Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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votes
1answer
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
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3answers
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 ...
0
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0answers
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 ...
2
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1answer
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
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0answers
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
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1answer
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 ...
2
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0answers
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
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0answers
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
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0answers
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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0answers
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 ...
0
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1answer
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
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1answer
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 ...
4
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0answers
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 $(\mathbb{D}^n)^o$ ...
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0answers
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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0answers
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
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1answer
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
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1answer
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
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0answers
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
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0answers
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
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2answers
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-...
5
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1answer
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 ...
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0answers
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}{...
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0answers
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....
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1answer
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
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2answers
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
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0answers
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
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1answer
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 ...
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1answer
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
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1answer
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
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1answer
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
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1answer
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
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1answer
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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0answers
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
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2answers
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
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1answer
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
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2answers
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 ...
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0answers
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
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1answer
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
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1answer
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
1answer
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 $0<r<1$....
1
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1answer
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 \...
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0answers
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 ...
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0answers
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 ...
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0answers
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
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1answer
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
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0answers
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-...