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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Capacity vs measure of a set - intuitive understanding

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given ...
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328 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). ...
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Potential measure of the product of (independent $\alpha$-stable) subordinators

For a nondecreasing Levy process $\mathbf{X}$ with values in $[0,\infty)$ (i.e. a subordinator) Jean Bertoin defines the potential measure of $\mathbf{X}$ in his book "Levy processes" as follows (p. ...
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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 ...
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138 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 ...
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51 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
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45 views

Potential theory for LCA groups

I was wondering if there is a potential theory for locally compact abelian groups. $\textbf{Edit:}$ What are the suitable analogs for logarithmic or Newtonian potentials in the context of LCA groups ...
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47 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} ...
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136 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 ...
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143 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 ...
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20 views

Riesz potential of a set and its complement

Let $F\subset [0,1]$ be a closed set, $G = [0,1]\setminus F$, $\alpha \in(1,2)$. Is there a simple condition on $F$ under which the integral $$ \int_F\int_G \frac{dx\,dy}{|x-y|^{\alpha}} $$ is finite? ...
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31 views

How does one find the coefficients in the solution to the Laplace equation?

I'm reading this. Equation (522) gives the general solution to the Laplace equation. What I'm stuck about, is how to determine the coefficients $a_m$, $\beta_m$, and $\theta_m$ for non-trivial ...
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70 views

Electrostatic capacity of two spheres with changing radii

Although I have read a lot of questions and answers here, this is my first time actually posting. Feel free to suggest needed edits. My question is the following (in a simplified setting). All this ...
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289 views

Harmonic functions in the upper half plane

It is a cautionary remark that is often made that solutions to the Dirichlet problem (with continuous boundary conditions) are not unique when the domain in question is the upper half plane. Yes, you ...
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63 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 ...
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335 views

Green function of the first quadrant

Find the Green function of the first quadrant $x_1>0, x_2>0$. HINT: Use the Green function of the half space $\Omega:=\left\{x\in\mathbb{R}^n : x_n > 0\right\}$ which is given by ...
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126 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 ...
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108 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 ...
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132 views

discrete harmonic extension (an exercise of Grimmett's “probability on graphs”)

I'm struggling with exercise 1.3 in Grimmett's book "probability on graphs". Take $G = (V,E)$ a finite connected graph with given positive conductances $(w_e)_{e \in E}$, and let $(x_v)_{v \in V}$ be ...
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59 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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106 views

Surface Integral over Ellipse

I want to prove this equality $$ ...
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20 views

Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D: $$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$ $$u = g ...
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65 views

Whittakers general solution to Laplace and its relation to separable variables

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates $u=x+iy$ and $v=x-iy$. This can be extended to n dimensions as long as the complex ...
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24 views

Helmholtz decomposition of $v\in (L^2(\Omega))^3$

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with Lipschitz boundary $\partial\Omega$ and outward unit normal $n$. I want to study the characterize whether a vector function defined on ...
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93 views

proof of Wiener’s criterion

I'm in my first course of PDE and I need to investigate the proof of Wiener's Criterion for Laplace Equation which says, if $\Omega \subset \mathbb{R}^n$$(n>2)$ is a bounded domain and $\partial ...
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23 views

Potentials and Markov Processes

Given a resistive electrical circuit $G$, i.e. a graph with nonzero weights attached to each edge whose reciprocal we call the "resistance," we can define a reversible markov chain on the graph, ...
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44 views

Newtonian potential of torus

Is there an explicit expression of the Newtonian potential for torus? As the expression for ball is clear by calculation. Definition of Newtonian potential of domain $\Omega$ at x is defined to be ...
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45 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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85 views

Estimation (Newton potential)

Let $f$ be a $C^2$-function on $\mathbb{R}^n$ with compact support. Let $N$ be the Newtonpotential on $\mathbb{R}^n$ and $u:=N\star f$ (i.e. a solution of the potential equation $\Delta u=f$). ...
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68 views

Inequality for harmonic extension : Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, is the following true ? (Question 1) Let $p(z,t) = \frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ ...
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Evaluate integral: $ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx$

Show that $$ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx = \log{\frac{|z+\sqrt{z^2-1}|}{2}},\quad z \in \mathbb{C} $$ How can I apply the Joukowski conformal map to this problem? Thanks.
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How to describe scalar fields over graphs

I am trying to find the section the literature that dwells on the propagation of scalar fields over random graphs. Think of a network of ideal resistors for example, with a voltage source at a ...
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The harmonic functions are smooth

Let $u \in C^2(\Omega)$ a harmonic function, then $u \in C^{\infty}(\Omega)$. Here $\Omega$ is a domain of $\mathbb{R}^{n}$. Step 1 for proof: $u \equiv u^{\epsilon}$ in $\Omega_{\epsilon}$, for ...
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Number of paths in a graph as a function of depth

This problem has been bugging me for weeks now. Consider a infinite graph, with a given degree distribution. Now, for the sake of intuition, consider that each vertex includes a match. We pick a ...
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Derivation of the Quadrilateral Source Element used in Fluid Potentials

Background: Potential method is used to solve linear fluid dynamics problems still to this date, which is based off reducing the Navier-Stokes equation to an inviscid, irrotational, incompressible ...
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23 views

Complex potential question

Consider the flow described by the complex potential $$w = Be^{−imπ}z^{m+1}$$ where $B > 0$ and $m ∈ (0,1]$. (i) Determine the stream function $ψ$ and the potential $φ$ in plane polar coordinates ...
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Boundedness of the solution of the integral equation associated to the heat kernel

(Cross-posting http://mathoverflow.net/questions/232720/boundedness-of-the-solution-of-the-integral-equation-associated-to-the-heat-kern ) Let $\Omega$ be a bounded open set of $\mathbb R^n$ ($n\geq ...
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21 views

Derivative I calculated does not match code (or intuition)?

I want to take the derivative with respect to the $x$ co-ordinate of a Hankel function with the norm of a 2d vector as its argument. Let $\mathbf{x} = (x_1, x_2) \in \mathbb{R}^2$. We have ...
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26 views

Fundamental solution of the Poisson equation with variable exponent

Let the variable exponent $p(x)$, where $p(x) \in C(\overline{\Omega})$, I want to know the fundamental solution of $$-(\Delta u)^{p(x)}=\delta_0.$$
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Non trivial boundaries for laplacian equation on rectangle

1) Can this Laplace equation, with its non trivial boundaries (on a rectangular domain), be solved analytically? $$\frac{d^2U}{dx^2}+\frac{d^2U}{dy^2}=0$$ $$U_x(0,y)=0\quad,\quad U_x(a,y)=f(y)$$ ...
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The Gauss theorem and discontinuity formulas for layer potentials

I'm studying electrostatics. I have solved, rigorously, the potential problem of volumetric distribution (I have proved continuity, potential existence in a point of charge distribution, and the ...
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31 views

Hamiltonian system - find gradient of vector

A particle velocity in $(x_1,x_2)$-plane is called $p=(x_1',x_2')$. The particles total energy can be written as $$H(x,p)= \frac{|p|^2}{2} + v(x).$$ A particle that moves a long the orbit $X(t)$ and ...
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How to get the equality on potential kernel?

Recently, I am reading the book ``Some random series of functions" by Jean-Pierre Kahane. I cann't understand an equality on page 134, Chapter 10. Namely, let $\mu$ be a probabiliy measure with ...
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19 views

General multipole solution to Laplace equation in polar coordinates

I am seeking the general solution for the Laplace equation in cylindrical coordinates or $\nabla^2 \omega = 0$. In several texts, the general solution can be found via separation of variables and ...
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External and internal multipole expansion for axisymmetric potential - the region of convergence.

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
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A question involving kernels on measurable spaces

Let $(E, \mathcal{B}(E)), (F, \mathcal{B}(F))$ measurable spaces. A $\it{kernel}$ from $(E, \mathcal{B}(E))$ to $(F, \mathcal{B}(F))$ is a map $N : p\mathcal{B}(E) \to p\mathcal{B}(F)$ such that: $$N ...
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51 views

The boundary integral equation

in which case we use the single layer potential and the double layer potential for the Laplace equation ? \begin{eqnarray}\tag{1} \Delta u = 0 \; \mathbb{R}^2\backslash\omega\\ u \to 0 \; at \; ...
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28 views

What is the axiomatic definition of potential kernel?

In the enter link description here, a kernel V is a Markov potential kernel if the operator I+V is invertible and its inverse is of the form I􀀀-N with N a sub-Markov kernel.The definition of Markov ...
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153 views

Reducing the Laplace equation with inhomogeneous BC's to the Poisson equation with homogeneous BC's

Given a domain $\Omega \subset \mathbb{R}^2$, one can reduce the Laplace equation $$\Delta u = 0, \qquad u = f \text{ on } \partial \Omega$$ to a Poisson equation $$\Delta v = g, \qquad v = 0 \text{ ...
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Two more little tasks concerning Dirichlet boundary value problem

Consider the Dirichlet boundary value problem for the Poisson-equation $$ -\Delta u=f\text{ in }B_R(0)\subset\mathbb{R}^3,~~~~~~~~~~u=0\text{ on }S_R(0) $$ with $f\in L^{\infty}(B_R(0))\cap ...