# Tagged Questions

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 ...
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### 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 ...
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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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### 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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### 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 ...
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### $f\in L^2(\mathbb{R}^3)$ 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 $v\in W^{2,2}(\mathbb{R}^3)$ ...
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### 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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### 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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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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. ...
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### 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 that the ...
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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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### 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 ...