Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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2
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1answer
47 views

How to prove $(\frac1\pi)\int_{-\pi}^\pi \left(\frac {1-r^2}{1-2r\cos\theta+r^2}\right)^2d\theta = \frac {2(1+r^2)}{1-r^2}$?

So for this problem I need to use the fact that $\frac {1-r^2}{1-2r\cos\theta+r^2}$=$1+2\sum_{n=1}^{\infty} r^n\cos n\theta$. I replaced the term in the integral but i ended up getting $\sum_{n=1}^{\...
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0answers
18 views

$u$ harmonic and $\lim_{|z| \to \infty} u(z) = 0$ imply that $u \equiv 0$ in $\mathbb{C}$.

If $u$ is harmonic and bounded in $\mathbb{C}$, then I've shown that $u$ is constant. I guess it can be helpful to show that $u \equiv 0$ if $u$ is harmonic and $\lim_{|z| \to \infty} u(z) = 0$, but ...
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0answers
11 views

implication, being $u$ harmonic

Let $D$ be a domain in $\mathbb{C}$ and let $u : D \to \mathbb{R}$ be an harmonic function. If $a \in D$ and $r \in (0 , \infty)$ are such that $\overline{D}(a , r_a) \subset D$ then I have to show ...
7
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0answers
74 views

Sufficient conditions for being a harmonic function

Let $f(x,y)$ be a real-valued function defined on an open subset $U \subset \mathbb R^2$. Suppose that $f$ is twice differentiable separately in each variable and satisfies Laplace's equation $f_{xx} +...
1
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0answers
19 views

Physical Intuition for Mean Value Property of Heat Equation.

I am trying to think of an intuitive explanation for the mean value property for the heat equation: $$u(x,t) = \frac{1}{4r^2} \int_{E(x,t,r)} u(y,s) \frac{|y|^2}{|s|^2} dy ds$$ where $E(x,t,r)$ is the ...
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1answer
38 views

show that $u$ is harmonic with a condition

Let $D$ be a domain in $\mathbb{C}$ and let $u : D \to \mathbb{R}$ be a continuous function. I suppose that for each $a \in D$ there exists $r_a > 0$ with $\overline{D}(a , r_a) \subset D$ and such ...
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0answers
18 views

show that $u(r e^{i \theta}) = \theta \log r$ is harmonic in $\mathbb{C} \setminus (- \infty , 0]$.

My attempt to show that $u(r e^{i \theta}) = \theta \log r$ is harmonic is the next: if $z = r e^{i \theta}$ we have $\log z = \log r + i \theta$, then ${(\log z)}^2 = {(\log r)}^2 + i 2 \theta \log r ...
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0answers
57 views

Determine all positive harmonic functions

If u is a positive harmonic function on $\mathbb{R^n}\setminus\{0\}$, show that there exists a,b which is non-negative, such that $$u(x)=a+b|x|^{2-n}\text{ for all }x\in\mathbb{R^n}\setminus\{0\}$$ ...
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0answers
62 views

Taking limits in a variational PDE with respect to a parameter

Let $\Omega \subset \mathbb{R}^n$ be a smooth bounded domain and $B \subset \Omega$ be an open ball, for a given $\epsilon>0$ consider $$a_\epsilon(x) = \begin{cases} 1, \qquad x \in \Omega \...
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0answers
30 views

Typo in this proof about harmonic functions?

In the proof of Lemma 1.1 of this article, we read The zeros of $f''(x+iy) = 0$ are critical points of $f(x+iy)$. This claim is essential to the proof yet appears to be wrong. For example, $f(z) = ...
2
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0answers
31 views

Expansion of a function in Spherical Harmonics

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a spherical harmonic expansion (https://en.wikipedia.org/wiki/Spherical_harmonics) of the function $f^n$.
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16 views

Prove the convolution of a summability kernel and an L^1 function converges uniformly to the function

For a function $f\in L^1(\mathbb{R})$ that is not necessarily continuous, I have proved that: $\lim_{\lambda \to \infty}||K_\lambda \ast f -f||_p=0,$ where $K_\lambda$ is a continuous summability ...
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0answers
19 views

Given $u$ harmonic find all matrices $Q$ s.t. $u \circ Q$ harmonic

More precisely, let $u \in C^{2}(\mathbb{R}^n,\mathbb{R})$ be harmonic, $Q \in \mathbb{R}^{n \times n}$ matrix. Classify $Q$ s.t. $u \circ Q$ harmonic. I had a few ideas to find positive or counter ...
1
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0answers
28 views

Lifting Minimal Surfaces

Let $f(z)=z+\frac{\overline{z}}{4}$ where $\left|z\right|<1$. Does the function lift to minimal surface? $h(z)=z$ , $g(z)=\frac{z}{4}$ since $f(z)=h(z)+\overline{g(z)}$ so $h\text{'}(z)=1$ and $g\...
1
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2answers
86 views

Double layer potential in 1d?

I would like to illustrate the double layer potential idea with a simple 1d example, but seem to run into a situation where the resulting integral equation is singular. The problem is $u''(x) = 0$ on $...
1
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1answer
48 views

Classify all harmonic $f$ s.t. $f=gh$

More specifically, $f$ harmonic s.t.: $f: (0,1)^{2} \rightarrow \mathbb{R}$ with $f(x,y)=g(x)h(y)$ and $g,h \in C^{2}((0,1))$ Bear in mind that I am at calc III level, having just started an ...
5
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1answer
77 views

Show that $H(x):=\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2})$ is harmonic if u is harmonic

(This is the $n$-dimensional analogue of the 3D case: Show that $H(x) := |x|^{-1} u(x/|x|^2) $ is harmonic if $u$ is harmonic) Suppose that $u$ is a harmonic function on $\mathbb{R}^n$. Prove that ...
1
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1answer
68 views

Maximum principle for $\varphi\in C^0(\overline{B},\mathbb{R}^2)\cap C^2(B,\mathbb{R}^2)$

Let $B:= \{w\in \mathbb{R}^2 : \lvert w\rvert < 1\}$ be the unit-circle. Suppose that $\varphi\in C^0(\overline{B},\mathbb{R}^2)\cap C^2(B,\mathbb{R}^2)$ is harmonic in $B$ (both component ...
0
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1answer
67 views

$u$ is harmonic, prove that $v(x)=\frac{1}{|x|^{n-2}}\cdot u\left(\frac{x}{|x|^2}\right)$ is harmonic. [closed]

$\Omega$ is open set in $\mathbb{R}^3, n\geq 3$. $u:\Omega\to\mathbb{R}$ is a harmonic function. Prove that $v(x)=\frac{1}{|x|^{n-2}}\cdot u\left(\frac{x}{|x|^2}\right)$ is harmonic. I knew that $\...
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0answers
22 views

Does the gradient of a harmonic function have any notable properties?

Of course $\mbox{div}(\nabla u) \equiv 0$ for harmonic $u$, but can anything else be said? In particular, I want to investigate the behaviour of directional derivatives of $u$.
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22 views

Confirming that an equation is harmonic

I have some problems in confirming that the following function is harmonic and thus satisfies the Laplace equation: $g(x,y)=ln\frac{1}{r}$, where $r=\sqrt{(x-x_0)^2+(y-y_0)^2}$, where the Laplace ...
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2answers
40 views

how can it be proved that $\text{Log}|z-z_0|$ is harmonic in $\mathbb{C}\setminus\{z_0\}$?

I know it can be proved in almost all of $\mathbb{C}\setminus\{z_0\}$, (exept the straight line from $z_0$ parallel to $[-\infty,0]$), as the real part of the holomorphic function $\text{Log}(z-z_0)$.
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0answers
27 views

How to estimate this trigonometric sum?

I am reading Gregory F. Lawler's Random Walk and the Heat Equation. In page 39-40 the author considers a set the following problem: Let $N$ be a positive integer, $N\geq 2$. Let $A_N=\{(x_1,x_2): x_i=...
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0answers
10 views

How can one interpret the steps of the procedure of converting the Laplace equation to polar coordinates?

As there are too many symbols to write, I take the liberty of uploading a screenshot of what I am wondering about. My question is , which differentiation operations are made to get the those terms ...
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1answer
33 views

Why does $\partial x/\partial r=x/r$ for $x^2+y^2+z^2=r^2$? (Mean Value Property for Harmonic Functions over the Sphere)

In Partial Differential Equations by Walter Strauss, Ch 7.1 Page 180, the author seeks to prove the mean value property that the average value of any harmonic function over any sphere equals its value ...
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0answers
19 views

Showing that trigonometric polynomials are dense in $C(\partial \mathbb{D} )$

I want to show that if $f \in C (\partial \mathbb{D})$ and $\epsilon >0$ then there exists a trigonometric polynomial $P$ such that: $$ |P(e^{it})-f(e^{it})| < \epsilon $$ for all $t \in \mathbb{...
1
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1answer
99 views

How do derivatives w/r to polar variables behave at the origin?

While argument is canonically undefined at the origin, since $u(r,\theta) = u(0)$ we could define $u(0, \theta):= u(0)$ [i.e., a constant function with respect to $\theta$]. So is $\frac{\partial^2 u}{...
0
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0answers
24 views

Showing that a particular harmonic function is constant [duplicate]

Let $u$ be a real-valued harmonic function in the complex plane such that $$ u(z) \leq b + a \log{|z|} $$ whenever $|z| \geq 1$(for some constants $a > 0$ and $b$). Show that $u$ is constant. I ...
0
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0answers
17 views

Finding a general harmonic function [duplicate]

Find the most general harmonic function of the form $f(|z|), z \in \mathbb{C} \smallsetminus {0} $. I know that harmonic functions satisfy: $$ \frac {\partial^{2} f} {\partial x^{2}} + \frac {\...
0
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1answer
29 views

average value of harmonic function around a circle

I am self-studying this class notes on MIT OCW https://ocw.mit.edu/courses/mathematics/18-112-functions-of-a-complex-variable-fall-2008/lecture-notes/lecture16.pdf . This question is taken from the ...
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0answers
27 views

Periodic function in polar coordinates

Hy friends! I am studying the Dirichlet problem for Laplace equation on disk $$ \begin{equation} \left\{ \begin{array}{rcll} v_{rr}+\frac{1}{r}v_r+\frac{1}{r^2}v_{\theta \theta}&= &0 & \...
1
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1answer
33 views

Poisson Integral for half plane approaches argument

I'm looking at an exercise in Ahlfors' Complex Analysis. It states the following: Assume $U(\zeta)$ is piecewise continuous and bounded for all real $\zeta$, and assume $U$ has a jump at $0$ (i.e. $U(...
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0answers
16 views

non-constant harmonic function on simply connected domain does not achieve maximum in the interior of the domain

Let $u$ be a harmonic function on a simply connected domain $\Omega$. Prove that $u$ does not achieve maximum in $\Omega$. What I have shown is that for any $u$ harmonic function on $\Omega$, $u(z)=\...
4
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3answers
162 views

Is it true that a smooth, harmonic functions with compactly supported gradient is trivial?

Let $u \in C^{\infty}(\mathbb{R}^n)$ be such that $\nabla u \in C^{\infty}_c(\mathbb{R}^3; \mathbb{R}^3)$ and $\Delta u=0$. I wish to prove or disprove that $u = c$ for some $c \in \mathbb{R}^n$; note ...
0
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0answers
18 views

Laplace equation in an annular fin of rectangular cross section in cylindrical coordinates

The goal is to find the solution of Laplace equation on a rectangular cross section of a cylindrical body with the following limits: $$R_i<r<R_o$$ $$Z_1<z<Z_2$$ The Laplace equation on ...
0
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0answers
21 views

Does there exist a 2D harmonic function which never goes to ∞ at any point, goes to 0 as r goes to ∞, and isn't f=0? [duplicate]

I need a 2D harmonic function which is always real, never diverges to $\infty$ anywhere, goes to 0 as one goes infinitely far away from the origin, and isn't f=0. $$\begin{matrix}0=\nabla^2f&0=\...
0
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0answers
32 views

Electrostatic Potential from Array of Equally Charged Strips

This is question 5.1 from Mathews and Walker , Mathematical Methods of Physics: it is very difficult for me to find which conformal transformation would simplify the spaced arrangement as in the ...
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0answers
38 views

Show that a continuous weakly harmonic function is harmonic.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq2$. An integrable function $u$ is said to be weakly harmonic in $\Omega$ if for all $\varphi\in C_{c}^{2}\left( \Omega\right) $, $$ \int_{\...
0
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0answers
40 views

Show that $ \lim_{x\rightarrow x_{0}}u\left( x\right) $ exists and $u$ is harmonic in $\Omega$ [duplicate]

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq 3$ and $x_{0}\in\Omega$. Suppose $u$ is harmonic in $\Omega\backslash\left\{ x_{0}\right\} $ and $$ \lim_{x\rightarrow x_{0}}\left\vert x-...
0
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0answers
20 views

The gradient of poisson's equation is unbounded at zero on the upper half space

Assume $g\in C\left( \mathbb{R}^{n-1}\right) \cap L^{\infty}\left( \mathbb{R}^{n-1}\right) $ and $g\left( x\right) =\left\vert x\right\vert $ for $\left\vert x\right\vert \leq1$. Let $ u\left( x\...
3
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0answers
102 views

Harmonic function such that $u =\partial_\nu u = 0$ on a surface must vanish.

I'm studying problems from some qualifiers and I came across a problem I couldn't solve: Let $\Omega\subseteq \mathbb{R}^3$ be a domain and suppose that $\Sigma\subseteq \Omega$ is a smooth (...
1
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1answer
45 views

Holomorphicity of $\sin(\theta)\cos(\theta)+i\cos(\theta)^2$ imply that $\theta$ is constant

Suppose $\theta(x,y)$ be a smooth function on a connected domain, taking values in $(0,2\pi]$, and define $ c=\cos(\theta),s=\sin(\theta). $ Suppose that the function $f(x,y)=sc+ic^2$ is holomorphic. ...
3
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1answer
133 views

Fourier Transform of a harmonic polynomial

Let $x \in S^n$ and $P_d$ be a homogenous harmonic polynomial of degree $d$ in $n+1$ variables. I would like to find Fourier transform of the following function: $$ \frac{1}{|x|^{n+d}}P_d(x). $$
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2answers
42 views

Prove: $\sum\limits_{j=1}^{2^n}\frac{1}{j} \geq 1 + \frac{n}{2}$ for all positive integers $n$.

Given the following proof. Proof: by induction on n Base Case: $n = 1$ is trivial. Done Inductive Step: suppose for $n \geq 1$ we have $\sum\limits_{j=1}^{2^n}\frac{1}{j} \geq 1 + \frac{n}{2}$. We ...
0
votes
1answer
23 views

Can I use the mean value property for a function that is harmonic on only the interior of a ball?

Let $f$ be a continuous function on the closure of $U$ where $U=\{ (x,y) \in \mathbb{R}^2 : x^2+y^2<1 \}$ and harmonic on $U$, If $$f(x,y)=x^2y^2 $$ on $\partial U$ I want to find $f(0,0)$. Now I ...
1
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1answer
25 views

Laplace's equation on the plane: how much boundary data must be specified to guarantee existence and uniqueness?

My question stemmed from a specific problem, so let's jump right in. I want to solve Laplace's equation in plane polars $(r, \theta)$ on the domain $(r,\theta) \in [1,\infty)\times[0,2\pi)$ subject to ...
3
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1answer
38 views

Two harmonic functions with positive ratio are multiple of each-other?

Suppose $f,g,h:\mathbb{R}^n \to \mathbb{R}$ are functions so that $f$ and $g$ are harmonic and not identically zero, $f=g\cdot h$ and $h\geq 0$. Is $h$ a constant function? EDIT: Someone voted to ...
1
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1answer
31 views

harmonic function in a 2d disc.

So my professor gave as the following question: Given the function u which is continuous in $\bar{D}$ , where $D = \left \{ (x,y) \in \mathbb{R}^{2} : x^{2}+y^{2}<1 \right \}$, and harmonic in $D$. ...
2
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1answer
42 views

Regularity conditions for curves in Riemann surfaces

So, I am studying complex analyisis in one variable from various books (Rudin, Forster and Gamelin) and each one of them uses a particular regularity condition to prove their theorems, and I am not ...
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0answers
18 views

pointwise limit of harmonic functions is harmonic on an open dense set

I'm trying to solve exercise 2.9 of Axler's book "Harmonic function theory". The exercise says: Show that if $u$ is a pointwise limit of a sequence of harmonic functions on $\Omega$, then $...

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