# Questions tagged [potential-theory]

Potential theory concerns solutions of elliptic partial differential equations (especially Laplace's equation) that are represented by integration against a measure or a more general distribution.

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### Integral representation of the pressure of the Stokes flow

I'm currently reading these three books: S. Kim, S. J. Karrila - Microhydrodynamics: Principles and Selected Applications O. A. Ladyzhenskaia -The Mathematical Theory of Viscous Incompressible Flow ...
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### Measurability of the exterior boundary of a set for a transition kernel

I am learning the potential theory for the Markov chains. And I've encountered a problem: Let $\pi$ be a transition kernel on a Polish space $(S,\mathcal{B})$, and let $D \in \mathcal{B}$. The ...
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### The coefficients of a positive current are radon measures

I am trying to understand the following proof: from page 9 of this notes. I do not quite follow the logic here. So I guess the point is that $T \wedge \omega_{s}$ is a radon measure. Why is this true,...
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### A property of potential function ($\lim_{z\rightarrow\infty}(p_{\mu}(z)-\mu(\mathbb{C})\log|z|)=O(1/|z|)$)

I am studying the potential theory from Ransford book. Let $\mu$ be a finite Borel measure on $\mathbb{C}$ with compact support and let $p_{\mu}:\mathbb{C}\rightarrow [-\infty,\infty)$. The result is ...
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### estimate for exterior harmonic function

Problem Let $D\subset \mathbb{R}^3$ be a smooth, connected, bounded domain, consider Dirichlet boundary problem \begin{equation} \left\{ \begin{aligned} &\Delta u=0, &\mathrm{in}\ \ \mathbb{R}^...
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### Null sets and the Riesz measure of a subharmonic function

Let $D$ be a bounded domain of $\mathbb{R}^{m}$ with $m>1$, and $u$ a subharmonic function on $D$. Let $u_{\epsilon}$ be a sequence of smooth subharmonic functions on $D_{\epsilon}$ (the set of ...
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### A question on integrability of harmonic functions

Suppose $D$ is a bounded domain of $\mathbb{R}^{m}$ ($m>1$). If $h$ is harmonic on $D$, do we have $$\int_{D}|h(x)|dx<\infty?$$
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### Is there a way to mathematically prove $\psi (\mathbf{r})$ varies continuously (using the intuitive arguments provided below)?

Electric potential at a point outside the charge distribution is: $\displaystyle \psi (\mathbf{r})= \int_{V'} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ where: $\mathbf{r}=(x,y,z)$ ...