The harmonic-functions tag has no usage guidance.

**4**

votes

**0**answers

40 views

### $G$-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 $\text{Harm}(\mathbb{R}^n, S_n)$?

**0**

votes

**0**answers

59 views

### Filled level sets of harmonic funtions

Let $f$ be an enitre function. Define the "filled level set of $f$ as follows:
$$A_M(f):=\{z\in{\mathbb C}:\ |f(z)|\le M\}$$
Theorem 1 in Topological Properties of Level-Sets of Entire Functions ...

**7**

votes

**0**answers

107 views

### Is $F_{f, c, \ell}$ a $G$-harmonic polynomial?

Let $G \subset \text{GL}_n(\mathbb{C})$ be a finite subgroup. The group $G$ acts naturally on $\mathbb{C}^1[\mathbb{C}^n]$ the space of degree $1$ homogeneous polynomials in $x_1, \dots, x_n$, i..e, ...

**2**

votes

**1**answer

103 views

### Expectation equation, harmonic functions, do not understand why equation is true

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| ...

**5**

votes

**1**answer

234 views

### harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...

**2**

votes

**3**answers

134 views

### Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces

By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet ...

**2**

votes

**2**answers

117 views

### Appropriate Recursion relations for Wigner 3j Symbols

I am attempting to code the Cosmic Microwave Lensed Temperature and Polarisation power spectra from first principles and have been told to code the relevant Wigner 3j symbols using recursion rather ...

**1**

vote

**0**answers

32 views

### Legendre equation with homogeneous boundary condition

I'm looking for solutions to the Legendre differential equation
$$
\frac{d}{dx}\left((1-x^2)\frac{d}{dx}f(x)\right)+(\alpha-1)\alpha f(x)=0
$$
with boundary conditions $f'(0)=0$ and $f(1)=0$, but ...

**0**

votes

**0**answers

44 views

### Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...

**6**

votes

**0**answers

110 views

### Harmonic map heat flow in positive curvature

Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...

**3**

votes

**0**answers

84 views

### Why a cone/parabolic set for the nontangential maximal function?

Suppose $f\in L^p(\mathbb{R}^d)$. Then the Dirichlet BVP for the Laplace equation $\Delta u = 0$ in the upper-half plane $\mathbb{R}^d\times\mathbb{R}_{>0}$ with boundary value $f$ can be solved by ...

**3**

votes

**0**answers

72 views

### Barycentric interpolation in hyperbolic triangles

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...

**2**

votes

**2**answers

143 views

### Growth of a harmonic function on the disc

Here is my question : I have a harmonic function $h$ on the open unit disc in $D \subset \mathbb{C}$, such that $\iint_D e^{2h} d\lambda(z) \leq A < \infty$ ($d\lambda$ is the Lebesgue measure on ...

**0**

votes

**1**answer

40 views

### 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 ...

**0**

votes

**0**answers

66 views

### Heredity of harmonic functions

We know that if $u(x,y,z)$ is harmonic in
$\mathbb{R}^3\text{ or }\mathbb{R}^3-\{0\} $, then
$v(x,y):=u(x,y,1)$ may not be harmonic in $\mathbb{R}^2$.
($u = 1/r$, for example)
I wonder what ...

**3**

votes

**0**answers

106 views

### Homogeneous polynomial

Can a homogeneous harmonic polynomial mapping $p(x)=(p_1(x),p_2(x),p_3(x))$, $p(0)=0$, of odd degree $m\ge 3$, of the unit ball $B^3$ into $R^3$ be injective. This means that $p_i$ are harmonic ...

**0**

votes

**0**answers

142 views

### How to find critical points of the following polynomial?

I am trying to find critical points of the following equation in $\mathbb{R}^n$:
$$F(x)=d-\sum_i a_{i}x_i^2+H(x).$$ Here, $a_{i}$ are positive real numbers, and $H(x)$ is a $\mathbf{harmonic}$ ...

**2**

votes

**0**answers

60 views

### Obstructions for existence of a Riemannian metric such that a given function is harmonic

Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic ...

**2**

votes

**0**answers

120 views

### Harmonic maps and centers of mass in Riemannian manifolds

Consider a smooth map $f : M \to N$ between two Riemannian manifolds $(M,g)$ and $(N,h)$. I would like to think of the tension field of $f$ and the harmonicity of $f$ in terms of centers of mass.
I ...

**4**

votes

**1**answer

124 views

### A free boundary problem

Do there exist Jordan analytic curves $J$ in the complex plane $C$, other than circles, with the following property:
There exists a harmonic function $u$ in the unbounded component of $C\backslash ...

**1**

vote

**3**answers

177 views

### Harmonic Function with special property

I would appreciate any help with the following problem:
Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...

**0**

votes

**1**answer

105 views

### Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$

I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that ...

**3**

votes

**1**answer

154 views

### Harmonic function osculating a given subharmonic function

Let $\Omega\subseteq\mathbb C$ be an open set and let $\phi:\Omega\to\mathbb R_{\geq 0}$ be an exhausting (i.e. proper) smooth subharmonic function. Fix $p\in\Omega$. Does there exist a harmonic ...

**-2**

votes

**1**answer

147 views

### Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question
We consider the following two classes of smooth maps on ...

**1**

vote

**0**answers

84 views

### Recovering Spherical Harmonics from Discrete Samples

Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...

**1**

vote

**1**answer

119 views

### Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ [closed]

Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant. My question is: Find sufficient and necessary ...

**2**

votes

**1**answer

144 views

### Extending a harmonic function in a ball to subharmonic in a larger ball

Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$:
$$
\begin{cases}
-\Delta u &= 0, \quad \text {in} \quad B(r), \\
\ \ \ \ \ \, u&= g, \quad \text {in}\quad ...

**1**

vote

**0**answers

46 views

### Equivalence of $L^p$ harmonic functions on the ball and a representation by harmonic homogeneous polynomials

In Harmonic function theory, there is a theorem which says that if $u$ is an harmonic function on $B\left(a,r\right)$, then there exist homogeneous harmonic polynomials $p_{m}$ in $\mathbb{R}^{n}$ ...

**3**

votes

**1**answer

82 views

### Extendability of $L^{p}$ harmonic functions

Let $u$ be a harmonic function on some open set $\Omega\subset\mathbb{R}^{n}$ and $u\in L^{p}\left(\Omega\right)$. Is there any reference on extending $u$ to harmonic function on a larger open set ...

**4**

votes

**1**answer

131 views

### Closed form for 3j-symbol ratios

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...

**1**

vote

**0**answers

66 views

### Counting frequencies of occurrence of patterns within a sequence using harmonic analysis?

Assume that we are given a sequence $\mathbf x := X_1,\dots,X_n \in \mathbb N^n$ for some $n \in \mathbb N$. I am interested in calculating the frequency of occurrence of some fixed sequence $\mathbf ...

**2**

votes

**1**answer

148 views

### Conjectures in classical harmonic function theory

Because I'm doing research in the area of harmonic function theory I would like to know are there any conjectures in the theory of harmonic functions in $\mathbb{R}^{n}$ still open. I know that there ...

**1**

vote

**0**answers

58 views

### Wavelets in the spaces of harmonic functions

I plan to do something with the theory of wavelets but in harmonic function theory. My question is about this interconnection between wavelets and harmonic functions. Can you recommend me some paper ...

**2**

votes

**1**answer

351 views

### Sufficient conditions for equality of measures related to harmonic functions

In Axler's book "Harmonic function theory" on this link http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Livres/Harmonic%20Function%20theory.pdf on page 112 ...

**0**

votes

**2**answers

123 views

### Subharmonic function on a twice punctured complex plane

is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function?
Thanks,

**20**

votes

**1**answer

1k views

### The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...

**1**

vote

**0**answers

82 views

### Existence of harmonic maps between loops

Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
...

**2**

votes

**0**answers

80 views

### What are the boundary asymptotics of harmonic symmetric transverse traceless rank-s tensors on $\mathbb{H}^n$ in the Poincare upper-half-space model? [closed]

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf
In this paper some of its most important results about the asymptotics of symmetric traceless ...

**1**

vote

**1**answer

213 views

### A proof of energy functional appearing in the regularity of elliptic and parabolic equations

I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...

**8**

votes

**0**answers

109 views

### Functions between Markov chains that preserve local harmonicity

Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...

**3**

votes

**1**answer

303 views

### on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...

**1**

vote

**2**answers

702 views

### refined Kato inequality

For any real-valued smooth function $u$, we have the Kato inequality
$|D|Du||^2\leq(det(Hess(u)))^2$,
which holds when $|Du|\neq0$.
If moreover $u$ is harmonic (in an open set in $\mathbb{R}^n$), ...

**1**

vote

**1**answer

144 views

### Biharmonic function with a constant modulus

A bi-harmonic function $u:U\to C$, where $U$ is an open subset of the complex plane $C$ is a solution of the equation $\Delta^2u=0$. Can a nonconstant bi-harmonic mapping have a constant modulus in an ...

**1**

vote

**2**answers

325 views

### Biharmonic function

Is a family of bounded bi-harmonic functions defined in the unit disk an equicontinuous family of functions on compacts? A bi-harmonic function $u$ is a solution of the equation $\Delta^2 u =0$.

**7**

votes

**2**answers

531 views

### Boundary regularity for the Dirichlet problem

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times \{0\}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator.
We wish to solve the Dirichlet problem ...

**4**

votes

**1**answer

613 views

### Harmonic functions on the plane

I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the question first. Is it ...

**9**

votes

**2**answers

388 views

### Signed factors of harmonic polynomials

Let ${\rm Harm}_n^d$ be the space of real harmonic polynomials in $n$ variables, homogeneous of degree $d$. If $P\in{\rm Harm}_n^d$, then
$$\left(\frac{\partial^2}{\partial ...