poisson kernel of the upper half-plane is the Cauchy distribution

Let $d=2$, and consider the domain $D=H$, the upper half-plane. Show that the Poisson kernel is the Cauchy distribution by following the steps below. (A) Let $W_t=(X_t,Y_t)$. Show that for any $\...
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1answer
54 views

Laplace equation problem: difficulty finding a particular solution given the boundary conditions.

Solve Laplace's equation: $$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$$ for the region $\{(x, y) : 0 < x < 1, 0 < y < 1\}$, given these boundary conditions: ...
0
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1answer
37 views

When finding a harmonic function, from a conformal mapping, is it ok to have imaginary numbers in the argument,

Say after finding a conformal mapping, using this conformal mapping, I find a harmonic function $$\arg \left[\frac{(z^2+1)}{(z^2-1)}\right]$$ that satisfies boundary conditions that I want. Is it ...