All Questions
7 questions
8
votes
2
answers
773
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 \...
- 4,135
8
votes
2
answers
622
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, ...
- 903
4
votes
1
answer
221
views
Is a specific product function orthogonal to all harmonic functions
Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
- 4,135
3
votes
0
answers
90
views
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
1
answer
186
views
Expectation equation, harmonic functions, do not understand why equation is true
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to \...
- 13
1
vote
0
answers
192
views
The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
0
votes
0
answers
116
views
Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane
We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...
- 5,063