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-4
votes
0answers
28 views

Harmonic function, inversion [migrated]

Let $u$ be a harmonic function defined on $B_1(0)$, the unit sphere. Define $v: \mathbb{R}^d \setminus B_1(0) \to \mathbb{R}$ by $$v(x) = u(x/|x|^2).$$ How does one show that $v$ is harmonic?
1
vote
2answers
105 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
1answer
34 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
0answers
44 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
0answers
101 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
0answers
122 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
0answers
58 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
0answers
95 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
1answer
115 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
3answers
160 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
0answers
44 views

harmonic functions on hyperbolic plane x Real line

I am looking for references of harmonic functions(no growth condition) on the product of hyperbolic plane $\mathbb H^2$ with the real line $\mathbb R$. The metric is just the product one. Especially ...
0
votes
1answer
96 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
1answer
140 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
1answer
143 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
0answers
71 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
1answer
116 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
1answer
130 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
0answers
23 views

Equivalence of Lp harmonic functions on the ball and a representation by harmonic homogeneous polynomials

In Harmonic function theory there is a theorem which say if $u$ is a harmonic function on $B\left(a,r\right)$, then there exist homogeneous harmonic polynomials $p_{m}$ in $\mathbb{R}^{n}$ such that ...
3
votes
1answer
77 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
1answer
116 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
0answers
64 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
1answer
134 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
0answers
55 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 ...
1
vote
1answer
277 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
2answers
109 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,
17
votes
1answer
799 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 ...
2
votes
0answers
73 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
1answer
207 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
0answers
106 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
1answer
286 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
2answers
609 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
1answer
143 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
2answers
308 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
2answers
475 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
1answer
574 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
2answers
378 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 ...