# Tagged Questions

For questions regarding harmonic functions.

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### Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 ...
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### Compact set of measure zero and sequence of Harmonic Functions with nice properties.

I was studying John B. Garnett's book Bounded Analytic Functions, and then I decided to try the following problem: Let $E \subset \mathbb{R}$ be a compact set, with $|E|=0$. Prove that there ...
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### How to check the barrier function is superharmonic?

Suppose $n\geq 3$ and $\Omega$ is a bounded domain. In the Perron's method to solve the PDE -\Delta u = 0 \text{ in } \Omega \quad \text{and } u = g \text{ on }\partial\Omega, ...
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### Fourier Transforms of hyperspherical harmonics

I am trying to compute the Fourier Transform of a function on a 3-sphere, $f(\hat{Q})$, where $\hat{Q}$ is a unit vector in four-dimensional space. The function $f(\hat{Q})$ is expressed as a series ...
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### How can I prove that $\partial\varphi\neq0$ implies $\bar\partial\partial\varphi>0$?

Let $\Omega\subseteq\Bbb C$ open and $\varphi:\Omega\to\Bbb R$ strongly subharmonic, $\varphi\in\mathcal{C}^2$ such that $\partial\varphi\neq0$. My problem is to prove that ...
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### Show an equation only has harmonic solution

I want to show $$\begin{cases} \Delta(\Delta u) - \nabla\cdot (\Delta u \cdot \nabla u)=0\\ \int \Delta u < \infty\\ \Delta u \ge0 \end{cases}$$ in $\mathbb{R}^2$ only has a solution such that ...
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### Are 1-D line sections of 2-D point source-invoked potential distributions positive definite?

Consider the 2-D potential distribution induced on the plane $y=0$ by a point source positioned at $(0, -y_0, 0)$ in the open halfspace below that plane. The material below the plane is assumed ...
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### Source of the “$\cosh$ trick” for Laplacian eigenfunctions or Helmholtz equation solutions?

Suppose a smooth function $f : \mathbb{R}^n \to \mathbb{R}$ satisfies the Helmholtz equation, the PDE $\Delta f + k^2 f = 0$. A while ago someone showed me a trick: Define a function ...
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### Must a function hold true for all (x,y) to be harmonic?

I've found lots of examples that show various functions that are harmonic but I still can't figure something out. Does a function have to hold true for all (x,y) to be considered harmonic or is it ...
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### By direct computation, check that the function u(x) = $|x|^{1/2}$ cos($\theta$/2) is harmonic in the upper half plane H := {x = (x1, x2) | x2 > 0}.

Okay so Partial Differential Equations make no sense to me. Don't know what to do. All I know is that if a function u is harmonic then $\Delta$u = 0
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### Determining if a Continuous $u:\mathbb{C}\to \mathbb{R}$ Satisfying some Property is Harmonic

If $u : \mathbb{C} \to \mathbb{R}$ satisfies $$u(x + iy) =\frac{1}{4}[u(x + a + iy) + u(x − a + iy) + u(x + i(y + a)) + u(x + i(y − a))]\tag{*}$$ for all $a$ then determine whether $u$ is harmonic, ...
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### Harmonic solutions

Assume that $\Omega\subset R^2$ is an open bounded set with a smooth boundary, $g:\partial\Omega\to R$ is a continuous map and $\{b_i \ | \ i=1,2,\ldots,d\}$ is a finite subset of $\Omega$. ...
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### Finding the harmonic conjugate of $u(x,y)=\sinh(x) \sin(y)$

I know this is already a harmonic function but I am having trouble finding its harmonic conjugate. My instructor did this: $v_{x}=\cosh x \sin y \implies v(x,y)=\sin y \sinh(x)+g_{1}(x)$ ...
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### 2D Taylor expansion of F(x,y) where F(x,y) is harmonic (a solution of Laplace equation)

I would like to know if, given F(x,y) a real function of 2 variables that obeys $\nabla^{2} \left(F \left( x,y \right)\right) = 0$ , is it true that F(x,y) always equals its Taylor expansion within ...
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### oblique derivative smoothness of harmonic functions

Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and ...
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### Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball ...
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### Conformal transformation of complement of disk in upper half plane

Let $U$ be the complement in the half-plane $\operatorname{Im} z > 0$ of a disk of radius $a<1$ centered at $i$. I am looking for a conformal transformation that maps $U$ onto an annulus. Since ...
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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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### 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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### 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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### 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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### 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 ...
$\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$. $J_1 = \{e^{i\theta}: \theta \in (0, \pi/2)\}, J_2 = \{e^{i\theta}: \theta \in (\pi/2, \pi)\}, J_3 = \{e^{i\theta}: \theta \in (\pi, 2\pi)\}$ It's ...