List questions from harmonic-functions

Published on 07 Jan 2017 - 16:16

228

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Prove $\int_{B(x,r)}|\nabla u|^2\leq \frac{C}{r^2}\int_{B(x,2r)}|u|^2$ if $-\Delta u(x)+f(x)u(x)=0.$

Published on 07 Jan 2017 - 17:17

202

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Harmonic function on a binary tree

Published on 07 Jan 2017 - 22:20

254

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Why $|\partial _i u|\leq \frac{C}{r^{d}}r^{d-1}\|u\|_{L^1(B(x,4r)}$ if $\partial _iu=\frac{1}{|B(x,r)|}\int_{\partial B(x,r)}u\nu_i$?

Published on 08 Jan 2017 - 16:42

377

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Analytic Functions - why 2 dimensions?

Published on 13 Jan 2017 - 8:48

196

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$f \in C^2(\mathbb{R}^2)$ is harmonic; $f$ is in one $L^p(\mathbb{R}^2)$ $\implies$ $f$ is constant

Published on 14 Jan 2017 - 0:11

162

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sum of generalized harmonic numbers

Published on 15 Jan 2017 - 23:31

289

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maximum principle for harmonic functions (equivalence)

Published on 16 Jan 2017 - 18:50

517

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Fundamental solution of elliptic pde with constant coefficient

Published on 26 Mar 2026 - 12:34

664

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Discrete Hölder inequality

Published on 23 Jan 2017 - 15:36

235

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Subharmonic functions with finite norm

Published on 23 Jan 2017 - 16:50

258

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Equivalent definition of Wiener capacity

Published on 28 Mar 2026 - 15:26

710

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Dirichlet Problem for the Half space: differentiating under the integral sign.

Published on 26 Jan 2017 - 0:31

243

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Converse of Kellogg Warschawski theorem

Published on 28 Jan 2017 - 21:02

265

Views

Harmonic function that is not pluriharmonic

Published on 26 Mar 2026 - 11:05

List Question

Find harmonic function s.t. $\int_{\mathbb R^d}|D^2 u|^2<\infty $

Prove $\int_{B(x,r)}|\nabla u|^2\leq \frac{C}{r^2}\int_{B(x,2r)}|u|^2$ if $-\Delta u(x)+f(x)u(x)=0.$

Harmonic function on a binary tree

Why $|\partial _i u|\leq \frac{C}{r^{d}}r^{d-1}\|u\|_{L^1(B(x,4r)}$ if $\partial _iu=\frac{1}{|B(x,r)|}\int_{\partial B(x,r)}u\nu_i$?

Analytic Functions - why 2 dimensions?

$f \in C^2(\mathbb{R}^2)$ is harmonic; $f$ is in one $L^p(\mathbb{R}^2)$ $\implies$ $f$ is constant

sum of generalized harmonic numbers

maximum principle for harmonic functions (equivalence)

Fundamental solution of elliptic pde with constant coefficient

Discrete Hölder inequality

Subharmonic functions with finite norm

Equivalent definition of Wiener capacity

Dirichlet Problem for the Half space: differentiating under the integral sign.

Converse of Kellogg Warschawski theorem

Harmonic function that is not pluriharmonic

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