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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Comparison of capacity of sets in $\mathbb{R}^n$

This is mainly in reference to this MSE post. Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \...
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Capacity of a set in $\mathbb{R}^n$

The $2$-capacity of a set $\Omega$ sitting inside an open set $V \subset \mathbb{R}^n$ is given by $$\text{cap}_2(\Omega, V) = \inf_{u \in C^\infty_0(V), u|_\Omega \equiv 1} \int_V |\nabla u|^2 dx.$$ ...
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How is called 1D and 2D analogy to saddle point

Consider n-dimensional function with multiple local minima (e.g. http://ac6la.com/aeopt5.png). There exist some basins of attraction for which particle fall to one or the other minimum. Boundary ...
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27 views

Physical interpretation of Poisson's equation and Dirichlet problem

I am a mathematics student trying to understand physics behind the Poisson equation and the Dirichlet problem. We know that the electric potential potential $u$ satisfies the Poisson Equation: $\...
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Neumann boundary condition, spherical shell.

The velocity of a fluid $\mathbf{u}$ is assumed to have the velocity potential $\Phi$ such that $\mathbf{u}= \nabla \Phi$. The fluid is contained in a rigid shell, of radius $a$, which is moving with ...
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Geometry of level sets of an harmonic function

Suppose you have an harmonic function on an exterior domain of $\mathbb{R}^n$, i.e., a function $u \colon \mathbb{R}^n \setminus \bar\Omega \to \mathbb{R}$, where $\Omega$ is a smooth and bounded open ...
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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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Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
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$2$-capacity of a set in $\mathbb{R}^n$

Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \mathbb{R}^n$ such that $F \subset \Omega$, define $...
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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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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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Well definition of newtonian potential

Given $f : \Omega \to \mathbb{R}$ an integrable function defined on the domain $\Omega \subset \mathbb{R}^n$ ($n > 2$), its newtonian potential $w : \mathbb{R}^n \to \mathbb{R}$ is defined by $$w(x)...
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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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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 $$\frac{\...
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Surface Integral over Ellipse

I want to prove this equality $$ \frac{ab}{2\pi}\frac{a^2\cos^2(\alpha)-a^2\cos(\beta)\cos(\alpha)+b^2\sin^2(\alpha)-b^2\sin(\beta)\sin(\alpha)}{a^2(\cos(\alpha)-\cos(\beta))^2+b^2(\sin(\alpha)-\sin(\...
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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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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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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 $(0,R)$. ...
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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 $\Omega$...
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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} = \nabla\...
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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 $V(\underline{r}...
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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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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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46 views

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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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. $$(1)...
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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 R^2:=\{(x_1,...
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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 u}{\...
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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 \...