# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

110
questions

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votes

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76 views

### regularity of p-harmonic functions

We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$.
Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...

**0**

votes

**3**answers

82 views

### Intersection of Subharmonic and Harmonic Functions (also: Extension of harmonic functions)

Let $\Omega \subset \mathbb{R}^n$ be open, convex and bounded with smooth boundary. Define
$$\mathcal{A}(\Omega) = \left\{ C \subset \Omega \ \big\vert \begin{array}{l} \text{for any subharmonic ...

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**0**answers

71 views

### What is the definition of Sobolev maps between surfaces used by Jurgen Jost in his book Compact Riemann Surfaces?

In the book Compact Riemann Surface by Jurgen Jost, the notion of $W^{1,2}(\Sigma_1,\Sigma_2)$, where $\Sigma_1,\Sigma_2$ are Riemann surfaces, is used. But I can't figure out what definition he is ...

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votes

**1**answer

136 views

### The positive solutions of the weighted Laplacian equation

Let $u$ be a positive function on $\mathbb R^n$ such that
$$
\Delta u-\partial_{x_1}u=0,
$$
where $\Delta$ is the Laplacian operator $\partial_{x_1}^2+\partial_{x_2}^2+\cdots+\partial_{x_n}^2$.
Can ...

**7**

votes

**0**answers

92 views

### implicit function theorem and harmonic mapping

We are given two Riemannian manifolds $M,N$ of dimension $m$ and $n$ and a function $G \colon M \times N \to \mathbb{R}^n$ which satisfies the assumptions of the implicit function theorem, meaning ...

**1**

vote

**1**answer

154 views

### Continuity of subharmonic functions

There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...

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votes

**0**answers

54 views

### Smooth function whose average value on any ball $B(x,r)$ is a polynomial of $r$

Let $f$ be a smooth function on $\mathbb R^N$, assume that for any $x \in \mathbb R^N$, $\frac{1}{Vol(B(x,r))} \int_{B(x,r)}f$ is a polynomial of $r$ (we denote the polynomial by $p_x(r)$), where $B(...

**5**

votes

**0**answers

117 views

### Continuity of the Green function with respect to the measure

Let $G$ be a finitely generated group and let $\mu$ be a finite measure on $G$. Define the Green function as
$$G(\mu)=\sum_{n\geq 0}\mu^{*n}(e),$$
where $\mu^{*n}$ is the $n$th convolution power of $\...

**1**

vote

**0**answers

64 views

### Existence of nonparabolic ends

Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the ...

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**0**answers

60 views

### Find limit of sequence defined by sum of previous terms and harmonics

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function?
I tried to ...

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votes

**1**answer

88 views

### How to create a function whose harmonic is a sine wave [closed]

How do I solve the following equation for $f(\cdot)$?
$f(x)+\frac{1}{n}f(nx)=\sin(x)$
That is, how do I create a function which, when combined with its nth harmonic, will be a sine wave?

**5**

votes

**1**answer

147 views

### Taylor-like expansion for a holomorphic function in non-simply-connected domain

Suppose $f$ is a holomorphic function in a simply connected open set $U$, and we know it's Taylor expansion at a point $p\in U$. We can then find a holomorphic map $g$ of $U$ to the unit disc which ...

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votes

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56 views

### Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?

Let $\mathbb{D}^n$ be the closed unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; More precisely, I assume that $f$ is real-analytic and harmonic on the interior $(\mathbb{D}^n)^o$ ...

**1**

vote

**0**answers

67 views

### Motivation and examples of parabolic manifolds

Let $(M^{n},g)$ be a Riemannian manifold, we say that $M$ is parabolic if the constant functions over $M$ are the only subharmonic functions which are bounded above, i.e, for a function $u \in C^{2}(M)...

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63 views

### On a property of harmonic functions of stochastic processes

I have a question which relates to an argument appearing in this paper 1.
Let $D$ be a domain of $\mathbb{R}^d$ and $X=(X_t, P_x)$ be a diffusion process on $D$.
Let $h : D \to [0,\infty]$ be a ...

**5**

votes

**1**answer

114 views

### A question about integration of spherical harmonics on $(S ^ 2, can)$

Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that
$$ \int_{\mathbb{...

**5**

votes

**1**answer

165 views

### harmonic coordinates on non-compact manifolds

Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...

**3**

votes

**0**answers

151 views

### Can the rank of harmonic maps decrease far from the boundary?

Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...

**0**

votes

**0**answers

77 views

### Hitting probabilities and harmonic functions

I have a question about harmonic functions and hitting probabilities.
Let $d \ge 3$ be an integer.
Let $D=\mathbb{R}^d \times (-1,1)$ and denote points $z \in D$ by $z=(r,\theta,y),$ where $(r,\...

**7**

votes

**2**answers

758 views

### Eigenvalues of Laplace-Beltrami on half sphere

Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...

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votes

**1**answer

123 views

### Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?

In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...

**1**

vote

**0**answers

87 views

### Is every “higher-order” harmonic morphism conformal?

$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\TstarM}{...

**2**

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48 views

### Does a map which preserve harmonic forms preserve co-closed forms (locally)?

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth.
Let $1 \le k \le d-1$ be fixed....

**1**

vote

**1**answer

714 views

### Partial derivatives of spherical harmonics

Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?

**8**

votes

**2**answers

375 views

### Obstructions for the wedge of coordinate differentials to be harmonic

Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist ...

**2**

votes

**0**answers

73 views

### On the solution of Laplace equation with mixed boundary condition

Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system
$$
\begin{...

**4**

votes

**1**answer

94 views

### Dynamics for approximating harmonic functions on graphs

A harmonic function on a graph is a function on its vertices such that the value at every vertex is the average of the values at its neighbors.
Consider the following method for approximating a ...

**1**

vote

**1**answer

93 views

### Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field

Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...

**0**

votes

**1**answer

91 views

### Harmonicity of the Martin kernels

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Consider the Green function $G(x,y)=\sum_{n\geq 0}\mu^{*n}(x^{-1}y)$, where $\mu^{*n}$ is the $n$th ...

**1**

vote

**1**answer

132 views

### Topological similarity of solutions to Dirichlet problem

Let $\varphi_{1},\varphi_{2}:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be two
smooth general position (Morse) functions having the same set of critical
points $\left\{ p_{1},...,p_{n}\right\} \subset\...

**4**

votes

**1**answer

118 views

### A question about harmonic function

Let $u$ be a harmonic function inside the unit ball $B(0, 1)$ in $\mathbb{R}^2$ so that $|u|\leqslant 1$. Does a function u which satisfies $|\nabla u(0)|>1$ exist? If not, please prove that $|\...

**1**

vote

**1**answer

249 views

### Functions orthogonal to harmonic functions

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose
$\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\...

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**0**answers

52 views

### Which planar smooth foliations are not smooth equivalent to a foliation arising from level sets of a harmonic function?

Is there an smooth foliation of the plane which is not smoothly equivalent to a foliation $dH=0$ where H is a harmonic function without critical values?
If the answer is negative then we conclude ...

**12**

votes

**2**answers

1k views

### Harmonic function properties on $\mathbb R^3$

Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\...

**1**

vote

**1**answer

134 views

### Optimal Holder regularity estimates for the single layer potential

I need a statement that must be classical but it's being very difficult (even after consulting experts) to find a reference. Let's try here!
Let $f$ be a function on the boundary of a bounded domain $...

**6**

votes

**1**answer

300 views

### Harmonic maps are light

Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map
and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected?
I hope that the answer is yes.
But actually I ...

**3**

votes

**0**answers

112 views

### Wolff's article: Note on counterexamples in strong unique continuation problems

I am reading Wolff's Note on counterexamples in strong unique continuation problems:
http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf
On Page 3, ...

**2**

votes

**1**answer

198 views

### The flow of Harmonic vector fields

A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions.
Motivated by conversations on this questions we ask:
...

**4**

votes

**1**answer

460 views

### Spherical Harmonics on $S^3$ [closed]

My understanding is that harmonic analysis on the circle ($S^1$) leads to Fourier Series/Integrals whereas harmonic analysis on the sphere ($S^2$) leads to Spherical Harmonics. If we take the next ...

**13**

votes

**2**answers

614 views

### Tweetable way to see Riemannian isometries are harmonic?

$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$
Smooth Riemannian isometries are harmonic. Can one conclude ...

**1**

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**0**answers

137 views

### Expansion of prolate spheroidal harmonics

For two coordinate frames $O'$ and $O''$ both offset along the $z$-axis by $\pm R$ respectively, with corresponding offset spherical coordinates $r'$, $\theta'$, $r''$ and $\theta''$, and with prolate ...

**5**

votes

**1**answer

179 views

### Are all the mappings which satisfy this equation scaled isometries?

Let $M,N$ be smooth oriented $d$-dimensional Riemannian manifolds, $\, f:M \to N$ a smooth map. Let $\Omega^1(M,f^*TN)=\Gamma(T^*M \otimes f^*TN)$ be the space of $f^*TN$-valued one-forms.
Let $d$ ...

**1**

vote

**1**answer

130 views

### Harmonic maps between surfaces

Suppose $M$ and $N$ are Riemannian manifolds (non compact) of dimension $2$ and $f$ is an harmonic map between $M$ and $N$. When is $f$ conformal?

**2**

votes

**1**answer

218 views

### Laplace equation in a domain with holes

Suppose $u_r(x_1,x_2)$ in $B_1(0) \setminus B_r(0)$ satisfy
\begin{align*}
\begin{cases}
\Delta u_r &=0\\
u_r(|x|=r)&=-\log r \\
u_r(|x|=1)&=0
\end{cases}
\end{align*}
with $0<r<1$....

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vote

**1**answer

165 views

### Is maximum principle valid in the case of non-smooth boundaries?

Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth
or Lipschitz, they may be very bad.
Denote $U=U_2 \...

**1**

vote

**0**answers

66 views

### Question concerning the proof of the Stein-Weiss interpolation theorem

Currently I am going through the proof of the Stein-Weiss interpolation theorem for a seminar paper and in particular through the proof of Lemma 1 which begins at page 320 (I will use a particular ...

**1**

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**0**answers

101 views

### Bounds for Discrete Poisson Kernel of a Square

I am having difficulty in proving the lower bound of the discrete Poisson kernel of a square denoted as $H$ described below. It is stated in Gregory F. Lawler's Randomm Walk and the Heat Equation as ...

**0**

votes

**0**answers

207 views

### Is the Lie derivative of a harmonic form also a harmonic form?

On Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" it is said that the Lie derivative along a left-invariant vector field of an harmonic form is again a harmonic form. This ...

**11**

votes

**1**answer

391 views

### Harmonic functions (eigenfunctions of the Laplace-Beltrami operator) of SO(2n)/U(n)

Have the eigenfunctions of the Laplace-Beltrami operator on $SO(2n)/U(n)$ been worked out explicitly? If not, how does one approach finding them?
(I'm thinking of this as in analogy with the ...

**4**

votes

**0**answers

214 views

### Geometrical point of view of the harmonic constraints ($\Delta g_{ij}=0$) in General Relativity

What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor's coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-...