The harmonic-functions tag has no usage guidance.

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### Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set ...

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74 views

### Is a harmonic function with injective boundary conditions an immersion outside a negligible set?

This question is a strengthening of this question.
Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f \colon M \to ...

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**1**answer

109 views

### The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...

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37 views

### The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...

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49 views

### Bounds on the the spherical harmonics on $S^{p-1}$

The only reference I could find in this regard is for upper bounding the n-homogeneous spherical harmonics on $S^{p-1}$ as in equation 4.29 here, ...

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2k views

### Nelson's proof of Liouville's theorem

The paper "A proof of Liouville's theorem" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic function in ...

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57 views

### Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...

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**1**answer

78 views

### PDE with harmonic function [closed]

I'm looking for the solution of the following equation
$$2cE^3=E_u^2+ E_v^2$$ where $c$ is a constant and $E$ is an harmonic function w.r. to the variables $u$ and $v$.

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100 views

### Harmonic function with injective boundary conditions is an immersion?

Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...

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**1**answer

119 views

### Harmonic/Subharmonic lifting of functions on an annulus

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in ...

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101 views

### Coefficient problem for univalent harmonic functions on unit disk

The Clunie Sheil Small conjecture for the second coefficient of a univalent harmonic function on the unit disk is as follows:
Suppose, $h(z)+\overline{g(z)}$ is a one-to-one harmonic function on the ...

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62 views

### Weak $H^1$-limit of “almost conformal” maps

Let us consider a sequence of maps $\phi_n : (M, g) \to \mathbb{R}^k$, where $M$ is a surface. Let $\phi_n \to \phi$ weakly in the sense of $H^1$-norm, and let $\phi$ be non-trivial. Let $\mathcal{H}$ ...

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**1**answer

163 views

### An example for affine function [closed]

I'm looking for an example of a non-Euclidean non-compact Riemannian manifold $(M,g)$ such that we could define a non-constant affine function $f:M\rightarrow \mathbb{R}$, namely its gradient vector ...

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**1**answer

115 views

### harmonic balance method for a 2-mass 3-spring system [closed]

I am trying to solve a nonlinear 3spring-2mass system under harmonic loading by using Fourier series expansion of states of the differential equation. The system is just basically two masses, two ...

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49 views

### Products Laplacian Eigen-Functions over a Kaehler Manifold

I've been trying to learn a little about Laplacians acting on the smooth functions of a compact Kaehler manifold, and made the following (possibly incorrect) observation:
Let $\{f_i\}$ be a set of ...

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**1**answer

119 views

### Generate harmonic polynomials for a finite group

Let $G$ be a finite group acting on a complex vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A ...

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52 views

### $G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$? [closed]

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

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66 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 ...

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194 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, ...

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114 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| ...

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**1**answer

238 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 $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...

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### 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 ...

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208 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 ...

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59 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 ...

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55 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 ...

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135 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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183 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 ...

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**1**answer

116 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 ...

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**1**answer

163 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 ...

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153 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 ...

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**1**answer

152 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 ...

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### 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 ...

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158 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 ...

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### 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 $\mathbb{R}^{n}$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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,

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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 ...