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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Error on Wikipedia: Nelson's proof of Liouville's theorem works only for bounded modulus?

On Wikipedia, it is stated: If $f$ is a harmonic function defined on all of $\mathbb{R}^n$ which is bounded above or bounded below, then $f$ is constant...Edward Nelson gave a particularly ...
8
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1answer
85 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
6
votes
1answer
345 views

Green's function in 2D

How does one compute the Green's function of the laplacian on $ \mathbb{R}^2 $? Can someone point me to a reference? In particular, the fourier transform: $$ \int_0^\infty \int_0^\infty \frac{e^{i m ...
6
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251 views

Potential theory: discrete-time Markov processes

Recently I've found lecture notes on "Analysis on Graphs" where the potential theory methods were used to study discrete-time, time-reversible Markov chains (i.e. the state space is countable). ...
5
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2answers
108 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
5
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1answer
122 views

The second Friedrichs' inequalities?

In paper On the Validity of Friedrichs' Inequalities,$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le ...
4
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1answer
476 views

Brownian motion on the circle

Let $\mathbb S^1$ be the unit circle and $\Delta$ be the Laplace-Beltrami operator on $\mathbb S^1$ which is an infinitesimal generator of the correspondent Markov semigroup $P_t$. Is the explicit ...
4
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1answer
328 views

Reference on Doob's h-transform

I am searching for a reference about conditioning a Markov process in the sense of Doob, i.e. using h-transforms. My particular concern is to condition a discrete-time Markov Process on a possibly ...
4
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207 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasi-linear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
4
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1answer
80 views

Definition of logarithmic capacity

In the definition of logarithmic capacity of a compact set $E$ in the plane, the Robin constant is defined to be $V(E)=\inf\int_E\int_E \log\frac{1}{|z-w|} d\mu(z)d\mu(w)$ where $\inf$ is taken over ...
4
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418 views

Finding the Green Function of the upper half ball

Find the Green function of $\Omega:=\left\{x\in\mathbb{R}^n:\lVert x\rVert<R, x_n>0\right\}$ and show that the function you've found is indeed a Green function! You are allowed to use ...
4
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110 views

Harmonic measure or harmonic kernel

In the theory of discrete-time stochastic processes on a measurable space $(\mathscr X,\mathscr B(\mathscr X))$ one usually starts with a Markov kernel $$ P:\mathscr X\times \mathscr B(\mathscr ...
3
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1answer
56 views

$f\in L^2(\mathbb{R}^3)$ implies $v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy\in W^{2,2}$?

Let $f\in L^2(\mathbb{R}^3)$ be a function with compact support and define $v:\mathbb{R}^3\to\mathbb{R}$ by $$v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy$$ Is true that $v\in W^{2,2}(\mathbb{R}^3)$ ...
3
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1answer
39 views

Determining the effective coefficient in a boundary value problem.

I'm trying to understand the setup given in Multiscale analytical solutions and homogenization of n-dimensional generalized elliptic equations, pg 41: Given the boundary value problem: ...
3
votes
1answer
161 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...
3
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0answers
82 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the simple and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
3
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0answers
41 views

Bounds on solutions of elliptic pde's in the whole plane

My question is rather simple. Given $f$ a $L^2(\mathbb{R}^2)$ function with zero mean but supported in the whole plane, is there a bound of the form $$ \|\nabla(\Delta)^{-1} ...
3
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88 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
3
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0answers
119 views

A finely open set, not open up to polar set?

Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, which is not open up to a polar set (i.e. zero capacity), i.e., there does ...
3
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1answer
105 views

A reference to study p-admissible functions

I am studying p-admissible functions. I am using the book of heinonen (nonlinear potential theory of degenerate elliptic equations). I am searching for a good proof of the result: Suppose that w ...
2
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2answers
112 views

Assumption on $u$ harmonic in disc making $u$ unique with particular (discontinuous) boundary data

From an old qualifier: Let $$J_1 = \{e^{i\theta}: 0 < \theta < \frac\pi2\}, \,\,J_2 = \{e^{i\theta}:\frac\pi2<\theta< \pi\},\,\, J_3 = \{e^{i\theta}:\pi<\theta < 2\pi\}$$ be ...
2
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1answer
44 views

Show $\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = \int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$

I read in an article about Laplace's equation that $$-\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = -\int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$$ Could someone explain to me why this is? I ...
2
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1answer
82 views

How would you prove $E = -\vec{\nabla} V$ from the electric potential's line integral?

Basically, we were given an equation: $$V_1 - V_2 = \int_{r_1}^{r_2}\vec{E}\cdot d\vec{r}$$ where $\vec{E}$ is the electric field distribution and $d\vec{r}$ is the displacement vector of the charge. ...
2
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1answer
166 views

Find the Green function of the upper half ball by using the Green function of the whole ball

Find the Green function of the upper half ball $\Omega:=\left\{x\in\mathbb{R}^n|\lVert x\rVert<R, x_n>0\right\}$ (for the Dirichlet boundary value problem of the Laplace equation). Show ...
2
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1answer
24 views

introduction to potential theory in $\mathbb{R}^3$ [on hold]

A differentiable function $g: \mathbb{R}^3 \to \mathbb{R}$ is said to be harmonic in a subset $B \subset \mathbb{R}^3$ if $\Delta^2 g = 0$ for all $p \in B$. Let $M \subset \mathbb{R}^3$ be a bounded ...
2
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1answer
45 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
2
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1answer
59 views

$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
2
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1answer
72 views

Irrotational fields and change of reference frame

Given a reference frame $(x_1,x_2,x_3)$ and a vector field $\overrightarrow{V}(x_1,x_2,x_3)$, in this frame, if $\overrightarrow{\nabla}\times\overrightarrow{ V}(x_1,x_2,x_3)=0$ the field is ...
2
votes
1answer
146 views

Removal of singularities for harmonic functions with finite energy

Denote by $B = B(0,1) \subset \mathbb{C}$ the open unit disc and by $B' = B \setminus \{ 0 \} \subset \mathbb{C}$ the punctured unit disc. Assume that $u : B' \rightarrow \mathbb{R}$ is a harmonic ...
2
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1answer
31 views

Taylor expansion of the electrostatic potential $1/\|\cdot \|$

I have stumbled over this problem several times in electrodynamics, and I just don't get the hang of it. The task is to do a Taylor expansion of $\,f(\vec{x},\,\vec{a}) = ...
2
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1answer
53 views

Computing Newtonian capacity of sets like intervals, discs?

For a metric space $(E,\rho)$ the $a$-capacity is defined as $$\mathrm{Cap}_{a}(E)=\left[\inf\left\{\int \int \frac{d\mu(x) \, d\mu(y)}{\rho(x,y)^{a}}:\mu\text{ probability measures on ...
2
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1answer
40 views

How to finish the computation of $u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
2
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0answers
24 views

Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry. Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined ...
2
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0answers
131 views

Potentials in Probability Theory

Could someone give an intuitive interpretation of potentials in the field of probability theory. How do they link to the theory of stochastic processes. And maybe link this with SEP. References are ...
2
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0answers
76 views

Interpretation of potential kernels for Markov processes

One can associate a strongly continuous contraction semi group (SCCSG) to a Markov process with state space $S$ through its transition function, say $P_t$. Now one can interpret $P_t$ as a linear ...
2
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0answers
80 views

Solving Dirichlet problem by means of potential theory

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$ Is there a way to solve this problem by using ...
2
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0answers
51 views

Tight bounds for harmonic measure

I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ...
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4answers
66 views

Evaluation of the integral $\int_{0}^{R}\int_{-1}^{1} r^2/\sqrt{r^2+L^2+2L\alpha}\,d\alpha dr$

I am trying to solve this integral $$ \int_{0}^{R}\int_{-1}^{1}\frac{r^{2}\,{\rm d}\alpha\,{\rm d}r}{\, \sqrt{\vphantom{\Large A}\,r^{2} + L^{2} + 2L\alpha\,}\,} $$ where $L$ is some positive ...
1
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1answer
145 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
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2answers
96 views

Line integral over a vector field using a potential function.

Let $X(t)=\left ( \pi\cos t, t, e^{\sin t} \right ), 0\leq t\leq \pi $. And let $\textbf{F}(x,y,z)=\left ( 2xy+\frac{-y}{x^2+y^2}, x^2+ze^{yz}+\frac x{x^2+y^2}, ye^{yz} \right )$ be a vector field. I ...
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1answer
48 views

Prove that $u \circ f $ is plurisubharmonic on $\Omega_1$

I'm trying to show that the theorem in my book: Let $f: \Omega_1 \to \Omega_2$ be a holomorphic map between open sub - sets $\Omega_1, \Omega_2$ of $\Bbb C^n$. If $u$ is plurisubharmonic on ...
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1answer
20 views

A polynomial can be written as the difference of sub-harmonic functions

Let $\Omega\subset \mathbb R^N$ open bounded be given, I am trying to prove that first any Polynomial can be written as difference of two sub-harmonic functions, and then for any continuous function ...
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1answer
48 views

How to show by example that existence of barrier function of any set $U\subset \mathbb{C}$ is dependent of its set?

How to show that there a set that has no barrier function? I mean that how to show by example that existence of barrier function of any set $U\subset \mathbb{C}$ is dependent of its set. Definition ...
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1answer
31 views

maps that preserve harmonic functions

Is there a theory of the type of maps between domains that preserve harmonic functions? For instance, in the 2-dimensional case, we know that conformal maps (or even just holomorphic ones) are such ...
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1answer
45 views

Riesz measure associated with a subharmonic function

In page 101, corollary 4.3.3., from Armitage and Gardiner's book on potential theory, the authors prove that any subharmonic function, can be identified with a positive measure (Riesz measure). In ...
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1answer
20 views

Showing there's no potential for a vector field on some region of the space?

Imagining we have a vector field $f$ for which $\operatorname{curl} f = (0,0,0)$, why can there be a potential to it on some region and not on some other. e.g. what kind of reasoning can prove that ...
1
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1answer
31 views

Harmonic inside with zero average

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where ...
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1answer
171 views

Derive Poisson integral formula in a ball

Trying to derive by myself the Poisson integral formula in a unit ball. I should get $$\Delta u=0 \,\text{ in } B(0,1), \,\,\, u(x)=\varphi(x)\,\,\text{at } \partial B(0,1) \Longrightarrow \\$$$$u(x) ...
1
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1answer
95 views

Show that the Kelvin transformation is a diffeomorphism and find the inverse

Let $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. We call $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ with $$ y=\phi(x):=\frac{R^2}{\lVert ...
1
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1answer
41 views

Show that $\widetilde{u} \in PSH(\Omega)$

EDITED I'm reading the book: (Oxford science publications._ London Mathematical Society monographs, new ser., no. 6) Maciej Klimek -Pluripotential theory -OUP (1992). I don't understand ...