For questions regarding harmonic functions.

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1answer
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Divergence structure equation

Consider Laplace's equation with potential function $c$: $$-\Delta u + cu = 0, \tag{$*$}$$ and the divergence structure equation $$-\operatorname{div}(aDv)=0, \tag{$**$}$$ where the function $a$ is ...
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40 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
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0answers
12 views

Why does from Perron-Frobenius follow that that constant functions are the only harmonic functions here?

in our reading we had the following example for a Markov chain. State Space $E=\left\{1,2,3,4\right\}$ and Transition Matrix $$ P=\frac{1}{3}\begin{pmatrix}0 & 1 & 1 & 1\\1 ...
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Calculate all harmonic functions

Let $E$ befinite and suppose that $P$ is irreducible and strictly sub-stochastic. Calculate all harmonic functions. To my understanding, $P$ strictly sub-stochastic means that $\sum_{y\in ...
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1answer
20 views

Extending a harmonic function satisfying a growth condition at an isolated singularity

Consider a harmonic function u on the punctured disc $\Delta(\rho)^*:= \{ z\in \mathbb C: 0 <|z|<\rho\}$ with $\lim\limits_{z \rightarrow 0} z*u(z) = 0$. Prove that $u$ can be written in the ...
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Equivalence for being subharmonic

I'm working with this definition of subharmonic function: Let $\Omega \subseteq \mathbb{R}^n$ be an open set. A function $u\in C(\Omega)$ is said to be subharmonic on $\Omega$ if for every ...
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1answer
30 views

Composition of harmonic and holomorphic function

Simmiliar to this question my problem is as following: If $u$ is harmonic, and $f$ is holomorphic function, are $u \circ f$ and $f \circ u$ harmonic? I tried to do it like this: $$\Delta (u \circ f)= ...
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1answer
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Green's function for Dirichlet problem on a half disk

Let $D=\{z=(x,y):x^2+y^2<R^2, y>0\}$ be the half disk with radius R. Then if we consider the Dirichlet problem on this domain, i.e., we want to find $$ \Delta u=0, ~~z\in D,\\ ...
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52 views

Poisson problem, green's function

I'm stuck at a Poisson integral problem and need some guidance. Assume that the Poisson integral is known as $$\int\int_S P(r,r')u_0(r')dr'$$ and gives the solution to the boundary value problems ...
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0answers
35 views

Harmonic maps and angle preservation

I have the following question: What are the angle preservation properties of harmonic maps? Conformal maps preserve angles exactly, but they distort lengths. In this sense a conformal map is the ...
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58 views

Solving Laplace equation in polar coordinates

I have some assignments to do and I don't even know where to start. The notes in the course aren't too good, so I didn't understand too much from them. Given $$ \Omega = \{(x, y) \in \mathbb{R}^2 , ...
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Proving $u(x) = \frac{1}{\omega_n r^{n-1}}\int_{\partial B(x,r)} u\, d\sigma = \frac{n}{\omega_n r^n}\int_{B (x,r)} u\, dV$ for harmonic $u$

I'm having a bit of a problem proving the equality: $$u(x) = \frac{1}{\omega_n r^{n-1}}\int_{\partial B(x,r)} u\, d\sigma = \frac{n}{\omega_n r^n}\int_{B (x,r)} u\, dV$$ Which is the mean value ...
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1answer
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Holomorphic function locally represented as $(\partial_{x} - i \partial_{y}) h(x, y)$, with $h$ a scalar harmonic function

While reading an article, I came across the following statement. Moreover, locally every holomorphic function $f(x + iy)$ may be written as $(\partial_{x} - i \partial_{y})h(x, y)$, for some ...
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2answers
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Limit of bounded harmonic functions is harmonic

I am trying to solve this old qual problem: Suppose $\{u_n\}_{n=1}^\infty$ is a sequence of functions harmonic on an open set $U \subset \mathbb{C}$ and uniformly bounded by 1. Suppose there is a ...
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2answers
61 views

Under which conditions is the harmonic function unique that has piecewise constant values on the boundary

$\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$. $J_1 = \{e^{i\theta}: \theta \in (0, \pi/2)\}, J_2 = \{e^{i\theta}: \theta \in (\pi/2, \pi)\}, J_3 = \{e^{i\theta}: \theta \in (\pi, 2\pi)\}$ It's ...
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1answer
41 views

Convexity of sub-harmonic functions in a sector

Let $F(z)$ be an analytic function in an open sector $\Sigma_{\gamma}=\{0<\arg z<\gamma<\frac{\pi}{2}\}$, and continuous to the boundary. Then $\log{|F(re^{i\theta})|}$, $z=re^{i\theta}\in ...
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1answer
24 views

If $u$ is harmonic, prove that $|Du|^2$ is subharmonic.

We say $v \in C^2(\bar{U})$ is subharmonic if $-\Delta v \le 0$ in $U \subset \mathbb{R^n}$. Prove that $v := |Du|^2$ is subharmonic, whenever $u$ is harmonic. This is Exercise 5, part d, ...
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1answer
24 views

Mean-value formulas

From PDE Evans, 2nd edition, pages 25-26. THEOREM 2 (Mean Value Formulas for Laplace's equation). If $u \in C^2(U)$ is harmonic, then $$u(x)=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} ...
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1answer
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Derivation of energy integral - harmonic functions

I am following the solution of the following problem on the topic of the energy integral of a surface. For a real-valued continuously differentiable function $u(x,y)$ on a closed domain $D$, the ...
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1answer
30 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
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Harmonic functions and null recurrence

Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions ...
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1answer
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Calculate $\int_{B_R(0)}u$ for a very weak solution $u$ of a Dirichlet boundary problem

Let $B_R:=B_R(0)\subseteq\mathbb{R}^n$ be the open ball with radius $R>0$ around $0$, $f\in\mathcal{L}^\infty(B_R)$ be radial and $u$ be a very weak solution² of $$\left\{\begin{matrix}-\Delta ...
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1answer
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Meanvalue formula of harmonic functions - translation invariance of surface measure

In almost every proof of the mean value formula for harmonic functions (e.g. Evans 2.Theorem 2) one finds the following calculation $$ \frac{1}{|{\partial B(0,1)}|}\int_{\partial B(0,1)} D ...
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1answer
24 views

Show that a harmonic function on an open connected set which is holomorphic on some open subset is in fact holomorphic everywhere.

Suppose $f$ is a harmonic function on a connected open set $\Omega$ in the complex plane, and suppose also that $f$ is holomorphic on some open subset $U$ of $\Omega$. Prove that $f$ is holomorphic on ...
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1answer
30 views

Show that if $g$ is nonconstant holomorphic and $f$ is harmonic such that $fg$ is harmonic, then $f$ is holomorphic.

Let $\Omega$ be an open and connected set in the complex plane and $g$ be a nonconstant holomorphic function on $\Omega$. Show that if $f$ is harmonic on $\Omega$ such that $fg$ is also harmonic on ...
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1answer
54 views

3-dim Brownian motion, harmonic function and its expectation

Given $f(x)=\frac{1}{|x+z|}$, a function from $\mathbb{R}^3\backslash \{z\}$ to $\mathbb{R}$, $z \in \mathbb{R}^3\backslash \{0\}$ and $B$ a 3-dim Brownian motion. I had succes showing that this ...
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1answer
56 views

Using conformal mapping to solve a boundary value problem,

Use conformal mapping to solve the following boundary value problem for $u=u(x,y)$ in the planar region $R=\{(x,y) \in \mathbb{R}^2: x^2 + y^2 > 1 \text{ and } y>0\}$: u solves ...
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0answers
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Harmonic function on a wedge

Find a harmonic function $\phi(r,\theta)$ in the wedge with three sides $\theta=0$, $\theta=\beta$ and $r=a$ and boundary conditions $\phi(r,0)=0=\phi(r,\beta)$, $0<r<a$ and ...
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1answer
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Harmonic / Analytic functions

Show that if φ(x, y) is harmonic in a domain D, then f(z) = φ$\\_{x}$(x, y) − iφ$\\_{y}$(x, y) is analytic in D. I figure that φ$\\_{xx}$ = - φ$\\_{yy}$ from "φ(x, y) is harmonic" so then that means ...
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0answers
84 views

Prove or disprove: If $h$ is harmonic on $E$, then $h$ is constant on each $C_i$

For a general finite Markov chain $(X_n)_{n\in\mathbb{N}_0}$ with state space $E$ and transition matrix $P=(p_{x,y})_{x,y\in E}$, not necessarily irreducible, we define the linear space of ...
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0answers
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Removable singularity of a harmonic function

Assume that $h$ is harmonic in the punctured unit disk $\mathbb D\backslash\{0\}$ such that $$ \lim_{r\to0}h(re^{it})=0 $$ for all $t\in\mathbb R$. Can $h$ be extended to a function harmonic in ...
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0answers
41 views

How can I mathematically proof an incoherent superposition of waves?

Let $\psi = A(t)\cos(\theta_1(t))$ and $\phi = B(t)\cos(\theta_2(t))$ two independent waves which phases and amplitudes depend on the time. Then it follows that the intensity of the superposition of ...
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Dirichlet with $C^1$ boundary data

Do you know an explicit example for an harmonic function $u$ in a regular domain $\Omega$ which satisfies the following two conditions? $u$ belongs to $C^0(\overline{\Omega})$ but not to ...
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1answer
23 views

An identity for the integral $ \int_{\partial B(0,1)}u(x_0+aw)u(x_0+cw)$ with a harmonic function $u$

This is Question 2.18 from Gilbarg and Trudinger, chapter 2. We are given that $\Omega$ is open bounded smooth boundary. Now fix $x_0\in \Omega$ and a constant $c>0$ such that ...
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1answer
41 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
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1answer
36 views

Dirichlet problem on rectangle with nonhomogeneous boundary conditions

I want to solve the following problem in Dirichlet Problem: Let $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ be a rectangle with boundary $\partial D$. I want to solve ...
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Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y ...
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1answer
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Positivity of Poisson kernel

Let $K(x,y)$ be the Poisson kernel for the Dirichlet problem of Laplace equation on a bounded domain $D$ with smooth boundary, i.e. for a harmonic function $u$ on $\bar D$ with $u|_{\partial D}=g$, we ...
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1answer
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Analytic function on the whole plane, positive imaginary part, what can it be?

Part (a): The function f is analytic in the whole plane with positive imaginary part. What can it be? Part (b): What if all you know is that the imaginary part of f tends to 0 at infinity? what we ...
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Harmonic function with boundary value 0 except one point

Let $\Delta=\{z\in\mathbb{C}: |z|<1\}$ Assume $u\in C(\overline{\Delta}\setminus \{1\})$ such that it is harmonic in $\Delta$ and $u(\xi)=0$ for $\xi\in S^1\setminus \{1\}$. (a) Find an example ...
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1answer
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You can't solve Laplace's equation with boundary conditions on isolated points. But why?

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of "holes" $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I'm pretty sure that in more than one dimension, ...
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Prove that harmonic numbers satisfy the equality. [duplicate]

The $k$th harmonic number is defined to be $H_k = 1 + 1/2+ 1/3 + · · · + 1/k .$ Prove that harmonic numbers satisfy the equality $H_1 + H_2 + · · · + H_n = (n + 1)H_n − n$ for all $n \in\Bbb N.$
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Pattern emerging from expansion of sphere in spherical harmonics

Seeing as one can express any function on the sphere in terms of the spherical harmonics, I am interested in what the sphere itself looks like when expanded. To get a function for the sphere, I use ...
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2answers
92 views

Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
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1answer
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Harmonic function with Neumann boundary condition in the disk

I have a problem where I denote by $D = B(0, 2)$ the disk in the plane with radius 2 centered at the origin. I have to find a harmonic function $u(r, \theta)$ in $D$ which satisfies the additional ...
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1answer
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Isolated singularity of harmonic function

I am working with a book by Axler, Bourdon and Ramey and find the following problem: Suppose $u$ is a harmonic function on $B \setminus \{0 \}$ such that $$ |x|^{n-2} u(x) \to 0, \qquad x \to 0 $$ ...
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1answer
26 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
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1answer
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Harmonicity of the expectation of a stopped Brownian Motion

Let $\mathbb{E}_x$ be the expectation associated with a probability measure such that $B_{t\geq0}$ is a Brownian motion started in x. I want to show that for $D\subset\mathbb{R}^2$ bounded, $y\in D, ...
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37 views

Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
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2answers
52 views

There is no holomorphic function in $\Omega=\{0<r<\lvert z\rvert <R\}$ with real part $u(x,y)=\frac{1}{2}\log(x^2+y^2)$

Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$ My attempt: I understand that $u$ ...