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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Why is electric potential function in free space infinitely differentiable?

Electric potential function in free space of a continuous charge distribution $\rho'$ distributed over volume $V' \subset \mathbb{R}^3$ is denoted by: $\psi (x,y,z): \mathbb{R}^3 \setminus{V'} \to \...
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12 views

Proof of the continuity and discontinuity of layer potentials

Let $\Phi$ be the Green's function of the Laplace equation: $$ \Delta \Phi(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}') $$ Then the single-layer and double-layer potential on a boundary $\...
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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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32 views

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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Restricting the range of Riesz Kernel: Does the inequality still remains true?

I am reading something containing the following statement: We have seen that if $u$ is a smooth function defined on a ball $В \subset \mathbb{R}^n$ (possibly with infinite radius so that $В = \...
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13 views

Definition of generalized laplacian by Green's theorem

I am reading a book on Potential Theory and they motivate the generalized Laplacian by saying that Green's theorem implies the following: $$ \int_D \phi \Delta u = \int_D u \Delta \phi dA $$ where $\...
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25 views

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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18 views

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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44 views

Is there characterization of critical points of the electric potential function?

Suppose $\vec{r}_1,\vec{r}_2,\cdots, \vec{r}_n \in \mathbb{R}^3$ and $Z_i \in \mathbb{Z}$ for each $i \in \{1,2,\cdots,n\}$. I would assume the electric/gravitational potential function $V : \mathbb{R}...
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31 views

Green's Function for the potential

I need to find a Green function that provides the value of U at any point within a circle from the values of U at the limit of the circle, knowing that the function U satisfies the two-dimensional ...
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37 views

Why is the $1/r^2$ factor retained in Laplace's equation in spherical coordinates?

Consider Laplace's equation in spherical coordinates $$\nabla^2 f = \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2 \frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\...
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38 views

Solenoidal and irrotational field [closed]

A radial field is described by $F=ar^n \hat r$ . 1) Find the values for n for which $F$ is solenoidal in regions where $r$ is not equal to zero. 2) Find the values of $n$ ($r$ not equal to zero) for ...
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regular point of open set under Brownian motion

Consider an open set $G$ and a boundary point $x$, for a standard Brownian motion, my question is, if $x$ is regular to the closure of $G$: $\overline{G}$, is $x$ also regular to $G$ itself? I am ...
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12 views

For $h,k$ harmonic prove that $hk$ is harmonic if and only if $h +ick$ is holomorphic.

This is an exercise from "Potential Theory in the Complex Plane" by Ransford. As in the title: Assume $h,k$ are harmonic and non constant on some domain $D$. Prove that $hk$ is harmonic if and only ...
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Far field asymptotic behavior for Strongly Localized Perturbed domain

Let $\Omega$ be a $3-D$ bounded domain with a hole of radius $\mathcal{O}(\epsilon)$ and centre $x_0$, say $\Omega_0$, that is removed from the $\Omega$. Define $v_c (y)$ such that it satisfies, \...
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Maximum principle for subharmonic functions on unbounded domains

Let $K$ be a compact of $\mathbb R^n$ ($n\geq2$) and $V_0$ and $V_1$ the complements of $K$ in $\mathbb R^n$ and $\mathbb R_\infty^n$ (one point compactification of $\mathbb R^n$) , respectively. ...
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A question on generalized Dirichlet problem

Let $W$ be an open set in $\mathbb{R}^{n}$ ($n>1$), and $u$ a subharmonic function on an open set containing the boundary $\partial W$ of $W$ (not necessarily continuous). Let $H_{u}(x)$ be the ...
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26 views

If $F(x + iy) = \frac{Q}{2\pi}\ln (x + iy)$, how to prove that $\phi(x,y) = \frac{Q}{2\pi}\ln (\sqrt{x^2 + y^2})$?

The potential flow theory states that $$ F(z) = \phi + i\psi $$ where $z = x + iy$ In case of a source/sink, we have $$ F(z) = \frac{Q}{2\pi}\ln z\tag{1} $$ $$ \phi(x,y) = \frac{Q}{2\pi}\ln (\sqrt{x^...
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How to find the vector potential of the field $F = (-y,x,0)$? [duplicate]

Given the vector field $F = (-y,x,0)$, how can I find a vector potential of this field? I know that we have to find a vector field $G$ such that $F = \nabla \times G$, but I am struggling with finding ...
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Potential calculation of irreductible and recurrent Markov chain

Take a MC in $\mathbb{N}$ with transition matrix $P(n,n+1)=p(n)$, $P(n,0)=1-p(n)$ and $\prod p(i)=0$ so it is recurrent. Show the solutions of the equation $h=\delta_i +Ph$ and use them to calculate ...
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How to prove the following potential function?

Consider a game where players can choose between three different product namely A, B, C. If have constructed the individual payoff function and potential function, but I am struggling to prove that ...
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Potential function for three competing products in networks with network effects

I have been working on my thesis and am currently stuck at making the potential function that is able to track unilateral deviation within the network. Suppose there are three different products $A, B,...
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143 views

Triple integral for electrostatic potential of a particle in a cubic quantum box

If we consider a cubic shaped quantum box and put a charged particle in the ground state there, we can compute the electrostatic potential at any point inside the cube using the integral: $$I=\int_{-...
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Regular set for a nearly Borel set is Borel (Brownian motion)?

I have a question based on book "Markov processes, Brownian motion and time symmetry", on page 113, proposition 10 says, under Hypothesis(L), for any nearly borel set A, there is a sequence of compact ...
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25 views

Sets of harmonic measure zero

Let $\Omega$ and $\Omega'$ be two bounded open sets of ${R}^{n}$ with $n\geq2$. Suppose $E$ is a common subset of the boundaries of both $\Omega$ and $\Omega'$. My question is: if the measure of $E$ ...
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21 views

Game theory, Potential games, creation of potential function.

How can a potential function be found in a three player game (prisoner's dilemma). I hope someone can help me understand how a potential function can be created for a tree player game.
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31 views

Does small capacity of a set imply that its measure is small too?

I would like to know if an open set with small capacity also has small measure. To be more specific, I would like to know if the following statement is true: For each $\epsilon>0$, there exists $\...
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166 views

The integral is not iterated integral. Will this fact prevent us from swapping the order of surface and volume integrals? Why? Why not?

Consider a continuous charge distribution in volume $V'$. Draw a closed surface $S$ inside the volume $V'$. Consider the following multiple integral: $$B= \iint_S \Biggl( \iiint_{V'} \...
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36 views

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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35 views

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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Does integrability implies integrability with respect to the harmonic measure?

Let $D$ be a bounded domain in $\mathbb{R}^{d}$ ($d>1$) and $f$ a measurable function on $D$. Suppose $K$ is a compact of the boundary of $ D$ and $\omega_{x}$ designates the harmonic measure of $...
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72 views

Necessary and sufficient for vector field to be conservative [duplicate]

I found the following statement in this youtube lecture: Let $A\subset \mathbb{R}^n$ be open and convex and let $f:A\to\mathbb{R}^n$ be continuously differentiable vector field. Then $f$ is ...
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83 views

The integral with respect to the Riesz measure over a compact with empty interior

Suppose $D$ is a bounded domain of $\mathbb{R}^{n}$ ($n\geq2$), $u$ a subharmonic function on $D$ and $\mu$ the Riesz measure associated with $u$. Let $F$ be a closed set with empty interior in $D$. ...
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26 views

Uniform behaviour of limit infimum

Let $f$ and $f_n,n=1,2,\dots$ be real valued function on domain $\Omega \subset \mathbb C$. Then what is mean by $$f(x) \leq \liminf_{n \to \infty}f_n(x)$$ uniformly on compact subsets of $\Omega?$ ...
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78 views

Poisson Equation on $\mathbb{R}^3$.

Let $f\in C^1(\mathbb{R}^3)\cap L^1 (\mathbb{R}^3)$, then there exists a solution $u\in C^2(\mathbb{R}^3)$ solving $-\Delta u = f$ on $\mathbb{R}^3$ given via $$u(x) = \frac{1}{4\pi}\int_{\mathbb{R}^3}...
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Do the following extra highlighted words in the $\epsilon-\delta$ definition of limit prevent us from concluding that the limit exists? Why? Why not?

This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary. ...
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How to bound this integral with the Hardy-Littlewood maximal function

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^d$ and consider the following situation. $$\phi(x) = \int_{\partial \Omega} \frac{x-y}{|x-y|^d} f(y) d\sigma(y)$$ Where $\partial \Omega$ ...
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Mollifier of a subharmonic function

Suppose $D$ is a bounded open set in $\mathbb{R}^{m}$ with $m\geq2$, and $u(x)$ is locally integrable on $D$. We know that if $\omega$ is an open set that is included with its closure to $D$, the ...
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Series expansion of general solution of Laplace equation in arbitrary bounded 3D regions

Let $X \subset \mathbb{R}^3$ be a connected and open 3-dimensional set. Furthermore $X$ is bounded \begin{equation} \exists r^\varnothing : \|\mathbf{r}\| < r^\varnothing \quad \forall \mathbf{r}...
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Bounded Subharmonic functions in the Plane are constant.

I would like to prove this result without any appeals to theorems like, say the three-lines theorem, but only through the use of the fundamental solution, $\log |z - w|$ in the plane. The proof I ...
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Equilibrium Measure is Harmonic Measure at Infinity

Consider $K\subset \mathbb{C}$ a compact domain. For any Borel Probability measure on $K$, we define the "energy" of $K$ to be: $$ I_K(\mu) = \int \int _{K \times K} \log |z-w| \mathrm{d}\mu (z) \...
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Dirichlet problem on a wire: a co-dimension 2 boundary condition

A wire may be thought of as a smooth compact curve $C \subset \mathbb{R}^3$ with boundary two endpoints. Suppose we are given a smooth $\phi: C \to \mathbb{R}$ (a potential on the wire), then can $\...
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63 views

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: $\mathbf{r}=(x,y,z)$ ...
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97 views

Potential Theory vs Harmonic Analysis

In layman's terms, what is the difference between Harmonic analysis and Potential Theory? Could you please give a quick synopsis of what each is trying to study?
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Why the relation $\mathbf{E}=-\nabla \psi$ holds at points $P \in V'$?

Let there be a continuous charge distribution in space having volume $V'$ and density $\rho$. Let: $\displaystyle \mathbf{E}=\int_{V'} \rho\ \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3}...
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$L^2$ norm of Riesz Potential of Disk

I am interested in evaluating, or more precisely finding asymptotics in terms of $s$, of the following integral: $$2^{-s}\pi^{-1}\frac{\Gamma(\frac{2-s}{2})}{\Gamma(\frac{s}{2})}\int_{|x|\leq (2\pi)^{-...
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112 views

Why are we not using Dirac delta and ignoring the contribution to the surface integral from the point $r=0$?

Let $V'$ be the volume of dipole distribution and $S'$ be the boundary. The potential of a dipole distribution at a point $P$ is: $$\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{...
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63 views

Deriving the inequalities in potential theory

I am reading the following pages from my book on "foundations of potential theory" page 150/151: I understand up to inequality $(4)$. I can't derive inequality $(5)$ from $(4)$. Please derive ...
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Integrability of exponential of the Logarithmic Potential

Let $u$ be a subharmonic function defined on some open set $G\subset \mathbb{C}$. BY definition, it is uppersemicontinuous and $\vartriangle u\geq 0$ in the sense of distribution. Therefore we can ...

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