All Questions
8
questions
1
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0
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116
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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 ...
- 543
1
vote
0
answers
61
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\...
- 63
5
votes
0
answers
117
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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+\...
- 113
1
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1
answer
460
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A harmonic function $\varphi$ with $D\varphi \in L^q(\mathbb R^n)$ is constant
Let $\varphi$ be an harmonic function such that $D\varphi \in L^q(\mathbb R^n)$ for $q \in (1, +\infty)$. I read in Partial Differential Equations of Quin Han and Fanghua Lin that for $q = 2$, $\...
- 420
4
votes
1
answer
214
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 ...
- 4,113
1
vote
1
answer
384
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The reproducing kernel for harmonics on compact manifolds
Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
- 555
1
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0
answers
173
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 ...
- 555
1
vote
1
answer
255
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A proof of energy functional appearing in the regularity of elliptic and parabolic equations
I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...
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