For questions regarding harmonic functions.

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1answer
311 views

Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
0
votes
1answer
82 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
2
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0answers
59 views

Tight bounds for harmonic measure

I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ...
1
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1answer
79 views

a special extension of a two variable function

We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in ...
0
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0answers
2 views

Optimizing value of discrete harmonic function at a given point

Let $n>0$, and let $C_n$ denote the discrete square $C_n=[|-n,n|]^2$ (so $C_n$ has $(2n+1)^2$ elements). Let $f$ be a nonnegative function on $C_n$, such that $f$ is harmonic on $C_{n-1}$ (i.e. ...
1
vote
1answer
23 views

$g: (U \times U - D) \to \mathbb{R}$ is continuous, $D$ diagonal? [on hold]

Do we have necessarily have that$$g: (U \times U - D) \to \mathbb{R},$$is continuous, where $D$ is the diagonal? Idea. Perhaps we want to apply the maximum-minimum principle to $G(z, z_0)$?
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0answers
41 views

Equivalent definition of harmonic functions

Let $\Omega \subset \mathbb{R}^n$ be a domain. And let $u \in \mathcal{C}(\Omega)$. Prove that: The function u is harmonic in $\Omega$, i.e. $u \in \mathcal{C}^2(\Omega)$ and $\Delta u = 0$ on ...
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1answer
18 views

Mean Value Property - Nonnegative Harmonic function

I want to prove that the mean value property $$u(\textbf{x}_0) = \frac{1}{\pi r^2} \int \int _{\left \{ \left | x_0-x \right < r| \right \}} u(\textbf{x})d\textbf{x}$$ for non-negative harmonic ...
0
votes
2answers
17 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
1
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1answer
23 views

Uniform Convergence: Poisson Kernel

If we fix $θ_∗ > 0$, then $P(r, θ) → 0$ uniformly on the set $ \left\lbrace θ : |θ| ≥ θ_∗ \right\rbrace $ as $r → a^-$ $$P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2}$$ $0\leq r ...
0
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1answer
28 views

Plotting Spherical Coordinates

I'm trying to plot the Poisson Kernel, where a = 1, so the resulting equation would be $$P(r,\theta) = \frac{1-r^2}{1-2r\cos(\theta)+r^2}$$ $0\leq r <a = 1 ,\textrm{ } -\pi \leq \theta \leq \pi$ ...
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0answers
36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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0answers
15 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
1
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1answer
19 views

Bounds on derivatives of harmonic functions on unit ball

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^n$. Show that $$\sup_{B_{1/2}} \lvert \nabla u \rvert \leq C(n) \sup_{\partial B_1} \lvert u \rvert$$ More generally, show that ...
0
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0answers
42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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0answers
12 views

Mean value property for harmonic functions (geometric idea)

Someone knows a bibliographic reference geometrically explore the idea of ​​the mean property for harmonic functions in domains of $\mathbb{R}^{n}$ ?
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2answers
36 views

Newton potential for Neumann problem on unit disk

Problem: Show that $$G(x, y) = \frac{1}{2\pi} \left( \log \lvert x - y \rvert + \log \left\lvert \frac{x}{\lvert x \rvert} - \lvert x \rvert y \right\rvert \right)$$ is a Green's function for the ...
0
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2answers
97 views

How to prove Liouville's theorem for subharmonic functions

I noticed this post and this paper, which gives a version of Liouville's theorem for subharmonic functions and the reference of its proof, but I think there must be an easier proof for the following ...
1
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1answer
15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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2answers
25 views

Laplace transform

Hello how can I solve this problem $ty''-y'=t^2$, $y (0)=0 $ I though about several ways like $y''-y'/t =t$ but I did not know how to solve the integral of $( y'/t) $. Then I thought about ...
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0answers
17 views

Stuck while trying to show fundamental solution $\frac{1}{2\pi}\ln|x|$ satisfies $\Delta \phi(x) = \delta(x)$ in 2d?

I am trying to show that the fundamental solution to the Laplacian in 2D satisfies $$\Delta \phi(x) = \delta(x)$$ where $x = (x_1, x_2) \in \mathbb{R}^2$. So the fundamental solution in 2D is ...
2
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1answer
55 views

Separation of variables for PDE: dividing by zero?

This feels like a question that is both simple and duplicate but I can't find an answer or a previous version of the question. Suppose we are given some PDE, for example Laplace's Equation in polar ...
3
votes
1answer
58 views

Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
2
votes
1answer
31 views

Properties of harmonic function on $\mathbb{R}^2$

Assume $f$ is harmonic on $\mathbb{R}^2$. I want to prove that if there exists a constant $M$ such that $f(x,y) \geq M$ for all $(x,y)\in \mathbb{R}^2$, then $f$ must be a constant fuction. I'm ...
3
votes
1answer
52 views

Computing $\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y$ using the mean value property.

I am asked to compute $$\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y,$$where $D = \{ (x,y) \mid (x+1)^2+y^2 \leq 9, \text{and }(x-1)^2+y^2 \geq 1 \}$. Granted, $u(x,y) = x^3-3xy^2$ is harmonic (it is the real ...
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0answers
27 views

What does it mean for a complex-valued function to be bounded above (or below)?

I was reading about the maximum-minimum principle for harmonic functions in my lecture notes, and it was formulated like this: Let $\phi$ be harmonic in a simply-connected domain $D$. If $\phi$ is ...
0
votes
1answer
21 views

Relation between Poisson kernel and harmonic measure

If $D$ is a domain in the complex plane bounded by a Jordan curve $J$, what's the relation between the harmonic measure and the Poisson kernel on the boundary? More specifically, if $z_0 \in D$ and ...
2
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1answer
25 views

The value of a harmonic function in the interior of a unit disk

Let $u(z)$ be a bounded harmonic function in $D$ such that the limit $$\lim_{r→1^-}u(re^{iφ})$$ is equal to 1 when for $0 < φ < π$ and to 0 for $π < φ < 2π$. Find $u(1/2)$. ...
1
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1answer
36 views

Does harmonic decomposition preserve immersions?

In a nutshell: If we start from an immersion, and look at its harmonic decomposition, are the components also immersions? Details: Let $(M,g)$ be an $n$-dimensional, compact Riemannian manifold with ...
0
votes
1answer
16 views

Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
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0answers
15 views

3 dimensional harmonic conjugates?

An $n$ dimensional harmonic function is defined to be a real valued function $f$ in $\mathbb{R}^n$ such that $\nabla^2 f = 0 $. Equivalently, $f$ is the scalar potential of a conservative vector field ...
1
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1answer
48 views

Show composition of harmonic function and analytic function is harmonic without calculating 2nd derivatives

This is not a duplicate. I realize similar questions to this have been asked, but what I am asking is slightly different: I need to prove the following, arguing by complex differentiability only, and ...
0
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0answers
11 views

Critical points of a harmonic function

Suppose $\phi$ is harmonic on some compact, connected region of $\mathbb{R}^3$. Is there an algorithm that is guaranteed to find all critical points of $\phi$? (Obviously, these will all be saddle ...
3
votes
1answer
37 views

Find a harmonic conjugate for the function $u(x,y)=x^{3}+Axy^{2}$ if one exists - $u(x,y)$ might not be harmonic?

For $u(x,y) = x^{3} + Axy^{2}$, where $A$ is a real number, I need to find a harmonic conjugate, or if one does not exist, show that it does not exist. I began my approach to this problem by first ...
0
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0answers
33 views

Square integrable functions on the unit ball

In one dimension, the space $L^{2}([0,1], dx)$ of complex-valued square integrable functions on $[0,1]$ is well known to be separable, and sine and cosine provide an explicit Hilbert basis for it. My ...
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0answers
21 views

Find the maximum of the function $u(x,y)=3xy+2$

If $u$ is a solution of $\begin{cases} -\Delta u=0&\text{in}\ B(0,2)\\ u(x,y)=3xy+2& \text{for} (x,y)\in \partial B(0,2)\end{cases}$ then find maximum of $u$ in $\overline{B(0,2)}$ and the ...
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1answer
32 views

Find a harmonic function which goes to $0$ on the boundary, which is not identically $0$

Find a function which is harmonic on the area bounded by positive x axis and the line $y=x$, which goes to $0$ on the boundary, which is not identically $0$. Why doesn't it violate the max/min ...
3
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3answers
146 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
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1answer
41 views

Intuition behind $\nabla \cdot \frac{1}{\rho}\nabla$?

Oftentimes instead of the Laplacian I notice the very similar operator $$\nabla \cdot \frac{1}{\rho}\nabla$$ What is the intuition behind this operator? How does it differ intuitively from the ...
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2answers
48 views

Fourier Transform to solve Laplace's equation in cylindrical coordinates

I am trying to solve $\nabla^2 u = 0$ in cylindrical polar coordinates (and radial symmetry) $$ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{\partial^2 ...
1
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1answer
1k views

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
0
votes
1answer
12 views

Laplace's equation 2 variable PDE/chain rule show function is a solution

The question is: 'Show that if f(x,y) is harmonic, then $f(x^2-y^2,2xy)$ is also harmonic using Laplace's equation: $\frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = 0 $. I end ...
0
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1answer
21 views

Inequality involving harmonic functions over the ball and half ball

Let $B\subset \mathbb R^2$ be a unit ball. Let $B^+:=B\cap \{x_2\geq 0\}$ where we set $x=(x_1,x_2)\in \mathbb R^2$. Let $\omega\in C^1(\partial B)$ be given such that $|\nabla \omega|>0$ for all ...
0
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1answer
58 views

Find Green's function of quarter-plane with method of images

Consider a domain $$D= \{(x,y) : x>0, y>0 \}$$ Let $\textbf x = (x,y)$ and $\xi =(\xi_x , \xi_y)$. Find the Green's function, $G(\textbf x , \xi)$ such that $$\nabla ^2 G=\delta (\textbf x - ...
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0answers
64 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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vote
1answer
28 views

Greens function for 2d laplace equation with neumann boundary conditions

I have a domain, $ D : {(x,y) : x>0 , y>0}$ Let $ \mathbf{x}= (x,y) $ and $\mathbf{\xi}= (\xi_x, \xi_y)$, The Greens function satisfying: $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
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0answers
26 views

Verify that Poisson's integral formula is harmonic on the unit disk

Let $D = \{ z : |z| < 1\}$ be the unit disk and let $z \in D$ and $e^{i\theta} \in \partial D$. Let $h(e^{i\theta})$ be a function defined on $\partial D$ and let $\tilde h(z)$ be defined in ...
0
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0answers
13 views

Is it posible to solve the Laplce equation on an open set with dirac delta boundary conditions?

By dirac deta boundary conditions, i mean there is an $y \in \partial D$ such that the value on that point is $\infty$ and 0 elsewhere. Intuitivelly i would think it should be true: the Laplace ...
0
votes
2answers
49 views

Show that $z^n+c$ is harmonic, $c\in\mathbb C$.

I would like to know how to show $$f(z) = z^n+c$$ for $c\in \mathbb C$ is harmonic over $D =\{|z|\leq r\}$. I know that if I express $z = x+iy$, then I can have $f=u+iv$, where $u$ and $v$ will be ...
3
votes
1answer
42 views

Intersections of the level curves of two (conjugate) harmonic functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ with $f(x,y)=e^{-x}(x\sin y-y\cos y)$. 1 Let $g$ be one of the conjugate harmonics of $f$ on $\mathbb{R}^2$ and assume the level curves of $f$ and $g$ ...