Questions tagged [harmonic-functions]
For questions regarding harmonic functions.
211 questions
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Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
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Bôcher's theorem for singularities on the boundary
Let $\Omega\subset\mathbb{R}^2$ be connected, open, bounded, and smooth. Suppose that $u\in C^0(\bar \Omega\setminus \{0\})\cap C^2(\Omega\setminus\{0\})$ is harmonic and positive in $\Omega$.
If $0\...
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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)...
CommunityBot
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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 (...
2
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1
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Prove the orthogonality of vector spherical harmonics
We define
$S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$
$Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$
to be the axial vector ...
- 136
2
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0
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85
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Dirichlet problem for an elliptic operator
consider de Dirichlet problem $Lu=0$ on the unit ball B of $\Bbb C^n$ and $u=f$ on the unit sphere $S^{2n-1}$, we suppose that $L$ is an elliptic operator.
My question is there is a formula of the ...
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Strong Liouville property of virtually abelian groups
Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
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Clarification about solvability of Dirichlet problem at infinity on a pinched negative curvature space
Let $M$ be a complete Riemannian manifold of pinched negative curvature $(-a^2 \leq K \leq -b^2 < 0)$. Let $M_\infty$ denote the ideal boundary and $\varphi \in C^0(M_\infty)$ be a prescribed "...
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A few points of clarification on the Martin boundary
Let $\Gamma$ be a finitely generated group, and let $M$ be the Martin boundary of $\Gamma$. I was reading the article on Martin boundary on Encyclopedia of Math, and I have a few questions about what ...
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Have strictly superharmonic functions on graphs been studied?
Given a graph $G$ and a function $f:G\to\mathbb R$, we say that $f$ is harmonic if
$$f(x)=\frac{1}{|N(x)|}\sum_{y\in N(x)}f(y)$$
for every $x\in G$, where $N(x)$ denotes the set of neighbors of $G$. ...
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1
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Harmonic functions and monotonic decay
I have a general question surrounding certain harmonic functions.
I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
- 8,992
4
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1
answer
202
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Green's kernel estimates on finitely generated groups
I was reading a paper by W. Hebisch and L. Saloff-Coste titled "Gaussian Estimates for Markov Chains and Random Walks on Groups" where I came to know about certain bounds on convolution ...
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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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1
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122
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how to construct a finite energy map
In the construction of harmonic maps by Eells and Sampson, one needs to start with a map with finite energy and use the heat equation to deform it into a harmonic map. The construction of such a ...
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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 $\Delta u=0$ ...
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1
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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 ...
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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 ...
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2
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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 $u$...
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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 \...
- 27.5k
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3
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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
$$\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\...
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Is a discrete harmonic function bounded below on a large portion of $\mathbb{Z}^2$ constant?
In the paper https://doi.org/10.1215/00127094-2021-0037, the main result is if we partition the plane $\mathbb{R}^2$ into unit squares (cells) so that the centers of squares have integer coordinates ...
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If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine
We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$
Therefore $u-u(...
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3
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Nelson's proof of Liouville's theorem
The paper "A proof of Liouville's theorem" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic ...
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Integrability (and hence regularity) of $\alpha$-harmonic maps
To prove the smoothness of an $\alpha$-harmonic map, Sachs and Uhlenbeck firstly show (in their paper "The existence of minimal immersions of 2-spheres") that it is in the Sobolev space $L^...
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An open problem of Hardy and Littlewood on $p$-integral means
In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ ...
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Fourier transforms of homogeneous functions [closed]
Compute Fourier transforms of homogeneous functions of the form,
$$
\frac{1}{|x|^{n+d}}P_d(x)
$$
where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.
- 16.2k
2
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1
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Pair of positive harmonic functions with negative inner product in Drury-Arveson space
Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by
$$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$
Call the corresponding real reproducing kernel ...
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3
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701
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Regularity of Newtonian potential along smooth boundary
Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
...
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Generalized harmonic map
Let $M, N$ be closed Riemannian manifolds and $c$ be a constant. For a map $f:M\to N$, define the energy as
$$E(f) = \frac{1}{2} \int_M\Big( \| df(x)\|^2 - c\| f(x) \|^2 \Big) d\mu(x).$$
When $c=0$, ...
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Ratio of measure of level region for harmonic functions
Let $u$ be a harmonic function defined on $B_1(0)\subset\mathbb{R}^2$, $u(0)=0$, and $\{x\in B_1(0):u(x)>0\}$ is simply connected. Is there a universal constant $c>0$ satisfying that
$$
c\leq \...
- 91.8k
2
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1
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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 $\liminf_{|y|\rightarrow 0}u(y)>0$?
- 91.8k
5
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1
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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 $\mathbb{B}^n$ ...
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3
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1
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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 ...
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2
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1
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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 $...
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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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120
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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
$$
Lu = \sum_{i,j} a_{ij}u_{x_i}u_{x_j}, \qquad \lambda |\xi|...
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Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
CommunityBot
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Direct proof of the global submean property for $\log |f|$
Given an entire function $f : \mathbb{C} \to \mathbb{C}$, $\log |f|$ is subharmonic. Globally, this means that for any disk $D_r(c)$ we have the submean property
$$\log |f(c)| \le \frac{1}{\mu(D_r(c))...
- 91.8k
4
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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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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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'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 ...
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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 ...
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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{...
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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 ...
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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 $\...
CommunityBot
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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 ...
14
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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 $\mathbb{R}^n$, ...
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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 ...
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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
$$
f(v) = \frac{1}{d_v}...
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0
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Characterization of elements of Hardy Space
Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that
$$
\forall f\in H^2(\partial\...
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