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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Surface Integral over Ellipse

Let $$E=\left\{(x_1,x_2)~\middle|~\frac{x_1^2}{ a^2}+\frac{x_2^2}{b^2}=1\right\}=\left\{X(\theta)=(a\cos(\theta),b\sin(\theta))\,\middle|\,0\leq \theta\leq 2\pi\right\},$$ be an ellipse. Let ...
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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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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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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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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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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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The average of a subharmonic function on a circle increases with radius

Let $u$ be a subharmonic on open set $\Omega$. Let $a\in\Omega,R>0$ such that $B(a,r)\subset \Omega$. Prove $$v(\rho)=\int_0^{2\pi}u(a+\rho e^{it})dt$$ is a monotone increasing function on ...
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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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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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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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The “inverse” of $\nabla\times$ operator

From physics, just to use a well known example, we know that the relationship between the magnetic induction $\mathbf{B}$ and the potential vector $\mathbf{A}$ is given by: $$\mathbf{B} = ...
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36 views

Convergence of measures and potential theory

The following implication should hold: $\mu_{n}, \mu$ are positive measures whose supports are included in a compact set $K\subset \mathbb{C}$ and $$\lim_{n\to\infty}U^{\mu_n}(z)=U^{\mu}(z)$$ ...
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Trouble evaluating spherical Fourier Transform in Quantum Field Theory

(This is purely for personal study - the exercise is 20.2(a) from Lancaster and Blundell (2014), Oxford Uni. Press - an excellent textbook btw.) "Confirm that the Fourier transform of ...
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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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22 views

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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Integrability of Riesz potential

Given $f\in L^1(\mathbb{R}^3)$, define $$\phi(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|y-x|}\,dy.$$ I was able to show that $\phi$ exists for almost all $x$ (I used the Lebesgue differentiation theorem). ...
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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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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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Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
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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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What is the difference between single and double layer potential

I want to know the difference between single layer and double layer potentials. Is there a link between the choice of single/double layer potential and the boundary condition of a PDE or an ...
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Jacobian of the Kelvin transform

The Kelvin transform of the circle in $\mathbb{R}^n$ with centre $\textbf{u}$ and radius $r$ is defined by $$\textbf{y} \mapsto \textbf{u} + r^2|\textbf{y} - \textbf{u}|^{-2}(\textbf{y}-\textbf{u}).$$ ...
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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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182 views

How to use Fourier Transform with non-trivial boundary conditions such as in potential flow around a plate?

I'd specifically like to be able to solve this PDE with boundary conditions corresponding to flow around a line (plate cross-section), otherwise known as flow-tangency, with integral transforms. ...
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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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35 views

Maximum of a subharmonic function on it's boundary.

I am trying to solve below problem of the book Partial Differential Equations(Third edition) written by jurgen just, problem 3.9. Can any one give an idea? Thanks in advance. Let $\Bbb ...
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Can a complete pluripolar set be a single point?

Let $f:\mathbb{C}^n\rightarrow\mathbb{R}\cup\{-\infty\}$ be a plurisubharmonic function which is not identically $-\infty$.The set $\mathcal{P}:=\{z\in\mathbb{C}^n:f(z)=-\infty\}$ is called a complete ...
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Fourier transform on fractional Sobolev spaces

We say that a tempered distribution $f$ satisfies $f \in H^s(\mathbb R)$ for some $s \in \mathbb R$ if $(1+|\xi|^2)^{s/2} \hat f \in L^2(\mathbb R)$. Here, $\hat f$ denotes the Fourier transform of ...
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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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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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Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 ...
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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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the polar set $V^o$ is identified isometrically with $(V^\perp)^*$

Let $X$ and $M$ be two Hilbert spaces and let $B:X\mapsto M^*$ denotes a continuous linear operator. We consider the set $$V=\ker B=\{ v\in X :Bv=0\}$$ we define the polar set of $V$ as: ...
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What does it mean to “admit” something in vector calculus?

Trying to understand the Helmholtz decomposition has lead me to the concept of a vector potential. From Wikipedia [1]: If a vector field v admits a vector potential A, then [...] I've searched ...
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How did Poisson discover his integral formula?

I am quite curious about the history behind it. His derivation should be different from those on today's textbooks.
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Normal derivative of the autonomous single layer potential

Let $G$ be the fundamental solution of the Laplace equation. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$ regular enough. It is known that: \begin{equation} \int_{\partial ...
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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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Newtonian capacity: do we have $\operatorname{cap}(A\cup B)+\operatorname{cap}(A\cup C)-\operatorname{cap}(A)-\operatorname{cap}(A\cup B\cup C)>0$

$\newcommand{\Cap}{\operatorname{cap}}$ For compact disjoint sets A,B,D each with positive Newtonian capacity do we have $$\Cap(A\cup B)+\Cap(A\cup C)-\Cap(A)-\Cap(A\cup B\cup C)>0\text{ ?}$$ ...
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62 views

$L^2$ regularity of a convolution with Newtonian potential

I am reading Vorticity and incompressible flow (Bertozzi, Majda) and on page 71-72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
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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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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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You can't solve Laplace's equation with boundary conditions on isolated points. But why?

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of "holes" $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I'm pretty sure that in more than one dimension, ...
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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 $0 \le|x|\le R$ ...
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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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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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The existence of a measure of finite energy implies a lower bound on Hausdorff dimension

What is the significance of $\mu(x)=0$ and the use of continuity this proof? I am not quite sure about the general direction in the second paragraph.
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Is the potential of a periodic conservative field periodic?

Let $Y = [0,1]^3$ and consider a conservative vector field $F$. Denote its scalar potential by $\varphi$, i.e. $$ \nabla \varphi = F. $$ If $\varphi$ is $Y$-periodic it is clear that $F$ is periodic, ...
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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 ...