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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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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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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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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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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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 ?...
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}{\...
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 \...