Questions tagged [harmonic-functions]
For questions regarding harmonic functions.
14 questions from the last 365 days
7
votes
1
answer
177
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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}...
- 1,101
0
votes
0
answers
34
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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\...
- 308
2
votes
0
answers
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 ...
- 21
2
votes
1
answer
202
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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 $G$...
- 1,407
2
votes
0
answers
86
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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 "...
- 1,407
3
votes
1
answer
209
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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 ...
- 1,407
1
vote
0
answers
129
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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 $G$. ...
- 111
0
votes
1
answer
172
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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, ...
- 11
2
votes
1
answer
147
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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 ...
- 117
4
votes
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 ...
- 131
3
votes
0
answers
90
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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 $[0,\lambda]$-...
1
vote
0
answers
43
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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 $\Delta u=0$ ...
- 11
1
vote
1
answer
152
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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 ...
- 637
2
votes
0
answers
75
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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 ...
- 71