Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

Filter by
Sorted by
Tagged with
3 votes
1 answer
76 views

An inequality for harmonic functions

In a paper that I am reading the author quotes the following result about harmonic functions. According to him this should be "easy to show" but I don't seem to be able to do so. Let $u:\...
  • 701
3 votes
1 answer
75 views

A harmonic function degenerate in one direction

Question. Let $u: B^3 \to \mathbf{R}$ be a harmonic function with $u(0) = 0$, $Du(0) = 0$, where its homogeneous harmonic blow-up is a polynomial $p = p(x,y)$ in two variables, so independent of $z$; ...
  • 3,684
2 votes
0 answers
45 views

Has the nodal map been studied?

Let $D \subset \mathbf{R}^n$ be the unit disc, and $\alpha \in (0,1)$. Let $f \in C^{0,\alpha}(\partial D)$, and $u \in C^{2,\alpha}(D)$ be the harmonic function with $u = f$ on the boundary. Define ...
  • 3,684
2 votes
1 answer
93 views

'Dirichlet problem' along axis for harmonic functions

Question. Let $f: \mathbf{R} \to \mathbf{R}$ be an analytic function. Is there a harmonic function $u$ on the circular cylinder $D \times \mathbf{R} \subset \mathbf{R}^3$ so that $u = f$ along the ...
  • 3,684
2 votes
1 answer
87 views

Reference for harmonic functions in cylinders

Question. What is a good reference (textbook, article etc.) to learn more about harmonic functions on finite (and infinite) cylinders? I am trying to gain a better understanding of the behavior of ...
  • 3,684
6 votes
1 answer
227 views

Limit of zero sets of harmonic functions

Let $u_n : \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions which converge uniformly on compact subsets. The limit function $u$ (which we assume to be not identically $0$) is clearly ...
4 votes
1 answer
295 views

Is there a harmonic function with just one singular point?

Let $D \subset \mathbf{R}^2$ be the unit disc, and $L > 0$. Let $u: D \times (-L,L) \to \mathbf{R}$ satisfy \begin{equation} \begin{cases} \Delta u = 0 \quad \text{ on $D \times (-L,L)$ } \\ \frac{...
  • 3,684
-1 votes
1 answer
122 views

Proof verification for a theorem about a harmonic function on the unit disc [closed]

On why this is here I tried posting on math stackexchange but I got no comments or answers. I even bountied the question but I am still not getting any responses. I am getting the sense that I wasn't ...
4 votes
1 answer
107 views

Harmonic functions as limits of harmonic functions on graphs?

I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this ...
  • 757
1 vote
0 answers
53 views

Characterization of elements of Hardy Space

Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that $$ \forall f\in H^2(\partial\...
  • 53
2 votes
0 answers
16 views

Do Szego Kernel in one variable by fixing another variable in a $C^{\infty}$ bounded domain is Bounded?

Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain. Let $ S(.,.)$ denotes the Szego Kenel of Holomorphic Hardy Space $H^2(\partial\Omega)$. Then for $w\in\Omega$ do $S(.,w)$ is a ...
  • 53
5 votes
0 answers
1k views

$\mathbb Z_k$-harmonic function that distinguishes two vertices of a graph

Let $G$ be a simple, undirected, connected graph on $n$ vertices, and let $A$ be an abelian group. A function $f:V(G)\rightarrow A$, on the vertices of the graph $V(G)$, is said to be $A$-harmonic if ...
4 votes
1 answer
125 views

Lipschitz harmonic functions on graphs?

Let $G$ be an (infinite) countable graph of bounded degree with vertex and edge sets $V(g)$ and $E(G)$, respectively. A function $f : V(G) \to \mathbb{R}$ is called harmonic if $$ f(v) = \frac{1}{d_v}...
  • 757
2 votes
1 answer
248 views

Difference equation satisfied by discrete harmonic functions on square lattice

A function $f:\mathbb Z^2 \rightarrow \mathbb R$ is said to be discrete harmonic if it satisfies the discrete Laplacian equation $$ \Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~. $$ ...
7 votes
1 answer
223 views

Harmonic functions on complete Riemannian manifolds

I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at ...
  • 71
14 votes
2 answers
661 views

Harmonic polynomials on the sphere

Let $\mathbb{S}=\{x\in\mathbb{R}^n|x_1^2+\ldots +x_n^2=1\}$ be the unit sphere in $\mathbb{R}^n$, $\mathbb{C}[x]=\mathbb{C}[x_1,\ldots ,x_n]$ the complex-valued polynomial functions on $\mathbb{R}^n$, ...
  • 21.9k
2 votes
0 answers
85 views

Bubble tree convergence: Why is it necessary to consider centre of mass of the energy measure?

In the paper “Bubble Tree Convergence for Harmonic Maps” by Thomas H. Parker, after the picking the energy concentration points, he proceeded by expanding the map around each energy concentration ...
6 votes
1 answer
142 views

Positive harmonic functions on nilpotent groups & Random walk on groups with a finite number of generators

I want to read the following papers in the English version which I could not find anywhere (the only papers I can get are the Russian versions). Kindly help me out. Gregory A. Margulis, Positive ...
  • 989
3 votes
1 answer
86 views

Component wise convergence of a sequence of complex harmonic functions

It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In ...
  • 169
3 votes
2 answers
253 views

A Sobolev embedding theorem for functions on spheres

$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds: $$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
7 votes
1 answer
601 views

Kernel of the Laplacian + a function

It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
  • 369
3 votes
0 answers
81 views

Differentiability of a weak solution

Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...
  • 499
5 votes
0 answers
141 views

Potential theory as a tool in extrinsic flows

Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
1 vote
1 answer
120 views

Liouville property of hyperbolic spaces

It seems classically known (and mentioned in several papers without reference) that there exist bounded non-constant harmonic functions on the hyperbolic space $\mathbb{H}^n, n \geq 2$. I am ...
  • 1,141
3 votes
0 answers
141 views

Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to \begin{equation} \begin{cases} -\Delta u=0 \quad &\mbox{in $\Omega$}\\ \frac{\partial ...
  • 1,238
2 votes
0 answers
55 views

Harmonic function over a square with linear Neumann boundary conditions

For a rectangle with height 1 and length 2, here is the unique numerical solution (showing contours of the equipotential from 0, defined by the bottom, to 0.54, the numerically-calculated maximum) to ...
  • 1,507
0 votes
0 answers
139 views

On a harmonic coordinate for a differential operator

I am looking for a harmonic coordinate for a degenerate differential operator on the unit ball. We write $\langle \cdot,\cdot \rangle$ for the standard inner product on $\mathbb{R}^d$. Set $\lvert\...
  • 499
8 votes
2 answers
729 views

Points where harmonic functions fail to give a coordinates system

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
  • 3,301
1 vote
0 answers
109 views

Are Poisson integrals uniquely determined by their radial limits?

Let $\mu$ be a complex Borel measure on the unit circle, and suppose its Poisson integral $u$ satisfies $\lim_{r\to 1-}u(re^{i\theta})=0$ for every $\theta$. Does it follow that $\mu=0$? This is of ...
7 votes
1 answer
283 views

Does the pointwise mean value property imply harmonicity?

Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(...
  • 763
3 votes
1 answer
266 views

Geometric flow by the level sets of a harmonic function

Let $u$ be an harmonic function in a cylindrical domain $B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets $\Gamma_t=\{u=t\}$ are graphs of functions on $B_2^{n-1}$. Consider a ...
1 vote
0 answers
59 views

Polar growth property for harmonic Maass forms

The definition of a harmonic Maass form consists of three properties; (1) that it is modular, (2) that it is harmonic, and (3) that it has at most polar growth at the cusps (ordered in accordance with ...
  • 880
-1 votes
1 answer
312 views

Harmonic function in infinite domain in $\mathbb{R}^3$, constant on the boundary and decaying as $1/r^2$

EDIT: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth connected boundary. Let $f\colon \mathbb{R}^3\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\...
  • 19.4k
0 votes
1 answer
199 views

Harmonic functions in infinite domain in Euclidean space

EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
  • 19.4k
3 votes
1 answer
108 views

About the proof of higher regularity boundary Harnack inequality

I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1. In the paper they used the Hopf lemma to show that $u_\nu>c&...
3 votes
0 answers
54 views

Matrix equation and spherical harmonics

I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$), $$ \eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi) $$ Similar to the ...
4 votes
0 answers
136 views

How to use blow-up to prove the boundary regularity for a harmonic function

While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem: Thm. 2.30. Let $\alpha \in (0,1)$ and $k \in N$ with $k \leq 2$, and let $\Omega$ be a bounded $C^{k, \alpha}$ ...
5 votes
0 answers
242 views

Are the two-valued homogeneous harmonic functions classified?

Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$? For reference, multi-valued functions are familiar objects in ...
  • 3,684
4 votes
2 answers
476 views

$\log |f|$ is subharmonic

It is known that the logarithm of the modulus of an analytic function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions: (1) Are there some weaker ...
  • 165
2 votes
0 answers
95 views

Dimension of critical set of p-harmonic function

Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$. Question: What is the Hausdorff dimension of the critical ...
8 votes
2 answers
504 views

Vanishing rate of a harmonic function near a boundary point

Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is, $$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$ for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
  • 883
5 votes
1 answer
236 views

A question on the monotonicity formula for minimal submanifolds

I'm reading the proof of monotonicity formula from A Course in Minimal Surfaces by Colding-Minicozzi. The theorem says Suppose $\Sigma^k \subset \mathbb{R}^n$ is a minimal submanifold and $x_0\in\...
3 votes
1 answer
232 views

Harmonic interpolation with analytic initial condition

Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold. Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function. Is there a Harmonic ...
5 votes
0 answers
115 views

Good (Sidon) Approximation of "Bumps"

Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...
3 votes
0 answers
199 views

Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
  • 109
0 votes
0 answers
88 views

Extending harmonic functions defined in the closure of a bounded smooth domain to some larger domain

Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N$ where $N\geq 2$. Consider the Laplace equation with a Neumann boundary condition $$ -\Delta u = 0 \quad\mbox{in } \Omega, \qquad \frac{\...
  • 1
6 votes
1 answer
133 views

Tangential harmonic $1$-forms are pullbacks of harmonic functions

This question has also been posted on MSE, but maybe here is the right place to obtain an answer. Let $(M^3,g)$ be a compact connected oriented Riemannian $3$-manifold with nonempty boundary. The ...
0 votes
0 answers
59 views

A question about super-harmonic functions

Lets call a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ to be super-harmonic if $\nabla ^2 f = \sum_{i=1}^n \partial_i^2 f \leq 0$. Now given such a $f$ as above I want to consider the ...
  • 2,026
1 vote
0 answers
99 views

Mean Value Inequality with Linear Term

I am having trouble proving this modified mean-value inequality. Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$ Prove that there exists constants $r_0,C>0$ depending only on $c$...
  • 11
0 votes
0 answers
121 views

Function Spaces on the Open Unit Disk defined by Hardy Space norms

I've been reading up on Hardy spaces and (sub)harmonic functions over the open unit disk $\mathbb{D}\subset\mathbb{C}$, and I've found myself working with atypical objects in mostly-typical situations....
  • 1,216