Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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Upcrossing lemma and subharmonic functions

I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $ \lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-...
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Behaviour of higher order Laplacian in punctured domain

Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
User 11111's user avatar
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1 answer
127 views

Carleman's Liouville theorem for entire functions bounded along every ray

There is a long history on constructing entire functions bounded along every direction. For example, we refer to Burckel's math review on Newman (Amer. Math. Monthly 1976 MR0387593) or this ...
Bo_Y's user avatar
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Autocovariance of harmonic oscillator in fluid (Langevin Equation)

I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...
SRB121's user avatar
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2 answers
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$C^1$ harmonic functions on a dense open set are globally harmonic

In a paper I am studying, at a certain point the authors introduce a function $u\in C^1(B_1,\mathbb{R})$ which is harmonic in a dense open subset $U$ of $B_1$. From this, they seem to conclude that $u$...
No-one's user avatar
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4 votes
1 answer
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Bounded covariant derivative of curvature tensor

Let $M$ be a complete Riemannian manifold. Suppose that there are positive constants $i_0$ and $K$ such that the injectivity radius of $M$ is at least $i_0$ and $|\mathrm{Rm}|\le K$ and $|\nabla \...
Anton Petrunin's user avatar
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1 answer
240 views

If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine

We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$ Therefore $u-u(...
mathex's user avatar
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Integrability (and hence regularity) of $\alpha$-harmonic maps

To prove the smoothness of an $\alpha$-harmonic map, Sachs and Uhlenbeck firstly show (in their paper "The existence of minimal immersions of 2-spheres") that it is in the Sobolev space $L^...
Wai's user avatar
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An open problem of Hardy and Littlewood on $p$-integral means

In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ ...
an_ordinary_mathematician's user avatar
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Fourier transforms of homogeneous functions [closed]

Compute Fourier transforms of homogeneous functions of the form, $$ \frac{1}{|x|^{n+d}}P_d(x) $$ where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.
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Generalized harmonic map

Let $M, N$ be closed Riemannian manifolds and $c$ be a constant. For a map $f:M\to N$, define the energy as $$E(f) = \frac{1}{2} \int_M\Big( \| df(x)\|^2 - c\| f(x) \|^2 \Big) d\mu(x).$$ When $c=0$, ...
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8 votes
3 answers
626 views

Regularity of Newtonian potential along smooth boundary

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define $$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$ Is it true that $V(z) \in C^{\infty}(\partial \Omega)$? ...
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Ratio of measure of level region for harmonic functions

Let $u$ be a harmonic function defined on $B_1(0)\subset\mathbb{R}^2$, $u(0)=0$, and $\{x\in B_1(0):u(x)>0\}$ is simply connected. Is there a universal constant $c>0$ satisfying that $$ c\leq \...
Sam and Jim's user avatar
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181 views

Super harmonic function

If $u>0$ in $\mathbb{R}^n\backslash\{0\}$ ($n\geq 2$) and $-\Delta u>0$ in $\mathbb{R}^n\backslash\{0\}$, is it true that $\liminf_{|y|\rightarrow 0}u(y)>0$?
Davidi Cone's user avatar
5 votes
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Newtonian potentials of balls and spheres

This is a simple question whose answer was probably known to Poisson, but I was not able to find it by searching. I need explicit formulas for the Newtonian potential of the unit ball $\mathbb{B}^n$ ...
Piero D'Ancona's user avatar
2 votes
1 answer
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Pair of positive harmonic functions with negative inner product in Drury-Arveson space

Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel ...
J. E. Pascoe's user avatar
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Global Hölder regularity

I am reading the book "Regularity theory for elliptic PDE" by Xavier Fernández-Real and Xavier Ros-Oton, and I saw this result on page 69 about solutions of $\Delta u = f$ in $\Omega$ with $...
Sean's user avatar
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Subharmonic distributions on the plane

A subharmonic (Schwartz) distribution on $\mathbf R^n$ is a distribution $u$ satisfying $\Delta u\ge0$. This implies $\Delta u$ is a positive Radon measure $\mu$, thus for any ball $B$ the convolution ...
Piero D'Ancona's user avatar
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Critical points of a strictly subharmonic function

Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant: \begin{equation} \Delta u = A > 0....
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Estimate value of harmonic function in the annulus

Let $D = B_{2r}(0)\backslash \overline{B}_r(0)$. Assume $Lu = 0$ in $D$ where $L$ is a uniform elliptic operator with constant coefficients $$ Lu = \sum_{i,j} a_{ij}u_{x_i}u_{x_j}, \qquad \lambda |\xi|...
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Is a discrete harmonic function bounded below on a large portion of $\mathbb{Z}^2$ constant?

In the paper https://doi.org/10.1215/00127094-2021-0037, the main result is if we partition the plane $\mathbb{R}^2$ into unit squares (cells) so that the centers of squares have integer coordinates ...
katago's user avatar
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Linear elliptic equation

Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
Samir's user avatar
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4 votes
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Direct proof of the global submean property for $\log |f|$

Given an entire function $f : \mathbb{C} \to \mathbb{C}$, $\log |f|$ is subharmonic. Globally, this means that for any disk $D_r(c)$ we have the submean property $$\log |f(c)| \le \frac{1}{\mu(D_r(c))...
Geoffrey Irving's user avatar
4 votes
1 answer
3k 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:\...
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3 votes
1 answer
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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 $z$; ...
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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 ...
Leo Moos's user avatar
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2 votes
1 answer
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'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 ...
Leo Moos's user avatar
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2 votes
1 answer
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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 ...
Leo Moos's user avatar
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6 votes
1 answer
352 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 ...
user492517's user avatar
4 votes
1 answer
326 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{...
Leo Moos's user avatar
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-1 votes
1 answer
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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 ...
Maths Wizzard's user avatar
4 votes
1 answer
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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 ...
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1 vote
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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\...
Naruto's user avatar
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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 ...
Naruto's user avatar
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5 votes
0 answers
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$\mathbb Z_k$-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 ...
Pranay Gorantla's user avatar
4 votes
1 answer
208 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}...
Sprotte's user avatar
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2 votes
1 answer
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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 $$ \Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~. $$ ...
Pranay Gorantla's user avatar
7 votes
1 answer
317 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 ...
Sakunee's user avatar
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14 votes
2 answers
777 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 $\mathbb{R}^n$, ...
Qfwfq's user avatar
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2 votes
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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 ...
Loafy Loafer's user avatar
6 votes
1 answer
215 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 ...
piper1967's user avatar
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3 votes
1 answer
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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 ...
Nik's user avatar
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3 votes
2 answers
327 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}\!}...
Dapao Zhang's user avatar
7 votes
1 answer
866 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 ...
Llohann's user avatar
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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 ...
sharpe's user avatar
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5 votes
0 answers
154 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 ...
maxematician's user avatar
1 vote
1 answer
164 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 ...
SMS's user avatar
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3 votes
0 answers
334 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 ...
student's user avatar
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2 votes
0 answers
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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 ...
bobuhito's user avatar
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8 votes
2 answers
758 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 \...
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