Articles of potential theory

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, then $u$ is constant in $U$. Proof: Let $x_0$ be an point in $U$ with $u(x_0)\geq u(x)$ for all $x$ in some neighbourhood […]

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, it doesn’t make sense to solve Laplace’s equation with Dirichlet boundary conditions on $X$. That is, there does not exist any “valid”* $f$ such that $$\begin{align} […]

How to evaluate$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D$ is the sphere in 3D?

I would like to evaluate this integral over the surface of a sphere in 3D: $$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy. $$ It seems there is a lot of symmetry in this integral so I imagine there is a good chance there is an explicit solution. However, I usually deal with 2D Helmholtz problems so […]

How to evaluate$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D_1$ and $D_2$ are spheres in 3D?

In my previous question I asked about evaluating the following integral over the surface of a sphere in 3D: $$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy, $$ and it turned out that the answer is $(4\pi)^2$ for a unit sphere. Now what about the case where $x$ and $y$ are not on the surface of the […]

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.

Calculation of the $s$-energy of the Middle Third Cantor Set

As the title suggests, I am trying to calculate the $s$-energy of the middle third Cantor set. I am reading Falconer’s Fractal Geometry book, available here: and this is an exercise (exercise 4.9 on page 45 of the pdf – 68 of the book). First, we define the $s$-potential at a point $x\in\mathbb{R}^{n}$ as […]

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: $$\mathbf{B} = \nabla\times\mathbf{A}$$ My question is: could/does exist an operator $\mathrm{\hat{O}}$ (or with a bad notation: “$(\nabla\times)^{-1}”$ such that $$(\nabla\times)^{-1}\mathbf{B} = \mathbf{A}$$ I mean: knowing the magnetic field […]

Phase Potentials

Looking to get a explanation of the following solution. $x’=sinx$ I understand that you must integrate $sinx$ and then take the negative of it making it $v(x)=cos(x) + c = cos(x)$ with $c=0$. However I am having problems understanding the equilibrium points. The equilibrium points of this system are, $$x^*=kz$$ $x*=kz$ k is even, this […]

Some properties of capacity

Let $\Omega\subset\mathbb{R}^N$. For compact $K\subset \Omega$ we can define the $p$-capacity, $p\in (1,\infty)$ as the number $$\operatorname{cap}_p(K)=\inf \int_\Omega |\nabla u|^p$$ where the infimum is taken over all $C_0^\infty(\Omega)$ with $u\ge 1$ in $K$. If $U\subset\Omega$ is open, set $$\operatorname{cap}_p(U)=\sup_{K\subset U}\operatorname{cap}_p (K)$$ where $K$ is compact. My question is: how to find a open set $U\subset […]